From owner-cse663-fa08-list@LISTSERV.BUFFALO.EDU Sun Sep 21 16:56:06 2008 Date: Sun, 21 Sep 2008 16:55:57 -0400 From: "William J. Rapaport" Subject: 663: Quantified Modal Logic II: Leibniz's Law To: CSE663-FA08-LIST@LISTSERV.BUFFALO.EDU ------------------------------------------------------------------------ Subject: Quantified Modal Logic II: Leibniz's Law ------------------------------------------------------------------------ Here's another problem with quantified modal logic that I didn't have time to discuss in class. Because of the limitations of an ASCII keyboard, I'll use the following notation: A for the universal quantifier > for the material conditional iff for the material biconditional P for an n-place predicate p(x) for P containing the free variable x L for the necessity "box"; 1. "Leibniz's Law" is an attempt (based on some writings of the 17th-century rationalist philosopher and co-inventor of calculus, G.W. Leibniz) to define equality: (LL) AxAy[x=y > P(x) iff P(y)], for any P I.e., if x and y are identical... (Digression: A question to ponder: How can TWO things be identical? If "they" are identical, there's only one of them :-) ... anyway: if x and y are identical, then: something is true about one iff it's true about the other. I.e., identical things share all and only the same properties.(*) 2. Now: Take "P" to be: being necessarily equal to x, i.e., L(x=__) Then an instance of (LL) is: (LL1) AxAy[x=y > L(x=x) iff L(x=y)] So far, so good. But: |- Ax[x=x] (i.e., "x=x" is a theorem of FOL) So, |-LAx(x=x) (by the Rule of Necessitation) So far, so good, again. But now it follows that: AxAy[x=y > L(x=y)] (from (LL1) by Universal Instantiation and MP) What does that say? It says that if "two" things just happen to be identical, then "they" are *necessarily* identical. Suppose x = the morning star and y = the evening star. As it happens, the morning star = the evening star. But, logically, we now see that this was *necessary*, which seems counterintuitive to say the least. ------------------------------------------------------------------------ (*) By the way, the converse of (LL) is: AxAy[P(x) iff P(y) > x=y] i.e., if "two" things share all and only the same properties, then "they" are identical, i.e., there is really only 1 thing, not 2. Here's a puzzle: The Two Spheres: Consider a possible universe that contains only two planets, both of which have all and only the same properties. Yet there are two, not one. How can this be? For discussion, see: Black, Max (1952), "The Identity of Indiscernibles", Mind 61: 153-164. http://www.jstor.org/stable/2252291