Beating the Dead Horse of the Thin Theory Some More

It is obviously true that something exists.  This is not only true, but known with certainty to be true:  I think, therefore I exist, therefore something exists.  That is my Grand Datum, my datanic starting point.  Things exist! 

Now it seems perfectly clear to me that 'Something exists' cannot be translated adequately as 'Something is self-identical' employing just the resources of modern predicate logic (MPL), i.e., first-order predicate logic  with identity.    But it seems perfectly clear to van Inwagen that it can.  See my preceding post on this topic. So one of us is wrong, and if it is me, I'd like to know exactly why.  Let me add that 'Something is self-identical' is the prime candidate for such a thin translation.  If there is a thin translation, this is it.  Van Inwagen comes into the discussion only as a representative of the thin theory, albeit as the 'dean' of the thin theorists.

Consider the following formula in first-order predicate logic with identity that van Inwagen thinks adequately translates 'There are objects' and 'Something exists':

1. (∃x) (x = x).

It seems to me that there is nothing in this formula but syntax: there are no nonlogical expressions, no content expressions, no expressions like 'Socrates' or 'cat' or placeholders for such expressions such as  'a' and 'C.'  The parentheses can be dropped, and van Inwagen writes the formula without them. This leaves us with '∃,' three bound occurrences of the variable 'x,' and the identity sign '=.' 

Now here is my main question:  How can the extralogical and extrasyntactical fact that something exists be a matter of pure logical syntax?  How can this fact be expressed by a string of merely syntactical symbols: '∃,' 'x,' '='?

It is not a logical truth that something exists; it is a matter of extralogical fact.  There's this bloody world out there and it certainly wasn't sired by the laws of logic.  Logically, there might not have been anything at all.  It is true, but logically contingent, that something exists.  Compare (1) with the universal quantification

2. (x)(x =x).

If (1) translates 'Something exists,' then (2) translates 'Everything exists.'  But (2) is a logical truth, and its negation a contradiction.  Since (1) follows from (2), (1) is a logical truth as well.  But (1) is not a logical truth as we have just seen.  We face an aporetic triad:

a. '(x)(x =x)' is logically true.
b. '(∃x) (x = x)' follows from '(x)(x = x).'
c. '(∃x) (x = x)' adequately translates 'Something exists.'

Each limb is plausible, but they cannot all be true.   The truth of any two linbs entails the falsehood of the remaining one. For example, the first two entail that '(∃x) (x = x)' is logically true. But then (c) is false:  One sentence cannot be an adequate translation of a second if the first fails to preserve the modal status of the second.  To repeat myself: 'Something exists' is logically contingent whereas the canonical  translation is logically necessary.

Now which of the limbs shall we reject?  It is obvious to me that the third limb must be rejected, pace van Inwagen.

Now consider 'Everything exists.'  Can it be translated adequately as '(x)(x = x)'?  Obviously not.  The latter is a formal-logical truth. and its negation is a formal-logical contradiction.  But the negation of 'Everything exists' -- 'Something does not exist' -- is not a formal logical contradiction.  Therefore, 'Everything exists' is not a formal-logical truth.  And because it is not, it cannot be given the canonical translation.

Finally, consider 'Nothing exists.'  This is false, but logically contingent: there is no formal-logical necessity that something exist.  One cannot infer the existence of anything (or at least anything concrete) from the principles of formal logic alone.  The canonical translation of 'Nothing exists,' however -- (x)~(x = x)' -  is not contingently false, but logically false.  Therefore, 'Nothing exists' cannot be translated adequately as 'Everything is not self-identical.'

Van Inwagen and his master Quine are simply mistaken when they maintain that existence is what 'existential' quantification expresses. 

Comments

[Edward Ockham]

>>How can the extralogical and extrasyntactical fact that something exists be a matter of pure logical syntax?

It isn't. Given that the sentence "Henry = Henry" expresses anything at all, it must be that 'Henry' signifies something. And the fact that it signifies something, i.e. an existing something, is not a matter of pure logical syntax. We have moved beyond syntax to semantics.

N.B. I am assuming here that 'Henry' signifies Henry himself, rather than a 'sense' or a 'meaning' separate from Henry.

[Jan]

Dr Vallicella,

You write:
"a. '(x)(x =x)' is logically true.
b. '(∃x) (x = x)' follows from '(x)(x = x).'
c. '(∃x) (x = x)' adequately translates 'Something exists.'"

(b) is however false; in MPL 'every' does not imply 'some'. I believe the inference holds in Traditional Logic. In MPL 'everything is self-identical' and 'everything is self-diverse' are not contradictories. Both are true iff the domain of quantification is empty. vI's logical formulas rest on this fact.

[David Brightly]

Bill, I have put up a longish critical comment here.

[Bill Vallicella]

Ed,

So you agree with me, as against van Inwagen, that 'There are objects' cannot be said in MPL?

[Bill Vallicella]

You're mistaken, Jan. (b) is true. Applying Universal Instantiation to '(x) (x = x)' we get: 'x = x.' And then by applying Existential Generalization we get: '(Ex)(x = x).'

Obviously, if every x is self-identical, then some x is self-identical. 'Some x is not self-identical' is necessarly false.

How did you like the aphorisms of Stanislaw Lec?

[Bill Vallicella]

David,

Thanks. I'll take a look.

[Jan]

We can pass from *∀x (x=x)* to *a=a* as long as a belongs to the domain of quantification, i.e. if the domain of quantification is nonempty. Every x is self-identical implies some x is self-identical as long as the former statement is about something (the domain of quantification is nonempty).

"'Some x is not self-identical' is necessarly false."

Yes. It translates to ~Ex ~(x = x) and is logically equivalent to ∀x (x=x) (everything is self-identical'). This is however not a contradictory of ∀x ~(x=x) ('everything is self-different').

Lec is almost a mainstream writer here. I find him not serious enough to be funny.

[Edward Ockham]

>>So you agree with me, as against van Inwagen, that 'There are objects' cannot be said in MPL?

It depends what you mean by 'said'. "Ex x=x" is true when there is at least one object, and false when there are no objects. If 'saying' something is communicating under what circumstances the sentence you are uttering is true, and under what circumstances it is false, then it does say that.

[Bill Vallicella]

Ed,

You still don't get my point that MPL cannot express its presupposition, namely, that the domain of quantification contains existing items.

[Bill Vallicella]

Jan writes, "We can pass from *∀x (x=x)* to *a=a* as long as a belongs to the domain of quantification, i.e. if the domain of quantification is nonempty."

So you now agree with me that my (b) above is true. Now solve my triad.

[Jan]

Repeating myself,

b*. (b) is true iff the domain of quantification is nonempty.

If (b) were true with an empty d.o.q, it would mean that when nothing exists there exists something identical to itself. Do you accept (b*)?

We are in this context quantifying over all existing objects. This means (b) is false in the possible world where nothing exists and true in every other possible world (including the actual world).

"For example, the first two entail that '(∃x) (x = x)' is logically true. "

(a) holds in all possible worlds, (b) in all but the empty one. Therefore '(∃x) (x = x)' holds in all possible worlds but the empty one. Its truth values agree with the truth values of 'Something exists' across all possible worlds.

I say solved.

[Hrodberht]

Jan,

Classical predicate logic demands the domain be non-empty.

[Bill Vallicella]

Hrod,

Yes. More later.

[Jan]

Hrodberht,

I've been aware traditional logic demands that. Upon doing some reading it seems MPL demands it too. (This is a surprise to me; in mathematics we regularly quantify over empty domains. I thought standard mathematics is done in MPL?)

If this is indeed the case I concede BV's argument.

[Edward Ockham]

>>You still don't get my point that MPL cannot express its presupposition, namely, that the domain of quantification contains existing items.

Yes I still don't get your point :) It seems to me perfectly clear that this is what it is expressing when it says 'Ex, x=x'. Exactly that.

[Bill Vallicella]

Ed,

So, according to you, 'Something is self-identical' says exactly what 'Something exists' says?

If that is right, then 'Everything is self-identical' says exactly what 'Everything exists' says. But the former is such that its negation is a contradiction, whereas the latter is not such that its negation is a contradiction. Ergo, etc.

[Edward Ockham]

>> But the former is such that its negation is a contradiction, whereas the latter is not such that its negation is a contradiction. Ergo, etc.

Is 'not everything self-identical' a contradiction? Or just false? A contradiction is two statements, one of which is the negation of the other. (I think - I haven't checked).

I was going to email you about Woody Guthrie, by the way.

[David Brightly]

Regarding the question whether standard first order logic prescribes a non-empty domain may I refer readers to Thomas Hofweber's SEP article Logic and Ontology? In section 4.2 he says

If we look more closely how it comes about that these existential statements are logical truths in these logical systems we see that it is only so because, by definition, a model for (standard) first order logic has to have a non-empty domain. It is possible to allow for models with an empty domain as well (where nothing exists), but models with an empty domain are excluded, again, by definition from the (standard) semantics in first order logic. Thus (standard) first order logic is sometimes called the logic of first order models with a non-empty domain. If we allow an empty domain as well we will need different axioms or rules of inference to have a sound proof system, but this can be done....For a sound and complete proof system for logic with an empty domain, see (Tennant 1990).

Tennant's book Natural Logic is out of print but is available online here (via Peter Smith)

My semi-educated guess as to why the standard presentation prescribes a non-empty domain is that in general a first order language has constant terms and each must be mapped by any interpretation to some domain element. But there is no requirement that a first order language have constants or function letters and in this case an empty domain does suffice. Of course, such a language is rather trivial, with, I think, just universally quantified formulas (always true) and existentially quantified formulas (always false) and logical combinations thereof.

Jan is absolutely right that mathematicians have to be constantly wary of quantifying over empty types. It's often the case that we can prove ∀x.Px-->Qx. If we restrict attention to the Ps only we have ∀x.Qx, but we daren't conclude ∃x.Qx because there may be no Ps at all, even when the 'outer' domain is non-empty.

[Bill Vallicella]

Ed,

As you well know, a proposition can be a contradiction by being a self-contradiction, in which case there is no need for two propositions. For example:

(Ex)(Fx & ~Fx)

and

(Ex) ~(x = x)

The latter is the negation of the logical truth '(x)(x = x).'

The negation of 'Everything exists' is 'Some thing does not exist' which is not formally self-contradictory.

'Some thing that exists does not exist' is formally self-contradictory. But 'thing' and 'existing thing' do not have the same meaning.

This may be the ultimate bone of contention between us.

But experience has shown that the location of the bone of contention is itself a bone of contention.

(I will refrain from riffing on the *dislocation* of the bone of contention.)

[Bill Vallicella]

David,

Thanks for that interesting comment.

But shouldn't we distinguish between an empty domain and one containing nonexistent objects?

'Vulcan' is a constant mappable to the nonexistent domain element, Vulcan

[David Brightly]

Part of our difficulty may be that we are abusing the technical sense of the term 'domain' as it appears in the semantic theory of first order languages. There it means a set of usually mathematical objects and properties onto which the constant terms and predicate symbols of the language can be mapped in order to give a precise account of the meaning of the logical symbols, especially the quantifiers. I think we are now using 'domain' in the sense of 'possible world', against which the truth of sentences regimented into first order form may be evaluated. One such possible world is empty, another contains an intra-Mercurial planet, another a flying horse, concepts not instantiated in the actual world. But I don't understand what it would be for a possible world to contain a special category of non-existent objects.