Abstract
Kripke’s theory of truth is arguably the most influential approach to self-referential truth and the semantic paradoxes. The use of a partial evaluation scheme is crucial to the theory and the most prominent schemes that are adopted are the strong Kleene and the supervaluation scheme. The strong Kleene scheme is attractive because it ensures the compositionality of the notion of truth. But under the strong Kleene scheme classical tautologies do not, in general, turn out to be true and, as a consequence, classical reasoning is no longer admissible once the notion of truth is involved. The supervaluation scheme adheres to classical reasoning but violates compositionality. Moreover, it turns Kripke’s theory into a rather complicated affair: to check whether a sentence is true we have to look at all admissible precisification of the interpretation of the truth predicate we are presented with. One consequence of this complicated evaluation condition is that under the supervaluation scheme a more proof-theoretic characterization of Kripke’s theory becomes inherently difficult, if not impossible. In this paper we explore the middle ground between the strong Kleene and the supervaluation scheme and provide an evaluation scheme that adheres to classical reasoning but retains many of the attractive features of the strong Kleene scheme. We supplement our semantic investigation with a novel axiomatic theory of truth that matches the semantic theory we have put forth.
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13 March 2019
The original version of the article unfortunately contained a mistake. In the Acknowledgments section of the original version of the article, the grant number of the Marie Sklodowska-Curie Individual Fellowship supporting the author���s work was misstated.
13 March 2019
The original version of the article unfortunately contained a mistake. In the Acknowledgments section of the original version of the article, the grant number of the Marie Sklodowska-Curie Individual Fellowship supporting the author���s work was misstated.
Notes
The choice of an arithmetical base language and theory is not essential to the proposal. We could work in an alternative framework and language as long as a sufficiently rich theory of syntax is available. Throughout the paper we assume \(\mathcal {L}_{T}\) to be a standard first-order arithmetical language that contains an additional unary predicate—the truth predicate. \(\mathcal {L}_{T}\) may also, in addition to S, + and ×, contain further function symbols for certain primitive recursive functions.
As a matter of fact there is not one particular supervaluation scheme but rather a family of different such schemes. For the sake of this introduction we ignore this complication and treat them as one. For the most important supervaluation schemes for theories of truth see our Section 3.1, McGee [19], Burgess [1] or Field [5].
Compositionality is sometimes taken to imply that truth, or the satisfaction relation, commutes with all logical connectives. However, since the strong Kleene scheme assumes three truth values but is formulated in a classical metatheory this will not generally hold for this scheme. While the scheme commutes with conjunction, disjunction and the quantifiers it does not commute with negation: if a sentence is not true in the metalinguistic sense according to the strong Kleene scheme this does not imply that its negation is true in the metalinguistic sense according to the strong Kleene scheme. The sentence may be neither true nor false. From this point of view the strong Kleene scheme is perhaps not fully compositional. But this notion of compositionality, which ties the idea of compositionality to the commutation of truth with the logical connectives, is at best a derived notion, which seems to be acceptable for classical logic but deeply misguided when applied to partial evaluation schemes.
Throughout this paper we say that a sentence is false iff its negation is true.
The term penumbral connections was introduced by Fine [6] in connection to vagueness.
See Halbach [11] for an exposition of both theories.
The theory VF developed by Cantini [3] is intended to capture aspects of Kripke’s supervaluational theory of truth. However, the theory cannot distinguish supervaluational truth from revision theoretic truth and in this sense falls short from providing a proof-theoretic account of Kripke’s supervaluational theory of truth.
See, e.g., Fine [6] for remarks along these lines. Note that a sentence is indeterminate iff it does not receive a classical truth value. For a true disjunction with indeterminate disjuncts the Nixon-example above does the trick. For the indeterminate case replace one disjunct of the Nixon-example by, e.g., the sentence “this sentence is not true”.
More precisely, the antiextension of the truth predicate S− is defined relative to the extension S in the following way
$$S^{-}:=\{\#\psi:\exists\phi(\phi\doteq\neg\psi\,\&\,\#\phi\in S\}\cup\{\#\neg\psi:\#\psi\in S\,\&\,\neg\exists\phi(\psi\doteq\neg\phi)\}.$$By #ϕ we denote the Gödel number of a sentence ϕ. In the context of Kripke’s theory of truth there is no loss of generality by taking the antiextension to be defined because for all customary evaluation schemes the Kripkean process of constructing the interpretation of the truth predicate guarantees the extension and the antiextension to be interdefinable in the above way.
is a name of the sentence ϕ. Throughout this paper we take 
to be the numeral of the Gödel number of ϕ. The Gödel number of ϕ will be denoted, as mentioned, by #ϕ.The two definiens are equivalent for consistent sets S only but the supervaluation scheme is usually only defined for consistent sets. So there is no loss of generality. The argument works by showing that for all S′ such that S ⊆ S′ and S−∩ S′ = ∅ and all sentences ϕ of \(\mathcal {L}_{T}\):
$$S\models_{\mathsf{sk}}\phi\Rightarrow S^{\prime}\models\phi. $$See Halbach [11] for a definition of the strong Kleene satisfaction relation.
The monotonicity of \(\mathcal {J}_{e}\) guarantees the existence of fixed points and the existence of a minimal fixed point: there are more ordinal numbers than sentences of the language so at some point we run out of sentences that we can add to the interpretation of the truth predicate—we have reached a fixed point. If we start from the empty set the fixed point we obtain must, by monotonicity, be the minimal fixed point.
There is a further fixed point which is of particular interest, namely, the maximal intrinsic fixed point. The maximal intrinsic fixed point contains all sentences for which the negation of the sentence is in no fixed point of the jump operation at stake. So the construction of the fixed point does not involve any arbitrary semantic decisions.
A set of sentences X is consistent if for no sentence ϕ: #ϕ ∈ X and #¬ϕ ∈ X.
MAXCONS is the set of maximally consistent sets of sentences. A set of sentences X is maximally consistent if it is consistent and, in addition, it is maximal, that is for all sentences ϕ: #ϕ ∈ X or #¬ϕ ∈ X. The scheme was, e.g., suggested by Kripke [16].
Trivialization is immediate in the case of the VC jump. In case of the VB there is actually no trivialization but for no inconsistent set S we have S ⊆VB(S) and thus all fixed points are reached starting from consistent sets.
See Fischer et al. [7] for a proof of this observation and further discussion of the different supervaluation schemes.
See, e.g., Halbach [11].
See Halbach [11, pp. 202-210] for a detailed exposition of this result.
It is perhaps worth mentioning that certain observations in this paper also hold for the mc scheme. For example, the results of Section 4 will carry over to the minimal fixed point of this scheme.
We implicitly assume the following equivalence:
Notice that the only purpose of the initial sequent (AX.C) of \(\mathsf {SV}^{c}_{\infty }\) (cf. Definition 4.5) is to account for the fact that in the scheme vc we only consider consistent precisifications of a given interpretation of the truth predicate. As a consequence all sentences of the form
are true in each such admissible precisification and are thus members of every VC-fixed point. (AX.C) guarantees that we can derive all sentences of this form and their consequences. Sentences of this form are, however, not always true in all admissible vb-precisifications and thus not always members of the VB-fixed points. For this reason we need to drop the initial sequent (AX.C) from the infinitary Tait-style calculus in this case. However, once we have dropped (AX.C) we can copy Cantini’s reasoning and establish Lemma 4.3.By SV∞⊩Γ(\(\mathsf {SV}^{c}_{\infty }\vdash {\Gamma }\))we denote that ⊩Γis a derivable sequent in SV∞(\(\mathsf {SV}^{c}_{\infty }\)).
One might wonder how the minimal strong Kleene fixed point relates to the minimal supervaluation-style truth fixed point. The former is, obviously, a subset of the latter. Moreover, the minimal supervaluation-style fixed point is not just the minimal strong Kleene fixed-point closed under PAT. The sentence
, where λ is a usual Liar sentence, is a member of the former but it is not a member of the minimal strong Kleene fixed point closed under PAT.See Burgess [2] for a proof of the former. The latter observation follows from the definition of the maximal intrinsic fixed point and the fact that the complexity of set of sentences true in some SSK fixed point is at most \({{\Sigma }^{1}_{1}}\), which follows immediately from the definition of the SSK scheme.
See, for instance, Visser [22, Theorem 2.18.2].
Welch’s observation is actually slightly stronger. It shows that there is a maximal VB-fixed point S such that S⫅̸SSK(S).
See Halbach [11] for a presentation and discussion of KF.
Here, and in the remainder of the paper 𝜃(x, T) (𝜃c(x, T)) and ξ(x, T) (ξc(x, T)) denote the formulas where all occurrences of the free second-order variable X have been replaced by the truth predicate.
See Fischer et al. [7] for further discussion and explanation.
The notion of truth definability was introduced by Fujimoto [9]. Roughly, it means thatthere is an unrelativized interpretation ofKF inIT which keeps the arithmetical vocabulary fixed.
See Friedman and Sheard [8, p. 15] for the argument leading to the inconsistency of (Ax6).
No set S of stable sentences can be a Θ-fixed point since for each such S there will be Liar sentences λ1 and λ2 such that #(λ1 ⇔ λ2) ∈ S but #(λ1 ⇔ λ2) cannot be a member of any Θ-fixed point. See Burgess [1] for further details.
The truth rank keeps track of the number of applications of the truth rules (T) and (¬T).
If the consequence relation is reflexive we can omit—like in the case of the skk scheme—the first disjunct, that is, the strong Kleene satisfaction relation in the formulation of the scheme.
Such an axiomatic theory of truth should also be possible if the relevant consequence relation is \({{\Delta }^{1}_{1}}\). However, if the consequence relation is of greater complexity, then this will only be possible if there is a simpler way to define the resulting truth sets.
A further interesting question that we have not addressed in this paper is whether the supervaluation-style truth schemes can be fruitfully applied to the study of vagueness. This would be interesting since vagueness was one of the principle fields of applications of the supervaluation schemes.
is a name of the sentence ϕ. Throughout this paper we take 
to be the numeral of the Gödel number of ϕ. The Gödel number of ϕ will be denoted, as mentioned, by #ϕ.
are true in each such admissible precisification and are thus members of every 
, where λ is a usual Liar sentence, is a member of the former but it is not a member of the minimal strong Kleene fixed point closed under References
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Acknowledgments
This work was supported by the DFG-funded research project Syntactical Treatments of Interacting Modalities, the German BA, and by the European Commission through a Marie Sklodowska-Curie Individual Fellowship (TREPISTEME, Grant No. 703529). I wish to thank Catrin Campbell-Moore, Martin Fischer, Carlo Nicolai, Philip Welch and two anonymous referees for helpful comments on and discussion of my paper. Early versions of this paper were presented at Bristol, Bath, and Munich. I thank the audiences for their feedback.
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Stern, J. Supervaluation-Style Truth Without Supervaluations. J Philos Logic 47, 817–850 (2018). https://doi.org/10.1007/s10992-017-9451-0
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DOI: https://doi.org/10.1007/s10992-017-9451-0