Monday, August 22, 2011

Translating ordinary language into predicate logic.

Vallicella has now apparently accepted my arguments against grammatically proper names being translatable into the ‘logical constants’ of modern predicate logic (MPL). But this leaves him with the problem that there are also strong arguments against translating them into the predicates of MPL.  I summarised these arguments earlier in June, as follows.

Argument 1 was that a proper name does not signify something that is repeatable, therefore does not signify a property. Therefore it signifies an object. A reply is here.

Argument 2 was that a name cannot be significant or intelligible to another unless the idea of what the name applies to is in the other person’s mind. But we can only have the idea of a particular thing by being acquainted with that thing, which is only possible if that thing actually exists. A reply is here.

Argument 3 was that definition proceeds by genus and specific difference. Therefore a proper name cannot be defined, for they name individuals, and individuals are not species. They have no specific difference, and can only be distinguished by the proper name itself.  A reply is here.

Argument 4 was that truth-conditional semantics rests on the assumption that the conditions for the truth of a sentence give the sentence’s meaning or significance. But there is no truth evaluable content when reference failure occurs. If there are no truth conditions, then there is no meaning or significance. I have not replied to this argument yet.

If Vallicella accepts these arguments as well as those he summarises in his post, he is apparently left with the problem that ordinary language cannot be translated into MPL at all. Is that a problem?

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10 Comments:

Blogger David Brightly said...

>> Vallicella has now apparently accepted my arguments against grammatically proper names being translatable into the ‘logical constants’ of modern predicate logic (MPL)

I'm not convinced. I left a comment at BV's here. (Correction: In the second last sentence I mean 'logical connectives' not 'constants')

Another thought: your argument seems too general. For it seems that a subset of OL sentences can be successfully translated into MPL. Would not the argument apply to this restricted subset? Contradiction.

10:43 am  
Blogger Edward Ockham said...

Which subset?

12:35 pm  
Blogger Edward Ockham said...

On your comment at BV's place, which 'quoted passage' were you referring to? You said "Which sense is operative in the quoted passage?"

Re the Pimms, yes, I had another. Best served in a pint glass.

12:53 pm  
Blogger David Brightly said...

Re: which subset?
How about {'Frodo carried the ring', 'Frodo is a hobbit', 'some hobbit carried the ring'}? These are meaningful, consistent under MPL, yet 'Frodo' doesn't refer in the strong sense.

Re: which quoted passage?
The quote from you that Bill places at the top of his post, viz,

"Now I claim that in systems where there is no distinction between predicate and sentence negation, we have ‘direct reference’. This is easily shown. Direct reference in a singular sentence is when the sentence is meaningless when the singular subject fails to refer. Assume that ‘a is F’ is not meaningless. If it is true, then there is a referent for ‘a’. If it is not true, the sentential negation ‘It is not the case that a is F’ is true. If sentential negation is equivalent to predicate negation, it follows that ‘a is non-F’ is true, and so a exists, and so, there is a referent for ‘a’. But (by excluded middle) either ‘a is F’ is true, or its contradictory (the sentential negation) is true. In either case, ‘a’ has a referent. Thus if ‘a is F’ is not meaningless, ‘a’ has a referent. Conversely if ‘a’ does not have a referent, ‘a is F’ is meaningless. But that is Direct Reference, as I have defined it."

1:22 pm  
Blogger Edward Ockham said...

>>How about {'Frodo carried the ring', 'Frodo is a hobbit', 'some hobbit carried the ring'}? These are meaningful, consistent under MPL, yet 'Frodo' doesn't refer in the strong sense.
<<

But the argument I gave is precisely the opposite. If 'carried_ring(Frodo)' is meaningful, either it is true, or its negation is. But both it and its negation imply that 'Frodo' refers to something. But 'Frodo' doesn't refer to anything. Ergo, the sentence is not meaningful.

That's the whole point of the argument.

Pimm's o'clock, I think.

3:02 pm  
Blogger David Brightly said...

This may be a bit picky. By what rule of inference do we make the metalinguistic step from

carried_ring(Frodo)

to

'Frodo' refers to something

Don't we need an extra premiss? And won't that look just like DR?

11:20 pm  
Blogger Edward Ockham said...

>>This may be a bit picky. By what rule of inference do we make the metalinguistic step from
carried_ring(Frodo)
to
'Frodo' refers to something
Don't we need an extra premiss? And won't that look just like DR?
<<

In standard MPL, Fa implies Ex Fx. One way of avoiding this would be to say that the existential quantifier in MPL is not really existential (Peter Lupu uses this dodge from time to time). But that is Meinongianism.

6:58 am  
Blogger David Brightly said...

Ed,

The sticking point for me in this topic is the apparent validity of non-existence proofs by reductio ad absurdum which I outlined at BV's here. These start by assuming the existence of some object that possesses the required property, giving it a name, and then deducing a contradiction. Surely the sentences involving the named object are perfectly meaningful, in which case the DR criterion fails. I can't see how your response addresses this at all. There is something more subtle going on.

I understand and can use reductio as a mathematician but I know little about its formal aspects. Perhaps the sentences within the scope of the existence assumption are in some sense 'formal'. Perhaps we do make a temporary excursion into a logically consistent Meinongian jungle and back again. It looks a little like the way the formal expressions for the real roots of certain cubics involve the square roots of negative numbers. This worried renaissance mathematicians but they carried on regardless. Likewise, some proofs in real analysis are shorter via Robinson's non-standard analysis (rigorously defined infinitesimals), than via standard analysis.

4:31 pm  
Blogger Edward Ockham said...

>>I can't see how your response addresses this at all.

Well surely it addresses it by means of logic. If you accept

1. meaningful('Fa')->(Fa or ~Fa)

2. [p or q and p->r and q->r] -> r

3. Fa -> 'a' has a referent and ~Fa -> 'a' has a referent

then this logically implies

4. meaningful('Fa')-> 'a' has a referent

which further implies

5. ~ 'a' has a referent -> ~meaningful('Fa')

which is Direct Reference. Do you accept the implication? If you do, which premiss of the above do you disagree with.

>>Perhaps we do make a temporary excursion into a logically consistent Meinongian jungle and back again.

This is consistent with my conclusion. If 'a' refers to a non-existent object, then it refers to something. That is Direct Reference. Albeit reference to a non-existent something.

My arguments against Meinong are in another place (in the discussion of Inwagen's theory, I think).

7:32 am  
Blogger David Brightly said...

>> Well surely it addresses it by means of logic. I was hoping you'd tell me what was wrong with my purported counter-example!

I've had a little while to think about this. Non-existence proofs in maths actually show that none of an existing set of objects has some property. For example, let LP(x) denote '(natural number) x is prime and no prime is larger'. For arbitrary x, a simple construction shows LP(x)-->~LP(x). Hence ∀x.~LP(x) and so ~∃x.LP(x). The quantifications are over the set of natural numbers.

The argument for the non-existence of Vulcan is subtly different. The hypothesis is that there is a planet within the orbit of Mercury, which accounts for the oddities of Mercury's orbit, but is small enough to not be visible:

A: ∃v. P(v) & M(v) & ~V(v), so

A ⊢ ~V(v).

Some assumptions, N, about Newtonian dynamics shows that M(v)-->V(v). Hence

N, A ⊢ V(v).

So N ⊢ ~A, as required. Now if all this makes sense the question is to what does 'v' refer? It can't be an external object that the strong sense of Direct Reference requires. It can only have the weak sense of reference that your theory of 'relative reference' allows. In other words it means 'the same thing as is claimed to exist in statement A'.

This looks to me a good argument for relative reference! I'd have to say that your argument shows only that MPL rests on weak reference, not strong reference.

7:07 pm  

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