Typical Ambiguity: Trying to Have Your Cake and Eat It Too Export

@inbook{feferman:tatthycaeit, abstract = {Ambiguity is a property of syntactic expressions which is ubiquitous in all languages-natural, scientific and mathematical; the efficient use of language depends to an exceptional extent on this feature. Disambiguation is the process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical if the process of disambiguation can be carried out in some systematic way. Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency ("having the cake") with the freedom of the informal untyped theory of classes and relations (and "eating it too"). I shall begin with a brief tour of his own uses of typical ambiguity. Then I will take up cases that he did not consider, namely statements of the form: A is a member of B, where B is a class expression and A is an expression prima facie of the same or higher type than B. Three versions of typical ambiguity for such statements will be treated, respectively in the simple theory of types, Zermelo-Fraenkel set theory, and explicit mathematics. I will show how the "naive" theory of categories and other global theories of structures can thereby be accounted for (to some extent) in the latter two frameworks.}, author = {Feferman, Solomon}, booktitle = {One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy}, citeulike-article-id = {3682156}, citeulike-linkout-0 = {http://books.google.com/books?id=Xg6QpedPpcsC&printsec=frontcover#PPR1,M1}, editor = {Hodges, W. A. and Jensen, R. and Lempp, S. and Magidor, M.}, keywords = {logic, maths, type-theory}, pages = {135--152}, posted-at = {2008-11-25 00:00:13}, priority = {4}, publisher = {Walter de Gruyter}, title = {Typical Ambiguity: Trying to Have Your Cake and Eat It Too}, url = {http://books.google.com/books?id=Xg6QpedPpcsC&printsec=frontcover#PPR1,M1}, year = {2004} }

TY - CHAP ID - feferman:tatthycaeit L3 - citeulike-article-id:3682156 N2 - Ambiguity is a property of syntactic expressions which is ubiquitous in all languages-natural, scientific and mathematical; the efficient use of language depends to an exceptional extent on this feature. Disambiguation is the process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical if the process of disambiguation can be carried out in some systematic way. Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency ("having the cake") with the freedom of the informal untyped theory of classes and relations (and "eating it too"). I shall begin with a brief tour of his own uses of typical ambiguity. Then I will take up cases that he did not consider, namely statements of the form: A is a member of B, where B is a class expression and A is an expression prima facie of the same or higher type than B. Three versions of typical ambiguity for such statements will be treated, respectively in the simple theory of types, Zermelo-Fraenkel set theory, and explicit mathematics. I will show how the "naive" theory of categories and other global theories of structures can thereby be accounted for (to some extent) in the latter two frameworks. EP - 152 TI - Typical Ambiguity: Trying to Have Your Cake and Eat It Too PB - Walter de Gruyter T2 - One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy SP - 135 KW - logic KW - maths KW - type-theory AU - Feferman, Solomon A2 - Hodges, WA A2 - Jensen, R A2 - Lempp, S A2 - Magidor, M PY - 2004/// UR - http://books.google.com/books?id=Xg6QpedPpcsC&printsec=frontcover#PPR1,M1 ER -