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

This paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.

The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from proof-theoretic or categorical concerns and, on the other, from a possible-worlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's proof-theoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.

In this paper we present a Gentzen system for reasoning with contrary-to-duty obligations. The intuition behind the system is that a contrary-to-duty is a special kind of normative exception. The logical machinery to formalise this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes. 1

Abstract not found

Abstract. Separation Algebras serve as models of Separation Logics; Share Accounting allows reasoning about concurrent-read/exclusive-write resources in Separation Logic. In designing a Concurrent Separation Logic and in mechanizing proofs of its soundness, we found previous axiomatizations of separation algebras and previous systems of share accounting to be useful but imperfect. We adjust the axioms of separation algebras; we demonstrate an operator calculus for constructing new separation algebras; we present a more powerful system of share accounting with a new, simple model; and we provide a reusable Coq development. 1

Building semantic models that account for various kinds of indirect reference has traditionally been a difficult problem. Indirect reference can appear in many guises, such as heap pointers, higher-order functions, object references, and shared-memory mutexes. We give a general method to construct models containing indirect reference by presenting a “theory of indirection”. Our method can be applied in a wide variety of settings and uses only simple, elementary mathematics. In addition to various forms of indirect reference, the resulting models support powerful features such as impredicative quantification and equirecursion; moreover they are compatible with the kind of powerful substructural accounting required to model (higher-order) separation logic. In contrast to previous work, our model is easy to apply to new settings and has a simple axiomatization, which is complete in the sense that all models of it are isomorphic. Our proofs are machine-checked in Coq.

In this paper we present a Gentzen system for reasoning with contraryto -duty obligations. The intuition behind the system is that a contraryto -duty is a special kind of normative exception. The logical machinery to formalize this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes.

Abstract. We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, non-classical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework. 1 Three mistakes Pure Mathematics is the class of all propositions of the form ‘p implies q’... And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member... [45, p.3]. Thus begins the precursor of Principia Mathematica, Russell’s Principles of Mathematics, and thus begins the sad and confusing twentieth century tale of implication.

Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multi-modal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) [16], and we spell things out by means of Display Calculi (DC) [3, 10]. The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logic system and the way these properties are projected into the logic operators. We will show how we can obtain Lambek residuated triple n, = and of binary operators, and how the operators 3 and 2 introduced by Moortgat in [16] are indeed their unary counterpart.