More Fun With Existential Generalization

Intuitively, if something is identical to Venus, it follows that something is identical to something.  In the notation of MPL, the following is a correct application of the inference rule, Existential Generalization (EG):

1. (∃x)(x = Venus)
2. (∃y)(∃x)(x = y) 1, EG

(1) is contingently true: true, but possibly false.  (2), however, is necessarily true.  Ought we find this puzzling?  That is one question.  Now consider the negative existential, 'Vulcan does not exist.' 

3. ~(∃x)( x = Vulcan)
4. (∃y)~(∃x)(x = y) 3, EG

(3) is contingently true while (4) is a logical contradiction, hence necessarily false.  The inference is obviously invalid, having taken us from truth to falsehood.  What went wrong? 

Diagnosis A: "You can't existentially generalize on a vacuous term, and 'Vulcan' is a vacuous term."

The problem with this diagnosis is that whether a term is vacuous or not is an extralogical (extrasyntactic) question.  Let 'a' be an arbitrary constant, and thus neither a place-holder nor a variable.  Now if we substitute 'a' for 'Vulcan' we get:

3* ~(∃x)( x = a)
4. (∃y)~(∃x)(x = y) 3*, EG

The problem with this inference is with the conclusion: we don't know whether 'a' is vacuous or not.  So I suggest

Diagnosis B:  Singular existentials cannot be translated using the identity sign as in (1) and (3).  This fact, pace van Inwagen, forces us to beat a retreat to the second-level analysis.  We have to analyze 'Venus exists' in terms of

5. (∃x)(Vx)

where 'V' is a predicate constant standing for the haecceity property, Venusity.  Accordingly, what (5) says is that Venusity is instantiated.  Similarly, 'Vulcan does not exist' has to be interpreted as saying that Vulcanity is not instantiated. Thus

6. ~(∃x)(Wx)

where 'W' is a predicate constant denoting Vulcanity.

It is worth noting that we can existentially generalize (6) without reaching the absurdity of (4) by shifting to second-order logic and quantifying over properties:

7. (∃P)~(∃x)Px.

That says that some property is such that it is not instantiated.  There is nothing self-contradictory about (7).

But of course beating a retreat to the second-level analysis  brings back the old problem of haecceities.  Not to mention the circularity problem. 

The thin theory is 'cooked' no matter how you twist and turn.

Comments

[Edward Ockham]

I think a long-standing problem connected with your discussion of this issue is the way you conflate the 'thin theory' with the theory of existential generalisation in the (standard) predicate calculus. When you say that the thin theory 'is that existence is what existential generalisation expresses', you presumably mean existential generalisation in MPL. Remember that 'something is a cat' is also an existential generalisation, but in ordinary language.

You say above that 'a' may be vacuous. But 'a' is a constant and cannot be vacuous. Of course a proper name like 'Pegasus' is vacuous. But that is because 'Pegasus' is not a constant. It is a term of ordinary language.

[Bill Vallicella]

You really need to pay attention to what I actually write. What I have written, on many occasions, quoting Quine, is that "Existence is what existential quantification expresses." That is a key plank in the thin platform. You are confusing that with "Existence is what existential generalization expresses" which doesn't even have a clear sense.

[Edward Ockham]

>>Existence is what existential quantification expresses.

Same issue. Do you mean existence is what is expressed by some sentence in the predicate calculus? Or do you mean it is what is expressed by a quantified sentence in English? For example, do you hold that (according to the thin theory) the existence of cats is what is expressed by

(1) Ex cat(x)

or by

(2) something is a cat ?

Either of these is technically quantification. If you hold that MPL is simply a formalisation of ordinary English, then this distinction doesn't matter of course.

There is is the connected issue of your claim that a constant 'a' can be 'vacuous'.

[Bill Vallicella]

No, they are not both *technically* quantification. (2) is, or is close to, ordinary English. (Not that many people go around saying 'Something is a cat.') (1) is a regimentation into canonical notation of (2). So only (2) is *technically* a quantified sentence.

According to the Quinean version of the thin theory (which is championed by van Inwagen, though the latter goes further), the existence of cats is expressed by (1).

Is MPL simply a formalization of ordinary English? That depends on what all is involved in a formalization. I prefer to say that MPL is a regimentation of ordinary English.

But you are ignoring the issues I raised in my post. The first is the move from (1) to (2). Is there not a problem with that inference -- even though it is a kosher application of EG -- inasmuch as it takes us from a contingent truth to a necessary one?

[Edward Ockham]

Ahem, the term 'quantifier' derives from the Latin 'signum quantitatis', 'sign of quantity'. The signs are 'all' (universal) and 'some' (particular). But I am being picky. In any case, you hold that MPL is a regimentation of ordinary language, rather than a language in its own right, so you have answered my question.

>>But you are ignoring the issues I raised in my post. The first is the move from (1) to (2). Is there not a problem with that inference -- even though it is a kosher application of EG -- inasmuch as it takes us from a contingent truth to a necessary one?

Note that my browser does not accept your site's quantifier signs. I am guessing it is the upside down E, however. I don't agree with your (1), and this comes back to the problem of 'regimentation' versus 'translation'. In MPL, (1) is necessarily true, isn't it? How could it not be true that something in the domain is identical to Venus, given that Venus is in the domain? It follows – given that 'something is identical to Venus' in ordinary English is contingently true, that MPL cannot be a regimentation of ordinary English.

Your (3) is not contingently true, but false. Given that 'Vulcan' refers to something in the domain, it follows that Ex x = Vulcan. Now of course 'something is identical to Vulcan' (in English) is false. Therefore the MPL sentence cannot be a regimentation of the corresponding English sentence.

[David Brightly]

Bill, Ed,

I have used this argument before but I think it's relevant. Consider the classic proof of the non-existence of a rational square root of 2 by reductio ad absurdum. It starts by hypothesising that there is such a root: ∃x.x*x=2. If so, we can give this root a name, 'r', say. This is existential instantiation: r*r=2. We go on to deduce a contradiction and hence infer that our hypothesis was false, ie, ~∃x.x*x=2. Informally we express this by saying that r does not exist. My point is that the name 'r' fails to refer to an element of the domain of rational numbers. This seems paradoxical: aren't all logical constants in MPL supposed to have referents? Yet the proof is perfectly acceptable. The answer, I think, is that the name 'r' has validity only within the scope of the hypothesis ∃x.x*x=2. When we find that this is false that use of the name 'r' must cease (though it can be reintroduced in a new scope under a subsequent hypothesis to mean something different).

Now this suggests to me that ordinary English and MPL can be brought closer together. When we translate/regiment Bill's (1)

∃x.x = Venus

we should make explicit the general existential under which the name 'Venus' was introduced:

∃x.Planet(x)&Other(x)
Planet(Venus)&Other(Venus).

By 'Other()' I mean no kind of haecceity. Rather it predicates those of our background beliefs about the planet Venus that suffice for the argument at hand. These beliefs, I say, are implicit in our use of the term 'Venus' in ordinary English, and can be made explicit where necessary. If we follow this strategy consistently we end up with a first order language with no logical constants for which we are required to find domain referents. All constants are introduced under general existential hypotheses. Of course, this means that any formula that we infer that bears a constant remains conditional on the general existential that introduced that constant. This makes it obviously contingent. But this seems to me to reflect our epistemic state anyway. A bunch of formulae in a conventional first order language that has constants can be translated into a constant-free language by wrapping the sentences in sufficiently many nested general existentials to 'declare' all the constants. The argument then proceeds at the innermost level of this nesting.

I apologise if this is rather terse but I hope it makes sense.

[Edward Ockham]

David,

Your argument for the irrationality of 2 amounts to this:

If for some x, y, z, integer(x) and integer(y) and z = x/y, then for some p, p is a true contradiction
But for no p is p a true contradiction
Ergo, not for some x, y, z etc.

This contains no vacuous constant. Now you could replace the variables by constants. But either they are really variables, or they are really constants. If the latter, you are outside standard MPL.

[David Brightly]

Ed,

I think you will find that if you try to formalise your conditional

If for some x, y, z, integer(x) and integer(y) and z = x/y, then for some p, p is a true contradiction

you will find yourself writing

(∃x,y. (x*x=2*y*y) & ...) --> ?

with the '?' replaced by some expression. But you will be unable to use symbols 'x' and 'y' within the ? expression because the variables 'x' and 'y' are bound by the ∃-expression within the parentheses on the left of the implication.

My claim is not that the argument cannot be made without introducing new constants. Rather that the argument in this form is perfectly valid informal MPL that any mathematician would accept and which could be completely formalised.

[Edward Ockham]

>>informal MPL

? Informal formal logic ?

[David Brightly]

Ed,

I've seen the word 'idiomatic' used to describe the highly stylised English in which mathematical proofs are usually written.

Bill,

Applying my suggestion of 'translating out' any undeclared logical constants to your (1) and (2) we get

-2.	∃x.Venusity(x)				hyp
-1.	Venusity(Venus)				witness
0.	Venus=Venus				defn of '='
1.	∃x.x=Venus				EG of (0)
2.	∃y.∃x.x=y				EG of (1)

There is now a problem with the modal status of (1) when it is lifted out of context. Let's say that a wff is contingently true(false) if there are worlds and contexts in which it is meaningful and true(false) but also worlds and contexts in which it is false(true) or meaningless. Then (1) is contingently true. But (2) is but contingently true too because it is false in the empty world and true elsewhere. So the move from (1) to (2) preserves modal status as well as truth.

Looking now at (3) and (4):

-2.	∃x.Vulcanity(x)				hyp
-1.	Vulcanity(Vulcan)			witness
0.	Vulcan=Vulcan				defn of '='
3'.	∃x.x=Vulcan				EG of (0)

Hence where it is meaningful (3) is false, ie, it's contingently false. (4), viz (∃y)~(∃x)(x = y), is indeed necessarily false, but the move from (3) to (4) by Existential Generalisation is still truth-preserving, albeit vacuously.

[David Brightly]

Bill,

One last comment on this if I may. You seem to be saying that there are circumstances in which the Existential Generalisation rule of inference in MPL leads us astray. I find this unlikely and have tried to show that the anomalies that you highlight melt away if we take care with 'undeclared' constants such as the 'Venus' and 'Vulcan' in your examples. However, the same analysis seems to show that '~(∃x)( x = Vulcan)' is an unsatisfactory translation/regimentation of the ordinary English 'Vulcan does not exist', since the former, when meaningful, comes out false, whereas we accept the latter as true. Much hinges, as Ed suggests, on the extent to which the sense of the English term 'Vulcan' maps onto the sense of the MPL constant 'Vulcan'. So I agree that the Quine/van Inwagen goose is cooked. However, I've suggested that 'Vulcan does not exist' can be seen as a kind of imperative. It tells us to rescind the general existential assertion under which the name 'Vulcan' was introduced and to discard 'facts' about Vulcan. This is analogous to what the MPL proof rules tell us to do when we infer falsehood inside a proof box opened under the hypothesis '∃x.Vulcanity(x)'. We discard the box and infer '~∃x.Vulcanity(x)' instead. But Vulcanity() need be no haecceity concept. It could be taken as 'is a planet within the orbit of Mercury large enough to account for Mercury's anomalous orbit'. So I think there is some room left for a thin theory to stand on.

[Bill Vallicella]

David,

I do appreciate your careful thought on this topic. Other things are occupying me at the moment, but I shall come back to it.

[David Brightly]

Thank you, Bill. I look forward to your next instalment.

[André]

Dear Bill,

I know that you in your book discuss the regress objection against your account of existence as a unification of its constituents. However I have been unable to find it. Could you please just restate your reply to this objection (or state the pages of the book).