## 5/13/2008

### Against negative free logic

'Free logic' is an abbreviation for 'logic whose terms are free of existential assumptions, both singular and general.' Free logics attempt to deal with languages containing singular terms that do not denote anything, such as 'Pegasus'.

Free logics come in 3 basic flavors, which differ over what truth-values should be assigned to (atomic) sentences containing non-denoting names.
- Negative free logics declare all such sentences false;
- Neutral free logics declare all such sentences neither true nor false; and
- Positive free logics declare at least some such sentences true (in particular, 'Pegasus=Pegasus').

Tyler Burge argued for negative free logic over its rivals in "Truth and Singular Terms," Nous (1975). I came up with a little argument against negative free logic; but I do not know the argumentative landscape for these 3 options particularly well, so this may be extant already. (Note: if any readers have references for arguments pro and con negative free logic, I'd be very interested. I've found a couple of nice articles by James Tomberlin, and a short response by Richard Grandy to Burge's piece, but not much else.)

According to the negative free logician, all atomic sentences containing non-denoting names are false. Some people reject this because calling 'Pegasus=Pegasus' false seems wrong; here's another problematic type of case. Consider the following three sentences (and assume for the sake of argument that 'Atlantis' is a non-denoting name):
(1) Atlantis is West of London.
(2) Atlantis is East of London.
(3) Atlantis and London have the same longitude.
In negative free logic, all three of these must be false. But for the three predicates 'is west of,' 'is east of,' and 'has the same longitude as,' any one of the three can be defined in terms of the other two using only negation and conjunction. E.g.:
'x is west of y' means 'x is not east of y, and x does not have the same longitude as y.'
But now we've got 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.

And this same problem will crop up in general when we have a set of predicates that are definable in terms of one another and negation; in the simplest case, P = ~Q. And this is not that rare: {'before', 'after', 'simultaneous'} is another example. The negative free logician could save her position by maintaining that two of the predicates are somehow really basic, and the other really derivative. But at least in these two cases, it doesn't look legitimate to hold that 'west' is somehow fundamental and 'east' merely derivative.

Does anyone see a good response to this objection on behalf of the proponent of negative free logic?

Anonymous said...

You suggest that we might define one of 'x is west of y', 'x is east of y', and 'x does has the same longitude as y' in terms of the other two. It seems to me that this expedient will only work for a suitably restricted universe of discourse. Take the pair London and the planet Mars. London does not seem to stand in any of these three relations to Mars. So the fact that Atlantis does not stand in any of those relations to London does not count as an objection to negative free logics.

I think that the point extends to other trichotomies: Symbolizing them using fewer than three relations is a shortcut that only works for a suitable universe of discourse, but free logics always allow for unsuitable UDs. The relationship between Eastness and Westness can be added as an axiom, but we'd want various axioms anyway-- for example, to assure reflexivity of 'same longitude'.

Greg Frost-Arnold said...

Hi P.D. --

Thanks for the comment. I need to mull it over before I say anything definite.

I can see how the other example I gave (before, after, simultaneous) fits your thought: those 3 apply only to events, so e.g. 'London is after Paris' is gobbeldygook (sp.?).

But I just haven't thought about it enough yet to be certain that a similar thing will happen in every case where we have a trichotomy based on greater/ lesser/ equal (or bigger/ smaller/ equal, or more/ less/ same.) But maybe you're right.

But in terms of the bigger picture, I'm not sure your point about trichotomies requiring restricted universes of discourse would save the negative free logician. For it sounds like (correct me if I'm wrong) you want to say that e.g. 'east' (and 'west' and 'equilongitudinal') are partially-defined predicates, so that 'Mars is west of London' lacks a truth-value. But the negative free logician wants to say 'Atlantis is west of London' is FALSE, not that it lacks a truth-value. (Hmm... I think I may have missed something.)

Unknown said...

I'm sure that many logicians would disagree with me, but I think that some statements in free logic can be true and meaningful in some particular domains of discourse. For example, if you are discussing mythical animals, I think the statement "a unicorn is a horse-like animal with a horn on its head" would be true. If you are discussing the book or movie trilogy "Lord of the Rings", I think the statement "Hobbits are shorter than most men" would be true. In ordinary language, you could specify that you are including some fictitious things in the domain of discourse within the statement itself, such as by saying "In mythology, the unicorn was a horse-like animal with a horn on its head".

The concept of the relative longitude of two places is only meaningful if the two places both exist on the surface of the same object. For example, it doesn't seem meaningful to discuss whether or not the core of the Earth is east or west of the center of the sun, since the two places are not on the surface of the same object. Similarly, it doesn't seem meaningful to discuss the longitude of a non-existent place to an existing place. The concept of longitude is also complicated by the fact that east and west are relative directions, not absolute -- if you travel far enough in an easterly direction from location A, you will eventually end up west of location A, since the Earth is a sphere.

Greg Frost-Arnold said...

Hi Dan --

Thanks for the comment.
1. I do think a fair number of really good philosophers say sentences like 'Santa brings presents to children in the Christmas myth' are true. But they want to resist that that implies 'Santa brings presents to children' is (in some important sense) true period. (Compare: '"2+2=6" is written down on this piece of paper' can be true, without '2+2=6' being true.)

2. Your point about east/west, which appears to me related to PD's earlier comment, is exactly right. But I'll ask you the same question I asked P.D.: does that really save the negative free logician, who wants to say that all atomic sentences containing non-referring terms are false? If so, how?

Anonymous said...

I hadn't meant to say that 'east' (and the rest) are partially-defined. I was supposing bivalence and, since 'Mars is west of London' certainly isn't true, allowing it to count as false. This seems consistent with the sentiment behind the negative free logic: Since 'Mars' (or 'Atlantis') does not refer to the sort of thing that could be west of London, then it is neither west of, east of, not at the same longitude as London. The reason 'Mars' does not is that it refers to a different planet. The reason Atlantis does not is that it doesn't refer to anything.

Colin said...

This strikes me as basically right. I had been thinking something similar, not in the context of free logic, but just regarding anyone who takes the line that empty names render sentences in which they appear false. This seems like a problem for anyone who wants to maintain excluded middle.

That said, what if we give up excluded middle?

Greg Frost-Arnold said...

P.D. --

Thanks, that helps clear things up. So might the following be recognizable as a re-statement of your view?
'The mistake in the original post is that the definition of 'west' (or whichever of the 3 terms is derived) is not just = 'not east and not equilongitudinal.' Rather, the definition needs another clause that restricts the ordered pairs of objects that satisfy 'x is west of y' to locations on the surface of a single planet (or something like that). The original definition is too broad. And furthermore, definitions in similar cases (e.g. {before, after, simultaneous}) will likewise be too broad.'

Am I in your general vicinity?

Colin --

I am not completely averse to giving up the law of excluded middle, at least on certain construals of the LEM.

1. If the LEM says:
'Every truth-valued proposition or its negation is true'
then I would definitely like to hold on to the LEM.

2. If the LEM says:
'Every proposition or its negation is true'
then I would still like to hold on to the LEM, but I am not completely opposed to the ideas that (i) there are propositions that are neither true nor false and (ii) such propositions would count as counterexamples against the LEM.

3. If the LEM says:
'Every (grammatically declarative) sentence or its negation is true'
then while I think it would be nice for your logic to have the LEM, I don't see preserving the LEM as a really important or substantive desideratum in constructing a logical/ semantic theory. For I think it's quite possible that some declarative sentences lack truth-values (but I'm not nearly so sure about propositions). Empty names, presupposition failure, and category mistakes are mechanisms that arguably generate (grammatically declarative) sentences that are neither true nor false.

And the LEM, I would guess, was never meant to apply to things that don't have truth-values (e.g., questions, commands, or even just rocks and my cat).

Anonymous said...

Hi Greg,

Nice blog! I've only just come across it.

I think the negative free logician is going to say that your definition of west is inadequate. Antlantis can fail to be west of London by being east of it, or by being the same longitude as it or by not existing at all.

Standard definitions come apart, but can easily be amended, by adding an existence clause.

Unknown said...
This comment has been removed by the author.
Unknown said...

It's not only the issue of being on the same plane or planet.
For instance, what's north of the north pole? North/South directionality is meaningless on certain points of the sphere. What if the questions were Atlantis is north of London, etc. Directionality is also meaningless in the Atlantis example, unless of course Atlantis is not a non-existant place.

Greg Frost-Arnold said...

Jon --

But the negative free logician won't want to say that 'Atlantis is north of London' is meaningless -- that's what the neutral (and in some cases positive) free logician wants to say. In other words, someone who claims to be a negative free logician who says 'Atlantis is north of London' is meaningless has given me everything I want.

Andrew -

Thanks for the comment. I'm a fan of possibly philosophy. I'm not sure I have a satisfying response.

It may be that I have chosen a bad example of the situation I have in mind (Px = ~Qx), and that is what everyone in the comment thread is reacting negatively to.

Or it could be that there are no 'real' or 'genuine' predicates that are interdefinable like (Px = ~Qx). But I think there probably are. And I'm guessing (/hoping?) that some of those don't have a requirement of existence. But maybe I'm wrong.

Greg Frost-Arnold said...

Andrew -

I just came up with a more satisfying (to me, at least) response.

You suggest writing existence requirements into the definitions. That struck me as a good response. But if you actually try to write out the definition in first order logic (plus appropriate descriptive symbols), it's not clear how to do it.
Let W = west
E = east
S = same longitude as

I originally suggested the definition would be

∀x ∀y [Exy ↔ (~ Wxy & ~Sxy)]

Then, colloquially, you suggested amending this to

∀x ∀y [Exy ↔ (~ Wxy & ~Sxy & x exists & y exists)]

But how could you actually translate those colloquial phrases into the language of first order logic with identity? Perhaps I'm being dense, but it looks impossible to me.

Edward Ockham said...

I just spotted this (and found your blog too). I posted about it on my blog (which I won't link to here as it will hit your spam filter), but you can look for the post called "Is Atlantis west of London". Essentially, my reply to your objection is that it relies, first, on the perfectly valid assumption that if p is equivalent to "~q and ~r", then ~p is equivalent to "q or r". And, 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. "Atlantis is West of London" is equivalent (according to our variety of NFL) to "something which is identical with Atlantis is West of London".

I.e. 'Atlantis is West of London' and 'Atlantis is not West of London' are not contradictories, but contraries, and the 'not' is the 'not' of predicate negation (not sentential negation).

Or equivalently

WestofLondon(a) = notEastofLondon(a) and notSamelongitudeasLondon(a).

which if correct does not give the unfortunate implication that the objection requires.