Tuesday, August 23, 2011

First Order Logic and an ontological argument

[I also posted this on prosblogion.]
I want to give this argument in part to provoke a bit of discussion of the role of FOL in philosophy. I don't think the argument carries great weight, in large part because of Objection 2 (see the end).
1. (Premise) The inferences allowed by classical First Order Logic (FOL) combined with a modal logic that includes Necessitation are valid.
2. (Premise) If every being is contingent, then possibly nothing exists. (A material conditional)
3. Necessarily something exists. (By 1)
4. So, there is a necessary being. (By 2 and 3)
The proof of (3) is as follows. Classical logic allows (Ex)(x=x) to be inferred from (x)(x=x). Since (x)(x=x) is a theorem, so is (Ex)(x=x), and hence by the rule of Necessitation, we have: Necessarily (Ex)(x=x). And thus (3) follows. And of course Necessitation is a part of standard modal systems like M, S4 and S5.
I think (2) is intuitively plausible. Here is one way to try to argue for it:
5. (Premise for reductio) Premise (2) is false.
6. (Premise) The non-existence of non-unicorns does not necessitate the existence of unicorns.
7. Every being is contingent and it is necessary that at least one thing exists. (By 5)
8. Necessarily, if no non-unicorns exist, then at least one thing exists. (By 7)
9. Necessarily, if no non-unicorns exist, then at least one unicorn exists. (By 8) 
Since (9) contradicts (6), our reductio argument for premise (2) is complete.
(I am grateful to Josh Rasmussen for simplifying my original argument.)
Now, the weak point in the argument, I think, is premise 1, and specifically the assumption of classical FOL which allows the derivation of (Ex)F(x) from (x)F(x). In a free logic, this wouldn't happen.
But it is still an interesting fact, and a real cost to contingentism (the view that all beings are contingent), that it requires one to abandon classical logic or modify Necessitation. After all, there is some non-negligible prior probability that classical logic and Necessitation license only valid inferences.
Moreover, there is the question of why one should go for a free logic? If one's reason for going for a free logic is precisely that FOL licenses the derivation of (Ex)F(x) from (x)F(x), then one runs the danger of begging the question against the anti-contingentist, in that the derivation is valid (in the sense that necessarily if the premise is true, so is the conclusion) if there is a necessary being.
Objection 1: There is likewise a cost to the non-contingentist who is prevented from adopting those logics on which it is provable that possibly nothing exists.
Response: The non-contingentist who accepts such a logic can still make the move of distinguishing metaphysical and narrowly logical necessity. She can then say that the logic gives an account of narrowly logical necessity. Therefore, all that is shown in such a logic is that it is narrowly logically possible that nothing exists, but not that it is metaphysically possible that nothing exists. On the other hand, it is much harder for the contingentist to make the analogous move of saying that (3) is true in the case of "narrowly logical necessity". For it is widely accepted that if there is a distinction between metaphysical and narrowly logical necessity, the narrowly logical necessity is stronger of the two. Thus, if one accepts (3) with "narrowly logical necessity", one accepts (3) with metaphysical necessity, too.
Objection 2: There are other good reasons to accept free logic, besides the fact that FOL licenses the derivation of (Ex)F(x) from (x)F(x). Specifically, FOL+Necessitation implies that:
10. Necessarily (Ex)(x=a)
is true for every name a.
Response: This objection almost convinces me and is one of the main reasons why I think that while my argument lowers the probability of contingentism, it is not very powerful.
I do think there are two speculative responses to the objection, which is why I think my argument still has some weight.
i. The truth of (10) for every "name" a shows that FOL's "names" do not correspond in function to names in natural languages. In particular, they show that when translating natural language sentences into FOL, one can only employ FOL's "names" for necessary beings. This shows a significant limitation of FOL--namely, that FOL has no way of translating sentences like "Socrates is mortal." However, the fact that a logic has no way of translating a sentence does not mean that the logic's inferences are invalid. There is probably no standard formal logic that can translate all sentences of natural language.
ii. Another move in defense of FOL+Necessitation is that we should see the inclusion of non-dummy names in a language L as embodying existential assumptions about the referents of these names. Consequently, when we give the Tarskian semantics for a modal logic built on top of FOL, the recursive clauses for "Necessarily s" and "Possibly s" in a language L under an interpretation J should respectively read:
- Necessarily: If e(L,J), then s.
- Possibly: e(L,J) and s.
Here, e(J,L) is the conjunction of all metalanguage claims of the form "a* exists" where "a*" is a metalanguage name for the entity that the L-name "a" refers to under J, if L contains any names, and is any tautology otherwise. Then my initial argument needs to be run in a language with no names.


Heath White said...

I suppose I see FOL, like the U.S. Constitution, as a construction for specific purposes and of a specific place and time, which remains highly useful for its intended purposes but which is liable to provoke too much reverence in its more enthusiastic adherents. Also, again like the Constitution, taking this position is liable to provoke accusations, e.g. of heresy and sacrilege, which are better reserved for religious questions.

FOL was designed to reduce arithmetic to logic. Modality is a fairly clean add-on. It is not designed to deal with empty universes (hence Ex(x=x) is a theorem), and it is not designed to deal with empty names (hence Ex(x=a) is a theorem). It is also not designed to deal with values which themselves have truth values, hence all the paradoxes of self-reference. I suspect it doesn’t handle conditionals correctly, leading to the paradoxes of material implication, and I have my doubts about universal quantification. Mass terms are hard to represent and so are fallacies of composition and distribution.

So, in short, I don’t think you can draw too many conclusions from strange consequences of FOL.

Alexander R Pruss said...

I guess I would distinguish between metaphysical soundness and expressive completeness worries. A logic is metaphysically sound iff all the inferences it licenses correspond to metaphysically necessary conditionals. Expressive completeness is something like this: the logic is such that for any proposition p, there is a language L governed by the logic, such that p can be represented in L without loss of "logical structure" (whatever that is). (The logical structure condition rules out trivial cases, such as languages that have nullary predicates that can be used to trivially represent arbitrary propositions.)

I don't think anybody thinks FOL is expressively complete. One example of that could be the inability to handle ordinary language conditionals--they are just not representable in FOL, perhaps. The logic of material conditionals in FOL is, however, metaphysically sound, as long as we interpret "a→b" as "not both: a and ~b", or in some similar way.

Likewise, my point (i) in response to Objection 2 could be taken as saying that FOL, while metaphysically sound, is not expressively complete in respect to names of contingent beings.

I think the self-reference stuff reaches beyond FOL. It requires a truth-predicate, for instance. So I don't know how much of an objection to the metaphysical soundness of FOL it is.

To get out of my argument, one needs reason to think that FOL is not merely expressively incomplete, which is uncontroversial, but also metaphysically unsound.

There is a similar distinction in regard to the Constitution: one needs to distinguish between the question whether the Constitution contains all the fundamental laws that should be there and the question whether the fundamental laws that are there are good ones. Never having studied your Constitution with any seriousness, I'll refrain from commenting.

Alexander R Pruss said...

For the record, I should say that I think that I am not of the opinion that FOL is metaphysically sound. But I am inclined to think a restriction of FOL is metaphysically sound. The restriction adds the non-formal (people will hate this) codicil to every rule of inference that the inference is only permitted when the inferred sentence successfully expresses a proposition. I think the way out of the Liar may involve denying the claim that truthfunctional combinations of sentences that successfully express propositions always successfully express propositions.

I think this modified version of FOL is sufficient for my argument here.

Douglas said...

Hi Alex,

Here are some speculations that might lead me to think your arguer can do without premise (2).

I might not have existed. We can shadow that in a variable domain modal logic by saying: d is in the domain of the actual world, but for some possible world w, d is not in the domain of w. ‘(x)(x=x)’ is true at every world, but ‘(Ex)(d=x)’ is not true at every world. So universal instantiation fails. So modal logics with domains that very from world to world are free logics (although they need not have all the features of some free logics). Point: varying domain modal logic, it might seem, can’t be married to classical FOL.

Perhaps to get classical FOL+Necessitation one has to go for a constant domain modal logic. But, someone might argue, in constant domain quantified modal logic this material conditional gets regimented by something valid: if necessarily something exists, then something necessarily exists. This is because constant domains seem to require the Barcan formula and its converse. So premise (2) of the argument might be an unnecessary detour for the friends of FOL+Necessitation.

Alexander R Pruss said...

I don't think you need a constant domain modal logic to have FOL+Necessitation.

You do get Necessarily Ex(d=x) for every name d given FOL+Necessitation. But that only implies anything like a constant domain if you assume that every object in the domain has a name. (And even then, I think it only implies that the actual domain is a subset of every domain) The lesson to be learned is perhaps that only the names of necessary beings can be handled in the way FOL handles names. But we can still do FOL+Necessitation with contingent beings--we just don't get to have names for them in the language. We can use Russellized definite descriptions, however.

Maybe there is some reason this wouldn't work? I really don't know enough about the technical side here.

Apparently, there are also free logics that are incompatible with an empty domain.

Mike Almeida said...

What makes (6) true? The person who denies (2) is going to deny (6), since any world in which everything that exists is not a non-unicorn will be one in which there is something (by the denial of (2)) and so a world in which there is a unicorn.

Here's a different reason to deny (2). Every world is accessible from every other world (S5). You cannot access a world in which there exists no contingent objects from a world in which there exist contingent objects. That is, you cannot access such a world without a miracle such as objects or their parts "popping" out of existence. This is the anti-cosmological argument which requires some explanation for there being nothing at all!

Alexander R Pruss said...


(6) is intuitively pretty plausible. To deny it is a cost. You can think of (6) as a very weak version of a Humean thesis about lack of necessary connections, so weak that standard counterexamples don't apply.

"You cannot access a world in which there exists no contingent objects from a world in which there exist contingent objects."

I don't see this at all. I can see trying to make an argument the other way--that you can't access a world that has objects that your world doesn't from your world. But I would deny that, too.

Note, by the way, that S5 does not say that all worlds are mutually accessible. Rather, S5 says that accessibility is an equivalence relation. This doesn't, I think, affect this argument of yours, though.

Mike Almeida said...

How do you get to a world with no objects--nothing at all--from a world with contingent objects? Fill in the intuition. I don't have it. We might as well start from our world: how do we get from here to a world in which there are no objects (and so no contigent objects) at all? The standard move s via what's called a 'subtraction argument'. But all of those arguments appeal to brute possibility: i.e. they offer no explanation for how this is possible. The other alternative is a miracle of some sort: but the miracle would itself have no explanation in the case we're considering. I can't think of another way off hand.

Small point. Right, on S5, you can have isolated worlds (and isolated sets of worlds). But it's hard to think of a world at which I utter 'every world is accessible from every other' and say something false. Those isolated worlds are not possible relative to t world of my utterance.

Wolfgang said...

I wonder what you think about this argument.