This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much (...) more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included. (shrink)

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; (...) these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science. (shrink)

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part II supplements (...) the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

Free logic, an alternative to traditional logic, has been seen as a useful avenue of approach to a number of philosophical issues of contemporary interest. In this collection, Karel Lambert, one of the pioneers in, and the most prominent exponent of, free logic, brings together a variety of published essays bearing on the application of free logic to philosophical topics ranging from set theory and logic to metaphysics and the philosophy of religion. The work of such distinguished philosophers as Bas (...) van Fraassen, Dana Scott, Tyler Burge, and Jaakko Hintikka is represented. Lambert provides an introductory essay placing free logic in the logical tradition beginning with Aristotle, developing it as the natural culmination of a trend begun in the Port Royal logic of the 1600s, and continuing through current predicate logic--the trend to rid logic of existence assumptions. His Introduction also provides a useful systematic overview of free logic, including both a standard syntax and some semantical options. (shrink)

The expression ‘free logic,’ coined by the author in 1960, is an abbreviation for ‘logic free of existence assumptions with respect to its terms, singular and general, but whose quantifiers are treated exactly as in standard quantifier logic.’ In more traditional language, such logics do not presume that either singular or general terms — the two distinct categories of terms emphasized in modern logical grammar — have existential import. A singular term ‘t’ has existential import just in case t exists (...) (or, equivalently, there exists an object the same as t) and a general term (or predicate) ‘G’ has existential import just in case G exist (or, equivalently, there exists an object that is G).1 Examples from colloquial English customarily taken to be singular terms are expressions such as ‘Socrates’, ‘the planet causing perturbations in the orbit of Mercury’, ‘5’, ‘5/0’, ‘the square of 3’ and ‘having a heart’. Some of these do not have existential import — in particular, ‘5/0’ and ‘the planet causing perturbations in the orbit of Mercury’. Examples from colloquial English customarily taken to be general terms are expressions such as ‘is a philosopher’, ‘is a planet causing perturbations in the orbit of Mercury’, ‘number’, ‘is divisible by 0’, and ‘has a heart’. Some of these general terms do not have existential import — in particular, ‘is a planet causing perturbations in the orbit of Mercury’ and ‘is divisible by 0’. To say that the quantifers are treated exactly as in standard quantifier logic is to say, roughly, that the operator symbol ‘∃’ (the existential quantifier) reads: ‘There exists an object’, and the operator symbol ‘∀’ (the universal quantifier) reads: ‘Every existent object’. (shrink)

The logic of partial terms (LPT) is a variety of negative free logic in which functions, as well as predicates, are strict. A companion paper focused on nonconstructive LPTwith definite descriptions, called LPD, and laid the foundation for tableaux systems by defining the concept of an LPDmodel system and establishing Hintikka's Lemma, from which the strong completeness of the corresponding tableaux system readily follows. The present paper utilizes the tableaux system in establishing an Extended Joint Consistency Theorem for LPDthat incorporates (...) the Robinson Joint Consistency Theorem and the Craig-Lyndon Interpolation Lemma. The method of proof is similar to that originally used in establishing the Extended Joint Consistency Theorem for positive free logic. Proof of the Craig-Lyndon Interpolation Lemma for formulas possibly having free variables is readily had in LPTand its intuitionistic counterpart. The paper concludes with a brief discussion of the theory of definitions in LPD. (shrink)