Axiomatizing Semantic Theories of Truth?

Fischer, Martin H.; Halbach, Volker; Kriener, Jönne; Stern, Johannes

doi:10.1017/s1755020314000379

Cited by 28 publications

(39 citation statements)

References 27 publications

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“…This was proved for the Strong Kleene scheme by Burgess [3] and Cantini [5] (the result was already announced by Kripke), for the van Fraassen scheme by Burgess [3], for the Cantini scheme by Cantini [6], 1 and for Leitgeb's [21] theory (which is based on his notion of semantic dependence) by Welch [40]. The proof in [40] was generalized by Fischer et al [8] to cover a range of further supervaluational schemes (for example, supervaluations based on maximal consistent extensions). Leitgeb's theory is not defined in the Kripkean way, but in Beringer and Schindler [1] it is shown that Leitgeb's theory coincides with the least fixed point of a particular 3-valued valuation scheme that we dubbed the Leitgeb valuation scheme, V L .…”

Section: Thus Whenever (N E A) Is a Fixed Point M E Coincides Witmentioning

confidence: 81%

“…There are some further philosophical issues on which our results may shed some light, such as the question of what constitues a good axiomatization of a semantic truth theory (Fischer et al [8]), or the question of the 'unsubstantiality' of truth (Horsten [17], Shapiro [31], Ketland [19], Halbach [13]). However, I won't attempt to draw any substantial philosophical conclusions in this paper and leave a proper assessment for another occasion.…”

Section: Introductionmentioning

confidence: 99%

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Some Notes on Truths and Comprehension

Schindler

2017

J Philos Logic

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In this paper we study several translations that map models and formulae of the language of second-order arithmetic to models and formulae of the language of truth. These translations are useful because they allow us to exploit results from the extensive literature on arithmetic to study the notion of truth. Our purpose is to present these connections in a systematic way, generalize some well-known results in this area, and to provide a number of new results. Sections 3 and 4 contain some recursion-and proof-theoretic results about Kripke-style fixed-point theories of truth. Section 5 shows how to derive full second-order arithmetic from principles of truth. Section 6 investigates the proof-theoretic strength of disquotation without an arithmetical base theory.

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Section: Thus Whenever (N E A) Is a Fixed Point M E Coincides Witmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Some Notes on Truths and Comprehension

Schindler

2017

J Philos Logic

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“…8 Since we work with a two-sided sequent calculus with a special treatment of negation, the usual initial sequents will come in two versions.…”

Section: Pkf: Adding Compositional Truth Sequents To Palmentioning

confidence: 99%

“…In Fischer et al [8] a criterion of adequacy for axiomatic theories trying to capture a semantic theory was discussed. There it was already mentioned that PKF is adequate with respect to the class of strong Kleene fixed-points based on the standard interpretation of arithmetic.…”

Section: Lemma 12mentioning

confidence: 99%

Truth, Partial Logic and Infinitary Proof Systems

Fischer

Gratzl

2017

Stud Logica

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In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of KripkeFeferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω-rule. MotivationThis paper is part of a systematic investigation of syntactical predicates based on proof theoretic methods. In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate, interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke-Feferman with its intended semantics. The method we apply is a variation of Schütte's based on infinitary proof systems.The conception of truth we investigate is a version of Kripke's theory. Kripke made use of inductive definitions in order to characterize fixed-points as intended interpretations of the truth predicate. Of special interest for Kripke is the minimal fixed-point. We will use an infinitary proof system for which the set of theorems coincides with the minimal fixed-point. One part of our investigation designs an infinitary proof system, SK ∞ . The system SK ∞ is a sequent calculus with an ω-rule and truth introduction rules. With SK ∞ we can characterize the minimal strong Kleene fixed-point, I sk . In a second part we consider an internal axiomatization of the semantical theory; the resulting thoery is called PKF. This theory is a good candidate and does not contain an infinitary rule, however it is not based on classical logic but on partial logic. The system PKF was introduced by Halbach & Horsten [11] for achieving a faithful axiomatization of Kripke's fixed-point construction of Strong Kleene. The theory PKF has several interesting features for our purposes. On the one hand PKF is close to the semantic construction as it directly incorporates the closure conditions of the fixed-points. Moreover PKF is true in and only in the fixed-point models as an adequacy result shows. On the other hand PKF is close to the infinitary system SK ∞ as it can be embedded into it in a straight way. A closer look at 1

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“…The second of these results shows we have developed what Fischer et al (2015) call an N-categorical axiomatisation, although they would not consider the ω-rule as a permissible axiom as they only consider recursively enumerable theories. Theorem 5.13 could also be taken to show that KFC ∪ InteractionAx, which is a recursively enumerable theory, is an N-categorical axiomatisation given the structure M.…”

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confidence: 99%

How to Express Self-Referential Probability. A Kripkean Proposal

Campbell-Moore

2015

The Review of Symbolic Logic

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We present a semantics for a language that includes sentences that can talk about their own probabilities. This semantics applies a fixed point construction to possible world style structures. One feature of the construction is that some sentences only have their probability given as a range of values. We develop a corresponding axiomatic theory and show by a canonical model construction that it is complete in the presence of the ω-rule. By considering this semantics we argue that principles such as introspection, which lead to paradoxical contradictions if naively formulated, should be expressed by using a truth predicate to do the job of quotation and disquotation and observe that in the case of introspection the principle is then consistent.

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Axiomatizing Semantic Theories of Truth?

Cited by 28 publications

References 27 publications

Some Notes on Truths and Comprehension

Some Notes on Truths and Comprehension

Truth, Partial Logic and Infinitary Proof Systems

How to Express Self-Referential Probability. A Kripkean Proposal

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