Cut Elimination Versus Logic of Paradox

1. Introduction

According to [14], the constant-free propositional empty-sequent-less fragment of $LK$ [4] endowed with rules inverse to logical ones is equivalent (in the sense of [11]) to the logic of paradox $LP$ [9], having the same theorems as the classical propositional logic $PC$, in view of [13], that has yielded both a novel semantic insight into Cut Elimination in $LK$ and the fact that the only proper consistent extension of the sequent calculus involved distinct from $LK$ is the one relatively axiomatized by Cut with minimal non-empty context (viz., having just a single formula either on the right or on the left but not on both sides). The primary objective of this work is to expand [14] to the full propositional fragment of $LK$ upon proper expanding underlying works [10,13].

The rest of the work is as follows. Section 2 is a brief summary of basis issues underlying the work. Section 3 is devoted to key universal issues, then used in the main part presented in Section 4.

2. General background

2.1. Set-theoretical background

Non-negative integers are identified with sets/ ordinals of lesser ones, their set/ordinal being denoted by $\omega$. Unless any confusion is possible, one-element/-component sets/sequences are identified with their elements/components. As usual, functions are treated as binary relations.

Given any sets A, B and an infix $\diamond :A^2\to A$, let ${\wp }_{\left(\omega \right)}\left([B,]A\right)$ be the set of all (finite) subsets of A [including B], ${ϵ}_A\triangleq \{\langle a,a\rangle \mid a\in A\}$ the equality relation on A, $A^{*|+}\triangleq \left({\bigcup }_{m\in (\omega \setminus (0\left|1\right))}{A}^m\right)$ and ${\diamond }_{+}:A^{+}\to A,\langle \{\langle \overline{a},b\rangle ,\}c\rangle \mapsto \left(\{{\diamond }_{+}\left(\langle \overline{a},b\rangle \right)\diamond \}c\right)$, while A-tuples 〈viz., functions with domain $A\rangle$ are written in the sequence form $\overline{t}$ with $t_a$, where $a\in A$, stands for ${\pi }_a\left(\overline{t}\right)$, as well as, in case $A=\left(\right[n+\left]1\right)$ [where $n\in \omega$] written also in the standard finite tuple/sequence form $\left[\left(\langle )\right]t_0[\{,\}\dots \{,\}{t}_n(\rangle )\right]$ [and identified with $\langle \overline{t}\upharpoonright n,t_n\rangle$] under identification of $B^{[n+]1}$ with $[B^n\times ]B$ whereas $*:{\left(A^{*}\right)}^2\to A^{*},\langle \overline{a},\overline{b}\rangle \mapsto (\overline{a}\cup ({(\left((+(dom\overline{a}))\right)\upharpoonright (dom\overline{b}))}^{-1}\circ \overline{b}))$ the concatenation binary operation.

An $X\in Y\subseteq \wp \left(A\right)$ is said to be meet-irreducible in Y, if $\forall Z\in \wp \left(Y\right):\left(\right(A\cap (\bigcap Z))=X)\Rightarrow (X\in Z)$, their set being denoted by $MI\left(Y\right)$. A $\mathcal{U}\subseteq \wp \left(A\right)$ is said to be upward-directed, if $\forall \mathcal{S}\in {\wp }_{\omega }\left(\mathcal{U}\right):\exists T\in (\mathcal{U}\cap \wp (\bigcup \mathcal{S},A))$, subsets of $\wp \left(A\right)$ closed under unions of upward directed subsets being called inductive. A [finitary] closure operator over A is any unary operation on $\wp \left(A\right)$ such that $\forall X\in \wp \left(A\right),\forall Y\in \wp \left(X\right):(X\cup C\left(C\left(X\right)\right)\cup C\left(Y\right))\subseteq C\left(X\right)[=(\bigcup C\left[{\wp }_{\omega }\left(X\right)\right])]$. A closure system over A is any {〈inductive〉} $\mathcal{C}\subseteq \wp \left(A\right)$ containing A and closed under intersections of subsets containing A, any $\mathcal{B}\subseteq \wp \left(A\right)$ {such that $\mathcal{C}=\{A\cap (\bigcap \mathcal{S})\mid \mathcal{S}\subseteq \mathcal{B}\}$ being called a (closure) basis of $\mathcal{C}$ and} determining the {〈finitary〉} closure operator $C_{\mathcal{B}}\triangleq \{\langle Z,A\cap (\bigcap (\mathcal{B}\cap \wp (Z,A)\left)\right)\rangle \mid Z\in \wp \left(A\right)\}\{=C_{\mathcal{C}}\}$ over A such that $\mathcal{B}\subseteq (img{C}_{\mathcal{B}})\{=\mathcal{C}\}$. Conversely, $imgC$ is a[n inductive] closure system over A such that $C_{imgC}=C$, C and $imgC$ being called dual to one another.

Remark 1.  

Due to Zorn Lemma, according to which any non-empty inductive set has a maximal element, $MI\left(\mathcal{C}\right)$ is a basis of any inductive closure system $\mathcal{C}$. □

A [dual] Galois connection/retraction between/of a poset $\mathcal{Q}=\langle Q,\precsim \rangle$ and/onto a poset $\mathcal{P}=\langle P,\lesssim \rangle$ is any $\langle f,g\rangle \in (Q^P\times P^Q)$ such that:

$$\begin{array}{ccc}\hfill (a\lesssim c)& \Rightarrow & \left(f\left(c\right){\precsim }^{[-1]}f\left(a\right)\right),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \left(b{\precsim }^{[-1]}d\right)& \Rightarrow & \left(g\right(d)\lesssim g(b\left)\right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill (a\lesssim g(b\left)\right)& \iff & (b/a){⪯}^{[-1]}/\lesssim \left(f\right(a)/g(b\left)\right),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \left(f\right(a)/b)& =/{⪯}^{[-1]}& f\left(g\right(f\left(a\right)/b\left)\right),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill g\left(f\right(g\left(b\right)/a\left)\right)& =& \left(g\right(b)/a),\hfill \end{array}$$

(5)

for all $a,c\in P$ and $b,d\in Q$, [dual] Galois retractions of $\mathcal{Q}$ onto $\mathcal{P}$ being exactly [dual] Galois connections between $\mathcal{Q}$ and $\mathcal{P}$ with either injective left or surjective right component.

2.2. Algebraic background

Unless otherwise specified, we deal with a fixed but arbitrary finitary algebraic (viz., functional) signature L, viewed as a propositional language consisting of (propositional) connectives, L-algebras/“their carriers|class” being denoted by “/respective capital Fraktur/Italic letters [with /same indices], unless otherwise specified”$|{\mathsf{A}}_L$. Then, ${Tm}_L^{\left\{\alpha \right\}}$ {where $\alpha \in \left(\right(\omega (\setminus 1)\left)\right|\left\{\omega \right\})$ (unless L has a constant)|} is the set of L-terms, viewed as 〈propositional〉L-formulas, with ⌈propositional⌉ variables in ${Var}_{\left\{\alpha \right\}}\triangleq (img{\overline{x}}_{\left\{\alpha \right\}})$, where ${\overline{x}}_{\left\{\alpha \right\}}\triangleq \{\langle i,x_i\rangle \mid i\in \left(\omega \{\cap \alpha \}\right)\}$, viz., the carrier of the absolutely-free L-algebra ${\mathfrak{Tm}}_L^{\left\{\alpha \right\}}$, freely-generated by ${Var}_{\left\{\alpha \right\}}$ |{whose endomorphisms are viewed as ⌊propositional⌋L-substitutions, their set being denoted by ${Sb}_L\ni {\sigma }_{+n}\triangleq {[x_j/x_{j+n}]}_{j\in \omega }$, where $n\in \omega$}. Any m-ary connective $\mathsf{c}\in L$, where $m\in \omega$, is identified with $\mathsf{c}\left({\overline{x}}_m\right)\in {Tm}_L^m$. As usual, the class of all “isomorphic copies”/subalgebras/“[ultra-]products of tuples” of members of a $\mathsf{K}\subseteq {\mathsf{A}}_L$ is denoted by $(\mathbf{I}/\mathbf{S}/{\mathbf{P}}^{[}\mathrm{U}])\mathsf{K}$.

2.3. Logical background

Here, we mainly follow [11] but allow infinitary logics and calculi as well as adopt more conventional terminology and notations.

Let $F=\langle L,P\rangle$ be a [first-order] language (viz., finitary signature), where $P\ne ⌀$ is a relational one, any $\sigma \in {Sb}_L$ being extended to the equally-denoted unary operation on the set ${Fm}_{F|L}^{|P}$ of F-formulas/-axioms (viz., first-order atomic formulas of the signature F with variables in Var) via setting $\sigma \left(\Phi \right)\triangleq \mathsf{p}(\overline{\phi }\circ \sigma )$, for all $\Phi =\mathsf{p}\left(\overline{\phi }\right)\in {Fm}_F$. Then, any $\mathcal{R}=\langle \Gamma ,\Psi \rangle \in {Ru}_F^{\left[\omega \right]}\triangleq ({\wp }_{\left[\omega \right]}\left({Fm}_F\right)\times {Fm}_F)$ is called a (non-axiomatic∥proper) [finitary] F-rule with “elements of $\Gamma$”/$\Psi$ called its premises/conclusion, written as (either ${\displaystyle \frac{{\displaystyle \Gamma }}{{\displaystyle \Phi }}}$ or $\Gamma ⇝)\Phi$ and identified with 〈the universal closure of〉$\left(\right(\bigwedge \Gamma )\to )\Phi$ (iff $\Gamma \ne ⌀$), those of the form $\Psi ⇝{\rm Y}$, where ${\rm Y}\in \Gamma$, being said to be inverse to $\mathcal{R}$, while any $f:{Fm}_F\to \lceil {\wp }_{\left[\omega \right]}(\rceil {Fm}_{F{\lfloor }^{\prime }\rfloor }\lceil )\rceil$ ⌊where $F^{\prime }=\langle L^{\prime },P^{\prime }\rangle$ is a language⌋ is extended to the equally-denoted $f:{Ru}_F^{\left[\omega \right]}\to \lceil {\wp }_{\left[\omega \right]}(\rceil {Ru}_{F{\lfloor }^{\prime }\rfloor }^{\left[\omega \right]}\lceil )\rceil$ via setting $f\left(\mathcal{R}\right)\triangleq \left(\right\{\lceil \bigcup \rceil f\left[\Gamma \right]\}\times f(\Phi \left)\right)$ under proper identifying singletons with their elements in the non-$\lceil \rceil$-optional case covering L-substitutions, whereas sets of {non-proper} [finitary] F-rules are called {axiomatic} “[finitary] F-calculi”[/“deductive bases over F”] [/[11].

A closure operator C [with non-one-element range-image] over ${Fm}_F$ is said to be structural, if, for all $\sigma \in {Sb}_L$ and $X\subseteq {Fm}_F$, $\sigma \left[C\right(X\left)\right]\subseteq C\left(\sigma \right[X\left]\right)$, i.e., $imgC$ is closed under inverse substitutions in the sense that, for all $\sigma \in {Sb}_L$ and $T\in (imgC)$, ${\sigma }^{-1}\left[T\right]\in (imgC)$, in which case it is called a [consistent] F-logic(al system) {satisfying an F-rule $\langle \Gamma ,\Phi \rangle$, if $\Phi \in C\left(\Gamma \right)$}, elements of $Thm\left(C\right)\triangleq C(⌀)$ being called its theorems, while any F-logic $C^{\prime }$ such that $(img{C}^{\prime })[=((imgC)\cap \wp (Thm\left(C^{\prime }\right),{Fm}_F))]\subseteq (imgC)\{\ne (img{C}^{\prime })\}$ is said to be an [axiomatic] {proper} extension of C, F-logics forming a complete lattice poset under extension partial ordering ≦, intersection of dual closure systems as join and point-wise intersection of F-logics as meet, whereas C is an extension of the theorem-less F-logic $C^{+0}$ dual to the closure system $(imgC)\cup 1$ over ${Fm}_F$, called the theorem-less version of C. Then, the least F-logic ${Cn}_{\mathcal{C}}^{\left[C\right]}$ [being an extension of C and] satisfying all rules in a (finitary) F-calculus $\mathcal{C}$ is said to be axiomatized by $\mathcal{C}$ [relatively toC] (${Cn}_{\mathcal{C}}$ being finitary), in which case C is axiomatized by the set of all {finitary} F-rules satisfied in it {if it is finitary}, and so C is finitary iff it is axiomatized by a finitary F-calculus, while axiomatic extensions of C are exactly its extensions relatively axiomatized by axiomatic F-calculi, whereas any F-rule $\langle \Gamma ,\Phi \rangle$ is satisfied in ${Cn}_{\mathcal{C}}$ iff it is derivable in$\mathcal{C}$ in the sense that there is a $\mathcal{C}$-derivation of Φ from $\Gamma$, i.e., a mapping $\overline{\Psi }$ from a (finite) ordinal $\alpha$, called its length, to ${Fm}_F$ such that $\Phi \in (img\overline{\Psi })$ and, for each $\beta \in \alpha$, either ${\Psi }_{\beta }\in \Gamma$ or there is some $\Xi \subseteq \overline{\Psi }\left[\beta \right]$ such that $\langle \Xi ,{\Psi }_{\beta }\rangle \in {Sb}_L\left[\mathcal{C}\right]\triangleq (\bigcup \{\sigma \left[\mathcal{C}\right]\mid \sigma \in {Sb}_L\})$, as well as:

$${Cn}_{\mathcal{C}}^{\left[C\right]+0}={Cn}_{{\mathcal{C}}^{+0}}^{\left[C^{+0}\right]},$$

(6)

where ${\mathcal{C}}^{+0}\triangleq \{\langle \Gamma \cup \left\{\Psi \right\},\Phi \rangle \mid \Psi =\mathsf{p}\left({\overline{x}}_n\right)\in {Fm}_{⌀}^P,n\in \omega ,\langle \Gamma ,\Phi \rangle \in {\sigma }_{+n}\left[\mathcal{C}\right]\}$. Given a sublanguage $F^{\prime }=\langle L^{\prime },P^{\prime }\rangle$ of F, where $L^{\prime }\subseteq L$ and $⌀\ne P^{\prime }\subseteq P$, the $L^{\prime }$-fragment ofC is the $L^{\prime }$-logic $(C\upharpoonright F^{\prime })\triangleq \{\langle X,C\left(X\right)\cap {Fm}_{L^{\prime }}\rangle \mid X\subseteq {Fm}_{L^{\prime }}\}$.

2.3.1. Basic kinds of languages

Sentential languages

Let D be a unary truth predicate relation symbol, $L^D\triangleq \langle L,D\rangle$ the sentential L-language and ${(Fm|Ru)}_L\triangleq {(Fm|Ru)}_{L^D}$ “identified with ${Tm}_L$ under identification of any $\phi \in {Tm}_L$ with $D\left(\phi \right)$”|, $L^D$-formulas/-rules/-axioms/-calculi/-logics being called [sentential] L-formulas/-rules/-axioms/-calculi/-logics.

First-order $L^D$-structures (viz., algebraic systems of the signature $L^D$; cf. [7]) [with truth predicate distinct from its carrier] are called [consistent] (logical) L-matrices (cf. [6]), identified with {the left components of} the couples constituted by their underlying algebras (viz., L-reducts) and truth predicates {whenever these are are empty, L-algebras being thus viewed as L-matrices with empty truth predicates} as well as denoted by capital Calligraphic letters 〈with indices〉, their underlying algebras being denoted by respective capital Gothic letters 〈with same indices〉. Any class $\mathsf{M}$ of L-matrices defines its L-logic ${Cn}_{\mathsf{M}}$ dual to the closure system over ${Fm}_L$ with closure basis $\{h^{-1}\left[D^{\mathcal{A}}\right]\mid \mathcal{A}\in \mathsf{M},h\in hom({\mathfrak{Tm}}_L,\mathfrak{A})\}$, satisfying any L-rule iff this is satisfied in $\mathsf{M}$ in the usual-model-theoretic sense, as well as being finitary, whenever both $\mathsf{M}$ and all its members are finite (cf. [6]), but, otherwise, not necessarily being so,1 such that

$${Cn}_{\mathsf{M}}^{+0}={Cn}_{\mathsf{M}\cup \{\langle \mathfrak{O},⌀\rangle \}},$$

(7)

where $\mathfrak{O}$ is any one-element L-algebra. Then, L-matrices defining extensions of an L-logic C are called its models, their class being denoted by $Mod\left(C\right)$.

Given L-matrices $\mathcal{A}$ and $\mathcal{B}$, a[n] [injective] /surjective |strict homomorphism from $\mathcal{A}$ to/onto$\mathcal{B}$ is any [injective] $h\in hom(\mathfrak{A},\mathfrak{B})$ such that $D^{\mathcal{A}}\subseteq |=h^{-1}\left[D^{\mathcal{B}}\right]$ and $h\left[A\right]\subseteq /=B$ (their set being denoted by ${hom}^{[}\mathrm{I}]/[\mathrm{I},]{\mathrm{S}}_{|}\mathrm{S}(\mathcal{A},\mathcal{B})$) |[also called an embedding/isomorphism of/from $\mathcal{A}$ into/onto$\mathcal{B}$, $\mathcal{A}$ being said to be embedable/isomorphic into/to $\mathcal{B}$underh as well as, in case $h={ϵ}_A$, called a∥the submatrix∥restriction of$\mathcal{B}$ ∥“onA with setting $(\mathcal{B}\upharpoonright A)\triangleq \mathcal{A}$”], in which case

$$hom({\mathfrak{Tm}}_L,\mathfrak{B})\supseteq /=\{g\circ h\mid g\in hom({\mathfrak{Tm}}_L,\mathfrak{A})\},$$

(8)

and so:

$${Cn}_{(}\mathcal{B}/\mathcal{A}\left)\right|\mathcal{A}(⌀|\Gamma )\subseteq /(\subseteq |=){Cn}_{\mathcal{B}}(⌀|\Gamma ),$$

(9)

for all $\Gamma \subseteq {Fm}_L$.

Equational languages

Let ≈ be an infix binary equality relation symbol, $L^{\approx }\triangleq \langle L,\approx \rangle$ the equational L-language and ${Eq}_L\triangleq {Fm}_{L^{\approx }}$ the set of L-equations/-identities identified with ${Fm}_L^2$ under identification of any $\varphi \approx \psi$ with $\langle \varphi ,\psi \rangle$, $L^{\approx }$-rules/-axioms/-calculi/-logics being called equational L-rules/-axioms/-calculi/-logics. Then, given a class $\mathsf{K}$ of L-algebras, we have the equational L-logic ${Cn}_{\mathsf{K}}^{\approx }$ dual to the closure system over ${Eq}_L$ with closure basis $\{kerh\mid \mathfrak{A}\in \mathsf{K},h\in hom({\mathfrak{Tm}}_L,\mathfrak{A})\}$, called the equational logic of $\mathsf{K}$, equal to that of

$$\mathbf{I}\left[\mathbf{SP}\right]\mathsf{K}[=\{\mathfrak{A}\in {\mathsf{A}}_L\mid (A^2\cap (\bigcap \{kerh\mid h\in hom(\mathfrak{A},\mathfrak{B}),\mathfrak{B}\in \mathsf{K}\}))={ϵ}_A\}],$$

(10)

satisfying any equational L-rule iff this is satisfied in $\mathsf{K}$ in the usual model-theoretic sense, as well as being finitary, whenever ${\mathbf{P}}^{\mathrm{U}}\mathsf{K}\subseteq \mathbf{I}\mathsf{K}$ (in view of the Compactness Theorem; cf. [7]) {in particular, both $\mathsf{K}$ and all its members are finite; cf. [3]}, but, otherwise, not necessarily being so,2 in which case L-algebras defining extensions of an equational logic C are called its models, their class being denoted by $Mod\left(C\right)$, and so the mappings $C^{\prime }\mapsto Mod\left(C^{\prime }\right)$ and $\mathsf{P}\mapsto {Cn}_{\mathsf{P}}^{\approx }$ form a Galois retraction of the poset of pre-varieties (in the sense of [15]; viz., classes closed under $\mathbf{I}$, $\mathbf{S}$ and $\mathbf{P}$, $\mathbf{ISP}\mathsf{K}$ being the least one including $\mathsf{K}$ and said to be generated by $\mathsf{K}$) of L-algebras onto the one of the equational logics of classes of L-algebras. Clearly, the latter poset is closed under axiomatic extensions, the reservation “axiomatic” appearing redundant, in view of the following observation:

Remark 2.  

Given any class $\mathsf{K}$ of L-algebras and any $\theta \in (img{Cn}_{\mathsf{K}}^{\approx })$, there are some set I, some $\overline{\mathfrak{A}}\in {\mathsf{K}}^I$ and some $\overline{h}\in ({\prod }_{i\in I}hom({\mathfrak{Tm}}_L,{\mathfrak{A}}_i))$ such that $\theta =\left({\bigcap }_{i\in I}(ker{h}_i)\right)$, in which case $h:{Fm}_L\to \left({\prod }_{i\in I}{A}_i\right),\phi \mapsto {\langle h_i\left(\phi \right)\rangle }_{i\in I}$ is a [surjective] homomorphism from ${\mathfrak{Tm}}_L$ [on]to $[{\mathfrak{A}}_{\theta }\triangleq ]\left(\left({\prod }_{i\in I}{\mathfrak{A}}_i\right)[\upharpoonright (imgh)]\right)[\in \mathbf{SP}\mathsf{K}]$ such that $(kerh)=\theta$, and so, by the right alternative of (8), every equational L-rule $\mathcal{R}$, such that each one in ${Sb}_L\left[\left\{\mathcal{R}\right\}\right]$ is true in ${\mathfrak{A}}_{\theta }$ under h, is true in ${\mathfrak{A}}_{\theta }$, any extension C of ${Cn}_{\mathsf{K}}^{\approx }$ being then the equational logic of $\{{\mathfrak{A}}_{\theta }\mid \theta \in (imgC)\}\subseteq \mathbf{SP}\mathsf{K}$. □

Sequential languages

Let ${⊢}_n^m$, where $\langle m,n\rangle \in [\ell \subseteq ]{\omega }^2$, be an $(m+n)$-ary sequent relation symbol, $P_{[\ell ]}^{⊢}\triangleq \{{⊢}_n^m\mid \langle m,n\rangle \in \left({\omega }^2[\cap \ell ]\right)\}$ the [ℓ-]sequent(ial) relation signature, $L_{[\ell ]}^{⊢}\triangleq \langle L,P_{[\ell ]}^{⊢}\rangle$ the [ℓ-]sequent(ial) L-language and ${Seq}_L^{[\ell ]}\triangleq {Fm}_{L_{[\ell ]}^{⊢}}$ the set of L-sequents [of rank ℓ], any one ${⊢}_n^m\left(\overline{\phi }\right)$ being written in the standard form $(\overline{\phi }\upharpoonright m)⊢(((+m)\upharpoonright n)\circ \overline{\phi })$. Then, $L_{[\ell ]}^{⊢}$-rules/-axioms/-calculi/-logics are called [ℓ-]sequent(ial) L-rules/-axioms/-calculi/-logics.

3. Preliminaries

3.1. Disjunctive sentential logics

Fix any $\Delta \in {\wp }_{\left[\omega \right]}\left({Tm}_L^2\right)$. Given any $X,Y\subseteq {Fm}_L$, set $\Delta (X,Y)\triangleq (\bigcup \{\Delta (\varphi ,\psi )\mid \varphi \in X,\psi \in Y\}$. Then, an L-logic C is said to be weakly/〈strongly〉 [finitely] Δ-disjunctive, if, for all $(X\cup \{\varphi ,\psi \})\subseteq {Fm}_L$, $C(X\cup \Delta (\varphi ,\psi \left)\right)\subseteq /=\left(C\right(X\cup \left\{\varphi \right\})\cap C(X\cup \left\{\psi \right\}\left)\right)$ /“in which case:

$$\begin{array}{ccc}\hfill C\left(\Delta \right(\varphi ,\psi \left)\right)& =& C\left(\Delta \right(\psi ,\varphi \left)\right),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill C\left(\Delta \right(\varphi ,\varphi \left)\right)& =& C\left(\right\{\varphi \left\}\right)’’.\hfill \end{array}$$

(12)

Likewise, it is said to be Δ-multiplicative, if, for all $(X\cup \left\{\phi \right\})\subseteq {Fm}_L$, $\Delta \left(C\right(X),\phi )\subseteq C\left(\Delta \right(X,\phi \left)\right)$. Finally, an L-matrix $\mathcal{A}$ is said to be Δ-disjunctive, if, for all $a,b\in A$, $((\{a,b\}\cap D^{\mathcal{A}})\ne ⌀)\iff (\{{\delta }^{\mathfrak{A}}(a,b)\mid \delta \in \Delta \}\subseteq D^{\mathcal{A}})$.

Theorem 1.  

A (finitary) L-logic C is Δ-disjunctive if(f) it is both weakly Δ-disjunctive and Δ-multiplicative, while both (11) and (12) hold {whereas:

$$C\left(\Delta \right(\phi ,\Delta (\varphi ,\psi )\left)\right)=C\left(\Delta \right(\Delta (\phi ,\varphi ),\psi \left)\right),$$

(13)

for all $\phi ,\varphi ,\psi \in {Fm}_L$} if(f) C is defined by a class of [consistent] Δ-disjunctive L-matrices.

Proof. 

{The second “if” part is immediate.} Now, assume C is both weakly $\Delta$-disjunctive and $\Delta$-multiplicative, while both (11) and (12) hold. Consider any $(X\cup \{\varphi ,\psi \})\subseteq {Fm}_L$ and any $\phi \in \left(C\right(X\cup \left\{\varphi \right\})\cap C(X\cup \left\{\psi \right\}\left)\right)$, in which case $\phi \in C\left(\Delta \right(\phi ,\phi \left)\right)\subseteq C\left(\Delta \right(C(X\cup \{\psi \left\}\right),\phi )\subseteq C(\Delta (X\cup \{\psi \},\phi ))\subseteq C(X\cup \Delta (\psi ,\phi ))=C(X\cup \Delta (\phi ,\psi ))\subseteq C(X\cup \Delta \left(C\right(\left\{\phi \right\}),\psi ))\subseteq C(X\cup C\left(\Delta \right(X,\psi )\cup \Delta (\varphi ,\psi \left)\right))=C(X\cup \Delta (\varphi ,\psi ))$, and so C, being weakly $\Delta$-disjunctive, is $\Delta$-disjunctive. (Finally, assume C is $\Delta$-disjunctive. Then, by Remark 1, it, being finitary and structural, is defined by $\mathsf{M}\triangleq (\left\{{\mathfrak{Tm}}_L\right\}\times MI(imgC))$. Consider any $T\in MI(imgC)\neg \ni {Fm}_L$ and any $\varphi ,\psi \in {Fm}_L$ such that $\Delta (\varphi ,\psi )\subseteq T$, in which case $T=C\left(T\right)=C(T\cup \Delta (\varphi ,\psi \left)\right)=\left(C\right(T\cup \left\{\varphi \right\})\cap C(T\cup \left\{\psi \right\}\left)\right)$, and so either $T=C(T\cup \{\varphi \left\}\right)\ni \varphi$ or $T=C(T\cup \{\psi \left\}\right)\ni \psi$, members of $\mathsf{M}$ being thus both consistent and $\Delta$-disjunctive, by the weak $\Delta$-disjunctivity of C.) □

3.1.1. Multiplicative sentential calculi

Given any $\mathcal{R}=\langle \Gamma ,\varphi \rangle \in {Ru}_L$ and $\psi \in {Fm}_L$, put $\Delta (\mathcal{R},\psi )\triangleq \{\langle \Delta (\Gamma ,\psi ),\phi \rangle \mid \phi \in \Delta (\varphi ,\psi \left)\right\}$, elements of $\Delta ({\sigma }_{+1}\left(\mathcal{R}\right),x_0)$ being called Δ-multiplications of$\mathcal{R}$. Then, an L-calculus $\mathcal{C}$ is said to be Δ-multiplicative, if each multiplication of every rule of it is derivable in it, in which case, by induction on the length of $\mathcal{C}$-derivations, ${Cn}_{\mathcal{C}}$ is $\Delta$-multiplicative, and so, by the structurality of L-logics and Theorem 1, we get:

Corollary 1.  

A [finitary] L-logic C is Δ-disjunctive if[f] it is both weakly Δ-disjunctive and axiomatized by a Δ-multiplicative L-calculus, while both (11) and (12) (as well as (13)) hold.

3.2. Extensions versus interpretations

Here, we entirely follow the conventions adopted in Chapter 2 but allowing not necessarily finitary logics as well as finitary translations (viz., those with finite values). Fix any propositional language L, first-order ones $F{[}^{\prime }]=\langle L,P{[}^{\prime }]\rangle$, translations $\tau |\rho$ from $P|P^{\prime }$ to $P^{\prime }|P$ over L and an $F{[}^{\prime }]$-logic $C{[}^{\prime }]$. Then, $\tau$ is said to be compatible with$C^{\prime }$, if the condition (ii) of Definition 2.1 of [11] holds, that is (in view of the structurality of $C^{\prime }$), $C^{\prime }\left(\sigma \left[\tau \left[\Gamma \right]\right]\right)=C^{\prime }\left(\tau \left[\sigma \left[\Gamma \right]\right]\right)$, for all $\sigma \in {Sb}_L$ and all [one-element] $\Gamma \subseteq {Fm}_F$ (cf. Proposition 2.2 therein), i.e., for all $\sigma \in {Sb}_L$ and all $\Theta \in (img{C}^{\prime })$, ${\sigma }^{-1}\left[{\tau }^{-1}\left[\Theta \right]\right]={\tau }^{-1}\left[{\sigma }^{-1}\left[\Theta \right]\right]$, in which case $\tau$ is compatible with any extension of $C^{\prime }$. In that case, $\tau$ is called an interpretation of C in $C^{\prime }$, if the condition (i) of Definition 2.1 of [11] holds too, that is, $(imgC)={\tau }^{-1}[img{C}^{\prime }]\triangleq \{{\tau }^{-1}\left[\Theta \right]\mid \Theta \in (img{C}^{\prime })\}$ (cf. Proposition 2.4 therein). We start from presenting the following almost immediate observation:

Lemma 1.  

Let $C^{\prime \prime }(\geqq C^{\prime })$ be an $F^{\prime }$-logic (and $C^{\prime \prime \prime }[\geqq C]$ an F-logic as well as $\mathcal{C}$ an F-calculus). Suppose τ is compatible with $C^{\prime \prime }$ (resp., with $C^{\prime }$ [in particular, τ is an interpretation of C in $C^{\prime }$ {more specifically, C and $C^{\prime }$ are equivalent with 〈respect to〉τ and ρ}]). Then, ${\tau }^{-1}[img{C}^{\prime \prime }]$ is a closure system over ${Fm}_F$ closed under inverse substitutions, in which case τ is an interpretation of the F-logic ${\tau }^{-1}\left(C^{\prime \prime }\right)$ dual to ${\tau }^{-1}[img{C}^{\prime \prime }]$ in $C^{\prime \prime }$ ([while $C\leqq {\tau }^{-1}\left(C^{\prime \prime }\right)$ {whereas $C^{\prime \prime }={\rho }^{-1}\left({\tau }^{-1}\left(C^{\prime \prime }\right)\right)$, and so ${\tau }^{-1}\left(C^{\prime \prime }\right)$ and $C^{\prime \prime }$ are equivalent with 〈respect to〉τ and ρ}]). (Conversely, ${\tau }_{C^{\prime }}[img{C}^{\prime \prime \prime }]\triangleq \{\Theta \in (img{C}^{\prime })\mid {\tau }^{-1}\left[\Theta \right]\in (img{C}^{\prime \prime \prime })\}$ is a closure system over ${Fm}_{F^{\prime }}$ closed under inverse substitutions, in which case the $F^{\prime }$-logic ${\tau }_{C^{\prime }}\left(C^{\prime \prime \prime }\right)$ dual to ${\tau }_{C^{\prime }}[img{C}^{\prime \prime \prime }]$ is an extension of $C^{\prime }$, while ${\tau }_{C^{\prime }}\left[{Cn}_{\mathcal{C}}^{C^{\prime \prime \prime }}\right]={Cn}_{\tau \left[\mathcal{C}\right]}^{{\tau }_{C^{\prime }}\left(C^{\prime \prime \prime }\right)}$ [and ${\tau }^{-1}\left({\tau }_{C^{\prime }}\left(C^{\prime \prime \prime }\right)\right)=C^{\prime \prime \prime }$, τ being an interpretation of $C^{\prime \prime \prime }$ in ${\tau }_{C^{\prime }}\left(C^{\prime \prime \prime }\right)$ {whereas ${\tau }_{C^{\prime }}\left(C^{\prime \prime \prime }\right)={\rho }^{-1}\left(C^{\prime \prime \prime }\right)$, $C^{\prime \prime \prime }$ and ${\tau }_{C^{\prime }}\left(C^{\prime \prime \prime }\right)$ being equivalent with 〈respect to〉τ and ρ}]).

This, first, immediately yields the following infinitary extension of [11]:

Theorem 2.  

Suppose C and $C^{\prime }$ are equivalent with (respect to) τ and ρ. Then, ${\rho }^{-1}|{\tau }_{C^{\prime }}$ and ${\tau }^{-1}\parallel {\rho }_C$ form inverse to one another isomorphisms between the complete lattices of [axiomatic] extensions of C and $C^{\prime }$, corresponding ones being equivalent with (respect to) τ and ρ.

And what is more, as an equally immediate consequence of Lemma 1, we have the following important result, being formally beyond the scopes of [11] but implicitly contained, though not explicitly presented, therein:

Theorem 3.  

Suppose τ is an interpretation of C in $C^{\prime }$. Then, ${\tau }_{C^{\prime }}$ and ${\tau }^{-1}$ form a dual Galois retraction of the poset of extensions of $C^{\prime }$ onto that of C, τ being an interpretation of any extension $C^{\prime \prime }$ of C (relatively axiomatized by an F-calculus $\mathcal{C}$) in the extension ${\tau }_{C^{\prime }}\left(C^{\prime \prime }\right)$ of $C^{\prime }$ (relatively axiomatized by $\tau \left[\mathcal{C}\right]$).

This, in its turn, by Remark 2, yields a more canonical insight into the main universal result of [13] in the spirit of the outstanding work [11] plagiarized (like [13]) more and more by such crooks as Font, Pigozzi, et al.:

Corollary 2 (cf. [13]) Let ∇ be a translation from $\left\{D\right\}$ to $\{\approx \}$ over L, C an L-logic, $\mathsf{K}$ a class of L-algebras, $\mathsf{P}\triangleq \mathbf{ISP}\mathsf{K}$ and $C^{\prime }\triangleq {Cn}_{\mathsf{K}}^{\approx }={Cn}_{\mathsf{P}}^{\approx }$. Suppose ∇ is an interpretation of C in $C^{\prime }$, i.e., C is defined by ${\mathsf{K}}^{\nabla }\triangleq \{\langle \mathfrak{A},\{a\in A\mid \mathfrak{A}\vDash (\bigwedge {\nabla }_D)[x_0/a]\}\rangle \mid \mathfrak{A}\in \mathsf{K}\}$, viz., by ${\mathsf{P}}^{\nabla }$. Then, the mappings $C^{\prime \prime }\mapsto (\mathsf{P}\cap Mod\left({\nabla }_{C^{\prime }}\left(C^{\prime \prime }\right)\right))$ and $\mathsf{S}\mapsto {Cn}_{{\mathsf{S}}^{\nabla }}$ form a Galois retraction of the poset of sub-pre-varieties of $\mathsf{P}$ onto the one of extensions of C such that, for any L-calculus $\mathcal{C}$, $(\mathsf{P}\cap Mod\left({\nabla }_{C^{\prime }}\left({Cn}_{\mathcal{C}}^C\right)\right))=(\mathsf{P}\cap Mod(\nabla \left[\mathcal{C}\right]))$, while, for any $\mathsf{C}\subseteq \mathsf{P}$, ${Cn}_{{\left(\mathbf{ISP}\mathsf{C}\right)}^{\nabla }}={Cn}_{{\mathsf{C}}^{\nabla }}$.

4. Main issues

Here, we deal with the propositional languages $L_{+[,01]}^{(-)}\triangleq \{\wedge ,\vee [,\perp ,\top ](,\neg )\}$, where ∧ and ∨ are binary [while ⊥ and ⊤ are nullary] (whereas ¬ is unary) with [bounded] lattices {cf. [1]} viewed as $L_{+[,01]}$-algebras, $\varphi ⪅\psi$ standing for $\varphi \approx (\varphi \wedge \psi )$. Then, a [bounded] (De)/ Morgan/Kleene lattice [traditionally called a (De)/ Morgan/Kleene algebra; cf., e.g., [1] is any $L_{+[,01]}^{-}$-algebra with [bounded] distributive lattice $L_{+[,01]}$-reduct, satisfying:

$$\begin{array}{ccc}\hfill \neg \neg x_0& \approx & x_0,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \neg (x_0\wedge x_1)& \approx & (\neg x_0\vee \neg x_1),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill (x_0\wedge \neg x_0)& ⪅& (x_{0/1}\vee \neg x_{0/1}),\hfill \end{array}$$

(16)

their variety being denoted by $\left[\mathsf{B}\right](\mathsf{M}/\mathsf{K})\mathsf{L}$. Let ${\mathfrak{L}}_{n[,01]}$ be the chain [bounded] lattice over $n\in (\omega \setminus 2)$, while ${\mathfrak{K}}_{n[,01]}$ the [bounded] Kleene lattice with [bounded] lattice reduct ${\mathfrak{L}}_{n[,01]}$ and ${\neg }^{{\mathfrak{K}}_{n[,01]}}\triangleq \{\langle i,n-1-i\rangle \mid i\in n\}$, whereas ${\mathfrak{M}}_{4[,01]}$ the [bounded] Morgan lattice with [bounded] lattice reduct ${\mathfrak{L}}_{2[,01]}^2$ and ${\neg }^{{\mathfrak{M}}_{n[,01]}}:2^2\to 2^2,\langle j,k\rangle \mapsto \langle 1-k,1-j\rangle$, the following standard notations of elements of $2^2$ being used in this connection $\left(\mathsf{f}\right|\mathsf{t})\triangleq \langle 0|1,0|1\rangle$ and $(\mathsf{n}\parallel \mathsf{b})\triangleq \langle 0\parallel 1,1\parallel 0\rangle$, as well as ${(FDE/\left(LP\right|K3)/PC)}_{\left[01\right]}$ the logic of ${(\mathcal{M}/\mathcal{K}/\mathcal{K})}_{(4/3/2)[,01]}^{/|\top /}\triangleq \langle {(\mathfrak{M}/\mathfrak{K}/\mathfrak{K})}_{(4/3/2)[,01]},\{\mathsf{b}/\left(1\right|2)/1,\mathsf{t}/2/1\}\rangle$, being [the bounded (version of the)] “{relevance} first-degree entailment”/(“logic of paradox”|“Kleene three-valued logic”)/ “classical logic” [2]/ ([9]|[5])/[8], in which case the truth predicate of ${\mathcal{K}}_{3/2[,01]}^{\left[\right|\top ]/}$ is equationally definable by the translation ${\nabla }^{\left[\right|\top ]}\triangleq \left\{\langle D,\{(\neg x_0\left[\right|\top ])⪅x_0\}\rangle \right\}$ from $\left\{D\right\}$ to $\{\approx \}$ over $L_{+[,01]}^{-}$ in the sense that:

$${\mathcal{K}}_{3/2[,01]}^{\left[\right|\top ]/}\vDash \forall x_0(D\left(x_0\right)\leftrightarrow {\nabla }_D^{\left[\right|\top ]}),$$

(17)

and so the universal elaboration of [13] is equally applicable to the bounded versions of both the logic of paradox and Kleene three-valued logic. Then, for any $L_{+[,01]}^{-}$-algebras $\mathfrak{A},\mathfrak{B}$ and any $L_{+[,01]}^{-}$-rule $\mathcal{R}$:

$$\begin{array}{ccc}\hfill {hom}^{(}\mathrm{I}\{,)\mathrm{S}\}(\mathfrak{A},\mathfrak{B})& \subseteq & {hom}^{(}\mathrm{I}\{,)\mathrm{S}{\left\}}_{\right(}\mathrm{S})({\mathfrak{A}}^{{\nabla }^{\left[\right|\top ]}},{\mathfrak{B}}^{{\nabla }^{\left[\right|\top ]}}),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {(\mathfrak{A}\times \mathfrak{B})}^{{\nabla }^{\left[\right|\top ]}}& =& ({\mathfrak{A}}^{{\nabla }^{\left[\right|\top ]}}\times {\mathfrak{B}}^{{\nabla }^{\left[\right|\top ]}}),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill (\mathfrak{A}\in Mod\left({\nabla }^{[\top ]}\left(\mathcal{R}\right)\right))& \iff & ({\mathfrak{A}}^{{\nabla }^{\left[\right|\top ]}}\in Mod\left(\mathcal{R}\right)),\hfill \end{array}$$

(20)

so, for any $\mathsf{K}\subseteq {\mathsf{A}}_{L_{+[,01]}^{-}}$:

$${Cn}_{{\mathsf{K}}^{{\nabla }^{\left[\right|\top ]}}}={Cn}_{{\left(\mathbf{ISP}\mathsf{K}\right)}^{{\nabla }^{\left[\right|\top ]}}}.$$

(21)

4.1. An axiomatization of the bounded version of FDE

Let ${\mathcal{C}}_{\mathrm{H}}[,01]$ be the $L_{+[,01]}^{-}$-calculus, constituted the $L_{+}^{-}$-rules given by [10]:

$$\begin{array}{cccccc}\hfill \left(R_1\right)& \hfill {\displaystyle \frac{{\displaystyle x_0\wedge x_1}}{{\displaystyle x_0}}}& \hfill \left(R_2\right)& {\displaystyle \frac{{\displaystyle x_0\wedge x_1}}{{\displaystyle x_1\wedge x_0}}}& \left(R_3\right)& {\displaystyle \frac{{\displaystyle x_0{x}_1}}{{\displaystyle x_0\wedge x_1}}}\\ \hfill \left(R_4\right)& \hfill {\displaystyle \frac{{\displaystyle x_0}}{{\displaystyle x_0\vee x_1}}}& \hfill \left(R_5\right)& {\displaystyle \frac{{\displaystyle x_0\vee x_1}}{{\displaystyle x_1\vee x_0}}}& \left(R_6\right)& {\displaystyle \frac{{\displaystyle x_0\vee (x_1\vee x_2)}}{{\displaystyle (x_0\vee x_1)\vee x_2}}}\\ \hfill \left(R_7\right)& \hfill {\displaystyle \frac{{\displaystyle x_0\vee (x_1\wedge x_2)}}{{\displaystyle (x_0\vee x_1)\wedge (x_0\vee x_2)}}}& \hfill \left(R_8\right)& {\displaystyle \frac{{\displaystyle (x_0\vee x_1)\wedge (x_0\vee x_2)}}{{\displaystyle x_0\vee (x_1\wedge x_2)}}}& \left(R_9\right)& {\displaystyle \frac{{\displaystyle x_0\vee x_0}}{{\displaystyle x_0}}}\\ \hfill \left(R_{10}\right)& \hfill {\displaystyle \frac{{\displaystyle x_0\vee x_2}}{{\displaystyle \neg \neg x_0\vee x_2}}}& \hfill \left(R_{11}\right)& {\displaystyle \frac{{\displaystyle \neg \neg x_0\vee x_2}}{{\displaystyle x_0\vee x_2}}}& \left(R_{12}\right)& {\displaystyle \frac{{\displaystyle (\neg x_0\wedge \neg x_1)\vee x_2}}{{\displaystyle \neg (x_0\vee x_1)\vee x_2}}}\\ \hfill \left(R_{13}\right)& \hfill {\displaystyle \frac{{\displaystyle \neg (x_0\vee x_1)\vee x_2}}{{\displaystyle (\neg x_0\wedge \neg x_1)\vee x_2}}}& \hfill \left(R_{14}\right)& {\displaystyle \frac{{\displaystyle (\neg x_0\vee \neg x_1)\vee x_2}}{{\displaystyle \neg (x_0\wedge x_1)\vee x_2}}}& \left(R_{15}\right)& {\displaystyle \frac{{\displaystyle \neg (x_0\wedge x_1)\vee x_2}}{{\displaystyle (\neg x_0\vee \neg x_1)\vee x_2}}}\end{array}$$

[and the following additional $L_{+,01}^{-}$-rules and -axioms:

$$\left(R_{16}\right){\displaystyle \frac{{\displaystyle \perp \vee x_0}}{{\displaystyle x_0}}}\left(R_{17}\right){\displaystyle \frac{{\displaystyle \neg \top \vee x_0}}{{\displaystyle x_0}}}\left(A_1\right)\top \left(A_2\right)\neg \perp ].$$

Let $\mathcal{EM}\triangleq (x_0\vee \neg x_0)$ be the Excluded Middle axiom, $\mathcal{RS}\triangleq (\{x_0\vee x_1,\neg x_0\vee x_1\}⇝x_1)$ the Resolution rule, ${\mathcal{C}}_{\mathrm{H}}^{\overline{\mathcal{R}}}[,01]\triangleq ({\mathcal{C}}_{\mathrm{H}}[,01]\cup (img\overline{\mathcal{R}}))$, where $\overline{\mathcal{R}}\in {\{\mathcal{EM},\mathcal{RS}\}}^{*}$, and ${\mathcal{M}}_{4-\overline{c}[,01]}\triangleq ({\mathcal{M}}_{4[,01]}\upharpoonright (2^2\setminus (img\overline{c})))$, where $\overline{c}\in {(2^2\setminus {ϵ}_2)}^{*}$.

Lemma 2.  

${\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]$ is ∨-multiplicative.

Proof. 

According to [10], $FDE$ is axiomatized by ${\mathcal{C}}_{\mathrm{H}}$. Then, since its defining matrix ${\mathcal{M}}_4$ is ∨-disjunctive, by Theorem 1, it is ∨-multiplicative, while (13) holds for it, in which case both the rule ${\overline{R}}_6$ inverse to $R_6$ and the ∨-multiplication of any rule of ${\mathcal{C}}_{\mathrm{H}}$, being satisfied in $FDE$, are derivable in ${\mathcal{C}}_{\mathrm{H}}$, and so in ${\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]$. Moreover, by $R_4$, the ∨-multiplication of any axiom in $⌀[\cup \{A_1,A_2\}](\cup \left\{\mathcal{EM}\right\})$ is derivable in ${\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]$. [Finally, due to the following demonstration:

• $((\perp |\neg \top )\vee x_1)\vee x_0$ — Hypothesis;
• $(\perp |\neg \top )\vee (x_1\vee x_0)$ — ${\overline{R}}_6[x_0/(\perp |\neg \top ),x_2/x_0]:\left(1\right)$;
• $x_1\vee x_0$ — $R_{16|17}[x_0/(x_1\vee x_0)]:\left(2\right)$;

the ∨-multiplication of $R_{16|17}$, being derivable in $\{R_{16|17},{\overline{R}}_6\}$, is so in ${\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}},01$.] {Likewise, due to the following one:

• $(x_1\vee x_2)\vee x_0$ — Hypothesis;
• $(\neg x_1\vee x_2)\vee x_0$ — Hypothesis;
• $x_1\vee (x_2\vee x_0)$ — ${\overline{R}}_6[x_0/x_1,x_1/x_2,x_2/x_0]:\left(1\right)$;
• $\neg x_1\vee (x_2\vee x_0)$ — ${\overline{R}}_6[x_0/\neg x_1,x_1/x_2,x_2/x_0]:\left(2\right)$;
• $x_2\vee x_0$ — $\mathcal{RS}[x_0/x_1,x_1/(x_2\vee x_0)]:\left(3\right),\left(4\right)$;

the ∨-multiplication of $\mathcal{RS}$, being derivable in $\{\mathcal{RS},{\overline{R}}_6\}$, is so in ${\mathcal{C}}^{(}\mathcal{EM}){\mathcal{RS}}_{\mathrm{H}}[,01]$.} □

An $L_{+[,01]}^{-}$-matrix $\mathcal{A}$ is said to be (∧-)conjunctive, if $\langle \mathfrak{A},A\setminus D^{\mathcal{A}}\rangle$ is ∧-disjunctive.

Theorem 4.  

${Cn}_{{\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]}={Cn}_{{\mathcal{M}}_{4-\left(\mathsf{n}\right)\left\{\mathsf{b}\right\}[,01]}}$.

Proof. 

Clearly, since ${\mathfrak{M}}_{4[,01]}\in \left[\mathsf{B}\right]\mathsf{ML}$, ${\mathcal{M}}_{4-\left(\mathsf{n}\right)\left\{\mathsf{b}\right\}[,01]}$, being both conjunctive and ∨-disjunctive, is a model of ${\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]$, i.e., $C\triangleq {Cn}_{{\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]}\leqq {Cn}_{{\mathcal{M}}_{4-\left(\mathsf{n}\right)\left\{\mathsf{b}\right\}[,01]}}$. Conversely, by Theorem 1, Corollary 1, Lemma 2 and the inclusion $\{R_i\mid i\in \{4,5,9\}\}\subseteq {\mathcal{C}}^{(}{\mathcal{EM})\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01]$, C, being both finitary and ∨-disjunctive, is defined by a class $\mathsf{M}$ of consistent ∨-disjunctive $L_{+[,01]}^{-}$-matrices. Consider any $\mathcal{A}\in \mathsf{M}\subseteq Mod\left({\mathcal{C}}^{(}\mathcal{EM}\right){\left\{\mathcal{RS}\right\}}_{\mathrm{H}}[,01])$ and take any $a\in (A\setminus D^{\mathcal{A}})\ne ⌀$, in which case, by the truth of $R_1$, $R_2$ and $R_3$ in $\mathcal{A}$, this is conjunctive [while, by that of $A_1$, ${\top }^{\mathfrak{A}}\in D^{\mathcal{A}}$, whereas by that of $R_{16}$ under $[x_0/a]$, ${\perp }^{\mathfrak{A}}\notin D^{\mathcal{A}}$]. Then, [by the truth of $A_2$ in $\mathcal{A}$, ${\perp }^{\mathfrak{A}}\notin E\triangleq (A\setminus {\left({\neg }^{\mathfrak{A}}\right)}^{-1}\left[D^{\mathcal{A}}\right])$, while, by that of $R_{17}$ under $[x_0/a]$, ${\top }^{\mathfrak{A}}\in E$, whereas] by that of $R_{12}$, $R_{13}$, $R_{14}$ and $R_{15}$ under assignments containing $\langle x_2,a\rangle$, $\langle \mathfrak{A},E\rangle$ is both conjunctive and ∨-disjunctive, for $\mathcal{A}$ is so. Finally, consider any $b\in A$, in which case, by the truth of $R_{10}$ and $R_{11}$ in $\mathcal{A}$ under $[x_0/b,x_2/a]$, $(b\in D^{\mathcal{A}})\iff ({\neg }^{\mathfrak{A}}{\neg }^{\mathfrak{A}}b\in D^{\mathcal{A}})$ (while, by that of $\mathcal{EM}$ in $\mathcal{A}$ under $[x_0/b]$, $(\{b,{\neg }^{\mathfrak{A}}b\}\cap D^{\mathcal{A}})\ne ⌀$) {whereas, by that of $\mathcal{RS}$ in $\mathcal{A}$ under $[x_0/b,x_1/a]$, $\{b,{\neg }^{\mathfrak{A}}b\}⊈D^{\mathcal{A}}$}, and so $e\triangleq {\chi }_A^{D^{\mathcal{A}}}\in hom(\mathfrak{A}\upharpoonright L_{+[,01]},{\mathfrak{L}}_{2[,01]})\ni f\triangleq {\chi }_A^E$, $\{\langle c,\langle e\left(c\right),f\left(c\right)\rangle \rangle \mid c\in A\}$ being in ${hom}_{\mathrm{S}}(\mathcal{A},{\mathcal{M}}_{4-\left(\mathsf{n}\right)\left\{\mathsf{b}\right\}[,01]})$, as required, in view of (9). □

This, by (9) and the fact that $((+|{\pi }_0)\upharpoonright M_{4-\left(\right(\mathsf{n}\left|\mathsf{b}\right)/\left(\mathsf{n}\mathsf{b}\right))})$ is an isomorphism from ${\mathcal{M}}_{4-\left(\right(\mathsf{n}\left|\mathsf{b}\right)/\left(\mathsf{n}\mathsf{b}\right)\left)\right[,01]}$ onto ${\mathcal{K}}_{(3/2)[,01]}^{|\top /})$, immediately yields:

Corollary 3.${(FDE/\left(LP\right|K3)/PC)}_{\left[01\right]}$ is axiomatized by ${\mathcal{C}}^{/}\left(\mathcal{EM}\right|\mathcal{RS})/{(\mathcal{EM},\mathcal{RS})}_{\mathrm{H}}[,01]$. In particular, $K{3}_{\left[01\right]}$ is the extension of $FD{E}_{\left[01\right]}$, relatively axiomatized by $\mathcal{RS}$, while ${(LP\parallel PC)}_{\left[01\right]}$ is the axiomatic extension of ${(FDE\parallel K3)}_{\left[01\right]}$ relatively axiomatized by $\mathcal{EM}$.

This subsumes [10].

4.2. Extensions of the bounded logic of paradox and Kleene’s three-valued logic versus pre-varieties of Kleene algebras

Key observations enabling one to expand [13] onto the bounded case almost immediately are as follows:

Lemma 3.  

Let $\mathfrak{A}$ and $\mathfrak{B}$ be bounded lattices and $h\in hom(\mathfrak{A}\upharpoonright L_{+},\mathfrak{B}\upharpoonright L_{+})$. Suppose ${(\perp |\top )}^{\mathfrak{B}}\in h\left[A\right]$ (in particular, $h\left[A\right]=B$). Then, $h\left({(\perp |\top )}^{\mathfrak{A}}\right)={(\perp |\top )}^{\mathfrak{B}}$.

Proof. 

Take any $\left(a\right|b)\in A$ such that $h\left(a\right|b)={(\perp |\top )}^{\mathfrak{B}}$, in which case $\left(\left(a\right|b){(\wedge |\vee )}^{\mathfrak{A}}{(\perp |\top )}^{\mathfrak{A}}\right)={(\perp |\top )}^{\mathfrak{A}}$, so $h\left({(\perp |\top )}^{\mathfrak{A}}\right)=\left({(\perp |\top )}^{\mathfrak{B}}{(\wedge |\vee )}^{\mathfrak{B}}h\left({(\perp |\top )}^{\mathfrak{A}}\right)\right)={(\perp |\top )}^{\mathfrak{B}}$. □

Lemma 4.  

For any {2-element} [bounded] Morgan lattice $\mathfrak{A}$ and any (distinct) $a,b\in A$, $\{\langle 0,\left(\left(a{\wedge }^{\mathfrak{A}}{\neg }^{\mathfrak{A}}a\right){\wedge }^{\mathfrak{A}}\left(b{\wedge }^{\mathfrak{A}}{\neg }^{\mathfrak{A}}b\right)\right)\left[{\wedge }^{\mathfrak{A}}{\perp }^{\mathfrak{A}}\right]\rangle ,\langle 1,\left(\left(a{\vee }^{\mathfrak{A}}{\neg }^{\mathfrak{A}}a\right){\vee }^{\mathfrak{A}}\left(b{\vee }^{\mathfrak{A}}{\neg }^{\mathfrak{A}}b\right)\right)\left[{\vee }^{\mathfrak{A}}{\top }^{\mathfrak{A}}\right]\rangle \in {hom}^{(}\mathrm{I}\{,\mathrm{S}\})({\mathfrak{K}}_{2[,01]},\mathfrak{A})$.

Since ${\mathfrak{K}}_3$ has no non-one-element subalgebra not retaining bounds, by (10), Lemma 3 and [12], we, first, have the following well-known fact (cf., e.g., [1]):

Corollary 4.  

$\mathsf{BKL}=\mathbf{ISP}\left({\mathbf{P}}^{\mathrm{U}}\right){\mathfrak{K}}_{3,01}$.

Let $N\left(L\right)P_{01}$ be the extension of $L{P}_{01}$ relatively axiomatized by the Ex Contradictione Quodlibet rule $\mathcal{NP}\triangleq (\{x_0,\neg x_0\}⇝x_1)$, viz., the least non-paraconsistent extension of $L{P}_{\left[01\right]}$. Then, a [bounded] Kleene lattice is said to be non-paraconsis-tent, if it satisfies $\nabla \left(\mathcal{NP}\right)$, i.e., satisfies [13]:

$$\{\neg x_0\approx x_0\}⇝(\nabla [x_0/x_1]),$$

(22)

in which it satisfies $\left(\right)[x_1/\neg x_1]$, and so, by the right alternative of (16), satisfies [12]:

$$\{\neg x_0\approx x_0\}⇝(x_0\approx x_1),$$

(23)

i.e., it is non-idempotent in the sense of [12]. Conversely, any [bounded] Kleene lattice, satisfying (23), satisfies $\left(\right)[x_1/\neg x_1]$, in which case it is non-paraconsistent, and so non-paraconsistent [bounded] Kleene lattices are exactly non-idempotent ones, their quasi-variety being denoted by $\mathsf{N}\left[\mathsf{B}\right]\mathsf{KL}$.

Let ${\mathfrak{NK}}_{6\left[\right(+2),01]}\triangleq \left(\left(({\mathfrak{K}}_{3[,01]}\times {\mathfrak{K}}_{2[,01]})\left[(\times {\mathfrak{K}}_{3,01})\right]\right)\left[(\upharpoonright (((3\times 2)\times \left\{1\right\})\cup \{\langle 2·i,i,2·i\rangle \mid i\in 2\}))\right]\right)$ and ${\mathcal{NK}}_{6,01}\triangleq ({\mathcal{K}}_{3,01}\times {\mathcal{K}}_{2,01})$. Then, since the only non-one-element subalgebra of ${\mathfrak{NK}}_6$ not retaining bounds is that with two-element carrier $\left\{1\right\}\times 2$, by (10), Lemmas 3, 4 and [12]/[13], we immediately have:

Corollary 5.  

$\mathsf{NBKL}=\mathbf{ISP}\left({\mathbf{P}}^{\mathrm{U}}\right){\mathfrak{NK}}_{6,01}$.

Let $M{P}_{\left[01\right]}$ be the extension of $\left(N\right)L{P}_{\left[01\right]}$ relatively axiomatized by the Modus Ponens rule for material implication $\mathcal{MP}\triangleq (\{x_0,\neg x_0\vee x_1\}⇝x_1)$ (in view of the ∨-disjunctivity of ${\mathcal{K}}_{3[,01]}$ and Theorem 1). Then, a [bounded] Kleene lattice is said to be regular/classical (cf. Definition 4.6/4.11 of [12]/[13]), if it satisfies $\nabla \left(\mathcal{MP}\right)$, i.e., satisfies (10/14) of [12]/[13]:

$$(\nabla \cup \{(x_0\wedge \neg x_1)⪅(\neg x_0\vee x_1)\})⇝(\nabla [x_0/x_1]),$$

(24)

their quasi-variety being denoted by $\mathsf{R}\left[\mathsf{B}\right]\mathsf{KL}\subseteq \mathsf{N}\left[\mathsf{B}\right]\mathsf{KL}$. Since the only non-one-element subalgebra of ${\mathfrak{K}}_4$ not retaining bounds is that with two-element carrier $\{1,2\}$, by (10), Lemmas 3, 4 and [12], we immediately get:

Corollary 6.  

$\mathsf{R}\left[\mathsf{B}\right]\mathsf{KL}=\mathbf{ISP}\left({\mathbf{P}}^{\mathrm{U}}\right){\mathfrak{K}}_{4[,01]}$.

From now on, we use (17), (18), (19), (20), (21) and Corollary 2 tacitly. Then, by Corollaries 4 and 5, we, first, have:

Theorem 5.(An arbitrary exrension C of) $\left[N\right]L{P}_{01}={Cn}_{\left[\mathcal{N}\right]{\mathcal{K}}_{3[+3],01}}$ is defined by $\left[\mathsf{N}\right]{\mathsf{BKL}}^{\nabla }(\cap Mod\left(C\right))$.

Theorem 6.$(Thm)\left(M{P}_{\left[01\right]}\right)=(Thm)\left(P{C}_{\left[01\right]}\right)(=Thm\left(L{P}_{\left[01\right]}\right))$.

Proof. 

Then, by Theorem 5 and Corollary 6, $M{P}_{\left[01\right]}$, being defined by $\mathsf{R}\left[\mathsf{B}\right]\mathsf{KL}$, is defined by ${\mathcal{K}}_{4[,01]}\triangleq {\mathfrak{K}}_{4[,01]}^{\nabla }=\langle {\mathfrak{K}}_{4[,01]},\{2,3\}\rangle$, while ${\chi }_4^{\{2,3\}}\in {hom}_{\mathrm{S}}^{\mathrm{S}}({\mathcal{K}}_{4[,01]},{\mathcal{K}}_{2[,01]})$, whereas $\left\{\langle i,{\chi }_4^{\left\{3\right\}}\left(i\right)+{\chi }_4^{4\setminus 1}\left(i\right)\rangle \right\}\in {hom}^{\mathrm{S}}({\mathcal{K}}_{4[,01]},{\mathcal{K}}_{3[,01]})$, (9) ending the proof.3 □

Theorem 7.  

Proper consistent extensions of $L{P}_{01}$ form the two-element chain $N{P}_{01}\lneqq P{C}_{01}$.

Proof. 

First, ${\mathcal{K}}_{3,01}\neg \vDash \mathcal{NP}{[x_i/(1-i)]}_{i\in 2}$, while ${\mathcal{NK}}_{6,01}\neg \vDash \mathcal{MP}{[x_j/\langle 1-j,1\rangle ]}_{j\in 2}$, whereas ${\mathcal{K}}_{2,01}\neg \vDash x_0[x_0/0]$. Then, by Theorems 5 and 6, $N{P}_{01}$ and $P{C}_{01}$ are proper consistent extensions of $L{P}_{01}$ forming the chain involved. Finally, consider any consistent extension C of $L{P}_{01}$, in which case, by Theorem 5, there is a non-one-element $\mathfrak{A}\in \mathsf{BKL}$ such that ${\mathfrak{A}}^{\nabla }\in Mod\left(C\right)$, and so, by (9) and Lemma 4, $C\leqq P{C}_{01}$. In particular, $C=P{C}_{01}$, whenever $P{C}_{01}\leqq C$. Otherwise, consider the following complementary cases:

• $N{P}_{01}\leqq C$,
  in which case, by Theorem 5, there is some $\mathfrak{A}\in (\mathsf{NBKL}\setminus \mathsf{RBKL})$ such that ${\mathfrak{A}}^{\nabla }\in Mod\left(C\right)$, and so $\mathfrak{B}\triangleq (\mathfrak{A}\upharpoonright L_{+}^{-})\in (\mathsf{NKL}\setminus \mathsf{RKL})$. Then, by the case 4/3 of the proof of Theorem 4.8/4.11 of [12]/[13], there is an $e\in {hom}^{\mathrm{I}}({\mathfrak{NK}}_6,\mathfrak{B})$. Consider the following complementary subcases:
  • $e\left(\langle 0,0\rangle \right)={\perp }^{\mathfrak{A}}$,
    in which case $e\left(\langle 2,1\rangle \right)={\top }^{\mathfrak{A}}$, and so $e\in {hom}^{\mathrm{I}}({\mathfrak{NK}}_{6,01},\mathfrak{A})$. Then, by (9), ${\mathcal{NK}}_{6,01}={\mathfrak{NK}}_{6,01}^{\nabla }\in Mod\left(C\right)$.
  • $e\left(\langle 0,0\rangle \right)\ne {\perp }^{\mathfrak{A}}$,4
    in which case $e\left(\langle 2,1\rangle \right)\ne {\top }^{\mathfrak{A}}$, so $((({\pi }_0\upharpoonright (N{K}_6\times \left\{1\right\}))\circ e)\cup \{\langle 0,0,0,{\perp }^{\mathfrak{A}}\rangle ,\langle 2,1,2,{\top }^{\mathfrak{A}}\rangle \})\in {hom}^{\mathrm{I}}({\mathfrak{NK}}_{8,01},\mathfrak{A})$. Also, $({\pi }_0\upharpoonright N{K}_{8,01})\in {hom}_{\mathrm{S}}^{\mathrm{S}}({\mathfrak{NK}}_{8,01}^{\nabla },{\mathcal{NK}}_{6,01})$. Then, by (9), ${\mathcal{NK}}_{6,01}\in Mod\left(C\right)$.
  Thus, anyway, ${\mathcal{NK}}_{6,01}\in Mod\left(C\right)$, in which case, by Theorem 5, $C\leqq N{P}_{01}$, and so $C=N{P}_{01}$.
• $N{P}_{01}\nleqq C$,
  in which case, by Theorem 5, there is some $\mathfrak{A}\in (\mathsf{BKL}\setminus \mathsf{NBKL})$ such that ${\mathfrak{A}}^{\nabla }\in Mod\left(C\right)$, and so there is some $a\in A\ne \left\{a\right\}$ such that ${\neg }^{\mathfrak{A}}a=a$. Then, $\{\langle 0,{\perp }^{\mathfrak{A}}\rangle ,\langle 1,a\rangle ,\langle 2,{\top }^{\mathfrak{A}}\rangle \}\in {hom}^{\mathrm{I}}({\mathfrak{K}}_{3,01},\mathfrak{A})$, in which case, by (9), ${\mathcal{K}}_{3,01}={\mathfrak{K}}_{3,01}^{\nabla }\in Mod\left(C\right)$, and so $C=L{P}_{01}$. □

□

Theorem 8.  

$P{C}_{01}$ is the only proper consistent extension of $K{3}_{01}$.

Proof. 

Consider a consistent extension C of $K{3}_{01}$ distinct from $P{C}_{01}$, in which case, by Theorem 7, $P{C}_{01}\nleqq C$, and so, by Corollaries 3 and 4, there is some $\mathfrak{A}\in \mathsf{BKL}$ such that ${\mathfrak{A}}^{{\nabla }^{\top }}\in (Mod\left(C\right)\setminus Mod\left(\mathcal{EM}\right))$. Then, there is some $a\in A$ such that $b\triangleq \left(a{\vee }^{\mathfrak{A}}{\neg }^{\mathfrak{A}}a\right)\ne {\top }^{\mathfrak{A}}$. Consider the following complementary cases:

• ${\neg }^{\mathfrak{A}}b=b$,
  in which case $\{\langle 0,{\perp }^{\mathfrak{A}}\rangle ,\langle 1,b\rangle ,\langle 2,{\top }^{\mathfrak{A}}\rangle \}\in {hom}^{\mathrm{I}}({\mathfrak{K}}_{3,01},\mathfrak{A})$, and so, by (9), ${\mathcal{K}}_{3,01}^{\top }\in Mod\left(C\right)$.
• ${\neg }^{\mathfrak{A}}b\ne b$,
  in which case $\{\langle 0,{\perp }^{\mathfrak{A}}\rangle ,\langle 1,{\neg }^{\mathfrak{A}}b\rangle ,\langle 2,b\rangle ,\langle 3,{\top }^{\mathfrak{A}}\rangle \}\in {hom}^{\mathrm{I}}({\mathfrak{K}}_{4,01},\mathfrak{A})$, and so, by (9), ${\mathcal{K}}_{3,01}^{\top }\in Mod\left(C\right)$, for $\{\langle i,{\chi }_4^{4\setminus 1}\left(i\right)+{\chi }_4^{4\setminus 3}\left(i\right)\rangle \mid i\in 4\}\in {hom}_{\mathrm{S}}^{\mathrm{S}}({\mathfrak{K}}_{4,01}^{{\nabla }^{\top }},{\mathcal{K}}_{3,01}^{\top })$.

Thus, in any case, ${\mathcal{K}}_{3,01}^{\top }\in Mod\left(C\right)$, i.e., $C\leqq K{3}_{01}$, C being equal to $K{3}_{01}$, as required, in view of Corollary 3, for ${\mathcal{K}}_{3,01}^{\top }\neg \vDash \mathcal{EM}[x_0/1]$. □

If, for any ${\nabla }^{\prime }\subseteq {\left({Tm}_L^1\right)}^2$, $\forall x_0(D\left(x_0\right)\leftrightarrow (\bigwedge {\nabla }^{\prime }))$ was true in ${\mathcal{K}}_3^{\top }$, then it would be true in ${\mathcal{K}}_1\triangleq ({\mathcal{K}}_3\upharpoonright \left\{1\right\})$, in which case, since ${\mathcal{K}}_1\vDash \forall x_0(\bigwedge {\nabla }^{\prime })$, $\forall x_{0}D\left(x_0\right)$ would be true in ${\mathcal{K}}_1$, and so $1\in K_1$ would be in $D^{{\mathcal{K}}_1}=⌀$. Nevertheless, though the universal algebraic approach developed in [13] is thus not applicable to $K_3$, Theorem 8 is still so as follows.

Lemma 5.  

Any extension C of $K3$ with(out) theorems is (the theorem-less version of) the $L_{+}^{-,D}$-fragment of an extension of $K{3}_{01}$.

Proof. 

Consider any $T\in \left(\right(imgC)\setminus 1)$, in which case there exists some $H_T\subseteq hom({\mathfrak{Tm}}_{L_{+}^{-}},{\mathfrak{K}}_3)$ such that $T=({Fm}_{L_{+}^{-}}\cap \left({\bigcap }_{h\in H_T}{h}^{-1}\left[\left\{2\right\}\right]\right))$, and so $g_T:{Fm}_{L_{+}^{-}}\to 3^{H_T},\phi \mapsto {\langle f\left(\phi \right)\rangle }_{f\in H_T}$ is a [surjective] strict homomorphism from $\langle {\mathfrak{Tm}}_{L_{+}^{-}},T\rangle$ [on]to $[{\mathcal{A}}_T\triangleq ]\left({\left({\mathcal{K}}_3^{\top }\right)}^{H_T}[\upharpoonright (img{g}_T)]\right)$. Take any $\varphi \in T\ne ⌀$, in which case $A_T\ni g_T\left(\varphi \right)=(H_T\times \left\{2\right\})$, and so $A_T\ni g_T(\neg \varphi )=(H_T\times \left\{0\right\})$. Then, $A_T$ forms a subalgebra of ${\mathfrak{D}}_T\triangleq {\mathfrak{K}}_{3,01}^{H_T}$, in which case, by (9), ${\mathcal{B}}_T\triangleq \langle {\mathfrak{D}}_T\upharpoonright A_T,D^{{\mathcal{A}}_T}\rangle$, being a submatrix of ${\left({\mathcal{K}}_{3,01}^{\top }\right)}^{H_T}$, is a model of $K{3}_{01}$, and so $K{3}_{01}\leqq C^{\prime }\triangleq {Cn}_{\{{\mathcal{B}}_T\mid T\in ((imgC)\setminus 1)\}}$. Thus, by (9), $C={Cn}_{\left\{{\mathfrak{Tm}}_{L_{+}^{-}}\right\}\times ((imgC)\setminus 1)}^{(+0)}={Cn}_{\{{\mathcal{A}}_T\mid T\in ((imgC)\setminus 1)\}}^{(+0)}={(C^{\prime }\upharpoonright L_{+}^{-,D})}^{(+0)}$. □

Let $IC$ be the inconsistent $L_{+}^{-}$-logic.

Corollary 7.  

Proper extensions of $K3$ form the diamond lattice, isomorphic to ${\mathfrak{L}}_2^2$ under ${\iota }^{-1}$, where $\iota :2^2\to \wp {\left({Fm}_{L_{+}^{-}}\right)}^{\wp \left({Fm}_{L_{+}^{-}}\right)},\langle 0|1,1[-1]\rangle )\mapsto \left(P\right|I)C^{[+0]}={Cn}_{\left\{[x_1⇝]\left(\mathcal{EM}\right|x_0)\right\}}^{LP\left|\right(LP)}={Cn}_{\left(\left\{{\mathcal{K}}_2\right\}\right|⌀)[\cup \left\{{\mathcal{K}}_1\right\}]}$ is injective.

Proof. 

Clearly, no axiom is true in the {sub}matrix ${\mathcal{K}}_1$ {of ${\mathcal{K}}_3^{\top }$} with empty truth predicate, while ${\mathcal{K}}_{3\parallel 2}^{\top \parallel }\neg \vDash (\langle x_1⇝\rangle (\mathcal{EM}\parallel x_0)[x_0/(1\parallel 0)\langle ,x_1/(2\parallel 1)\rangle ]$, (6), (7), (9), Corollary 3, Theorem 8 and Lemma 5 completing the argument. □

4.3. Cut-free versions of Gentzen calculus

Let $L{K}^{(}-\mathrm{R}/{\langle \mathrm{R}\rangle \mathrm{C}\left]\right)}_{(}/\{+\mathrm{CFC}\})\left[01\right]$ be the $({\omega }^2\setminus {\left(1[-1]\right)}^2)$-sequent $L_{+[,01]}^{-}$-calculus constituted by the following strucural rules (except for Reflexivity/“〈both Reflexivity and〉 Cut {with non-$\left(1\right[-1\left]\right)$-ary sequent predicate in conclusion}”):

$$\begin{array}{ccc}Reflexivity\hfill & \multicolumn{2}{c}{x_0⊢x_0}\\ (/\{Context-Free\left\}\right)Cut\hfill & \multicolumn{2}{c}{{\displaystyle \frac{{\displaystyle \{\Lambda ,x_0⊢\Xi ;\Lambda ⊢\Xi ,x_0\}}}{{\displaystyle \Lambda ⊢(\Xi *({\overline{x}}_{\mathbb{k}}\circ {\sigma }_{+1}))}}}}\\ & Left& Right\\ Enlargement\hfill & {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta }}{{\displaystyle \Gamma ,x_0⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta }}{{\displaystyle \Gamma ⊢\Delta ,x_0}}}\\ Contraction\hfill & {\displaystyle \frac{{\displaystyle \Gamma ,x_0,x_0⊢\Delta }}{{\displaystyle \Gamma ,x_0⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta ,x_0,x_0}}{{\displaystyle \Gamma ⊢\Delta ,x_0}}}\\ Permutation\hfill & {\displaystyle \frac{{\displaystyle \Gamma ,x_0,x_1,\Theta ⊢\Delta }}{{\displaystyle \Gamma ,x_1,x_0,\Theta ⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta ,x_0,x_1,\Theta }}{{\displaystyle \Gamma ⊢\Delta ,x_1,x_0,\Theta }}}\end{array}$$

where $\Gamma ,\Delta ,\Theta \in {Var}^{*}\ni \Lambda ,\Xi (\notin (⌀/(⌀\{\cup {Var}^{+}\})))$ and $\mathbb{k}\triangleq \left(\right(1-min(1,(dom\Lambda )+(dom\Xi \left)\right)\left)\right[·0\left]\right)$, together with the following logical rules [and axioms]:

$$\begin{array}{ccc}& Left& Right\\ & [\perp ⊢& ⊢\top ]\\ (\neg )\hfill & {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta ,x_0}}{{\displaystyle \Gamma ,\neg x_0⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \Gamma ,x_0⊢\Delta }}{{\displaystyle \Gamma ⊢\Delta ,\neg x_0}}}\\ (\wedge )\hfill & {\displaystyle \frac{{\displaystyle \Gamma ,x_0,x_1⊢\Delta }}{{\displaystyle \Gamma ,x_0\wedge x_1⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \{\Gamma ⊢\Delta ,x_0;\Gamma ⊢\Delta ,x_1\}}}{{\displaystyle \Gamma ⊢\Delta ,x_0\wedge x_1}}}\\ (\vee )\hfill & {\displaystyle \frac{{\displaystyle \{\Gamma ,x_0⊢\Delta ;\Gamma ,x_1⊢\Delta \}}}{{\displaystyle \Gamma ,x_0\vee x_1⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta ,x_0,x_1}}{{\displaystyle \Gamma ⊢\Delta ,x_0\vee x_1}}}\end{array}$$

where $\Gamma ,\Delta \in {Var}^{*}$ (as well as [both] the rules inverse to logical ones [and the following constant elimination rules:

$$\begin{array}{cc}Left& Right\\ {\displaystyle \frac{{\displaystyle \Gamma ,\top ⊢\Delta }}{{\displaystyle \Gamma ⊢\Delta }}}& {\displaystyle \frac{{\displaystyle \Gamma ⊢\Delta ,\perp }}{{\displaystyle \Gamma ⊢\Delta }}}\end{array}$$

where $\Gamma ,\Delta \in {Var}^{*}$]) its rules being derivable in $L{K}_{\left[01\right]}$.

Let

$${\tau }_{\left[01\right]}\triangleq \{\langle {⊢}_n^m,\left\{{\wedge }_{+}\left([\langle ]{\overline{x}}_m*({\overline{x}}_n\circ {\sigma }_{+m})[,\perp \rangle ]\right)\right\}\rangle \mid \langle m,n\rangle \in ({\omega }^2\setminus {\left(1[-1]\right)}^2)\}$$

and $\rho \triangleq \left\{\langle D,\{⊢x_0\}\rangle \right\}$ be parameter-less translations from $P_{{\omega }^2\setminus {\left(1[-1]\right)}^2}^{⊢}$ to $\left\{D\right\}$ and vice versa over $L_{+[,01]}^{-}$ in the sense of the fundamental work [11] we follow here tacitly.

Lemma 6.  

${Cn}_{L{K}^{-}{\mathrm{RC}}_{\left[01\right]}}$ and $FD{E}_{\left[01\right]}$ are equivalent with (respect to) ${\tau }_{\left[01\right]}$ and ρ.

Proof. 

First, for any $(\Gamma ⇝\Psi )\in L{K}^{-}{\mathrm{RC}}_{\left[01\right]}$ and any $\overline{\Phi }\in {\Gamma }^{\left|\Gamma \right|}$ with range-image $\Gamma$, $\left({\wedge }_{+}\left([\langle ]\overline{\Phi }\circ {\tau }_{\left[01\right]}[,\top \rangle ]\right)\right)⪅{\tau }_{\left[01\right]}\left(\Psi \right)$ is true in $\left[\mathsf{B}\right]\mathsf{ML}\ni {\mathfrak{M}}_{4[,01]}$, so ${\tau }_{\left[01\right]}\left[\Gamma \right]⇝{\tau }_{\left[01\right]}\left(\Psi \right)$ is true in ${\mathcal{M}}_{4[,01]}$, for $D^{{\mathcal{M}}_{4[,01]}}[\ni {\top }^{{\mathfrak{M}}_{4,01}}]$ is a filter of ${\mathfrak{M}}_{4[,01]}\upharpoonright L_{+}$. Likewise, $x_0\approx \rho \left({\tau }_{\left[01\right]}\left(D\right)\right)$ is true in $\left[\mathsf{B}\right]\mathsf{ML}\ni {\mathfrak{M}}_{4[,01]}$, so both $x_0⇝{\tau }_{\left[01\right]}\left(\rho \left(D\right)\right)$ and ${\tau }_{\left[01\right]}\left(\rho \left(D\right)\right)⇝x_0$ are true in ${\mathcal{M}}_{4[,01]}$. Conversely, for any $(\Gamma ⇝\phi )\in {\mathcal{C}}_{\mathrm{H}}[,01]$, $\rho \left[\Gamma \right]⇝\rho \left(\phi \right)$ is derivable in $L{K}^{-}{\mathrm{RC}}_{\left[01\right]}$. Finally, for all $\langle m,n\rangle \in ({\omega }^2\setminus {\left(1[-1]\right)}^2)$, both ${\rm Y}⇝\rho \left({\tau }_{\left[01\right]}\left({⊢}_n^m\right)\right)$ and $\rho \left({\tau }_{\left[01\right]}\left({⊢}_n^m\right)\right)⇝{\rm Y}$, where ${\rm Y}\triangleq ({\overline{x}}_m⊢({\overline{x}}_n\circ {\sigma }_{+m}))$, are derivable in $L{K}^{-}{\mathrm{RC}}_{\left[01\right]}$. Then, Theorems 2.24 of [11] and 4 complete the argument. □

Theorem 9.  

${Cn}_{L{K}_{\left[01\right]}^{-}(\mathrm{C}/\mathrm{R})}$ and ${(LP/K3)}_{\left[01\right]}$ are equivalent with (respect to) ${\tau }_{\left[01\right]}$ and ρ.

Proof. 

Clearly, ${\tau }_{\left[01\right]}(\mathrm{Reflexivity}/\mathrm{Cut})$ is true in ${\mathcal{K}}_{3[,01]}^{/\top }$, while $\rho (\mathcal{EM}/\mathcal{RS})$ is derivable in $L{K}_{\left[01\right]}^{-}(\mathrm{C}/\mathrm{R})$, Corollaries 2.27 of [11], 3 and Lemma 6 ending the proof. □

Corollary 8 (cf. [13] for the non-[]-optional case) Proper consistent extensions of ${Cn}_{L{K}_{\left[01\right]}^{-}\mathrm{C}}$ form the two-element chain ${Cn}_{L{K}^{-}{\mathrm{C}}_{+}\mathrm{CFC}[,01]}\lneqq {Cn}_{L{K}_{\left[01\right]}}$, the lesser/greater being equivalent to ${(NP/PC)}_{\left[01\right]}$ with (respect to) ${\tau }_{\left[01\right]}$ and ρ.

Proof. 

Clearly, ${\tau }_{\left[01\right]}\left[\{x_0⊢;⊢x_0\}\right]⇝{\tau }_{\left[01\right]}\left(⊢({\overline{x}}_{1[·0]}\circ {\sigma }_{+1})\right)$ is true in ${\mathcal{NK}}_{6[,01]}$. Conversely, $\rho \left[\{x_0,\neg x_0\}\right]⇝\rho \left(x_1\right)$ is derivable in $L{K}^{-}{\mathrm{C}}_{+}\mathrm{CFC}[,01]$. Likewise, for any instance $\Gamma ⇝\Phi$ of Cut ${\tau }_{\left[01\right]}\left[\Gamma \right]⇝{\tau }_{\left[01\right]}\left(\Phi \right)$ is true in ${\mathcal{K}}_{2[,01]}$. Finally, $\rho \left[\{x_0,\neg x_0\vee x_1\}\right]⇝\rho \left(x_1\right)$ is derivable in $L{K}_{\left[01\right]}$. Then, Theorems 2, 5, 6, 7,13], 9 and [11] complete the argument. □

Corollary 9.  

${Cn}_{L{K}_{01}}$ is the only proper consistent extension of ${Cn}_{L{K}_{01}^{-}\mathrm{C}}$.

Proof. 

Clearly, ${\tau }_{01}\left(\mathrm{Reflexivity}\right)$ is true in ${\mathcal{K}}_{2,01}$. Conversely, $\rho \left(\mathcal{EM}\right)$ is derivable in $L{K}_{01}$. Then, Theorems 2, 8, 9 and [11] complete the argument. □

Likewise, by Theorems 2 and 9 as well as Corollaries [11] and 7, we eventually get:

Corollary 10.  

Proper extensions of ${Cn}_{L{K}^{-}\mathrm{R}}$ form the diamond lattice, isomorphic to the one of those of $K3$ under $\kappa \triangleq \{\langle {Cn}_{L{K}^{-}\mathrm{R}\cup \{(⊢x_1)⇝(⊢x_0\vee \neg x_0)\}},P{C}^{+0}\rangle ,\langle {Cn}_{L{K}^{-}\mathrm{R}\cup \{(⊢x_1)⇝(⊢x_0)\}},I{C}^{+0}\rangle ,\langle {Cn}_{L{K}^{-}\mathrm{R}\cup \{⊢x_0\}},IC\rangle ,\langle {Cn}_{LK},PC\rangle \}$, any $C\in (dom\kappa )$ and $\kappa \left(C\right)$ being equivalent with [respect to] τ and ρ.