Logics of Statements in Context—Deduction in First-Order Logic

1. Introduction

Logics of Statements in Context (LoSiCs) have been introduced in [1] as a general framework enabling us to describe, analyze and relate a variety of diverse concepts, results and constructions from a broad spectrum of logics and other formalisms including First-Order Logic [2,3,4], Universal Algebra [5], Algebraic Specifications [6,7,8], Ehresmann Sketches [9], Generalized Sketches [10,11,12], Description Logic [13], Graph Transformations [14] and Software Modeling [15,16], for example.

When it comes to open formulas, we are faced with the strange situation that deduction in traditional first-order logic is, in fact, deduction of open formulas while open formulas have no "official meaning"! Nevertheless, the deduction calculi turn out to be sound and complete for closed formulas. Logics of Statements in Contexts propose a uniform way to define logics also comprising open formulas and to potentially resolve this paradoxical phenomenon in a precise formal way. As a first step, it has been shown in [1] that we can define arbitrary first-order open formulas and their semantics in arbitrary categories.

The crucial idea behind the concept of context is to establish an additional conceptual layer between syntax (signatures, constant symbols, variables, terms, formulas) and semantics (structures and their carriers). Especially, we want to offer a home for those entities, being around in logic, that neither properly belong to syntax nor to semantics.

Examples of contexts, we do have in mind, are sets of generators in Group Theory, sets of literals in PROLOG, sets of individual names (nominals, objects) in Description Logic, sets of Henkin constants, underlying graphs of Ehresmann sketches, underlying graph structures of Software Models and so on. Contexts can also accommodate Skolem constants as we will exemplify in Section 7 and Section 8.

An associated vision is, that contexts establish the "technological spaces" where deduction takes place, can be formalized and analyzed. Sets of (names for) elements in carriers, for example, can also take the role of contexts as outlined in the discussion of elementary diagrams in Section 5.1.

From the qualified perspective of traditional first-order logic the concept of LoSiCs boils down to the following principles and proposals:

1.

Instead of implicitly relying on sets of free variables syntactically appearing in open formulas, we propose to rather work with explicit declarations of sets of available free variables. This principle enables us to define arbitrary first-order open formulas in arbitrary categories and is especially relevant for many-sorted first-oder logic [17]!

2.

Treat open formulas as first-class citizens! This principle has, at least, three facets:

(a)

Any syntactic entity and, especially, any term and any open formula should have a semantics in any first-order structure. A term syntactically represents a derived operation build up from the basic operations, i.e., the semantics of the operation symbols (in a signature $\Sigma$) fixed by the given $\Sigma$-structure $\mathcal{U}$. Correspondingly, an open formula syntactically represents a derived predicate assembled, with help of the derived operations, from the basic predicates in $\mathcal{U}$.

(b)

Introducing contexts implicates that the informal concept open formula splits into two formal concepts, namely expression and statement in context. Open formulas as such, i.e., only relying on free variables, are formalized as expressions. An expressions together with an embedding (binding) of its free variables into a context is called a statement in context.

(c)

The principle facilitates a shift of perspective: The actual subjects of first-order logic are not first-order structures as such but single (!) interpretations of contexts in structures. A closed formula simply is a statement in the empty context, i.e., a statement about the only interpretation of the empty context in a first-order structure, and thus a statement about the structure as a whole!

3.

The elements of contexts are, in principal, neither constant symbols nor variables nor elements of carrier sets. Or, to put it the other way around: Of course we can use one and the same entity in different roles, but we should always announce the corresponding role as well as changes of roles!

Ehresmann Sketches [9] and Generalized Sketches in the sense of Makkai [10] or Diskin [11,12] are two of the main historical roots of LoSiCs and provide paradigmatic examples (comprising, however, only atomic statements in context). Therefore, we decided to reuse in LoSiCs the term sketch to denote a pair consisting of a context and a set of arbitrary first-order statements in this context. In [1] we also adapted Makkai’s concept of sketch implication [10]. In LoSiCs a sketch implication is given by two sketches and a morphism between the two underlying contexts. In this paper we demonstrate, for the first time, how sketch implications also can be employed as rules to deduce new sketch implications from given ones in a sound way.

In view of LoSiCs, traditional notations in first-order logic are lazy in the sense that there are neither explicit variable declarations nor are contexts explicitly considered. Accordingly, there are no explicit bindings or context morphisms either. The only morphisms, implicitly present in traditional first-order logic, are inclusion maps.

By introducing explicit variable and context declarations as well as explicit inclusion maps, we are fortunately able to recast the traditional lazy approach to first-order logic in a LoSiC-compatible way based on a special kind of sketches, called Σ-specifications, and a special kind of sketch implications, called Σ-specification implications. The price, we have to pay to achieve this, is compliance with Barendregt’s variable convention [18].

After this has been settled, we are able to accomplish the main objective of the paper, namely to certify the LoSiC-approach, once more, by demonstrating that we can formalize and re-validate traditional first-order deduction only relying on the novel concepts context, Σ-specifications and Σ-specification implication. We analyze and formalize three different deduction calculi for traditional first-order logic - a Hilbert System $\mathcal{H}$, a Gentzen’s system $\mathcal{L}\mathcal{J}$ and a corresponding Natural Deduction System $\mathcal{N}\mathcal{D}$. We elucidate the structural essence of reasoning in each of the systems and, thus, the crucial conceptual differences as well as their commonalities. Especially, we are able to formally grasp the different meanings of the turnstile symbol ⊢ and the inference line, respectively.

We reckon that the highlight of the paper, from the perspective of traditional first-order logic, is that the LoSiC-approach leads to the invention of open Σ-specification implications. This innovation enables us to syntactically represent the outcome of hypothetical proofs in Natural Deduction Calculi in a precise formal way and independent of the concrete hypothetical proof! Complementary, we demonstrate that open Σ-specification implications give us "deduction rules in waiting position locally bound to a context" at hand. We show how those locally bound deduction rules can be employed in a semantical sound way. In summary, we somehow finally license hypothetical reasoning as such.

Triggered by an intensive and inspiring discussion with Nicolas Behr and his group in spring 2025, we decided to be this time, in contrast to [1] and [17], more rigorous and to fully back up our exposition by a hom-set based formalization of the semantics of quantification as elaborated in the Appendix A of the paper.

The paper is organized as follows. Section 2 recapitulates necessary concepts and constructions from category theory and fixes corresponding notational conventions.

Relying on hom-sets as categorical products, instead of Cartesian products, Section 3 gives a rigorous and novel description of basic concepts in unsorted first-order logic and their semantics – signatures, terms and expressions.

Section 4 demonstrates how to construct for any unsorted first-order signature $\Sigma$ a corresponding institution ${\mathcal{FL}}_{\Sigma }$ of $\Sigma$-statements in context not only comprising closed formulas, as it is traditionally the case [19,20], but also open formulas! In addition, we study syntax and semantics of substitutions and establish a corresponding satisfaction condition.

In Section 5 we introduce for any institution ${\mathcal{FL}}_{\Sigma }$ corresponding $\Sigma$-sketches $\mathbb{P}=(P,S{t}^{\mathbb{P}})$ and sketch arrows given by two sketches $\mathbb{P}$, $\mathbb{S}$ and a map $\psi :P\to C$ between the two underlying contexts. Sketch arrows are subject of two different, but dual, semantical conditions. First, any institution gives us a corresponding concept of presentation morphism at hand [19,20]. We discuss the corresponding morphism condition and, as a terminological sleight of hand, we use the term sketch morphism to indicate that a sketch arrow is subject of the morphism condition. The implication condition is the second semantical condition for sketch arrows and is addressed in [10]. We use the term sketch implication to indicate that a sketch arrow is subject of the implication condition. As a new result, we show in SubSection 5.3 that sketch implications can be applied, via sketch morphisms as matches, to deduce new sketch implications and we prove that this kind of deduction is sound.

Section 6 shows that traditional first-order logic can be equivalently described by restricting the institutions ${\mathcal{FL}}_{\Sigma }$ to inclusion maps. As a suitable special kind of sketches we introduce Σ-specifications and also Σ-specification implications as a corresponding special kind of sketch implications. In Section 6.4 we prove that we can deduce $\Sigma$-specifications, in a sound way, by applying $\Sigma$-specification implications as rules via substitutions as matches.

In Section 7 we analyze and exhaustively re-validate a Hilbert System $\mathcal{H}$ and a Gentzen’s system $\mathcal{L}\mathcal{J}$ relying on the concepts context, Σ-specifications and Σ-specification implication. In Section 8 we do the same for a corresponding Natural Deduction System $\mathcal{N}\mathcal{D}$ and introduce the novel concept of open Σ-specification implication enabling us to finally license hypothetical reasoning in traditional first-order logic.

We conclude the paper with Section 9.

2. Notations and Preliminaries

The paper assumes some knowledge on category theory. We summarize in this section the basic concepts, constructions and results from category theory, utilized in the paper, and our corresponding notational conventions. We recommend [9] as an introduction into category theory for logicians and computer scientists.

Categories and Functors:$\left|\mathsf{C}\right|$ denotes the collection of objects of a category $\mathsf{C}$ and ${\mathsf{C}}_{Mor}$ the collection of morphisms of $\mathsf{C}$, respectively. $\mathsf{C}(A,B)$ is the collection of all morphisms from object A to object B in $\mathsf{C}$. If the category $\mathsf{C}$ is clear from the context, we will often use the more compact notation $B^A$ instead of $\mathsf{C}(A,B)$. We use the diagrammatic notation $f;g:A\to C$ for the composition of morphisms $f:A\to B$ and $g:B\to C$ in $\mathsf{C}$.

$\mathsf{Set}$ denotes the category of all sets and all (total) maps. For any inclusion $A\subseteq B$ of sets there is a corresponding inclusion map$i{n}_{A,B}:A\hookrightarrow B$ with $i{n}_{A,B}\left(a\right)=a$ for all $a\in A$. The cardinality of a finite set A is denoted by $\left|A\right|$.

An object $\mathbf{0}$ in a category $\mathsf{C}$ is called initial if there exist for any object A in $\mathsf{C}$ exactly one morphism from $\mathbf{0}$ into A often denoted by ${!}_A:{}^{\mathbf{0}}\to A$ and called initial morphism for A. The empty set ∅ is the initial object in $\mathsf{Set}$ while for any set A the corresponding initial morphism for A is simply the inclusion map ${!}_A=i{n}_{A,B}:\varnothing \to A$.

A category $\mathsf{C}$ is small if the collection ${\mathsf{C}}_{Mor}$, and thus also the collection $\left|\mathsf{C}\right|$, is a set. $\mathsf{Cat}$ is the category of all small categories. $\mathsf{Cat}$ and $\mathsf{Set}$ are not small! A category $\mathsf{C}$ is locally small if $\mathsf{C}(A,B)$ is a set for all objects A and B in $\mathsf{C}$. $\mathsf{Cat}$ and $\mathsf{Set}$ are locally small! $\mathsf{CAT}$ is the category with all small categories and categories like $\mathsf{Cat}$ and $\mathsf{Set}$ as objects.

For a category $\mathsf{C}$ we denote by $gr\left(\mathsf{C}\right)$ the underlying reflexive graph, i.e., in $gr\left(\mathsf{C}\right)$ we simply forget that there is a composition operation in $\mathsf{C}$. Obviously, each functor $\mathbf{F}:\mathsf{C}\to \mathsf{D}$ comprises a corresponding reflexive graph homomorphism$\mathbf{F}:gr\left(\mathsf{C}\right)\to gr\left(\mathsf{D}\right)$ between the underlying reflexive graphs. Not every reflexive graph homomorphism $\mathbf{G}:gr\left(\mathsf{C}\right)\to gr\left(\mathsf{D}\right)$ establishes, however, also a functor $\mathbf{G}:\mathsf{C}\to \mathsf{D}$ because a reflexive graph homomorphism $\mathbf{G}:gr\left(\mathsf{C}\right)\to gr\left(\mathsf{D}\right)$ is not required to be compatible with composition.

We consider also reflexive graph homomorphisms from reflexive graphs H into the underlying reflexive graph $gr\left(\mathsf{D}\right)$ of categories $\mathsf{D}$ and denote them by $\mathbf{F}:H\to \mathsf{D}$ (compare the concept of a model of a graph in a category in [9]). The traditional definition of natural transformations between functors can be reused to define natural transformations $\alpha :\mathbf{F}\Rightarrow \mathbf{G}$ between those reflexive graph homomorphisms $\mathbf{F},\mathbf{G}:H\to \mathsf{D}$.

Natural Transformations: If $\alpha :\mathbf{F}\Rightarrow \mathbf{G}:\mathsf{A}\to \mathsf{B}$ and $\beta :\mathbf{G}\Rightarrow \mathbf{H}:\mathsf{A}\to \mathsf{B}$ are natural transformations, the vertical composition of $\alpha$ and $\beta$ is denoted $\alpha ;\beta :\mathbf{F}\Rightarrow \mathbf{H}:\mathsf{A}\to \mathsf{B}$ such that for each $A\in \left|\mathsf{A}\right|$, ${(\alpha ;\beta )}_A:={\alpha }_A;{\beta }_A$.

Also, if $\mathbf{F}:\mathsf{A}\to \mathsf{B}$, $\mathbf{G},{\mathbf{G}}^{\prime }:\mathsf{B}\to \mathsf{C}$, $\mathbf{H}:\mathsf{C}\to \mathsf{D}$ are functors and $\alpha :\mathbf{G}\Rightarrow {\mathbf{G}}^{\prime }:\mathsf{B}\to \mathsf{C}$ is a natural transfomation, the horizontal compositions of $\mathbf{F}$ with $\alpha$, and $\alpha$ with $\mathbf{H}$ are represented as $\mathbf{F}\alpha :\mathbf{F};\mathbf{G}\Rightarrow \mathbf{F};{\mathbf{G}}^{\prime }:\mathsf{A}\to \mathsf{C}$ and $\alpha \mathbf{H}:\mathbf{G};\mathbf{H}\Rightarrow {\mathbf{G}}^{\prime };\mathbf{H}:\mathsf{B}\to \mathsf{D}$ such that for each $C\in \left|\mathsf{C}\right|$, $A\in \left|\mathsf{A}\right|$, ${\left(\mathbf{F}\alpha \right)}_C:={\alpha }_{\mathbf{F}\left(C\right)}$ whereas ${\left(\alpha \mathbf{H}\right)}_A:=\mathbf{H}\left({\alpha }_A\right)$.

The traditional definition of natural transformations $\alpha :\mathbf{F}\Rightarrow \mathbf{G}:\mathsf{A}\to \mathsf{B}$ between functors can be reused to define natural transformations $\alpha :\mathbf{F}\Rightarrow \mathbf{G}:gr\left(\mathsf{A}\right)\to gr\left(\mathsf{B}\right)$ for all (!) reflexive graph homomorphisms between the underlying reflexive graphs of categories (compare [9]).

Institutions: The concept of Institution introduced by Goguen and Burstall [19] formally captures the notion of logical systems. A similar proposal of an abstract concept of a logic had been given already by Barwise [21]. The following definition can be found in [19,20]: An institution $\mathcal{I}=(\mathsf{Sig},\mathbf{Sen},\mathbf{Mod},\vDash )$ is given by a category $\mathsf{Sig}$ of (abstract) signatures, a functor $\mathbf{Sen}:\mathsf{Sig}\to \mathsf{Set}$, a functor $\mathbf{Mod}:{\mathsf{Sig}}^{op}\to \mathsf{CAT}$ and a $\left|\mathsf{Sig}\right|$-indexed family ${\vDash }^{\Sigma }\subseteq \left|\mathbf{Mod}\left(\Sigma \right)\right|\times \left|\mathbf{Sen}\left(\Sigma \right)\right|$ of satisfaction relations such that for each morphism $\phi :\Sigma \to {\Sigma }^{\prime }$ in $\mathsf{Sig}$, the Satisfaction Condition

$$M^{\prime }{\vDash }^{{\Sigma }^{\prime }}\mathbf{Sen}\left(\phi \right)\left(e\right)ifandonlyif\mathbf{Mod}\left(\phi \right)\left(M^{\prime }\right){\vDash }^{\Sigma }e$$

holds for each (abstract) model $M^{\prime }\in \left|\mathbf{Mod}\left({\Sigma }^{\prime }\right)\right|$ and (abstract) sentence$e\in \left|\mathbf{Sen}\right(\Sigma \left)\right|$.

Name spaces: It is under-communicated that the definition of morphisms between concrete institutions often relies on the implicit assumption that the syntactic entities of both institutions are build upon the same stocks of symbol identifiers. We restrict ourselves to finitary syntactic entities. Therefore, we assume that all considered first-order institutions ${\mathcal{FL}}_{\Sigma }$ are build upon a certain choice of three enumerable and disjoint name spaces:

• A set $N_{fun}$ of names for operation symbols.
• A set $N_{pred}$ of names for predicate symbols.
• A set $N_{var}$ of names for variables which is equipped with a fixed total order.

We choose $N_{var}$ to be the totally ordered set $\{x,x_1,x_2,\dots ,y,y_1,y_2,\dots ,z,z_1,z_2,\dots \}$.

In the general LoSiC-terminology, introduced in [1], all institutions ${\mathcal{FL}}_{\Sigma }$ share the same base category$\mathsf{Base}:=\mathsf{Set}$.

3. Basic Concepts and Constructions in First-Order Logic

Relying on hom-sets, instead of Cartesian products, we recast in this section the definitions of the basic syntactic and semantic concepts of unsorted first-order logic.

3.1. Signatures and Structures

In this paper about first-order logic we use the traditional term signature instead of the neologism footprint coined for the general setting in [1]. Since we will neither misuse constant symbols to encode variables nor to encode elements of carriers of first-order structures, we can restrict ourselves to finite signatures only.

To achieve category-independent definitions, we require in [1] that the arities of predicate and operation symbols are defined by means of "variable declarations". In LoSiC-terminology all ${\mathcal{FL}}_{\Sigma }$ share the same category $\mathsf{Var}$ of "variable declarations". For unsorted first-order logic we simply define $\mathsf{Var}$ to be the set $Va{}^r:={\wp }_{fin}\left(N_{var}\right)$ of all finite subsets of $N_{var}$ (considered as a discrete subcategory of $\mathsf{Base}=\mathsf{Set}$). We will simply use the term set of variables, instead of "variable declaration", for those finite subsets $X\subset N_{var}$.

Definition 1

(Signature). Asignature$\Sigma =(P,F,ar,in,out)$ is given by

• a finite set $P\subset N_{pred}$ of predicate (relation) symbols,
• a finite set $F\subset N_{fun}$ of operation (function) symbols, and
• arity functions $ar:P\to Var$, $in:F\to Var$, $out:F\to Var$ where $out\left(\mathtt{op}\right)$ is a singleton for all $\mathtt{op}\in F$. We assume that $in\left(\mathtt{op}\right)$ and $out\left(\mathtt{op}\right)$ are disjoint for all $\mathtt{op}\in F$ (compare [22]).

We intend to be as precise as possible concerning notations. Therefore, we include in the paper many remarks devoted to bridge the notational gap between the category-independent notations in [1] and the traditional notations in first-order logic. We tried our best, but still the notations may become too heavy at some places and we apologize for this. Another essential part of the many remarks is devoted to explain the new or modified concepts and to discuss their relationships to traditional concepts.

Remark 1

(Representation and Syntactification of Sets of Variables). Any set X of variables inherits a total order from $N_{var}$. This allows us to represent X as an n-tuple $(v_1^X,\dots ,v_n^X)$ of variables with $n=\left|X\right|$ thecardinalityof X and $v_i^X$ denoting the i-th variable in the totally ordered set X for all $1\le i\le n$.

In such a way, we also can syntactically encode the set X of variables by the string of symbols $\mathbf{syn}\left(X\right):=[v_1^X,\dots ,v_n^X]$. We use the delimiter signs … to indicate that the expression between the delimiters is a string of symbols. So, the delimiter signs $\dots$ are not constituents of syntatctic entities and we may just drop them if convenient.

Example 1

(Signature). We define a signature $\Sigma =(P,F,ar,in,out)$ to describe and reason about structures with an irreflexive order relation, constants zero and one as well as binary operations addition and multiplication as follows:

• $P=\left\{\mathtt{less}\right\}$ with arity $ar\left(\mathtt{less}\right)=\{x_1,x_2\}$; and
• $F=\{\underline{0},\underline{1},\underline{+},\underline{*}\}$ with input arities $in\left(\underline{0}\right)=in\left(\underline{1}\right)=\varnothing$, $in\left(\underline{+}\right)=in\left(\underline{*}\right)=\{x_1,x_2\}$ and output arities $out\left(\underline{0}\right)=out\left(\underline{1}\right)=out\left(\underline{+}\right)=out\left(\underline{*}\right)=\left\{y\right\}$.

In unsorted first-order logic we traditionally assume that all structures do have a non-empty carrier set! This is a crucial prerequisite to achieve soundness of the traditional lazy approach to first-order deduction that avoids, as much as possible, to explicitly handle (sets of) variables! So, in LoSiC-terminology all institutions ${\mathcal{FL}}_{\Sigma }$ share the same category $\mathsf{Carr}:=\mathsf{Set}\setminus \{\varnothing \}$ of potential carriers of structures. After we this is clarified, we are going to define the category of structures for a given signature $\Sigma$.

Definition 2

(Structure). For a signature $\Sigma =(P,F,ar,in,out)$ a$\Sigma$-Structure$\mathcal{U}=(U,P^{\mathcal{U}},F^{\mathcal{U}})$ is given by

• a set U, called thecarrierof $\mathcal{U}$,
• a family $P^{\mathcal{U}}=\{{\mathtt{p}}^{\mathcal{U}}\mid \mathtt{p}\in P\}$ ofpredicates, i.e., subsets ${\mathtt{p}}^{\mathcal{U}}\subseteq U^{ar\left(\mathtt{p}\right)}=\mathsf{Set}(ar\left(\mathtt{p}\right),U)$ ofvalid assignmentsfor the predicate symbol $\mathtt{p}$ in $\mathcal{U}$, and
• a family $F^{\mathcal{U}}=\{{\mathtt{op}}^{\mathcal{U}}\mid \mathtt{op}\in F\}$ ofoperations (functions)${\mathtt{op}}^{\mathcal{U}}:U^{in\left(\mathtt{op}\right)}\to U^{out\left(\mathtt{op}\right)}$ in $\mathcal{U}$.

Remark 2

(Notation of Structures). We draw attention to a notational subtlety: We denote a structure and its underlying carrier set with the same letter but in different fonts. For the Σ-structure we use the calligraphic font $\mathcal{U}$ and for the carrier set we are using the normal font U. Note, that the superscripts for the predicates and operations are therefore also calligraphic!

Remark 3

(Hom-set vs. Cartesian Product). Attention! We don’t follow the traditionally way to use Cartesian products to define predicates and operations. We use hom-sets $\mathsf{Set}(X,U)$ instead!

The category $\mathsf{Set}$ is small thus the hom-set $\mathsf{Set}(X,U)$ is a set and can be treated, in a "self-referential way", as an object in $\mathsf{Set}$! Besides many other specialties of $\mathsf{Set}$, we do have that $\mathsf{Set}$(X,U) can be considered to "be the same" as the set $U^X$ of all maps from X to U, which is, in turn, anexponential objectin $\mathsf{Set}$. The exponential object $U^X$ is, however, also a (categorical) product in $\mathsf{Set}$. Especially, it is isomorphic to the Cartesian product $U^n=U\times \dots \times U$ with $n=\left|X\right|$!

For nearly any other category the equivalence between the concepts hom-set, exponential object, and (categorical) product vanishes! It took us a while to realize the disappearance of this "self-evident equivalence" in other categories and to find out, that only the hom-set perspective allows us to generalize first-order logic to arbitrary categories in the way it is outlined in [1].

Remark 4

(Representation of Variable Assignments). We consider a set X of variables with $n=\left|X\right|$. For a given set U, a map $\mathbf{a}:X\to U$ will be also called(variable) assignment of X in U.

There is a bijection between the set $U^X$ of all assignments of X in U and the Cartesian product $U^n$. That is, anyassignment$\mathbf{a}:X\to U$ can equivalently be represented as an n-tuple $\mathbf{a}=(a_1,\dots ,a_n)\in U^n$ with $a_i=\mathbf{a}\left(v_i^X\right)$ for all $1\le i\le n$ where the $v_i^X$ are defined as in Remark 1.

We utilize both notations $\mathbf{a}:X\to U$ and $\mathbf{a}=(a_1,\dots ,a_n)$ synonymously. Note, that in case of the empty set $X=\varnothing$ the only element ${!}_U:\varnothing \to U$ in $U^{\varnothing }$ is represented by theempty tuple$\left(\right)$.

Due to the hom-set perspective the translation of variable assignments along homomorphisms can be expressed by simple composition thus the homomorphism conditions appear in an unconventional but quite adequate form.

Definition 3

(Homomorphism). A$\Sigma$-homomorphism$\varsigma :\mathcal{U}\to \mathcal{V}$ between two Σ-structures $\mathcal{U}$ and $\mathcal{V}$ is given by a map $\varsigma :U\to V$ such that

1.

$\mathbf{a}\in {\mathtt{p}}^{\mathcal{U}}$ implies $\mathbf{a};\varsigma \in {\mathtt{p}}^{\mathcal{V}}$ for all $\mathtt{p}\in P$ and all assignments $\mathbf{a}:ar\left(\mathtt{p}\right)\to U$, i.e., the post-composition map $(\_;\varsigma ):U^{ar\left(\mathtt{p}\right)}\to V^{ar\left(\mathtt{p}\right)}$, with $(\_;\varsigma )\left(\mathbf{a}\right):=\mathbf{a};\varsigma$ for all $\mathbf{a}\in U^{ar\left(\mathtt{p}\right)}$, restricts to a map $(\_;\varsigma ):{\mathtt{p}}^{\mathcal{U}}\to {\mathtt{p}}^{\mathcal{V}}$.

2.

${\mathtt{op}}^{\mathcal{U}}\left(\mathbf{a}\right);\varsigma ={\mathtt{op}}^{\mathcal{V}}(\mathbf{a};\varsigma )$ for all $\mathtt{op}\in F$ and all input assignments $\mathbf{a}:in\left(\mathtt{op}\right)\to U$, i.e., ${\mathtt{op}}^{\mathcal{U}};(\_;\varsigma )=(\_;\varsigma );{\mathtt{op}}^{\mathcal{V}}$.

Remark 5

(Notation of homomorphisms). The convention to denote a Σ-structure and its carrier set by the same letter but in differents fonts, allows us to distinguish as well between different kinds of morphisms: When we write $\varsigma :U\to V$, we mean a map. When we write, however, $\varsigma :\mathcal{U}\to \mathcal{V}$, we mean a Σ-homomorphism. In such a way, the notation $\varsigma :\mathcal{U}\to \mathcal{V}$ carries the implicit information/assumption that ς is not only a map but satisfies, in addition, the homomorphism conditions.

Example 2

(Structures and Homomorphisms). For the sample signature $\Sigma =(P,F,ar,in,out)$ in Example 1 with $P=\left\{\mathtt{less}\right\}$, $F=\{\underline{0},\underline{1},\underline{+},\underline{*}\}$ we consider three Σ-structures:

1.

$\mathcal{N}=(\mathbb{N},P^{\mathcal{N}},F^{\mathcal{N}})$ with $\mathbb{N}$ the set of all natural numbers as carrier.

• ${\mathtt{less}}^{\mathcal{N}}\subseteq {\mathbb{N}}^{\{x_1,x_2\}}$ is the usual irreflexive order on $\mathbb{N}$: $\mathbf{a}\in {\mathtt{less}}^{\mathcal{N}}$ iff $\mathbf{a}\left(x_1\right)<\mathbf{a}\left(x_2\right)$.
• ${\underline{0}}^{\mathcal{N}}:{\mathbb{N}}^{\varnothing }\to {\mathbb{N}}^{\left\{y\right\}}$ is the natural number zero: ${\underline{0}}^{\mathcal{N}}\left({!}_{\mathbb{N}}\right)\left(y\right):=0$ for the only element ${!}_{\mathbb{N}}\in {\mathbb{N}}^{\varnothing }$. ${\underline{1}}^{\mathcal{N}}:{\mathbb{N}}^{\varnothing }\to {\mathbb{N}}^{\left\{y\right\}}$ is the natural number one: ${\underline{1}}^{\mathcal{N}}\left({!}_{\mathbb{N}}\right)\left(y\right):=1$.
• ${\underline{+}}^{\mathcal{N}}:{\mathbb{N}}^{\{x_1,x_2\}}\to {\mathbb{N}}^{\left\{y\right\}}$ is addition of natural numbers, i.e., for all $\mathbf{a}\in {\mathbb{N}}^{\{x_1,x_2\}}$ we do have ${\underline{+}}^{\mathcal{N}}\left(\mathbf{a}\right)\left(y\right):=\mathbf{a}\left(x_1\right)+\mathbf{a}\left(x_2\right)$. ${\underline{*}}^{\mathcal{N}}:{\mathbb{N}}^{\{x_1,x_2\}}\to {\mathbb{N}}^{\left\{y\right\}}$ is multiplication of natural numbers: ${\underline{*}}^{\mathcal{N}}\left(\mathbf{a}\right)\left(y\right):=\mathbf{a}\left(x_1\right)*\mathbf{a}\left(x_2\right)$ for all $\mathbf{a}\in {\mathbb{N}}^{\{x_1,x_2\}}$.

2.

$\mathcal{Q}=(\mathbb{Q},P^{\mathcal{Q}},F^{\mathcal{Q}})$ with $\mathbb{Q}$ the set of all rational numbers as carrier.

• ${\mathtt{less}}^{\mathcal{Q}}\subseteq {\mathbb{Q}}^{\{x_1,x_2\}}$ is the usual irreflexive order on $\mathbb{Q}$: $\mathbf{a}\in {\mathtt{less}}^{\mathcal{Q}}$ iff $\mathbf{a}\left(x_1\right)<\mathbf{a}\left(x_2\right)$.
• ${\underline{0}}^{\mathcal{Q}}:{\mathbb{Q}}^{\varnothing }\to {\mathbb{Q}}^{\left\{y\right\}}$ is the rational number zero: ${\underline{0}}^{\mathcal{Q}}\left({!}_{\mathbb{Q}}\right)\left(y\right):=0$ for the only element ${!}_{\mathbb{Q}}\in {\mathbb{Q}}^{\varnothing }$. ${\underline{1}}^{\mathcal{Q}}:{\mathbb{Q}}^{\varnothing }\to {\mathbb{N}}^{\left\{y\right\}}$ is the rational number one: ${\underline{1}}^{\mathcal{Q}}\left({!}_{\mathbb{Q}}\right)\left(y\right):=1$.
• ${\underline{+}}^{\mathcal{Q}}:{\mathbb{Q}}^{\{x_1,x_2\}}\to {\mathbb{Q}}^{\left\{y\right\}}$ is addition of rational numbers, i.e., for all $\mathbf{a}\in {\mathbb{Q}}^{\{x_1,x_2\}}$ we do have ${\underline{+}}^{\mathcal{Q}}\left(\mathbf{a}\right)\left(y\right):=\mathbf{a}\left(x_1\right)+\mathbf{a}\left(x_2\right)$. ${\underline{*}}^{\mathcal{Q}}:{\mathbb{Q}}^{\{x_1,x_2\}}\to {\mathbb{Q}}^{\left\{y\right\}}$ is multiplication of rational numbers: ${\underline{*}}^{\mathcal{Q}}\left(\mathbf{a}\right)\left(y\right):=\mathbf{a}\left(x_1\right)*\mathbf{a}\left(x_2\right)$ for all $\mathbf{a}\in {\mathbb{Q}}^{\{x_1,x_2\}}$.

3.

$\mathcal{P}=(\wp \left(\mathbb{N}\right),P^{\mathcal{P}},F^{\mathcal{P}})$ with the power set $\wp \left(\mathbb{N}\right)$ as carrier.

• ${\mathtt{less}}^{\mathcal{P}}\subseteq \wp {\left(\mathbb{N}\right)}^{\{x_1,x_2\}}$ is the irreflexive inclusion order: $\mathbf{a}\in {\mathtt{less}}^{\mathcal{P}}$ iff $\mathbf{a}\left(x_1\right)\subset \mathbf{a}\left(x_2\right)$.
• ${\underline{0}}^{\mathcal{P}}:\wp {\left(\mathbb{N}\right)}^{\varnothing }\to \wp {\left(\mathbb{N}\right)}^{\left\{y\right\}}$ is the empty set: ${\underline{0}}^{\mathcal{P}}\left({!}_{\wp \left(\mathbb{N}\right)}\right)\left(y\right):=\varnothing$ for the only element ${!}_{\wp \left(\mathbb{N}\right)}\in \wp {\left(\mathbb{N}\right)}^{\varnothing }$. ${\underline{1}}^{\mathcal{P}}:\wp {\left(\mathbb{N}\right)}^{\varnothing }\to \wp {\left(\mathbb{N}\right)}^{\left\{y\right\}}$ is the set $\mathbb{N}$: ${\underline{1}}^{\mathcal{P}}\left({!}_{\wp \left(\mathbb{N}\right)}\right)\left(y\right):=\mathbb{N}$.
• ${\underline{+}}^{\mathcal{P}}:\wp {\left(\mathbb{N}\right)}^{\{x_1,x_2\}}\to \wp {\left(\mathbb{N}\right)}^{\left\{y\right\}}$ is union of sets, i.e., for all $\mathbf{a}\in \wp {\left(\mathbb{N}\right)}^{\{x_1,x_2\}}$ we do have ${\underline{+}}^{\mathcal{P}}\left(\mathbf{a}\right)\left(y\right):=\mathbf{a}\left(x_1\right)\cup \mathbf{a}\left(x_2\right)$. ${\underline{*}}^{\mathcal{P}}:\wp {\left(\mathbb{N}\right)}^{\{x_1,x_2\}}\to \wp {\left(\mathbb{N}\right)}^{\left\{y\right\}}$ is intersection of sets: ${\underline{*}}^{\mathcal{P}}\left(\mathbf{a}\right)\left(y\right):=\mathbf{a}\left(x_1\right)\cap \mathbf{a}\left(x_2\right)$ for all $\mathbf{a}\in \wp {\left(\mathbb{N}\right)}^{\{x_1,x_2\}}$.

The inclusion map $i{n}_{\mathbb{N},\mathbb{Q}}:\mathbb{N}\to \mathbb{Q}$ obviously defines a Σ-homomorphism $i{n}_{\mathbb{N},\mathbb{Q}}:\mathcal{N}\to \mathcal{Q}$, while the embedding map $\{\_\}:\mathbb{N}\to \wp \left(\mathbb{N}\right)$ assigning to each natural number $n\in \mathbb{N}$ the singleton $\left\{n\right\}$ does not provide a Σ-homomorphism from $\mathcal{N}$ to $\mathcal{P}$.

For any $\Sigma$-structure $\mathcal{U}$ the identity map $i{d}_U:U\to U$ obviously establishes a $\Sigma$-homomorphism $\varsigma :\mathcal{U}\to \mathcal{U}$. Moreover, it can easily be shown that for any $\Sigma$-homomorphisms $\varsigma :\mathcal{U}\to \mathcal{V}$, $\epsilon :\mathcal{V}\to \mathcal{W}$ the composition $\varsigma ;\epsilon :U\to W$ of the underlying maps becomes also a $\Sigma$-homomorphism $\varsigma ;\epsilon :\mathcal{U}\to \mathcal{W}$. Identity and associativity law for composition are inherited from the category $\mathsf{Set}$, thus we obtain the category of all $\Sigma$-structures and all $\Sigma$-homomorphisms denoted by $\mathsf{Str}\left(\Sigma \right)$.

3.2. Terms and First-Order Expressions: Syntax

We consider terms and first-order expressions as syntactic entities and define them as finite strings of symbols. To distinguish terms from meta-level expressions, such as ${\mathtt{op}}^{\mathcal{U}}(a_1,\dots ,a_n)$, we will use angle bracket symbols $"\langle ","\rangle "$, instead of parenthesis $"(",")"$, to build terms. As discussed in Remark 1, we will use delimiter signs … to indicate that the expression between the delimiters is a string of symbols.

The following is a traditional inductive definition of terms similar to [5,6,7] and relies on the notational conventions in Remark 4:

Definition 4

(Terms: Syntax). The set $T_{\Sigma }\left(X\right)$ of all$\Sigma$-termsover a set X of variables is the smallest set of strings of symbols such that

Variables:

$x\in T_{\Sigma }\left(X\right)$ for all $x\in X$;

Constants:

$\mathtt{c}\langle \rangle \in T_{\Sigma }\left(X\right)$ for all $\mathtt{c}\in F$ with $in\left(\mathtt{c}\right)=\varnothing$;

Operations:

$\mathtt{op}\langle t_1,\dots ,t_n\rangle \in T_{\Sigma }\left(X\right)$ for all $\mathtt{op}\in F$ with $n=\left|in\right(\mathtt{op}\left)\right|\ge 1$ and all assignments $\mathbf{t}=(t_1,\dots ,t_n)$ in $T_{\Sigma }{\left(X\right)}^{in\left(\mathtt{op}\right)}$.

Note that the assignments $x\mapsto x$, assigning to each element in X the string constituted exactly of this single symbol, define an injective map ${\eta }_X$: $X\to T_{\Sigma }\left(X\right)$.

Note further, that each operation symbol $\mathtt{op}\in F$ with $n=\left|in\right(\mathtt{op}\left)\right|\ge 1$ is reborn as the $\Sigma$-term $\mathtt{op}\langle v_1^{in\left(\mathtt{op}\right)},\dots ,v_n^{in\left(\mathtt{op}\right)}\rangle \in T_{\Sigma }{\left(in\left(\mathtt{op}\right)\right)}_{srt\left(\mathtt{op}\right)}$ with the $v_i^{in\left(\mathtt{op}\right)}$ as in Remark 4.

Examples of terms can be found in Example 3 (Expressions: Syntax).

Remark 6

(Terms: Inclusions). We do have $T_{\Sigma }\left(Y\right)\subseteq T_{\Sigma }\left(X\right)$ whenever $Y\subseteq X$. For a set X of variables not every Σ-term in $T_{\Sigma }\left(X\right)$ "syntactically contains" all the variables in X. The set $fv\left(t\right)$ of all variables in X, "syntactically appearing" in a Σ-term $t\in T_{\Sigma }\left(X\right)$, is the smallest subset $Y\subseteq X$ such that $t\in T_{\Sigma }\left(Y\right)$. Inductively, we can define $fv\left(t\right)$ as follows: $fv\left(x\right):=\left\{x\right\}$, $fv(\mathtt{c}\langle \rangle ):=\varnothing$ and $fv\left(\mathtt{op}\langle t_1,\dots ,t_n\rangle \right):=fv\left(t_1\right)\cup \dots \cup fv\left(t_n\right)$.

Remark 7

(Syntactification of Substitutions). A variable assignment of the form $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ will be also called a$\Sigma$-substitution (declaration). Due to Remark 4, we can uniquely represent any Σ-substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ by an n-tuple $(t_1,\dots ,t_n)$ of terms where $n=\left|X\right|$.

What we implicitly did in Definition 4, is to introduce the correspondingsyntactic encoding$\mathbf{syn}\left(\mathbf{t}\right):=\langle t_1,\dots ,t_n\rangle$ of the substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ as a string of symbols.

For the injective map ${\eta }_X$: $X\to T_{\Sigma }\left(X\right)$ we get $\mathbf{syn}\left({\eta }_X\right):=\langle v_1^X,\dots ,v_n^X\rangle$ with variables $v_i^X$ due to Remark 1. In case of the empty set $X=\varnothing$ the only element ${!}_{T_{\Sigma }\left(Y\right)}:\varnothing \to T_{\Sigma }\left(Y\right)$ in $T_{\Sigma }{\left(Y\right)}^{\varnothing }$ is represented by theempty tuple$\left(\right)$ thus we have $\mathbf{syn}\left(\right(\left)\right)=\langle \rangle$.

We introduce first-order expressions as the basic syntactic entities in our formalization of the informal concept of an "open formula". As syntactic entities we define them as strings of symbols relying on the notational conventions in Remark 1 concerning the syntactic representation of finite sets of variables and in Remark 7 concerning the syntactic representation of substitution declarations.

Definition 5

(Expressions: Syntax). For a signature $\Sigma =(P,F,ar,in,out)$, we define inductively and in parallel a family $F{E}_{\Sigma }$ of sets $F{E}_{\Sigma }\left(X\right)$ of(first-order) $\Sigma$-expressions $Ex$ on X, $X▹Ex$ in symbols, where X varies over all sets of variables, i.e., all elements in $Var$:

1.

Atomic expressions:

(a)

Equation: $X▹(t_1=t_2)$ for any Σ-terms $t_1,t_2\in T_{\Sigma }\left(X\right)$.

(b)

Relational Atom: $X▹\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)$ for any $\mathtt{p}\in P$ and any $\mathbf{t}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(X\right)$.

2.

Everything: $X▹\top \langle \rangle$ for any set X of variables.

3.

Void: $X▹\perp \langle \rangle$ for any set X of variables.

4.

Conjunction: $X▹(E{x}_1\wedge E{x}_2)$ for any expressions $X▹E{x}_1$ and $X▹E{x}_2$.

5.

Disjunction: $X▹(E{x}_1\vee E{x}_2)$ for any expressions $X▹E{x}_1$ and $X▹E{x}_2$.

6.

Implication: $X▹(E{x}_1\to E{x}_2)$ for any expressions $X▹E{x}_1$ and $X▹E{x}_2$.

7.

Negation: $X▹\neg Ex$ for any expression $X▹Ex$.

8.

Quantification: $X▹\exists \left(\mathbf{syn}\right(Y\setminus X):Ex)$ and $X▹\forall \left(\mathbf{syn}\right(Y\setminus X):Ex)$ for any expression $Y▹Ex$ and any proper inclusion $X\subset Y$.

Remark 8

(Expressions: Indexed vs Fibred Syntax). We construct in Definition 5 a $Var$-indexed family of sets, i.e., a map $F{E}_{\Sigma }:Va{}^r\to \left|\mathsf{Set}\right|$. The elements of $F{E}_{\Sigma }\left(X\right)$ are expressions $Ex$ on X. Obviously, there are for any expression $Ex$ infinite many different sets X of variables such that $Ex\in F{E}_{\Sigma }\left(X\right)$! The notation $X▹Ex$ simply encodes the "meta-level statement" $Ex\in F{E}_{\Sigma }\left(X\right)$ and serves, at the same time, as a notational means to describe the disjoint union of all the sets $F{E}_{\Sigma }\left(X\right)$ as the set $\uplus F{E}_{\Sigma }:=\{X▹Ex\mid X\in Va{}^r,Ex\in F{E}_{\Sigma }\left(X\right)\}$.

So, the syntax of expressions is described in two equivalent versions - theindexed version$F{E}_{\Sigma }:Va{}^r\to \left|\mathsf{Set}\right|$ and thefibred versiongiven by $\uplus F{E}_{\Sigma }$ and the projection ${\Pi }^V:\uplus F{E}_{\Sigma }\to Var$ with ${\Pi }^V(X▹Ex):=X$ for all $X▹Ex\in \uplus F{E}_{\Sigma }$. We will utilize both versions in parallel!

Remark 9

(Syntax of expressions). Every predicate symbol $\mathtt{p}\in P$ is reborn as the Σ-expression $ar\left(\mathtt{p}\right)▹\mathtt{p}\langle v_1^{ar\left(\mathtt{p}\right)},\dots ,v_n^{ar\left(\mathtt{p}\right)}\rangle$ since $\mathbf{syn}\left({\eta }_{ar\left(\mathtt{p}\right)}\right)=\langle v_1^{ar\left(\mathtt{p}\right)},\dots ,v_n^{ar\left(\mathtt{p}\right)}\rangle$ with $n=\left|ar\right(\mathtt{p}\left)\right|$ according to Remark 7.

There are three changes/corrections compared to the category-independent Definition 8 of expressions in [1] where quantification is defined by means of morphisms since morphisms are the only ingredients we do have available in arbitrary categories.

First, we do not use arbitrary maps between sets of variables but only proper inclusion maps $i{n}_{X,Y}:X\to Y$ in quantifications. Second, these inclusion maps are not considered to be morphisms in the category $\mathsf{Var}$ of "variable declarations" for the institutions of statements ${\mathcal{FL}}_{\Sigma }$ defined in Section 4.1. (This is a correction!) Third, we adapt the traditional notation for quantification, i.e., we represent the proper inclusion maps $i{n}_{X,Y}:X\to Y$ by the complements $Y\setminus X$ in $\mathsf{Set}$. Be aware, that the sets X, $Y\setminus X$ are disjoint and that Y can be reconstructed as their union $Y=X\cup (Y\setminus X)$.

Remark 10

(Free and Bound Variables). Relying on Definition 4 and 5, we can inductively define for any expression $Ex$ the set $fv\left(Ex\right)$ of allfree variables"syntactically appearing" in $Ex$ and the set $bv\left(Ex\right)$ of allbound variablesin $Ex$. The basic case "Atomic expressions" is given by Remark 6 for free variables and the fact that terms do not have bound variables. Induction passes trivially through the con nectives. The essential deduction step is "Quantification".

$$fv(\exists (\mathbf{syn}(Y\setminus X):Ex\left)\right)=fv(\forall (\mathbf{syn}(Y\setminus X):Ex\left)\right):=fv\left(Ex\right)\setminus (Y\setminus X)$$

(1)

$$bv(\exists (\mathbf{syn}(Y\setminus X):Ex\left)\right)=bv(\forall (\mathbf{syn}(Y\setminus X):Ex\left)\right):=bv\left(Ex\right)\cup (Y\setminus X)$$

(2)

Attention, the definition of quantification in Definition 5 ignores the free variables in $Ex$. So, it may happen that none of the variables in $Y\setminus X$ actually appears as a free variable in $Ex$!

We witness here a kind of "methodological accident" in the sense that the instantiation of our category independent approach to quantification in [1] reinvents, in the special case of traditional first-order logic, a well-known principle.

Our approach, to explicitly declare first all the variables $X▹\dots$, which are available, before actually constructing expressions using only the variables in X, together with our requirement $X\subset Y$ for quantification has a subtle but important consequence. If $X▹Ex$ then all the free variables in $Ex$ are contained in X, i.e., $fv\left(Ex\right)\subseteq X$, and none of the variables in X appears as a "bound variable" in $Ex$, i.e., $bv\left(Ex\right)\cap X=\varnothing$. Other variables than the ones in X can appear, however, multiple times as bound variables in $Ex$ but only in non-overlapping "sub-expressions". These restrictions correspond toBarendregt’s variable conventionfor the λ-calculus (see [18], p. 26):

• bound variables are distinct from free variables, i.e., $bv\left(Ex\right)\cap fv\left(Ex\right)=\varnothing$, and
• all binders bind variables not already in scope, i.e., $(Y\setminus X)\cap bv\left(Ex\right)=\varnothing$ in case of quantification with $Y▹Ex$ and $X\subset Y$.

To say it the other way around: Any traditional "well-formed formula", in the sense of [4] for example, complying with Barendregt’s variable convention corresponds to an expression in the sense of Definition 5 since the convention ensures that we can unambiguously construct the proper inclusions $X\subset Y$ we need to define the case Quantification in Definition 5 (see also Remark (9)).

Compliance with Barendregt’s variable convention ensures that we can characterize $fv\left(Ex\right)$ as the smallest set Z of variables such that $Z▹Ex$.

Lemma 1

(Reducing Variable Declarations). For any set X of variables and any Σ- expression $Ex\in F{E}_{\Sigma }\left(X\right)$, i.e., $X▹Ex$, we also have $Ex\in F{E}_{\Sigma }\left(Z\right)$, i.e., $Z▹Ex$, for all sets Z of variables with $fv\left(Ex\right)\subseteq Z\subset X$.

Proof. 

By induction on expressions according to Definition 5. For atomic expressions the claim is obvious since $t\in T_{\Sigma }\left(fv\left(t\right)\right)\subseteq T_{\Sigma }\left(Z\right)\subset T_{\Sigma }\left(X\right)$ for any $t\in T_{\Sigma }\left(X\right)$ if $fv\left(t\right)\subseteq Z\subset X$, due to Remark 6. Induction passes trivially through the connectives.

In case of quantification $X▹Q\left(\mathbf{syn}\right(Y\setminus X):Ex)$ with $Q\in \{\exists ,\forall \}$ let Z be arbitrary with $fv\left(Q\right(\mathbf{syn}(Y\setminus X):Ex\left)\right)\subseteq Z\subset X$. Due to Definition 5 and the definition of free variables in (1), we get $fv\left(Ex\right)\subseteq Z\cup (Y\setminus X)\subset X\cup (Y\setminus X)=Y$ with $Z\cap (Y\setminus X)=\varnothing$ and thus $(Z\cup (Y\setminus X\left)\right)\setminus Z=Y\setminus X$. By Induction Hypothesis, $(Z\cup (Y\setminus X\left)\right)▹Ex$ and thus $Z▹Q\left(\mathbf{syn}\right(Y\setminus X):Ex)$ due to Definition 5. □

Remark 11

(Everything and Void). We consider ⊤ and ⊥ not aslogical constantsbut asbuilt-in nullary predicate symbols, i.e., $ar(\top )=ar(\perp )=\varnothing$. Therefore, we are not using traditional constant symbols likeTorF, for example. Built-in, means, especially, that ⊤ and ⊥ do have a fixed semantics for any Σ-structure $\mathcal{U}=(U,P^{\mathcal{U}},F^{\mathcal{U}})$, namely ${\perp }^{\mathcal{U}}=\varnothing$ and ${\top }^{\mathcal{U}}=U^{\varnothing }=\{{!}_U:\varnothing \to U\}$. Be aware, that the string $\langle \rangle$ in $X▹\top \langle \rangle$ and $X▹\perp \langle \rangle$ encodes the initial morphism ${!}_{T_{\Sigma }\left(X\right)}:\varnothing \to T_{\Sigma }\left(X\right)$. Analogously, the equation symbol "=" represents a binary built-in predicate symbol with arity $ar(=)=\{x_1,x_2\}$ and a fixed semantics in any Σ-structure. Finally, built-in means that none of the symbols $=,\perp ,\top$ appears in any of the sets $N_{pred}$, $N_{fun}$ or $N_{var}$.

Remark 12

(Closed expressions: Syntax). Σ-expressions of the form $\varnothing ▹Ex$ will be calledclosed $\Sigma$-expressions. They represent nullary derived predicates that can be either valid or not valid in a given Σ-structure (compare Remark 11 and Remark 16).

Example 3

(Expressions: Syntax). We discuss some Σ-expressions for our sample signature $\Sigma =(P,F,ar,in,out)$ with $P=\left\{\underline{<}\right\}$, $F=\{\underline{0},\underline{1},\underline{+},\underline{*}\}$ in Example 1.

The following closed Σ-expression, which we identify by the "auxiliary name" $\mathtt{madd}$,

$\mathtt{madd}:=\varnothing ▹\forall (\{x,y,z\}:\underline{<}\langle x,y\rangle ⟶\underline{<}\langle \underline{+}\langle x,z\rangle ,\underline{+}\langle y,z\rangle \rangle )$

claims that the order relation is strict monoton w.r.t. addition. We can also express De Morgan laws

$\mathtt{demo}:=\varnothing ▹\forall (\{x,y,z\}:(\underline{+}\langle x,\underline{*}\langle y,z\rangle \rangle =\underline{*}\langle \underline{+}\langle x,y\rangle ,\underline{+\langle x,z\rangle \rangle })$

Our main methodological point is, however, to consider expressions as syntactic representations ofderived predicates,enabling us to denote properties in an anonymous way (analogously to "anonymous functions" in functional programming).

We may, for example, define the property "even" by the following Σ-expression

$\mathtt{even}:=\left\{x\right\}▹\exists (\left\{y\right\}:x=\underline{*}\langle y,\underline{+}\langle \underline{1},\underline{1}\rangle \rangle )$,

which we consider as a derived predicate with arity $\left\{x\right\}$ and "auxiliary name" $\mathtt{even}$. We may be also interested to define the property "minimum"

$\mathtt{min}:=\left\{x\right\}▹\forall (\left\{y\right\}:(x=y)\vee \underline{<}\langle x,y\rangle )$

and the property "local minimum"

$\mathtt{lmin}:=\left\{x\right\}▹\neg (\exists (\left\{y\right\}:\underline{<}\langle y,x\rangle ))$.

Or, we want to have the concept "immediate successor" at hand

$\mathtt{succ}:=\{x,y\}▹(\underline{<}\langle x,y\rangle \wedge \neg (\exists (\left\{z\right\}:\underline{<}\langle x,z\rangle \wedge \underline{<}\langle z,y\rangle )))$.

We may be even interested to talk about "prime numbers"

$\mathtt{prim}:=\left\{x\right\}▹(\underline{<}(\underline{1},x)\wedge \forall (\{y,z\}:x=\underline{*}\langle y,z\rangle ⟶((y=\underline{1}\wedge z=x)\vee (y=x\wedge z=\underline{1}))))$.

We may also introduce a tertiary predicate "sum" representing the graph of the addition operation

$\mathtt{sum}:=\{x,y,z\}▹(\underline{+}\langle x,y\rangle =z)$.

Remark 13

(Role of Auxiliary Names). The auxiliary names $\mathtt{madd}$, $\mathtt{demo}$, $\mathtt{even}$, $\mathtt{min}$, $\mathtt{lmin}$, $\mathtt{succ}$, $\mathtt{sum}$ in Example 3 are not auxiliary predicate symbols! They are shorthands for Σ-expressions and live on the "meta-level". They are something likemacrosin assembly programming. The auxiliary names bring us closer to the praxis of "working mathematicians":

• We define new properties/concepts and give them a name to be able to formulate our statements in a more natural language style, like "the sum of two even numbers is also even" for example.
• To formally prove those statements, we have to unfold our definitions. In assembly programming they use the termmacro expansionfor this kind of unfolding.
• On the other hand, we need a mechanism to "fold" our proofs into those natural language like statements.

A first variant of such a "folding of proofs" is discussed in Section 8.2.

3.3. Terms and First-Order Expressions: Semantics

As mentioned in Section 1, the semantics of a $\Sigma$-term in a given $\Sigma$-structure should be a derived operation. To define those operations, we consider the evaluation of $\Sigma$-terms in $\Sigma$-structures: Let a $\Sigma$-structure $\mathcal{U}=(U,P^{\mathcal{U}},F^{\mathcal{U}})$ be given. Based on Definition 4 and employing the fixed semantics ${\mathtt{op}}^{\mathcal{U}}$ of all operation symbols in F, we can inductively extend any assignment $\mathbf{a}:X\to U$ of a set X of variables in the carrier U to a map ${\mathbf{a}}^{\circ }:T_{\Sigma }\left(X\right)\to U$ such that

This allows us to define the semantics $t^{\mathcal{U}}$ of a $\Sigma$-term$t\in T_{\Sigma }\left(X\right)$ in a $\Sigma$-structure $\mathcal{U}$ as the map $t^{\mathcal{U}}:U^X\to U$ defined by $t^{\mathcal{U}}\left(\mathbf{a}\right):={\mathbf{a}}^{\circ }\left(t\right)$ for all $\mathbf{a}:X\to U$. We call $t^{\mathcal{U}}$ the derived operation in $\mathcal{U}$ represented by t. Each variable $x\in T_{\Sigma }\left(X\right)$ represents a projection$x^{\mathcal{U}}:U^X\to U$. We can go even further: Any $\Sigma$-substitution $\mathbf{t}:Y\to T_{\Sigma }\left(X\right)$ defines a derived operation${\mathbf{t}}^{\mathcal{U}}:U^X\to U^Y$ in $\mathcal{U}$ (in the opposite direction!) with

$${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right):=\mathbf{t};{\mathbf{a}}^{\circ }forall\mathbf{a}\in U^X.$$

(4)

Since $U^{\varnothing }$ is a singleton, the inclusion map ${!}_{T_{\Sigma }\left(X\right)}:\varnothing \hookrightarrow T_{\Sigma }\left(X\right)$ represents a constant operation ${!}_{T_{\Sigma }\left(X\right)}^{\mathcal{U}}:U^X\to U^{\varnothing }$. Moreover, the canonical injective map ${\eta }_X$: $X\to T_{\Sigma }\left(X\right)$ represents the identity map on $U^X$:

$${\eta }_X^{\mathcal{U}}=i{d}_{U^X}:U^X\to U^{X}dueto\left(\right)since{\eta }_X;{\mathbf{a}}^{\circ }=\mathbf{a}.$$

(5)

Following a popular and convenient tradition in logic (see [3], p. 66) we define a satisfaction relation $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹Ex$ between assignments $\mathbf{a}:X\to U$ of a set X of variables in a $\Sigma$-structure $\mathcal{U}$ and $\Sigma$-expressions $X▹Ex$ on X. We say that "$(\mathbf{a},\mathcal{U})$ is a valid assignments for $X▹Ex$" or "$X▹Ex$ is valid for $(\mathbf{a},\mathcal{U})$".

Given an inclusion $X\subseteq Y$, we say that an assignment $\mathbf{b}:Y\to U$ is an extension of an assignment $\mathbf{a}:X\to U$ to Y if, and only if, $i{n}_{X,Y};\mathbf{b}=\mathbf{a}$.

Definition 6

(Satisfaction Relation: Assignments and Expressions). For a signature $\Sigma =(P,F,ar,in,out)$, we define inductively and in parallel for an arbitrary, but fixed, a Σ-structure $\mathcal{U}$ asatisfaction relation$(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹Ex$ between assignments $\mathbf{a}\in U^X$ of a set X of variables in $\mathcal{U}$ and Σ-expressions $X▹Ex$ on X. X varies over all sets of variables, i.e., all elements in $Var$.

1.

Atomic expressions:

(a)

Equation: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹(t_1=t_2)$ iff $t_1^{\mathcal{U}}\left(\mathbf{a}\right)=t_2^{\mathcal{U}}\left(\mathbf{a}\right)$ for $t_1,t_2\in T_{\Sigma }\left(X\right)$.

(b)

Relational Atom: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)$ iff ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)\in {\mathtt{p}}^{\mathcal{U}}$ for $\mathbf{t}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(X\right)$.

2.

Everything: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹\top \langle \rangle$ for all $\mathbf{a}\in U^X$

3.

Void: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹\perp \langle \rangle$ for none $\mathbf{a}\in U^X$

4.

Conjunction: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹(E{x}_1\wedge E{x}_2)$ iff $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹E{x}_1$ and $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹E{x}_2$

5.

Disjunction: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹(E{x}_1\vee E{x}_2)$ iff $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹E{x}_1$ or $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹E{x}_2$

6.

Implication: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹(E{x}_1\to E{x}_2)$ iff $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹E{x}_1$ implies $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹E{x}_2$

7.

Negation: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹\neg Ex$ iff not $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹Ex$

8.

Existential quantification: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹\exists (\mathbf{syn}(Y\setminus X):Ex)$ iff there exists an extension $\mathbf{b}:Y\to U$ of $\mathbf{a}:X\to U$ to Y such that $(\mathbf{b},\mathcal{U}){\vDash }^{\Sigma }Y▹Ex$

Universal quantification: $(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹\forall (\mathbf{syn}(Y\setminus X):Ex)$ iff for all extensions

$\mathbf{b}:Y\to U$ of $\mathbf{a}:X\to U$ to Y we have $(\mathbf{b},\mathcal{U}){\vDash }^{\Sigma }Y▹Ex$

We follow the "methodological imperative" that every syntactic entity should have a semantics in any structure. The semantics of a set X of variables in a $\Sigma$-structure $\mathcal{U}$$\mathcal{U}=(U,P^{\mathcal{U}},F^{\mathcal{U}})$ is simply the set ${\left[\left[X\right]\right]}^{\mathcal{U}}:=U^X$ of all assignments of X in U.

We consider first-order $\Sigma$-expressions as syntactic representations of derived predicates, i.e., the semantics of a $\Sigma$-expression $X▹Ex$ in $\mathcal{U}$ should be a subset of ${\left[\left[X\right]\right]}^{\mathcal{U}}$. The satisfaction relation defined in Definition 6 can be equivalently represented by those subsets. For all sets X of variables, all $\Sigma$-structures $\mathcal{U}$ and all $\Sigma$-expressions $X▹Ex$ we set

$${\left[[X▹Ex]\right]}^{\mathcal{U}}={\left[\left[Ex\right]\right]}_X^{\mathcal{U}}:=\{\mathbf{a}\in U^X\mid (\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }X▹Ex\}\subseteq U^X.$$

(6)

We can, however, also define the hom-set predicates ${\left[\left[Ex\right]\right]}_X^{\mathcal{U}}\subseteq U^X=\mathsf{Set}(X,U)$ independent of Definition 6, in a more compact and structured way, relying on the partial orders $(\wp \left(U^X\right),\subseteq )$, the respective Boolean algebras $(\wp \left(U^X\right),\cap ,\cup ,\overline{0ex1.5ex\_0ex1.5ex},U^X,\varnothing )$ and the monotone shift, existential and universal operators introduced and elaborated in Appendix A.

Definition 7

(Expressions: Semantics). The semantics of Σ-expressions in an arbitrary, but fixed, Σ-structure $\mathcal{U}=(U,P^{\mathcal{U}},F^{\mathcal{U}})$ is defined inductively:

1.

Atomic expressions:

(a)

Equation: ${\left[[t_1=t_2]\right]}_X^{\mathcal{U}}:=\{\mathbf{a}\in U^X\mid t_1^{\mathcal{U}}\left(\mathbf{a}\right)=t_2^{\mathcal{U}}\left(\mathbf{a}\right)\}$ for $t_1,t_2\in T_{\Sigma }\left(X\right)$ is nothing but the equalizer of the derived operations $t_1^{\mathcal{U}},t_2^{\mathcal{U}}:U^X\to U$.

(b)

Relational Atom: ${\left[\left[\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)\right]\right]}_X^{\mathcal{U}}{:=\downarrow }_{\mathbf{t}}^X\left({\mathtt{p}}^{\mathcal{U}}\right)={\left({\mathbf{t}}^{\mathcal{U}}\right)}^{-1}\left({\mathtt{p}}^{\mathcal{U}}\right)$ for $\mathbf{t}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(X\right)$.

2.

Everything: ${\left[[\top \langle \rangle ]\right]}_X^{\mathcal{U}}:={\left[\left[X\right]\right]}^{\mathcal{U}}=U^X$

3.

Void: ${\left[[\perp \langle \rangle ]\right]}_X^{\mathcal{U}}:=\varnothing$

4.

Conjunction: ${\left[\left[(E{x}_1\wedge E{x}_2)\right]\right]}_X^{\mathcal{U}}:={\left[\left[E{x}_1\right]\right]}_X^{\mathcal{U}}\cap {\left[\left[E{x}_2\right]\right]}_X^{\mathcal{U}}$

5.

Disjunction: ${\left[\left[(E{x}_1\vee E{x}_2)\right]\right]}_X^{\mathcal{U}}:={\left[\left[E{x}_1\right]\right]}_X^{\mathcal{U}}\cup {\left[\left[E{x}_2\right]\right]}_X^{\mathcal{U}}$

6.

Implication: ${\left[[E{x}_1\to E{x}_2]\right]}_X^{\mathcal{U}}:=\overline{{\left[\left[E{x}_1\right]\right]}_X^{\mathcal{U}}}\cup {\left[\left[E{x}_2\right]\right]}_X^{\mathcal{U}}$

7.

Negation: ${\left[[\neg Ex]\right]}_X^{\mathcal{U}}:=\overline{{\left[\left[Ex\right]\right]}_X^{\mathcal{U}}}$

8.

Existential quantification: ${\left[[\exists (\mathbf{syn}(Y\setminus X):Ex)]\right]}_X^{\mathcal{U}}:={\exists }_{i{n}_{X,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}$

Universal quantification: ${\left[[\forall (\mathbf{syn}(Y\setminus X):Ex)]\right]}_X^{\mathcal{U}}:={\forall }_{i{n}_{X,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}$.

Remark 14

(Expressions: Indexed vs Fibred Semantics). In view of Remark 8, we construct in Definition 7 an $\left|{\mathsf{Str}}_{\Sigma }\right|$-indexed family of "natural transformations". For each Σ-structure $\mathcal{U}$ we define a $Var$-indexed family of maps ${\left[[\_]\right]}_X^{\mathcal{U}}:F{E}_{\Sigma }\left(X\right)\to \wp \left(U^X\right)$ which establishes a "natural transformation" between the map $F{E}_{\Sigma }:Va{}^r\to \left|\mathsf{Set}\right|$ and the map $\wp \left(\mathsf{Set}(\_,U)\right):Va{}^r\to \left|\mathsf{Set}\right|$ assigning to each set X of variables the power set $\wp \left(\mathsf{Set}(X,U)\right)=\wp \left(U^X\right)$

An equivalent, more fibred view is that the assignments $X▹Ex\mapsto {\left[[X▹Ex]\right]}^{\mathcal{U}}$ define for each Σ-structure $\mathcal{U}$ a map ${\left[[\_▹\_]\right]}^{\mathcal{U}}$ from $\uplus F{E}_{\Sigma }$ into $\bigcup \{\wp \left(U^X\right)\mid X\in Va{}^r\}$ with ${\Pi }^V={\left[[\_▹\_]\right]}^{\mathcal{U}};{\Pi }^U$ where ${\Pi }^U$ is the projection from $\bigcup \{\wp \left(U^X\right)\mid X\in Va{}^r\}$ into $Var$. Keep in mind that all the hom-sets in a category are assumed to be disjoint!

Remark 15

(Expressions: Semantics). As mentioned in Remark 9, every predicate symbol $\mathtt{p}\in P$ is reborn as the Σ-expression $ar\left(\mathtt{p}\right)▹\mathtt{p}\mathbf{syn}\left({\eta }_{ar\left(\mathtt{p}\right)}\right)$ with ${\eta }_{ar\left(\mathtt{p}\right)}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(ar\left(\mathtt{p}\right)\right)$. Definition 6 or 7 and Equation (5) ensure that also their semantics coincides ${\left[\left[\mathtt{p}\mathbf{syn}\left({\eta }_{ar\left(\mathtt{p}\right)}\right)\right]\right]}_{ar\left(\mathtt{p}\right)}^{\mathcal{U}}={\mathtt{p}}^{\mathcal{U}}$.

The universal quantification $\forall \left(\mathbf{syn}\right(Y\setminus X):Ex)$ is trivially valid for $\mathbf{a}$ if there is no extension of $\mathbf{a}$ to Y at all, while the existential quantification $\exists \left(\mathbf{syn}\right(Y\setminus X):Ex)$ is not valid, in this case.

Two expressions $X▹E{x}_1$ and $X▹E{x}_2$ aresemantical equivalent, $X▹E{x}_1\equiv E{x}_2$ in symbols, if, and only if, ${\left[\left[E{x}_2\right]\right]}_X^{\mathcal{U}}={\left[\left[E{x}_2\right]\right]}_X^{\mathcal{U}}$ for all Σ-structures $\mathcal{U}$. Definition 7 ensures that we do have the usual semantic equivalences available. In particular, conjunction and disjunction are associative; thus we can drop, for convenience, the corresponding parenthesis.

Remark 16

(Closed expressions: Semantics). We consider a closed expression $\varnothing ▹Ex$ (see Remark 12). ${\left[[\varnothing ]\right]}^{\mathcal{U}}=U^{\varnothing }$ is a singleton with the initial morphism ${!}_U:\varnothing \to U$ as the only element. In such a way, we have either ${\left[\left[Ex\right]\right]}_{\varnothing }^{\mathcal{U}}={\left[[\top \langle \rangle ]\right]}_{\varnothing }^{\mathcal{U}}=\left\{{!}_U\right\}$, i.e., $({!}_U,\mathcal{U}){\vDash }^{\Sigma }\varnothing ▹Ex$, or ${\left[\left[Ex\right]\right]}_{\varnothing }^{\mathcal{U}}={\left[[\perp \langle \rangle ]\right]}_{\varnothing }^{\mathcal{U}}=\varnothing$, i.e., $({!}_U,\mathcal{U}){⊭}^{\Sigma }\varnothing ▹Ex$.

Example 4

(Expressions: Semantics). For the sample signature $\Sigma =(P,F,ar,in,out)$ in Example 1 with $P=\left\{\mathtt{less}\right\}$, $F=\{\underline{0},\underline{1},\underline{+},\underline{*}\}$ we consider the three Σ-structures $\mathcal{N}$, $\mathcal{Q}$ and $\mathcal{P}$ in Example 2 and the derived predicates in Example 3.

For the closed Σ-expression $\mathtt{madd}=\varnothing ▹\forall (\{x,y,z\}:\underline{<}\langle x,y\rangle ⟶\underline{<}\langle \underline{+}\langle x,z\rangle ,\underline{+}\langle y,z\rangle \rangle )$ we do have ${\left[\left[\mathtt{madd}\right]\right]}_{\varnothing }^{\mathcal{N}}=\left\{{!}_{\mathbb{N}}\right\}$, ${\left[\left[\mathtt{madd}\right]\right]}_{\varnothing }^{\mathcal{Q}}=\left\{{!}_{\mathbb{Q}}\right\}$ and ${\left[\left[\mathtt{madd}\right]\right]}_{\varnothing }^{\mathcal{P}}=\varnothing$, i.e., addition is strict monoton in $\mathcal{N}$ and $\mathcal{Q}$ but not in $\mathcal{P}$.

Conversly, we have for $\mathtt{demo}=\varnothing ▹\forall (\{x,y,z\}:(\underline{+}\langle x,\underline{*}\langle y,z\rangle \rangle =\underline{*}\langle \underline{+}\langle x,y\rangle ,\underline{+\langle x,z\rangle \rangle })$ that ${\left[\left[\mathtt{madd}\right]\right]}_{\varnothing }^{\mathcal{N}}=\varnothing$, ${\left[\left[\mathtt{madd}\right]\right]}_{\varnothing }^{\mathcal{Q}}=\varnothing$ and ${\left[\left[\mathtt{madd}\right]\right]}_{\varnothing }^{\mathcal{P}}=\left\{{!}_{\wp \left(\mathbb{N}\right)}\right\}$, i.e., addition distributes over multiplication only in $\mathcal{P}$.

For $\mathtt{even}=\left\{x\right\}▹\exists (\left\{y\right\}:x=\underline{*}\langle y,\underline{+}\langle \underline{1},\underline{1}\rangle \rangle )$ we do have ${\left[\left[\mathtt{even}\right]\right]}_{\left\{x\right\}}^{\mathcal{Q}}={\mathbb{Q}}^{\left\{x\right\}}$, ${\left[\left[\mathtt{even}\right]\right]}_{\left\{x\right\}}^{\mathcal{P}}=\wp {\left(\mathbb{N}\right)}^{\left\{x\right\}}$ and ${\left[\left[\mathtt{even}\right]\right]}_{\left\{x\right\}}^{\mathcal{N}}=\{\mathbf{a}\in {\mathbb{N}}^{\left\{x\right\}}\mid \mathbf{a}\left(x\right)iseven\}$, . So in $\mathcal{Q}$ and $\mathcal{P}$ all elements are "even".

For $\mathtt{min}=\left\{x\right\}▹\forall (\left\{y\right\}:(x=y)\vee \underline{<}\langle x,y\rangle )$ and $\mathtt{lmin}=\left\{x\right\}▹\neg (\exists (\left\{y\right\}:\underline{<}\langle y,x\rangle ))$ we do have ${\left[\left[\mathtt{min}\right]\right]}_{\left\{x\right\}}^{\mathcal{N}}={\left[\left[\mathtt{lmin}\right]\right]}_{\left\{x\right\}}^{\mathcal{N}}=\left\{(x\mapsto 0)\right\}\subset {\mathbb{N}}^{\left\{x\right\}}$, ${\left[\left[\mathtt{min}\right]\right]}_{\left\{x\right\}}^{\mathcal{Q}}={\left[\left[\mathtt{lmin}\right]\right]}_{\left\{x\right\}}^{\mathcal{Q}}=\varnothing$ and ${\left[\left[\mathtt{min}\right]\right]}_{\left\{x\right\}}^{\mathcal{P}}={\left[\left[\mathtt{lmin}\right]\right]}_{\left\{x\right\}}^{\mathcal{N}}=\left\{(x\mapsto \varnothing )\right\}\subset \wp {\left(\mathbb{N}\right)}^{\left\{x\right\}}$.

For $\mathtt{succ}=\{x,y\}▹(\underline{<}\langle x,y\rangle \wedge \neg (\exists (\left\{z\right\}:\underline{<}\langle x,z\rangle \wedge \underline{<}\langle z,y\rangle )))$ we do have ${\left[\left[\mathtt{succ}\right]\right]}_{\{x,y\}}^{\mathcal{N}}=\{\mathbf{a}\in {\mathbb{N}}^{\{x,y\}}\mid \mathbf{a}\left(y\right)=\mathbf{a}\left(x\right)+1\}$, ${\left[\left[\mathtt{succ}\right]\right]}_{\{x,y\}}^{\mathcal{Q}}=\varnothing$ and ${\left[\left[\mathtt{succ}\right]\right]}_{\{x,y\}}^{\mathcal{P}}=\{\mathbf{a}\in \wp {\left(\mathbb{N}\right)}^{\{x,y\}}\mid \mathbf{a}\left(x\right)\subset \mathbf{a}\left(y\right)and\mathbf{a}\left(y\right)\setminus \mathbf{a}\left(x\right)issingleton\}$.

Moreover, we have ${\left[\left[(\underline{+}\langle x,y\rangle =z)\right]\right]}_{\{x,y,z\}}^{\mathcal{N}}=\{\mathbf{a}\in {\mathbb{N}}^{\{x,y,z\}}\mid \mathbf{a}\left(x\right)+\mathbf{a}\left(y\right)=\mathbf{a}\left(z\right)\}$, ${\left[\left[(\underline{+}\langle x,y\rangle =z)\right]\right]}_{\{x,y,z\}}^{\mathcal{Q}}=\{\mathbf{a}\in {\mathbb{Q}}^{\{x,y,z\}}\mid \mathbf{a}\left(x\right)+\mathbf{a}\left(y\right)=\mathbf{a}\left(z\right)\}$ and ${\left[\left[(\underline{+}\langle x,y\rangle =z)\right]\right]}_{\{x,y,z\}}^{\mathcal{P}}=\{\mathbf{a}\in \wp {\left(\mathbb{N}\right)}^{\{x,y,z\}}\mid \mathbf{a}\left(x\right)\cup \mathbf{a}\left(y\right)=\mathbf{a}\left(z\right)\}$.

4. First-Order Logics of Statements in Context

We are not addressing signature morphisms in this paper thus we do not recapitulate the corresponding parts from [17]. To make the paper sufficiently self-contained and accessible, we adapt and revise, however, in this section those parts from Section 5 and 6 in [17] to the case of traditional unsorted first-order logic which are relevant for the analysis of deduction calculi. Compared to [1,17], we elaborate, moreover, in addition and in more detail the hom-set predicate perspective on definitions, constructions and results, as we already started it with Definition 6 versus Definition 7 (see Appendix A).

4.1. Institutions of Statements

In this subsection we construct for each signature $\Sigma =(P,F,ar,in,out)$ an institution of statements ${\mathcal{FL}}_{\Sigma }=(\mathsf{Cxt},{\mathbf{Stm}}_{\Sigma },{\mathbf{Int}}_{\Sigma },{\vDash }^{\Sigma })$. Compared to the general case in [1], we consider only the variant with the full spectrum $F{E}_{\Sigma }$ of first-order expressions and with the whole category $\mathsf{Str}\left(\Sigma \right)$ of $\Sigma$-structures as semantics $\mathsf{Sem}\left(\Sigma \right)$. Therefore, we only need a simplified version of the construction of institution of statements as visualized in Figure 1.

Contexts are the (abstract) signatures in an institutions of statements. In the case of unsorted first-order logic all ${\mathcal{FL}}_{\Sigma }$ share the same base category and the same category of contexts which simply are the category of sets: $\mathsf{Base}=\mathsf{Cxt}:=\mathsf{Set}$. As common category $\mathsf{Var}$ of variable declarations we choose the set $Va{}^r:={\wp }_{fin}\left(N_{var}\right)$ of all finite subsets of $N_{var}$ (considered as a discrete subcategory of $\mathsf{Base}=\mathsf{Set}$).

The soundness of deduction calculi for traditional unsorted first-order logic relies on the assumption that all $\Sigma$-structure do have a non-empty carrier. Therefore, all institutions ${\mathcal{FL}}_{\Sigma }$ share the same category $\mathsf{Carr}:=\mathsf{Set}\setminus \{\varnothing \}$ of potential carriers of structures.

Note, that the proper inclusions $\mathsf{Var}⊏\mathsf{Cxt}⊐\mathsf{Carr}$ reflect the intended role of contexts to establish a bridge between syntax and semantics.

Remark 17

(Contexts in the Wild). Contexts are meant to establish a bridge between meta-level syntax (signatures, constant symbols, variables, terms, expressions) and semantics (structures and their carriers). In addition, we want to offer a home for those entities, being around in logic, which neither properly belong to meta-level syntax nor to semantics.

Demonstrating the bridge feature, we will utilize in the Section 7 and Section 8 sets of variables in the role of contexts. On the other hand, sets of (names for) elements in carriers can also take the role of contexts as outlined in the discussion ofelementary diagramsin Section 5.1.

Other examples of contexts, we do have in mind, are sets G ofgeneratorsin Group Theory, sets O ofliteralsin PROLOG, sets $N_O$ ofindividual names (nominals, objects)in Description Logic [13], sets ofHenkin constantsand so on. Contexts can also accommodateSkolem constants.

4.1.1. Institutions of Statements: Sentence and Model Functor

Statements in context are the (abstract) sentences in an institutions of statements.

Definition 8

(Statement). A$\Sigma$-statement $(X▹Ex,\gamma )$ in a context $K\in \left|\mathsf{Cxt}\right|$is given by a set X of variables, a Σ-expression $Ex\in F{E}_{\Sigma }\left(X\right)$ and abinding map$\gamma :X\to K$. $(X▹Ex,\gamma )$ is calledatomicif $Ex$ is an atomic expression, i.e., an Equation or a Relational Atom.

By $St{m}_{\Sigma }\left(K\right)$, we denote the set of all Σ-statements in K and by ${{\rm Y}}_K:St{m}_{\Sigma }\left(K\right)\to Var$ the obvious projection map. $St{m}_{\Sigma }\left(K\right)\left(X\right)={{\rm Y}}_K^{-1}\left(X\right)\subset St{m}_{\Sigma }\left(K\right)$ is the set of all Σ-statements $(X▹Ex,\gamma )$ in K for an arbitrary, but fixed, set X of variables.

Remark 18

(General statements and closed formulas). For any closed Σ-expression $\varnothing ▹Ex$ (see Remarks 12 and 16) there is a unique initial morphism $\gamma ={!}_K:\varnothing \to K$; thus, we have $(\varnothing ▹Ex,{!}_K)\in St{m}_{\Sigma }\left(K\right)(\varnothing )\subset St{m}_{\Sigma }\left(K\right)$ for any context K and all Σ-expressions $Ex\in F{E}_{\Sigma }(\varnothing )$. We call $(\varnothing ▹Ex,{!}_K)$ ageneral statement in K.

From all the general statements $(\varnothing ▹Ex,{!}_K)$ sharing the same closed expression $\varnothing ▹Ex$, only the general statement $(\varnothing ▹Ex,i{d}_{\varnothing })$ in the empty context ∅ is the proper formal counterpart of traditionalclosed formulasin institutions of statements!

Example 5

(Context and Statement). For the sample signature Σ in Example 1, we define the contexts $K:=\{a,b,c,d,e\}$ and $G:=\{n\in \mathbb{N}\mid 0\le n\le 100\}$. We can consider the elements in G as "number literals" in the sense of PROLOG or simply as elements of the carrier $\mathbb{N}$ of the Σ-structure $\mathcal{N}$.

We adapt the encoding of assignments from Remark 4 and use for Σ-statements $(X▹Ex,\gamma )$ the shorthand notation $Ex({\gamma }_1,\dots ,{\gamma }_n)$. Instead of expressions $Ex$, we will also use auxiliary names for expressions, as introduced in Example 3. If $Ex$ is a “reborn predicate symbol” (see Remark 9), we may use the traditional notation $\mathtt{p}({\gamma }_1,\dots ,{\gamma }_n)$ instead of $\mathtt{p}\langle v_1^{ar\left(\mathtt{p}\right)},\dots ,v_n^{ar\left(\mathtt{p}\right)}\rangle ({\gamma }_1,\dots ,{\gamma }_n)$.

$\underline{<}(a,b)$ and $\underline{<}\langle \underline{+}\langle x,y\rangle ,z\rangle (a,b,e)$ are Σ-statements in K with relational atoms. Σ-statements in K with equations are $(\underline{+}\langle x,y\rangle =z)(a,c,d)$, $(\underline{+}\langle x,y\rangle =z)(b,c,e)$, $(\underline{+}\langle x,\underline{1}\rangle =y)(b,c)$, and Σ-statements in K with auxiliary names are, for example, $\mathtt{even}\left(a\right)$, $\mathtt{even}\left(c\right)$, $\mathtt{succ}(b,c)$, $\neg \mathtt{min}\left(a\right)$.

Attention!We are not substituting variables by elements of contexts! Therefore, we can not write, for example, $(\underline{+}\langle b,\underline{1}\rangle =c)$ instead of $(\underline{+}\langle x,\underline{1}\rangle =y)(b,c)$.

Remark 19

(Statements in the Wild). We continue Remark 17. In Group Theory a statement in context G is called adefining relationbetween the generators in G and is, in fact, an equation bound to G. In Logic Programming a statement in context O is an atomic relational statement and usually called afactabout the atoms in O. In Description Logics they use the termrole assertionfor statements in context.

The "binding trick" starts to pay off. It allows us to encapsulate the relatively intricate construction of first-order expressions in the sense that we don’t need "substitutions" to define the translation of statements.

Any morphism $\psi :K\to G$ in $\mathsf{Cxt}$ induces a map ${\mathbf{Stm}}_{\Sigma }\left(\psi \right):St{m}_{\Sigma }\left(K\right)\to St{m}_{\Sigma }\left(G\right)$ defined by simple post-composition for all statements $(X▹Ex,\gamma )$ in K:

$${\mathbf{Stm}}_{\Sigma }\left(\psi \right)(X▹Ex,\gamma ):=(X▹Ex,\gamma ;\psi ).$$

(7)

We do have ${{\rm Y}}_K={\mathbf{Stm}}_{\Sigma };{{\rm Y}}_G$ and thus, equivalently said, that ${\mathbf{Stm}}_{\Sigma }\left(\psi \right)$ restricts to a map ${\mathbf{Stm}}_{\Sigma }\left(\psi \right)\left(X\right):St{m}_{\Sigma }\left(K\right)\left(X\right)\to St{m}_{\Sigma }\left(G\right)\left(X\right)$ for every set X of variables.

Associativity of composition in $\mathsf{Base}=\mathsf{Set}$ ensures that the assignments $K\mapsto St{m}_{\Sigma }\left(K\right)$ and $\psi \mapsto {\mathbf{Stm}}_{\Sigma }\left(\psi \right)$ define a functor ${\mathbf{Stm}}_{\Sigma }:\mathsf{Cxt}\to \mathsf{Set}$. This is the sentence functor of the institution ${\mathcal{FL}}_{\Sigma }$.

Example 6

(Translation of Statements). For the contexts $G=\{n\in \mathbb{N}\mid 0\le n\le 100\}$ and $K=\{a,b,c,d,e\}$ from Example 5, we consider the map $\psi :K\to G$ given by the assignments $a\mapsto 2,b\mapsto 3,c\mapsto 4,d\mapsto 6,e\mapsto 7$.

The statements in K from Example 5 are simply translated by means of the map $\psi :K\to G$ to the following statements in context G. $\underline{<}(a,b)$ translates to $\underline{<}(2,3)$ and $\underline{<}\langle \underline{+}\langle x,y\rangle ,z\rangle (a,b,e)$ to $\underline{<}\langle \underline{+}\langle x,y\rangle ,z\rangle (2,3,7)$.

$(\underline{+}\langle x,y\rangle =z)(a,c,d)$ translates to $(\underline{+}\langle x,y\rangle =z)(2,4,6)$, $(\underline{+}\langle x,y\rangle =z)(b,c,e)$ to $(\underline{+}\langle x,y\rangle =z)(3,4,7)$ and $(\underline{+}\langle x,\underline{1}\rangle =y)(b,c)$ to $(\underline{+}\langle x,\underline{1}\rangle =y)(3,4)$.

Finally, the Σ-statements in context K$\mathtt{even}\left(a\right)$, $\mathtt{even}\left(c\right)$, $\mathtt{succ}(b,c)$ and $\neg \mathtt{min}\left(a\right)$ with auxiliary names, are translated to $\mathtt{even}\left(2\right)$, $\mathtt{even}\left(4\right)$, $\mathtt{succ}(3,4)$ and $\neg \mathtt{min}\left(2\right)$, respectively.

Interpretations of contexts are the (abstract) models in an institution of statements.

Definition 9

(Context interpretations). A$\Sigma$-interpretation $(\iota ,\mathcal{U})$ of a context$K\in \left|\mathsf{Cxt}\right|$ is given by a Σ-structure $\mathcal{U}=(U,P^{\mathcal{U}},F^{\mathcal{U}})$ in $\mathsf{Str}\left(\Sigma \right)$ and a map $\iota :K\to U$.

A morphism $\varsigma :(\iota ,\mathcal{U})\to (\vartheta ,\mathcal{V})$ between Σ-interpretations of K is given by a Σ-homomorphism $\varsigma :\mathcal{U}\to \mathcal{V}$ such that $\iota ;\varsigma =\vartheta$ for the underlying map $\varsigma :U\to V$.

For any context K in $\mathsf{Cxt}$, we denote by ${\mathsf{Int}}_{\Sigma }\left(K\right)$ the category of all Σ-interpretations of K and all morphisms between them and by ${\Pi }_K:{\mathsf{Int}}_{\Sigma }\left(K\right)\to \mathsf{Str}\left(\Sigma \right)$ the obvious projection functor.

Note, that ${\Pi }_{\varnothing }:{\mathsf{Int}}_{\Sigma }(\varnothing )\to \mathsf{Str}\left(\Sigma \right)$ is an isomorphism and compare the later Remark 20. Any morphism $\psi :K\to G$ in $\mathsf{Cxt}$ induces a functor ${\mathbf{Int}}_{\Sigma }\left(\psi \right):{\mathsf{Int}}_{\Sigma }\left(G\right)\to {\mathsf{Int}}_{\Sigma }\left(K\right)$ defined by simple pre-composition for all $\Sigma$-interpretations $(\vartheta ,\mathcal{V})$ of G:

$${\mathbf{Int}}_{\Sigma }\left(\psi \right)(\vartheta ,\mathcal{V}):=(\psi ;\vartheta ,\mathcal{V}).$$

(8)

For any morphism $\varsigma :(\iota ,\mathcal{U})\to (\vartheta ,\mathcal{V})$ between two $\Sigma$-interpretations of G the same underlying map $\varsigma :U\to V$ establishes a morphism ${\mathbf{Int}}_{\Sigma }\left(\psi \right)\left(\varsigma \right):=\varsigma :(\psi ;\iota ,\mathcal{U})\to (\psi ;\vartheta ,\mathcal{V})$ between the corresponding two $\Sigma$-interpretations of K, thus we have

$${\mathbf{Int}}_{\Sigma }\left(\psi \right);{\Pi }_K={\Pi }_G:{\mathsf{Int}}_{\Sigma }\left(G\right)\to \mathsf{Str}\left(\Sigma \right)\left(seethediagramaboveontheright\right).$$

(9)

Equivalently said, the functor ${\mathbf{Int}}_{\Sigma }\left(\psi \right):{\mathsf{Int}}_{\Sigma }\left(G\right)\to {\mathsf{Int}}_{\Sigma }\left(K\right)$ restricts to a functor ${\mathbf{Int}}_{\Sigma }\left(\psi \right)\left(\mathcal{U}\right):{\mathsf{Int}}_{\Sigma }\left(G\right)\left(\mathcal{U}\right)\to {\mathsf{Int}}_{\Sigma }\left(K\right)\left(\mathcal{U}\right)$ with the pre-composition map $(\psi ;\_):U^G\to U^K$ as the underlying map on objects.

Associativity of composition in $\mathsf{Set}$ ensures that the assignments $K\mapsto {\mathsf{Int}}_{\Sigma }\left(K\right)$ and $\psi \mapsto {\mathbf{Int}}_{\Sigma }\left(\psi \right)$ define a functor ${\mathbf{Int}}_{\Sigma }:{\mathsf{Cxt}}^{op}\to \mathsf{CAT}$. This is the model functor of the institution ${\mathcal{FL}}_{\Sigma }$.

4.1.2. Institutions of Statements: Satisfaction Relation and Satisfaction Condition

The last two steps, in establishing an institution, are the definition of a satisfaction relation for interpretations and statements and the proof of the so-called satisfaction condition. The satisfaction relation is simply provided by the satisfaction relation for assignments and expressions, as defined in Definition 6.

Definition 10

(Satisfaction Relation: Interpretations and Statements). For any context K, any Σ-statement $(X▹Ex,\gamma )$ in K and any Σ-interpretation $(\iota ,\mathcal{U})$ of context K we define:

If $(\iota ,\mathcal{U}){\vDash }_K^{\Sigma }(X▹Ex,\gamma )$, we say that ι is avalid interpretationfor $(X▹Ex,\gamma )$ in $\mathcal{U}$ or that $(X▹Ex,\gamma )$ is valid for the interpretation ι in $\mathcal{U}$.

Analogously to (6), the satisfaction relation in Definition 10 allows us to define the semantics of statements in $\Sigma$-structures. For all sets X of variables, all $\Sigma$-structures $\mathcal{U}$ and all $\Sigma$-statements $(X▹Ex,\gamma )$ in a context K we set

$${\left[\left[(X▹Ex,\gamma )\right]\right]}_K^{\mathcal{U}}:=\{\iota \in U^K\mid (\iota ,\mathcal{U}){\vDash }_K^{\Sigma }(X▹Ex,\gamma )\}\subseteq U^K.$$

(11)

Definition 10 implicitly relies on pre-composition maps $(\gamma ;\_):U^K\to U^X$ thus we can define for all $\Sigma$-structures $\mathcal{U}$ the semantics of $\Sigma$-statements $(X▹Ex,\gamma )$ in a context K independent of Definition 10 by

$${\left[\left[(X▹Ex,\gamma )\right]\right]}_K^{\mathcal{U}}{=\downarrow }_{\gamma }^U{\left[[X▹Ex]\right]}^{\mathcal{U}}\subseteq U^K$$

(12)

utilizing the shift operator ${\downarrow }_{\gamma }^U={(\gamma ;\_)}^{-1}:\wp \left(U^X\right)\to \wp \left(U^K\right)$ according to Appendix A.2.

Remark 20

(Validity of General Statements). If $X=K=\varnothing$, we do have for any Σ-structure $\mathcal{U}$ exactly one interpretation $({!}_U,\mathcal{U})$ thus for any "closed formula" $(\varnothing ▹Ex,i{d}_{\varnothing })$ (see Remark 18) $({!}_U,\mathcal{U}){\vDash }_{\varnothing }^{\Sigma }(\varnothing ▹Ex,i{d}_{\varnothing })$ means nothing but that the closed formula $(\varnothing ▹Ex,i{d}_{\varnothing })$ isvalid in $\mathcal{U}$in the traditional sense.

Moreover, the validity of general statements issemantically context independentin the following sense: For any context K, any Σ-structure $\mathcal{U}$, any Σ-interpretation $(\iota ,\mathcal{U})$ of K in $\mathcal{U}$ and any closed expressions $\varnothing ▹Ex$, we have due to the uniqueness if intitial morphisms:

$$(\iota ,\mathcal{U}){\vDash }_K^{\Sigma }(\varnothing ▹Ex,{!}_K)iff{!}_K;\iota ={!}_U=i{d}_{\varnothing };{!}_U\in {\left[\left[Ex\right]\right]}_{\varnothing }^{\mathcal{U}}iff({!}_U,\mathcal{U}){\vDash }_{\varnothing }^{\Sigma }(\varnothing ▹Ex,i{d}_{\varnothing }).$$

Example 7

(Interpretation and Satisfaction). The reader may check with help of Example 4, that the map $\psi :K\to G$ in Example 6 is defined in such away that all the Σ-statements in context $K=\{a,b,c,d,e\}$ from Example 6 are valid for the interpretation $\psi ;i{n}_{G,\mathbb{N}}:K\to \mathbb{N}$ of the context K in the Σ-structure $\mathcal{N}$ from Example 2!

In contrast, only the Σ-statements $\mathtt{even}\left(a\right)$, $\mathtt{even}\left(c\right)$ and $\neg \mathtt{min}\left(a\right)$ in K are valid for the interpretation $\psi ;\{\_\}:K\to \wp \left(\mathbb{N}\right)$ in $\mathcal{N}$ where $\{\_\}:G\to \wp \left(\mathbb{N}\right)$ assigns to each natural number $n\in G$ the singleton $\left\{n\right\}$. Note, that the Satisfaction Condition in Corollary 1 allows us to show this claim by checking that the translated statements $\mathtt{even}\left(2\right)$, $\mathtt{even}\left(4\right)$, $\neg \mathtt{min}\left(2\right)$ in context G are valid for the interpretation $\{\_\}:G\to \wp \left(\mathbb{N}\right)$ of G in $\mathcal{N}$.

After we developed everything in a systematic modular way, we obtain the satisfaction condition "for free", namely by associativity of composition in $\mathsf{Set}$.

Corollary 1

(Satisfaction Condition). For any morphism $\psi :K\to G$ in $\mathsf{Cxt}$, any Σ-statement $(X▹Ex,\gamma )$ in context K and any interpretation $(\vartheta ,\mathcal{U})$ of context G in a Σ-structure $\mathcal{U}$, we have:

$${\mathbf{Int}}_{\Sigma }\left(\psi \right)(\vartheta ,\mathcal{U}){\vDash }_K^{\Sigma }(X▹Ex,\gamma )iff(\vartheta ,\mathcal{U}){\vDash }_G^{\Sigma }{\mathbf{Stm}}_{\Sigma }\left(\psi \right)(X▹Ex,\gamma ).$$

(13)

Proof. 

Due to the definition of the functors ${\mathbf{Int}}_{\Sigma }:{\mathsf{Cxt}}_S^{op}\to \mathsf{CAT}$ and ${\mathbf{Stm}}_{\Sigma }:{\mathsf{Cxt}}_S\to \mathsf{Set}$, we obtain the commutative diagram, above on the right, thus the satisfaction condition follows immediately from Definition 10 (Satisfaction Relation: Interpretations and Statements) and associativity of composition. in $\mathsf{Set}$ □

We can also independently prove the satisfaction condition on the level of hom-sets. Exemplifying a general result about an equivalent characterizations of the satisfaction condition for institutions (see the concept institution frame in [23,24]) and due to the properties of the maps ${\mathbf{Stm}}_{\Sigma }\left(\psi \right)$ and ${\mathbf{Int}}_{\Sigma }\left(\psi \right)$ concerning restrictions (see (7)), the satisfaction condition in (13) is equivalent to the requirement that for any context morphism $\psi :K\to G$, any $\Sigma$-structure $\mathcal{U}$ and any set X of variables the diagram below on the right commutes.

For any $\Sigma$-statement $(X▹Ex,\gamma )\in St{m}_{\Sigma }\left(K\right)\left(X\right)$ we get indeed

$$\begin{array}{cccc}\hfill {\downarrow }_{\psi }^U{\left[\left[(X▹Ex,\gamma )\right]\right]}_K^{\mathcal{U}}& =& {\downarrow }_{\psi }^U\left({\downarrow }_{\gamma }^U{\left[[X▹Ex]\right]}^{\mathcal{U}}\right)\hfill & \left(definition\right(12\left)\right)\hfill \\ & =& {\downarrow }_{\gamma ;\psi }^U{\left[[X▹Ex]\right]}^{\mathcal{U}}\hfill & \left(equation\right(\mathrm{A}23\left)\right)\hfill \\ & =& {\left[\left[(X▹Ex,\gamma ;\psi )\right]\right]}_G^{\mathcal{U}}\hfill & \left(definition\right(12\left)\right)\hfill \\ & =& {\left[\left[{\mathbf{Stm}}_{\Sigma }\left(\psi \right)(X▹Ex,\gamma )\right]\right]}_G^{\mathcal{U}}\hfill & \left(definition\right(7\left)\right)\hfill \end{array}$$

Summarizing all definitions and results, we obtain for each signature $\Sigma$ the Institution of $\Sigma$-Statements in Context${\mathcal{FL}}_{\Sigma }=(\mathsf{Cxt},{\mathbf{Stm}}_{\Sigma },{\mathbf{Int}}_{\Sigma },{\vDash }^{\Sigma })$ not only comprising "closed formulas", as it is traditionally the case [19,20], but also "open formulas"!

Since we only consider, in the remaining part of the paper, one single arbitrary but fixed signature $\Sigma$ we may drop the superscript in ${\vDash }^{\Sigma }$ whenever it is convenient.

4.2. Substitutions

Substitutions are not relevant for utilizing a logic as a specification formalism. As also mentioned in [20], substitutions are, however, an important logical device for deduction. In this section we investigate substitutions and develop corresponding extensions of constructions and results necessary and sufficient for deduction.

4.2.1. Categories of Substitutions

The simple natural idea is to extend the family $Var$ of sets of variables by substitutions. For any signature $\Sigma =(P,F,ar,in,out)$ the small category ${\mathsf{Var}}_{\Sigma }$ is defined as follows:

Objects:

are all sets X of variables, i.e., $|{\mathsf{Var}}_{\Sigma }|:=Var$;

Morphisms:

$\mathbf{t}:X\to Y$ in ${\mathsf{Var}}_{\Sigma }$ are given by $\Sigma$-substitutions $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$;

Identities:

on sets X are given by the canonical maps ${\eta }_X$: $X\to T_{\Sigma }\left(X\right)$; and the

Composition:

$\mathbf{t}{;}^{\Sigma }\mathbf{r}:X\to Z$ of two morphisms $\mathbf{t}:X\to Y$, $\mathbf{r}:Y\to Z$ in ${\mathsf{Var}}_{\Sigma }$ is given by the $\Sigma$-substitution $\mathbf{t};{\mathbf{r}}^{*}:X\to T_{\Sigma }\left(Z\right)$ where ${\mathbf{r}}^{*}:T_{\Sigma }\left(Y\right)\to T_{\Sigma }\left(Z\right)$ is the inductive extension of $\mathbf{r}:Y\to T_{\Sigma }\left(Z\right)$ such that

Note, that ${\mathbf{r}}^{*}:T_{\Sigma }\left(Y\right)\to T_{\Sigma }\left(Z\right)$ describes nothing but the application of the $\Sigma$-substitution $\mathbf{r}:Y\to T_{\Sigma }\left(Z\right)$ to all $\Sigma$-terms over Y! We can inductively prove that

$${\eta }_X^{*}=i{d}_{T_{\Sigma }\left(X\right)}and{(\mathbf{t};{\mathbf{r}}^{*})}^{*}={\mathbf{t}}^{*};{\mathbf{r}}^{*}$$

(16)

thus the composition of substitutions is associative, and we get indeed a category.

Remark 21

(Lawvere Theories). The tentative reader may have realized that the categories ${\mathsf{Var}}_{\Sigma }$ are nothing but so-calledsyntactic Lawvere theories[25,26,27]. We will, however, not walk further into the realm of "Categorical Algebra" and "Functorial Semantics". We will neither utilize the fact that the categories ${\mathsf{Var}}_{\Sigma }$ are finite product categories nor reconstruct $\mathsf{Str}$(Σ) as a subcategory of the functor category $[{\mathsf{Var}}_{\Sigma }\to \mathsf{Set}]$ in case of signatures Σ without predicate symbols!

Remark 22

(Kleisli Morphisms). It is a common practice to describe substitutions as morphisms of the Kleisli category of a "term algebra adjunction". In such a way, we would get the extension of assignments in (3) and the application of substitutions in (15) as well as their compatibility "for free". See the discussion of "internalization of terms" and of substitution calculi in [22].

To stay closer to traditional presentations of logics, we rely instead in this paper on explicit inductive definitions (located on the meta-level). Another reason is, that in the many-sorted case some of the necessary constructions and results, as the translation of terms along signature morphisms and the composition of term translations for example, can anyway not be obtained by means of term algebra adjunctions (see [17])!

4.2.2. Substitution Application for Expressions

Analogously to terms, we would also like to extend the map $F{E}_{\Sigma }:Va{}^r\to \left|\mathsf{Set}\right|$, constructed in Definition 5, to a functor ${\mathbf{FE}}_{\Sigma }:{\mathsf{Var}}_{\Sigma }\to \mathsf{Set}$. We can indeed define for all $\Sigma$-substitutions $\mathbf{r}:X\to T_{\Sigma }\left(Z\right)$ a map $\overline{\mathbf{r}}:F{E}_{\Sigma }\left(X\right)\to F{E}_{\Sigma }\left(Z\right)$ describing the application of the substitution to $\Sigma$-expressions on X. Due to the presence of quantification, the assignments $\mathbf{r}\mapsto \overline{\mathbf{r}}$ will be, however, not functorial on the nose as we will discuss below.

There seems to be no way out! As long as we do not utilize something like de Bruijn indices [28] to avoid named bound variables, we have to deal, in one or another way, with the problem that substitution application may cause an unintended interaction between free and bound variables. One approach to avoid such an unintended interaction is to inductively define a binary meta-level predicate stating when a substitution is legal for a certain expression as it is done, for example, in [4]. We will, however, not adapt this approach, since we would end up with partial substitution maps .

Another approach is to use α-conversion, i.e., renaming of bound variables, in such a way, that we obtain total substitution maps. We will define the application of a substitution to expressions, according to the inductive definition of expressions in Definition 5, thereby establishing a stepwise inductive variant of an implicit $\alpha$-conversion.

To meet the proper inclusion condition for Quantification in Definition 5, we must use non-symmetric sums when defining substitution application in case of Quantification. That is, for any two sets $X_1,X_2\in Var$ of variables we must choose a set $X_1\overrightarrow{+}{X}_2\in Var$ which is a sum in $\mathsf{Set}$ such that the left injection ${\kappa }_1:X_1\to X_1\overrightarrow{+}{X}_2$ is an inclusion $X_1\subseteq X_1\overrightarrow{+}{X}_2$.

General Assumption: If $X_1$ and $X_2$ are disjoint, we simply choose $X_1\overrightarrow{+}{X}_2:=X_1\cup X_2$.

Definition 11

(Substitution Application for Expressions). For any signature Σ the family of maps $\overline{\mathbf{r}}:F{E}_{\Sigma }\left(X\right)\to F{E}_{\Sigma }\left(Z\right)$ with $\mathbf{r}:X\to T_{\Sigma }\left(Z\right)$ a Σ-substitution is defined inductively and in parallel by the following assignments:

1.

Atomic expressions:

(a)

Equation: $X▹(t_1=t_2)\mapsto Z▹({\mathbf{r}}^{*}\left(t_1\right)={\mathbf{r}}^{*}\left(t_2\right))$ for any $t_1,t_2\in T_{\Sigma }\left(X\right)$ with ${\mathbf{r}}^{*}:T_{\Sigma }\left(X\right)\to T_{\Sigma }\left(Z\right)$ given by (15).

(b)

Relational Atom: $X▹\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)\mapsto Z▹\mathtt{p}\mathbf{syn}(\mathbf{t};{\mathbf{r}}^{*})$ for any $\mathtt{p}\in P$ and any substitution $\mathbf{t}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(X\right)$ with ${\mathbf{r}}^{*}:T_{\Sigma }\left(X\right)\to T_{\Sigma }\left(Z\right)$ given by (15).

2.

Everything: $X▹\top \langle \rangle \mapsto Z▹\top \langle \rangle$

3.

Void: $X▹\perp \langle \rangle \mapsto Z▹\perp \langle \rangle$

4.

Conjunction: $X▹(E{x}_1\wedge E{x}_2)\mapsto Z▹(\overline{\mathbf{r}}\left(E{x}_1\right)\wedge \overline{\mathbf{r}}\left(E{x}_2\right))$

5.

Disjunction: $X▹(E{x}_1\vee E{x}_2)\mapsto Z▹(\overline{\mathbf{r}}\left(E{x}_1\right)\vee \overline{\mathbf{r}}\left(E{x}_2\right))$

6.

Implication: $X▹(E{x}_1\to E{x}_2)\mapsto Z▹(\overline{\mathbf{r}}\left(E{x}_1\right)\to \overline{\mathbf{r}}\left(E{x}_2\right))$

7.

Negation: $X▹\neg Ex\mapsto Z▹\neg \overline{\mathbf{r}}\left(Ex\right)$

8.

Quantification: $X▹\exists (\mathbf{syn}(Y\setminus X):Ex)\mapsto Z▹\exists (\mathbf{syn}(W\setminus Z):\overline{[\mathbf{r};\subset ,{\kappa }_2;{\eta }_W]}\left(Ex\right))$ and $X▹\forall (\mathbf{syn}(Y\setminus X):Ex)\mapsto Z▹\forall (\mathbf{syn}(W\setminus Z):\overline{[\mathbf{r};\subset ,{\kappa }_2;{\eta }_W]}\left(Ex\right))$ for any expression $Y▹Ex$ and any proper inclusion $X\subset Y$ where $W:=Z\overrightarrow{+}(Y\setminus X)$ and thus $Z\subset W$. The extended substitution $[\mathbf{r};\subset ,{\kappa }_2;{\eta }_W]:Y\to T_{\Sigma }\left(W\right)$ is constructed as a cotuple (by case distinction). Remind Remark 6 and that Y is the sum of X and $Y\setminus X$ in $\mathsf{Set}$ since $X\subset Y$.

There seems to be no choice of non-symmetric sums in $Var$ such that the assignments $X\mapsto F{E}_{\Sigma }\left(X\right)$ and $\mathbf{r}\mapsto \overline{\mathbf{r}}$ define a functor from ${\mathsf{Var}}_{\Sigma }$ into $\mathsf{Set}$ since for any $\Sigma$-substitutions ${\mathbf{r}}_1:X_1\to T_{\Sigma }\left(X_2\right)$, ${\mathbf{r}}_2:X_2\to T_{\Sigma }\left(X_3\right)$ the maps $\overline{{\mathbf{r}}_1};\overline{{\mathbf{r}}_2}:F{E}_{\Sigma }\left(X_1\right)\to F{E}_{\Sigma }\left(X_3\right)$ and $\overline{{\mathbf{r}}_1;{\mathbf{r}}_2^{*}}:F{E}_{\Sigma }\left(X_1\right)\to F{E}_{\Sigma }\left(X_3\right)$ coincide, in general, only for $\Sigma$-expressions without quantifications. For a $\Sigma$-expression $Ex$ on $X_1$ containing quantifications, the two $\Sigma$-expressions $\overline{{\mathbf{r}}_1};\overline{{\mathbf{r}}_2}\left(Ex\right)$ and $\overline{{\mathbf{r}}_1;{\mathbf{r}}_2^{*}}\left(Ex\right)$ may be syntactically different. They are, however, "the same up to $\alpha$-conversion". Especially, it can be shown that they are semantically equivalent.

Lemma 2

(Semantic Equivalence). For Σ-substitutions ${\mathbf{r}}_1:X_1\to T_{\Sigma }\left(X_2\right)$, ${\mathbf{r}}_2:X_2\to T_{\Sigma }\left(X_3\right)$ the maps $\overline{{\mathbf{r}}_1};\overline{{\mathbf{r}}_2}:F{E}_{\Sigma }\left(X_1\right)\to F{E}_{\Sigma }\left(X_3\right)$ and $\overline{{\mathbf{r}}_1;{\mathbf{r}}_2^{*}}:F{E}_{\Sigma }\left(X_1\right)\to F{E}_{\Sigma }\left(X_3\right)$ are semantically equivalent in the sense that

$${\left[[\overline{{\mathbf{r}}_1};\overline{{\mathbf{r}}_2}\left(Ex\right)]\right]}_{X_3}^{\mathcal{U}}={\left[\left[\overline{{\mathbf{r}}_1;{\mathbf{r}}_2^{*}}\left(Ex\right)\right]\right]}_{X_3}^{\mathcal{U}}forallEx\in F{E}_{\Sigma }\left(X_1\right)andall\Sigma -structures\mathcal{U}.$$

We will define and investigate $\alpha$-conversion in Section 6.3. To describe deduction in first-order logic we will need $\alpha$-conversion anyway, independent of the fact if the assignments $X\mapsto F{E}_{\Sigma }\left(X\right)$ and $\mathbf{r}\mapsto \overline{\mathbf{r}}$ define a functor from ${\mathsf{Var}}_{\Sigma }$ into $\mathsf{Set}$ or not. So, let us formally grasp what we have at hand besides Lemma 2.

Due to the General Assumption concerning non-symmetric sums of sets of variables and (16), we obviously have $\overline{{\eta }_X}=i{d}_{F{E}_{\Sigma }\left(X\right)}:F{E}_{\Sigma }\left(X\right)\to F{E}_{\Sigma }\left(X\right)$ for the canonical substitutions ${\eta }_X$: $X\to T_{\Sigma }\left(X\right)$, thus the assignments $X\mapsto F{E}_{\Sigma }\left(X\right)$ and $\mathbf{r}\mapsto \overline{\mathbf{r}}$ define a reflexive graph homomorphism ${\mathbf{FE}}_{\Sigma }:gr\left({\mathsf{Var}}_{\Sigma }\right)\to \mathsf{Set}$ (see Section 2).

4.2.3. Substitution Application vs Derived Operations

According to (3) and (4), any $\Sigma$-substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ represents for any $\Sigma$-structure $\mathcal{U}$ a derived operation ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^X$ in the opposite direction with ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right):=\mathbf{t};{\mathbf{a}}^{\circ }$ for all assignments $\mathbf{a}:Y\to U$. For any $\Sigma$-substitution $\mathbf{r}:Y\to T_{\Sigma }\left(Z\right)$ and any assignment $\mathbf{b}:Z\to U$ it can also inductively shown that

This entails

$${\left(\mathbf{t}{;}^{\Sigma }\mathbf{r}\right)}^{\mathcal{U}}={(\mathbf{t};{\mathbf{r}}^{*})}^{\mathcal{U}}={\mathbf{r}}^{\mathcal{U}};{\mathbf{t}}^{\mathcal{U}}:U^Z\to U^X.$$

(18)

That is, composition in ${\mathsf{Var}}_{\Sigma }$ syntactically represents the composition of derived operations! The equations (18) and (5) ensure that for any $\Sigma$-structure $\mathcal{U}$ the assignments $X\mapsto U^X$ and $\mathbf{t}\mapsto {\mathbf{t}}^{\mathcal{U}}$ define a contra-variant functor ${\mathbf{Op}}_{\Sigma }^{\mathcal{U}}:{\mathsf{Var}}_{\Sigma }^{op}\to \mathsf{Set}$ as the semantic counterpart of the reflexive graph homomorphisms ${\mathbf{FE}}_{\Sigma }:gr\left({\mathsf{Var}}_{\Sigma }\right)\to \mathsf{Set}$.

${\mathbf{Op}}_{\Sigma }^{\mathcal{U}}$ and ${\mathbf{FE}}_{\Sigma }$ are not fully matching the "original institution pattern". Nevertheless, we do have a corresponding satisfaction condition: For any $\Sigma$-substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$, any $\Sigma$-expression $X▹Ex$, any $\Sigma$-structure $\mathcal{U}$ and any assignment $\mathbf{a}\in U^Y$ it holds that

$$({\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right),\mathcal{U}){\vDash }^{\Sigma }X▹Exiff(\mathbf{a},\mathcal{U}){\vDash }^{\Sigma }Y▹\overline{\mathbf{t}}\left(Ex\right).$$

(19)

This is our reconstruction of the statement in Proposition 5.8 in [20].

Once more, we can independently describe the traditional satisfaction condition in (19) by means of hom-sets. We extended the map $F{E}_{\Sigma }:Va{}^r\to \left|\mathsf{Set}\right|$ in Remark 8 to a reflexive graph homomorphism ${\mathbf{FE}}_{\Sigma }:gr\left({\mathsf{Var}}_{\Sigma }\right)\to \mathsf{Set}$. Correspondingly, we can extend for each $\Sigma$-structure $\mathcal{U}$ the map $\wp \left(\mathsf{Set}(\_,U)\right):Va{}^r\to \left|\mathsf{Set}\right|$ in Remark 14 to a functor ${\mathbf{Hom}}_{\Sigma }^{\mathcal{U}}:{\mathsf{Var}}_{\Sigma }\to \mathsf{Set}$ obtained by composing the contravariant functor ${\mathbf{Op}}_{\Sigma }^{\mathcal{U}}:{\mathsf{Var}}_{\Sigma }^{op}\to \mathsf{Set}$ with the contravariant powerset functor that relies on the formation of pre-images.

The satisfaction condition in (19) is then equivalent to the requirement that for any $\Sigma$-structure $\mathcal{U}$ the $Var$-indexed family of maps ${\left[[\_]\right]}_X^{\mathcal{U}}:F{E}_{\Sigma }\left(X\right)\to \wp \left(U^X\right)$ establishes a natural transformation ${\left[[\_]\right]}^{\mathcal{U}}:{\mathbf{FE}}_{\Sigma }\Rightarrow {\mathbf{Hom}}_{\Sigma }^{\mathcal{U}}$ : For any $\Sigma$-substitution and $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$, any $\Sigma$-expression $X▹Ex$ and any $\Sigma$-structure $\mathcal{U}$ it holds that

$${\downarrow }_{\mathbf{t}}^{\mathcal{U}}{\left[[X▹Ex]\right]}^{\mathcal{U}}{=\downarrow }_{\mathbf{t}}^{\mathcal{U}}{\left[\left[Ex\right]\right]}_X^{\mathcal{U}}={\left({\mathbf{t}}^{\mathcal{U}}\right)}^{-1}{\left[\left[Ex\right]\right]}_X^{\mathcal{U}}={\left[\left[\overline{\mathbf{t}}\left(Ex\right)\right]\right]}_Y^{\mathcal{U}}={\left[[Y▹\overline{\mathbf{t}}\left(Ex\right)]\right]}^{\mathcal{U}}.$$

(20)

5. Sketches and Arrows between Sketches

Any institution gives us a corresponding category of presentations and an extension of the model functor of the institution to the category of presentations at hand [19,20]. We outline, adapt and extend this construction for Institutions of Statements.

5.1. Sketches and Diagrams

In the area of Algebraic Specifications those presentations are, for example, pairs of an algebraic signature and a set of (conditional) equations and are called specifications. [6,7,8]. In the different institutions of algebraic specifications and in the traditional institution of first-order logic [19,20] the "models" are whole structures.

In contrast, LoSiCs shift the perspective from whole structures to single interpretations of contexts in structures. Note, however, that this is not only a shift but also an enhancement of perspective since the "whole structure perspective" can be reconstructed by the restriction to empty contexts only (see Remark 20). Incarnations of this enhanced perspective are, for example, Ehresmann Sketches [9] and Generalized Sketches in the sense of Makkai [10] or in the sense of Diskin [11,12]. Generalized Sketches in the sense of Diskin have been shown to be an appropriate conceptual tool to formalize software models and to pave a way for a theoretical foundation of Model Driven Software Engineering [15,16]. To underline this enhanced perspective and the "historical roots" of LoSiCs, we use the term sketch, instead of presentation or specification.

Definition 12

(Sketch). For any signature $\Sigma =(P,F,ar,in,out)$ a$\Sigma$- sketch$\mathbb{K}=(K,S{}_t)$ is given by a context $K\in \left|\mathsf{Cxt}\right|=\left|\mathsf{Set}\right|$ and a set $S{}_t\subseteq St{m}_{\Sigma }\left(K\right)$ of Σ-statements in context K. $\mathbb{K}=(K,S{}_t)$ is calledatomicif all the Σ-statements $(X▹Ex,\gamma )$ in $St$ are atomic meaning that $Ex$ is an atomic expression, i.e., an equation or a relational atom according to Definition 5.

Remark 23

(Sketches in the Wild). We continue Remark 19. In Group Theory we meet sketches as pairs $(G,R)$ of a set G of generators and a set R of defining relations between the generators in G. A sketch $(O,F)$ in Logic Programming is given by a set O of atoms and a set F of facts about the atoms in O. Sometimes one may also meet the termlogical database.

Finally, we find sketches in Description Logics as pairs $(N_O,\mathcal{A})$ with $N_O$ a set of individual names (nominals, objects) and $\mathcal{A}$ a so-calledABoxof assertional axioms.

Definition 13

(Interpretations of Sketches). An interpretation $\iota :K\to U$ of context K in a Σ-structure $\mathcal{U}$ isa valid interpretationof the sketch $\mathbb{K}=(K,S{}_t)$ in $\mathcal{U}$, $(\iota ,\mathcal{U}){\vDash }_K^{\Sigma }St$ in symbols, if, and only if, $(\iota ,\mathcal{U}){\vDash }_K^{\Sigma }(X▹Ex,\gamma )$, due to (10), for all Σ-statements $(X▹Ex,\gamma )$ in $St$. Be aware that the Σ-statements in $St$ may have different sets X of variables!

Analogously to the semantics of statements in (11), we can define the semantics of a $\Sigma$-sketch $\mathbb{K}=(K,S{}_t)$ in a $\Sigma$-structure $\mathcal{U}$ as the set of all valid interpretations of $\mathbb{K}$ in $\mathcal{U}$:

$${\left[\left[\mathbb{K}\right]\right]}^{\mathcal{U}}={\left[\left[(K,S{}_t)\right]\right]}^{\mathcal{U}}={\left[[S{}_t]\right]}_K^{\mathcal{U}}:=\{\iota \in U^K\mid (\iota ,\mathcal{U}){\vDash }_K^{\Sigma }S{}_t\}\subseteq U^K.$$

(21)

Relying on (11) or (12), we can obviously reformulate Definition 13, more abstractly, by

$${\left[\left[\mathbb{K}\right]\right]}^{\mathcal{U}}={\left[\left[(K,S{}_t)\right]\right]}^{\mathcal{U}}={\left[[S{}_t]\right]}_K^{\mathcal{U}}:=\bigcap \{{\left[\left[(X▹Ex,\gamma )\right]\right]}_K^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{}_t\}\subseteq U^K.$$

(22)

For later references it is convenient to lift up the characterization of the satisfaction condition in (14) to arbitrary subsets of statements and, thus, to fully exemplify the concept of institution frame in [23,24]. For any context morphism $\psi :K\to G$ and any $\Sigma$-structure $\mathcal{U}$ the diagram below commutes.

For any set $S{}_t\subseteq St{m}_{\Sigma }\left(K\right)$ of $\Sigma$-statements in context K we get indeed

$$\begin{array}{cccc}\hfill {\left[\left[{\mathbf{Stm}}_{\Sigma }\left(\psi \right)(S{}_t)\right]\right]}_G^{\mathcal{U}}& =& \bigcap \{{\left[\left[(X▹Ex,\gamma ;\psi )\right]\right]}_G^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{}_t\}\hfill & \left(definition\right(22\left)\right)\hfill \\ & =& \bigcap \{{\downarrow }_{\psi }^U{\left[\left[(X▹Ex,\gamma )\right]\right]}_K^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{}_t\}\hfill & \left(condition\right(14\left)\right)\hfill \\ & =& {\downarrow }_{\psi }^U(\bigcap \{{\left[\left[(X▹Ex,\gamma )\right]\right]}_K^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{}_t\})\hfill & ({\downarrow }_{\psi }^{U}distrib.over\cap )\hfill \\ & =& {\downarrow }_{\psi }^U{\left[[S{}_t]\right]}_K^{\mathcal{U}}\hfill & \left(definition\right(22\left)\right)\hfill \end{array}$$

(Elementary) diagrams play an essential role in completeness proofs. We will not address completeness proofs in this paper. We want, however, shortly outline and discuss this additional motivation for introducing contexts.

We became familiar with diagrams not via Model Theory [2,3,20] but as a technique used in the construction of free algebras [6,7,29]. From this perspective diagrams arise as syntactic representations of structures. To obtain such a representation, we need, first, a mechanism to syntactically represent the elements of a given structure.

A widely used mechanism is to add for a given $\Sigma$-structure $\mathcal{U}$ "copies" of all the elements in the carrier U of $\mathcal{U}$ as new auxiliary constant symbols to the signature $\Sigma$ (thus blowing up $\Sigma$ to an infinite entity in most cases) [2,6,20]. In contrast, Logics of Statements in Context generalize the approach in [7,29] which adapts the idea of generators and defining relations from Group Theory.

The elements of contexts are for us, in principal, neither constant symbols nor variables nor elements of carrier sets. Or, to put it the other way around: Of course we can use one and the same entity in different roles, but we should always be aware of the different roles and changes of roles! Contexts establish a bridge between syntax (constant symbols, variables) and semantics (elements of carrier sets), and we use them to define diagrams.

The first step in transforming a $\Sigma$-structure $\mathcal{U}$ into a $\Sigma$-sketch is to transform the carrier U of $\mathcal{U}$ into a context by introducing a syntactic duplicate for each element in U or by simply declaring that "we consider U in the role of a context".

In the literature we essentially find two variants of diagrams - an atomic variant [3,20] and a full variant [2]. Obviously, we can define two corresponding variants of sketch encodings of a $\Sigma$-structure $\mathcal{U}$. The atomic variant is given by the $\Sigma$-sketch

$${\mathbb{S}}_{at}^{\mathcal{U}}:=(U,S{t}_{at}^{\mathcal{U}})withS{t}_{at}^{\mathcal{U}}:=\{(X▹Ex,\gamma )\mid Ex\mathrm{atomic},(i{d}_U,\mathcal{U}){\vDash }_U^{\Sigma }(X▹Ex,\gamma )\}$$

(24)

for the canonical interpretation $(i{d}_U,\mathcal{U})$ of context the U in the $\Sigma$-structure $\mathcal{U}$.

The full variant ${\mathbb{S}}^{\mathcal{U}}=(U,S{t}^{\mathcal{U}})$ of the $\Sigma$-sketch encoding of $\mathcal{U}$ is obtained by droping the restriction "atomic". The interest reader may have a look at [1] for more concepts and results around sketch encodings. Especially, we discuss there a universal property of the canonical interpretation which is strongly related to Proposition 4.10 in [20].

5.2. Arrows between Sketches

In this subsection we revisit, revise, adapt and extend the discussions, definitions and results of the Subsections 5.2.3, 5.2.4 and 5.2.5 in [1]. Each $\Sigma$-sketch has his own local context declaration thus arrows between sketches are established by context morphisms.

Definition 14

(Sketch Arrow). A$\Sigma$-sketch arrowbetween two Σ-sketches $\mathbb{P}=(P,S{t}^{\mathbb{P}})$ and $\mathbb{C}=(C,S{t}^{\mathbb{C}})$ is given by a context morphism $\psi :P\to C$ and denoted by $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$. $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ isatomicif all the Σ-statements in $S{t}^{\mathbb{P}}$ and $S{t}^{\mathbb{C}}$ are atomic. The category of all Σ-sketches and all Σ-sketch arrows is denoted by $\mathsf{Sk}\left(\Sigma \right)$.

A Σ-sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ is calledstrict, $\psi :\mathbb{P}⤏\mathbb{C}$ in symbols, if, and only if, $S{t}^{\mathbb{C}}\supseteq {\mathbf{Stm}}_{\Sigma }\left(\psi \right)\left(S{t}^{\mathbb{P}}\right)$ with the statement translation ${\mathbf{Stm}}_{\Sigma }\left(\psi \right)$ defined in (7). ${\mathbf{Stm}}_{\Sigma }:\mathsf{Cxt}\to \mathsf{Set}$ is a functor thus the strict Σ-sketch arrows constitute a subcategory $\mathsf{Sks}\left(\Sigma \right)$ of $\mathsf{Sk}\left(\Sigma \right)$.

In case $P=C$ and $\psi =i{d}_P$, we simply write $\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ and $\mathbb{P}⤏\mathbb{C}$, respectively.

5.2.1. Sketch Morphisms

Sketch arrows are subject of two different, but dual, semantical conditions. First, any institution gives us a corresponding concept of presentation morphism at hand [19,20]. We discuss the corresponding morphism condition and, as a terminological sleight of hand, we will use the term sketch morphism and the notation $\mathbb{P}\stackrel{\psi }{⟶}\mathbb{C}$ to indicate that a sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ is subject of the morphism condition.

Definition 15

(Satisfaction of Morphism Condition). A Σ-sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ satisfies themorphism conditionfor a Σ-structure $\mathcal{U}$, $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟶}\mathbb{C}$ in symbols, if the "forgetful map" ${\mathbf{Int}}_{\Sigma }\left(\psi \right)=(\psi ;\_):U^C\to U^P$ in (14) restricts to a map $(\psi ;\_):{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\to {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$, i.e., if ${\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=(\psi ;\_){\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\subseteq {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$ or, equivalently due to Corollary A4, if ${\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}{\subseteq \downarrow }_{\psi }^U{\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$.

Identity $\Sigma$-sketch arrows trivially satisfy the morphism condition and satisfaction of the morphism condition for a $\Sigma$-structure $\mathcal{U}$ is, due to (A23) and the monotonicity of the operators ${\exists }_{\psi }^U$, closed under composition in $\mathsf{Sk}\left(\Sigma \right)$ thus all $\Sigma$-sketch arrows satisfying the morphism condition for a $\Sigma$-structure $\mathcal{U}$ constitute a subcategory $\mathsf{SkM}\left(\Sigma \right)\left(\mathcal{U}\right)$ of $\mathsf{Sk}\left(\Sigma \right)$.

The syntactic strictness condition ensures the satisfaction of the morphism condition.

Corollary 2

(Morphism Condition by Strictness). We have $\mathsf{Sks}\left(\Sigma \right)⊑\mathsf{SkM}\left(\Sigma \right)\left(\mathcal{U}\right)$ for any Σ-structure $\mathcal{U}$ , i.e., for any strict Σ-sketch arrow $\psi :\mathbb{P}⤏\mathbb{C}$ it holds that $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟶}\mathbb{C}$.

Proof. 

For any strict $\Sigma$-sketch arrow $\psi :\mathbb{P}⤏\mathbb{C}$ and any $\Sigma$-structure $\mathcal{U}$ we get indeed

$$\begin{array}{cccc}\hfill {\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}& =& \bigcap \{{\left[\left[(X▹Ex,\rho )\right]\right]}_C^{\mathcal{U}}\mid (X▹Ex,\rho )\in S{t}^{\mathbb{C}}\}\hfill & \left(definition\right(22\left)\right)\hfill \\ & \subseteq & \bigcap \{{\left[\left[(X▹Ex,\gamma ;\psi )\right]\right]}_C^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{t}^{\mathbb{P}}\}\hfill & (ass.S{t}^{\mathbb{C}}\supseteq {\mathbf{Stm}}_{\Sigma }\left(\psi \right)\left(S{t}^{\mathbb{P}}\right))\hfill \\ & =& \bigcap \{{\downarrow }_{\psi }^U{\left[\left[(X▹Ex,\gamma )\right]\right]}_P^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{t}^{\mathbb{P}}\}\hfill & (satisfaction condition(14\left)\right)\hfill \\ & =& {\downarrow }_{\psi }^U(\bigcap \{{\left[\left[(X▹Ex,\gamma )\right]\right]}_P^{\mathcal{U}}\mid (X▹Ex,\gamma )\in S{t}^{\mathbb{P}}\})\hfill & ({\downarrow }_{\psi }^{U}distributes over\cap )\hfill \\ & =& {\downarrow }_{\psi }^U{\left[\left[\mathbb{P}\right]\right]}_P^{\mathcal{U}}\hfill & \left(definition\right(22\left)\right)\hfill \end{array}$$

□

5.2.2. Sketch Implications

The implication condition is the second semantical condition for sketch arrows. To motivate and elucidate this second condition we informally discuss the paradigmatic specification framework of conditional equations. An analogous discussion applies to the specification framework of Horn clauses (PROLOG).

The first crucial methodological observation is that conditional equations $(X:t_1=r_1,\dots ,t_n=r_n⟶t_0=r_0)$ are not, and never have been, a lazy notation for closed formulas $\forall (X:(t_1=r_1\wedge \dots \wedge t_n=r_n)⟶t=r)$! Conditional equations have been introduced and utilized as a conceptual tool of its own independent of first-order logic! One of the many motivations to coin the concept of LoSiCs was to develop a conceptual framework enabling us to precisely grasp the particular nature of conditional equations. In LoSiC-terminology conditional equations are a special kind of atomic sketch arrows

$$i{d}_X:(X,\{(X▹(t_1=r_1),i{d}_X),\dots (X▹(t_n=r_n),i{d}_X)\})@.>{}_{\left[r\right]}(X,\left\{(X▹(t_0=r_0),i{d}_X)\right\}).$$

A conditional equation is satisfied in a $\Sigma$-structure $\mathcal{U}$ if, and only if, each solution of the premise $(X,\{(X▹(t_1=r_1),i{d}_X),\dots (X▹(t_n=r_n),i{d}_X)\})$ in $\mathcal{U}$ also is a solution of the conclusion $(X,\left\{(X▹(t_0=r_0),i{d}_X)\right\})$.

The second observation is that we do have, at least, three well distinguished (!) syntactic entities that one could call "equation".

Expression:

First, we do have atomic expressions $Y▹(t=r)$ with their individual semantics in any $\Sigma$-structure $\mathcal{U}$. This semantics is the set of all variable assignments $\mathbf{a}:Y\to U$ solving the equation $(t=r)$.

Statements in Context:

Second, there are for each atomic expression $Y▹(t=r)$ infinite many single statements $(Y▹(t=r),\gamma :Y\to K)$ in contexts where the semantics of $(Y▹(t=r),\gamma )$ in a $\Sigma$-structure $\mathcal{U}$ is the set of all interpretations $\iota :K\to U$ such that $\gamma ;\iota :Y\to U$ solves the equation $(t=r)$.

Unconditional Assertion:

Third, we do have for each atomic expression $Y▹(t=r)$ a sketch arrow $(Y,\varnothing )@.>{}_{\left[r\right]}(Y,\left\{(Y▹(t=r),i{d}_Y)\right\})$ which is valid in a $\Sigma$-structure $\mathcal{U}$ if, and only if, each solution of the "empty premise" $(Y,\varnothing )$, i.e., all variable assignments $\mathbf{a}:Y\to U$, solve the equation $(t=r)$ or, in other words, if t and r represent the same derived opperation $t^{\mathcal{U}}=r^{\mathcal{U}}$ in $\mathcal{U}$.

This conceptional triad "expression - statements in context - unconditional assertion" we do have in LoSiCs for any expression at hand!

Analogous to the morphism condition, we will use the term sketch implication and the notation $\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$ to indicate that a sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ is subject of the implication condition.

Definition 16

(Satisfaction of Implication Condition). A Σ-sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ satisfies theimplication conditionfor a Σ-structure $\mathcal{U}$, $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$ in symbols, if, and only if, for all interpretations $(\iota ,\mathcal{U})$ of P in $\mathcal{U}$ it holds that $(\iota ,\mathcal{U}){\vDash }_P^{\Sigma }S{t}^{\mathbb{P}}$ implies the existence of an interpretation $(\vartheta ,\mathcal{U})$ of context C in $\mathcal{U}$ with $\psi ;\vartheta =\iota$ such that $(\vartheta ,\mathcal{U}){\vDash }_C^{\Sigma }S{t}^{\mathbb{C}}$.

Obviously, we trivially have $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$ if there is no interpretation of context P in $\Sigma$-structure $\mathcal{U}$ at all. As discussed before, this may be the case in many-sorted logics due to the potential presence of empty sorts. In case $P=C$ and $\psi =i{d}_P$, we have $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}⟹\mathbb{C}$ if, and only if, each valid interpretation of the premise $\mathbb{P}$ is also a valid interpretation of the conclusion $\mathbb{C}$.

By means of the existential operators from Appendix A.2, we can equivalently define the satisfaction of the implication condition as follows:

$$\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}iff{\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=(\psi ;\_){\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}iff\overline{{\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}}\cup {\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=U^P.$$

(25)

So, the morphism condition in Definition 15 and the implication condition are indeed dual semantical conditions! Analogously to the morphism condition, the characterization (25) ensures that all the $\Sigma$-sketch arrows satisfying the implication condition for a $\Sigma$-structure $\mathcal{U}$ constitute a subcategory $\mathsf{SkI}\left(\Sigma \right)\left(\mathcal{U}\right)$ of $\mathsf{Sk}\left(\Sigma \right)$.

We use the notation $@3->{}_{\left[r\right]}$ if both conditions are satisfied.

$$\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}@3->{\left[r\right]}^{\psi }\mathbb{C}iff{\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=(\psi ;\_){\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}={\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}.$$

(26)

Due to Corollary 2, the equality in (26) especially holds for all strict $\Sigma$-sketch implication!

Sketch implications enable us to specify and/or describe properties of $\Sigma$-structures. The implication condition is the essential specification tool in specification frameworks of conditional equations [7,29], in specification frameworks of Horn clauses as well as in Makkai’s General Sketch framework [10]. In case of first-order logic, however, sketch implications do not add any "specification power" since the satisfaction of a $\Sigma$-sketch implication $\mathbb{P}@3->{\left[r\right]}^{\psi }\mathbb{C}$ in $\Sigma$-structures can obviously be simulated by the satisfaction of a corresponding closed formula as long as $\mathbb{P}$ and $\mathbb{C}$ are finite (compare Section 5.2.4 in [1]).

5.2.3. Sketch Arrows and Injectivity

Besides specifying and/or describing properties of $\Sigma$-structures, sketch arrows can be also utilized to specify and/or describe the structure of $\Sigma$-sketches.

Definition 17

(Injectivity). A Σ-sketch $\mathbb{K}=(K,S{t}^{\mathbb{K}})$ isstrictly injective w.r.t. a $\Sigma$-sketch arrow$\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ if, and only if, there exists for each strict Σ-sketch arrow $\mu :\mathbb{P}⤏\mathbb{K}$ a strict Σ-sketch arrow $\varrho :\mathbb{C}⤏\mathbb{K}$ such that $\mu =\psi ;\varrho$.

In Definition 23 in [1] we used, instead of "strictly injective", the attribute "closed" based on the intuition that $\mathbb{K}$ is closed w.r.t. the application of $\psi$ as a deduction rule (compare the next subsection and [29]). Afterwards, we became more acquainted with the abstract concept of injectivity [30,31] and adapted therefore Diaconescu’s idea to indicate the restriction of the all-quantification in the definition of injectivity to a certain class of arrows by adding an adverb - "simply" on page 17 in [20] for the restriction to monic arrows. Since we only consider one type of injectivity in this paper, we simply drop the adverb "strictly" in most cases.

The satisfaction condition in ${\mathcal{FL}}_{\Sigma }$ ensures that the satisfaction of the implication condition in structures can equivalently be expressed by the strictly injectivity of sketch encodings of structures (24).

Proposition 1

(Implication Condition ≅ Injectivity). For any Σ-sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ and any Σ-structure $\mathcal{U}$ the following two statements are equivalent:

1.

$\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$, i.e., $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$ satisfies the implication condition for $\mathcal{U}$.

2.

The Σ-sketch encoding ${\mathbb{S}}^{\mathcal{U}}=(U,S{t}^{\mathcal{U}})$ of $\mathcal{U}$ is strictly injective w.r.t. $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$.

Results, analogously to Proposition 1, are the corner stone for completeness proofs in specification frameworks of conditional equations [7,29]. In Makkai’s Generalized Sketch framework [10] there is no distinction between structures and sketch encodings and injectivity w.r.t. sketch arrows is the only conceptual tool for specification and/or description. Moreover, sketch arrows are the only tool for deduction thus results, analogously to Proposition 1, also are the corner stone for completeness results in [10]. Probably, related results also play a crucial role in completeness proofs in traditional first-order logic!?

In Section 7 and Section 8 we will demonstrate that sketch arrows together with the morphism and implication condition are, at least, appropriate conceptional tools to analyze, describe, validate and relate traditional deduction calculi for first-order logic.

5.3. Deducing Sketch Implications

There are, at least, three semantically sound mechanisms to deduce sketch implications from given sketch implications. First, we can simply obtain new (strict) sketch implications from given (strict) sketch implications by means of composition in $\mathsf{Sk}\left(\Sigma \right)$. This procedure is sound since we have $\mathsf{SkI}\left(\Sigma \right)\left(\mathcal{U}\right)⊑\mathsf{Sk}\left(\Sigma \right)$ and thus also $\mathsf{Sks}\left(\Sigma \right)\cap \mathsf{SkI}\left(\Sigma \right)\left(\mathcal{U}\right)⊑\mathsf{Sk}\left(\Sigma \right)$.

Second, we can apply a given sketch implication as a rule via sketch morphisms to deduce sketch implications. We discuss and validate this mechanism in this subsection. Third, we can apply a given sketch implication as a rule via substitutions to deduce sketchs implication. Rule application via substitutions is, however, only feasible for sketches where all bindings are identities. We postpone the discussion of this mechanism to Section 6.4.

We consider an arbitrary, but fixed, $\Sigma$-sketch arrow $\psi :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{C}$. For any given $\Sigma$-sketch $\mathbb{G}$ and any $\Sigma$-sketch arrow $\mu :\mathbb{P}@.>{}_{\left[r\right]}\mathbb{G}$, called a match, we can construct a pushout of the underlying context morphisms $\psi :P\to C$, $\mu :P\to G$ in $\mathsf{Cxt}=\mathsf{Set}$.

As usual, the interpretation of a "syntactic pushout" in a chosen "universe" provides a "semantic pullback". In our case, we get for any carrier set U a pullback of hom-sets and pre-composition maps, as visualized in the middle square above, since for any interpretations $\iota :G\to U$, $\vartheta :C\to U$ with $\mu ;\iota =\psi ;\vartheta$ there exist a unique interpretation $\kappa :R\to U$ such that ${\psi }^{*};\kappa =\iota$ and ${\mu }^{*};\kappa =\vartheta$ due to the "syntactic pushout" in the left hand square above. Note, that this is nothing but model amalgamation in the sense of the theory of institutions [20]. Compare also the amalgamation of algebras in [6].

We extend, in a minimal way, the pushout of context morphisms to a commutative square of $\Sigma$-sketch arrows by defining $\mathbb{R}:=(R,S{t}^{\mathbb{R}})$ with $S{t}^{\mathbb{R}}:={\mathbf{Stm}}_{\Sigma }\left({\mu }^{*}\right)\left(S{t}^{\mathbb{C}}\right)$.

By definition, we obtain a strict $\Sigma$-sketch arrow ${\mu }^{*}:\mathbb{C}⤏\mathbb{R}$ which always satisfies, according to Corollary 2, the morphism condition.

Corollary 3

(Morphism Condition for Instances 1). For any Σ-structure $\mathcal{U}$ it holds that $\mathcal{U}{\vDash }^{\Sigma }\mathbb{C}\stackrel{{\mu }^{*}}{⟶}\mathbb{R}$, i.e., we have ${\exists }_{{\mu }^{*}}^U{\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}=({\mu }^{*};\_){\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}\subseteq {\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}$.

In light of sketch implications, we construct for a given sketch implication $\psi$ and any sketch morphism $\mu$ a corresponding sketch implication ${\psi }^{*}$ which we may call the result of applying the rule ψ to $\mathbb{G}$ via the match μ. We show the soundness of this procedure.

Proposition 2

(Rule via Sketch Morphism). For any Σ-structure $\mathcal{U}$ with $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$, i.e., ${\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=(\psi ;\_){\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$, and $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\mu }{⟶}\mathbb{G}$, i.e., ${\exists }_{\mu }^U{\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}=(\mu ;\_){\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}$$\subseteq {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$, it holds that $\mathcal{U}{\vDash }^{\Sigma }\mathbb{G}\stackrel{{\psi }^{*}}{⟹}\mathbb{R}$, i.e., ${\exists }_{{\psi }^{*}}^U{\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}=({\psi }^{*};\_){\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}$.

Proof. 

If ${\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}=\varnothing$, the inclusion trivially holds. If ${\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}\ne \varnothing$, we consider an arbitrary $\iota \in {\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}\subseteq U^G$. By assumption $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\mu }{⟶}\mathbb{G}$, we get $\mu ;\iota \in {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$ thus there exists a $\vartheta \in {\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\subseteq U^C$ with $\mu ;\iota =\psi ;\vartheta$ due to assumption $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$. For the unique $\kappa :R\to U$ with ${\psi }^{*};\kappa =\iota$ and ${\mu }^{*};\kappa =\vartheta$ the fact $\vartheta \in {\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}$ and the satisfaction condition (23) for ${\mu }^{*}:C\to R$ ensure $\kappa \in {({\mu }^{*};\_)}^{-1}{\left[\left[S{t}^{\mathbb{C}}\right]\right]}_C^{\mathcal{U}}={\left[\left[{\mathbf{Stm}}_{\Sigma }\left({\mu }^{*}\right)\left(S{t}^{\mathbb{C}}\right)\right]\right]}_R^{\mathcal{U}}={\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}$ with $({\psi }^{*};\_)\left(\kappa \right)={\psi }^{*};\kappa =\iota$ and thus $\iota \in {\exists }_{{\psi }^{*}}^U{\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}=({\psi }^{*};\_){\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}$, as required. □

It is maybe worth to mention, that Proposition 2 is related, in a certain sense, to the so-called "Extension Lemma" on page 223 in [6].

The semantic condition $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\mu }{⟶}\mathbb{G}$ is not really feasible. However, Corollary 2 allows us to work, instead, with the stronger but pure syntactic condition "strict".

Corollary 4

(Rule via Sketch Morphism). For any Σ-structure $\mathcal{U}$ with $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$, i.e., ${\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=(\psi ;\_){\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$, and any strict Σ-sketch arrow $\mu :\mathbb{P}⤏\mathbb{G}$, i.e., $S{t}^{\mathbb{G}}\supseteq {\mathbf{Stm}}_{\Sigma }\left(\mu \right)\left(S{t}^{\mathbb{P}}\right)$, it holds that $\mathcal{U}{\vDash }^{\Sigma }\mathbb{G}\stackrel{{\psi }^{*}}{⟹}\mathbb{R}$, i.e., ${\exists }_{{\psi }^{*}}^U{\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}=({\psi }^{*};\_){\left[\left[\mathbb{R}\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}$.

In case $S{t}^{\mathbb{G}}={\mathbf{Stm}}_{\Sigma }\left(\mu \right)\left(S{t}^{\mathbb{P}}\right)$, we may call $\mathbb{G}\stackrel{{\psi }^{*}}{⟹}\mathbb{R}$ the instance of $\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$ with respect to the context morphism (!) $\mu :P\to G$.

In practice, we may be interested to also deduce strict sketch implications. For the strict variant ${\mathbb{R}}_{st}:=(R,S{t}^{{\mathbb{R}}_{st}})$ with $S{t}^{{\mathbb{R}}_{st}}:={\mathbf{Stm}}_{\Sigma }\left({\psi }^{*}\right)\left(S{t}^{\mathbb{G}}\right)\cup {\mathbf{Stm}}_{\Sigma }\left({\mu }^{*}\right)\left(S{t}^{\mathbb{C}}\right)$ of the resulting $\Sigma$-sketch, we also get $\mathcal{U}{\vDash }^{\Sigma }\mathbb{G}\stackrel{{\psi }^{*}}{⟹}{\mathbb{R}}_{st}$ for any $\Sigma$-structure $\mathcal{U}$. To prove this, we need only to extend the last sentence in the proof of Proposition 2 as follows: For the unique $\kappa :R\to U$ with ${\psi }^{*};\kappa =\iota$ and ${\mu }^{*};\kappa =\vartheta$ the satisfaction condition (23) for ${\psi }^{*}:G\to R$ ensures $\kappa \in {\left[\left[{\mathbf{Stm}}_{\Sigma }\left({\psi }^{*}\right)\left(S{t}^{\mathbb{G}}\right)\right]\right]}_R^{\mathcal{U}}$ while the satisfaction condition for ${\mu }^{*}:C\to R$ ensures $\kappa \in {\left[\left[{\mathbf{Stm}}_{\Sigma }\left({\mu }^{*}\right)\left(S{t}^{\mathbb{C}}\right)\right]\right]}_R^{\mathcal{U}}$. In such a way, we have $\kappa \in {\left[\left[{\mathbb{R}}_{st}\right]\right]}^{\mathcal{U}}={\left[\left[{\mathbf{Stm}}_{\Sigma }\left({\psi }^{*}\right)\left(S{t}^{\mathbb{G}}\right)\right]\right]}_R^{\mathcal{U}}\cap {\left[\left[{\mathbf{Stm}}_{\Sigma }\left({\mu }^{*}\right)\left(S{t}^{\mathbb{C}}\right)\right]\right]}_R^{\mathcal{U}}$ with $({\psi }^{*};\_)\left(\kappa \right)={\psi }^{*};\kappa =\iota$ and thus $\iota \in {\exists }_{{\psi }^{*}}^U{\left[\left[{\mathbb{R}}_{st}\right]\right]}^{\mathcal{U}}=({\psi }^{*};\_){\left[\left[{\mathbb{R}}_{st}\right]\right]}^{\mathcal{U}}$, as required. By Corollary 2 we get, in such a way, a variant of Corollary 4 for the strict case.

Corollary 5

(Rule via Sketch Morphism - Strict). For any Σ-structure $\mathcal{U}$ with $\mathcal{U}{\vDash }^{\Sigma }\mathbb{P}\stackrel{\psi }{⟹}\mathbb{C}$, i.e., ${\exists }_{\psi }^U{\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}=(\psi ;\_){\left[\left[\mathbb{C}\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[\mathbb{P}\right]\right]}^{\mathcal{U}}$, and any strict Σ-sketch arrow $\mu :\mathbb{P}⤏\mathbb{G}$, i.e., $S{t}^{\mathbb{G}}\supseteq {\mathbf{Stm}}_{\Sigma }\left(\mu \right)\left(S{t}^{\mathbb{P}}\right)$, it holds that $\mathcal{U}{\vDash }^{\Sigma }\mathbb{G}@3->{\left[r\right]}^{{\psi }^{*}}{\mathbb{R}}_{st}$, i.e., ${\exists }_{{\psi }^{*}}^U{\left[\left[{\mathbb{R}}_{st}\right]\right]}^{\mathcal{U}}=({\psi }^{*};\_){\left[\left[{\mathbb{R}}_{st}\right]\right]}^{\mathcal{U}}={\left[\left[\mathbb{G}\right]\right]}^{\mathcal{U}}$.

6. Licensing Laziness

In view of LoSiCs, notations in traditional first-order logic are "lazy" in the sense that there are neither explicit variable declarations nor are contexts explicitly considered. In this subsection we discuss how and under what conditions the traditional lazy notations can be recasted in a LoSiC-compatible way. In first-order deduction calculi we only deal with finite sets of finite formulas thus we will only need finite sets of variables in $Va{r}_{\Sigma }$ as variable declarations and as contexts for our analysis.

Let $\Gamma$ be a finite set of "well-formed formulas" in the traditional sense, as in [4] for example. The expressions in LoSiCs are well-formed formulas satisfying some syntactic restrictions. First, LoSiCs require that each well-formed formula has its own local declaration of free variables establishing its "interface" with the outside world. We could say that the traditional approach implicitly provides this "interface" in form of the set $fv\left(B\right)$ of all free variables syntactically appearing in B. Due to the "methodological accident", discussed in Remark 10, expressions in LoSiCs must comply with Barendregt’s variable convention thus we will, in general, not have $fv\left(B\right)▹B$, i.e., $B\in F{E}_{\Sigma }\left(X\right)$, in the sense of Definition 5. By a renaming of bound variables we can, however, transform any well-formed formula B into an equivalent expression $B^{\prime }$ with $fv\left(B\right)=fv\left(B^{\prime }\right)$ and $fv\left(B\right)▹B^{\prime }$.

In logic textbooks, as in [4] for example, they say that deduction calculi for first-order logic can inherit axiom schemata and rules from deduction calculi for Propositional Logic. There is, however, no formal explanation in what sense an open formula represents a proposition. In LoSiCs we try to repair this flaw by introducing statements in context which are statements about single (!) interpretations of contexts in a given structure and, thus, propositions in the sense of Propositional Logic.

Due to the lack of explicit variable declarations and explicit declarations of contexts, there also are no explicit bindings and context morphisms in traditional first-order logic. The only implicitly present morphisms are inclusion maps. We may consider $X=fv\left(\Gamma \right):=\bigcup \left\{fv\right(B)\mid B\in \Gamma \}$ as the implicit context of $\Gamma$ thus we could transform $\Gamma$ into a sketch with context X and the set $\{(fv\left(B\right)▹B,i{n}_{fv\left(B\right),X})\mid B\in \Gamma \}$ of statements in context X. In alignment with traditional first-order logic, we can, however, in LoSiCs only combine expressions by Boolean operators but not statements in context! That is, we cannot represent an implication, like "$A\to B$" in traditional notation with $A,B\in \Gamma$, by a phrase like "$(fv\left(A\right)▹A,i{n}_{fv\left(A\right),X})\to (fv\left(B\right)▹B,i{n}_{fv\left(B\right),X})$" but only by a statement in context like $(Y▹A\to B,i{n}_{Y,X})$ with $fv\left(A\right)\cup fv\left(B\right)\subseteq Y\subseteq X$. Due to Barendregt’s variable convention this statement is only syntactically correct if the sets Y and $bv\left(A\right)\cup bv\left(B\right)$ are disjoint!

To be able to mimic the traditional free Boolean combination of well-formed formulas, we decide, therefore, to utilize a specific class of $\Sigma$-sketches $(X,S{}_t)$ in our analysis of traditional first-order deduction calculi. We use only sets X of variables as contexts. We only consider finite $\Sigma$-sketches with simple statements, i.e., statements of the form $(X▹B,i{d}_X)$. In such a way, we adapt a strict separation of free and bound variables in the sense that we require $X\cap bv(S{}_t)=\varnothing$. So, compared to the traditional approach, which simply works with arbitrary finite sets $\Gamma$ of arbitrary well-formed formulas, we propose three changes:

1.

We declare an explicit context X with $fv\left(\Gamma \right)\subseteq X$.

2.

We require $\Gamma \subseteq F{E}_{\Sigma }\left(X\right)$ and adapt, in such a way, Barendregt’s variable convention.

3.

We require, moreover, $X\cap bv\left(\Gamma \right)=\varnothing$, i.e., a strict separation of free and bound variables.

We call a pair $(X,\Gamma )$ of a set X of variables and a finite set $\Gamma$ of well-formed formulas LoSiC-compatible or, synonymously, a $\Sigma$-specification if it satisfies the conditions above, i.e., if $(X,\Gamma )$ can be seen as a representation of the simple $\Sigma$-sketch$(X,S{t}_{\Gamma }^X)$ with statements $S{t}_{\Gamma }^X:=\{(X▹B,i{d}_X)\mid B\in \Gamma \}$. The semantics of a $\Sigma$-specification $(X,\Gamma )$ is defined by

$${\left[\left[(X,\Gamma )\right]\right]}^{\mathcal{U}}:={\left[\left[(X,S{t}_{\Gamma }^X)\right]\right]}^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}=\bigcap \{{\left[\left[B\right]\right]}_X^{\mathcal{U}}\mid B\in \Gamma \}$$

(28)

for any $\Sigma$-structure $\mathcal{U}$. The syntactic requirements 2. and 3. above are only of technical nature and cause no loss of generality! We have to ensure, however, that all constructions, relevant for deduction, preserve LoSiC-compatibility!

6.1. Reduction of Contexts

A first simple construction is the reduction of contexts. Given a $\Sigma$-specification $(X,\Gamma )$ and a set Z of variables with $fv\left(\Gamma \right)\subseteq Z\subset X$, the pair $(Z,\Gamma )$ is expected to also be a $\Sigma$-specification and both pairs should represent the same semantic information.

Lemma 1 ensures that $(Z,\Gamma )$ is a $\Sigma$-specification too. That both $\Sigma$-specifications $(X,\Gamma )$ and $(Z,\Gamma )$ represent the same semantic information is ensured by Lemmma 3 below formally grasping the observation that the semantics of an open formula only depends on the free variables (compare Lemma 9.7 in [4] for example).

To prepare the proof of Lemma 3, we fix a crucial observation (that, unfortunately, vanishes in the many-sorted case). According to (A25), we do have for any carrier U that the projection map $(in;\_):U^G\to U^K$ for any inclusion map $in:=i{n}_{K,G}:K\to G$ between contexts is surjective thus we also have by (A24)

$${\downarrow }_{in}^U;{\forall }_{in}^U{=\downarrow }_{in}^U;{\exists }_{\psi }^U=i{d}_{\wp \left(U^K\right)}.$$

(29)

for the operators ${\downarrow }_{in}^U={(in;\_)}^{-1}:\wp \left(U^K\right)\to \wp \left(U^G\right)$, ${\exists }_{in}^U=(in;\_):\wp \left(U^G\right)\to \wp \left(U^K\right)$ and ${\forall }_{in}^U:\wp \left(U^G\right)\to \wp \left(U^K\right)$ defined and elaborated in Appendix A.2.

Lemma 3

(Inclusion of Contexts). For any Σ-expression B and any sets Z, X of variables with $fv\left(B\right)\subseteq Z\subset X$ and $Z▹B$, $X▹B$ the following two equivalent conditions for the inclusion map $in:=i{n}_{Z,X}:Z\to X$ hold for all Σ-structures $\mathcal{U}$

$${\left[\left[B\right]\right]}_X^{\mathcal{U}}{=\downarrow }_{in}^U{\left[\left[B\right]\right]}_Z^{\mathcal{U}}and{\forall }_{in}^U{\left[\left[B\right]\right]}_X^{\mathcal{U}}={\exists }_{in}^U{\left[\left[B\right]\right]}_X^{\mathcal{U}}={\left[\left[B\right]\right]}_Z^{\mathcal{U}}.$$

Proof. 

According to Corollary A3, the right condition always implies the left condition while the implication in the other direction is ensured by (29).

We prove the claim by induction on expressions according to the inductive definition of syntax of expressions in Definition 5 and of semantics of expressions in Definition 7.

For the case Relational Atom, we consider the substitution $in;{\eta }_X:Z\to T_{\Sigma }\left(X\right)$. The map ${(in;{\eta }_X)}^{*}:T_{\Sigma }\left(Z\right)\to T_{\Sigma }\left(X\right)$, due to (15), is nothing but the inclusion $T_{\Sigma }\left(Z\right)\subseteq T_{\Sigma }\left(X\right)$ claimed in Remark 6. The definition of derived operations by (3) and (4) entails ${(in;{\eta }_X)}^{\mathcal{U}}\left(\mathbf{a}\right)=in;\mathbf{a}$ for all $\mathbf{a}\in U^X$ and thus ${(in;{\eta }_X)}^{\mathcal{U}}=(in;\_):U^X\to U^Z$.

$Z▹\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)$ for $\mathbf{t}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(X\right)$ means that there is a unique $\mathbf{r}:ar\left(\mathtt{p}\right)\to T_{\Sigma }\left(Z\right)$ with $\mathbf{r};{(in;{\eta }_X)}^{*}=\mathbf{t}$ and thus $\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)=\mathtt{p}\mathbf{syn}\left(\mathbf{r}\right)$. Due to (17) and (18), this ensures that ${\mathbf{t}}^{\mathcal{U}}={(in;{\eta }_X)}^{\mathcal{U}};{\mathbf{r}}^{\mathcal{U}}=(in;\_);{\mathbf{r}}^{\mathcal{U}}$. In such a way, we get by Definition 7 and (A36). ${\left[\left[\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)\right]\right]}_Z^{\mathcal{U}}={\left[\left[\mathtt{p}\mathbf{syn}\left(\mathbf{r}\right)\right]\right]}_Z^{\mathcal{U}}{=\downarrow }_{\mathbf{r}}^U\left({\mathtt{p}}^{\mathcal{U}}\right)$ and, therefore, as required ${\left[\left[\mathtt{p}\mathbf{syn}\left(\mathbf{t}\right)\right]\right]}_X^{\mathcal{U}}={{\downarrow }_{\mathbf{t}}^U\left({\mathtt{p}}^{\mathcal{U}}\right)=\downarrow }_{\mathbf{r};{(in;{\eta }_X)}^{*}}^U\left({\mathtt{p}}^{\mathcal{U}}\right)=$${\downarrow }_{\mathbf{r}}^U{;\downarrow }_{in;{\eta }_X}^U\left({\mathtt{p}}^{\mathcal{U}}\right)={{\downarrow }_{in;{\eta }_X}^U\left({\downarrow }_{\mathbf{r}}^U\left({\mathtt{p}}^{\mathcal{U}}\right)\right)=\downarrow }_{in}^U{\left[\left[\mathtt{p}\mathbf{syn}\left(\mathbf{r}\right)\right]\right]}_Z^{\mathcal{U}}$.

The case Equation can be shown in the same way since the equation symbol "=" can be seen to represent a built-in predicate with arity $ar(=)=\{1,2\}$, as discussed in Remark 11, and a fixed semantics ${=}^{\mathcal{U}}=\{\mathbf{a}\in U^{\{1,2\}}\mid \mathbf{a}\left(1\right)=\mathbf{a}\left(2\right)\}$ in any $\Sigma$-structure $\mathcal{U}$.

Case Void is trivial since the preimage of the empty set is always the empty set and case Everything simply reflects that $(in;\_):U^X\to U^Z$ is a total (!) map.

The cases Conjunction, Disjunction, Implication and Negation are ensured since shift operators distribute over intersections as well as over unions and since they also are compatible with complements (see Appendix A.1 and (A13)).

We show case Quantification by means of the right condition in the lemma. For $X▹Q\left(\mathbf{syn}\right(Y\setminus X):Ex)$ with $Q\in \{\exists ,\forall \}$ let Z be arbitrary with $fv\left(Q\right(\mathbf{syn}(Y\setminus X):Ex\left)\right)\subseteq Z\subset X$. In such a way, we do have two chains of inclusions $Z\subset X\subset Y$ and $Z\subset W\subset Y$ with $W=Z\cup (Y\setminus X)$ and thus $Q_{i{n}_{Z,Y}}^U=Q_{i{n}_{X,Y}}^U;Q_{i{n}_{Z,X}}^U=Q_{i{n}_{W,Y}}^U;Q_{i{n}_{Z,W}}^U$ due to (A23).

By Induction Hypothesis, $Q_{i{n}_{W,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}={\left[\left[Ex\right]\right]}_W^{\mathcal{U}}$, thus we get, due to Definition 7, for the inclusion chains $Q_{i{n}_{Z,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}=Q_{i{n}_{Z,X}}^U\left(Q_{i{n}_{X,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}\right)=Q_{i{n}_{Z,X}}^U{\left[\left[Q(\mathbf{syn}(Y\setminus X):Ex)\right]\right]}_X^{\mathcal{U}}$ and $Q_{i{n}_{Z,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}=Q_{i{n}_{Z,W}}^U\left(Q_{i{n}_{W,Y}}^U{\left[\left[Ex\right]\right]}_Y^{\mathcal{U}}\right)=Q_{i{n}_{Z,W}}^U{\left[\left[Ex\right]\right]}_W^{\mathcal{U}}={\left[\left[Q(\mathbf{syn}(Y\setminus X):Ex)\right]\right]}_Z^{\mathcal{U}}$ and therefore $Q_{i{n}_{Z,X}}^U{\left[\left[Q(\mathbf{syn}(Y\setminus X):Ex)\right]\right]}_X^{\mathcal{U}}={\left[\left[Q(\mathbf{syn}(Y\setminus X):Ex)\right]\right]}_Z^{\mathcal{U}}$ as required. □

We can straightforwardly generalize Lemma 3 to arbitrary $\Sigma$-specifications.

Corollary 6

(Inclusion of Contexts). For any set Γ of Σ-expressions and any sets Z, X of variables with $fv\left(\Gamma \right)\subseteq Z\subset X$ such that $(Z,\Gamma )$ and $(X,\Gamma )$ are Σ-specifications the following two equivalent conditions for the inclusion map $in:=i{n}_{Z,X}:Z\to X$ hold for all Σ-structures $\mathcal{U}$

$${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}{=\downarrow }_{in}^U{\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}and{\forall }_{in}^U{\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}={\exists }_{in}^U{\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}.$$

Proof. 

Since ${\downarrow }_{in}^U$ distributes over arbitrary intersections, we get by (28) and Lemma 3 ${\downarrow }_{in}^U{\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}{=\downarrow }_{in}^U(\bigcap \{{\left[\left[B\right]\right]}_Z^{\mathcal{U}}\mid B\in \Gamma \})=\bigcap \{{\downarrow }_{in}^U{\left[\left[B\right]\right]}_Z^{\mathcal{U}}\mid B\in \Gamma \}=\bigcap \{{\left[\left[B\right]\right]}_X^{\mathcal{U}}\mid B\in \Gamma \}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}$. From this equality we can deduce the two equalities on the right by means of (29). □

6.2. Extension of Contexts

A second simple construction are context extensions. Let $(X,\Gamma )$ be a $\Sigma$-specification and $(X,S{t}_{\Gamma }^X)$ be the corresponding simple $\Sigma$-sketch with $S{t}_{\Gamma }^X:=\{(X▹B,i{d}_X)\mid B\in \Gamma \}$. We consider a set Y of variables with $X\subset Y$.

The simplest idea is, obviously, to consider $(Y,\Gamma )$ as the results of transforming $(X,\Gamma )$ along the inclusion map $i{n}_{X,Y}:X\to Y$. However, to make this idea work we must, at least, require that $(Y,\Gamma )$ is LoSiC-compatible. By assumption, we have $fv\left(\Gamma \right)\subseteq X\subset Y$ thus the first condition of LoSiC-compatibility is trivially satisfied.

$fv\left(\Gamma \right)\subseteq X\subset Y$ trivially implies $(Y\setminus X)\cap fv\left(\Gamma \right)=\varnothing$. By assumption we also do have $X\cap bv\left(\Gamma \right)=\varnothing$, thus the third condition $Y\cap bv\left(\Gamma \right)=\varnothing$ of LoSiC-compatibility will be satisfied if, and only if, in addition to $(Y\setminus X)\cap fv\left(\Gamma \right)=\varnothing$ also $(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$ holds. To say it with other words: Each variable in $Y\setminus X$ is required to be a notorious fresh variable in the traditional sense! That the "strong fresh variable condition" $(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$ even entails the crucial LoSiC-compatibility condition $\Gamma \subseteq F{E}_{\Sigma }\left(Y\right)$, is ensured by the next lemma that is a complement to Lemma 1.

Lemma 4

(Extending Variable Declarations). For any set X of variables and any Σ- expression $Ex\in F{E}_{\Sigma }\left(X\right)$, i.e., $X▹Ex$, we also have $Ex\in F{E}_{\Sigma }\left(V\right)$, i.e., $V▹Ex$, for all sets V of variables with $X\subset V$ and $(V\setminus X)\cap bv\left(Ex\right)=\varnothing$.

Proof. 

By induction on expressions. For atomic expressions the claim is obvious since $T_{\Sigma }\left(X\right)\subset T_{\Sigma }\left(V\right)$ if $X\subset V$. Induction passes trivially through the connectives.

In case of Quantification $X▹Q\left(\mathbf{syn}\right(Y\setminus X):Ex)$ with $Q\in \{\exists ,\forall \}$ let V be arbitrary with $X\subset V$ and $(V\setminus X)\cap bv\left(Q\right(\mathbf{syn}(Y\setminus X):Ex\left)\right)=\varnothing$. By Definition 5, we have $Y▹Ex$ and thus $Y\cap bv\left(Ex\right)=\varnothing$ due to Remark 10. By the second assumption and (2), we get $(V\setminus X)\cap \left(bv\right(Ex)\cup (Y\setminus X\left)\right)=(V\setminus X)\cap bv\left(Ex\right)\cup (V\setminus X)\cap (Y\setminus X)=\varnothing$ and thus $(V\setminus X)\cap bv\left(Ex\right)=\varnothing$ and $(V\setminus X)\cap (Y\setminus X)=\varnothing$. $(V\setminus X)\cap (Y\setminus X)=\varnothing$ and $X\subset V$ imply $V\cap (Y\setminus X)=\varnothing$. Therefore, we obtain $W\setminus (Y\setminus X)=V$ for $W:=V\cup (Y\setminus X)$ besides $Y\subset W$. This ensures $W\setminus Y=W\setminus \left(\right(Y\setminus X)\cup X)=(W\setminus (Y\setminus X\left)\right)\setminus X=V\setminus X$. This gives us $(W\setminus Y)\cap bv\left(Ex\right)=\varnothing$ since $(V\setminus X)\cap bv\left(Ex\right)=\varnothing$ as shown before.

Induction Hypothesis for $Y\subset W$ and $Y▹Ex$ provides $W▹Ex$. Due to $W\setminus (Y\setminus X)=V$, this means, according to Definition 5, that $V▹Q\left(\mathbf{syn}\right(Y\setminus X):Ex)$ as required. □

This was the syntactic side of the story. Let us now discuss the semantic side. The translation of statements along context morphisms in (7) has been chosen to guaranty the satisfaction condition which is the essential prerequisite to achieve a smooth and correct interplay of syntactic manipulations and semantic constructions within a formalism.

The binding morphism in a statement in context declares for what part of the context we impose restrictions concerning the interpretations of the whole context in structures. The translation of statements, defined in (7), preserves this information. In the lazy approach we must identify conditions ensuring such a preservation of "local restrictions" too.

Translating the $\Sigma$-sketch $(X,S{t}_{\Gamma }^X)$ along the inclusion map $in=i{n}_{X,Y}:X\to Y$ results in the $\Sigma$-sketch $(Y,{\mathbf{Stm}}_{\Sigma }\left(in\right)\left(S{t}_{\Gamma }^X\right))$ with ${\mathbf{Stm}}_{\Sigma }\left(in\right)\left(S{t}_{\Gamma }^X\right)=\{(X▹B,in)\mid B\in \Gamma \}$ which only imposes restrictions on the part X of Y. Above we learned that the strong fresh variable condition $(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$ entails the LoSiC-compatibility condition $\Gamma \subseteq F{E}_{\Sigma }\left(Y\right)$ ensuring, in such a way, that $(Y,\Gamma )$ is a $\Sigma$-specification representing the simple $\Sigma$-sketch $(Y,S{t}_{\Gamma }^Y)$ with $S{t}_{\Gamma }^Y:=\{(Y▹B,i{d}_Y)\mid B\in \Gamma \}$.

Fortunately, $\Gamma \subseteq F{E}_{\Sigma }\left(Y\right)$ implies, in parallel, that the interpretation of $Y\setminus X$ in a structure is independent of the interpretation of X, i.e., the semantic equivalence of the statements $(X▹B,in)$ and $(Y▹B,i{d}_Y)$. Due to the semantics of statements in (12) and Lemma 3 we do have for any $B\in \Gamma \subseteq F{E}_{\Sigma }\left(Y\right)$ and any $\Sigma$-structure $\mathcal{U}$

$${\left[\left[(X▹B,in)\right]\right]}^{\mathcal{U}}={{\downarrow }_{in}^U{\left[\left[B\right]\right]}_X^{\mathcal{U}}={\left[\left[B\right]\right]}_Y^{\mathcal{U}}=\downarrow }_{i{d}_Y}^U{\left[\left[B\right]\right]}_Y^{\mathcal{U}}={\left[\left[(Y▹B,i{d}_Y)\right]\right]}^{\mathcal{U}}.$$

and, in such a way, the following semantic equivalences of $\Sigma$-sketches and $\Sigma$-specifications by means of (22), (23) and (28)

$${\left[\left[(Y,\Gamma )\right]\right]}^{\mathcal{U}}={\left[\left[(Y,S{t}_{\Gamma }^Y)\right]\right]}^{\mathcal{U}}={\left[\left[(Y,{\mathbf{Stm}}_{\Sigma }\left(in\right)\left(S{t}_{\Gamma }^X\right))\right]\right]}^{\mathcal{U}}={{\downarrow }_{in}^U{\left[\left[(X,S{t}_{\Gamma }^X)\right]\right]}^{\mathcal{U}}=\downarrow }_{in}^U{\left[\left[(X,\Gamma )\right]\right]}^{\mathcal{U}}$$

(30)

So, the idea to consider the $\Sigma$-specification $(Y,\Gamma )$ as the translation of the $\Sigma$-specification $(X,\Gamma )$ along the inclusion map $in=i{n}_{X,Y}:X\to Y$ can be licensed if all variables in $Y\setminus X$ are fresh variables in the strong sense, i.e., $(Y\setminus X)\cap fv\left(\Gamma \right)=(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$.

In summary: If the pair $(X,\Gamma )$ of a set X of variables and a finite set $\Gamma$ of $\Sigma$-expressions is LoSiC-compatible (see page Section 6), $(X,\Gamma )$ can unambiguously utelized as a shorthand representation of the $\Sigma$-sketch $(X,S{t}_{\Gamma }^X)$ with $S{t}_{\Gamma }^X:=\{(X▹B,i{d}_X)\mid B\in \Gamma \}$. We coined the concept Σ-specification to denote LoSiC-compatible pairs $(X,\Gamma )$ while the semantics of a $\Sigma$-specification $(X,\Gamma )$ is given by (28).

For any proper extension $X\subset Y$ the strong fresh variable convention, to also require $(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$, in addition to $(Y\setminus X)\cap fv\left(\Gamma \right)=\varnothing$, ensures that $(Y,\Gamma )$ becomes a $\Sigma$-specification as well and that the corresponding simple $\Sigma$-sketch $(Y,S{t}_{\Gamma }^Y)$ is semantically equivalent to the non-simple $\Sigma$-sketch $(Y,{\mathbf{Stm}}_{\Sigma }\left(i{n}_{X,Y}\right)\left(S{t}_{\Gamma }^X\right))$ which we obtain, according to the LoSiC-methodology, by translating the statements in $(X,S{t}_{\Gamma }^X)$ along the inclusion map $i{n}_{X,Y}$, i.e., ${\mathbf{Stm}}_{\Sigma }\left(i{n}_{X,Y}\right)\left(S{t}_{\Gamma }^X\right)=\{(X▹B,i{n}_{X,Y})\mid B\in \Gamma \}$.

This suggests to introduce the following concept of Σ-specification implication as an adequate tool to describe and analyze traditional first-order deduction calculi.

Definition 18

(Specification Implication). A$\Sigma$-specification implication$(X,\Gamma )⟹(Y,\Delta )$ is given by two Σ-specifications $(X,\Gamma )$ and $(Y,\Delta )$ such that $X\subseteq Y$ and $(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$. $(X,\Gamma )⟹(Y,\Delta )$ is calledstrict, $(X,\Gamma )@3->{}_{\left[r\right]}(Y,\Delta )$ in symbols, if, and only if, $\Gamma \subseteq \Delta$.

$(X,\Gamma )⟹(Y,\Delta )$ issatisfiedin a Σ-structure $\mathcal{U}$, $\mathcal{U}{\vDash }^{\Sigma }(X,\Gamma )⟹(Y,\Delta )$ in symbols, if ${\exists }_{i{n}_{X,Y}}^U{\left[\left[(Y,\Delta )\right]\right]}^{\mathcal{U}}=(i{n}_{X,Y};\_){\left[\left[(Y,\Delta )\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[(X,\Gamma )\right]\right]}^{\mathcal{U}}$ for the inclusion map $i{n}_{X,Y}:X\to Y$.

Note, that the condition $(Y\setminus X)\cap bv\left(\Gamma \right)=\varnothing$ ensures that we can simply transform a $\Sigma$-specification implication $(X,\Gamma )⟹(Y,\Delta )$ into a corresponding strict $\Sigma$-specification implication $(X,\Gamma )@3->{}_{\left[r\right]}(Y,\Delta \cup \Gamma )$! On the other side, the condition entails for any strict $\Sigma$-specification implication $(X,\Gamma )@3->{}_{\left[r\right]}(Y,\Delta )$ the semantic equivalence of the $\Sigma$-sketches $(Y,S{t}_{\Delta }^Y)$ and $(Y,{\mathbf{Stm}}_{\Sigma }\left(i{n}_{X,Y}\right)\left(S{t}_{\Gamma }^X\right)\cup S{t}_{\Delta \setminus \Gamma }^Y)$ thus the concept "strict $\Sigma$-specification implication" represents a special case of the concept "strict $\Sigma$-sketch implication"! In such a way, we get by Corollary 2

$$\mathcal{U}{\vDash }^{\Sigma }(X,\Gamma )@3->{}_{\left[r\right]}(Y,\Delta )implies{\exists }_{i{n}_{X,Y}}^U{\left[\left[(Y,\Delta )\right]\right]}^{\mathcal{U}}=(i{n}_{X,Y};\_){\left[\left[(Y,\Delta )\right]\right]}^{\mathcal{U}}={\left[\left[(X,\Gamma )\right]\right]}^{\mathcal{U}}$$

(31)

As in the case of $\Sigma$-sketches, (A23) ensures that the composition $(X,\Gamma )⟹(Z,\Omega )$ of two $\Sigma$-specification implications $(X,\Gamma )⟹(Y,\Delta )$ and $(Y,\Delta )⟹(Z,\Omega )$ is also satisfied in a $\Sigma$-structure $\mathcal{U}$ if both $(X,\Gamma )⟹(Y,\Delta )$ and $(Y,\Delta )⟹(Z,\Omega )$ are satiesfied in $\mathcal{U}$. Obviously, composition also preserves strictness!

6.3. $\alpha$-Conversion

To come up, in analogy to Section 5.3, with a reasonable concept of "strict match" for the application of $\Sigma$-specification implications as deduction rules we must explicitly deal with $\alpha$-conversion, i.e., semantic preserving renaming of bound variables. We borrow the term α-equal from [32] to state that two well-formed formulas or $\Sigma$-expressions, respectively, are equal up to semantic preserving renaming of bound variables.

In the literature we found two approaches to achieve capture-avoiding substitution application. The "legal substitution approach" formally characterizes capture-avoiding substitutions and allows only the application of those legal substitutions [4]. The "concurrence class approach" doesn’t work with single formulas but classes of $\alpha$-equal formulas. The fact that for a given substitution there is in any equivalence class a formula for which the application of the given substitution is capture avoiding, allows us to define the application of any substitution to any equivalence class by choosing the right representative (see [32]).

The "methodological accident" that the LoSiC-approach inforces compliance with Barendregt’s variable convention as well as a strict separation of free and bound variables leads us, however, to a third possibility to achieve capture-avoiding substitution application. We elaborate this third possibility in this subsection.

In Definition 11 we assumed an arbitrary, but fixed, choice of non-symmetric sums of sets of variables to be able to define substitution application for all (!) $\Sigma$-expressions in a uniform way. To preserve compliance with Barendregt’s variable convention we have been enforced, therefore, to inductively define, in parallel with substitution application, a renaming of bound variables implicitly determined by the chosen overall non-symmetric sums. In the realm of deduction it is, however, more advantageous to individually adjust renaming of bound variables to single expressions. Fortunately, compliance with Barendregt’s variable convention and the strict separation of free and bound variables exactly enables such an individual adjustment of renaming. In addition to the declaration of a substitution for the free variables in an expression, we can also declare a renaming for the bound variables, and then recursively apply substitutions and the renamings in parallel.

Due to the induction scheme in Definition 5, the syntatctic structure of a $\Sigma$-expressions can be described by a binary tree where the proper nodes are either "Boolean nodes" or "quantifier nodes" while the leaves are either "everything/void nodes" or "atomic nodes". Each quantifier node is labeled with the set of variables that have been bound by the corresponding quantification. Two $\Sigma$-expressions can only be $\alpha$-equal if both trees do have the same structure and if for any pair of corresponding quantifier nodes the respective two sets of bound variables do have the same cardinality.

For each path in the tree from the root to a leaf there is a sequence of quantifier nodes where the corresponding sets of bound variables are all disjoint, i.e., no variable is quantified twice in the sequence. To ensure that renaming preserves compliance with Barendregt’s variable convention, a renaming should not identify any variables in the same sequences. Since quantifications in disjoint subtrees are semantically independent, two variables appearing in two of those independent quantifications can be, however, identified by a renaming! The discussion is summarized by the following definition.

Definition 19

(Renaming Substitution). Arenaming substitution$(\mathbf{r},\alpha )$ for a Σ-expression $X▹Ex$ is given by a substitution $\mathbf{r}:X\to T_{\Sigma }\left(Z\right)$ and a map $\alpha :bv\left(Ex\right)\to V$ satisfying the following requirements: $Z\cap V=\varnothing$ and for any path in the structure tree of $Ex$ from the root to a leaf the restriction of $\alpha :bv\left(Ex\right)\to V$ to the set of all variables, bound in the quantifier nodes along the path, is injective. If $X=Z$ and $({\eta }_X,\alpha )$ is a renaming substitution for $X▹Ex$, we call $\alpha :bv\left(Ex\right)\to V$ alegal renaming (of bound variables)for $X▹Ex$.

If $V=bv\left(Ex\right)$ and $\alpha =i{d}_{bv\left(Ex\right)}$ the renaming substitution condition reduces to $Z\cap bv\left(Ex\right)=\varnothing$, and we may call $\mathbf{r}:X\to T_{\Sigma }\left(Z\right)$ a legal substitution for $X▹Ex$.

The result $Z▹\overline{(\mathbf{r},\alpha )}\left(Ex\right)$ of applying a renaming substitution $(\mathbf{r},\alpha )$ to $X▹Ex$ is defined by adapting Definition 11 and reusing it as a recursion scheme. The basic cases 1-3 can be reused as they are. In the Boolean cases 4-6, we replace the two "recursion calls" $\overline{\mathbf{r}}$ on the right hand sides by $\overline{(\mathbf{r},{\alpha }_i)}$, $i\in \{1,2\}$ with the restrictions ${\alpha }_i:bv\left(E{x}_i\right)\to V$ of $\alpha$. In case 7, we simply replace the "recursion call" $\overline{\mathbf{r}}$ by $\overline{(\mathbf{r},\alpha )}$.

In case 8, i.e., $Ex=Q(\mathbf{syn}(Y\setminus X):E{x}^{\prime })$ with $Q\in \{\exists ,\forall \}$ and $Y▹E{x}^{\prime }$, we define

$$Z▹\overline{(\mathbf{r},\alpha )}\left(Q(\mathbf{syn}(Y\setminus X):E{x}^{\prime })\right):=Z▹Q(\mathbf{syn}(W\setminus Z):\overline{({\mathbf{r}}^{\prime },{\alpha }^{\prime })}\left(E{x}^{\prime }\right))$$

(32)

with $W▹\overline{({\mathbf{r}}^{\prime },{\alpha }^{\prime })}\left(E{x}^{\prime }\right)$ and $W:=Z\cup \alpha (Y\setminus X)$. We construct the recursion call $({\mathbf{r}}^{\prime },{\alpha }^{\prime })$ as follows: The simple idea is to move the assignments $(y\mapsto \alpha (y\left)\right)$ with $y\in Y\setminus X\subseteq bv\left(Ex\right)$ from the "renaming side" to the "substitution side". Fortunately, we can reconstruct from the given information the original set Y as the union $Y=X\cup (Y\setminus X)$. The injectivity requirement in Definition 19 for $\alpha$ ensures that the restriction ${\alpha }_Q:Y\setminus X\to \alpha (Y\setminus X)$ of $\alpha$ becomes bijective. The condition $Z\cap V=\varnothing$ entails $Z\cap \alpha (Y\setminus X)=\varnothing$ thus we obtain $W\setminus Z=\alpha (Y\setminus X)$ besides $T_{\Sigma }\left(Z\right)\subset T_{\Sigma }\left(W\right)$ and $\alpha (Y\setminus X)\subset T_{\Sigma }\left(W\right)$. This allows us to define the extended substitution ${\mathbf{r}}^{\prime }:Y\to T_{\Sigma }\left(W\right)$ by case distinction

$${\mathbf{r}}^{\prime }\left(y\right):=\left\{\begin{array}{cc}\hfill \mathbf{r}\left(y\right)& \hfill ify\in X\\ \hfill \alpha \left(y\right)& \hfill ify\in Y\setminus X.\end{array}\right.$$

Since $bv\left(Ex\right)=(Y\setminus X)\cup bv\left(E{x}^{\prime }\right)$ and $Y\cap bv\left(E{x}^{\prime }\right)=\varnothing$, we have $bv\left(E{x}^{\prime }\right)=bv\left(Ex\right)\setminus (Y\setminus X)$ thus the restriction ${\alpha }^{\prime }:bv\left(E{x}^{\prime }\right)\to V^{\prime }$ of $\alpha$ with $V^{\prime }:=V\setminus \alpha (Y\setminus X)$ is well-defined since $\alpha$ injective ensures ${\alpha }^{-1}\left(\alpha (Y\setminus X)\right)=Y\setminus X$. $({\mathbf{r}}^{\prime },{\alpha }^{\prime })$ is indeed a renaming substitution in the sense of Definition 19 since $Z\cap V=\varnothing$ entails $W\cap V^{\prime }=\varnothing$ by definition of W and $V^{\prime }$ while ${\alpha }^{\prime }$ inherits the satisfaction of the injectivity condition from $\alpha$.

Renaming of bound variables doesn’t change the semantics of $\Sigma$-expressions! For any isomorphism $i:X\to Y$ between sets of variables and any carrier U the pre-composition map $(i;\_):U^Y\to U^X$ is also an isomorphism with the inverse $(i^{-1};\_)$. This ensures for any renaming substitution $(\mathbf{r},\alpha )$, $\mathbf{r}:X\to T_{\Sigma }\left(Z\right)$, $\alpha :bv\left(Ex\right)\to V$ for a $\Sigma$-expression $X▹Ex$ and any legal renaming $\alpha :bv\left(Ex\right)\to V$ for $X▹Ex$ that

$${\left[\left[\overline{(\mathbf{r},\alpha )}\left(Ex\right)\right]\right]}_Z^{\mathcal{A}}={\left[\left[\overline{\mathbf{r}}\left(Ex\right)\right]\right]}_Z^{\mathcal{A}}and{\left[\left[Ex\right]\right]}_X^{\mathcal{A}}={\left[\left[\overline{\alpha }\left(Ex\right)\right]\right]}_X^{\mathcal{A}}$$

(33)

with $\overline{\alpha }\left(Ex\right):=\overline{({\eta }_X,\alpha )}\left(Ex\right)$ and $Z▹\overline{\mathbf{r}}\left(Ex\right)$ obtained according to Definition 11 for a certain fixed choice of non-symmetric sums for sets of variables..

As already discussed, the same variable can appear in different quantifications located in disjoint subtrees of the structure tree of a $\Sigma$-expression. The pattern of this "horizontal reuse" of bound variables may not match the corresponding pattern in another $\alpha$-equal expressions. Therefore, we must use spans of legal renamings to fully describe $\alpha$-equality for all $\Sigma$-expressions, i.e., for all well-formed formulas, in the traditional sense, complying with Barendregt’s variable convention.

In terms of the concurrence class approach to capture-avoiding substitutions [32], we can fortunately represent classes of $\alpha$-equal $\Sigma$-expressions by $\Sigma$-expressions in a certain normal form: We call a $\Sigma$-expression $X▹Ex$quantification unique if, and only if, each variable in $bv\left(Ex\right)$ appears in exactly one quantification in $Ex$. In other words: The quantification unique $\Sigma$-expressions are exactly those $\Sigma$-expressions, we can build by means of Definition 5 with the additional requirement $bv\left(E{x}_1\right)\cap bv\left(E{x}_2\right)=\varnothing$ in the cases Conjunction, Disjunction and Implication. Utilizing quantification unique $\Sigma$-expressions, we can give the following complete characterization of $\alpha$-equality for $\Sigma$-expressions: Two $\Sigma$-expressions $X▹E{x}_1$ and $X▹E{x}_2$ are $\mathbf{\alpha }$-equal if, and only if, there exist a quantification unique $\Sigma$-expresssion $X▹Ex$ and legal renamings ${\alpha }_1:bv\left(Ex\right)\to V_1$, ${\alpha }_2:bv\left(Ex\right)\to V_2$ for $X▹Ex$ such that $\overline{{\alpha }_1}\left(Ex\right)=E{x}_1$ and $\overline{{\alpha }_2}\left(Ex\right)=E{x}_2$.

6.4. Deducing Specification Implications

In Section 5.3 we investigated the deduction of new sketch implications by applying given sketch implications as deduction rules via sketch morphisms. We unexpectedly found out that rule application via substitutions can not be defined for arbitrary $\Sigma$-sketches but only for simple $\Sigma$-sketches, since we utilize only simple maps and not arbitrary substitutions to bind expressions to contexts. We introduced $\Sigma$-specifications $(X,\Gamma )$ as a representation of simple $\Sigma$-sketches $(X,S{t}_{\Gamma }^X)$, $S{t}_{\Gamma }^X:=\{(X▹B,i{d}_X)\mid B\in \Gamma \}$ to become better aligned with the traditional notations in first-order logic. Maps between sets of variables can be encoded by substitutions thus it suffices to consider rule application via substitutions!

We will investigate how to apply a $\Sigma$-specification implication $(P,\Pi )⟹(C,\Gamma )$, as a rule, to deduce for a given $\Sigma$-specification $(G,\mathrm{\Phi })$ a strict $\Sigma$-specification implication $(G,\mathrm{\Phi })@3->{}_{\left[r\right]}(R,\mathrm{\Psi })$ thereby ensuring soundness analogously to Corollary 5. We use the constructions and results in Section 5.3 as a blueprint for this subsection. Renaming substitutions, defined in Section 6.3, are the appropriate tool to define matches for rule application in analogy to the strict $\Sigma$-sketch arrow in Corollary 5.

Definition 20

(Match). Amatch$(\mathbf{t},\mathbf{\alpha })$ of a Σ-specification $(P,\Pi )$ in a Σ-specification $(G,\mathrm{\Phi })$, $(\mathbf{t},\mathbf{\alpha }):(P,\Pi )⤏(G,\mathrm{\Phi })$ in symbols, is given by a substitution $\mathbf{t}:P\to T_{\Sigma }\left(G\right)$ and a Π-indexed family $\mathbf{\alpha }$ of renaming substitutions $(\mathbf{t},{\alpha }_B)$, ${\alpha }_B:bv\left(B\right)\to V_B$ with $V_B\subseteq bv\left(\mathrm{\Phi }\right)$ such that $\overline{(\mathbf{t},{\alpha }_B)}\left(B\right)\in \mathrm{\Phi }$ for all $B\in \Pi$.

Note, that the $\Sigma$-expressions in $\Pi$ should preferably be quantification unique to ensure that our concept of match completely comprises $\alpha$-equality!

Definition (28) and the equations (33), (20) ensure that one can prove, analogously to Corollary 2, the following proposition.

Proposition 3

(Semantics of Matches). For any match $(\mathbf{t},\mathbf{\alpha }):(P,\Pi )⤏(G,\mathrm{\Phi })$ and any Σ-structure $\mathcal{U}$ it holds that ${\exists }_{\mathbf{t}}^U{\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}={\mathbf{t}}^{\mathcal{U}}{\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\subseteq {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$ and, equivalently due to Corollary A6, ${\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}{\subseteq \downarrow }_{\mathbf{t}}^U{\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$.

Let be given a $\Sigma$-specification implication $(P,\Pi )⟹(C,\Gamma )$, a $\Sigma$-specification $(G,\mathrm{\Phi })$ and a match $(\mathbf{t},\mathbf{\alpha }):(P,\Pi )⤏(G,\mathrm{\Phi })$. We use the abbreviation $\iota :=i{n}_{P,C}:P\to C$ for the corresponding inclusion map. To ensure the strong fresh variable condition for the $\Sigma$-implication, we are going to construct, we choose for $C\setminus P$ an isomorphic set $CP$ of variables together with an isomorphism $is:C\setminus P\to CP$ such that $CP\cap G=CP\cap bv\left(\mathrm{\Phi }\right)=\varnothing$. We set $R:=G\cup CP$ and get an inclusion map $j:=i{n}_{G,R}:G\to R$. Since $C=P\cup (C\setminus P)$, this allows us to extend the substitution $\mathbf{t}:P\to T_{\Sigma }\left(G\right)$ to a substitution $\mathbf{r}:C\to T_{\Sigma }\left(R\right)$:

$$\mathbf{r}\left(x\right):=\left\{\begin{array}{cc}\hfill \mathbf{t}\left(x\right)& \hfill ifx\in P\\ \hfill is\left(x\right)& \hfill ifx\in C\setminus P\end{array}\right.$$

(34)

To characterize our construction by means of a commutative square, we can not rely on the composition of maps since substitutions are involved! We have to consider, instead, composition of substitutions, i.e., composition in the category ${\mathsf{Var}}_{\Sigma }$ as defined in Section 4.2.1. The inclusion map $\iota :P\to C$ gives rise to the substitution $\iota ;{\eta }_C:P\to T_{\Sigma }\left(C\right)$ while $j:G\to R$ gives us the substitution $j;{\eta }_R:G\to T_{\Sigma }\left(R\right)$. Our constructions provide, in such a way, the left commutative diagram below in the category ${\mathsf{Var}}_{\Sigma }$.

Any substitution represents a derived operation in a $\Sigma$-structure $\mathcal{U}$ as defined in (4). As discussed in Section 4.2.3, composition of substitutions in the category ${\mathsf{Var}}_{\Sigma }$ represents the composition of corresponding derived operations for any $\Sigma$-structure $\mathcal{U}$, thus the commutative square of substitutions on the left in (35) transforms for any $\mathcal{U}$ into the commutative square of derived operations in the middle of (35). Note, that the derived operations ${(\iota ;{\eta }_C)}^{\mathcal{U}}=(\iota ;\_)$ and ${(j;{\eta }_R)}^{\mathcal{U}}=(j;\_)$ are simple projections! We show that this commutative diagram of derived operations is, in analogy to (27), also a pullback in $\mathsf{Set}$.

We have to prove that there is for any pair $\iota :G\to U$, $\vartheta :C\to U$ of interpretations with ${\mathbf{t}}^{\mathcal{U}}\left(\iota \right)=\iota ;\vartheta$ a unique interpretation $\kappa :R\to U$ with $j;\kappa =\iota$ and ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)=\vartheta$. The two uniqueness conditions enforce the following definition of $\kappa :R\to U$

$$\kappa \left(x\right):=\left\{\begin{array}{cc}\hfill \iota \left(x\right)& \hfill ifx\in G\\ \hfill \vartheta \left(i{s}^{-1}\left(x\right)\right)& \hfill ifx\in CP\end{array}\right.$$

(36)

where the case $x\in CP$ is entailed by the definition of $\mathbf{r}:C\to T_{\Sigma }\left(R\right)$ in (34) and the definition of derived operations in (4): That is, we have ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)=\mathbf{r};{\kappa }^{\circ }$ where ${\kappa }^{\circ }:T_{\Sigma }\left(R\right)\to U$ is the unique extension of $\kappa$ with ${\eta }_R;{\kappa }^{\circ }=\kappa$ thus for all $y\in C\setminus P$ the condition ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)=\vartheta$ forces ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)\left(y\right)={\kappa }^{\circ }\left(\mathbf{r}\left(y\right)\right)={\kappa }^{\circ }\left(is\left(y\right)\right)=\kappa \left(is\left(y\right)\right)=\vartheta \left(y\right)$ and, thus equivalently, $\kappa \left(x\right)=\vartheta \left(i{s}^{-1}\left(x\right)\right)$ for all $x\in CP$.

It remains to show that ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)\left(y\right)=\vartheta \left(y\right)$ for all $y\in P\subseteq C$. Due to the definition of $\kappa$ in (36), we have $j;\kappa =\iota$ thus the diagram below is commutative with ${(j;{\eta }_R)}^{*}$ an inclusion map and ${(j;{\eta }_R)}^{*};{\kappa }^{\circ }={\iota }^{\circ }$ according to (17). In such a way, we get for any $y\in P$ due to the assumption ${\mathbf{t}}^{\mathcal{U}}\left(\iota \right)=\iota ;\vartheta$ that ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)\left(y\right)={\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)\left(\iota \left(y\right)\right)={\mathbf{t}}^{\mathcal{U}}\left(\iota \right)\left(y\right)=\vartheta \left(\iota \left(y\right)\right)=\vartheta \left(y\right)$.

Analogously to Section 5.3, we extend the context $R=G\cup CP$ to a $\Sigma$-specification $(R,\mathrm{\Psi })$ with $\mathrm{\Psi }$ constructed out of the $\Sigma$-expressions in $\mathrm{\Phi }$ and $\Gamma$. To achieve strictness of the resulting $\Sigma$-specification implication, we simply declare $\mathrm{\Phi }\subseteq \mathrm{\Psi }$. Keep in mind that the condition $CP\cap G=CP\cap bv\left(\mathrm{\Phi }\right)=\varnothing$ ensures $R▹B$ for all $B\in \mathrm{\Phi }$ according to Lemma 4.

To include the $\Sigma$-expressions from $(C,\Gamma )$, we have first to extend the substitution $\mathbf{r}:C\to T_{\Sigma }\left(R\right)$ to a $\Gamma$-indexed family $(\mathbf{r},{\mathbf{\alpha }}^{\prime })$ of renaming substitutions. If $B\in \Gamma \cap \Pi$, we can simply set ${\alpha }_B^{\prime }:={\alpha }_B:bv\left(B\right)\to V_B$ since $bv\left(\mathrm{\Phi }\right)\subseteq bv\left(\mathrm{\Psi }\right)$. Due to the definition of $\mathbf{r}$ in (34), we get $\overline{(\mathbf{r},{\alpha }_B^{\prime })}\left(B\right)=\overline{(\mathbf{t},{\alpha }_B)}\left(B\right)\in \mathrm{\Phi }\subseteq \mathrm{\Psi }$ in this case. If $B\in \Gamma \setminus \Pi$, we can freely choose a set $V_B$ of fresh bound variables, i.e., with $V_B\cap R=V_B\cap bv\left(\mathrm{\Phi }\right)=\varnothing$, thus there exists an isomorphism ${\alpha }_B^{\prime }:bv\left(B\right)\to V_B$. Using the shorthand $\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right):=\{\overline{(\mathbf{r},{\alpha }_B^{\prime })}\left(B\right)\mid B\in \Gamma \}$, we can summarize our definitions by $(R,\mathrm{\Psi }):=(R,\mathrm{\Phi }\cup \overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$.

Our definitions ensure, that the pair $(R,\mathrm{\Psi })$ complies with the condition of a strict separation of free and bound variables, i.e., $(R,\mathrm{\Psi })$ is indeed a $\Sigma$-specification. We also get a strict $\Sigma$-specification implication $(G,\mathrm{\Phi })@3->{}_{\left[r\right]}(R,\mathrm{\Psi })$ (see the right square in (35)). Moreover, the pair $(\mathbf{r},{\mathbf{\alpha }}^{\prime })$ defines, by construction, a match $(\mathbf{r},{\mathbf{\alpha }}^{\prime }):(C,\Gamma )⤏(R,\mathrm{\Psi })$ thus Proposition 3 entails

$${\exists }_{\mathbf{r}}^U{\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}={\mathbf{r}}^{\mathcal{U}}{\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}\subseteq {\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}and{\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}{\subseteq \downarrow }_{\mathbf{r}}^U{\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}.$$

(37)

In full analogy to Corollary 5, we can prove that the application of a $\Sigma$-specification implication as a deduction rule via substitutions is semantically sound.

Proposition 4

(Specification Implications as Rules: Soundness). For any Σ-structure $\mathcal{U}$ with $\mathcal{U}{\vDash }^{\Sigma }(P,\Pi )⟹(C,\Gamma )$, i.e., ${\exists }_i^U{\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}=(i;\_){\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$ for $i:=i{n}_{P,C}$, and any match $(\mathbf{t},\mathbf{\alpha }):(P,\Pi )⤏(G,\mathrm{\Phi })$ it holds that $\mathcal{U}{\vDash }^{\Sigma }(G,\mathrm{\Phi })@3->{}_{\left[r\right]}(R,\mathrm{\Psi })$, i.e., ${\exists }_j^U{\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}=(j;\_){\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}={\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$ for $j:=i{n}_{G,R}$.

Proof. 

The strict $\Sigma$-specification implication $(G,\mathrm{\Phi })@3->{}_{\left[r\right]}(R,\mathrm{\Psi })$ gives rise to the trivial renaming substitution $(i;{\eta }_R,\mathbf{id})$, $\mathbf{id}=\{i{d}_{bv\left(B\right)}\mid B\in \mathrm{\Phi }\}$ with $\overline{(i;{\eta }_R,\mathbf{id})}\left(B\right)=B$ for all $B\in \mathrm{\Phi }$ thus the inclusion "⊆" is ensured by Proposition 3. It remains to show "⊇".

If ${\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}=\varnothing$, the inclusion trivially holds. If ${\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\ne \varnothing$, we consider an arbitrary $\iota \in {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\subseteq U^G$. Since $(\mathbf{t},\mathbf{\alpha }):(P,\Pi )⤏(G,\mathrm{\Phi })$ is a match, we get ${\mathbf{t}}^{\mathcal{U}}\left(\iota \right)\in {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$, due to Proposition 3, thus there exists a $\vartheta \in {\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}\subseteq U^C$ with ${\mathbf{t}}^{\mathcal{U}}\left(\iota \right)=i;\vartheta$ due to assumption $\mathcal{U}{\vDash }^{\Sigma }(P,\Pi )⟹(C,\Gamma )$.

For the unique $\kappa :R\to U$ with $j;\kappa =\iota$ and ${\mathbf{r}}^{\mathcal{U}}\left(\kappa \right)=\vartheta$ the fact $\vartheta \in {\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}$, (20) and (33) ensure $\kappa \in {\left({\mathbf{r}}^{\mathcal{U}}\right)}^{-1}{\left[\left[\Gamma \right]\right]}_C^{\mathcal{U}}={\left[\left[\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right)\right]\right]}_R^{\mathcal{U}}$ while $j;\kappa =\iota$ and $\iota \in {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}={\left[\left[\mathrm{\Phi }\right]\right]}_G^U$ entail, due to Lemma (3), $\kappa \in {\left[\left[\mathrm{\Phi }\right]\right]}_R^{\mathcal{U}}{=\downarrow }_j^U{\left[\left[\mathrm{\Phi }\right]\right]}_G^{\mathcal{U}}={(j;\_)}^{-1}{\left[\left[\mathrm{\Phi }\right]\right]}_G^{\mathcal{U}}$. In such a way, we have $\kappa \in {\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}={\left[\left[\mathrm{\Phi }\right]\right]}_R^{\mathcal{U}}\cap {\left[\left[\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\right]\right]}_R^{\mathcal{U}}$ with $(j;\_)\left(\kappa \right)=j;\kappa =\iota$ and thus $\iota \in {\exists }_j^U{\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}=(j;\_){\left[\left[(R,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}$, as required. □

In analogy to Section 5.3, we can also define a non-strict variant of rule application by simply defining $(R,\mathrm{\Psi }):=(R,\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$ instead of $(R,\mathrm{\Psi }):=(R,\mathrm{\Phi }\cup \overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$. A corresponding slight variation of the proof of Proposition 4 provides the following Corollary.

Corollary 7

(Non-strict Variant of Rule Application: Soundness). For any Σ-structure $\mathcal{U}$ with $\mathcal{U}{\vDash }^{\Sigma }(P,\Pi )⟹(C,\Gamma )$, i.e., ${\exists }_i^U{\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}=(i;\_){\left[\left[(C,\Gamma )\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$ for $i:=i{n}_{P,C}$, and any match $(\mathbf{t},\mathbf{\alpha }):(P,\Pi )⤏(G,\mathrm{\Phi })$ it holds that $\mathcal{U}{\vDash }^{\Sigma }(G,\mathrm{\Phi })⟹(R,\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$, i.e., ${\exists }_j^U{\left[\left[(R,\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))\right]\right]}^{\mathcal{U}}=(j;\_){\left[\left[(R,\overline{(\mathbf{r},{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))\right]\right]}^{\mathcal{U}}\supseteq {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$ for $j:=i{n}_{G,R}$.

Remark 24

(Non-proper Substitutions). In case that $\mathbf{t}:P\to T_{\Sigma }\left(G\right)$ is anon-proper substitution, i.e., can be described as the composition $\mathbf{t}=\mu ;{\eta }_G$ of maps $\mu :P\to G$ and ${\eta }_G:G\to T_{\Sigma }\left(G\right)$, the definition in (34) of $\mathbf{r}:C\to T_{\Sigma }\left(R\right)$ with $R:=G\cup CP$ entails that $\mathbf{r}$ is also a non-proper substitution, i.e., can be factorized into $\mathbf{r}={\mu }^{*};{\eta }_R$ with maps ${\mu }^{*}=\mu +is:C\to R$ and ${\eta }_R:R\to T_{\Sigma }\left(R\right)$. So, in this case, our construction of $\mathbf{r}:C\to T_{\Sigma }\left(R\right)$ boils down to the construction of a pushout $G\stackrel{j}{\to }R\stackrel{{\mu }^{*}}{\leftarrow }C$ in $\mathsf{Set}$ of the span $G\stackrel{\mu }{\leftarrow }P\stackrel{i}{\to }C$ of maps representing the commutative diagram in the category ${\mathsf{Var}}_{\Sigma }$ depicted left in (35). In other words: There is no need to adapt Section 5.3 to Σ-specification implications since this adaptation simply is a special case of the constructions we presented in this subsection!

Remark 25

(Extension of Premises). What we will mainly need for our analysis of traditional deduction calculi is the extension of the premise of a given specification implication by new free variables and/or new expressions. To perform this simple task, we have to apply the given specification implication as a rule and, in some cases, it may be necessary to take additional measurements to enable rule application.

For a Σ-specification implication $(P,\Pi )⟹(C,\Gamma )$ let be given an inclusion map $in:P\to G$ and a Σ-specification $(G,\Delta )$. If the set $I:=bv\left(\Pi \right)\cap (G\setminus P)$ is empty, no renaming of bound variables is necessary! $(G,\Pi \cup \Delta )$ is a legal Σ-specification and $in:P\to G$ establishes a match $(P,\Pi )⤏(G,\Pi \cup \Delta )$ in the sense of Definition 20, according to Remark 24.

In case $I\ne \varnothing$, we chose a set V of fresh variables with $V\cap (C\cup G\cup bv(\Pi )\cup bv(\Gamma \left)\right)=\varnothing$ together with a bijective map $\iota :I\to V$, and define a Π-indexed family $\mathbf{\alpha }$ of bijective renaming substitutions $(in,{\alpha }_B)$, ${\alpha }_B:bv\left(B\right)\to V_B$ with $V_B=(bv\left(B\right)\setminus I)\cup \iota (bv\left(B\right)\cap I)$ for all $B\in \Pi$ and ${\alpha }_B\left(x\right)=x$ if $x\in bv\left(B\right)\setminus I$ and ${\alpha }_B\left(x\right)=\iota \left(x\right)$ if $x\in bv\left(B\right)\cap I$.

First, this ensures that $(in,\mathbf{\alpha }):(P,\Pi )⤏(G,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta )$ is a match in the sense of Definition 20. Second, we can apply the rule $(P,\Pi )⟹(C,\Gamma )$, in a strict and non-strict way, via the match $(in,\mathbf{\alpha })$ and it is ensured that we obtain legal Σ-specification implications $(G,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta )⟹(R,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta \cup \overline{(in+is,{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$ and $(G,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta )⟹(R,\overline{(in+is,{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$ with $R=G\cup CP$, $is:C\setminus P\to CP$ a bijection and $in+is:C\to R$.

Finally, $\mathcal{U}\vDash (P,\Pi )⟹(C,\Gamma )$ entails $\mathcal{U}\vDash (G,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta )⟹(R,\overline{(in+is,{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$ and $\mathcal{U}\vDash (G,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta )⟹(R,\overline{(in,\mathbf{\alpha })}\left(\Pi \right)\cup \Delta \cup \overline{(in+is,{\mathbf{\alpha }}^{\prime })}\left(\Gamma \right))$ for any Σ-structure $\mathcal{U}$ according to Corollary 7 and Proposition 4, respectively.

If $P=C$, it still may happen that $I=bv\left(\Pi \right)\cap (G\setminus P)\ne \varnothing$, thus the only simplification is that $R=G$ and that $(in+is,{\mathbf{\alpha }}^{\prime })=(in,\mathbf{\alpha })$.

In contrast to the extension of contexts, the reduction of contexts doesn’t require any additional efforts. Be aware that this only works in the unsorted variant of first-order logic!

Corollary 8

(Reduction of Contexts). For any Σ-specification implication $(G,\Pi )⟹(R,\Gamma )$ and any sets P, C of variables with $P\subseteq C$, $fv\left(\Pi \right)\subseteq P\subseteq G$, $fv\left(\Gamma \right)\subseteq C\subseteq R$ we get a legal Σ-specification implication $(P,\Pi )⟹(C,\Gamma )$ and for all Σ-structure $\mathcal{U}$

$$\mathcal{U}\vDash (G,\Pi )⟹(R,\Gamma )implies\mathcal{U}\vDash (P,\Pi )⟹(C,\Gamma ).$$

Proof. 

The assumption $\mathcal{U}\vDash (G,\Pi )⟹(R,\Gamma )$, i.e., ${\left[\left[\Pi \right]\right]}_G^{\mathcal{U}}\subseteq {\exists }_{i{n}_{G,R}}^U{\left[\left[\Gamma \right]\right]}_R^{\mathcal{U}}$, implies ${\exists }_{i{n}_{P,G}}^U{\left[\left[\Pi \right]\right]}_G^{\mathcal{U}}$ $\subseteq {\exists }_{i{n}_{P,G}}^U\left({\exists }_{i{n}_{G,R}}^U{\left[\left[\Gamma \right]\right]}_R^{\mathcal{U}}\right)$. First, we have ${\exists }_{i{n}_{P,G}}^U{\left[\left[\Pi \right]\right]}_G^{\mathcal{U}}={\left[\left[\Pi \right]\right]}_P^{\mathcal{U}}$ due to Corollary 6. Second, we obtain by (A23) and Corollary 6: ${\exists }_{i{n}_{P,G}}^U\left({\exists }_{i{n}_{G,R}}^U{\left[\left[\Gamma \right]\right]}_R^{\mathcal{U}}\right)={\exists }_{i{n}_{P,R}}^U{\left[\left[\Gamma \right]\right]}_R^{\mathcal{U}}={\exists }_{i{n}_{P,C}}^U\left({\exists }_{i{n}_{C,R}}^U{\left[\left[\Gamma \right]\right]}_R^{\mathcal{U}}\right)$ $={\exists }_{i{n}_{P,C}}^U{\left[\left[\Gamma \right]\right]}_C^{\mathcal{U}}$, and thus ${\left[\left[\Pi \right]\right]}_P^{\mathcal{U}}\subseteq {\exists }_{i{n}_{P,C}}^U{\left[\left[\Gamma \right]\right]}_C^{\mathcal{U}}$, i.e., $\mathcal{U}\vDash (P,\Pi )⟹(C,\Gamma )$. □

We hopefully provided enough evidence that we can transform, by renaming of bound variables, any set $\Gamma$ of traditional first-order formulas into an equivalent $\Sigma$-specification $(X,{\Gamma }^{\prime })$ given by a set X of variables with $fv\left(\Gamma \right)=fv\left({\Gamma }^{\prime }\right)\subseteq X$ and ${\Gamma }^{\prime }\subseteq F{E}_{\Sigma }\left(X\right)$ a set of $\Sigma$-expressions on X, i.e., of traditional well-formed first-order formulas complying with Barendregt’s variable convention. Moreover, we can transform any pair $(\Gamma ,\Delta )$ of sets of traditional first-order formulas into a $\Sigma$-specification implication $(X,{\Gamma }^{\prime })⟹(Y,{\Delta }^{\prime })$, in the sense of Definition 18, with $X\subseteq Y$, $fv\left({\Gamma }^{\prime }\right)=fv\left(\Gamma \right)\subseteq X$, $fv\left({\Delta }^{\prime }\right)=fv\left(\Delta \right)\subseteq Y$ complying with the strong fresh variable condition $(Y\setminus X)\cap fv\left({\Gamma }^{\prime }\right)=(Y\setminus X)\cap bv\left({\Gamma }^{\prime }\right)=\varnothing$.

After this is settled, we will assume in our analysis of traditional deduction calculi in the following two sections that those necessary transformations are tacitly done in the background. Moreover, we assume that the necessary renamings of bound variables for extensions of premises, as described in Remark 25, are also tacitly done.

7. Deduction in Traditional First-Order Logic

We investigate three different deduction calculi for traditional first-order logic - in this section a Hilbert System $\mathcal{H}$ and a Gentzen’s system $\mathcal{L}\mathcal{J}$ and, then, a Natural Deduction System $\mathcal{N}\mathcal{D}$ in Section 8.

We are interested to analyze, understand and formalize, by means of the novel concepts context and specification implication, the structural essence of reasoning in each of the systems and, thus, the crucial conceptual differences as well as their commonalities. Especially, we are interested to formally grasp the different meanings and semantics of the turnstile symbol ⊢ and the inference line.

7.1. Hilbert System $\mathcal{H}$

We study the proof system for first-order logic presented in [4]. We learned from [33] that this system can still be seen as one of the very many Hilbert Systems and will, therefore, use the name $\mathcal{H}$ instead of the name $\mathcal{N}$ used in [4]. According to [4], the proof system $\mathcal{H}$ is devised to define a relation ${⊢}_{\mathcal{H}}\subseteq \wp \left(WF{F}_{FOL}^{\Sigma }\right)\times WF{F}_{FOL}^{\Sigma }$ with $WF{F}_{FOL}^{\Sigma }$ the set of all well-formed first-order formulas for a signature $\Sigma$. It consists of

Axioms : Ā0: $\Gamma {⊢}_{\mathcal{H}}B$, for all $B\in \Gamma$ ;

A1: $\Gamma {⊢}_{\mathcal{H}}A\to (B\to A)$ ;

A2: $\Gamma {⊢}_{\mathcal{H}}(A\to (B\to C))\to ((A\to B)\to (A\to C))$ ;

A3: $\Gamma {⊢}_{\mathcal{H}}(\neg B\to \neg A)\to (A\to B)$ ;

A4: $\Gamma {⊢}_{\mathcal{H}}{A}_t^x\to \exists xA$ if $A_t^x$ is legal ;

Rules :: MP: $\Gamma {⊢}_{\mathcal{H}}A$$\Gamma {⊢}_{\mathcal{H}}A\to B$ $\Gamma {⊢}_{\mathcal{H}}B$ ;

∃I: $\Gamma {⊢}_{\mathcal{H}}A\to B$if $x\notin fv\left(B\right)$$\Gamma {⊢}_{\mathcal{H}}\exists xA\to B$ At first glance, it may look like that the elements in $\wp \left(WF{F}_{FOL}^{\Sigma }\right)\times WF{F}_{FOL}^{\Sigma }$ can be encoded as $\Sigma$-specification implications. The relation ${⊢}_{\mathcal{H}}$ is meant, however, to reflect the semantic entailment relation ⊩ between sets of formulas and single formulas. Note, that we use the symbol ⊩ for semantic entailment instead of the symbol ⊧, as it is quite common in the literature. We prefer to exclusively reserve the symbol ⊧ to denote the validity (satisfaction) relation between structures and formulas (of different kind).

$\Gamma ⊩A$ means that $\mathcal{U}\vDash \Gamma$ implies $\mathcal{U}\vDash A$ for all $\Sigma$-structures $\mathcal{U}$, where $\mathcal{U}\vDash \Gamma$ (or $\mathcal{U}\vDash A$) is valid for a $\Sigma$-structure $\mathcal{U}$ if $\Gamma$ (or A) becomes true for all (!) variable assignments in $\mathcal{U}$. The so-called universal closure (compare [4], p. 232 and [33], p. 372) transforms an open formula A into a closed formula $\forall \left(A\right)$ and a set $\Gamma$ of open formulas into a set $\forall \left(\Gamma \right)$ of closed formulas, respectively. We have $\mathcal{U}\vDash \Gamma$ (or $\mathcal{U}\vDash A$) if, and only if, $\mathcal{U}\vDash \forall \left(\Gamma \right)$ (or $\mathcal{U}\vDash \forall \left(A\right)$) thus $\Gamma ⊩A$ can be equivalently expressed by the condition that $\mathcal{U}\vDash \forall \left(\Gamma \right)$ implies $\mathcal{U}\vDash \forall \left(A\right)$ for all $\Sigma$-structures $\mathcal{U}$.

At the beginning of Section 5.2.2 we exemplified that $\Sigma$-sketch implications (and thus also $\Sigma$-specification implications) offer the novel device of unconditional $\Sigma$-assertions to formally grasp the traditional definition of the satisfaction relation ⊧ (also applicable for formalisms lacking a universal quantification operator!). We can utilize the satisfaction of $\Sigma$-specification implications in $\Sigma$-structures, due to Definition 18, and reformulate that $\Gamma ⊩A$ if, and only if, $\mathcal{U}\vDash \left(fv\right(\Gamma ),\varnothing )⟹\left(fv\right(\Gamma ),\Gamma )$ implies $\mathcal{U}\vDash \left(fv\right(A),\varnothing )⟹\left(fv\right(A),\{A\left\}\right)$ for all $\Sigma$-structures $\mathcal{U}$. Remind, that $\mathcal{U}\vDash \left(fv\right(\Gamma ),\varnothing )⟹\left(fv\right(\Gamma ),\Gamma )$ means nothing but $U^{fv\left(\Gamma \right)}={\left[[\varnothing ]\right]}_{fv\left(\Gamma \right)}^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_{fv\left(\Gamma \right)}^{\mathcal{U}}$. So, in view of LoSiCs the turnstile symbol ${⊢}_{\mathcal{H}}$ represents implications between $\Sigma$-specification implications of the form $(X,\varnothing )⟹(X,\Gamma )$ with $fv\left(\Gamma \right)\subseteq X$ and $\Sigma$-specification implications of the form $(Y,\varnothing )⟹(Y,\{A\left\}\right)$ with $fv\left(A\right)\subseteq Y$. For convenience, we will mostly write $(Y,A)$ instead of $(Y,\{A\left\}\right)$. We provide evidence that this fresh and novel view on system $\mathcal{H}$ is consistent with the traditional view by re-validating the soundness of system $\mathcal{H}$ in terms of specification implications.

7.1.0.1. Axiom A0

This is the only axiom schemata or rule where $\Gamma$ is relevant! For any $\Sigma$-structure $\mathcal{U}$ we assume $\mathcal{U}\vDash (X,\varnothing )⟹(X,\Gamma )$ with $X=fv\left(\Gamma \right)$. By definition, we have ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}=\bigcap \{{\left[\left[A\right]\right]}_X^{\mathcal{U}}\mid A\in \Gamma \}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}$ and thus $\mathcal{U}\vDash (X,\Gamma )⟹(X,B)$. Since the composition of $\Sigma$-specification implications preserves satisfaction, we can conclude $\mathcal{U}\vDash (X,\varnothing )⟹(X,B)$ and thus also $\mathcal{U}\vDash \left(fv\right(B),\varnothing )⟹\left(fv\right(B),B)$, due to Corollary 8.

Obviously, the $\Sigma$-specification implication $(X,\Gamma )⟹(X,B)$ is the essence of this axiom. It will re-emerge as an axiom in the Gentzen System $\mathcal{L}\mathcal{J}$.

7.1.0.2. Axioms A1 - A3

All three axiom schemata are independent of $\Gamma$! One can argue that these axiom schemata can be directly inherited from propositional logic. As already discussed at the beginning of Section 6 this argumentation may be not convincing for everyone since it is not clear in what sense open formulas in the traditional approach represent propositions.

As alternative, we can validate these axiom schemata by Boolean Algebra reasoning. We exemplify this for Axiom A1. With $X=fv\left(A\right)\cup fv\left(B\right)$ we obtain, due to Definition 7:

$$\begin{array}{ccccccc}\hfill {\left[[A\to (B\to A)]\right]}_X^{\mathcal{U}}& =& \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}\cup {\left[[B\to A]\right]}_X^{\mathcal{U}}\hfill & =& \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}\cup (\overline{{\left[\left[B\right]\right]}_X^{\mathcal{U}}}\cup {\left[\left[A\right]\right]}_X^{\mathcal{U}})\hfill & & \\ & =& \overline{{\left[\left[B\right]\right]}_X^{\mathcal{U}}}\cup (\overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}\cup {\left[\left[A\right]\right]}_X^{\mathcal{U}})\hfill & =& \overline{{\left[\left[B\right]\right]}_X^{\mathcal{U}}}\cup U^X\hfill & =& U^X\hfill \end{array}$$

and thus $\mathcal{U}\vDash (X,\varnothing )⟹(X,A\to (B\to A\left)\right)$ as required.

7.1.0.3. Axiom A4

To turn "$A_t^x\to \exists xA$" into a legal $\Sigma$-specification, we require $fv\left(A\right)\cap bv\left(A\right)=\varnothing$ and, instead of "$A_t^x$ is legal", that $fv\left(t\right)\cap bv\left(A\right)=\varnothing$. This can be achieved by an appropriate renaming of bound variables. To allow $x\in fv\left(t\right)$, we rename , in addition, the bound variable x in $\exists xA$ by a fresh variable z with $z\notin fv\left(A\right)\cup bv\left(A\right)\cup fv\left(t\right)$. That is, we replace $\exists xA$ by $\exists z{A}_z^x$. Also this axiom schemata is independent of $\Gamma$. For all $\Sigma$-structures $\mathcal{U}$ we show that $\mathcal{U}\vDash (X,\varnothing )⟹(X,A_t^x\to \exists z{A}_z^x)$ with $X:=fv\left(A_t^x\right)=fv\left(t\right)\cup fv(\exists xA)$ where $fv(\exists xA)=fv(\exists z{A}_z^x)=fv\left(A\right)\setminus \left\{x\right\}$ due to (1) and ${\left[[\exists xA]\right]}_X^{\mathcal{U}}={\left[[\exists z{A}_z^x]\right]}_X^{\mathcal{U}}$ due to (33).

We don’t work with substitutions of a single variable by a single term but only with maps from sets of variables into sets of $\Sigma$-terms. To grasp the situation in Axiom A4, we also consider the set $Y:=fv\left(A_z^x\right)=X\cup \left\{z\right\}$ of variables. We define a substitution $\mathbf{t}:Y\to T_{\Sigma }\left(X\right)$ with $\mathbf{t}\left(z\right):=t$ and $\mathbf{t}\left(x\right):=x$ for all $x\in X$. We obtain $\overline{\mathbf{t}}\left(A_z^x\right)=A_t^x$, according to Definition 11, and $in;\mathbf{t}={\eta }_X$ for the inclusion map $in:X\to Y$.

We must show $\mathbf{a}\in {\left[[\overline{\mathbf{t}}\left(A_z^x\right)\to \exists z{A}_z^x]\right]}_X^{\mathcal{U}}$ for any $\mathbf{a}\in {\left[[\varnothing ]\right]}_X^{\mathcal{U}}=U^X$, i.e., that $\mathbf{a}\in {\left[\left[\overline{\mathbf{t}}\left(A_z^x\right)\right]\right]}_X^{\mathcal{U}}$ implies $\mathbf{a}\in {\left[[\exists z{A}_z^x]\right]}_X^{\mathcal{U}}$. According to (3), the substitution $\mathbf{t}:Y\to T_{\Sigma }\left(X\right)$ defines a derived operation ${\mathbf{t}}^{\mathcal{U}}:U^X\to U^Y$ with ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right):=\mathbf{t};{\mathbf{a}}^{\circ }$. We have $\mathbf{a}\in {\left[\left[\overline{\mathbf{t}}\left(A_z^x\right)\right]\right]}_X^{\mathcal{U}}$ if, and only if, ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)\in {\left[\left[A_z^x\right]\right]}_Y^{\mathcal{U}}$, due to (20), and we get ${\exists }_{in}^U\left({\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)\right)=in;{\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)=in;\mathbf{t};{\mathbf{a}}^{\circ }={\eta }_X;{\mathbf{a}}^{\circ }=\mathbf{a}$ and thus $\mathbf{a}\in {\left[[\exists z{A}_z^x]\right]}_X^{\mathcal{U}}={\left[[\exists xA]\right]}_X^{\mathcal{U}}$ by the semantics of existential quantification in Definition 7.

7.1.0.4. Rule MP

Rules in System $\mathcal{H}$ introduce a third level of implications since inference lines in System $\mathcal{H}$ establish implications between implications relating $\Sigma$-specification implications. Such a kind of twofold nesting of implications looks indeed a bit "unnatural".

Rule MP is, however, independent of $\Gamma$ thus the validation of the rule means to show for any $\Sigma$-structure $\mathcal{U}$ that $\mathcal{U}\vDash (X,\varnothing )⟹(X,A)$ and $\mathcal{U}\vDash (X,\varnothing )⟹(X,A\to B)$ implies $\mathcal{U}\vDash (X,\varnothing )⟹(X,B)$ where $X:=fv\left(A\right)\cup fv\left(B\right)$. This can be easily done.

$\mathcal{U}\vDash (X,\varnothing )⟹(X,A)$ and $\mathcal{U}\vDash (X,\varnothing )⟹(X,A\to B)$ means $U^X={\left[\left[A\right]\right]}_X^{\mathcal{U}}$ and $U^X={\left[[A\to B]\right]}_X^{\mathcal{U}}$ thus we also get $U^X={\left[\left[A\right]\right]}_X^{\mathcal{U}}\cap {\left[[A\to B]\right]}_X^{\mathcal{U}}={\left[\left[\{A,A\to B\}\right]\right]}_X^{\mathcal{U}}$, i.e., $\mathcal{U}\vDash (X,\varnothing )⟹(X,\{A,A\to B\left\}\right)$. Relying on the semantics of implications in Definition 7, we can easily show $\mathcal{U}\vDash (X,\{A,A\to B\left\}\right)⟹(X,B)$ by Boolean Algebra reasoning:

$${\left[\left[\{A,A\to B\}\right]\right]}_X^{\mathcal{U}}={\left[\left[A\right]\right]}_X^{\mathcal{U}}\cap {\left[[A\to B]\right]}_X^{\mathcal{U}}={\left[\left[A\right]\right]}_X^{\mathcal{U}}\cap (\overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}\cup {\left[\left[B\right]\right]}_X^{\mathcal{U}})={\left[\left[A\right]\right]}_X^{\mathcal{U}}\cap {\left[\left[B\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}$$

Since composition of $\Sigma$-specification implications preserves satisfaction, we can finally conclude $\mathcal{U}\vDash (X,\varnothing )⟹(X,B)$ as desired.

The $\Sigma$-specification implication $(X,\{A,A\to B\left\}\right)⟹(X,B)$ is the essence of this rule and will re-emerge as a rule in the Natural Deduction System $\mathcal{N}\mathcal{D}$.

7.1.0.5. Rule ∃I

Also this rule is independent of $\Gamma$ thus it suffices to show for any $\Sigma$-structure $\mathcal{U}$ that $\mathcal{U}\vDash (Y,\varnothing )⟹(Y,A\to B)$ implies $\mathcal{U}\vDash (X,\varnothing )⟹(X,\exists xA\to B)$ with $Y=fv(A\to B)$, $X=fv(\exists xA\to B)$ and thus $X=Y\setminus \left\{x\right\}$ since $x\notin fv\left(B\right)$.

We consider the inclusion map $in:X\to Y$ and turn the soundness proof by contraposition for Rule ∃I in [4], p. 261 into a direct proof of the following lemma.

Lemma 5

(Rule ∃I: Sem.). ${\left[[\exists xA\to B]\right]}_X^{\mathcal{U}}={\forall }_{in}^U{\left[[A\to B]\right]}_Y^{\mathcal{U}}$ for all Σ-structures $\mathcal{U}$ if $x\notin fv\left(B\right)$.

Proof. 

Due to the definition of universal operators in (A21), we have to show for any $\mathbf{a}\in {\left[[\exists xA\to B]\right]}_X^{\mathcal{U}}$ that $\mathbf{b}\in {\left[[A\to B]\right]}_Y^{\mathcal{U}}$ for all $\mathbf{b}\in U^Y$ with $in;\mathbf{b}=\mathbf{a}$. $\mathbf{b}\in {\left[\left[A\right]\right]}_Y^{\mathcal{U}}$ entails $\mathbf{a}=in;\mathbf{b}\in {\left[[\exists xA]\right]}_X^{\mathcal{U}}={\exists }_{in}^U{\left[\left[A\right]\right]}_Y^{\mathcal{U}}=(in;\_){\left[\left[A\right]\right]}_Y^{\mathcal{U}}$ thus we get $\mathbf{a}=in;\mathbf{b}\in {\left[\left[B\right]\right]}_X^{\mathcal{U}}$ by assumption $\mathbf{a}\in {\left[[\exists xA\to B]\right]}_X^{\mathcal{U}}$. Due to Lemma 3, this implies $\mathbf{b}\in {(in;\_)}^{-1}{\left[\left[B\right]\right]}_X^{\mathcal{U}}{=\downarrow }_{in}^U{\left[\left[B\right]\right]}_X^{\mathcal{U}}={\left[\left[B\right]\right]}_Y^{\mathcal{U}}$ thus we have shown $\mathbf{b}\in {\left[[A\to B]\right]}_Y^{\mathcal{U}}$, as required. □

Finally, we assume $\mathcal{U}\vDash (Y,\varnothing )⟹(Y,A\to B)$, i.e., $U^Y={\left[[A\to B]\right]}_Y^{\mathcal{U}}$. By Lemma 5, Corollary 6 and equation (29) we obtain ${\left[[\exists xA\to B]\right]}_X^{\mathcal{U}}={\forall }_{in}^U{\left[[A\to B]\right]}_Y^{\mathcal{U}}={\forall }_{in}^U\left(U^Y\right)={\forall }_{in}^U\left({(in;\_)}^{-1}\left(U^X\right)\right)=U^X={\left[[\varnothing ]\right]}_X^{\mathcal{U}}$, i.e., $\mathcal{U}\vDash (X,\varnothing )⟹(X,\exists xA\to B)$, thus Rule ∃I is indeed validated.

7.2. Gentzen’s System $\mathcal{L}\mathcal{J}$

There are many variants of System $\mathcal{L}\mathcal{J}$ around. At the end, we decided against the variants presenting Sequent Calculi by a set of rules cleanly devided in left rules and right rules [34]. We decided to rather rely on [32] instead, since the strong relation between System $\mathcal{L}\mathcal{J}$ and Natural Deduction is coherently addressed and explained in [32]. After a Natural Deduction calculus is presented and discussed in [32], the authors write:

This is the classical formulation of the calculus of natural deduction. To prepare the things we want to do later (and to get around the somewhat un-licensed extension by hypothetical reasoning in the calculus), we will reformulate the calculus by lifting it to the “judgements level”. Instead of postulating rules that make statements about the validity of propositions, we postulate rules that make statements about derivability. This move allows us to make the respective local hypotheses in Natural Deduction derivations into syntactic parts of the objects (we call them “sequents”) manipulated by the inference rules.

We like this didactical approach very much, but will, nevertheless, first discuss System $\mathcal{L}\mathcal{J}$ and only afterwards Natural Deduction. Our didactical reason is that we want to emphasize the stepwise reduction of "nesting levels" of implications. The second, more practical, reason is, that we must develop another chunk of theory to formally grasp and finally license hypothetical reasoning. We will only introduce and elaborate the necessary novel concept of open specification implications in Section 8.

Moving from System $\mathcal{H}$ to System $\mathcal{L}\mathcal{J}$, we get rid of one level of nesting of implications! Sequents $\Gamma {⊢}_{\mathcal{LJ}}A$ in System $\mathcal{L}\mathcal{J}$ are nothing but $\Sigma$-specification implications of the form $(X,\Gamma )⟹(Y,A)$ with $X\subseteq Y$ thus the inference lines in System $\mathcal{L}\mathcal{J}$ represent implications between a finite set of $\Sigma$-specification implications of this form and a single $\Sigma$-specification implication of this form. We present first the propositional rules of System $\mathcal{L}\mathcal{J}$ according to [32] where we use the symbols "→", "⊥" instead of the symbols "⇒", "F" in [32]. To safe space, we will also simply write ⊢ instead of ${⊢}_{\mathcal{LJ}}$.

*6ex Axiom $\Gamma ,A⊢A$ $\Gamma ⊢B$ weaken $\Gamma ,A⊢B$ TND $\Gamma ⊢A\vee \neg A$

$\Gamma ⊢A$$\Gamma ⊢B$$\wedge I$$\Gamma ⊢A\wedge B$$\Gamma ⊢A\wedge B$$\wedge E_l$$\Gamma ⊢A$$\Gamma ⊢A\wedge B$$\wedge E_r$$\Gamma ⊢B$

$\Gamma ⊢A\vee B$$\Gamma ,A⊢C$$\Gamma ,B⊢C$$\vee E$$\Gamma ⊢C$$\Gamma ⊢A$$\vee I_l$$\Gamma ⊢A\vee B$$\Gamma ⊢B$$\vee I_r$$\Gamma ⊢A\vee B$

$\Gamma ⊢A$$\Gamma ⊢A\to B$$\to E$$\Gamma ⊢B$$\Gamma ,A⊢B$$\to I$$\Gamma ⊢A\to B$

$\Gamma ⊢\neg A$$\Gamma ⊢A$$\perp I$$\Gamma ⊢\perp$$\Gamma ⊢\perp$$\perp E$$\Gamma ⊢A$$\Gamma ,A⊢\perp$$\neg I$$\Gamma ⊢\neg A$$\Gamma ⊢\neg \neg A$$\neg E$$\Gamma ⊢A$ For the remaining part of this subsection let $\mathcal{U}$ be an arbitrary $\Sigma$-structure. We discuss the first two rules. The rule Axiom corresponds to Axiom A0 in System $\mathcal{H}$ and that $\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)\Rightarrow (X,A)$ with $X=fv\left(\Gamma \right)\cup fv\left(A\right)$ has been already shown in Section 7.1. Analogously, we can argue that $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,B)$ with $X=fv\left(\Gamma \right)\cup fv\left(A\right)\cup fv\left(B\right)$, i.e., ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}$ , entails ${\left[[\Gamma \cup \left\{A\right\}]\right]}_X^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}$ , i.e., $\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)⟹(X,B)$, thus also rule weaken is validated.

Relying on Remark 25 and the fact that the validity of $\Sigma$-specification implications is closed under composition, each propositional rule, where all involved sequents share the same antecedent, can be validated by proving that a certain single underlying essential Σ-specification implication is valid in any $\Sigma$-structure $\mathcal{U}$. These essential $\Sigma$-specification implications appear later as rules in the Natural Deduction System $\mathcal{N}\mathcal{D}$! We list all those rules together with the corresponding essential $\Sigma$-specification implication.

$$\begin{array}{ccc}\mathrm{Rule}& \mathrm{Essential}\mathrm{Implication}\hfill & \mathrm{Variable}\mathrm{Declaration}\hfill \\ TND& (Y,\varnothing )⟹(Y,A\vee \neg A)\hfill & Y=fv\left(A\right)\hfill \\ \wedge I& (Y,\{A,B\left\}\right)⟹(Y,A\wedge B)\hfill & Y=fv\left(A\right)\cup fv\left(B\right)\hfill \\ \wedge E_l& (Y,A\wedge B)⟹(Y,A)\hfill & Y=fv\left(A\right)\cup fv\left(B\right)\hfill \\ \wedge E_r& (Y,A\wedge B)⟹(Y,B)\hfill & Y=fv\left(A\right)\cup fv\left(B\right)\hfill \\ \vee I_l& (Y,A)⟹(Y,A\vee B)\hfill & Y=fv\left(A\right)\cup fv\left(B\right)\hfill \\ \vee I_r& (Y,B)⟹(Y,A\vee B)\hfill & Y=fv\left(A\right)\cup fv\left(B\right)\hfill \\ \to E& (Y,\{A,A\to B\left\}\right)⟹(Y,B)\hfill & Y=fv\left(A\right)\cup fv\left(B\right)\hfill \\ \perp I& (Y,\{\neg A,A\left\}\right)⟹(Y,\perp )\hfill & Y=fv\left(A\right)\hfill \\ \perp E& (Y,\perp )⟹(Y,A)\hfill & Y=fv\left(A\right)\hfill \\ \neg E& (Y,\neg \neg A)⟹(Y,A)\hfill & Y=fv\left(A\right)\hfill \end{array}$$

(38)

We validate rule TND. We do have $\mathcal{U}\vDash (Y,\varnothing )\Rightarrow (Y,A\vee \neg A)$ for the essential implication, due to the semantics of disjunction, and thus $\mathcal{U}\vDash (X,\varnothing )\Rightarrow (X,A\vee \neg A)$ for $X=Y\cup fv\left(\Gamma \right)$ by Remark 25. $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,\varnothing )$ trivially holds since ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq U^X$ thus we obtain $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,A\vee \neg A)$ by composition of $\Sigma$-specification implications.

Rule$\to \mathbf{E}$ corresponds to rule MP in System $\mathcal{H}$. In Section 7.1 we have already shown $\mathcal{U}\vDash (Y,\{A,A\to B\left\}\right)\Rightarrow (Y,B)$ for the essential implication thus we have also $\mathcal{U}\vDash (X,\{A,A\to B\left\}\right)\Rightarrow (X,B)$ for $X=Y\cup fv\left(\Gamma \right)$ by Remark 25. ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[A\right]\right]}_X^{\mathcal{U}}$ and ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[[A\to B]\right]}_X^{\mathcal{U}}$ trivially imply ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[A\right]\right]}_X^{\mathcal{U}}\cap {\left[[A\to B]\right]}_X^{\mathcal{U}}={\left[\left[\{A,A\to B\}\right]\right]}_X^{\mathcal{U}}$ , i.e., $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,\{A,A\to B\left\}\right)$, thus we finally obtain $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,B)$ by composition of $\Sigma$-specification implications.

All the other essential $\Sigma$-specification implications in the above listing are valid in any $\Sigma$-structure $\mathcal{U}$ according to the semantics of Boolean connectives, thus the corresponding rules can easily be validated analogously to the rules we explicitly validated.

It remains to validate the three rules where the involved sequents don’t share the same antecedent. For rule $\neg \mathbf{I}$ we set $X=fv\left(\Gamma \right)\cup fv\left(A\right)$. $\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)\Rightarrow (X,\perp )$ means ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}={\left[[\perp ]\right]}_X^{\mathcal{U}}=\varnothing$ thus we obtain ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap U^X={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap ({\left[\left[A\right]\right]}_X^{\mathcal{U}}\cup \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}})={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}$ and, equivalently, ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}={\left[[\neg A]\right]}_X^{\mathcal{U}}$, i.e., $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,\neg A)$.

For rule $\vee \mathbf{E}$ we set $X=fv\left(\Gamma \right)\cup fv\left(A\right)\cup fv\left(B\right)\cup fv\left(C\right)$. The assumption $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,A\vee B)$, i.e., ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[A\right]\right]}_X^{\mathcal{U}}\cup {\left[\left[B\right]\right]}_X^{\mathcal{U}}$, implies ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap ({\left[\left[A\right]\right]}_X^{\mathcal{U}}\cup {\left[\left[B\right]\right]}_X^{\mathcal{U}})={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\cup {\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[B\right]\right]}_X^{\mathcal{U}}$ and thus ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[C\right]\right]}_X^{\mathcal{U}}$ since by assumption $\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)\Rightarrow (X,C)$, i.e., ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[C\right]\right]}_X^{\mathcal{U}}$, and $\mathcal{U}\vDash (X,\Gamma \cup \{B\left\}\right)\Rightarrow (X,C)$, i.e., ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[B\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[C\right]\right]}_X^{\mathcal{U}}$. This proves $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,C)$.

To validate rule $\to \mathbf{I}$ we need to prove the following semantic deduction theorem for specification implications with $X=fv\left(\Gamma \right)\cup fv\left(A\right)\cup fv\left(B\right)$.

Lemma 6.

$\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)\Rightarrow (X,B)$ iff $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,A\to B)$

Proof. 

"⇒": $\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)\Rightarrow (X,B)$, i.e., ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}$ entails, according to the semantics of the connective →, ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap U^X={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap ({\left[\left[A\right]\right]}_X^{\mathcal{U}}\cup \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}})={\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\cup {\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}\cup \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}={\left[[A\to B]\right]}_X^{\mathcal{U}}$, i.e., $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,A\to B)$.

"⇐": $\mathcal{U}\vDash (X,\Gamma )\Rightarrow (X,A\to B)$, i.e., ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\subseteq \overline{{\left[\left[A\right]\right]}_X^{\mathcal{U}}}\cup {\left[\left[B\right]\right]}_X^{\mathcal{U}}$, implies ${\left[\left[\Gamma \right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}\cap {\left[\left[A\right]\right]}_X^{\mathcal{U}}\subseteq {\left[\left[B\right]\right]}_X^{\mathcal{U}}$, i.e., $\mathcal{U}\vDash (X,\Gamma \cup \{A\left\}\right)\Rightarrow (X,B)$. □

After the propositional rules we now examine the four quantifier rules of System $\mathcal{L}\mathcal{J}$.

*2ex $\Gamma ⊢A$ $\forall I$$x\notin fv\left(\Gamma \right)$ $\Gamma ⊢\forall xA$ $\Gamma ⊢\forall xA$ $\forall E$$A_t^x$ is legal $\Gamma ⊢A_t^x$ $\Gamma ⊢A_t^x$ $\exists I$$A_t^x$ is legal $\Gamma ⊢\exists xA$

$\Gamma ⊢\exists xA$$\Gamma ,A_c^x⊢B$$\exists E$c a new Skolem constant $\Gamma ⊢B$

We postpone the validation of rule $\forall I$ to Section 8. In contrast to rule $\exists E$, the two rules $\exists I$ and $\forall E$ can be validated by proving the soundness of a corresponding underlying essential $\Sigma$-specification implication which re-emerges later as a rule in the Natural Deduction System $\mathcal{N}\mathcal{D}$.

Rule$\exists \mathbf{I}$ is the sequent variant of Axiom A4 in System $\mathcal{H}$. We adapt the argumentation in paragraph Axiom A4 on page 39: To turn the sequent $\Gamma ⊢A_t^x$ into a legal $\Sigma$-specification implication $(X,\Gamma )⟹(X,A_t^x)$ with $X=fv\left(\Gamma \right)\cup fv\left(A_t^x\right)$, we require $\left(fv\right(A)\cup fv(\Gamma \left)\right)\cap \left(bv\right(A)\cup bv(\Gamma \left)\right)=\varnothing$ and $fv\left(t\right)\cap \left(bv\right(A)\cup bv(\Gamma \left)\right)=\varnothing$, instead of "$A_t^x$ is legal". This can be achieved by an appropriate renaming of bound variables.

To allow $x\in fv\left(t\right)$, we rename , in addition, the bound variable x in $\exists xA$ by a fresh variable z with $z\notin fv\left(\Gamma \right)\cup fv\left(A\right)\cup fv\left(t\right)$ and $z\notin bv\left(A\right)$. That is, we replace $\exists xA$ by $\exists z{A}_z^x$. where $fv(\exists xA)=fv(\exists z{A}_z^x)=fv\left(A\right)\setminus \left\{x\right\}$, due to (1), and ${\left[[\exists xA]\right]}_Y^{\mathcal{U}}={\left[[\exists z{A}_z^x]\right]}_Y^{\mathcal{U}}$ with $Y:=fv\left(A_t^x\right)=fv\left(t\right)\cup fv(\exists xA)$ for any $\Sigma$-structure $\mathcal{U}$, due to (33).

We show that $\mathcal{U}\vDash (X,\Gamma )⟹(X,A_t^x)$ implies $\mathcal{U}\vDash (X,\Gamma )⟹(X,\exists z{A}_z^x)$ for any $\Sigma$-structure $\mathcal{U}$. $\mathcal{U}\vDash (Y,\varnothing )⟹(Y,A_t^x\to \exists z{A}_z^x)$ has been proven in Section 7.1, thus we obtain, by Lemma 6, $\mathcal{U}\vDash (Y,A_t^x)⟹(Y,\exists z{A}_z^x)$ for the essential $\Sigma$-specification implication $(Y,A_t^x)⟹(Y,\exists z{A}_z^x)$ underlying rule $\exists I$. $\mathcal{U}\vDash (X,A_t^x)⟹(X,\exists z{A}_z^x)$ is ensured by Remark 25, thus we finally get $\mathcal{U}\vDash (X,\Gamma )⟹(X,\exists z{A}_z^x)$, as desired, since validity of $\Sigma$-specification implications is preserved by composition.

Rule$\forall \mathbf{E}$ can be validated analogously to rule $\exists I$. To turn the sequent $\Gamma ⊢A_t^x$ into a legal $\Sigma$-specification implication $(X,\Gamma )⟹(X,A_t^x)$ with $X=fv\left(\Gamma \right)\cup fv\left(A_t^x\right)$, we impose the same restrictions as for rule "$\exists I$", and we correspondingly replace $\forall xA$ by $\forall z{A}_z^x$ to allow $x\in fv\left(t\right)$ where $fv(\forall xA)=fv(\forall z{A}_z^x)=fv\left(A\right)\setminus \left\{x\right\}$ and ${\left[[\forall xA]\right]}_Y^{\mathcal{U}}={\left[[\forall z{A}_z^x]\right]}_Y^{\mathcal{U}}$ with $Y:=fv\left(A_t^x\right)=fv\left(t\right)\cup fv(\forall xA)$ for any $\Sigma$-structure $\mathcal{U}$.

We want to show that $\mathcal{U}\vDash (X,\Gamma )⟹(X,\forall z{A}_z^x)$ implies $\mathcal{U}\vDash (X,\Gamma )⟹(X,A_t^x)$ for any $\Sigma$-structure $\mathcal{U}$. This can be done by proving $\mathcal{U}\vDash (Y,\forall z{A}_z^x)⟹(Y,A_t^x)$ for the essential $\Sigma$-specification implication $(Y,\forall z{A}_z^x)⟹(Y,A_t^x)$ underlying rule $\forall E$. Analogously to the case of Axiom A4 on page 39, we consider the set $Z=fv\left(A_z^x\right)=Y\cup \left\{z\right\}$ of variables and define a substitution $\mathbf{t}:Z\to T_{\Sigma }\left(Y\right)$ with $\mathbf{t}\left(z\right):=t$ and $\mathbf{t}\left(y\right):=y$ for all $y\in Y$, thus we have $\overline{\mathbf{t}}\left(A_z^x\right)=A_t^x$ and $in;\mathbf{t}={\eta }_Y$ for the inclusion map $in:Y\to Z$. Applying the derived operation ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^Z$ to any $\mathbf{a}\in {\left[[\forall z{A}_z^x]\right]}_Y^{\mathcal{U}}$, we get ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right):Z\to U$ given by ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)=\mathbf{t};{\mathbf{a}}^{\circ }$, due to (3), and thus $in;{\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)=in;\mathbf{t};{\mathbf{a}}^{\circ }={\eta }_Y;{\mathbf{a}}^{\circ }=\mathbf{a}$. This entails ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)\in {\left[\left[A_z^x\right]\right]}_Z^{\mathcal{U}}$ due to the semantics of universal quantification. According to (20), ${\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)\in {\left[\left[A_z^x\right]\right]}_Z^{\mathcal{U}}$ means, however, nothing but $\mathbf{a}\in {\left[\left[\overline{\mathbf{t}}\left(A_z^x\right)\right]\right]}_Y^{\mathcal{U}}={\left[\left[A_t^x\right]\right]}_Y^{\mathcal{U}}$, as required.

$\mathcal{U}\vDash (Y,\forall z{A}_z^x)⟹(Y,A_t^x)$ implies $\mathcal{U}\vDash (X,\forall z{A}_z^x)⟹(X,A_t^x)$, due to Remark 25, thus $\mathcal{U}\vDash (X,\Gamma )⟹(X,A_t^x)$ is indeed entailed by $\mathcal{U}\vDash (X,\Gamma )⟹(X,\forall z{A}_z^x)$ since composition of $\Sigma$-specification implications preserves validity.

In case of rule $\exists \mathbf{E}$, we choose $Y=fv\left(\Gamma \right)\cup fv(\exists xA)\cup fv\left(B\right),Z=Y\cup \left\{c\right\}$ and turn the sequents $\Gamma ⊢\exists xA$ and $\Gamma ,A_c^x⊢B$ into $\Sigma$-specification implications $(Y,\Gamma )⟹(Y,\exists xA)$ and $(Z,\Gamma \cup \left\{A_c^x\right\})⟹(Z,B)$, respectively.

For $I=fv(\exists xA)$, $P=I\cup \left\{x\right\}$ we have $\mathcal{U}\vDash (I,\exists xA)⟹(P,A)$ for any $\Sigma$-structure $\mathcal{U}$ since ${\left[[\exists xA]\right]}_I^{\mathcal{U}}={\exists }_i^U{\left[\left[A\right]\right]}_P^{\mathcal{U}}$, due to Definition 7, for the proper inclusion map $i:I\to P$. That is, this implication simply represents the semantics of existential quantification.

"c a new Skolem constant" means, especially, $c\notin Y$ and $c\notin bv\left(\Gamma \right)\cup bv\left(A\right)\cup bv\left(B\right)$ thus we can utilize c to describe the pushout of the span $Y\stackrel{in}{\leftarrow }I\stackrel{i}{\to }P$ of inclusion maps and the corresponding extension of the context I of the $\Sigma$-specification $(I,\exists xA)$ via the inclusion map $in:I\to Y$ (compare Section 6.4 and, especially, Remarks 24 and 25).

Corollary 7 ensures $\mathcal{U}\vDash (Y,\exists xA)⟹(Z,A_c^x)$ thus $\mathcal{U}\vDash (Y,\Gamma )⟹(Y,\exists xA)$ entails $\mathcal{U}\vDash (Y,\Gamma )⟹(Z,A_c^x)$ by composition of $\Sigma$-specification implications. So, for each $\iota \in {\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}$ there exists an $\vartheta \in {\left[\left[A_c^x\right]\right]}_Z^{\mathcal{U}}$ such that $j;\vartheta =\iota$ and therefore, due to Corollary 6, also $\vartheta \in {\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}$ as well as $\vartheta \in {\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}\cap {\left[\left[A_c^x\right]\right]}_Z^{\mathcal{U}}={\left[[\Gamma \cup \left\{A_c^x\right\}]\right]}_Z^{\mathcal{U}}$. In such a way, we get $\mathcal{U}\vDash (Y,\Gamma )⟹(Z,\Gamma \cup \left\{A_c^x\right\})$. So, if $\mathcal{U}\vDash (Z,\Gamma \cup \left\{A_c^x\right\})⟹(Z,B)$, we also have $\mathcal{U}\vDash (Y,\Gamma )⟹(Z,B)$ by composition of $\Sigma$-specification implications. It remains to get rid of the "new Skolem constant c".

We use the same trick as in the validation of Rule ∃I in System $\mathcal{H}$. $(Y,B)$ is a legal $\Sigma$-specification and we can factorize $(Y,\Gamma )⟹(Z,B)$ into $(Y,\Gamma )⟹(Y,B)⟹(Z,B)$. $\mathcal{U}\vDash (Y,\Gamma )⟹(Z,B)$ if, and only if, ${\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\subseteq {\exists }_j^U{\left[\left[B\right]\right]}_Z^{\mathcal{U}}$. By Corollary 6 and equation (29), we obtain ${\exists }_j^U{\left[\left[B\right]\right]}_Z^{\mathcal{U}}={\exists }_j^U\left({\downarrow }_j^U{\left[\left[B\right]\right]}_Y^{\mathcal{U}}\right)={\left[\left[B\right]\right]}_Y^{\mathcal{U}}$ and thus $\mathcal{U}\vDash (Y,\Gamma )⟹(Y,B)$. If desired, we can even conclude $\mathcal{U}\vDash \left(fv\right(\Gamma )\cup fv(B),\Gamma )⟹\left(fv\right(\Gamma )\cup fv(B),B)$, due to Corollary 8.

Remark 26

(Gentzen’s System $\mathcal{L}\mathcal{K}$). Gentzen’s System $\mathcal{L}\mathcal{K}$[34] and its variant System $\mathcal{G}$ in [4] work with sequents $\Gamma {⊢}_{\mathcal{L}\mathcal{K}}\Delta$ where both Γ and Δ are finite sets of formulas. Those sequents correspond to implications of the form $\bigwedge \Gamma \to \bigvee \Delta$.

Until now, we considered only "conjunctive Σ-specifications" $(X,\Gamma )$ with a semantics given by intersection ${\left[\left[(X,\Gamma )\right]\right]}^{\mathcal{U}}=\bigcap \{{\left[\left[B\right]\right]}_X^{\mathcal{U}}\mid B\in \Gamma \}$ for any Σ-structure $\mathcal{U}$. To treat sequents $\Gamma {⊢}_{\mathcal{L}\mathcal{K}}\Delta$ within the LoSiC-approach, we must, however, also introduce "disjunctive Σ-specifications" $(Y,{\Delta }^d)$ with semantics given by union ${\left[\left[(Y,{\Delta }^d)\right]\right]}^{\mathcal{U}}=\bigcup \{{\left[\left[D\right]\right]}_Y^{\mathcal{U}}\mid D\in \Delta \}$. Then, we can formalize sequents $\Gamma {⊢}_{\mathcal{L}\mathcal{K}}\Delta$ as Σ-specification implications $(X,\Gamma )⟹(Y,{\Delta }^d)$ with $\mathcal{U}\vDash (X,\Gamma )⟹(Y,{\Delta }^d)$ if, and only if, ${\left[\left[(X,\Gamma )\right]\right]}^{\mathcal{U}}\subseteq {\exists }_{in}^U{\left[\left[(Y,{\Delta }^d)\right]\right]}^{\mathcal{U}}$ for the inclusion map $in:X\to Y$. Note, that ${\exists }_{in}^U{\left[\left[(Y,{\Delta }^d)\right]\right]}^{\mathcal{U}}=\bigcup \{{\exists }_{in}^U{\left[\left[D\right]\right]}_Y^{\mathcal{U}}\mid D\in \Delta \}$ since ${\exists }_{in}^U$ distributes over arbitrary unions!

We are convinced that deduction systems like $\mathcal{L}\mathcal{K}$ and $\mathcal{G}$ can be validated by means of Boolean Algebra reasoning analogously to System $\mathcal{L}\mathcal{J}$. Due to limited time, space as well as motivation on our side, we postpone a detailed exploration of those systems to another occasion.

8. Natural Deduction and Open Specification Implications

Moving from System $\mathcal{L}\mathcal{J}$ to the Natural Deduction System $\mathcal{N}\mathcal{D}$, we finally reduce the level of nesting of implications to one for all rules where we have been able to identify an essential underlying $\Sigma$-specification implication. In System $\mathcal{N}\mathcal{D}$ these essential implications are simply represented as rules, i.e., the horizontal implication arrows "⟹" are transformed into vertical interference lines. In other words, we simply delete the "same antecedent" $\Gamma ⊢$ in the corresponding rules in System $\mathcal{L}\mathcal{J}$ to obtain a rule in System $\mathcal{N}\mathcal{D}$. The reader may forgive us for not repeating those rules in their corresponding $\mathcal{N}\mathcal{D}$-shape.

Attention, in case of rule $\forall I$ we also delete both $\Gamma ⊢$, but the condition "$x\notin fv\left(\Gamma \right)$" transforms into the somehow puzzling condition "A does not depend on any hypothesis in which x is free" in (49).

In the remaining four rules we also drop $\Gamma$ in all the antecedents thus the conclusions of the four rules become plain $\Sigma$-expressions. The premises of the rules remain, however, with a kind of implicational residue. The problem is to describe this implicational residue in a, more or less, precise way! The three propositional rules $\to I$, $\neg I$, $\vee E$ in System $\mathcal{N}\mathcal{D}$ are presented in [32] as follows:

As the rules may be nested, both the rules and the corresponding assumptions are decorated with a marker (here the number 1). We will discuss one nested proof in Section 8.4.

The semi-formal notation "${\left[A\right]}^1\dots B$" is usually read as "there is a proof showing that B holds under the additional hypothesis that A holds". Turning upside down the argumentation in [32], cited at the beginning of Section 7.2, we pose the question if it is possible to describe the result of such a hypothetical proof "${\left[A\right]}^1\dots B$" in a precise formal way and independent of the concrete proof. The answer is that we can indeed do this by introducing the concept of open specification implication. We will define, elaborate and apply this novel concept in the remaining part of this section.

8.1. Open Specification Implications

We combine two lines of observations and insights. First, we observe that assertions of the form "we can prove B under the additional hypothesis that A holds" are widely used as so-called universal properties to define concepts and to formalize findings in Category Theory. Second, the LoSiC-approach is based on the insight that the difference between closed formulas and open formulas essentially boils down to the difference between empty contexts and non-empty contexts, respectively. Statements in context are statements about single interpretations of contexts in a $\Sigma$-structure $\mathcal{U}$. For the empty context there is, however, only exactly one interpretation in a $\Sigma$-structure $\mathcal{U}$, thus a statement in the empty contexts turns into a statement about the $\Sigma$-structure $\mathcal{U}$ as a whole!

According to Definition 18, specification implications can serve as another tool to formulate assertions about $\Sigma$-structures as a whole and correspond, in this sense, to closed formulas. In analogy to $\Sigma$-expressions, we simply propose to equip specification implications with a potentially non-empty interfaces.

Definition 21

(Open Specification Implication: Syntax). Anopen $\Sigma$-specification implication$I\to (P,\Pi )\Rightarrow (K,\Delta )$ is given by a Σ-specification implication $(P,\Pi )⟹(K,\Delta )$, in the sense of Definition 18, and aninterface$I\subseteq P$.

In analogy to $\Sigma$-expressions, we can also define a semantics for open specification implications in any structure.

Definition 22

(Open Specification Implication: Semantics). For a Σ-structure $\mathcal{U}$ the semantics ${\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}\subseteq U^I$ of an open Σ-specification implication $I\to (P,\Pi )\Rightarrow (K,\Delta )$ in $\mathcal{U}$ is defined as follows: For all $\mathbf{a}\in U^I$ it holds that $\mathbf{a}\in {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$ if, and only if, ${\downarrow }_i^U\left(\left\{\mathbf{a}\right\}\right)\cap {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}\subseteq {\exists }_{in}^U{\left[\left[(K,\Delta )\right]\right]}^{\mathcal{U}}$ for the inclusion maps $i:I\to P$ and $in:P\to K$, i.e., if, and only if, for all extensions $\mathbf{b}:P\to U$ of $\mathbf{a}:I\to U$ along $i:I\to P$ satisfying Π there exists an extension $\mathbf{c}:K\to U$ along $in:P\to K$ satisfying Δ.

We can also equivalently reformulate Definition 22 in a more compact way

$${\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}:={\forall }_i^U\left(\overline{{\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}}\cup {\exists }_{in}^U{\left[\left[(K,\Delta )\right]\right]}^{\mathcal{U}}\right).$$

(41)

In case $I=\varnothing$, i.e., $i={!}_P:\varnothing \to P$, we do have only one assignment $\mathbf{a}={!}_U:\varnothing \to U$. Since ${\downarrow }_{{!}_P}^U\left(\left\{{!}_U\right\}\right)=U^P$, we indeed reconstruct the validity condition for $\Sigma$-specification implications in Definition 18 (compare also (25)):

$$\mathcal{U}{\vDash }^{\Sigma }(P,\Pi )⟹(K,\Delta )iff{!}_U\in {\left[[\varnothing \to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_{\varnothing }^{\mathcal{U}}.$$

In case $I=P$, we immediately obtain the following corollary.

Corollary 9

(Open Specification Implication: Semantics). For all Σ-structures $\mathcal{U}$ it holds that

1.

${\left[[I\to (I,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}=\overline{{\left[\left[(I,\Pi )\right]\right]}^{\mathcal{U}}}\cup {\exists }_{in}^U{\left[\left[(K,\Delta )\right]\right]}^{\mathcal{U}}$ and thus

2.

${\left[[I\to (I,\varnothing )\Rightarrow (I,\Delta )]\right]}_I^{\mathcal{U}}={\left[\left[(I,\Delta )\right]\right]}^{\mathcal{U}}$ and

3.

${\left[[I\to (I,A)\Rightarrow (I,B)]\right]}_I^{\mathcal{U}}={\left[\left[(I,A\to B)\right]\right]}^{\mathcal{U}}$.

In addition to Corollary 9.3, we can also prove a semantic deduction theorem for open specification implications by slightly adapting the proof of Lemma 6.

Lemma 7.

${\left[[I\to (P,\Pi \cup \left\{A\right\})\Rightarrow (P,B)]\right]}_I^{\mathcal{U}}={\left[[I\to (P,\Pi )\Rightarrow (P,A\to B)]\right]}_I^{\mathcal{U}}$

The declaration "$I\to$" for open $\Sigma$-specification implications has the same purpose as the declaration "$X▹$" for $\Sigma$-expressions. It declares the set of variables that are considered to be the potentially available free variables in $I\to (P,\Pi )\Rightarrow (K,\Delta )$. The set of free variables, actually appearing in $I\to (P,\Pi )\Rightarrow (K,\Delta )$, is the set $I\cap \left(fv\right(\Pi )\cup fv(\Delta \left)\right)$ while the set of all bound variables is $(K\setminus I)\cup bv\left(\Pi \right)\cup bv\left(\Delta \right)$. Note, that, due to Definition 22, the variables in $P\setminus I$ are implicitly universal quantified while the variables in $K\setminus P$ are implicitly existential quantified!

We propose to formalize assertions of the form "${\left[A\right]}^1\dots B$" as open $\Sigma$-specification implications thus the premises in the rules $\to I$, $\neg I$, $\vee E$ in (40) become $\Sigma$-specifications enriched by open $\Sigma$-specification implications as a new kind of statements in context.

In the traditional lazy approach all bindings are implicitly given by inclusion maps. So, if $I\to (P,\Pi )\Rightarrow (K,\Delta )$ is an element of $\mathrm{\Phi }$ for an enriched $\Sigma$-specification $(G,\mathrm{\Phi })$, we must have $I\subseteq G$ and, analogously to definition of the semantics of $\Sigma$-expressions in (12), we define for a $\Sigma$-structure $\mathcal{U}$ the semantics of the open $\Sigma$-specification implications $I\to (P,\Pi )\Rightarrow (K,\Delta )$ in context G by

$${\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_G^{\mathcal{U}}{:=\downarrow }_j^U{\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$$

(42)

for the inclusion map $j:I\to G$. The semantics of an enriched $\Sigma$-specification $(G,\mathrm{\Phi })$ is defined analogously to (28) thus we also obtain a corresponding variant of Corollary 6 for enriched $\Sigma$-specifications.

To approve our proposal two things should be at place. First, we must clarify what role open $\Sigma$-specification implications in enriched $\Sigma$-specifications play in deduction. Second, it is necessary to demonstrate that open $\Sigma$-specification implications enable us indeed to formally grasp the results of hypothetical reasoning.

What is our view on a single process of deduction? We start with a set of hypotheses given by an ordinary $\Sigma$-specification $(X,\Gamma )$. Than we successively extend this initial $\Sigma$-specification by applying the available deduction rules in a strict way. Following our proposal, we will also generate enriched $\Sigma$-specifications. In such a way, each stage of an actual deduction process gives rise to a strict $\Sigma$-specification implication $(X,\Gamma )@3->{}_{\left[r\right]}(G,\mathrm{\Phi })$ from the initial ordinary $\Sigma$-specification $(X,\Gamma )$ to the present enriched $\Sigma$-specification $(G,\mathrm{\Phi })$. Note, that $(G,\mathrm{\Phi })$ especially contains all the "hypotheses in which x is free" mentioned in the condition for the $\mathcal{N}\mathcal{D}$-version of rule $\forall I$ in (49)!

An open $\Sigma$-specification implication $I\to (P,\Pi )\Rightarrow (K,\Delta )$ in an enriched $\Sigma$-specification $(G,\mathrm{\Phi })$ is nothing but a deduction rule in waiting position locally bound to the context G by the inclusion map $j:I\to G$. Once we are able to extend the binding $j:I\to G$ to a match $\mu :(P,\Pi )⤏(G,\mathrm{\Phi })$ for a map $\mu :P\to G$ with $j=i;\mu$, we can apply the $\Sigma$-specification implication $(P,\Pi )⟹(K,\Delta )$ as a strict deduction rule as described in Section 6.4 (compare especially Remark 24).

We shortly discuss the soundness of such a deduction step. For any $\Sigma$-structure $\mathcal{U}$ we do have ${\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\subseteq {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_G^{\mathcal{U}}{=\downarrow }_j^U{\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$ since $I\to (P,\Pi )\Rightarrow (K,\Delta )$ is an element of $\mathrm{\Phi }$. Due to Proposition 3 and Definition 22, this ensures $\mu ;\iota \in {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$ for any $\iota \in {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$. Slightly varying the proof of Proposition 4, we can show, in such a way, that $\mathcal{U}\vDash (G,\mathrm{\Phi })@3->{}_{\left[r\right]}(R,\mathrm{\Psi })$ .

8.2. Licensing Hypothetical Reasoning

We evidence that open $\Sigma$-specification implications are indeed an appropriate tool to formally describe and finally license hypothetical reasoning. Especially, we demonstrate that the effect of hypothetical reasoning can be described by adding locally bound open $\Sigma$-specification implications to the present enriched $\Sigma$-specification $(G,\mathrm{\Phi })$ representing the actual status of all the things that have been deduced and/or introduced during a concrete deduction process.

How to transform a semi-formal notation like "${\left[A\right]}^1\dots B$" into an open $\Sigma$-specification implication $I\to (P,\Pi )\Rightarrow (K,\Delta )$? "A" is transformed into $\Pi =\left\{A\right\}$ and "B" into $\Delta =\left\{B\right\}$. The appropriate choice of the sets I, P, K of variables depends on the concrete situation but, at least, we should have the following inclusions $I\subseteq G$, $fv\left(A\right)\subseteq P$, $fv\left(B\right)\subseteq K$ in addition to the inclusions $I\subseteq P\subseteq K$. The inclusion map $j:I\to G$ describes to what part of G the open $\Sigma$-specification implication will be bound. The result of a sound hypothetical reasoning step can be only grasp by adding $I\to (P,\Pi )\Rightarrow (K,\Delta )$ to $\mathrm{\Phi }$ if for all $\Sigma$-structures $\mathcal{U}$ the following soundness condition is satisfied:

$${\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\subseteq {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_G^{\mathcal{U}}.$$

(44)

What means the step from "A" to "${\left[A\right]}^1$", i.e., the introduction of A as a fresh hypothesis? In LoSiC-terms it means to apply, in a strict manner, the $\Sigma$-specification implication $(I,\varnothing )⟹(P,\Pi )$ via the trivial match $j:(I,\varnothing )⤏(G,\mathrm{\Phi })$ as described in Section 6.4 (see the right diagram in (45)). In other words, we add the hypothesis $j^{*}\left(\Pi \right)$ to $\mathrm{\Phi }$! Keep in mind, that $R:=G\cup CP$ and that the underlying pushout of maps in the left diagram in (45) replaces the variables in $P\setminus I$ by entirely fresh variables in $CP$, referred to as new Skolem constants in [32], via a bijective map $is:P\setminus I\to CP$.

In some texts like [35], using the Fitch style to present proofs by natural deduction, the step from "A" to "${\left[A\right]}^1$" is accompanied by writing a note like "want B". Correspondingly, we may mimic this step by also drawing the arrow "$⟹(K,\Delta )$" in (45).

In the next modus "${\left[A\right]}^1\dots$", we extend $(R,\mathrm{\Phi }\cup j^{*}\left(\Pi \right))$ by successively applying, in a strict manner, sound general deduction rules or locally bound deduction rules in waiting position from $\mathrm{\Phi }$. In such a way, we have $\mathcal{U}\vDash (R,\mathrm{\Phi }\cup j^{*}\left(\Pi \right))@3->{}_{\left[r\right]}(H,\mathrm{\Psi })$ at any stage $(H,\mathrm{\Psi })$ of extension. The objective is to finally deduce an enriched $\Sigma$-specification $(H,\mathrm{\Psi })$ such that there exists a match $\tau :(K,\Delta )⤏(H,\mathrm{\Psi })$ with $in;\tau =j^{*};i{n}^{*}$ for the underlying maps. Note, that such a match $\tau :(K,\Delta )⤏(H,\mathrm{\Psi })$ provides for all the implicitly existentially quantified variables in $K\setminus P$ corresponding witnesses in H!

In case of success, we reach the stage "${\left[A\right]}^1\dots B$" and the "auxiliary deduction" (called "subproof" in [35]) has served its purpose. What is called discharging the assumptions of that subproof in [35], p. 108, can be formalized by adding $I\to (P,\Pi )\Rightarrow (K,\Delta )$ to $\mathrm{\Phi }$ and replacing the right diagram in (45) by the strict $\Sigma$-specification implication

$$(G,\mathrm{\Phi })@3->{}_{\left[r\right]}(G,\mathrm{\Phi }\cup \{I\to (P,\Pi )\Rightarrow (K,\Delta )\left\}\right).$$

(46)

This final step of discharging is semantically sound if, and only if, the condition in (44) is satisfied. The following proposition shows that this is indeed the case.

Proposition 5

(Hypothetical Reasoning: Soundness). Let the Σ-specification implication $(R,\mathrm{\Phi }\cup j^{*}\left(\Pi \right))@3->{}_{\left[r\right]}(H,\mathrm{\Psi })$ in (45) be strict and valid in the Σ-structure $\mathcal{U}$, i.e., we assume ${\exists }_{i{n}^{*}}^U{\left[\left[(H,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}={\left[\left[(R,\mathrm{\Phi }\cup j^{*}\left(\Pi \right))\right]\right]}^{\mathcal{U}}$. Then we do have for any match $\tau :(K,\Delta )⤏(H,\mathrm{\Psi })$, with $in;\tau =j^{*};i{n}^{*}$ for the underlying maps, that ${\exists }_j^U{\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\subseteq {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$, i.e., $j;\iota \in {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$ for all $\iota \in {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$.

Proof. 

We assume $\iota \in {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$. If ${\downarrow }_i^U\left(\{j;\iota \}\right)=\varnothing$, we trivially have, due to Definition 22, that $j;\iota \in {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$. If ${\downarrow }_i^U\left(\{j;\iota \}\right)\ne \varnothing$, let ${\beta \in \downarrow }_i^U\left(\{j;\iota \}\right)\cap {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}\subseteq U^P$ be arbitrary thus we especially have $i;\beta =j;\iota$. Due to the pushout in (45), there is a unique $\kappa :R\to U$ with $i^{*};\kappa =\iota$ and $j^{*};\kappa =\beta$. The assumptions $\iota \in {\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$, $\beta \in {\left[\left[(P,\Pi )\right]\right]}^{\mathcal{U}}$ entail $\kappa \in {\left[\left[(R,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}$, $\kappa \in {\left[\left[(R,j^{*}\left(\Pi \right))\right]\right]}^{\mathcal{U}}$ and thus $\kappa \in {\left[\left[(R,\mathrm{\Phi }\cup j^{*}\left(\Pi \right))\right]\right]}^{\mathcal{U}}$.

Since ${\exists }_{i{n}^{*}}^U{\left[\left[(H,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}={\left[\left[(R,\mathrm{\Phi }\cup j^{*}\left(\Pi \right))\right]\right]}^{\mathcal{U}}$, there exists a $\vartheta :H\to U$ with $i{n}^{*};\vartheta =\kappa$ and $\vartheta \in {\left[\left[(H,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}$ while the assumption ${\exists }_{\tau }^U{\left[\left[(H,\mathrm{\Psi })\right]\right]}^{\mathcal{U}}\subseteq {\left[\left[(K,\Delta )\right]\right]}^{\mathcal{U}}$ ensures $\tau ;\vartheta \in {\left[\left[(K,\Delta )\right]\right]}^{\mathcal{U}}$. This finally proves $\beta \in {\exists }_{in}^U{\left[\left[(K,\Delta )\right]\right]}^{\mathcal{U}}$, as required by Definition 22, since $\beta =j^{*};\kappa =j^{*};i{n}^{*};\vartheta =in;(\tau ;\vartheta )$. □

Due to Corollary A4 and (42), the condition ${\exists }_j^U{\left[\left[(G,\mathrm{\Phi })\right]\right]}^{\mathcal{U}}\subseteq {\left[[I\to (P,\Pi )\Rightarrow (K,\Delta )]\right]}_I^{\mathcal{U}}$ is equivalent to the soundness condition in (44) thus we indeed have been able to license hypothetical reasoning.

8.3. Validating Hypothetical Reasoning Rules

Backed up by Proposition 5, each of the rules $\to I^1$, $\neg I^1$, $\vee E^1$ in (40) can be transformed into a semantically equivalent implication from an enriched $\Sigma$-specification into an ordinary $\Sigma$-specification given by a single $\Sigma$-expression. In such a way, the validation of the three rules can be done by validating these corresponding enriched essential implications.

For rule $\to {\mathbf{I}}^{\mathbf{1}}$ the diagrams in (45) specialize with $I=fv\left(A\right)\cup fv\left(B\right)$ to

thus the rule transforms into the implication $(I,I\to (I,A)\Rightarrow (I,B\left)\right)⟹(I,A\to B)$. By Corollary 9.3 we do have ${\left[[I\to (I,A)\Rightarrow (I,B)]\right]}_I^{\mathcal{U}}={\left[\left[(I,A\to B)\right]\right]}^{\mathcal{U}}$ for all $\Sigma$-structures $\mathcal{U}$. This proves the soundness of the rule!

For rule $\neg {\mathbf{I}}^{\mathbf{1}}$ we need two independent auxiliary deductions as in (47), that can be performed in any order (or maybe even in parallel), resulting in an enriched $\Sigma$-specification $(I,\Omega )$ with $I=fv\left(A\right)\cup fv\left(B\right)$ and $\Omega =\{I\to (I,A)\Rightarrow (I,B),I\to (I,A)\Rightarrow (I,\neg B\left)\right\}$.

In such a way, the rule transforms into the implication $(I,\Omega )⟹(I,\neg A)$. Due to Corollary 9.1 and the semantics of negation we obtain by some Boolean Algebra reasoning ${\left[\left[(I,\Omega )\right]\right]}^{\mathcal{U}}=\left(\overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}\cup {\left[\left[B\right]\right]}_I^{\mathcal{U}}\right)\cap \left(\overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}\cup \overline{{\left[\left[B\right]\right]}_I^{\mathcal{U}}}\right)=\overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}\cup \overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}\cap \overline{{\left[\left[B\right]\right]}_I^{\mathcal{U}}}\cup {\left[\left[B\right]\right]}_I^{\mathcal{U}}\cap \overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}=\overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}={\left[[\neg A]\right]}_I^{\mathcal{U}}$ for all $\Sigma$-structures $\mathcal{U}$ thus the rule is validated.

By two independent auxiliary deductions as in (47) rule $\vee {\mathbf{E}}^{\mathbf{1}}$ transforms into the implication $(I,\Omega )⟹(I,C)$ with $\Omega =\{A\vee B,I\to (I,A)\Rightarrow (I,C),I\to (I,B)\Rightarrow (I,C\left)\right\}$ where $I=fv\left(A\right)\cup fv\left(B\right)\cup fv\left(C\right)$. Due to Corollary 9.1, we obtain by some Boolean Algebra reasoning for any $\Sigma$-structure $\mathcal{U}$

$$\begin{array}{ccc}\hfill {\left[\left[(I,\Omega )\right]\right]}^{\mathcal{U}}& =& \left({\left[\left[A\right]\right]}_I^{\mathcal{U}}\cup {\left[\left[B\right]\right]}_I^{\mathcal{U}}\right)\cap \left(\overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}\cup {\left[\left[C\right]\right]}_I^{\mathcal{U}}\right)\cap \left(\overline{{\left[\left[B\right]\right]}_I^{\mathcal{U}}}\cup {\left[\left[C\right]\right]}_I^{\mathcal{U}}\right)\hfill \\ & =& \left({\left[\left[A\right]\right]}_I^{\mathcal{U}}\cap {\left[\left[C\right]\right]}_I^{\mathcal{U}}\cap \overline{{\left[\left[B\right]\right]}_I^{\mathcal{U}}}\right)\cup \left({\left[\left[A\right]\right]}_I^{\mathcal{U}}\cap {\left[\left[C\right]\right]}_I^{\mathcal{U}}\right)\hfill \\ & & \cup \left({\left[\left[B\right]\right]}_I^{\mathcal{U}}\cap \overline{{\left[\left[A\right]\right]}_I^{\mathcal{U}}}\cap {\left[\left[C\right]\right]}_I^{\mathcal{U}}\right)\cup \left({\left[\left[B\right]\right]}_I^{\mathcal{U}}\cap {\left[\left[C\right]\right]}_I^{\mathcal{U}}\right)\hfill & \subseteq & {\left[\left[C\right]\right]}_I^{\mathcal{U}}\hfill \end{array}$$

and thus $\mathcal{U}\vDash (I,\Omega )⟹(I,C)$ as required.

Next we analyze the quantifier rule $\exists {\mathbf{E}}^{\mathbf{1}}$ in System $\mathcal{N}\mathcal{D}$ as presented in [32]

Instead of new Skolem constant also the term proxy is used in the literature (see [35], p. 119).

In this case, we get a non-trivial instance of (45). We set $I=fv(\exists xA)$, $P=I\cup \left\{x\right\}$ and $K=P\cup fv\left(B\right)$. As already seen in (39), the new Skolem constant c can be utilized to construct the necessary pushout of inclusion maps, i.e., we can set and $R:=G\cup \left\{c\right\}$.

A corresponding auxiliary deduction transforms the rule $\exists E^1$ into the implication $(I,\Omega )⟹(K,B)$ with $\Omega =\{\exists xA,I\to (P,A)\Rightarrow (K,B\left)\right\}$. We show $\mathcal{U}\vDash (I,\Omega )⟹(K,B)$, i.e., ${\left[\left[(I,\Omega )\right]\right]}^{\mathcal{U}}\subseteq {\exists }_i^U\left({\exists }_{in}^U{\left[\left[B\right]\right]}_K^{\mathcal{U}}\right)$, for any $\Sigma$-structure $\mathcal{U}$. Let be $\mathbf{a}\in {\left[\left[(I,\Omega )\right]\right]}^{\mathcal{U}}={\left[[\exists xA]\right]}_I^{\mathcal{U}}\cap {\left[[I\to (P,A)\Rightarrow (K,B)]\right]}_I^{\mathcal{U}}$. We have ${\left[[\exists xA]\right]}_I^{\mathcal{U}}={\exists }_i^U{\left[\left[A\right]\right]}_P^{\mathcal{U}}$ thus $\mathbf{a}\in {\left[[\exists xA]\right]}_I^{\mathcal{U}}$ implies that there exists a $\mathbf{b}\in {\left[\left[A\right]\right]}_P^{\mathcal{U}}$ with $i;\mathbf{b}=\mathbf{a}$ and, in such a way, also ${\mathbf{b}\in \downarrow }_i^U\left(\left\{\mathbf{a}\right\}\right)\cap {\left[\left[A\right]\right]}_P^{\mathcal{U}}$. On the other hand, $\mathbf{a}\in {\left[[I\to (P,A)\Rightarrow (K,B)]\right]}_I^{\mathcal{U}}$ means, according to Definition 22, that ${\downarrow }_i^U\left(\left\{\mathbf{a}\right\}\right)\cap {\left[\left[A\right]\right]}_P^{\mathcal{U}}\subseteq {\exists }_{in}^U{\left[\left[B\right]\right]}_K^{\mathcal{U}}$. Therefore, there exists a $\mathbf{c}\in {\left[\left[B\right]\right]}_K^{\mathcal{U}}$ with $in;\mathbf{c}=\mathbf{b}$. This finally proves $\mathbf{a}=i;\mathbf{b}=i;(in;\mathbf{c})\in {\exists }_i^U\left({\exists }_{in}^U{\left[\left[B\right]\right]}_K^{\mathcal{U}}\right)$ as required.

Attention! Due to the commutativity requirement $in;\tau =j^{*};i{n}^{*}$, the match $\tau$ translates the variable x into the Skolem constant c! We do not know if it is a flaw of the LoSiC-approach, that we actually require "${\left[A_c^x\right]}^1\dots B_c^x$", or if this is simply overseen in [32]? In [35], p. 119, it is explicitly required that "the constant c must not appear in $\exists xA$, in B, or in any undischarged assumption". So, at least compared to [35], the LoSiC-approach to hypothetical reasoning seems to be a bit more flexible since we allow that c appears in B. Keep in mind, that we can replace $\exists xA$ by the semantically equivalent formula $\exists z{A}_z^x$ as we did it for the validation of Axiom A4 in System $\mathcal{H}$, for example.

To deliver on our promise from Section 7.2, we finally validate rule $\forall \mathbf{I}$.

* means that A does not depend on any hypothesis in which x is free.

The informal version Intuition of rule $\forall I$ indicates how we can validate rule $\forall I$ by means of Proposition 5. For the $\mathcal{L}\mathcal{J}$-version of the rule we set $Z=fv\left(\Gamma \right)\cup fv\left(A\right)\cup \left\{x\right\}$ and $Y=Z\setminus \left\{x\right\}$. Condition $x\notin fv\left(\Gamma \right)$ ensures that as well $(Z,\Gamma )$ as $(Y,\Gamma )$ are $\Sigma$-specifications where we have ${\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}{=\downarrow }_{i^{*}}^U{\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}$, due to Corollary 6, for the inclusion map $i^{*}:Y\to Z$ and any $\Sigma$-structure $\mathcal{U}$. We show that $\mathcal{U}\vDash (Z,\Gamma )⟹(Z,A)$ implies $\mathcal{U}\vDash (Y,\Gamma )⟹(Y,\forall xA)$.

Let us assume $\mathcal{U}\vDash (Z,\Gamma )⟹(Z,A)$, i.e., ${\left[\left[\Gamma \right]\right]}_Z^{\mathcal{U}}={\left[[\Gamma \cup \left\{A\right\}]\right]}_Z^{\mathcal{U}}$. This gives rise to the following instance of (45) with $P=fv\left(A\right)\cup \left\{x\right\}$, $I=P\setminus \left\{x\right\}$

Proposition 5 entails ${\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\subseteq {\left[[I\to (P,\varnothing )\Rightarrow (P,A)]\right]}_Y^{\mathcal{U}}$ thus we get for any $\mathbf{a}\in {\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}$ also $\mathbf{a}\in {\left[[I\to (P,\varnothing )\Rightarrow (P,A)]\right]}_Y^{\mathcal{U}}{=\downarrow }_j^U{\left[[I\to (P,\varnothing )\Rightarrow (P,A)]\right]}_I^{\mathcal{U}}$ and, in such a way, $j;\mathbf{a}\in {\left[[I\to (P,\varnothing )\Rightarrow (P,A)]\right]}_I^{\mathcal{U}}$. According to Definition 22 this is equivalent to ${\downarrow }_i^U\{j;\mathbf{a}\}\subseteq {\left[\left[A\right]\right]}_P^{\mathcal{U}}$. This gives us the inclusion ${\downarrow }_i^U\left((j;\_){\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\right){=\downarrow }_i^U\left({\exists }_j^U{\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\right)\subseteq {\left[\left[A\right]\right]}_P^{\mathcal{U}}$.

Applying ${\forall }_i^U$ to both sides of the inclusion as well as utilizing (29) and the semantics of universal quantification we arrive at ${\exists }_j^U{\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}={\forall }_i^U\left({\downarrow }_i^U\left({\exists }_j^U{\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\right)\right)\subseteq {\forall }_i^U{\left[\left[A\right]\right]}_P^{\mathcal{U}}={\left[[\forall xA]\right]}_I^{\mathcal{U}}$. This finally entails ${\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\subseteq {{\downarrow }_j^U\left({\exists }_j^U{\left[\left[\Gamma \right]\right]}_Y^{\mathcal{U}}\right)\subseteq \downarrow }_j^U\left({\left[[\forall xA]\right]}_I^{\mathcal{U}}\right)={\left[[\forall xA]\right]}_Y^{\mathcal{U}}$, as required, due to Corollary A4 and Lemma 3.

The $\mathcal{N}\mathcal{D}$-version of rule $\forall I$ can be analogously validated by simply replacing $(Z,\Gamma )$ by $(G,\mathrm{\Phi })$. While $\Gamma$ is supposed to only contain $\Sigma$-expressions (formulas), we allow that $\mathrm{\Phi }$ also contains open specification implications and that those locally bound deduction rules in waiting position can be applied, in the spirit of (43), to arrive at $(G,\mathrm{\Phi }\cup \{A\left\}\right)$!

8.4. Nested Hypothetical Reasoning

To illustrate the LoSiC-approach to deduction, we discuss the deduction of the $\Sigma$-specification implication $(X,\varnothing )⟹(X,A\to (B\to A\left)\right)$ with $X:=fv\left(A\right)\cup fv\left(B\right)$ (compare axiom A1 in Section 7.1). We will go step by step through the corresponding proof in System $\mathcal{N}\mathcal{D}$ visualized by a proof tree and in Fetch notation as follows:

First, we introduce the two hypotheses and reach in System $\mathcal{N}\mathcal{D}$ the stage

Also representing the two want-phrases, we obtain in the LoSiC-approach the diagram

In the next stage

we realize that we do have the "wanted A" available and get the diagram

Closing the subproof and discharging the assumption ${\left[B\right]}^2$ gives us in System $\mathcal{N}\mathcal{D}$

In the LoSiC-approach, we close the subproof and discharge the assumption ${\left[B\right]}^2$ by turning $(X,\{A,B\left\}\right)$ into the enriched $\Sigma$-specification $(X,\{A,X\to (X,B)\Rightarrow (X,A)\left\}\right)$. Due to Proposition 5, we proved, in such a way, the enriched $\Sigma$-specification implication $(X,A)⟹(X,\{A,X\to (X,B)\Rightarrow (X,A)\left\}\right)$. Fortunately, Corollary 9 allows us to use instead the equivalent pure $\Sigma$-specification implication $(X,A)⟹(X,\{A,B\to A\left\}\right)$.

Finally, we do have the "wanted $B\to A$". We can close the whole proof and also discharge the assumption ${\left[A\right]}^1$. In view of LoSiCs, we deduce the enriched $\Sigma$-specification $(X,\{A,X\to (X,A)\Rightarrow (X,B\to A)\left\}\right)$ under the assumption $(X,\varnothing )$, thus we have finally proved the pure $\Sigma$-specification implication $(X,\varnothing )⟹(X,A\to (B\to A\left)\right)$ according to Proposition 5 and Corollary 9.

9. Conclusions and Further Work

We accomplished the first of the projects envisaged in the concluding section of [17]. We developed a conservative extension of the traditional institution of unsorted first-order logic to an unsorted first-order Logic of Statements in Context, comprising not only "closed" but also "open formulas". A crucial novelty, compared to [17], is that we in detail worked out an appropriate variant of a hom-set based semantic calculus of quantification and consequently used this calculus in our analysis of deduction in first-order logic.

We showed that the traditional presentation of first-order logic, where neither variable declarations nor contexts are explicitly considered, is equivalent to a first-order Logic of Statements in Context where all bindings of formulas to contexts and all morphisms between contexts are restricted to (implicit) inclusion maps.

By a kind of "methodological accident" this equivalence reinvents, however, Barendregt’s variable convention [18]. So, from a more technical point of view, we had to find out how to integrate Barendregt’s variable convention into traditional first-order logic in a syntactic and semantical correct way. It is maybe worth to mention that Barendregt’s variable convention allowed us to introduce renaming substitutions executing substitutions of free variables by terms and renamings of bound variables in parallel.

Relying on this preparatory efforts, we succeeded in exhaustively analyzing, formalizing and validating three of the basic kinds of deduction calculi for traditional first-order logic by means of the LoSiC-concepts context and specification implication, namely a Hilbert System $\mathcal{H}$ and a Gentzen’s system $\mathcal{L}\mathcal{J}$ and a Natural Deduction System $\mathcal{N}\mathcal{D}$.

The crucial innovation of the paper is the introduction of open specification implications and the related idea of deduction rules in waiting position. This innovation enabled us to syntactically represent the outcome of hypothetical proofs in a precise formal way and independent of the concrete hypothetical proof. In such a way, we have been able to finally license hypothetical reasoning in a proper syntactic and a semantical sound way.

To extend, consolidate and further develop the open framework of Logics of Statements in Context, we propose four projects:

Open Injectivity

Following [30,31], we decided to replace the term "Horn Logics", used in [17], by open injectivity. Open specification implications, introduced in this paper, are a first example of open injectivity. Obviously one could iterate the definition of open specification implications. We are, however, not sure that there really is a reason and/or need to do this at the moment. On the other side, nested open injectivity can only unfold its full potential in the presence of explicit variable declarations and contexts. This full potential is needed to describe and formalize reasoning in Category Theory, for example.

Since Category Theory relies on universal properties, there is no need for first-order formulas. Instead, one can use nested open sketch implications based on relational atoms to formalize Category Theory (compare [10]). Therefore, we decided to elaborate in the next paper a full account of this kind of nested open injectivity for arbitrary Logics of Statements in Context with Category Theory as running example. Note, that the habit of "want phrases" in the Fitch notation for Natural Deduction is strongly related to the heuristics of "diagram chasing" in Category Theory!

LoSiC-Deduction in the Unsorted Case

Writing this paper, we finally realized that our initial idea to simply borrow deduction calculi from traditional first-order logic was by far too naive. We also learned that substitutions should be used as morphisms between contexts. To also have in this case Institutions of Statements in Context at hand, we have, however, also to allow substitutions as bindings! What has to be done is to upgrade the Institutions ${\mathcal{FL}}_{\Sigma }$ of Statements in Context in the present paper to institutions with substitutions as bindings as well as context morphisms. Based on this, the necessary bigger effort should be spend to adapt, at least, one of the traditional deduction calculi to the corresponding upgraded first-order Logics of Statements in Context. We are convinced that Gentzen’s system $\mathcal{L}\mathcal{J}$ and its twin Natural Deduction are reasonable candidates to get this job done.

Traditional Deduction in the Many-Sorted Case

Another bigger project would be to extend the ideas, constructions and findings of the present paper to traditional many-sorted first-order logic. The remarks concerning the many-sorted case, scattered over the paper, together with [17] should establish a reasonable starting point to attack such a project. In a nutshell, the crucial problem with many-sortedness is that we can not get rid of variables in $X\setminus fv\left(\Gamma \right)$ for a $\Sigma$-specification $(X,\Gamma )$ by simply deleting them! That is, assertions as in (29) and Corollary 8, for example, vanish in the many-sorted case. The only chance to get rid of a variable $x\in (X\setminus fv(\Gamma \left)\right)$, in a sound way, is to find a $\Sigma$-term t over $X\setminus \left\{x\right\}$ (compare rule "Concretion" in [36]). Note, that the concept of context enables us to address and potentially deal with this issue without utilizing the crutch of universal quantification as it is done in [36].

Gentzen’s System $\mathcal{L}\mathcal{K}$

Finally, we could extend LoSiCs by disjunctive sketches and validate Gentzen’s System $\mathcal{L}\mathcal{K}$[34] and its variant System $\mathcal{G}$ in [4] by means of Boolean Algebra reasoning analogously to System $\mathcal{L}\mathcal{J}$ as discussed in Remark (26).

We will retire in autumn 2026 but will continue with our research. Everyone is invited to work on the last three projects - alone or in cooperation with us.

Appendix A.1. Shift, Existential and Universal Operators for Maps

For any set A the powerset construction gives us a partial order $(\wp (A),\subseteq )$ and a respective Boolean Algebra $(\wp \left(A\right),\cap ,\cup ,\overline{0ex1.5ex\_0ex1.5ex},A,\varnothing )$ at hand, where $\overline{M}:=A\setminus M$ is the compliment of $M\in \wp \left(A\right)$. It is quite common to call a subset $M\subseteq A$ a "property" of elements in A or a "predicate" for A. Keep in mind that any partial order can equivalently be seen as a partial order category, i.e., a category with at most one morphism between two objects and with identities the only isomorphisms.

For any map $f:A\to B$ the formation of pre-images

$$f^{-1}\left(N\right):=\{a\in A\mid f\left(a\right)\in N\}=\bigcup \{f^{-1}\left(b\right)\mid b\in N\}forallN\subseteq B$$

(A1)

defines a monotone function (a functor) from $(\wp (B),\subseteq )$ into $(\wp (A),\subseteq )$

$${\downarrow }_f:=f^{-1}:(\wp \left(B\right),\subseteq )\to (\wp \left(A\right),\subseteq ).$$

(A2)

For historical reasons [38,39,40] we call this function the shift operator given by f. The reader may think about a "shift of perspective" from B to A by returning to A along $f:A\to B$.

Dually, the formation of images defines for any map $f:A\to B$ a monotone function (a functor) from $(\wp (A),\subseteq )$ into $(\wp (B),\subseteq )$ defined for all $M\in \wp \left(A\right)$ by

$$f\left(M\right):=\left\{f\right(a)\mid a\in M\}=\{b\in B\mid existsa\in A:f(a)=banda\in M\}.$$

(A3)

As one can observe, the formation of images is related to existential quantification in the sense that $f\left(M\right)$ contains exactly all those elements from B for which there exists a pre-image w.r.t. f ( a "witness/antecedent") in M. Therefore, we call this function the existential operator given by f and adapt from [38,39] the notation

$${\exists }_f:=f:(\wp \left(A\right),\subseteq )\to (\wp \left(B\right),\subseteq ).$$

(A4)

The first crucial observation is that the operators ${\exists }_f$ and ${\downarrow }_f$ establish a monotone Galois Connection (an adjunction) between the partial orders (categories) $(\wp (A),\subseteq )$ and $(\wp (B),\subseteq )$.

Corollary A1

(Existential Galois Connection 1). For any $M\in \wp \left(A\right)$ and $N\in \wp \left(B\right)$ we have

unit:

${M\subseteq \downarrow }_f\left({\exists }_f\left(M\right)\right)$

counit:

${\exists }_f\left({\downarrow }_f\left(N\right)\right)\subseteq N$

adjunction:

${M\subseteq \downarrow }_f\left(N\right)$ iff ${\exists }_f\left(M\right)\subseteq N$

Note, that ${\exists }_f$ is called "left" adjoint to ${\downarrow }_f$ (and, correspondingly, ${\downarrow }_f$ "right" adjoint to ${\exists }_f$), ${\exists }_f{⊣\downarrow }_f$ in symbols, since ${\exists }_f$ appears in the adjunction statement on left of the "arrow" ⊆ while ${\downarrow }_f$ appears on the right!

For all $N\in \wp \left(B\right)$ we do have $N\setminus {\exists }_f\left({\downarrow }_f\left(N\right)\right)=N\setminus f\left(A\right)$. $f:A\to B$ is surjective if, and only if, ${\exists }_f\left(A\right)=f\left(A\right)=B$ thus $f:A\to B$ surjective implies ${\exists }_f\left({\downarrow }_f\left(N\right)\right)=N$ for all $N\in \wp \left(B\right)$. The other way around, if $B=\varnothing$ there exists only a map $f:A\to B$ if also $A=\varnothing$ thus we have $f=i{d}_{\varnothing }$ which is trivially surjective. If $B\ne \varnothing$, the condition ${\exists }_f\left({\downarrow }_f\left(N\right)\right)=N$ implies for all singletons $N=\left\{b\right\}\in \wp \left(B\right)$ that $b\in f\left(A\right)$, thus we get the following characterization of surjectivity:

$${f:A\to Bsurjectiveiff\downarrow }_f;{\exists }_f=i{d}_{\wp \left(B\right)}.$$

(A5)

Obviously, $f:A\to B$ is injective if, and only if, ${\downarrow }_f\left(\left\{b\right\}\right)=f^{-1}\left(b\right)$ is either a singleton or empty for all $b\in B$. ${M=\downarrow }_f\left({\exists }_f\left(M\right)\right)$ means that $M=\bigcup \{{\downarrow }_f\left(\left\{b\right\}\right)\mid b\in {\exists }_f\left(M\right)\}=\bigcup \{f^{-1}\left(b\right)\mid b\in f\left(M\right)\}$. This is the case for all $M\in \wp \left(A\right)$ if $f:A\to B$ is injective thus we have the following implication:

$$f:A\to Binjectiveimpliesi{d}_{\wp \left(A\right)}={\exists }_f{;\downarrow }_f.$$

(A6)

The second, more striking observation is that universal quantification is dual to existential quantification in a rigorous categorical sense. Universal quantification turns out to be nothing but the right adjoint to the shift operator.

For any map $f:A\to B$ we can define a monotone function (a functor)

$${\forall }_f:(\wp \left(A\right),\subseteq )\to (\wp \left(B\right),\subseteq )$$

(A7)

called the the universal operator given by f and defined for all $M\in \wp \left(A\right)$ by

$$\begin{array}{ccc}\hfill {\forall }_f\left(M\right)& :=& \{b\in B\mid foralla\in A:f(a)=bimpliesa\in M\}\hfill \\ \hfill & =& \{b\in B\mid {\downarrow }_f\left(b\right)=f^{-1}\left(b\right)\subseteq M\}.\hfill \end{array}$$

(A8)

${\forall }_f$ formalizes universal quantification in the sense that ${\forall }_f\left(M\right)$ contains exactly all those elements from B for which all (!) pre-images w.r.t. f are in M. Be aware, that we always have $\overline{{\exists }_f\left(A\right)}=\overline{f\left(A\right)}={\forall }_f(\varnothing )\subseteq {\forall }_f\left(M\right)$ since ${\downarrow }_f\left(b\right)=f^{-1}\left(b\right)=\varnothing$ for all $b\in \overline{f\left(A\right)}$.

We get indeed a monotone Galois Connection (an adjunction) between the partial orders (categories) $(\wp (B),\subseteq )$ and $(\wp (A),\subseteq )$.

Corollary A2

(Universal Galois Connection 1). For any $N\in \wp \left(B\right)$ and $M\in \wp \left(A\right)$ we have

unit:

$N\subseteq {\forall }_f\left({\downarrow }_f\left(N\right)\right)$

counit:

${\downarrow }_f\left({\forall }_f\left(M\right)\right)\subseteq M$

adjunction:

$N\subseteq {\forall }_f\left(M\right)$ iff ${\downarrow }_f\left(N\right)\subseteq M$

The interplay of the Universal and the Existential Galois Connection gives us a simple, but useful, Corollary at hand.

Corollary A3

(Shift vs Existential and Universal). For any map $f:A\to B$ and all $M\in \wp \left(A\right)$, $N\in \wp \left(B\right)$ it holds that $N\subseteq {\forall }_f\left(M\right)$ and ${\exists }_f\left(M\right)\subseteq N$ implies ${M=\downarrow }_f\left(N\right)$.

For all $N\in \wp \left(B\right)$ we do have ${\forall }_f\left({\downarrow }_f\left(N\right)\right)\setminus N=\overline{f\left(A\right)}\setminus N$ thus $f:A\to B$ surjective, i.e., $\overline{f\left(A\right)}=\varnothing$, implies $N={\forall }_f\left({\downarrow }_f\left(N\right)\right)$ for all $N\in \wp \left(B\right)$. The other way around, the condition $\varnothing ={\forall }_f\left({\downarrow }_f(\varnothing )\right)$ for $\varnothing \in \wp \left(B\right)$ implies $\overline{f\left(A\right)}=\varnothing$. In such a way, we get another characterization of surjectivity in terms of universal operators and, thus, an overlapping of existential and universal quantification due to (A5):

$$f:A\to B\mathrm{surjective}\mathrm{iff}i{d}_{\wp \left(B\right)}={\downarrow }_f;{\forall }_f\mathrm{iff}{\downarrow }_f;{\exists }_f=i{d}_{\wp \left(B\right)}.$$

(A9)

For all $M\in \wp \left(A\right)$ we do have ${\downarrow }_f\left({\forall }_f\left(M\right)\right)=M$ iff ${\exists }_f\left(M\right)=f\left(M\right)={\forall }_f\left(M\right)\setminus \overline{f\left(A\right)}$ iff ${\downarrow }_f\left({\exists }_f\left(M\right)\right)=M$. This means that we do have another overlapping of existential and universal quantification:

$$i{d}_{\wp \left(A\right)}={\exists }_f;{\downarrow }_{f}iff{\forall }_f;{\downarrow }_f=i{d}_{\wp \left(A\right)}.$$

(A10)

thus $f:A\to B$ injective also implies ${\forall }_f{;\downarrow }_f=i{d}_{\wp \left(A\right)}$ according to (A6).

Due to Corollary A2, the monotone function (the functor) ${\downarrow }_f$ is left adjoint to the monotone function (the functor) ${\forall }_f$, ${\downarrow }_f⊣{\forall }_f$ in symbols. We can summarize the basic findings of this section by the following diagram.

Fortunately, the construction of shift, existential and universal operators is compatible with composition. For any maps $f:A\to B$ and $g:B\to C$ we do have

$${\downarrow }_{f;g}={\downarrow }_g;{\downarrow }_f{\exists }_{f;g}={\exists }_f;{\exists }_g{\forall }_{f;g}={\forall }_f;{\forall }_g.$$

(A12)

The first two equations are ensured by the fact that both constructions - the formation of pre-images and the formation of images - define functors from $\mathsf{Set}$ into $\mathsf{Set}$, namely a contravariant and a covariant powerset functor, respectively. The third equation can also be shown straightforwardly.

We should mention that ${\forall }_f$ and ${\downarrow }_f$ as right adjoints distributes over arbitrary products, i.e., intersections, while ${\exists }_f$ and ${\downarrow }_f$ as left adjoints distribute over arbitrary sums, i.e. unions! Moreover, ${\downarrow }_f$ is compatible with respect to complements, i.e., for any map $f:A\to B$ we have

$${\downarrow }_f\left(\overline{N}\right)=\overline{{\downarrow }_f\left(N\right)}forallN\in \wp \left(B\right)$$

(A13)

since ${\downarrow }_f$ distributes over arbitrary unions, ${\downarrow }_f$ preserves disjointness and we consider only total maps.

Attention! As usual, the direction of "arrows" is a matter of tradition, taste, ideology and/or methodology. We work here with inclusions $A\subseteq B$ in accordance with the direction of the corresponding inclusion maps $i{n}_{A,B}:A\to B$. In [38,39] they also use inclusions ⊆ but denote them by ⊧ even if the direction of logic entailment would rather indicate ⊇! Note, that the sequence of adjunctions ${\exists }_f{⊣\downarrow }_f⊣{\forall }_f$ matches the well-known sequence of adjunctions $\Sigma ⊣\Delta ⊣\Pi ⊣\lambda$ characterizing the four basic constructions sum-copy-product-exponent in mathematics and category theory.

Appendix A.2. Shift, Existential and Universal Operators for Pre-composition Maps

This special kind of maps for arbitrary categories is addressed in [38,39]. Here we consider this special kind of maps for the category $\mathsf{Set}$.

To ease accessibility of the paper, we instantiate definitions, constructions and results in Appendix A.1 for hom-sets $U^K=\mathsf{Set}(K,U)$ and pre-composition maps $(\psi ;\_):U^G\to U^K$ induced by maps $\psi :K\to G$ and defined for any set U, by

$$(\psi ;\_)\left(\mathbf{b}\right):=\psi ;\mathbf{b}forall\mathbf{b}:G\to Uin{U}^G.$$

(A14)

In LoSiCs, U varies over all potential carriers $\left|\mathsf{Carr}\right|$ of structures, i.e., we assume U to be a non-empty set since $\left|\mathsf{Carr}\right|=\left|\mathsf{Set}\right|\setminus \{\varnothing \}$ in this paper!

As exponents we use sets $X\in Var$ of variables or contexts $K\in \left|\mathsf{Cxt}\right|=\left|\mathsf{Set}\right|$ while $Va{}^r\subset \left|\mathsf{Cxt}\right|$. Elements in $U^X$ are also called (variable) assignments and elements in $U^K$ interpretations. In our analysis of traditional first-order induction calculi, we mainly meet inclusion maps $i{n}_{X,Y}:X\to Y$ while $(i{n}_{X,Y};\_):U^Y\to U^X$ is usually callled a projection. Nevertheless, we consider in this section arbitrary maps $\psi :K\to G$.

In contrast to [38,39], we focus in LoSiCs on "syntactically represented" (hom-set) predicates. Any syntactic entity, like a $\Sigma$-expression, a $\Sigma$-statement in context or a finite $\Sigma$-sketch represents for each (!) $\Sigma$-structure $\mathcal{U}$ a corresponding predicate in $\mathcal{U}$. Different $\Sigma$-structures can, however, share the same carrier! So, in case we have to define (hom-set) predicates ${\mathrm{\Phi }}_K$ over K in the spirit of [38,39], we could consider $\left|\mathsf{Str}\right(\Sigma \left)\right|$-indexed families of subsets ${\mathrm{\Phi }}_K^{\mathcal{U}}\subseteq U^K$ instead of $\left|\mathsf{Set}\right|$-indexed families.

We adapt the notation in Appendix A.1 to the special case of pre-composition maps. First, we don’t use the identifiers of pre-composition maps $(\psi ;\_)$ but the identifiers of the underlying maps $\psi$ as subscripts for operators. Be aware that this reverses the direction of maps and thus the order of composition: For any maps $\psi :K\to G$, $\mu :G\to H$ and any carrier U we have the pre-composition maps $(\psi ;\_):U^G\to U^K$, $(\mu ;\_):U^H\to U^G$ where

$$(\mu ;\_);(\psi ;\_)=(\psi ;\mu ;\_):U^H\to U^K.$$

(A15)

Second, we use identifiers U of carriers as superscripts for operators.

For any map $\psi :K\to G$ and any carrier U the formation of pre-images for the corresponding pre-composition map $(\psi ;\_):U^G\to U^K$

$${(\psi ;\_)}^{-1}\left(\mathrm{\Phi }\right):=\{\mathbf{b}\in U^G\mid \psi ;\mathbf{b}\in \mathrm{\Phi }\}=\bigcup \{{(\psi ;\_)}^{-1}\left(\mathbf{a}\right)\mid \mathbf{a}\in \mathrm{\Phi }\}\mathrm{for}\mathrm{all}\mathrm{\Phi }\subseteq U^K$$

(A16)

defines a monotone function (a functor)

$${\downarrow }_{\psi }^U:={(\psi ;\_)}^{-1}:(\wp \left(U^K\right),\subseteq )\to (\wp \left(U^G\right),\subseteq ).$$

(A17)

which we call the shift operator for U induced by ψ. In case of an inclusion $X\subset Y$ of sets of variables, ${(i{n}_{X,Y};\_)}^{-1}\left(\mathbf{a}\right)$ is exactly the set of all extensions$\mathbf{b}:Y\to U$ of the variable assignment $\mathbf{a}:X\to U$ to Y.

Dually, the formation of images defines a monotone function (a functor)

$${\exists }_{\psi }^U:=(\psi ;\_):(\wp \left(U^G\right),\subseteq )\to (\wp \left(U^K\right),\subseteq )$$

(A18)

called the existential operator for U induced by ψ and defined for all $\mathrm{\Psi }\in \wp \left(U^G\right)$ by

$${\exists }_{\psi }^U\left(\mathrm{\Psi }\right):=\{\psi ;\mathbf{b}\mid \mathbf{b}\in \mathrm{\Psi }\}=\{\mathbf{a}\in U^K\mid exists\mathbf{b}\in U^G:\psi ;\mathbf{b}=\mathbf{a}and\mathbf{b}\in \mathrm{\Psi }\}.$$

(A19)

We instantiate the adjunction from Corollary A1 for pre-composition maps.

Corollary A4

(Existential Galois Connection 2). For any $\mathrm{\Psi }\in \wp \left(U^G\right)$ and $\mathrm{\Phi }\in \wp \left(U^K\right)$

unit:

$\mathrm{\Psi }{\subseteq \downarrow }_{\psi }^U\left({\exists }_{\psi }^U\left(\mathrm{\Psi }\right)\right)$

counit:

${\exists }_{\psi }^U\left({\downarrow }_{\psi }^U\left(\mathrm{\Phi }\right)\right)\subseteq \mathrm{\Phi }$

adjunction:

$\mathrm{\Psi }{\subseteq \downarrow }_{\psi }^U\left(\mathrm{\Phi }\right)$ iff ${\exists }_{\psi }^U\left(\mathrm{\Psi }\right)\subseteq \mathrm{\Phi }$

Note, that ${\exists }_{\psi }^U$ is called "left" adjoint to ${\downarrow }_{\psi }^U$ (and, correspondingly, ${\downarrow }_{\psi }^U$ "right" adjoint to ${\exists }_{\psi }^U$), ${\exists }_{\psi }^U{⊣\downarrow }_{\psi }^U$ in symbols, since ${\exists }_{\psi }^U$ appears in the adjunction statement on left of the "arrow" ⊆ while ${\downarrow }_{\psi }^U$ appears on the right!

For any map $\psi :K\to G$ and any carrier U we obtain a monotone function (a functor)

$${\forall }_{\psi }^U:(\wp \left(U^G\right),\subseteq )\to (\wp \left(U^K\right),\subseteq )$$

(A20)

called the the universal operator for U induced by ψ and defined for all $\mathrm{\Psi }\in \wp \left(U^G\right)$ by

$$\begin{array}{ccc}\hfill {\forall }_{\psi }^U\left(\mathrm{\Psi }\right)& :=& \{\mathbf{a}\in U^K\mid forall\mathbf{b}\in U^G:\psi ;\mathbf{b}=\mathbf{a}implies\mathbf{b}\in \mathrm{\Psi }\}\hfill \\ \hfill & =& \{\mathbf{a}\in U^K\mid {\downarrow }_{\psi }^U\left(\mathbf{a}\right)={(\psi ;\_)}^{-1}\left(\mathbf{a}\right)\subseteq \mathrm{\Psi }\}.\hfill \end{array}$$

(A21)

Be aware, that we always have $\overline{{\exists }_{\psi }^U\left(U^G\right)}=\overline{(\psi ;\_)\left(U^G\right)}={\forall }_{\psi }^U(\varnothing )\subseteq {\forall }_{\psi }^U\left(\mathrm{\Psi }\right)$ since ${\downarrow }_{\psi }^U\left(\mathbf{a}\right)={(\psi ;\_)}^{-1}\left(\mathbf{a}\right)=\varnothing$ for all $\mathbf{b}\in \overline{(\psi ;\_)\left(U^G\right)}$.

We instantiate the adjunction from Corollary A2 for pre-composition maps.

Corollary A5

(Universal Galois Connection 2). For any $\mathrm{\Phi }\in \wp \left(U^K\right)$ and $\mathrm{\Psi }\in \wp \left(U^G\right)$

unit:

$\mathrm{\Phi }\subseteq {\forall }_{\psi }^U\left({\downarrow }_{\psi }^U\left(\mathrm{\Phi }\right)\right)$

counit:

${\downarrow }_{\psi }^U\left({\forall }_{\psi }^U\left(\mathrm{\Psi }\right)\right)\subseteq \mathrm{\Psi }$

adjunction:

$\mathrm{\Phi }\subseteq {\forall }_{\psi }^U\left(\mathrm{\Psi }\right)$ iff ${\downarrow }_{\psi }^U\left(\mathrm{\Phi }\right)\subseteq \mathrm{\Psi }$

Due to Corollary A5, the monotone function (the functor) ${\downarrow }_{\psi }^U$ is left adjoint to the monotone function (the functor) ${\forall }_{\psi }^U$, ${\downarrow }_{\psi }^U⊣{\forall }_{\psi }^U$ in symbols. We can summarize the basic statements of this section by the following diagram with U an arbitrary non-empty set.

For pre-composition maps the compositionality statements in (A12) are transformed as follows. Keep in mind (A15). For any maps $\psi :K\to G$, $\mu :G\to H$ and any carrier U we do have the following equations:

$${\downarrow }_{\psi ;\mu }^U={{\downarrow }_{\psi }^U;\downarrow }_{\mu }^U{\exists }_{\psi ;\mu }^U={\exists }_{\mu }^U;{\exists }_{\psi }^U{\forall }_{\psi ;\mu }^U={\forall }_{\mu }^U;{\forall }_{\psi }^U.$$

(A23)

We round this section with a systematic analysis of surjectivity and injectivity of pre-composition maps. Since we only allow non-empty carriers U, all the hom-sets $U^K$ are non-empty for arbitrary K! By (A9), we do have for any map $\psi :K\to G$ and any carrier U:

$$(\psi ;\_):U^G\to U^{K}surjectiveiffi{d}_{\wp \left(U^K\right)}={{\downarrow }_{\psi }^U;{\forall }_{\psi }^{U}iff\downarrow }_{\psi }^U;{\exists }_{\psi }^U=i{d}_{\wp \left(U^K\right)}.$$

(A24)

For $K=\varnothing$ we have $U^{\varnothing }=\{{!}_U=i{n}_{\varnothing ,U}\}$ and $({!}_G;\_):U^G\to U^{\varnothing }$ is surjective for the only injective map $\psi ={!}_G=i{n}_{\varnothing ,G}:\varnothing \to G$ since ${!}_G;\mathbf{b}={!}_U$ for all $\mathbf{b}\in U^G$.

For $K\ne \varnothing$ the pre-composition map $(\psi ;\_):U^G\to U^K$ is surjective for any carrier U if $\psi :K\to G$ is injective, since all (!) injective maps $\psi :K\to G$ with $K\ne \varnothing$ are split monomorphisms (sections) in $\mathsf{Set}$, i.e., there exists a map ${\psi }^{-}:G\to K$ such that $\psi ;{\psi }^{-}=i{d}_K$. For any $\mathbf{a}\in U^K$ we obtain $(\psi ;\_)\left(\mathbf{b}\right)=\psi ;\mathbf{b}=\mathbf{a}$ for $\mathbf{b}:={\psi }^{-};\mathbf{a}\in U^G$ thus $(\psi ;\_)$ is indeed surjective. Note, that we actually argue $({\psi }^{-};\_);(\psi ;\_)=(i{d}_K;\_)=i{d}_{U^K}$, i.e., $(\psi ;\_):U^G\to U^K$ is a split epimorphism (retraction) in $\mathsf{Set}$ for any carrier U.

In this paper we have $\mathsf{Carr}⊏\mathsf{Cxt}=\mathsf{Set}$ thus we can choose $U=K$. The split monic requirement "there exists a map ${\psi }^{-}:G\to K$ such that $\psi ;{\psi }^{-}=i{d}_K$" is equivalent to the requirement that ${(\psi ;\_)}^{-1}\left(i{d}_K\right)$ is non-empty which is ensured if $(\psi ;\_):K^G\to K^K$ is surjective. In such a way we get the following equivalence completing (A24).

$$(\psi ;\_):U^G\to U^{K}surjectiveforallU\ne \varnothing iff\psi :K\to Ginjective.$$

(A25)

Deduction calculi in traditional unsorted first-order logic utilize inclusion maps and rely heavily on the equivalences in (A24) and (A25)!

According to (A10), we do have for all (!) maps $\psi :K\to G$ and all carriers U:

$$i{d}_{\wp \left(U^G\right)}={\exists }_{\psi }^U;{{\downarrow }_{\psi }^{U}iff{\forall }_{\psi }^U;\downarrow }_{\psi }^U=i{d}_{\wp \left(U^G\right)}.$$

(A26)

$\psi :K\to G$ surjective entails that $(\psi ;\_):U^G\to U^K$ is injective for all carriers U since surjective maps are exactly the epimorphisms in $\mathsf{Set}$. In such a way, we obtain from (A6) and (A26) for all carriers U:

$$\psi :K\to Gsurjectiveimpliesi{d}_{\wp \left(U^G\right)}={\exists }_{\psi }^U;{{\downarrow }_{\psi }^{U}and{\forall }_{\psi }^U;\downarrow }_{\psi }^U=i{d}_{\wp \left(U^G\right)}.$$

(A27)

Remark A1

(Many-sortedness: Operators). There are two related reasons that the "lazy approach" to deduction in unsorted first-order logic does not work for many-sorted first-order logic.

Let S be a set of sort symbols with cardinality $\left|S\right|\ge 2$. First, for any S-sets $U=(U_s\mid s\in S)$, $K=(K_s\mid s\in S)$ the condition $U\ne {\mathbf{0}}_S$, with ${\mathbf{0}}_S=(\varnothing \mid s\in S)$ the initial object in the category ${\mathsf{Set}}^S$ of S-sets and S-maps, doesn’t ensure that the set $U^K$ of all S-maps $f=(f_s:K_s\to U_s\mid s\in S)$ from K into U is non-empty. $U^K$ becomes empty if there is an $s\in S$ with $K_s\ne \varnothing$ but $U_s=\varnothing$!

Second, the condition $K\ne {\mathbf{0}}_S$ doesn’t ensure that every injective S-map $\psi :K\to G$ is split monic. $\psi :K\to G$ is not split monic if there is an $s\in S$ with $K_s=\varnothing$ but $G_s\ne \varnothing$!

We can repair the equivalence in (A25) by replacing "injective" by "split monic", but we have to overcome our laziness anyway and must work with explicit variable declarations since the projections $(i{n}_{X,Y};\_):U^Y\to U^X$ for $X\subset Y$ are not anymore surjective by default!

Appendix A.3. Shift, Existential and Universal Operators for Substitutions

As shown in Section 3.3 any $\Sigma$-substitution (declaration) $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ defines a derived operation ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^X$ in every $\Sigma$-structure $\mathcal{U}$ (in the opposite direction).

To ease accessibility of the paper, we instantiate definitions, constructions and results in Appendix A.1 for the special hom-sets $U^X=\mathsf{Set}(X,U)$, i.e., sets of variable assignments, and derived operation ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^X$ represented by $\Sigma$-substitutions $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$. Note, that there is a certain small overlap with Appendix A.2 since every map $\psi :X\to Y$ can be equivalently represented by the $\Sigma$-substitution $\psi ;{\eta }_Y:X\to T_{\Sigma }\left(Y\right)$.

We adapt the notation in Appendix A.1 to the special case of derived operations analogously to the adaptations in Appendix A.2. First, we use the identifiers of $\Sigma$-substitutions as subscripts for operators. Be aware that this again reverses the direction of maps and thus the order of composition. Second, we must use identifiers $\mathcal{U}$ of $\Sigma$-structures, instead of identifiers U of carriers as in Appendix A.2, as superscripts for operators.

In Section 4.2.1 we discussed that any $\Sigma$-substitution (declaration) can be extended to a map ${\mathbf{t}}^{*}:T_{\Sigma }\left(X\right)\to T_{\Sigma }\left(Y\right)$ with $\mathbf{t}={\eta }_X;{\mathbf{t}}^{*}$ where ${\mathbf{t}}^{*}:T_{\Sigma }\left(X\right)\to T_{\Sigma }\left(Y\right)$ describes the application of the $\Sigma$-substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ to all $\Sigma$-terms over X! This enabled us to define the composition of two $\Sigma$-substitutions $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$, $\mathbf{r}:Y\to Z$ as the $\Sigma$-substitution $\mathbf{t};{\mathbf{r}}^{*}:X\to T_{\Sigma }\left(Z\right)$. Moreover, (18) shows that this composition of $\Sigma$-substitutions reflects the composition of the corresponding derived operations ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^X$, ${\mathbf{r}}^{\mathcal{U}}:U^Z\to U^Y$

$${(\mathbf{t};{\mathbf{r}}^{*})}^{\mathcal{U}}={\mathbf{r}}^{\mathcal{U}};{\mathbf{t}}^{\mathcal{U}}:U^Z\to U^X.$$

(A28)

For any $\Sigma$-substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ and any $\Sigma$-structure $\mathcal{U}$ the formation of pre-images for the corresponding derived operation ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^X$

$${\left({\mathbf{t}}^{\mathcal{U}}\right)}^{-1}\left(\mathrm{\Phi }\right):=\{\mathbf{b}\in U^Y\mid {\mathbf{t}}^{\mathcal{U}}\left(\mathbf{b}\right)\in \mathrm{\Phi }\}=\bigcup \{{\left({\mathbf{t}}^{\mathcal{U}}\right)}^{-1}\left(\mathbf{a}\right)\mid \mathbf{a}\in \mathrm{\Phi }\}\mathrm{for}\mathrm{all}\mathrm{\Phi }\subseteq U^X$$

(A29)

defines a functor, called the shift operator for $\mathcal{U}$ induced by $\mathbf{t}$,

$${\downarrow }_{\mathbf{t}}^{\mathcal{U}}:={\left({\mathbf{t}}^{\mathcal{U}}\right)}^{-1}:(\wp \left(U^X\right),\subseteq )\to (\wp \left(U^Y\right),\subseteq ).$$

(A30)

Dually, the formation of images defines a functor

$${\exists }_{\mathbf{t}}^{\mathcal{U}}:={\mathbf{t}}^{\mathcal{U}}:(\wp \left(U^Y\right),\subseteq )\to (\wp \left(U^X\right),\subseteq )$$

(A31)

called the existential operator for $\mathcal{U}$ induced by $\mathbf{t}$ and defined for all $\mathrm{\Psi }\in \wp \left(U^Y\right)$ by

$${\exists }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Psi }\right):=\{{\mathbf{t}}^{\mathcal{U}}\left(\mathbf{b}\right)\mid \mathbf{b}\in \mathrm{\Psi }\}=\{\mathbf{a}\in U^X\mid exists\mathbf{b}\in U^Y:{\mathbf{t}}^{\mathcal{U}}\left(\mathbf{b}\right)=\mathbf{a}and\mathbf{b}\in \mathrm{\Psi }\}.$$

(A32)

We instantiate the adjunction from Corollary A1 for derived operations.

Corollary A6

(Existential Galois Connection 3). For any $\mathrm{\Psi }\in \wp \left(U^Y\right)$ and $\mathrm{\Phi }\in \wp \left(U^X\right)$

unit:

$\mathrm{\Psi }{\subseteq \downarrow }_{\mathbf{t}}^{\mathcal{U}}\left({\exists }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Psi }\right)\right)$

counit:

${\exists }_{\mathbf{t}}^{\mathcal{U}}\left({\downarrow }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Phi }\right)\right)\subseteq \mathrm{\Phi }$

adjunction:

$\mathrm{\Psi }{\subseteq \downarrow }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Phi }\right)$ iff ${\exists }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Psi }\right)\subseteq \mathrm{\Phi }$

By Corollary A6 the functor ${\exists }_{\mathbf{t}}^{\mathcal{U}}$ is left adjoint to ${\downarrow }_{\mathbf{t}}^{\mathcal{U}}$, ${\exists }_{\mathbf{t}}^{\mathcal{U}}{⊣\downarrow }_{\mathbf{t}}^{\mathcal{U}}$ in symbols. For any $\Sigma$-substitution $\mathbf{t}:X\to Y$ and any $\Sigma$-structure $\mathcal{U}$ we obtain a functor

$${\forall }_{\mathbf{t}}^{\mathcal{U}}:(\wp \left(U^Y\right),\subseteq )\to (\wp \left(U^X\right),\subseteq )$$

(A33)

called the the universal operator for $\mathcal{U}$ induced by $\mathbf{t}$ and defined for all $\mathrm{\Psi }\in \wp \left(U^Y\right)$ by

$$\begin{array}{ccc}\hfill {\forall }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Psi }\right)& :=& \{\mathbf{a}\in U^X\mid \mathit{for}\mathbf{all}\mathbf{b}\in U^Y:\mathbf{t}\left(\mathbf{b}\right)=\mathbf{a}implies\mathbf{b}\in \mathrm{\Psi }\}\hfill \\ \hfill & =& \{\mathbf{a}\in U^X\mid {\downarrow }_{\mathbf{t}}^{\mathcal{U}}\left(\mathbf{a}\right)={\left({\mathbf{t}}^{\mathcal{U}}\right)}^{-1}\left(\mathbf{a}\right)\subseteq \mathrm{\Psi }\}.\hfill \end{array}$$

(A34)

We instantiate the adjunction from Corollary A2 for derived operations.

Corollary A7

(Universal Galois Connection 3). For any $\mathrm{\Phi }\in \wp \left(U^X\right)$ and $\mathrm{\Psi }\in \wp \left(U^Y\right)$

unit:

$\mathrm{\Phi }\subseteq {\forall }_{\mathbf{t}}^{\mathcal{U}}\left({\downarrow }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Phi }\right)\right)$

counit:

${\downarrow }_{\mathbf{t}}^{\mathcal{U}}\left({\forall }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Psi }\right)\right)\subseteq \mathrm{\Psi }$

adjunction:

$\mathrm{\Phi }\subseteq {\forall }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Psi }\right)$ iff ${\downarrow }_{\mathbf{t}}^{\mathcal{U}}\left(\mathrm{\Phi }\right)\subseteq \mathrm{\Psi }$

Due to Corollary A7, the functor ${\downarrow }_{\mathbf{t}}^{\mathcal{U}}$ is left adjoint to the monotone function (the functor) ${\forall }_{\mathbf{t}}^{\mathcal{U}}$, ${\downarrow }_{\mathbf{t}}^{\mathcal{U}}⊣{\forall }_{\mathbf{t}}^{\mathcal{U}}$ in symbols. We can summarize the basic statements of this section by the following diagram with $\mathcal{U}$ an arbitrary $\Sigma$-structure $\mathcal{U}$.

For $\Sigma$-substitutions the compositionality statements in (A12) are transformed as follows. Keep in mind (A28). For any $\Sigma$-substitutions $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$, $\mathbf{r}:Y\to T_{\Sigma }\left(Z\right)$ and any $\Sigma$-structure $\mathcal{U}$ we do have the following equations:

$${\downarrow }_{\mathbf{t};{\mathbf{r}}^{*}}^{\mathcal{U}}={{\downarrow }_{\mathbf{t}}^{\mathcal{U}};\downarrow }_{\mathbf{r}}^{\mathcal{U}}{\exists }_{\mathbf{t};{\mathbf{r}}^{*}}^{\mathcal{U}}={\exists }_{\mathbf{r}}^{\mathcal{U}};{\exists }_{\mathbf{t}}^{\mathcal{U}}{\forall }_{\mathbf{t};{\mathbf{r}}^{*}}^{\mathcal{U}}={\forall }_{\mathbf{r}}^{\mathcal{U}};{\forall }_{\mathbf{t}}^{\mathcal{U}}.$$

(A36)

Properties of derived operations ${\mathbf{t}}^{\mathcal{U}}:U^Y\to U^X$ can heardly be traced back to properties of the $\Sigma$-substitution $\mathbf{t}:X\to T_{\Sigma }\left(Y\right)$ since the properties of ${\mathbf{t}}^{\mathcal{U}}$ heavily depend on the special properties of the operations in $\mathcal{U}$. Therefore, the findings concerning surjectivity and injectivity in Appendix A.1 for arbitrary maps can not be refined for derived operations.