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This chapter weaves together two themes in the work of Nuel Belnap. The earlier theme was to propose conditions (such as conservativity and uniqueness) under which logical rules determine the meanings of the connectives they regulate. The later theme was the employment of semantics for the open future in the foundations of logics of agency. This chapter shows that on the reasonable criterion for fixing meaning of a connective by its rule governed deductive behavior, the natural deduction rules for classical propositional logic do not fix the interpretation embodied in the standard truth tables, but instead express an open future semantics related to Kripke's possible worlds semantics for intuitionistic logic, called natural semantics. The basis for this connection has already been published, but this chapter reports new results on disjunction, and explores the relationships between natural semantics and supervaluations. A possible complaint against natural semantics is that its models may disobey the requirement that there be no branching in the past. It is shown, however, that the condition may be met by using a plausible reindividuation of temporal moments. The chapter also explains how natural semantics may be used to locate what is wrong with fatalistic arguments that purport to close the door on a open future. The upshot is that the open future is not just essential to our idea of agency, it is already built right into the foundations of classical logic. [ABSTRACT FROM AUTHOR]

Database design aims to find a database schema that permits the efficient processing of common types of queries and updates on future database instances. Full first-order decompositions constitute a large class of database constraints that can provide assistance to the database designer in identifying a suitable database schema. We establish a finite axiomatisation of full first-order decompositions that reflects best database design practice: an inference engine derives all potential candidates of a database schema, but the final choice remains with the database designer. [ABSTRACT FROM AUTHOR]

In this paper, we study the relations that exist between Dempster-Shafer Theory and one of its extensions named DSmT. In particular we show, by using propositional logic, that DSmT can be reformulated in the classical framework of Dempster-Shafer theory and that any combination rule defined in the DSmT framework corresponds to a rule in the classical framework. The interest of DSmT rather concerns the compacity of expression it manipulates. [ABSTRACT FROM AUTHOR]

Typically, before association rules are mined, a user needs to determine a support threshold in order to obtain only the frequent item sets. Having users to determine a support threshold attracts a number of issues. We propose an association rule mining framework that does not require a pre-set support threshold. The framework is developed based on implication of propositional logic. The experiments show that our approach is able to identify meaningful association rules within an acceptable execution time. [ABSTRACT FROM AUTHOR]

Introduction In looking at Wittgenstein's philosophy of mind up to 1945 we are attempting to survey a period that covers virtually the whole of Wittgenstein's philosophical development, from the Notebooks, 1914–1916 to the end of Part 1 of the Philosophical Investigations. One of the central interpretative questions raised by the large body of work that is produced in this period is whether we should see it as the more or less continuous development of a reasonably unified philosophical vision, or view it as containing one or more important discontinuities or radical breaks. It is a question on which interpreters of Wittgenstein fundamentally disagree. There can be no question of doing justice to this dispute in this brief introduction to Wittgenstein's thought. I shall therefore limit myself to attempting to develop one clear line of interpretation, in which I side with those who see Wittgenstein's later philosophy as a development, rather than a rejection, of his early work. From the very beginning, Wittgenstein characterises philosophy as a ‘critique of language’ (1921 [1922] Tractatus Logico-Philosophicus (TLP) 4.0031) and associates philosophical problems with ‘our failure to understand the logic of our language’ (TLP: 4.003). We should, therefore, expect his view of the mind to be grounded in his conception of language and how it functions. Similarly, we should expect any development in his view of the mind to be traceable ultimately to developments in his view of language and of how the task of achieving a clarified understanding of it is to be accomplished. Equally, the suggestion that we can trace a continuous development in Wittgenstein's philosophy of mind from the early to the later work commits us to the claim that there is an important, underlying continuity between his early and his late philosophy of language. [ABSTRACT FROM AUTHOR]

As we look back to the philosophy of the period from 1914 to 1945, we tend to think of this as a time when ‘analytic philosophy’ flourished, though of course many other types of philosophy also flourished at this time (idealism, phenomenology, pragmatism, etc.). But what was this ‘analytic philosophy’ of which John Wisdom wrote when he opened his book Problems of Mind and Matter (1934) by saying ‘It is to analytic philosophy that this book is intended to be introduction’ (1934: 1)? Wisdom makes a start at answering this question by contrasting analytic philosophy with ‘speculative’ philosophy: the contrast is that speculative philosophy aims to provide new information (for example, by proving the existence of God), whereas analytic philosophy aims only to provide clearer knowledge of facts already known. Much the same contrast is to be found in the ‘statement of policy’ which opens the first issue of the journal Analysis in 1933: papers to be published will be concerned ‘with the elucidation or explanation of facts…the general nature of which is, by common consent, already known; rather than with attempts to establish new kinds of fact about the world’ (Vol. I: 1). As we shall see, the thesis that philosophy does not aim to provide new knowledge is indeed a central theme of many ‘analytic’ philosophers of this period. But first we need to investigate the relevant conception of analysis – ‘philosophical’ analysis. [ABSTRACT FROM AUTHOR]

Introduction In the most narrow sense, the Polish school of logic may be understood, as the Warsaw school of mathematical logic with Jan Łukasiewicz, Stanisław Leśniewski, and Alfred Tarski as the leading figures. However, valuable contributions to mathematical logic were also made outside Warsaw, in particular by Leon Chwistek. Thus, the Polish school of logic sensu largo also comprises logicians not belonging to the Warsaw school of logic. The third interpretation is still broader. If logic is not restricted only to mathematical logic, several Polish philosophers who were strongly influenced by formal logical results, for example Kazimierz Ajdukiewicz and Tadeusz Kotarbiński, can be included in the Polish school of logic sensu largissimo. Polish work on logic can therefore encompass a variety of topics, from the ‘hard’ foundations of mathematics (e.g. inaccessible cardinals, the structure of the real line, or equivalents of the axiom of choice) through formal logic, semantics, and philosophy of science to ideas in ontology and epistemology motivated by logic or analysed by its tools. Since the development of logic in Poland is a remarkable historical phenomenon, I shall first discuss its social history, especially the rise of the Warsaw school. Then I shall describe the philosophical views in question, the most important and characteristic formal results of Polish logicians, their research in the history of logic, and applications of logic to philosophy. My discussion will be selective: in particular I will omit most results in the ‘hard’ foundations of mathematics. [ABSTRACT FROM AUTHOR]

Logical atomism is a complex doctrine comprising logical, linguistic, ontological, and epistemological elements, associated with Russell and Wittgenstein early in the twentieth century. The first appearance of a form of logical atomism (though not explicitly identified as such) is in Russell's philosophical introduction to Principia Mathematica (1910a; see esp. 43–5). Russell had acquired elements of this position from his earlier studies of Leibniz (who is a clear precursor of logical atomism), from his reaction against absolute idealism (where the influence of G. E. Moore's early atomism, as in Moore 1899, was important), and from his analysis of knowledge. A year later Russell used the term ‘logical atomism’ for the first time (though in French) in his lecture ‘Le réalisme analytique’, where he says of his analytic realism ‘this philosophy is the philosophy of logical atomism’ (1911 [1984– : VI, 135]). Russell's conception of logical atomism developed further in the course of his discussions and correspondence with Wittgenstein during the period from 1912 to 1914. These were primarily concerned with the foundations of logic, but the lessons learnt there were applied by Russell and Wittgenstein to other areas. The term ‘logical atomism’ then became known in English through Russell's 1918 lectures ‘The Philosophy of Logical Atomism’ which provide the fullest presentation of his position (1918 [1984– : VIII]). Though Russell there describes his views as ‘very largely concerned with explaining ideas which I learnt from my friend and former pupil Ludwig Wittgenstein’ (1918 [1984– : VIII, 160]), there are significant differences between their versions of logical atomism. Wittgenstein's logical atomism is set out in his Tractatus Logico-Philosophicus (1921). [ABSTRACT FROM AUTHOR]

Introduction It is uncontroversial to say that the period in question saw more important changes in the philosophy of mathematics than any previous period of similar length in the history of philosophy. Above all, it is in this period that the study of the foundations of mathematics became partly a mathematical investigation itself. So rich a period is it, that this survey article is only the merest sketch; inevitably, some subjects and figures will be inadequately treated (the most notable omission being discussion of Peano and the Italian schools of geometry and logic). Of prime importance in understanding the period are the changes in mathematics itself that the nineteenth century brought, for much foundational work is a reaction to these, resulting either in an expansion of the philosophical horizon to incorporate and systematise these changes, or in articulated opposition. What, in broad outline, were the changes? First, traditional subjects were treated in entirely new ways. This applies to arithmetic, the theory of real and complex numbers and functions, algebra, and geometry. (a) Some central concepts were characterised differently, or properly characterised for the first time, for example, from analysis, those of continuity (Weierstrass, Cantor, Dedekind) and integrability (Jordan, Lebesgue, Young), from geometry, that of congruence (Pasch, Hilbert), and geometry itself was recast as a purely synthetic theory (von Staudt, Pasch, Hilbert). (b) Theories were treated in entirely new ways, for example, as axiomatic systems (Pasch, Peano and the Italian School, Hilbert), as structures (Dedekind, Hilbert), or with entirely different primitives (Riemann, Cantor, Frege, Russell). [ABSTRACT FROM AUTHOR]