This book presents modern logic as the formalization of reasoning that needs and deserves a semantic foundation. Chapters on propositional logic; parsing propositions; and meaning, truth and reference give the reader a basis for establishing criteria that can be used to judge formalizations of ordinary language arguments. Over 120 worked examples illustrate the scope and limitations of modern logic, as analyzed in chapters on identity, quantifiers, descriptive names, and functions. The chapter on second-order logic shows how different conceptions of predicates (...) and propositions do not lead to a common basis for quantification over predicates, as they do for quantification over things. Notable for its clarity of presentation and supplemented by many exercises, this volume will be invaluable for philosophers, linguists, mathematicians, and computer scientists who wish to better understand the tools they use in formal reasoning. (shrink)

This book is dedicated to a classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal (...) logic to give a full development of Gödel's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics". (shrink)

The contributions in the book examine the historical and contemporary manifestations of organized crime, the symbiotic relationship between legitimate and ...

The most popular method of incorporating time into a formal logic is based on the work of Arthur Prior. It treats tenses as operators on sentences. In this essay I show a serious problem with that approach, a confusion of scheme versus proposition, which makes any system built in that way incoherent. I will compare how other formal logics deal with the scheme versus proposition distinction and find that only for formal modal logics does the same problem arise. I then (...) compare Prior’s approach to other ways of taking time into account in formal logics. (shrink)

This book grew out of my confusion. If logic is objective how can there be so many logics? Is there one right logic, or many right ones? Is there some underlying unity that connects them? What is the significance of the mathematical theorems about logic which I've learned if they have no connection to our everyday reasoning? The answers I propose revolve around the perception that what one pays attention to in reasoning determines which logic is appropriate. The act of (...) abstracting from our reasoning in our usual language is the stepping stone from reasoned argument to logic. We cannot take this step alone, for we reason together: logic is reasoning which has some objective value. For you to understand my answers, or perhaps better, conjectures, I have retraced my steps: from the concrete to the abstract, from examples, to general theory, to further confirming examples, to reflections on the significance of the work. (shrink)

What do we mean by ‘countable’?Export citation.

MOTIVATION The constructivization of ; => D(_£^') poses several problems. For some of these the tools of MD can be modified; for others new methods will need to be established. What must we do to make a full approximation to ; * D? We ...

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Theories of Truth provides a clear, critical introduction to one of the most difficult areas of philosophy. It surveys all of the major philosophical theories of truth, presenting the crux of the issues involved at a level accessible to nonexperts yet in a manner sufficiently detailed and original to be of value to professional scholars. Kirkham's systematic treatment and meticulous explanations of terminology ensure that readers will come away from this book with a comprehensive general understanding of one of philosophy's (...) thorniest set of topics. -/- Included are discussions of the correspondence, coherence, pragmatic, semantic, performative, redundancy, appraisal, and truth-as-justification theories. There are also chapters or sections of chapters on the liar paradox, three-valued logic, Field's critique of Tarski, Davidson's program, Dummett's theory of linguistic competence, satisfaction, recursion, the extension/intension distinction, and an explanation of how theories of justification, properly understood, differ from theories of truth. -/- A persistent theme is that philosophers have too often failed to recognize that not all theories of truth are intended to answer the same question. When the various questions are made distinct, it is apparent that many of the "debates" in this field are really cases of philosophers talking past one another. There is much less disagreement within the field than has commonly been thought. (shrink)

In a number of essays published over the last decade or so, W. V. Quine has made some interesting suggestions concerning the ontology of theories. If I understand him correctly, one of his principal objects has been to formulate a criterion by means of which one can correctly decide what are the ontological commitments of any given theory. My aim in this paper is to reveal what I think are inadequacies in Quine's criterion and to indicate the direction in which (...) a proper formulation lies. (shrink)