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This paper compares the roles classical and intuitionistic logic play in restricting the free use of truth principles in arithmetic. We consider fifteen of the most commonly used axiomatic principles of truth and classify every subset of them as either consistent or inconsistent over a weak purely intuitionistic theory of truth.

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard [11]. There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised… (More)

This paper explores the interface between principles of self-applicable truth and classical logic. To this end, the proof-theoretic strength of a number of axiomatic theories of truth over intuitionistic logic is determined. The theories considered correspond to the maximal consistent collections of fifteen truth-theoretic principles as isolated in Leigh… (More)

Recently a new connection between proof theory and formal language theory was introduced. It was shown that the operation of cut elimination for proofs in first-order predicate logic involving Π 1-cuts corresponds to computing the language of a particular class of regular tree grammars. The present paper expands this connection to the level of Π 2-cuts.… (More)

We present a structural representation of the Herbrand content of LK-proofs with cuts of complexity prenex Π 2 /Σ 2. The representation takes the form of a typed non-deterministic tree grammar G of order 2 which generates a finite language, L(G), of first-order terms that appear in the Herbrand expansions obtained through cut-elimination. In particular, for… (More)

We present a cut elimination argument that witnesses the conservativity of the compositional axioms for truth (without the extended induction axiom) over any theory interpreting a weak subsystem of arithmetic. In doing so we also fix a critical error in Halbach's original presentation. Our methods show that the admission of these axioms determines a… (More)

The closure ordinal of a formula of modal µ-calculus µXϕ is the least ordinal κ, if it exists, such that the denotation of the formula and the κ-th iteration of the monotone operator induced by ϕ coincide across all transition systems (finite and infinite). It is known that for every α < ω 2 there is a formula ϕ of modal logic such that µXϕ has closure… (More)

The closure ordinal of a formula of modal µ-calculus µXϕ is the least ordinal κ, if it exists, such that the denotation of the formula and the κ-th iteration of the monotone operator induced by ϕ coincide across all transition systems (finite and infinite). It is known that for every α < ω 2 there is a formula ϕ of modal logic such that µXϕ has closure… (More)

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