How to Determine Whether there Might Have Been Nothing
January 27, 2014 — 21:40

Author: Kenny Pearce  Category: Prosblogion Reviews  Tags: , , , , , , , ,   Comments: 9

Even those of us who think that necessary truths often need (and have) non-trivial explanations generally think that these explanations tend to look different from the explanations of contingent truths. Furthermore, one might well think that showing that p is necessary explains why p, even if one thinks that it is possible to show that necessarily p without explaining why necessarily p. Additionally, of course, there are those who hold that once one has shown a certain proposition to be a necessary truth, there are no further ‘why’ questions to be asked. Thus if one wants to know whether the question ‘why is there something rather than nothing?’ is a well-formed question that might have an answer, and what form such an answer might take, one would do well to start by determining whether it is possible that there should be nothing. In their contribution to The Puzzle of Existence, David Efird and Tom Stoneham set out to explain how one might go about making such a determination.

For starters, why would anyone doubt that there could be nothing? Well, for one thing, as is well known, standard logics actually turn out to entail that something exists. The argument goes like this. It is an axiom of first-order predicate logic with identity that:

∀x(x=x)

But by universal instantiation we then have:

a=a

where ‘a’ is an arbitrary constant. By existential generalization this gives us:

∃x(x=x)

which is how you say that something exists. But by the necessitation rule of K (and stronger modal logics, such as S4 and S5), we now have:

□∃x(x=x)

i.e., necessarily, something exists. (Note that in standard systems this is not equivalent to something necessarily exists.)

Now, Efird and Stoneham understand the ‘puzzle of existence’ as the question of why there are concrete things rather than not, so their question is whether it is possible that there should be no concrete things. I can think of three ways of responding to this argument without conceding that necessarily there is something concrete:

1. Revisionism. The argument above shows that there is something wrong with standard logics. We should endorse a free logic (or something else) instead.
2. Platonist Abstractionism. The argument is perfectly sound, since the empty set (for instance) could not have failed to exist, but this sheds no light on the question of whether there might not have been any concrete things.
3. Nominalist Abstractionism. There are many things different types of nominalists might say, and all of them are going to be complicated. Here’s the beginning of one possible nominalist response: ‘∃x(φ(x))’ does not in fact mean ‘something that satisfies φ exists’, for the formal quantifiers are most at home in mathematics, and it is certainly a truth of mathematics that ∃x(x > 5). This claim is true despite the fact that numbers do not exist, and hence nothing that is greater than 5 exists (since only a number could be greater than 5). As a result, the argument’s conclusion, □∃x(x=x), is not correctly interpreted as saying that, necessarily, something exists.

There are other reasons one might deny the possibility of nothing. For instance, one might doubt whether it is genuinely conceivable that there should be no concrete entities, or one might have a metaphysical theory of possible worlds which rules out the existence of an empty world. On the other hand, most of us find it to be at least apparently conceivable that there should be no concrete entities. Perhaps on further reflection I will think that I was really conceiving empty space, and space is a concrete entity, or, worse, that I was conceiving myself floating in empty space. (Depending on exactly what’s meant by ‘conceive,’ I’m not sure I can conceive my own non-existence; if I can’t, then clearly inconceivability is poor evidence of impossibility.) Alternatively, one might say: other than the above argument and ontological arguments for the existence of God, both of which many philosophers suspect of being sophistical and/or mere logical curiosities, no one has identified any contradiction in the proposition there are no concrete beings, hence it should be regarded as possible. (I think this last line of thought is not very compelling, since I think the reason most philosophers are so confident that there must be something wrong with these arguments is a prior conviction that it is possible that there should be nothing.)

To sum up what I’ve said so far: if we are interested in the question why there is something rather than nothing, then it will be important to us to answer the question whether there might have been nothing. The answer to this last question is, however, far from clear. Efird and Stoneham’s paper is an attempt to say how we might go about answering it.

Efird and Stoneham defend two main claims about this, which they label Methodological Separatism and Modal Pluralism. Methodological Separatism is the view that theories about the nature of possibility (e.g., a metaphysics of possible worlds) and theories about the extent of possibility (i.e., what things are possible) can and should be evaluated separately. This is not to deny that we may need to consider how theories of the nature of possibility interact with theories of the extent of possibility; rather, the idea is that we should first evaluate them separately and only later consider whether, for instance, a particular theory of the nature of possibility has the cost of foreclosing on our best theory of the extent of possibility, or other things of that sort.

Modal Pluralism is the view that a theory of the extent of possibility should consist of a series of claims of the form: if p, then ◊q. This is supposed to lead to a ‘regulative principle’ Efird and Stoneham call ‘Leibniz’s Principle of the Presumption of Possibility’:

One has the right to assume ◊p until someone proves the contrary. (pp. 162-163)

This regulative principle is supposed to follow because it is always (epistemically) possible that there is a principle one has missed (i.e., some other criterion of possibility not on one’s list.)

Let me pause to make a few remarks on this. The assumption here is that p is possible, not that p is contingent. It is crucial for Leibniz’s purposes that this be so, for Leibniz is running an ontological argument for the existence of God: he wants to presume that God possibly exists and show from this that God necessarily exits, and so actually exists. Clearly a presumption of contingency won’t do here. But in fact the presumption of contingency is rather more plausible than the presumption of possibility, as Robert Adams has argued. In fact, John Heil and C. B. Martin, whose criticisms Efird and Stoneham discuss, are actually criticizing the more plausible presumption of contingency, not the less plausible presumption of possibility. I find it odd that Efird and Stoneham do not remark on this important difference. Perhaps the reason for this is that the presumption of contingency looks like a strict strengthening of the presumption of possibility, since ‘p is contingent’ is written in modal logic ‘◊p & ◊~p.’ Nevertheless, there is this important difference: it is possible to have evidence against the contingency of a proposition without having any evidence that bears on the question whether it is necessarily true or necessarily false. In this scenario, the presumption of contingency would be rebutted, but the presumption of possibility would not. Thus if one were entitled to presume possibility, rather than contingency, one would be entitled, in such a scenario, to infer that the proposition was necessarily true. But here things get all screwy, since one could just as easily have started from the negation of the proposition one in fact started from, and presumed that proposition to be possible, in which case one would come to the opposite result. This seems bad.

In any event, Efird and Stoneham report that in other work they defend certain principles of possibility which entail that it is possible that there are no concrete beings. The main focus of this paper, is, however, on the methodological point about how this is done. The answer is that one is entitled to presume the possibility of nothing right off the bat. If one wants to provide a positive argument for the possibility of nothing what one needs to do is to identify a plausible principle of possibility which entails it.

I have two questions about Efird and Stoneham’s approach. First, I am unclear on what the justification for the asymmetry between possibility and necessity is supposed to be. Second, I’m not totally sure what they mean by ‘theory.’

Taking the first issue first, Efird and Stoneham’s approach sure seems to stack the deck in favor of the possibility of nothing. This is a result of their claim that a modal theory should contain a plurality of principles of possibility. Their argument for this claim is that no one has succeeded in identifying a single criterion of possibility that, all by itself, captures enough of our modal intuitions. But what seems to me to be left unjustified is their focus on principles of possibility rather than principles of necessity. After all, we have intuitions about necessity as well. In fact, Efird and Stoneham seem willing to admit at least two principle of necessity: everything is necessarily self-identical, and every proposition knowable a priori is a necessary truth (pp. 161-162; I say they ‘seem willing to admit’ these principles, although they do not unambiguously endorse them). But if there is a plurality of principles of necessity, then why presume possibility rather than necessity? After all, we might have overlooked a principle of necessity just as easily as a principle of possibility!

On the second point, Efird and Stoneham say, “we can regard any assertion or belief which organizes some data, usually by categorizing it or deriving it from some variables, as theoretical with respect to that data” (p. 146). They are quite clear that the theory/data distinction is a relative one. But, oddly, they say they are going to define ‘theory’ and then instead tell us what it means for an assertion or belief to be theoretical. Perhaps a theory of some data is the set of beliefs one holds which are theoretical with respect to that data. Now, it seems to me that they want these beliefs to be justified by their use to ‘organize’ the data. Is this justification an inference to the best explanation? If so, then this ‘organization’ must be some kind of explanation by unification. Yet it is far from clear that a theory of the extent of possibility of the sort Efird and Stoneham envision explains anything. Certainly one can explain an individual modal intuition by appealing to some more general principle that entails it, but would a hodgepodge of such general principles, with nothing to tie them together, really count as an explanation of our modal intuitions in general? I’m dubious.

(Cross-posted at blog.kennypearce.net.)

• standard logics actually turn out to entail that something exists

Classical first-order logic presupposes that something exists, by requiring the domain to be non-empty. As a reason for doubting ‘that there could be nothing’, this is unpersuasive.

January 28, 2014 — 9:05
• Kenny Pearce

It is hard to say exactly what the difference between an indispensable presupposition and an entailment is. I take it what you have in mind is something like this: when we set up the model theory for first-order logic, we at some point say ‘let the domain D be a non-empty set.’ But note (1) it is possible to perform the proof by mere syntactic manipulation without any model-theory, only worrying about the interpretation after the fact, and (2) it is not possible to dispense with the assumption and still retain the inference rules of classical logic, since the series of inferences I gave is valid.

Of course I agree that this is not a persuasive reason against the possibility of nothing, but that’s precisely what is philosophically interesting about this line of thought: ordinarily, ‘it is a theorem of first-order logic that p‘ is an excellent reason for believing that p! Exactly why is that not the case here? I don’t think your presupposition line gets to the heart of the matter, for one could just as easily say that classical first-order logic presupposes the Principle of Non-Contradiction, yet the fact that dialetheism requires one to give up classical logic seems to me to be a major strike against it. Perhaps what is really going on is just that we have an intuition that dialetheism is going against what’s right about classical logic, while metaphysical nihilism (the view that there might have been nothing) simply runs up against an unfortunate limitation of classical logic.

January 28, 2014 — 10:43
• It’s worse than FOL+necessitation just implying that necessarily there is something. If that’s all FOL did, then we might say that we’ve learned that necessarily there is something. But FOL+necessitation implies the absurdity that necessarily Alex Pruss exists. Just take pruss=pruss as your instance of the claim that everything is identical to itself, and then use existential generalization to get: (Ex)(pruss = x), and then necessitation makes that necessary.

The rules of FOL assume that the referent of each name exists. Thus, that something is a theorem of FOL only means that it is entailed by the conjunction of all the claims of the form (Ex)(a=x), where a is a name, not that it is necessary.

January 28, 2014 — 18:46
• That problem, though, arises only if we allow the introduction of constants for arbitrary actually existing entities. One can’t get the result just by using universal instantiation, for instance.

January 28, 2014 — 18:57
• “One can’t get the result just by using universal instantiation, for instance.”

I don’t follow. If “pruss” is a name in the language, then
pruss = pruss
follows by universal instantiation from (x)(x=x). And then by existential generaliation:
(Ex)(pruss = x)

Now, it’s true that this only works for those entities that have a genuine name in the language. So you can avoid this if your dialect of FOL has only dummy names. But if you restrict your FOL this far, then you limits its usefulness radically.

January 29, 2014 — 11:15
• Yes, I agree. All I meant was that if you only allow dummy names you don”t get the result. I’m not sure how much that limits our language. I guess it depends on some metaphysical issues. If you believe in the identity of indiscernibles, then you can introduce a predicate ‘Pruss,’ which abbreviates some definite description uniquely identifying you, without getting the bad result. That is, we’ll get:
∃x(Pruss(x))
But not:
□∃x(Pruss(x))

Similarly, if you believe in haecceities you can designate your haecceity with a predicate. Going the latter route, I think one can even say:
∀x(Pruss(x) -> □∀y(Pruss(y) -> y=x))

So I think the only case where we really need to allow those kinds of constants is the case where we’ve got distinct indiscernibles without haecceities, and that’s a pretty weird case. (We’re to imagine, I guess, that there is a fact of the matter about how many identical iron balls there are, but no fact of the matter about which one is which?) But maybe I’m missing something.

January 29, 2014 — 11:28
• Wait, I WAS missing something. From:
∃x(Pruss(x))
We have:
Pruss(a)
For dummy variable a. Then we have, by the usual argument:
□∃x(x=a)
Which means that you exist necessarily. So using the apparatus of predicates brings the problem back, which means that it’s only a radically impoverished FOL that can avoid the problem. Huh.

January 29, 2014 — 11:47
• I think you can deny the inference from:
∃x(Pruss(x))
to:
Pruss(a).
In some FOL systems, you only get to do that in an ∃-elim subproof.

(Of course, you’ll need a modal logic to go along with the FOL that doesn’t let you derive the Barcan or converse Barcan formulas if you want to avoid absurdity.)

January 30, 2014 — 8:57
• Dianelos Georgoudis

I agree that standard logic entails that something exists. On the other hand, clearly, standard logic does not hold in the empty world. After all the empty world is a world in which nothing exists, no matter, no space, no time, no numbers, no propositions, no truth values. And thus nothing for logic to apply to, or make sense about.

Is such an empty world logically possible? I don’t see why not. We call possible those worlds the book of which does not entail contradictory propositions. (By “book” of a world I understand the set of all propositions that are true in that world.) The book of the empty world is the empty set, which by definition does not entail any contradictions. Therefore the empty world is logically possible. In that sense then, it might have been nothing.

February 3, 2014 — 14:31