Torben Braüner

Hybrid Logic and its Proof-Theory

Preface,.- 1 Introduction to Hybrid Logic.- 2 Proof-Theory of Propositional Hybrid Logic .- 3 Tableaus and Decision Procedures for Hybrid Logic .- 4 Comparison to Seligmans Natural Deduction System .- 5 Functional Completeness for a Hybrid Logic .- 6 First-Order Hybrid.- 7 Intensional First-Order Hybrid Logic.- 8 Intuitionistic Hybrid Logic.- 9 Labelled Versus Internalized Natural Deduction .- 10 Why does the Proof-Theory of Hybrid Logic Behave soWell? - References .- Index.

Tableau-based Decision Procedures for Hybrid Logic

Based on tableau systems, we in this chapter prove decidability results for hybrid logic using tableau systems. The chapter is structured as follows. In the first section of the chapter we sketch the basics of tableau systems. In the second section we give a tableau-based decision procedure for a very expressive hybrid logic including the universal modality. In the third section we show how the decision procedure of the second section can be modified such that simpler tableau-based decision procedures (that is, without loop-checks) are obtained for a weaker hybrid logic where the universal modality is not included. In the fourth section we reformulate the tableau systems of the second and the third sections as Gentzen systems and we discuss how to reformulate the decision procedures. In the fifth section we discuss the results. The results of the second, fourth, and fifth sections of this chapter are taken from Bolander and Brauner (2006). The material in the third section is new (but the tableau systems considered in the third section are obtained by directly modifying the tableau system given in the second section, inspired by a tableau-based decision procedure given in Bolander and Blackburn (2007)).

Natural Deduction for Hybrid Logic

In this paper we give a natural deduction formulation of hybrid logic. Our natural deduction system can be extended with additional inference rules corresponding to conditions on the accessibility relations expressed by so-called geometric theories. Thus, we give natural deduction systems in a uniform way for a wide class of hybrid logics which appears to be impossible in the context of ordinary modal logic. We prove soundness and completeness and we prove a normalization theorem. We finally prove a result which says that normal derivations in the natural deduction system correspond to derivations in a cut-free Gentzen system.

Intuitionistic Hybrid Logic

In this chapter we introduce intuitionistic hybrid logic and its proof-theory. Intuitionistic hybrid logic is hybrid modal logic over an intuitionistic logic basis instead of a classical logical basis. The chapter is structured as follows. In the first section of the chapter we introduce intuitionistic hybrid logic (this is taken from Brauner and de Paiva (2006)). In the second section we introduce a natural deduction system for intuitionistic hybrid logic (taken from Brauner and de Paiva (2006)) and in the third and fourth sections we introduce axiom systems for intuitionistic and paraconsistent hybrid logic (taken from Brauner and de Paiva (2006)). In the last section we discuss certain other work, namely a Curry-Howard interpretation of intuitionistic hybrid logic.

Natural Deduction for First-Order Hybrid Logic

This is a companion paper to Braüner (2004b, Journal of Logic and Computation14, 329–353) where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order hybrid logic.

A cut-free Gentzen formulation of the modal logic S5

The goal of this paper is to introduce a new Gentzen formulation of the modal logic S5. The history of this problem goes back to the fifties where a counter-example to cut-elimination was given for an otherwise natural and straightforward formulation of S5. Since then, several cut-free Gentzen style formulations of S5 have been given. However, all these systems are technically involved, and furthermore, they differ considerably from Gentzen’s original formulation of classical logic. In this paper we give a new sequent system for S5 which is a straightforward and technically simple extension of Gentzen’s original sequent system for classical logic. A characteristic feature is the notion of a connection in a proof. The new system satisfies cut-elimination as well as the subformula property. Cut-elimination is proved by giving an algorithm for eliminating cuts. The fact that we give an algorithm for eliminating cuts makes it clear that our system is susceptible to a formulae-as-types computational interpretation.1

Two Natural Deduction Systems for Hybrid Logic: A Comparison

In this paper two different natural deduction systems forhybrid logic are compared and contrasted.One of the systems was originally given by the author of the presentpaper whereasthe other system under consideration is a modifiedversion of a natural deductionsystem given by Jerry Seligman.We give translations in both directions between the systems,and moreover, we devise a set of reduction rules forthe latter system bytranslation of already known reduction rules for the former system.

A Formulation of Linear Logic Based on Dependency-Relations

In this paper we describe a solution to the problem of proving cut-elimination for FILL, a variant of exponential-free and multiplicative Linear Logic originally introduced by Hyland and de Paiva. In the work of Hyland and de Paiva, a term assignment system is used to describe the intuitionistic character of FILL and a proof of cut-elimination is barely sketched. In the present paper, as well as correcting a small mistake in their work and extending the system to deal with exponentals, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut-elimination theorem. The formal system is based on a dependency-relation between formulae occurrences within a given proof and seems of independent interest. The procedure for cut-elimination applies to (multiplicative and exponential) Classical Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut-elimination proofs, is a little involved and we have not seen it published anywhere.

9 First-order modal logic

Publisher Summary First-order modal logics are modal logics in which the underlying propositional logic is replaced by a first-order predicate logic. They pose some of the most difficult mathematical challenges. This chapter surveys basic first-order modal logics and examines recent attempts to find a general mathematical setting in which to analyze them. A number of logics that make use of constant domain, increasing domain, and varying domain semantics is discussed, and a first-order intensional logic and a first-order version of hybrid logic is presented. One criterion for selecting these logics is the availability of sound and complete proof procedures for them, typically axiom systems and/or tableau systems. The first-order modal logics are compared to fragments of sorted first-order logic through appropriate versions of the standard translation. Both positive and negative results concerning fragment decidability, Kripke completeness, and axiomatizability are reviewed. Modal hyperdoctrines are introduced as a unifying tool for analyzing the alternative semantics. These alternative semantics range from specific semantics for non-classical logics, to interpretations in well-established mathematical framework. The relationship between topological semantics and D. Lewiss counterpart semantics is investigated and an axiomatization is presented.

Determinism and the Origins of Temporal Logic

The founder of symbolic temporal logic, A. N. Prior was to a great extent motivated by philosophical concerns. The philosophical problem with which he was most concerned was determinism versus free will. The aim of this paper is to point out some crucial interrelations between this philosophical problem and temporal logic. First, we sketch how Prior’s personal reasons for studying the problems related to determinism were philosophical — initially, indeed theological. Second, we discuss his reconstruction of the classical Master Argument, which has since Antiquity been considered a strong argument for determinism. Furthermore, the treatment of determinism in two of Prior’s proposed temporal systems, namely the Ockhamistic and the Peircean systems, is investigated. Third, we illustrate the fundamental role of the very same issue in more recent discussions of some tempo-modal systems: The ‚Leibniz-system‘ based on ideas of Nishimura (1979) as well as Belnap and Green’s argument (! 1994), to which we add some necessary revisions.