One of the key issues in artiicial intelligence is the representation and processing of knowledge. The selection of an appropriate formalism for doing so is a crucial step in building a usable system. In this selection process one tries to nd an optimal balance between expressive power of the system and adequacy of the representation process on the one hand and eeciency and simplicity of the re...

Modal logic is an enormous subject, and so any discussion of it must limit itself according to some set of principles. Modal logic is of interest to mathematicians, philosophers, linguists and computer scientists, for somewhat different reasons. Typically a philosopher may be interested in capturing some aspect of necessary truth, while a mathematician may be interested in characterizing a clas...

The question at issue is to develop a computational interpretation of Linear Logic [S] and to establish exactly its expressive power. We follow the bottom-up approach. This involves starting with the simplest of the systems we are interested in, and then expanding them step-by-step. We begin with the !-Horn fragment of Linear Logic, which uses only positive literals, the linear implication -, t...

A powerful syntactic theory as well as expressive modal logics have to deal with self-referentiality. Self-referentiality and paradoxes seem to be close neighbours and depending on the logical system, they have devastating consequences, since they introduce contradictions and trivialise the logical system. There is a large amount of diierent attempts to tackle these problems. Some of them are c...

The logic of bunched implications (BI) of O’Hearn and Pym [5] is a substructural logic which freely combines additive connectives ⊃ , ∧, ∨ from propositional logic and multiplicative connectives −?, ? from linear logic. Because of its concise yet rich representation of states of resources, BI is regarded as a logic suitable for reasoning about resources. For example, by building a model for BI ...

Fibring has been shown to be useful for combining logics endowed with truthfunctional semantics. However, the techniques used so far are unable to cope with fibring of logics endowed with non-truth-functional semantics as, for example, paraconsistent logics. The first main contribution of the paper is the development of a suitable abstract notion of logic, that may also encompass systems with n...

A modal reduction principle of the form [i1] . . . [in]p ⇒ [j1] . . . [jn′ ]p can be viewed as a production rule i1 · . . . · in → j1 · . . . · jn′ in a formal grammar. We study the extensions of the multimodal logic Km with m independent K modal connectives by finite addition of axiom schemes of the above form such that the associated finite set of production rules forms a regular grammar. We ...

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E . In contrast to the wellknown interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE , but rather by a suitable complete Heyting algebra H . The canonical map relating H and ΩE both serves to interpret equality and provides a modal...

Propositional modal logic is a standard tool in many disciplines, but first-order modal logic is not. There are several reasons for this, including multiplicity of versions and inadequate syntax. In this paper we sketch a syntax and semantics for a natural, well-behaved version of first-order modal logic, and show it copes easily with several familiar difficulties. And we provide tableau proof ...

In lecture 7, we have seen how axiomatics and semantics of modal logic fit together in soundness proofs and correspondence proofs. We have seen several examples of classes of Kripke frames that are characterized by formulas of propositional modal logic. These were several special cases. But we are looking for a general correspondence result. Can we find a full correspondence result? For any for...