On 10/21/2011 7:23 AM, Steve Vickers wrote:
What Vaughan called the "clean-shaven gods" argument is, I think, the normal logicians' excuse for barring empty carriers.
That's my alternative to St. Augustine's proof of the existence of god, with "clean-shaven" thrown in to draw attention away from the real fallacy (one would no more portray God without a beard than Santa Claus). What I don't understand is why the emphasis on existence to the exclusion of uniqueness, given that monotheism is the intellectual descendant not of atheism but polytheism. The only explanation I've been able to come up with is that uniqueness is an equational property (g = g') and therefore admissible as an axiom, say as part of the definition of god. Postulating existence in a definition is an obvious cheat since the definition might be inconsistent. A minimal requirement for an existence proof is a consistency proof, which are tough to begin with and get increasingly tougher with increasing scientific understanding.
The logical deduction
(all) x. P(x) ------------- P(a) ------------- (exists) x. P(x)
Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure predicate calculus, which is "clearly" false in the empty universe when P is taken to be the identically true predicate.
The fix was discovered by Mostowski (I believe). To use a free variable such as a is to hypothesize a denotation for it in the carrier. Therefore in each deduction one should be explicit about the context of free variables that may be used. This deals cleanly with empty carriers.
If you use cylindric algebras B as the subalgebras of the Boolean algebra 2^D^V of all V-ary relations on a domain D, there is nothing to fix. Given a first order language L of wffs over a set X of variables, for each subset V of X, let L_V denote those wffs of L using variables from V, and for each D and each B as above let I_B: L_V --> B interpret the wffs of L_V in B subject only to requirement T0 below. Define the theory T(L) to consist of those wffs P of L such that every I_B having P in its domain map P to the top element of B. Requirement T0 is that the intersection of T(L) and L_0 consist of the propositional tautologies. (Every reasonable notion of classical pure predicate calculus must surely satisfy these extremely mild conditions, which shouldn't be too hard to phrase appropriately for intuitionist logic as well.) Claim. T(L) is independent of whether D = 0 is forbidden. Proof. Allowing D = 0 can only decrease T(L). There are two cases. 1. V > 0. In this case D^V = 0 whence B has only one element and so I_B maps *every* wff to the top element of B. This cannot decrease T(L). 2. V = 0. This is propositional calculus, for which requirement T0 makes the choice of D irrelevant. QED I noticed this property of cylindric algebras when I first ran across them in mid-1979 and assumed it must be well-known since Tarski and Henkin had been playing around with them since the 1950s and could hardly have overlooked such an elementary fact. A month later I attended the sixth International Congress of Logic, Methodology, and Philosophy of Science in Hannover where I met a number of algebraic logicians for the first time: Routley, Dunn, Nemeti, Andreka, Maksimova, etc. When Nemeti and Andreka told me they were working with Henkin, Monk and Tarski on Vol. 2 of Cylindric Algebras, I asked them if they were aware that cylindric algebra semantics solved the logic-of-the-empty-universe problem. They said no so I showed them the argument. Since they were much better connected to the cylindric community than I, I figured they'd pass it on, relieving me of any obligation to write up what seemed to me little more than a passing remark. I was hoping it might appear in Vol. 2 but I don't see it there. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]