Now in its fourth edition, this classic work clearly and concisely introduces the subject of logic and its applications. The first part of the book explains the basic concepts and principles which make up the elements of logic. The author demonstrates that these ideas are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The second part of the book shows the applications of logic in mathematical theory building with concrete examples that draw (...) upon the concepts and principles presented in the first section. Numerous exercises and an introduction to the theory of real numbers are also presented. Students, teachers and general readers interested in logic and mathematics will find this book to be an invaluable introduction to the subject. (shrink)

The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.

Starting from certain metalogical results, I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. Consequently, each philosophical approach to these (...) truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate. (shrink)

Abstract: Luciano Floridi has impressively applied the concept of information to problems in semantics and epistemology, among other areas. In this essay, I briefly review two areas where I think one may usefully raise questions about some of Floridi's conclusions. One area is in the project to naturalize semantics and Floridi's use of the derived versus nonderived notion of semantic content. The other area is in the logic of information and knowledge and whether knowledge based on information necessarily supports closure, (...) in every instance. I suggest that it does not and, thereby, raise a challenge to Floridi's logic of being informed. (shrink)

Understandably absorbed in technical details, discussion of the semantic paradoxes risks losing sight of broad methodological principles. This chapter sketches a general approach to the comparison of rival logics, and applies it to argue that revision of classical propositional logic has much higher costs than its proponents typically recognize.

According to John Haugeland, the capacity for “authentic intentionality” depends on a commitment to constitutive standards of objectivity. One of the consequences of Haugeland’s view is that a neurocomputational explanation cannot be adequate to understand “authentic intentionality”. This paper gives grounds to resist such a consequence. It provides the beginning of an account of authentic intentionality in terms of neurocomputational enabling conditions. It argues that the standards, which constitute the domain of objects that can be represented, reflect the statistical structure (...) of the environments where brain sensory systems evolved and develop. The objection that I equivocate on what Haugeland means by “commitment to standards” is rebutted by introducing the notion of “florid, self-conscious representing”. Were the hypothesis presented plausible, computational neuroscience would offer a promising framework for a better understanding of the conditions for meaningful representation. (shrink)

This paper traces the application of information theory to philosophical problems of mind and meaning from the earliest days of the creation of the mathematical theory of communication. The use of information theory to understand purposive behavior, learning, pattern recognition, and more marked the beginning of the naturalization of mind and meaning. From the inception of information theory, Wiener, Turing, and others began trying to show how to make a mind from informational and computational materials. Over the last 50 years, (...) many philosophers saw different aspects of the naturalization of the mind, though few saw at once all of the pieces of the puzzle that we now know. Starting with Norbert Wiener himself, philosophers and information theorists used concepts from information theory to understand cognition. This paper provides a window on the historical sequence of contributions made to the overall project of naturalizing the mind by philosophers from Shannon, Wiener, and MacKay, to Dennett, Sayre, Dretske, Fodor, and Perry, among others. At some time between 1928 and 1948, American engineers and mathematicians began to talk about `Theory of Information' and `Information Theory,' understanding by these terms approximately and vaguely a theory for which Hartley's `amount of information' is a basic concept. I have been unable to find out when and by whom these names were first used. Hartley himself does not use them nor does he employ the term `Theory of Transmission of Information,' from which the two other shorter terms presumably were derived. It seems that Norbert Wiener and Claude Shannon were using them in the Mid-Forties. (shrink)

This paper is about the putative theoretical virtue of strength, as it might be used in abductive arguments to the correct logic in the epistemology of logic. It argues for three theses. The first is that the well-defined property of logical strength is neither a virtue nor a vice, so that logically weaker theories are not—all other things being equal—worse or better theories than logically stronger ones. The second thesis is that logical strength does not entail the looser characteristic of (...) scientific strength, and the third is that many modern logics are on a par—or can be made to be on a par—with respect to scientific strength. (shrink)

While Karl Popper’s philosophy of science has only few followers among modern philosophers, it is easily the view of science with the biggest impact on practicing scientists. According to Peter Medawar, Nobel laureate and eminent physiologist, Popper was the greatest authority ever on the scientific method. He praised the “great strength of Karl Popper’s conception of the scientific process,” a main reason for the praise being “that it is realistic—it gives a pretty fair picture of what goes on in real (...) life laboratories” (Medawar 1990, p. 100). Either explicitly or (more often) implicitly, many scientists subscribe to some version of simplified Popperianism, usually by adopting the demarcation criterion that a .. (shrink)

In "Semantic paradoxes and abductive methodology", Williamson presents a new Quinean argument based on central ingredients of common pragmatism about theory choice (including logical theory, as is common). What makes it new is that, in addition to avoiding Quine's unfortunate charge of mere terminological squabble, Williamson's argument explicitly rejects at least for purposes of the argument Quine's key conservatism premise. In this paper I do two things. First, I argue that Williamson's new Quinean argument implicitly relies on Quine's conservatism principle. (...) Second, by way of answering his charges against nonclassical logic I directly defend a particular subclassical account of logical consequence. (shrink)

First, we consider an argument due to Popper for maximal strength in choice of logic. We dispute this argument, taking a lead from some remarks by Susan Haack; but we defend a set of contrary considerations for minimal strength in logic. Finally, we consider the objection that Popper presupposes the distinctness of logic from science. We conclude from this that all claims to logical truth may be in equal epistemological trouble.