Friday, January 13, 2012

Men and non-men

Anthony rightly raised the problem of my ‘some men are not men’. He suggested (rightly I think) that this is a problem for ‘Brentano equivalence’, the thesis that ‘Some A is B’ is equivalent to ‘An A-B exists’, and that ‘some A is not B’ is equivalent to ‘An A-not-B exists’.

He is right. If Brentano is right, then ‘some men are not men’ is equivalent to ‘men that are not men exist’, which is clearly wrong. So what’s the problem? I suggest that Brentano is wrong. Clearly we can say that some of the men who landed on the moon have now died ( for example, Alan Shepard, the one who played golf on the moon). So, some men such as Shepard are no longer men. If Brentano is right, that implies that men who are non-men exist, which is false. Non valet consequentia, so Brentano is wrong.

The late thirteenth century philosophers of language, such as the early Scotus, were acutely aware of this problem. Many of them distinguished between so-called indefinite negation of the form ‘A is a non-B’, and pure negation ‘A is not B’. Indefinite negation is affirmative. It affirms the existence of an A that is non-B. In this sense ‘some man is a non-man’ is false. By contrast, pure negation denies everything, including the affirmation of existence. In that sense ‘some man is a not a man’ is true, pace Brentano.

The problem is to render this in predicate logic. The formal sentence ‘for some x, Ax and not Bx’ is affirmative in the traditional sense: it asserts that some x is both A and non-B. However, the pure negative for ‘not for some x, Ax and Bx’ is not equivalent to the medieval ‘some A is not B’. The predicate logic version simply denies the existence of anything that is both A and B, whereas the medievals understood it in the sense we understand ‘some men (such as Alan Shepard) are not men (i.e. are men no longer)’.

Brentano’s thesis was the first formulation of one of the key assumptions of the modern predicate calculus. It is wrong for the same reason the calculus is wrong. It does not translate the meaning of a standard English sentence in the way we want to translate it. So what is the meaning of the sentence, and into what formal language can we translate it?

Labels: , ,

33 Comments:

Blogger Anthony said...

I'd like to see some more about this "pure negation". I can't make sense of it.

It seems what you are trying to say is that some men are not currently-existing-men (and further, that "some men are not currently-existing-men" does not mean the same thing as "some men are not-currently-existing-men", which you would say is false, nor the same thing as "it is not the case that some men are currently-existing-men" which we both agree is false).

>> If Brentano is right...
>> ...so Brentano is wrong.

Brentano is not right. Brentano is not wrong. Brentano is not Brentano. Right? ;)

Sorry, I couldn't resist.

2:13 am  
Blogger David Brightly said...

Hang on. Again, I think you are being pessimistic. Let's not chuck out Brentano just yet. Your argument, I think, is that from

(*) some men such as Shepard are no longer men,

Brentano allows us to infer

men who are non-men exist,

which is false. But surely (*) can only be made true by equivocation on 'men'. This is clear if we see it as

one of the men (ie, Shepard) is no longer one of the men.

which implies

one of the men (ie, Shepard) is not one of the men.

Since the extension of 'the men' changes over time we can make this true (and mean what we want it to mean) by making explicit the times that are implicit:

one of the then-men is not one of the now-men
or
one of the quondam-men is not one of the now-men.

This is not so far from my current problem, I think, where the extension or referent of 'the lump', 'the statue, 'the hand', 'the fist' changes over time.

12:09 pm  
Blogger Edward Ockham said...

>>Since the extension of 'the men' changes over time we can make this true (and mean what we want it to mean) by making explicit the times that are implicit<<<

We discussed this solution before, and yes it will work, but the problem of 'x' or 'thing' remains. 'Some things are no longer things' has the same problem. In order to have our word 'thing' or 'x' range over past objects, it must do just that, which is impossible. Perhaps you could say the variable used to range over such things, but it doesn't now. I made exactly this point somewhere way back, but can't find it.

12:27 pm  
Blogger Edward Ockham said...

In fact, I make the point in the very post you link to above. I don't see the reply about sets as a solution, either.

12:28 pm  
Blogger David Brightly said...

>> In order to have our word 'thing' or 'x' range over past objects, it must do just that, which is impossible.

I don't see this. What is wrong with saying 'some then-things are not 'now-things', with the extension of the concept 'thing' changing over time? After all, 'Caesar' refers quite happily to a non 'now-thing' thing and you are happy, I think, to allow this in your theory of reference. Can't MPL variables be similar referring terms?

>> I don't see the reply about sets as a solution, either.

Could you expand on this? I will look back at the old comments and replies.

1:05 pm  
Blogger Edward Ockham said...

>>What is wrong with saying 'some then-things are not 'now-things'
<<

It is very wrong, because of the present tense. It presumes that some x's are then-things. But of course nothing is a then thing. The only things are now-things.

1:08 pm  
Blogger David Brightly said...

I stand by my second comment on that post, regarding reference and satisfaction. Maybe we need another understanding of substitution. Not 'textual' or 'objectual', but 'referential', perhaps. I sense that you feel there is a problem with the 'objectual' notion of substitution. How can we substitute a object when it no longer exists? Something like that?

1:18 pm  
Blogger Edward Ockham said...

>>I sense that you feel there is a problem with the 'objectual' notion of substitution. How can we substitute a object when it no longer exists?
<<

Absolutely. If my theory of reference is correct, then there is no semantic relation between language and the world. Thus the notion of 'satisfaction' is bad, as is the notion of objectual substitution.

This theory is very far from being developed, as you know.

1:23 pm  
Blogger David Brightly said...

I see the object-ion.

I'm not so sure about 'satisfaction' but I agree about 'objectual' substitution. Very Russellian. Mont Blanc, etc.

I agree that the only things are the now-things, but we can refer to the then-things, surely. BTW I'm seeing 'now-things' and 'then-things' as sets used to explain what we mean rather than alternative concepts that could be used in the object sentences. Hence the 'one of the x-things' wording. And sets I see as referring devices, of course.

I think the theory is heading in the right direction. But I think you need sets.

1:40 pm  
Blogger Edward Ockham said...

Sets are evil. Or at least they would be, if they existed.

2:32 pm  
Blogger Anthony said...

"But of course nothing is a then thing."

You say of course, but I disagree. My choices seem to be between believing that some things are not things, and believing that past-things exist. I choose the latter.

Of course every thing is a thing.

---

I am intrigued by the claim that medievals had a notion of "some things are not things" which was true, though. I haven't yet read your latest post, which hopefully explains this.

10:51 pm  
Blogger Anthony said...

Edward, who is your favorite medieval philosopher?

12:20 am  
Blogger Edward Ockham said...

>>Edward, who is your favorite medieval philosopher?

Er, are you serious?

10:21 am  
Blogger Anthony said...

Yes, I'm serious. Why wouldn't I be?

1:21 pm  
Blogger David Brightly said...

Let's try 'Ockham sets' rather than those devilish mathematical sets. If 'the First Triumvirate' refers to Caesar, Pompey, and Crassus then 'the then-men' can refer to Caesar and Tiberius, say. And 'the now-men' can refer to DB and EO. And surely we can say 'Neither of the then-men is or was either of the now-men?

We can also say 'Suppose these are the only men ever to have existed.' The meaning of this is quite clear despite the use of both past and present tenses, neither of which is wholly appropriate to all four intended objects.

In fact, I don't think tense matters at all here. We are stating the referents of our terms quite explicitly. We are not using a concept term with the attendant risk that ambiguity is introduced by its extension varying over time. It's this latter effect that seems to be the critical factor in generating paradox.

3:24 pm  
Blogger Edward Ockham said...

But Ockham sets reduce to simple reference. Thus "DB is not Caesar and DB is not Pompey and DB is not Crassus and EO is not Caesar and EO is not Pompey and EO is not Crassus" which does explicitly signify any set. And then you have the problem of explaining "DB is not Caesar". This converts to "Caesar is not DB" and you have the problem of predicating "-is not DB" of something which doesn't exist. If predication is a relation between predicate and subject, then it must be a relation between predicate and existing subject. Which it can't be in the case of non-existent Caesar.

>>We are not using a concept term with the attendant risk that ambiguity is introduced by its extension varying over time.

But 'extension' is still a problem, as I have said. A term can have only one extension, namely the present one.

3:33 pm  
Blogger Anthony said...

So you don't have a favorite medieval philosopher? Your answer is to ignore the question?

Who is the world record holder for highest career batting average?

3:51 pm  
Blogger David Brightly said...

Didn't we agree sometime back that identity over objects is co-reference over terms? So 'DB is not Caesar' is 'DB and Caesar do not co-refer'.

So there is no predication here. But even so, if the claim is that predication is a relation between predicate and existing subject, then 'Frodo is a hobbit' is hardly predication. But surely we'd want to say that it is?

>> A term can have only one extension, namely the present one.
Are you sure? What about 'Caesar ruled the Romans'?

4:14 pm  
Blogger David Brightly said...

>>But Ockham sets reduce to simple reference.

Unfortunately, yes. They run out of steam when we need a referring term like 'the Romans' when we don't have a list of names, or just don't want to repeat such a list. Hence not much cop for maths. We need something like an 'anonymous referring term', which if not oxymoronic, is what MPL variables offer.

4:22 pm  
Blogger Edward Ockham said...

>>Yes, I'm serious. Why wouldn't I be?

Well it is so obvious that I thought you might be making fun of me!

4:27 pm  
Blogger Edward Ockham said...

>>what about 'Caesar ruled the Romans'?

Well obviously it makes sense, but to explain its semantics is the problem. It can't possibly consist in a relation between the predicate '-ruled the Romans' and some existing thing, i.e. something. It could only be true if Caesar still exists, but once ruled the Romans, or still does.

4:29 pm  
Blogger David Brightly said...

Sure, I think we agree that meaning is not about relations between words and the world. I'm not suggesting the semantics involve the extension. But to get to truth conditions for a sentence we need to get to extensions of terms, and the term 'the Romans' has different extensions in 'Caesar ruled the Romans' and 'The Romans like espresso'.

5:03 pm  
Blogger Anthony said...

>> Well it is so obvious that I thought you might be making fun of me!

You don't have a favorite medieval philosopher, because there are no medieval philosophers, because medieval philosophers don't exist? Is that what is obvious?

7:28 pm  
Blogger David Brightly said...

Morning Ed,

I have studied the post again. You are saying that the medievals understood 'some man is not a man' in the same way we understand 'some man is a man no longer', but you claim this can't be rendered in MPL. As it's equivalent to 'some man is a dead man' why not try 'for some x, Man(x) and not Lives(x)' ?

Brentano turns this into 'a Man-not-Living exists' which is proven by the instance Alan Shepard.

If you object that this merely gets us back to quantification over apparent non-existents then maybe this is the real nub of our problem and we need to investigate it in more detail. This surely is the bullet that has to be bitten.

12:26 am  
Blogger Edward Ockham said...

>>Is that what is obvious?

No, what is obvious is that the name of my favourite medieval philosopher is visible to you in at least two places on this very blog.

Edward Ockham

2:39 pm  
Blogger Edward Ockham said...

>>quantification over apparent non-existents ... maybe ... is the real nub of our problem

Yes. On the reasonable assumption that there is an implicit tense in the statement "for some x, Man(x) and not Lives(x)".

There is an uncanny resemblance between this question and the medieval dispute about whether 'Caesar is dead' is true or false. Here, e.g.

2:44 pm  
Blogger Anthony said...

How can your favorite philosopher be someone who doesn't exist?

Edward Ockham is not a medieval philosopher, is he?

11:57 pm  
Blogger Edward Ockham said...

Which is your favourite character in Lord of the Rings? Alternatively, do you have a favourite fictional character?

10:28 am  
Blogger Anthony said...

I guess I must have a favorite fictional character. I have no idea what character that is, though.

Anyway, enough with this thread.

2:31 am  
Blogger David Brightly said...

Hi Ed,
Regarding quantification, I have a suggestion here.

11:32 am  
Blogger Edward Ockham said...

I like your account as a sort of psychological account of narration. But how does it explain truth-conditions? What would have to be true in order for 'someone crossed the Rubicon 2,000 years ago' to be true?

1:00 pm  
Blogger David Brightly said...

I feel we have almost done this question to death but here is yet another take. We go from 'AS, who was a man, has died' to 'AS is no longer a man'. The latter is too readily seen as the indefinite negation 'AS is a non-man'. Rather, I suggest that the grammatically present-tensed 'is no longer' is a superficial contraction of the logical 'AS was a man and not (AS is a man)', where the negation in the second conjunct is pure. This, and its existential generalisation, are readily rendered in MPL using distinct tensed predicates 'was a man' and 'is a man' and are not problematic, I submit, except in so far as the latter quantifies over non-existents.

More on semantic graphs soon, I hope.

12:00 pm  
Blogger David Brightly said...

Here

4:16 pm  

Post a Comment

<< Home