Kripke Semantics and Strange Truths
On a standard Tarskian account of FOPL a model is taken to be an ordered pair where D is a non-empty domain of individuals. Given the non-empty constraint, there is no model on which some sentence (x)Fx expresses a vacuous truth. It is true in M iff every member of D in M (and there has to be at least one) satisfies the open formula Fx.
Now consider a Kripkean account of MFOPL. On this account a model includes a a set of world W, a domain D and a function Q from members of w to subsets of D. There is no constaint that says the subset must be non-empty. So, there are models with worlds that have an empty domain of quantification. This is wierd.
Why? Because on such a model the sentence <>(x)Fx will express a truth. Moreover, given a plausible extension of the semantics to allow for second order quantification (MSOPL) we will have the following turn out true in models that have empty world: <>(x)(F)Fx. This is very strange. Moreover, the situation becomes more strange when we realize that in the empty world it must also be true that (x)(F)~Fx. Why? Because the denial of (x)(F)~Fx is existentially commiting. But the empty world is empty. So, there are models on which the following is true <>[(x)(F)Fx & (x)(F)~Fx]. But that entails that <>[(x)(F) Fx & ~Fx]. But that can't be true. So what the hell is going on? Should there be a constraint on Q; should it be a function from worlds to non-empty subsets of D? Moreover, if that is the case, then do we have reasons from the semantics of modal discourse for the conclusion that necessarily, something exists?
Now consider a Kripkean account of MFOPL. On this account a model includes a a set of world W, a domain D and a function Q from members of w to subsets of D. There is no constaint that says the subset must be non-empty. So, there are models with worlds that have an empty domain of quantification. This is wierd.
Why? Because on such a model the sentence <>(x)Fx will express a truth. Moreover, given a plausible extension of the semantics to allow for second order quantification (MSOPL) we will have the following turn out true in models that have empty world: <>(x)(F)Fx. This is very strange. Moreover, the situation becomes more strange when we realize that in the empty world it must also be true that (x)(F)~Fx. Why? Because the denial of (x)(F)~Fx is existentially commiting. But the empty world is empty. So, there are models on which the following is true <>[(x)(F)Fx & (x)(F)~Fx]. But that entails that <>[(x)(F) Fx & ~Fx]. But that can't be true. So what the hell is going on? Should there be a constraint on Q; should it be a function from worlds to non-empty subsets of D? Moreover, if that is the case, then do we have reasons from the semantics of modal discourse for the conclusion that necessarily, something exists?
posted by Joshua at 8:42 PM
4 Comments:
It doesn't seem so bad to me to say that there could have been nothing and if that possibility were to have obtained, then possibly, what exists there could have had any property you like, though what exists there in fact has no properties. Also, if there's a problem here, MSOPL isn't to blame; presumably one could prove the corresponding schema of a metatheorem for MFOPL that would in effect establish that for whatever F you like, possibly all things there have it.
Is the worry that the possibility claim will be true only if some world accessible from the empty world is one in which everything has every property? That's satisfied by the empty world itself. And, besides, it does not seem too odd to me to assert, de re, of things that exist at the empty world (i.e., no things), that they could have been any way you like.
Am I missing the point here?
My worry had nothing to do with the introduction of MSOPL. A similarly distressing problem can arise with MFOPL. The worry is simply that if Kripkean Semantics is correct, then ther eis a model in which <>[(x)Fx & ~Fx) is true. But that doesn't seem like it should come out true in any model.
It just doesn't seem that weird to me if we keep in mind that it only holds at empty worlds. Can you say more about why I should be distressed?
I think that I am confusing a metaphysical point with a logical point. I think this because the only reasons I can think of to bolster the seeming wierdness of this have to do with metaphysics rather than logic.
Here is what I have been thinking since your last comment. Suppose someone holds a radical view according to which necessarily, if there is anything, then everything is contingent. If the person also holds plausible principles of recombination, then he will also believe that possibly, there is nothing. But then he is committed to the strange sounding claim that possibly, everything both has and lacks every property.
Of course, this view that everything is contingent is false. Moreover, it seems to be insconsistent with a possible worlds analysis of modality. If necessarily, 'possibly, Phi' expresses a truth in English just in case 'in some world it is true that, phi', and if 'necessarily, it is possible that something exists' expresses a truth', then we have the conclusion that necessarily something exists.
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