Well, I’ve answered the specifics of Libertarian’s preferred modal proof in at least two of the earlier threads, so I will attempt to not repeat myself here. I will summarize that objections to the proof can include both issues with the form of the definition, the assumption of “God’s” possible existence, the applicability of S5 to modal questions regarding non-rigid expressions, the appropriate reading of the symbol (see Quine and especially Prior, who insists upon a distinction between A and ~<>~A for possibly uninstantiated variables.) If anyone really wishes a recapitulation of these points please ask (or wad through the earlier threads.)
I would like to address Lib’s use of RN to prove that “necessary existence” is true in S5. I want to highlight hits because it is very illustrative of the potential confusions that can arise in interpreting modal arguments. The proof appears to be valid in S5, even sound, but quantifiers in modal logic are not straightforward and I disagree with Lib’s interpretation of what has been demonstrated. The key(s) lie in the interpretation of the identity axiom, the implications of quantifiers, and in the Rule of Necessitation itself.
x = x
Simple, right? Often expressed (x)(x=x). In Lib’s quantified notation, Ax(x=x). Here’s the question: Does the identity axiom imply the existence of x?
Most modern logicians, following Boole, would say no. Universal quantifiers do not carry existential import, though existential quantifiers do (intuitively). So, this means that in Lib’s proof we have not demonstrated that “x” exists. No problem, really. Yet.
Step 2 – looks good, still no implication of existential import.
Step 2 – looks good, still no implication of existential import.
Step 4 – oops. Now we are implying Ey(y=x). Somehow we have managed to insert an implication of actual existence into our logical structure. Let’s look at how that happened.
Step 4 relies upon the definition of E {Ex(A) -> ~Ax(~A)}. This is a classically correct treatment of quantifiers, but that can be a problem in modal logics–it leads to many conclusions that are so counter-intuitive as to convince many logicians that teh rules of FL (Free Logic) should be used instead. Interestingly enough, one of the classic (so stanford.edu assures me) problems encountered when applying classical quantifiers to a fixed-domain modal logic (which is the best place to apply them, most agree) is that it leads to exactly the conclusion of Libertarian’s proof. More on that in a second.
In the quantifier rules of FL, put simply, introduce a new predicate, let’s call it E, for “exists actually”. In this case, the existential predicate of classical quantification (“E” in Lib’s notation, a reversed “E” symbol classically, but I don’t have the patience to figure out what ascii code will produce that on the board) is read as “some” and understood to not imply “real” existence. In FL, the inference from Ax(Rx) (Everything is real) to Rn (some specific thing is real) is not valid. Coincidentally, one of the charges sometimes leveled against the FL approach is that it makes existence a predicate, which ties back very nicely to Lib’s OP. I’ll get back to that soon.
First, a return to Lib’s conclusion. One of the reasons that it is considered a damning conclusion for fixed-domain applications of classical quantifiers is that the classical interpretation is not usually interpreted “necessary existence is true”. It is read: everything exists necessarily. To break it down: AxEy(y = x)
[ul]li – y is identical to x[/li][li]Ey(y = x) – x exists[/li][li]Ey(y = x) – x exists necessarily[/li][li]AxEy(y = x) – Every x exists necessarily[/ul][/li]Now, this can be variously read as “all things that exist exist necessarily” or “necessarily all things exist”. Both of these conclusions, though, conflict plainly with the intuition that some things really do exist conditionally. Interestingly enough, this very conclusion lends credence to the arguments that “G” could be [insert mundane or ridiculous being here]. I have no idea whether those raising that barb intended it to call attention to the use of classical quantifiers in a fixed modal domain, but I think that that is the only way it really carries weight. Barring such context, it is pretty hard to argue that all existence is unconditional.
So, this brings us to the Rule of Necessitation itself. The interesting thing about RN is that it shows us exactly how X -> X is generally understood within a modal logic. X is a theorem of the logic in question. That is all. Any theorem exists unconditionally in the domain defined by the logic of which it is a member. Barring the interjection of some other predicate for “G” in Lib’s original ontological proof, there is absolutely no reason to bind the symbol to a “supreme being” of any definition rather than to the theorem “… = ”. Now, I admit that it’s a pretty cool theorem, but I don’t think I am going to start worshipping it.
Lib
I have to apologize, now. I had hopes to dispense with the modal arguments more quickly than this since I was trying not to repeat old ground, but brevity has never been my strength. Then the board went down and I couldn’t even post this reply “on time”. I’ll return to discuss the idea of predicated existence as soon as I am able.
Later.
