A New and Improved Argument for the Necessity of Something

Previous versions were long-winded.  Herewith, an approach to the lapidary.

1) If nothing exists, then something exists.
2) If something exists, then something exists.
3) Either nothing exists or something exists.
Therefore
4) Necessarily, something exists.

The argument is valid. The second two premises are tautologies. The conclusion is interesting, to put it mildly: it is equivalent to the proposition that it is impossible that there be nothing at all.  But why accept (1)?

Argument for (1)

5) If p, then the proposition expressed by 'p' is true.
Therefore
6) If nothing exists, then nothing exists is true.
7) The consequent of (6) commits us to the existence of at least one proposition.
Therefore
1) If nothing exists, then something exists.

Surely (5) is unproblematic, being one half of the disquotational schema,

DS. P iff the proposition expressed by 'p' is true.

For example, snow is white if and only if snow is white is true. The semantic ascent on the right-hand side of the biconditional involves the application of the predicate 'true' to a proposition. So it is not the case that the left and right hand sides of the biconditional say the same thing or express the same proposition. The LHS says that snow is white; the RHS says something different, namely, that the proposition expressed by 'snow is white' is true. The RHS has an ontological commitment that the LHS does not have: the RHS commits us to a proposition. Since the RHS is true, the proposition exists. (Cf. Colin McGinn, Logical Properties, Oxford UP 2000, 92-93. I am taking from McGinn only the insight that the LHS and RHS of (DS) do not say the same thing.)

But what about the inference from (5) to (6)? Can it be questioned? Yes, if we are willing to countenance counterexamples to (5) and thereby call into question Bivalence, the semantic principle that every proposition is either true or false, but not both. I'll pursue this in a later post. If, however, one accepts Bivalence and its syntactic counterpart, Excluded Middle, then it looks as if I've got me a rigorous a priori argument for the necessity of something and the impossibility of there being nothing at all.

Comments

[The Bad Ostrich]

One woman's modus ponens etc. This is a good argument against ‘propositions’.

If your argument is sound, then necessarily something exists. But it is absurd and false that necessarily something exists. Therefore (modus tollens) your argument is not sound.

It seems to be valid. Therefore one or more of your premisses is false. The likely culprit is the major premiss (1). You defend this with a second argument. The likely culprit is (5) ‘If p, then the proposition expressed by 'p' is true’. This implies there is always something that ‘the proposition expressed by 'p'’ refers to. As you go on to say ‘The RHS has an ontological commitment that the LHS does not have: the RHS commits us to a proposition.’

I hold, with Aristotle, that a proposition is a special kind of sentence, one which (unlike a prayer or a command or a question) is capable of truth or falsity. Ergo, if nothing exists, nothing is a sentence, and if every a proposition is a sentence, nothing is a proposition. Ergo etc. It could be true that nothing exists, without there having to exist anything that has the ‘property’ of being true. Corollary: the account of truth must be deflationary.

You have proved many excellent things here, Bill.

[BV]

>>But it is absurd and false that necessarily something exists.<<

'Absurd' in a logical context means self-contradictory. Or are you using 'absurd' in some emotive way?

So if you are not emoting, you are saying that the following is self-contradictory: Necessarily, something exists. You thereby must be affirming that the following is a logical truth: Possibly, nothing exists.

Do you really want to say that? At a minimum, you are begging the question against me. I gave a rigorous argument. The only way to block it is by denying (5), and rejecting Bivalence. But you accept that principle, although I think you confuse it with Excluded Middle.

So you ought to accept my argument and admit that necessarily, something exists.

It is worth noting that from 'Necessarily, something exists' it does not immediately follow that 'Something is such that it necessarily exists.' That would be a further step, one I haven't taken above.

[The Bad Ostrich]

>>The only way to block it is by denying (5), and rejecting Bivalence.

Not at all. I reject the ontological commitment of the RHS that you mention. I already gave the argument above. Shall I repeat it?

OK I will repeat it.

This implies there is always something that ‘the proposition expressed by 'p'’ refers to.

>> But you accept that principle, although I think you confuse it with Excluded Middle.

In no way do I ever confuse bivalence with EM.

[The Bad Ostrich]

Perhaps there is another confusion here. You write

5) If p, then the proposition expressed by 'p' is true.
Therefore

6) If nothing exists, then nothing exists is true.

7) The consequent of (6) commits us to the existence of at least one proposition.

Why does the consequent of (6)commit us to the existence of at least one proposition? If I read (6) as saying

(6*) If nothing exists, then it is true that nothing exists.

then there is no existential implication (on a deflationary interpretation). If on the other hand I read it as

(6**) If nothing exists, then there exists a true proposition *nothing exists*.

Your argument works on the second interpretation, but justify the second interpretation.

[David Bagwill]

Do we read 6) as : If nothing exists, then 'nothing exists' is true?David

[BV]

Dave,

(6) says that if nothing exists, then the proposition expressed by 'nothing exists' is true.

You can say the same thing in different languages. The same thing is the proposition.

By the way, your comment got sent to the dreaded spam corral for some reason.

[David Gudeman]

BV, I was surprised to see you distinguishing between bivalence and the LEM. As far as I can tell, in the traditional and most common formulations, they are identical. My knowledge of the literature on logic is far from exhaustive, but as far as I can tell, the notion that the excluded middle involves the "not" connective is a modern innovation designed to make the LEM a dual to the Law of Contradiction. Symbolically:

LEM: p or ~p
LC: ~(p and ~p)

This is cute, but doesn't fit the tradition I was taught, where the LEM was stated as "every proposition is true or false" that is, bivalence.

I'm curious whether you have some more profound reason to prefer the modern retreading of the LEM other than this clever duality, especially given that the new formulation does not exclude a middle. That is,

p or ~p

only says that p is true or ~p is true, it does not say that neither p nor ~p can be something other than true or false.