Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource management . . . . . . . . . . . . . . . . . . . . . . 13 3 The syntax-semantics interface: proofs and readings 16 3.1 Term assignment for categorial deductions . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Natural language interpretation: the deductive view . . . . . . . . . . . . . . . . . . . 21 4 Grammatical composition: multimodal systems 26 4.1 Mixedinference:themodesofcomposition........................ 26 4.2 Grammaticalcomposition:unaryoperations ....................... 30 4.2.1 Unary connectives: logic and structure . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Applications: imposing constraints, structural relaxation

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resource management properties of the ffl connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The first type is obtained by combining a number of unimodal systems into one multimodal logic. The...

The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, extending previous work by Masini on a two-dimensional generalization of Gentzen's sequents (2-sequents). The modal rules closely match the standard rules for an universal quantifier and different logics are obtained with simple conditions on the elimination rule for 2. We give an explicit term calculus corresponding to proofs in these systems and, after defining a notion of reduction on terms, we prove its confluence and strong normalization. 1. Introduction Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent ex...

abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is ad-hoc. While we can formulate several modal logics in the sequent calculus that enjoy cut-elimination, their formalisation arises through system-bysystem fine tuning to ensure that the cut-elimination holds, and the correspondence to the axioms of the Hilbert-Lewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the Hilbert-Lewis axiomatisation. We show that the calculus possesses a cut-elimination property directly analogous to cut-elimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1

. We extend Belnap's Display Logic to give a cut-free Gentzen-style calculus for relation algebras. The calculus gives many axiomatic extensions of relation algebras by the addition of further structural rules. It also appears to be the first purely propositional Gentzen-style calculus for relation algebras. 1 Introduction Given a non-empty set U , the universal relation U \Theta U is the set of all ordered pairs (a; b) where a 2 U and b 2 U . Any subset of U \Theta U is a binary relation over U , and the set of all subsets of U \Theta U is the set of all binary relations over U . Thus any two binary relations R and S are each just a set of ordered pairs, and we can use the set-theoretic operations of complement, intersection and union to build other relations. The identity relation is f(a; a) j a 2 Ug while the "relative" analogues of complement, intersection and union are converse (` R) = f(b; a) j (a; b) 2 Rg, composition (R ffi S) = f(a; b) j 9c; (a; c) 2 R and (c; b) 2 Sg and ...

The paper presents a set-theoretic translation method for polymodal logics that reduces the derivability problem of a large class of propositional polymodal logics to the derivability problem of a very weak firstorder set theory\Omega\Gamma Unlike most existing translation methods, the one we proposed applies to any normal complete finitely-axiomatizable polymodal logic, regardless if it is first-order complete or if an explicit semantics is available for it. Moreover, the finite axiomatizability of\Omega makes it possible to implement mechanical proof search procedures via the deduction theorem or more specialized and efficient techniques. In the last part of the paper, we briefly discuss the application of set T -resolution to support automated derivability in (a suitable extension of) \Omega\Gamma This work has been supported by funds MURST 40% and 60%. The second author was supported by a grant from the Italian Consiglio Nazionale delle Ricerche (CNR). 1 Introduction The paper...

. We describe an implementation of the Display Logic calculus for relation algebra as an Isabelle theory. Our implementation is the first mechanisation of any display calculus, but also provides a useful interactive proof assistant for relation algebra. The inference rules of Display Logic are coded directly as Isabelle theorems, thereby guaranteeing the correctness of all derivations. We describe various tactics and derived rules developed for simplifying proof search, including an automatic cutelimination procedure, and example theorems proved using Isabelle. We show how some relation algebraic theorems proved using our system can be put in the form of structural rules of Display Logic, facilitating later re-use. We then show how the implementation can be used to prove results comparing alternative formalizations of relation algebra from a proof-theoretic perspective. Keywords: logical frameworks, higher-order logic, relation algebra, display logic 1 Introduction Relation algebras a...

. We define cut-free display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)-(C7). Finally, we show a weak Sahlqvist-style theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cut-free display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...

Introduction The present paper explores applications of Display Logic as defined in [ Belnap, 1982 ] to modal logic. Acquaintance with that paper is presupposed, although we will give all necessary definitions. Display Logic is a rather elegant proof-theoretic system that was developed to explore in depth the possibility of total Gentzenization of various propositional logics. By Gentzenization I understand the strategy to replace connectives by structures. Gentzenization is something of an ingenious optical trick because it uses a single symbol to mean different things depending on the place it occupies in the sequent. In the original Gentzen system it was the comma that had to be interpreted as and when to the left of the turnstile and as or when to the right. The interpretation of the structures oscillates between two logical symbols depending on whether it is in the antecedent or in the consequ