At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties, and shows how the logical results impinge on the philosophical topics related to truth. For instance, he shows how the discussion of topics such as deflationism depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate and professional philosopher in theories of truth. volker halbach is professor in philosophy at the University of Oxford and a fellow of New College.
We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF. namely the strength of the system ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system of ramified analysis up to ε0.
Abstract.A Gödel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence of arithmetic to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed points for the formulae are obtained. This paper is the first of two papers. In the present paper we focus on provability. In part II, we will consider other properties like Rosser provability and partial truth predicates. §1. Introduction. 'We thus have a sentence before us that states its own unprovability.' This is how Gödel describes the kind of sentence that has come to bear his name. 1 Ever since, sentences constructed by Gödel's method have been described in lectures and logic textbooks as saying about themselves that they are not provable or as ascribing to themselves the property of being unprovable.The only 'self-reference-like' feature of the Gödel sentence γ that is used in the usual modern proofs of Gödel's first incompleteness theorem is the derivability of the equivalence γ ↔ ¬Bew( γ ); in other words, the only feature needed is the fact that γ is a (provable) fixed point of ¬Bew(x). 2 But the fact that a sentence is a fixed point of a certain formula expressing a certain property does by no means guarantee that the sentence ascribes that property to itself, as we shall argue in what follows; and whether γ is also Received: Dec. 23, 2013. 1 The German original reads: 'Wir haben also einen Satz vor uns, der seine eigene Unbeweisbarkeit behauptet.' Of course Gödel (1931, p. 175) refers to provability in a specific system. 2 In what follows, we will often talk about fixed points when we mean fixed points that can be shown to be fixed points in the relevant theory, that is, provable fixed points. As a referee correctly pointed out, Gödel did not prove the equivalence γ ↔ ¬Bew( γ ) in his original paper.
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