A simple sequent calculus for nominal logic

Nominal logic is a variant of first-order logic that provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the new-quantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made to develop convenient rules for reasoning with the new-quantifier, but we argue that none of these attempts is completely satisfactory. In this article we develop a new sequent calculus for nominal logic in which the rules for the new- quantifier are much simpler than in previous attempts. We also prove several structural and metatheoretic properties, including cut-elimination, consistency, and equivalence to Pitts' axiomatization of nominal logic

Ambiguity and reasoning

In this paper, reasoning with ambiguous representations is explored in a formal way, with ambiguities at the level of propositions in propositional logic and predicate logic, and ambiguous representations of scopings in predicate logic as the main examples. First a version of propositional logic with propositional ambiguities is presented and a sequent axiomatization for it is given. This is then extended to predicate logic. Next, predicate logic with scope ambiguities is introduced and discussed, and again a sequent calculus for it is proposed. The conclusion connects the results to natural language semantics, and briefly compares them with existing logics of ambiguity. An appendix gives completeness proofs for our versions of ambiguous propositional and predicate logic

Grammatical structures and logical deductions

The three essays presented here concern natural connections between grammatical derivations and structures provided by certain standard grammar formalisms, on the one hand, and deductions in logical systems, on the other hand. In the first essay we analyse the adequacy of Polish notation for higher-order languages. The Ajdukiewicz algorithm (Ajdukiewicz 1935) is discussed in terms of generalized MP-deductions. We exhibit a failure in Ajdukiewicz’s original version of the algorithm and give a correct one; we prove that generalized MP-deductions have the frontier property, which is essential for the plausibility of Polish notation. The second essay deals with logical systems corresponding to different grammar formalisms, as e.g. Finite State Acceptors, Context-Free Grammars, Categorial Grammars, and others. We show how can logical methods be used to establish certain linguistically significant properties of formal grammars. The third essay discusses the interplay between Natural Deduction proofs in grammar oriented logics and semantic structures expressible by typed lambda terms and combinators

Modal Hybrid Logic

This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory

A Simple Logical Matrix and Sequent Calculus for Parry’s Logic of Analytic Implication

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints