In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

This paper offers a response to William’s Hanson’s criticism of Sher’s formal-structural conception of logical consequence and logical constants.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

Perhaps the most important work of philosophy written in the twentieth century, the Tractatus Logico-Philosophicus first appeared in 1921 and was the only philosophical work that Ludwig Wittgenstein (1889-1951) published during his lifetime. Written in short, carefully numbered paragraphs of extreme compression and brilliance, it immediately convinced many of its readers and captivated the imagination of all. Its chief influence, at first, was on the Logical Positivists of the 1920s and 30s, but many other philosophers were stimulated by its philosophy (...) of language, finding attractive, if ultimately unsatisfactory, its view that propositions were pictures of reality. Perhaps most of all, Wittgenstein himself, after his return to philosophy in the late 1920s, was fascinated by its vision of an inexpressible, crystalline world of logical relationships. The posthumous publication of other writings has, therefore, only served to reawaken interest in the Tractatus and to illuminate its more neglected aspects. David Pears and Brian McGuinness revised their translation in the light of Wittgenstein's own suggestions and comments in his correspondence with C.K. Ogden about the first translation. In addition, this edition contains the introduction by Bertrand Russell which appeared in the original English translation. (shrink)

The paper argues that a philosophically informative and mathematically precise characterization is possible by (i) describing a particular proposal for such a characterization, (ii) showing that certain criticisms of this proposal are incorrect, and (iii) discussing the general issue of what a characterization of logical constants aims at achieving.

In a recent paper, “The Concept of Logical Consequence,” W. H. Hanson criticizes a formal-structural characterization of logical consequence in Tarski and Sher. Hanson accepts many principles of the formal-structural view. Relating to Sher 1991 and 1996a, he says.

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

In this paper I present an outline of a model of knowledge that complements, and is complemented by, my the conception of logic delineated in The Bounds of Logic. The Bounds of Logic had as its goal a critical, systematic and constructive understanding of logic. As such it aimed at maximum neutrality vis-a-vis epistemic, metaphysical and meta-mathematical controversies. But a conception of logic does not exist in a vacuum. Eventually our goal is to produce an account of logic that answers (...) the needs of, contributes to the development of, and is supported by, a broader epistemology. In this paper I make first steps in this direction. I begin with an outline of a model of knowledge whose basic principles are based on the early Quine. I identify, and offer independent justification for, the special requirements this model sets on an adequate conception of logic. Finally, I show how, by satisfying these requirements, the conception of logic delineated in The Bounds of Logic can naturally be incorporated in this epistemic model. Logic, in this model, is both central and peripheral, is guided by both veridical and pragmatic norms, is both grounded in the world and assigned a central role in unifying our body of knowledge. (shrink)

In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which (...) follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentenceσis false. Moreover, … we are concerned here with the concept of logical, i.e.,formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since. (shrink)

One of the cornerstone books of Western philosophy, Critique of Pure Reason is Kant's seminal treatise, where he seeks to define the nature of reason itself and builds his own unique system of philosophical thought with an approach known as transcendental idealism. He argues that human knowledge is limited by the capacity for perception and attempts a logical designation of two varieties of knowledge: a posteriori, the knowledge acquired through experience; and a priori, knowledge not derived through experience. This accurate (...) translation by J. M. D. Meiklejohn offers a simple and direct rendering of Kant's work that is suitable for readers at all levels. (shrink)