From holmes@catseye.idbsu.edu Tue Mar 23 13:42 MST 1999 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA29916; Tue, 23 Mar 1999 13:40:36 -0700 Date: Tue, 23 Mar 1999 13:40:36 -0700 From: Randall Holmes Return-Path: To: csilver@sophia.smith.edu, holmes@catseye.idbsu.edu Subject: Re: FOM: sterility? Cc: fom@math.psu.edu Status: R I understand by "second-order logic" first-order logic with the addition of quantifiers over predicates (there's more to it, of course). In this language, the issue of identity conditions for second-order "objects" does not arise for purely grammatical reasons. One could introduce P = Q (where P and Q are second-order variables) as an abbreviation for (forall x.Px iff Qx) if one favored an extensional approach to equality; there doesn't seem to be a way to introduce any other notion of equality. I think that some formal treatments of second-order logic do allow equations between second-order variables; in a such a treatment, I suppose one could adopt extensionality or not. I'm partial myself to the view that equations between properties are either senseless (this would go along with their exclusion by the grammar of one's formal language) or to be understood extensionally, but this is a substantial philosophical question which requires careful consideration. I don't think that it affects the mathematical applications of second-order logic considered here; I would be interested to hear from anyone who thinks it does bear on the mathematics. In third-order logic, equality for second order objects becomes formally definable! (x^2 = y^2 =def for all P^3, P^3(x^2) iff P^3(y^2)). Philosophers might or might not like this definition? But I prefer to eliminate third order logic in favor of second-order logic with two first-order sorts; I don't think that nth order logic for n>2 adds anything essential to the logical expressiveness of 2nd order logic (nth order logic is stronger than 2nd order logic for n>2, but is equivalent to 2nd order systems with additional "nonlogical" hypotheses). And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes the gates of Cantor's paradise, that the | Boise State U. (disavows all) slow-witted and the deliberately obtuse might | holmes@math.idbsu.edu not glimpse the wonders therein. | http://math.idbsu.edu/~holmes