W. Van Orman Quine. Mathematical Logic. Norton, New York, 1940.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....both the simple theory of types and ZF (which includes (FA) give rise to an iterative concept of set. That is, both require the elements of a set be present before a new set can be constructed [7] We cannot however stop our discussion of set theory here. Quine s strati cation in NF [56] and ML[57], two non iterative set theories) are suciently type like to merit some discussion. Quine restricted the axiom of comprehension, to obtain the following: SCP) 9x8y[ y 2 x) y) where x is not free in (y) and (y) is strati ed. Quine s NF has attracted a lot of research. Specker [69] ....

....that in each atomic formula F (x i 1 ; x i n ) which appears in , and each integer 1 j n, we have (x i j ) x i 1 ) F j F1 . ML was liked both for the manipulative convenience we regain in it and the symmetrical universe it furnishes. The earlier version of the rst edition of [57] was subject to the Burali Forti paradox The well ordered set of all ordinals has an ordinal which is greater than any member of and hence is greater than In the second edition of [57] Quine corrected the axiomatization of ML (following a suggestion of Hao Wang) so that it is demonstrably ....

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W. Van Orman Quine. Mathematical Logic. Norton, New York, 1940.

....name is not mentioned in the article. Cellucci also reports that this was Quine s first exposure to natural deduction. We can see the background influence that Quine had from the very earliest statements of natural deduction (Jaskowski s) through various pedagogical efforts of Cooley s (and Quine s own) in the 1940s, cumulating with the (1950a) publication and its various editions. 6 Fitch About the same time that Quine was writing his textbook, Frederic Fitch was working on one too. It was published in 1952, and the method in it was also called A History of Natural Deduction and Elementary Logic ....

W.V. Quine. Mathematical Logic. NY: Norton & Co.

....needs to be made explicit. In the previous discussion, a representation of the symbols is given and then a representation of the terms and formulas is constructed in the standard way. However, it is often more convenient to hide the details of the representation from the programmer. Quine (1951) introduced the quasi quotes already used in this chapter to indicate a representation of some unspecified object expression. This (or similar syntax) can be used instead of an explicit representation. For example, dCons(x; N il)e would correspond (using the above representation) to the term ....

Quine, W. V. O. (1951), Mathematical Logic (Revised Edition), Harvard University Press.

....type) Then it is possible to show that WF(EMB) is a provable formula, that (8R)WF(R) EMB(R;EMB) and that (8R)WF(R) EMB(R;R) We have then a contradiction, since EMB(EMB;EMB) and EMB(EMB;EMB) are both provable. It is interesting to note that the notion of stratified formula of Quine [24] is very close to this notion of generic statement. Indeed, the test for checking that a given formula is stratified [24] is a particular case of the ML type checking algorithm. See also the derivation of Rosser [25] of the Burali Forti paradox in an earlier version of the system of Quine. 6 ....

....: We have then a contradiction, since EMB(EMB;EMB) and EMB(EMB;EMB) are both provable. It is interesting to note that the notion of stratified formula of Quine [24] is very close to this notion of generic statement. Indeed, the test for checking that a given formula is stratified [24] is a particular case of the ML type checking algorithm. See also the derivation of Rosser [25] of the Burali Forti paradox in an earlier version of the system of Quine. 6 Consistent extensions of Church s calculus 6.1 Predicative polymorphism It is worth to note that the language HOL of M. ....

W.O. Quine. "Mathematical logic." Harvard University Press (1940).

....That the next toss is a member of (the next toss [ tosses resulting in heads) is not an adequate ground for believing that heads will occur on the next toss. We shall construe these constraints as syntactical: grue(x) emeralds(x) 0:64; 0:86) is 4 We use single quotation marks for Quine s [Qui51] quasi quotation, and double quotation marks for naming expressions. Thus :OE(ff) is not the name of the string consisting of a negation sign, the Greek letter OE, etc. but a function that yields a name on the replacement of OE by a formula of our language, and of ff, wherever it occurs in ....

W. V. O. Quine. Mathematical Logic. Harvard Press, Cambridge, 1951.

....the right translation. Quine [ 1949, p. 27 ] shows how Russell s method can be extended to include proper names, but this is done in a framework where To say that something does not exist, or that is something which is not is clearly a contradiction in terms; hence (x) x :exists) must be true [ Quine, 1947, p. 150 ] Hence, Quine s solution cannot accommodate the cancellation of presuppositions. Strawson s [ 1950 ] attack in the early fifties was only the beginning of the denial of the Russellian approach to presuppositions. Today, the paraphrase theory is no longer widely accepted. 5.2 Meinongian ....

W.V.O. Quine. Mathematical Logic. Cambridge, 1947.

....introduced in [4] The only other axiom of NF is the axiom of extensionality. 1 He suggested the same definition of equality which we will give below, but he did not address the question as to whether the defined notion could be shown to have the correct logical properties; in his later [5] he gives a different definition of equality (from which the logical properties of equality could not be derived) then postulates an axiom from which substitutivity of identity can be derived. The consistency of NF remains an open question. The axiom scheme of stratified comprehension is known to ....

W. V. O. Quine, Mathematical Logic, second edition, Harvard, 1951. 4

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Willard V. O. Quine, Mathematical logic, Norton, New York.

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