Monday, June 06, 2011

Is Atlantis West of London?

I just spotted this excellent post from May 2008, by philosopher Greg Frost-Arnold. He says that, in negative free logic (a variety of which I have argued for here, across too many posts to mention), all atomic sentences containing non-denoting names are false. But this leads to the following problem, he says. Assume, as seems probable, that 'Atlantis' is a non-denoting name.  Then in negative free logic, all three of the following sentences must be false.

(1) Atlantis is West of London.

(2) Atlantis is East of London.

(3) Atlantis and London have the same longitude.

But 'x is west of y' means the same as 'x is not east of y, and x does not have the same longitude as y', and now we have a problem.
If 'Atlantis is west of London' is false (as the free logician says), then at least one of 'Atlantis is east of London' or 'Atlantis and London the same longitude' has to be true -- but that contradicts the earlier assumption (of the negative free logician) that all of (1)-(3) are false.
He asks "Does anyone see a good response to this objection on behalf of the proponent of negative free logic?"  I answer: the objection relies, first, on the perfectly valid assumption that if p is equivalent to "~q and ~r", then ~p is equivalent to "q or r".  Second, it relies on the invalid assumption that "Atlantis is West of London" is equivalent to a conjunction of two negatives.  On the contrary, it is equivalent to the conjunction of two affirmatives. It is equivalent (according to our variety of NFL) to "something which is identical with Atlantis is West of London".  This in turn is equivalent to "something which is identical with Atlantis is neither East of London nor has the same longitude as London".  And the negation of that is

(4) It is not the case that something identical with Atlantis is not East of London, or it is not the case that something identical with Atlantis does not have the same longitude as London.

Now if (1) is false then (4) is true, and if Atlantis exists, it follows that Atlantis is East of London, or that Atlantis and London have the same longitude (I will omit the proof, which is elementary).  But if Atlantis does not exist, both (2) - which is equivalent to "something which is identical with Atlantis is East of London" - and (3) - which is equivalent to "something which is identical with Atlantis has the same longtitude as London" - are false.  For the presumption is that nothing is identical with Atlantis. 

This requires the negative free logician to bite a bullet, but it is not the unpalatable one that Frost-Arnold suggests.  We have to concede that sentences like "Atlantis is large" and "Atlantis is not large" are not true contradictories, but contraries.  The second is not the negation of the first. The true negation of the first is "notthing is Atlantis or Atlantis is not large".  But there cannot be simple solutions in philosophy.  When we have exhausted everything that is wrong, what remains must be right.

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5 Comments:

Blogger Greg Frost-Arnold said...

Hello --

Thanks for linking to my post, and more importantly, thanks for the very good response to it! I'll need to think about this a little, but this strikes me as a very good response. Three quick questions:

1. In the post, you refer to "our variety of NFL." Could you tell me what that variety is? (A reference to a publication would be plenty.)

2. What do you do (if anything) if there is no predicate for identity in the language?

3. Finally, is the following proposal equivalent (or equivalent enough) to yours?

GF-A assumes that

(*) ∀x[WestOfLondon(x) v EastofLondon(x) v SameLongitude(x)]

is true. However, GF-A is wrong about this assumption. Really, what's true is the following:

(**) ∀x[WestOfLondon(x) v EastofLondon(x) v SameLongitude(x) v ~∃y(y=x)]

And if one can only use (**) and not (*), then the contradiction does not follow (since all of your (1)-(3) can be false, but (**) will still be true).

9:18 pm  
Blogger Edward Ockham said...

And thanks for your reply! Welcome to Beyond Necessity.

1. On your first point – I haven’t yet published anything on this. I call it the ‘Frodo’-Frodo theory of reference, or the relativity theory of reference. I have discussed ideas about it with Mark Sainsbury since the 1980s; he has proposed a similar but less radical thesis in various places, notably Reference with Referents. I did complete a book in 2005 but was dissatisfied with it. I started again recently and have been posting bits of it here. I will pick out some key posts at some point and link to them. You could start with this The ‘Frodo’-Frodo theory of reference.

2. I didn’t understand your second point.

3. I disagree with (**), which is self-contradictory, and do in fact agree that

(x) [WestOfLondon(x) v EastofLondon(x) v SameLongitude(x)]

Now if ‘Atlantis is WestOfLondon’ is false that could be (I argue) for two reasons. (1) Because for some x, x = Atlantis and x is not WestOfLondon. In this case it follows from your (*) that it is either EastofLondon or SameLongitude. (2) Or because there is no such x. In which case we can’t infer anything from your (*).

Hope that makes sense. The key thing to get your head round is that, according to the ‘Frodo’-Frodo theory, there are certain uses of proper names that you can’t ‘quantify into’. Negative existential statements are one example. We can’t infer from “Pegasus does not exist” that “something does not exist”. Nor from “Greg is thinking about Atlantis” to “something is such that Greg is thinking about it”. Although we can infer, as it happens “Greg is thinking about something”

What’s the point of all this? (1) A theory of meaning for fictional names and (2) A non-externalist theory of mind. More later.

8:06 am  
Blogger Edward Ockham said...

Oops the Sainsbury title should be Reference without Referents. Amazon review.

Mark's version differs from mine in at least two respects. (1) He holds that ‘Pegasus does not fly’ is a true contradictory of ‘Pegasus flies’, and so both cannot be false. I hold, as you have seen, that they are only contraries (i.e. both cannot be true, but both can be false). (2) He holds that “ ‘Pegasus’ refers to Pegasus” is false, whereas I hold that it is true. (But n.b. I also hold that “ ‘Pegasus’ refers to Pegasus” is consistent with ‘nothing is Pegasus’).

10:10 am  
Blogger David Brightly said...

1) I don't know where Atlantis might be relative to London but judging from your picture it certainly has a worse plumbing problem.

2) I we are trying to free our logic of assumptions then the situation is more complicated than even free logic allows. The space of possibilities decomposes as follows:

---Nothing is Atlantis
---More than one thing is Atlantis (cf Charley's Ant)
---Atlantis exists uniquely
   ---Atlantis is not the kind of thing that has size
   ---Atlantis is the kind of thing that has size
      ---Atlantis is large
      ---Atlantis is not large = Atlantis is small

In other words, 'Atlantis is large' can fail to be true in three distinct ways:

Atlantis exists uniquely, has size, but happens to be small
Atlantis exists uniquely, but is not at all the kind of thing that has size
Atlantis fails to exist uniquely; either there is no Atlantis or many Atlantises.

3) I am uncomfortable (and I suspect Greg also, judging by his query (2)) with your translation into 'something which is identical with Atlantis is large'. This seems to sweep the existence question under the musty old carpet of identity. I'd hoped, with Wittgenstein, we could do without identity. The only way I can make sense of your formulation is to see it as 'something which instantiates the concept of Atlantis is large'. But this appears to embrace a description theory right at the start and runs foul of your strictures against seeing 'Atlantis doesn't exist' as about the idea of Atlantis.

10:58 am  
Blogger Edward Ockham said...

>>3) I am uncomfortable (and I suspect Greg also, judging by his query (2)) with your translation into 'something which is identical with Atlantis is large'. This seems to sweep the existence question under the musty old carpet of identity. I'd hoped, with Wittgenstein, we could do without identity. The only way I can make sense of your formulation is to see it as 'something which instantiates the concept of Atlantis is large'. But this appears to embrace a description theory right at the start and runs foul of your strictures against seeing 'Atlantis doesn't exist' as about the idea of Atlantis.
<<

I did discuss why we can't do without identity in an earlier post. Roughly, a referential chain forms when it is part of the semantics of the terms in the chain that they are verifiable of a single thing (no identity statement needed). But when two such chains form, we might want to state that they are also verifiable of a single thing. Then you need identity.

On existence claims, we could have 'Atlantis exists' (but beware of Meinongians), or 'there is no such thing as Atlantis'. I don't see a problem.

And yes we could treat proper names as a sort of predicate: Ex Atlantis(x). But there are fairly strong arguments (see my four arguments for direct reference) that they aren't predicates, in any reasonable sense of the word.

2:34 pm  

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