Tuesday, April 01, 2008

Englebretsen on the Liar Paradox

George Englebretsen has a good summary in the most recent edition of The Reasoner of his propositional depth account of the Liar Paradox. I discussed this approach in August 2006. As I noted there, a problem with this approach is that, if it is true and taken as a general solution to Liars, whether a statement is meaningful or not will sometimes depend entirely on the existence of other sentences; and another is that you can generate sentences that certainly seem to be false, but which would really be meaningless if the propositional depth account is correct. These are indeed problems, or at least puzzles; but I think one can bite the bullet on both and still be entirely reasonable.

There are several different approaches you might take to a Liar:

(1) It is true.
(2) It is false.
(3) It is true and false.
(4) It is meaningless.

On (1) you'd say that (for example) the following sentence is simply true:

This sentence is false.

Needless to say, it is not generally regarded as viable (although it does seem to have been held once or twice in history). (3) is a dialetheist position; it has never been popular, although thanks to Graham Priest and a small handful of others it is not a completely dead approach to the paradoxes, either. If we set aside (1) and (3), and there are reasons to set them both aside, then that leaves (2) and (4) as the primary approaches. (2) has occasionally had its partisans, although it's not the dominant view these days. I'm a fairly solid (2) man myself; I'm inclined to think the paradoxes are generated by dubious assumptions about what can be inferred from the falsity of a statement. That is, the problem is inference, not meaning. But I have no knock-down argument for this, and the propositional depth account still seems to me to be the best version of (4) I've ever come across. I like the apparatus of the account, or I would if I knew of any independent reasons for it, any real uses for it outside of Liars.

Incidentally, the Wikipedia article on the Liar paradox has a somewhat odd claim about Prior's version of (2):

Moreover, if all sentences are really hidden conjunctions, then some rules of propositional logic, such as the rule that one can derive any conjunct immediately and the rule that from any two propositions one can immediately derive their conjunction, are called into question. If we can derive this statement is false from This statement is true and this statement is false, then the paradox is back. And if we are not allowed to make such a derivation, then Prior has, in effect, invented a new kind of conjunction whose truth value characteristics are so mysterious, we cannot really say with any confidence that the paradox has been dissolved.


But surely no one holds that we can use conjunction elimination to conclude to a conjunct if the conjunction is false? For instance, suppose I take a contradiction:

p & ~p

I cannot from this infer either p or ~p unless I take it to be true. And this does not change if p is understood as follows:

p = p & ~p

And if I have just an ordinary conjunction, e.g.,

The sky is red and grass is green

and if it's clear that it's false, as we would ordinarily take this one to be, then I cannot infer either of its conjuncts. I can only infer a conjunct if I take it as true -- e.g., treat it as a premise. To be sure, every conjunction implies its conjuncts, but that doesn't mean that from every conjunction you can conclude the truth of its conjuncts. Because then you could take any false conjunction, e.g.,

2+2=5 and everything is false

and conclude that everything is false (or that 2+2=5, which is not an improvement). So if there is a problem with Prior's solution, it isn't that he has "invented a new kind of conjunction" with mysterious truth value characteristics. Just look at a truth table. If p and q are your conjuncts, and the conjunction is false, that may mean any of the following:

p is false and q is true
p is true and q is false
p is false and q is false

So which are you going to pick? Nothing about the conjunction itself requires any of these inferences; so nothing about p or q can be inferred from the conjunction itself.

(I'm also a little puzzled at the claim that if we interpret the Liar not as a conjunction but as an equation, A = (A = false), the paradox returns. But from

(1) A = (A = false)

we get by associativity

(2) (A = A) = false

which by identity is

(3) true = false

which, of course, is by definition the same as

(4) true = ~true

which anyone who takes Prior's view would obviously regard as just plain false, and for exactly the same reasons.)

[From the discussion page, incidentally, I notice that the Prior section has given the editors quite a bit of trouble; apparently it was muddled and in violation of Wikipedia standards from the get-go, and getting it into shape has been a struggle. Since the above passage has no citation, I would suggest that by WP:NOR and WP:V the above passage should be removed and only restored if it can be re-written, with citations, so as to describe an actual discussion in the literature. Perhaps there is one; but if so, it's clear the section needs both clarification and citation.]

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