New Foundations for Mathematical Logic

1937 ◽  
Vol 44 (2) ◽  
pp. 70 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Mathematical Logic ◽  
New Foundations

Willard van Orman Quine. Foreword, 1980. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. vii–ix. - Willard Van Orman Quine. On what there is. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 1–19. - Willard Van Orman Quine. Two dogmas of empiricism. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 20–46. - Willard Van Orman Quine. The problem of meaning in linguistics. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 47–64. - Willard Van Orman Quine. Identity, ostension, and hypostasis. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 65–79. - Willard Van Orman Quine. New foundations for mathematical logic. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 80–101. - Willard Van Orman Quine. Logic and the reification of universals. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 102–129. - Willard Van Orman Quine. Notes on the theory of reference. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 130–138. - Willard Van Orman Quine. Reference and modality. A revised reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 139–159. - Willard Van Orman Quine. Meaning and existential inference. A reprint of XXXIII 149. From a logical point of view, 9 logico-philosophical essays, by Willard Van Orman Quine, second edition, revised, Harvard University Press, Cambridge, Mass., and London, 1980, pp. 160–167.

1982 ◽  
Vol 47 (1) ◽  
pp. 230-231 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Mathematical Logic ◽  
Harvard University ◽  
Point Of View ◽  
Logical Point ◽  
New Foundations

On the consistency of Quine's New foundations for mathematical logic

1939 ◽  
Vol 4 (1) ◽  
pp. 15-24 ◽  
Author(s):  
Barkley Rosser
Keyword(s):  
Mathematical Logic ◽  
New Foundations ◽  
Binary Scale

In the present paper, the various means used by the author in attempting to find a contradiction in Quine's New foundations are sketched, and the reason why each method failed is indicated. It was not Quine's system itself which was tested, but a stronger one obtained by adding a rule of Kleene's type to Quine's system. Reasons are presented why a contradiction in the stronger system should be as damning to Quine's system as a contradiction in Quine's system itself. Two other features of the system are worthy of note. One is the fact that only two symbols are used in building up the formulas of the system. Other systems have been built using only two symbols, but the particular method used in this paper is very flexible and simple, and is peculiarly adapted to the use of the Gödel technique. It was suggested by the consideration that the formulas of any system can be written by the use of only two symbols by first assigning Gödel numbers to the formulas and then writing those numbers in the binary scale of notation. The second feature is that ι (to be used in ιxp, meaning “the x such that p”) is an explicit and integral part of the system, and the axioms and rules governing its use are such as to make it very simple to handle. By a process similar to “Die Eliminierbarkeit der ι-Symbole” of Hilbert-Bernays, it is shown that the system involving the ι can be reduced to one not involving it. The points of difference with the Hilbert-Bernays technique make possible an especially unhampered use of ι.

Stratified systems of logic

1943 ◽  
Vol 39 (2) ◽  
pp. 69-83 ◽  
Author(s):  
M. H. A. Newman
Keyword(s):  
Mathematical Logic ◽  
Partially Ordered Set ◽  
General Class ◽  
Ordered Set ◽  
Partially Ordered ◽  
New Foundations

The suffixes used in logic to indicate differences of type may be regarded either as belonging to the formalism itself, or as being part of the machinery for deciding which rows of symbols (without suffixes) are to be admitted as significant. The two different attitudes do not necessarily lead to different formalisms, but when types are regarded as only one way of regulating the calculus it is natural to consider other possible ways, in particular the direct characterization of the significant formulae. Direct criteria for stratification were given by Quine, in his ‘New Foundations for Mathematical Logic’ (7). In the corresponding typed form of this theory ordinary integers are adequate as type-suffixes, and the direct description is correspondingly simple, but in other theories, including that recently proposed by Church(4), a partially ordered set of types must be used. In the present paper criteria, equivalent to the existence of a correct typing, are given for a general class of formalisms, which includes Church's system, several systems proposed by Quine, and (with some slight modifications, given in the last paragraph) Principia Mathematica. (The discussion has been given this general form rather with a view to clarity than to comprehensiveness.)

A set of axioms for logic

1944 ◽  
Vol 9 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Theodore Hailperin
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Predicate Calculus ◽  
Set Membership ◽  
New Foundations

One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.

On Cantor's theorem

1937 ◽  
Vol 2 (3) ◽  
pp. 120-124 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Mathematical Logic ◽  
Relation Theory ◽  
Logical Constants ◽  
New Foundations ◽  
Cantor’S Theorem ◽  
Universal Quantification ◽  
New System

Among various unnatural and technically burdensome effects of the theory of types, one is the unstable character of meaningfulness: a mere permutation of variables is capable of reducing a significant context to meaninglessness. Another effect, and perhaps the most conspicuous, is the systematic reduplication to which the logical constants are subjected; the calculi of classes and relations and even arithmetic lose their unity and generality, and are reproduced anew within each type. The elaborate compensatory manoeuvres which are thus made necessary are familiar to all readers of Principle, mathematica.In a recent publication, to be cited henceforth as New foundations, I proposed an alternative course which avoids these consequences, but which would seem to offer less assurance of consistency. My efforts to derive a contradiction have delivered none, but they have lent a strange aspect to Cantor's proof that every class has more subclasses than members. This result is the topic of the present paper.Preparatory to sketching the new system, which I shall call S′, I shall sketch a very similar but contradictory system S. By way of primitives S involves just membership, universal quantification, and alternative denial (Sheffer's stroke function), together with general variables “x”, “y”, …; the adequacy of this equipment as a basis for mathematical logic is made evident by Wiener's and Kuratowski's discovery of methods of constructing relation theory in terms of classes. Thus the formulae or statements and statement forms of S are describable recursively as follows: if a variable is put in each blank of “(ϵ)”, the result is a formula; if a variable enclosed in parentheses is prefixed to a formula, the result is a formula; and if a formula is put in each blank of “(∣)”, the result is a formula. The theorems of S are determined by a postulate and five rules, called P1 and R1-5 in New foundations. P1 and R1-3 specify various formulae as initial theorems, and R4-5 specify inferential connections for deriving further theorems. R1–2 and R4–5 are so fashioned as to provide the “theory of deduction”: they provide as theorems all those formulae which are valid by virtue merely of their structure in terms of alternative denial and quantification.

Willard Van Orman Quine. On what there is. A reprint of XIX 134(1). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 1–19. - Willard Van Orman Quine. Two dogmas of empiricism. A reprint of XIX 134(2). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 20–46. - Willard Van Orman Quine. The problem of meaning in linguistics. A reprint of XIX 134(3). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 47–64. - Willard Van Orman Quine. Identity, ostension, and hypostasis. A reprint of XIX 134(4). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 65–79. - Willard Van Orman Quine. New foundations for mathematical logic. A reprint of XIX 134(5). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 80–101. - Willard Van Orman Quine. Logic and the reification of universals. A revised reprint of XIX 135. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 102–129. - Willard Van Orman Quine. Notes on the theory of reference. A reprint of XIX 136. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 130–138. - Willard Van Orman Quine. Reference and modality. A revised reprint of XIX 137. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 139–159. - Willard Van Orman Quine. Meaning and existential inference. A reprint of XIX 138. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 160–167.

1968 ◽  
Vol 33 (1) ◽  
pp. 149-150
Author(s):  
Frederic B. Fitch
Keyword(s):  
New York ◽  
Mathematical Logic ◽  
Harvard University ◽  
Point Of View ◽  
Logical Point ◽  
New Foundations
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