Graham E. Leigh

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.

Herbrand Disjunctions, Cut Elimination and Context-Free Tree Grammars

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 Pi_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 Pi_2-cuts. Given a proof pi of a Sigma_1 formula with cuts only on Pi_2 formulae, we show there is associated to pi a natural context-free tree grammar whose language is finite and yields a Herbrand disjunction for pi.

Cut-free completeness for modal mu-calculus

We present two finitary cut-free sequent calculi for the modal μ-calculus. One is a variant of Kozens axiomatisation in which cut is replaced by a strengthening of the induction rule for greatest fixed point. The second calculus derives annotated sequents in the style of Stirlings ‘tableau proof system with names’ (2014) and features a generalisation of the ν-regeneration rule that allows discharging open assumptions. Soundness and completeness for the two calculi is proved by establishing a sequence of embeddings between proof systems, starting at Stirlings tableau-proofs and ending at the original axiomatisation of the μ-calculus due to Kozen. As a corollary we obtain a new, constructive, proof of completeness for Kozens axiomatisation which avoids the usual detour through automata and games.

On the Herbrand content of LK.

We present a structural representation of the Herbrand content of LK-proofs with cuts of complexity prenex Sigma-2/Pi-2. The representation takes the form of a typed non-deterministic tree grammar of order 2 which generates a finite language of first-order terms that appear in the Herbrand expansions obtained through cut-elimination. In particular, for every Gentzen-style reduction between LK-proofs we study the induced grammars and classify the cases in which language equality and inclusion hold.

Some Weak Theories of Truth

In this article we present a number of axiomatic theories of truth which are conservative extensions of arithmetic. We isolate a set of ten natural principles of truth and prove that every consistent permutation of them forms a theory conservative over Peano arithmetic.