Truth as Consistent Assertion

1. Introduction

So much has been written about the notion of truth and the importance we attach to it in our daily lives, yet our common understanding of it remains poor. Grasping all the major accounts of truth considered today is not a simple task, and even if one eventually succeeds, they are faced with a choice between the so-called ‘substantial’ and ‘deflationary’ accounts. One may argue whether the former actually provide any substantial information about truth, and whether the latter provide any information at all. There is an urgent need for our societies to find a common ground in terms of an account of truth, especially in light of pervasive relativism, or the ‘Who is to say?’ and ‘That’s just your opinion’ mantras, as Blackburn put it (Blackburn, 2005).

With this article, I aim to contribute to changing this pessimistic picture. I aim to provide a sketch of a theory of truth that is intuitive and straightforward to apply. To this end, I will draw upon some lesser-known ideas of the Polish analytic philosopher Kazimierz Ajdukiewicz.

2. Assertional Rules

Ajdukiewicz, in (1931) and (1934), developed theories of meaning based on what he called ‘language directives’, ‘meaning directives’, or ‘meaning rules’.1 These rules establish a relationship between the forms of language sentences on the one hand and specific circumstances on the other. They are meant to reflect the fact that the language user, in certain circumstances, feels motivated to assert sentences of a certain form.

Ajdukiewicz specified three types of meaning rules, with the reservation that the list may not be exhaustive:

1.

An empirical rule makes the assertion of a sentence dependent on empirical data. For example, if an English speaker feels pain, the speaker asserts the sentence ‘it hurts’.2 This is an example of an empirical rule in English. Another example: if an English speaker sees a fire, the speaker asserts the sentence ‘fire!’. It is worth noting that the asserted sentence need not be a grammatically complete declarative sentence, but must be a sentence in the logical sense (cf. 1985, pp. 126, 148–149).

2.

A deductive rule makes the assertion of a sentence dependent on the assertion of other sentences. For example: if ‘2’ has been defined in the language of arithmetic as ‘1 + 1’, and a user of this language asserts a sentence S including ‘2’, the user asserts the sentence built by replacing the ‘2’ in S with ‘1 + 1’.

3.

An axiomatic rule makes the assertion of a sentence independent of any circumstances, i.e., the sentence is to be unconditionally asserted. For example, an English speaker unconditionally asserts every sentence of the form ‘every A is A’.

The meaning rules have interesting characteristics (cf. 1985, pp. 129–130, 149–154):

i.

They do not determine nor establish the truth conditions of a sentence, but only the assertion conditions.

ii.

They are closely related to the meaning of a sentence, but they are not the meaning itself.

iii.

The assertion of sentences according to the meaning rules ‘is marked by strict obviousness and irrevocable decisiveness’3 (because of that, it is not possible to indicate meaning rules for every sentence of a natural language).

iv.

Together with the vocabulary and syntax rules, they co-define the language. If someone does not follow the meaning rules of a given language, they cannot be said to speak that language.

v.

The user of a language does not have to know the meaning rules of that language, they just have to follow them.

The features i. and ii. make the meaning rules a suitable basis for defining truth for a given language, as they seem to be a simple, low-level concept.

From now on, I will modify the concept of meaning rules to suit the task of defining truth for a natural language spoken by any particular person. To avoid confusion with Ajdukiewicz’s original concept, I will rename it ‘assertional rules’.

We can allow assertional rules to indicate not only the assertion conditions of a sentence, but also its denial conditions. Additionally, we would like assertional rules to be indicated for any sentence of a natural language in any context. When no decisive conditions can be found for the assertion or denial of a given sentence—as feature iii. suggests—then we will stipulate that the assertional rules governing assertion and denial of that sentence are not consistent.

3. Partial Assertion and Consistent Assertion

Let us consider an individual who believes that eating meat is morally neutral, neither right nor wrong. However, they are vulnerable to the argument that ‘Eating meat causes suffering to innocent sentient beings, therefore eating meat is morally wrong’. As a result, the individual tends to deny the sentence ‘Eating meat is morally wrong’ most of the time, but when exposed to this argument, they are unable to refute it and are therefore somewhat motivated to assert the sentence ‘Eating meat is morally wrong’. In this scenario, their attitude can be described as ambivalent, as they are motivated to both deny and assert the sentence ‘Eating meat is morally wrong’, albeit for different reasons.

To model this ambivalent attitude using assertional rules, we can postulate the presence of two such rules within the individual’s language. The first rule dictates denying the sentence ‘Eating meat is morally wrong’ irrespective of context (or in all contexts), while the second requires its assertion under specific circumstances, namely when the individual is directly presented with the aforementioned ethical argument. Consequently, in the context of being presented with the ethical argument, the individual is subject to two inconsistent rules. To describe such a condition in a generic manner, we would say the individual partially asserts and partially denies the sentence ‘Eating meat is morally wrong’ in the given context.

If an individual partially asserts a sentence without partially denying it in a given context, we would say they consistently assert it. Conversely, if they partially deny a sentence without partially asserting it, we would say they consistently deny it. A sentence that is both partially asserted and partially denied in a given context will be called equivocal in that context.

It is the combination of all assertional rules related to a sentence that determines whether assertion or denial of that sentence, in a given context, is consistent or not. A single assertional rule, responsible either for assertion or denial of a sentence, only determines partial assertion or denial of that sentence.

4. Tarski’s Strive for Consistency

Alfred Tarski, in (1935), proposed a general scheme that represents the necessary and sufficient conditions for a sentence to be true. The sentences that instantiate this scheme are referred to as ‘T-sentences’ or ‘unrestricted Tarski-biconditionals’ (Kirkham, 1992; Horsten, 2011).4 A commonly used version of the scheme, sometimes called the ‘Tarski schema’ (Field, 2008), although slightly different from Tarski’s original proposal, is:

$$\begin{array}{c}\hfill ‘p’ is true if and only if p,\end{array}$$

(Tarski Scheme)

where the variable ‘p’ ranges through all sentences in a given language, i.e. the language for which we are defining the truth predicate.5 For the English sentence: ‘Snow is white’, the corresponding T-sentence would be: ‘“Snow is white” is true if and only if snow is white’.

One widely held interpretation of the Tarski Scheme, often referred to as the ‘disquotational’ interpretation, is that the truth of a sentence consists merely in its assertion (cf. Wright, 1992, p. 14; Horsten, 2011, p. 17).6 Based on this interpretation, the Tarski Scheme can be rewritten as follows:

$$\begin{array}{c}\hfill x is true if and only if x is asserted.\end{array}$$

(Truth as Mere Assertion)

The Truth as Mere Assertion interpretation does not specify who should make the assertion or in what circumstances it should be made. In this way, it is similar to the original Tarski Scheme, which does not indicate who should determine whether snow is white. In both cases, an explanation that appeals to the rules of language provides the most plausible account.

Tarski believed it was impossible to give a correct definition of the truth predicate for everyday language due to the latter’s inherent inconsistency (Corcoran, 1983, pp. 164–165). This inconsistency is demonstrated by the existence of the so-called Liar sentence:

$$\begin{array}{c}\hfill LS is not true.\end{array}$$

(LS)

If we assert LS, or consider it true according to the Truth as Mere Assertion, we are saying that it is not true, which forces us to deny it, or consider it not true. Conversely, if we deny LS, or consider it not true, we are denying that LS is not true, thereby forcing ourselves to assert LS, or consider it true. The ambiguities involved in the logical analysis of LS are referred to as the Liar paradox. LS is an example of an equivocal sentence, but expressing this fact unequivocally is not straightforward. Later I will provide a special metalanguage that will allow us to do so.

Tarski’s approach to address the inconsistency introduced by the Liar sentence and other similar paradoxical sentences was to make it impossible to formulate them. As he could not ban everyday language from creating such sentences, Tarski announced that providing a consistent definition of the truth predicate for everyday language was not possible and focused his efforts on devising consistent definitions for formal languages instead.

There is no doubt that the set of true sentences in a given language must be consistent, or that a true sentence cannot contradict other true sentences. Formal languages are highly resistant to inconsistency due to the way they are constructed: only sentences that have been indicated as their axioms, or can be inferred from those axioms using strict, carefully selected rules, are accepted as their own sentences. Tarski uses the terms ‘accepted’, ‘provable’, and ‘asserted’ synonymously when referring to such derived sentences (Corcoran, 1983, pp. 166, 182).

By incorporating the Tarski Scheme into the inference rules of a formal language, a serious security breach is created. This is because applying the scheme to the Liar sentence results in the following self-contradictory biconditional:

$$\begin{array}{c}\hfill LS is true if and only if LS is not true.\end{array}$$

(error)

As an inference rule, the Tarski Scheme requires the acceptance of TS_LS, thereby causing a lack of consistency in the language.7

To counter the threat of inconsistency that the scheme raises when added to the inference rules of a formal language, Tarski restricted the scheme by differentiating between the metalanguage and the object language. The metalanguage contains the names of expressions in the object language, and employs the truth predicate to ascribe truth or falsity only to the sentences in the latter. The object language itself has no truth predicate, as well as no names for its own expressions.

The sentences that instantiate the restricted scheme:

$$\begin{array}{c}\hfill x is true if and only if p,\end{array}$$

(Restricted Tarski Scheme)

belong to the corresponding metalanguage. The variable ‘x’ ranges over the names of sentences of the object language that are present in the metalanguage. The variable ‘p’ represents the translation of the sentence denoted by ‘x’ into the metalanguage (Corcoran, 1983, pp. 167, 187–188). The Liar sentence and other paradoxical sentences that mix up the two languages cannot be sentences in either of them, and without being sentences, they cannot be considered accepted sentences.

5. Giving Up on the Tarski Scheme

When appropriately restricted, the Tarski Scheme may be effective for formal languages. However, these restrictions do not apply to everyday language. The primary critique is that self-referential sentences are not rare in everyday language. Therefore, the prohibition of a specific class of such sentences, namely those that attribute truth or falsity to themselves, appears unjustified (Beall et al., 2018, p. 69).

It seems that among logicians and philosophers of language, there is a widespread belief that, although paradoxical, sentences like the Liar sentence have a rightful place in everyday language. However, their paradoxical consequences are a different matter, and while the Liar sentence is acceptable, the Liar paradox is not so welcome. To address this issue, various solutions have been put forward, as listed by Field (2008) in his exploration of approaches to the Liar paradox (p. 117). If we choose not to exclude the Liar sentence from everyday language, we are faced with two options: either to restrict some laws of classical logic or to abandon, at least in part, the Tarski Scheme.

Some scholars contend that everyday language cannot be completely governed by classical logic. This perspective is supported by the fact that two widely recognized formal accounts of truth, which enable the creation of Liar sentences, are not fully classical (Kripke, 1975; Gupta & Belnap, 1993).8 Beall (2007) openly acknowledges: ‘We know that, due to paradoxical sentences, there’s no truth predicate in, and for, our “real language” if our real language is fully classical’ (p. 8). I aim to challenge this perspective in the sections on Effective Assertional Valuations and Truth Valuations.

Having differentiated the two facets of sentence assertion—partial assertion and consistent assertion—it should now be apparent that mere assertion, or partial assertion, is insufficient for making a sentence true. This is because some sentences that are partially asserted are also partially denied, and such equivocal sentences are not suitable as true sentences. The Tarski Scheme not only falls short of guaranteeing consistency within the set of true sentences, but even by itself requires the acceptance of equivocal sentences as true. Thus, giving up on the Tarski Scheme in the hope of preserving classical logic in everyday language appears to be a reasonable solution.

The next step is to explicitly define a true sentence as one that is consistently asserted. Consequently, a false sentence will be one that is at least partially denied. To take into account contexts, which are crucial for assertional rules, the definition of a true sentence will be context-relative: in a given language, a sentence is considered true with respect to a given context when it is consistently asserted in that context.

6. Assertional Language

I will now introduce the concept of an epistemic model of language, called the ‘assertional language’. The purpose of this model is to provide a formal framework in which various languages can be analysed in their epistemic aspect, including the truth valuation of their sentences.

The idea of assertional language is straightforward. It enables testing particular sentences—or sentence-shapes, to be exact—against one context or another. The result of such a test is called the ‘effective assertional valuation’ (EAV) of the given sentence-shape in relation to the given context. An EAV is always a set of assertional values, such as assertion or denial, available in the language. If we model a language where only these two assertional values are available, an EAV can be a singleton of assertion (for consistently asserted sentence-shapes), a singleton of denial (for consistently denied sentence-shapes), a set containing both assertion and denial (for equivocal sentence-shapes), or an empty set.9

Definition 1

(Assertional Language). An assertional language is defined as a tuple $(V,P,S,W,E)$, where:

• V is a set of assertional values that contains at least two featured elements, referred to as $\langle +\rangle$ (assertion) and $\langle -\rangle$ ( denial )
• P is a set that includes V; it is referred to as the patterns or known patterns
• S is a non-empty subset of P; it is referred to as the sentence-shapes
• W is a set of worlds, which are mutually exclusive subsets of $2^P$, excluding the empty set; the elements of a given world, w, are referred to as contexts in w
• E is a function $S\times C\to 2^V$, where C is the set of all contexts in all worlds; it is referred to as the effective assertional valuation.

At the core of this definition lies the notion of ‘pattern’, which encompasses everything that a language user is capable of recognizing or distinguishing.10 This all-inclusive term empowers the user to communicate any conceivable idea. The choice of ‘pattern’ as the descriptor for this concept stems from two facts. First, the expression ‘pattern recognition’ has been widely embraced in recent years (cf. Bishop, 2006), which suggests that the term ‘pattern’ has become readily associated with the notion of something to be recognized. Second, patterns can be recognized by both humans and machines, making the proposed language definition applicable to languages used by either or both. Still, from a formal perspective, any set that contains the two featured elements can qualify as the set of patterns.

The definition allows for additional assertional values beyond the mandatory two. Sentence-shapes are considered known patterns. They are represented by a separate set because it is crucial for the user of a given language to recognize certain patterns as sentences of that language. Contexts are represented as arbitrary, non-empty sets of known patterns. Different worlds, or mutually exclusive families of contexts, are introduced to make it easier to devise assertional rules that operate on separate sets of contexts. For example, a different collection of rules would be necessary for our everyday world than the ones applicable in a specific professional domain or applicable to the world portrayed in a novel.

The model defined above does not use the concept of an assertional rule directly. Instead, it encapsulates this concept in the effective assertional valuation function. This allows it to represent the epistemic aspect of languages, including natural languages, without being tightly coupled with their structure (the problem Tarski deemed insuperable, cf. Corcoran, 1983, pp. 164–165, 267).

Therefore, I will not present a precise definition of an assertional rule. The effective assertional valuation of an assertional language can be directly provided, and in such a case, a set of assertional rules can be derived from it.11 However, this method of defining an assertional language may be deemed, at most, as a supplementary approach. A more pragmatic approach involves furnishing a set of assertional rules in conjunction with an algorithm capable of computing the effective assertional valuation from those rules.12

7. Effective Assertional Valuations

Definition 2

(Effective Assertional Valuations). Let (·, ·, S, W, E) be an assertional language, $s\in S$, $w\in W$, and $c\in w$.

(a)

s is partially asserted with respect to ciff. $\langle +\rangle \in E(s,c)$.

(b)

s is partially denied with respect to ciff. $\langle -\rangle \in E(s,c)$.

(c)

s is consistently asserted with respect to ciff. $E(s,c)=\{\langle +\rangle \}$.

(d)

s is consistently denied with respect to ciff. $E(s,c)=\{\langle -\rangle \}$.

(e)

s is equivocal with respect to ciff. $\langle +\rangle \in E(s,c)and\langle -\rangle \in E(s,c)$.

The definition above creates a specific metalanguage that allows for reporting the basic epistemic status of sentence-shapes of a given assertional language in a consistent manner. The consistency of this metalanguage only depends on whether the effective assertional valuation function in the given assertional language, or in the relevant object language, is well-defined, which is already guaranteed by Def. 1. Specifically, if the Liar sentence (previously referred to as ‘LS’) belongs to the sentence-shapes of the object language, and if some intuitive assertional rules dictate both assertion and denial of LS in any scenario, as previously discussed, then this metalanguage enables LS to be consistently classified as equivocal with respect to any context. This way, we avoid the Liar paradox.

8. Truth Valuations

Definition 3

(Local Truth). Let (·, ·, S, W, ·) be an assertional language, $s\in S,w\in W$, and c $\in w$.

(a)

s is (locally) true with respect to ciff. s is consistently asserted with respect to c.

(b)

s is (locally) false with respect to ciff. s is partially denied with respect to c.

Definition 4

(Global Truth). Let (·, ·, S, W, ·) be an assertional language, $s\in S,andw\in W$.

(a)

s is (globally) true in wiff. for every $c\in w,s$ is true with respect to c.

(b)

s is (globally) false in wiff. for every $c\in w,s$ is false with respect to c.

Def. 3 presents a formal account of the concept of truth described in a previous section, while Def. 4 provides a generalization of it. Both definitions are useful extensions of the metalanguage in Def. 2 and allow for the consistent reporting of the Liar sentence, as well as other equivocally self-referencing sentences, as false.

9. Consistent Assertions as Facts

The correspondence theory is considered ‘perhaps the most important of the neo-classical theories [of truth] for the contemporary literature’ (Glanzberg, 2018). This section seeks to bridge the correspondence theory and the concept of truth proposed in this paper.

According to a commonly held version of the correspondence theory, a sentence is true if it corresponds to a fact of a kind. We can reasonably assume that the user of a given assertional language is capable of recognizing a consistent assertion of any sentence-shape within that language, resulting in such a consistent assertion becoming a known pattern in the language. It is tempting to equate patterns of this kind with facts. For example, if the user recognizes the sentence-shape ‘Snow is white’ and is aware of this sentence-shape being consistently asserted, we can say that the user is aware of the fact that snow is white.

Furthermore, an assertional language can recognize known patterns not only from the effective valuations of its own sentence-shapes but also from the effective valuations in other assertional languages. For instance, if the user of an assertional language recognizes the sentence-shape ‘2 + 2 = 4’ being consistently asserted in the language of arithmetic, we can say that the user recognizes an arithmetic fact, that is, 2 + 2 = 4. In this interpretation, Defs. 1–4 can be seen as a formal account of the correspondence theory.

10. Conclusions

Whether or not the proposed account of truth is accepted, the concept of an assertional language, as defined in Defs. 1–2, is worth considering in its own right. It offers a simple and versatile framework for determining and reporting the epistemic status of sentences in languages of various kinds, both context-dependently and context-independently. The framework is versatile because it can be applied to languages of any definition, provided that the definition can be translated into an effective assertional valuation function.

Another feature, which is hard to overestimate, is that it can handle languages that, in terms of their epistemic aspect, may be perceived as inconsistent.

Moreover, due to its broad notion of a pattern, the framework allows for the combination of different languages in a meaningful way. An assertional language or part of it can be nested within another assertional language by including effective assertional valuations of the former within the known patterns of the latter.