An argument for a necessary being
August 23, 2011 — 9:47

Author: Alexander Pruss  Category: Existence of God  Tags: ,   Comments: 13

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.

• Marcus

Maybe I’m wrong about this, but I don’t think I am. You say, in defense of (3), that classical logic allows us to infer the existential claim that there is some x such that x=x from the universally quantified truth that for all x, x=x. I’m pretty certain that classical logic does not allow that inference. After all, the universally quantified claim is vacuously true if there are no x’s at all. Anyway, that’s my memory of the semantics of universal quantifiers, and although I haven’t done formal logic in a long time, it seems pretty clearly right. Anyway, if this is right – if the universally quantified claim is true if there are no x’s – you can’t validly infer the existential claim.

August 24, 2011 — 9:21
• Dianelos Georgoudis

Alex,
Isnât any argument of this type actually begging the question? After all, premises like (1) assume that there is some epistemology which applies to (is valid in) all possible worlds. But then such premises assume that some X exists in all possible worlds, namely that which makes that epistemology valid. Thus (1) already assumes that there is a necessary being.
I think that epistemology always follows ontology, for ontology is about what is and epistemology is about how one should think about what is. When doing metaphysics one should start with a particular ontological hypothesis, which in turn implies which epistemology is appropriate when thinking about it. I hold this to be an axiom of philosophy. As an illustration letâs consider the question of whether on theism it is valid to use modal logic. Since a person must consider (metaphysically) possible worlds when reasoning about how to act, and since on theism reality is ultimately personal, it follows that modal logic is appropriate on theism. As far as I can see though, it is not clear to me that this justification also covers modality in the narrowly logical sense.

August 24, 2011 — 11:28
• “But then such premises assume that some X exists in all possible worlds, namely that which makes that epistemology valid.”
Only if something like Truthmaker Maximalism is necessarily true.
And if Truthmaker Maximalism is necessarily true, then there can’t be any empty worlds anyway. Maybe there is something weaker than Truthmaker Maximalism that gives you the claim, though.

August 24, 2011 — 11:47
• Jeremy Gwiazda

On a quick reading, it seems that one might defend the claim that every being is contingent by saying that if 3 is true, then 2 is false. That is, every being is contingent. So if necessarily something exists, then 2) is false. The non-existence of non-unicorns necessitates the existence of unicorns. Why? Because somethings got to exist.

August 24, 2011 — 14:24
• Marcus

I went back at my old logic textbook and I was right. You can’t validly infer the existential claim from the universally quantified claim. The universally quantified claim is vacuously true when there are no x’s, so you can’t infer the existential claim that there is at least one x such that x=x.

August 24, 2011 — 22:04
• David

Couldn’t someone who accepts your premise (7) consistently hold that non-unicorns exist in every possible world? He could then say that (8) and (9) have impossible antecedents, so that he need not accept them. (6) would also be problematic if non-unicorns exist in every possible world.

August 24, 2011 — 22:10
• Marcus:
Which textbook is that?
I’d say that in the terminology of the free/inclusive logic discussion, your textbook has a non-classical logic, and they must have some restriction on universal elimination.
I use Barwise and Etchemendy when teaching logic, and their formal definition of validity is in terms of first order structures, and they explicitly define those as having a non-empty domain.
Jeremy:
Yes, one can say that, but there is a cost: conditionals like that if there were no contingent non-living things, there would be a contingent living thing have pretty low prior probability. So if a theory commits one to them, that should lower the theory’s probability.

August 25, 2011 — 7:23
• Kenny Pearce

Bergmann, Moor, and Nelson’s The Logic Book includes an inference rule for quantifier logic they call ‘universal decomposition’ which allows you to introduce an arbitrary constant satisfying the formula inside the universal quantifier.
Enderton’s Mathematical Introduction to Logic contains an inference rule called ‘substitution’ which does the same thing (pp. 112-114).

August 25, 2011 — 9:56
• Dianelos Georgoudis

Alex,
I am not using anything as strong as âTruthmaker Maximalismâ. Rather I am pointing out that any valid epistemological tool (such as xyz logic) must fit the domain to which it is applied, which entails that that domain must be such that the tool fits. Thus, if one assumes that the same tool can be applied to all possible worlds then one is already assuming some necessary existence, namely that which at a minimum is required for the tool to fit.
Take for example a basic epistemological principle which almost everybody accepts as valid, namely the principle of parsimony, or Occamâs razor. Theism entails that all reality is the creation of a perfect person, who would plausibly want to create an elegant world. Thus theism supports well the principle of parsimony, and one should therefore not violate that principle when thinking about theism. It is not clear what in the naturalistic hypothesis would imply the validity of that principle, but nevertheless virtually all naturalists are inclined to accept it too. So should we accept that this principle is valid in all possible worlds? I think clearly not. I find it is not difficult to conceptualize an intrinsically complex naturalistic world in which itâs turtles and different other animals all the way down (and in which incidentally intelligent organisms may naturalistically evolve). In such a world the opposite of the principle of parsimony applies. Therefore we may violate the principle of parsimony when thinking about naturalism.
Finally, is there some epistemological tool that applies to all possible worlds? Since (I say) the empty world is possible, and that world fits no epistemological tool (simply because it

August 25, 2011 — 10:28
• Dianelos Georgoudis

Oops, the final paragraph got truncated. It should read as follows:
Finally, is there some epistemological tool that applies to all possible worlds? Since (I say) the empty world is possible, and that world fits no epistemological tool (simply because it contains nothing for any tool to fit with), the answer seems to be no.

August 25, 2011 — 10:58
• Jeremy Gwiazda

“Yes, one can say that, but there is a cost: conditionals like that if there were no contingent non-living things, there would be a contingent living thing have pretty low prior probability. So if a theory commits one to them, that should lower the theory’s probability.”
But if 3) is the case, wouldn’t those sorts of conditionals have a prior of 1?
That asked, there does seem something a bit odd about such conditionals.

August 25, 2011 — 12:32
• Jeremy:
Necessary truths shouldn’t all get priors of 1. I certainly shouldn’t assign a prior of 1 to the Riemann zeta conjecture, even if (as seems likely) it is in fact true.

August 25, 2011 — 13:31
• CliveStaples

I agree with Marcus. In an empty domain of discourse, universal quantifications are vacuously true, and existential quantifications are trivially false.
If as Alexander says we presuppose a non-empty domain of discourse, then isn’t it trivial that there is no “empty” world?

August 31, 2011 — 16:06