Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context (...) of negation-less logics, _positive logics_, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. (shrink)

We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results.

This chapter offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the chapter begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains how it overcomes the weaknesses of (...) the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The chapter describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance. (shrink)

Alfred Tarski’s influence on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is Tarski’s work on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, model-theoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up.

This paper addresses the theoretical notion of a game as it arisesacross scientific inquiries, exploring its uses as a technical andformal asset in logic and science versus an explanatory mechanism. Whilegames comprise a widely used method in a broad intellectual realm(including, but not limited to, philosophy, logic, mathematics,cognitive science, artificial intelligence, computation, linguistics,physics, economics), each discipline advocates its own methodology and aunified understanding is lacking. In the first part of this paper, anumber of game theories in formal studies are critically (...) surveyed. Inthe second part, the doctrine of games as explanations for logic isassessed, and the relevance of a conceptual analysis of games tocognition discussed. It is suggested that the notion of evolution playsa part in the game-theoretic concept of meaning. (shrink)

We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.