This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.

I propose a formalisation of knowledge sharing scenarios that aims at capturing the crucial role played by an existing duality between ontological theories one wants to merge and particular situations that need to be linked. I use diagrams in the Chu category and colimits over these diagrams to account for the reliability and optimality of knowledge sharing systems. Furthermore, I show how we may obtain a deeper understanding of a system that shares knowledge between a probabilistic logic program and Bayesian belief networks by re-analysing the scenario in terms of the present approach.

Using the programming-language concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an improved account of quantificational binding, weak crossover, wh-questions, superiority, and polarity licensing.

Sufficient conditions for first order based sequent calculi to admit cut elimination by a Schütte-Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.

We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truth-values, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [38], according to which explicit negation entails default negation. We then de ne Coherent Answer Sets, and the Paraconsistent Well-founded Model semantics, generalising many paraconsistent semantics for logic programs. In particular, Paraconsistent Well-Founded Semantics with eXplicit negation (WFSXp ) [3, 11]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program.

The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are order-preserving; in this paper instead, we consider the addition of order-reversing, Galois connected operators. In x2 we do the basic model-theoretic and proof-theoretic groundwork. In x3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quanti er expressions, and to describe a typology of polarity items.

Abstract. Modular cut-elimination is a particular notion of ”cut-elimination in the presence of non-logical axioms ” that is preserved under the addition of suitable rules. We introduce syntactic necessary and sufficient conditions for modular cut-elimination for standard calculi, a wide class of (possibly) multipleconclusion sequent calculi with generalized quantifiers. We provide a ”universal” modular cut-elimination procedure that works uniformly for any standard calculus satisfying our conditions. The failure of these conditions generates counterexamples for modular cut-elimination and, in certain cases, for cut-elimination. 1

this paper I show that these hints and gestures are true. And perhaps truer than those that made them thought at the time

In many frameworks for reasoning under inconsistency, it is implicitly assumed that the formulae from the belief base are connected using a weak form of conjunction. When it is consistent, a belief base B = {ϕ1,..., ϕn}, where the ϕi are propositional formulae, is logically equivalent to the base {ϕ1 ∧... ∧ ϕn}. However, when it is not consistent, both bases typically lead to different conclusions. This illustrates the fact that the comma used in base B has to be considered as an additional, genuine connective, and not as a simple conjunction. In this work we define and investigate a propositional framework with such a “comma connective”. We give it a semantics and show how it generalizes several approaches for reasoning from inconsistent beliefs. 1

We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate context-free languages. In [1] it has been shown that nite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages; here we give a new proof of this fact. We also obtain similar results for systems with permutation and n ary operations.