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" <rapaport@cse.Buffalo.EDU>
Subject: 663: Quantified Modal Logic II:  Leibniz's Law
To: CSE663-FA08-LIST@LISTSERV.BUFFALO.EDU

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Subject:  Quantified Modal Logic II:  Leibniz's Law
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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.
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(*) 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