Volker Halbach

Axiomatic Theories of Truth

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.

A System of Complete and Consistent Truth

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation φ/Tφ and conecessitation Tφ/φ and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is ω-inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give an analysis of its proof theory. 1 Preliminaries Let L be the first-order language of arithmetic with symbols for all primitive recursive functions; that is, if [e] is a primitive recursive function with index e, a function symbol fe for [e] is available in L . We suppose that L has =, ¬, → and ∃ as logical symbols. If we expand L by adding the new predicate constant T we obtain the language LT. Throughout the whole paper we shall identify every expression ofLT with its Gödel number (under a standard gödelnumbering). Because we also identify languages with the set of their formulas, a language will be a set of natural numbers. All theories we shall discuss are extensions of Peano arithmetic: PA is the theory containing all defining equations of the primitive recursive functions and all the induction axioms in the full language LT. The index e of a primitive recursive function [e] will provide the defining equation(s) for the symbol fe associated with the index e. If a primitive recursive function h is explicitly given by some equations, we have a natural index e for this function which is again associated with a function symbol fe in the language L . Usually we shall denote this function symbol for h by h. . So h. naturally represents h in PA in the language L . It is useful to conceive of the logical connectives as functions of expressions (i.e., of natural numbers). So we have for negation a function symbol ¬. representing the operation of prefixing a negation symbol to an expression (and similarly for material implication and the existential quantifier). Hence we can show for every formula φ ∈ LT that: PA

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

¬. φ = ¬φ Received April 24, 1994; revised October 11, 1994

Principles of Truth

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □⌜A⌝ holds at a world w∈W if and only if A holds at every world v∈W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several ‘paradoxes’ (like Montagues Theorem, Gödels Second Incompleteness Theorem, McGees Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.

Truth and reduction

On the one hand, the concept of truth is a major research subject in analytic philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. Recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction and collaboration between philosophers and logicians investigating the concept of truth than has existed so far. Aside from an extended introductory overview of recent work in the theory of truth, the volume consists of articles by leading philosophers and logicians on subjects and debates that are situated on the interface between logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians. Contributors: Volker Halbach, Leon Horsten, John P Burgess, Paul Horwich, Steward Shapiro, Hannes Leitgeb, Vann McGee, Michael Sheard and Andrea Cantini.

AXIOMATIZING SEMANTIC THEORIES OF TRUTH

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.

Disquotational Truth and Analyticity

We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ℕ-categoricity and discuss its usefulness and limits.

Tarskian and Kripkean truth

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences. yet not in the scope of negation, the system with the reflection schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF. and thus of ramified analysis up to e 0 .

On the Costs of Nonclassical Logic

A theory of the transfinite Tarskian hierarchy of languages is outlined and compared to a notion of partial truth by Kripke. It is shown that the hierarchy can be embedded into Kripke’s minimal fixed point model. From this results on the expressive power of both approaches are obtained.

The Henkin Sentence

Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: (a variant) of the system KF of Feferman (The Journal of Symbolic Logic, 56, 1–49, 1991) formulated in classical logic, and (a variant of) the system PKF of Halbach and Horsten (The Journal of Symbolic Logic, 71, 677–712, 2006), formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true.