Barkley Rosser. Definition by induction in Quine's New foundations for mathematical logic. The journal of symbolic logic, vol. 4 (1939), pp. 80–81.

1940 ◽  
Vol 5 (1) ◽  
pp. 33-33
Author(s):  
Paul Bernays
Keyword(s):  
Mathematical Logic ◽  
Symbolic Logic ◽  
New Foundations

scholarly journals Relative Necessity and Propositional Quantification

2019 ◽  
Vol 49 (4) ◽  
pp. 703-726
Author(s):  
Alexander Roberts
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
Recent Article ◽  
Symbolic Logic ◽  
Philosophical Logic ◽  
Alternative Account ◽  
Standard Account ◽  
Current Article ◽  
Modal Propositional Logic ◽  
Propositional Quantification

AbstractFollowing Smiley’s (The Journal of Symbolic Logic, 28, 113–134 1963) influential proposal, it has become standard practice to characterise notions of relative necessity in terms of simple strict conditionals. However, Humberstone (Reports on Mathematical Logic, 13, 33–42 1981) and others have highlighted various flaws with Smiley’s now standard account of relative necessity. In their recent article, Hale and Leech (Journal of Philosophical Logic, 46, 1–26 2017) propose a novel account of relative necessity designed to overcome the problems facing the standard account. Nevertheless, the current article argues that Hale & Leech’s account suffers from its own defects, some of which Hale & Leech are aware of but underplay. To supplement this criticism, the article offers an alternative account of relative necessity which overcomes these defects. This alternative account is developed in a quantified modal propositional logic and is shown model-theoretically to meet several desiderata of an account of relative necessity.

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.

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