On the Costs of Nonclassical Logic

Halbach, Volker; Nicolai, Carlo

doi:10.1007/s10992-017-9424-3

Cited by 29 publications

(41 citation statements)

References 13 publications

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“…The soundness direction of Lemma 1 is straighforward. For a proof of the completeness direction we refer to [2,17].…”

Section: Logic Arithmetic Truthmentioning

confidence: 99%

Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

Nicolai

2017

Stud Logica

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We study the relationships between two clusters of axiomatizations of Kripke's fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call 'PKF-like' theories, originating in recent work Halbach and Horsten, whose axioms and rules (in Basic De Morgan Logic) are all valid in fixed-point models; the second by 'KF-like' theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke's construction is carried out. We show that to any natural system in one cluster -corresponding to natural variations on induction schemata -there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth.

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“…The soundness direction of Lemma 1 is straighforward. For a proof of the completeness direction we refer to [2,17].…”

Section: Logic Arithmetic Truthmentioning

confidence: 99%

Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

Nicolai

2017

Stud Logica

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“…By performing the appropriate substitution in (10), we have (12) Pr TS 0 (sub(tr([Av]), num(x)), [Ax]) and therefore, by the properties of substitution, also (13) Pr TS0 (sub(tr([Av]), Av ), num(x)).…”

Section: Prmentioning

confidence: 99%

“…As a consequence, they also deem true more sentences that belong to the intended extension of the truth predicate than the corresponding internal axiomatizations; for some authors (cf. [10,12]), this is considered a clear advantage over internal theories.…”

Section: Introductionmentioning

confidence: 99%

Iterated reflection over full disquotational truth

Fischer

Nicolai

Horsten

2017

Journal of Logic and Computation

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Iterated reflection principles have been employed extensively to unfold epistemic commitments that are incurred by accepting a mathematical theory. Recently this has been applied to theories of truth. The idea is to start with a collection of Tarski-biconditionals and arrive by finitely iterated reflection at strong compositional truth theories. In the context of classical logic it is incoherent to adopt an initial truth theory in which A and 'A is true' are inter-derivable. In this article we show how in the context of a weaker logic, which we call Basic De Morgan Logic, we can coherently start with such a fully disquotational truth theory and arrive at a strong compositional truth theory by applying a natural uniform reflection principle a finite number of times.

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“…12 They use a proof-theoretical analysis of Cantini to show that the arithmetical theorems provable in PKF are those provable in RT ω ω . But Cantini has another result which states that if KF INT T( A ), then #A ∈ Γ ω ω and if KF T( A ), then #A ∈ Γ 0 .…”

Section: An Upper Bound For Pkf Theoremsmentioning

confidence: 99%

“…13 See Reinhardt [16]. 14 For more on this topic see Halbach and Nicolai [12] and Fischer et al [9].…”

Section: An Upper Bound For Pkf Theoremsmentioning

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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On the Costs of Nonclassical Logic

Cited by 29 publications

References 13 publications

Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

Iterated reflection over full disquotational truth

Truth, Partial Logic and Infinitary Proof Systems

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