Monday, November 9, 2009

Liar Family Reunion?

Most philosophers are familiar with the Liar Paradox:

L: This sentence is not true.

If L is true, then things are the way it reports, namely, L is not true.
If L is not true, then thing are not the way it reports, namely, L is true.

So, we seem to be able to conclude that L is true iff L is not true. Paradox.

But let's consider some related sentences:

P: I promise not to fulfill this promise.
C: Do not comply with this command.
Q1: What is an incorrect answer to this question?
Q2: What is not an answer to this question?

Now, P, C, Q1 and Q2 seem to admit of similarly paradoxical results.
An action fulfills the promise made by P just in case it does not fulfill the promise.
An action complies with the command issued by C just in case it does not comply with the command.
Something is the correct answer to Q1 just in case it is not the correct answer to Q1.
Something is an answer to Q2 just in case it is not an answer to Q2.

It is worth mentioning that, like L, each of the above can be reformulated in non-directly self-referential terms (replacing name for the sentence with "the the Nth labeled sentence in such-and-such blog post" and rephrasing slightly).

One thing that I want to note is that, if these are genuinely Liar-like paradoxes, they might be taken to suggest that focusing on "truth" in the Liar paradox is something of a red-herring. Prima facie, none of these invokes the truth-predicate, but seem to be of a piece with the Liar paradox.

One thing I want to ask is whether these paradoxical sentences have been discussed in the literature. I've read a fair amount about the Liar and the truth predicate, but haven't come across any discussion of these sorts of sentences. The closest thing I know of is Markosian's paradox of the question ("What is the pair <Q, A> such that Q is the most useful question, and A is its correct answer?").

Any thoughts?


Joshua said...

I think it is a mistake to conclude form these examples that it is a red herring to focus on "truth" when addressing the liar paradox. Rather, it think what we learn is that there are a class of entities that behave in a structurally similar way. This class includes the meanings of 'is not true', 'will not fufill this promise', 'will not comply with this command' and 'is not an answer to this question'.

For example, one might think that all these predicates refer to properties that are non-exclusive. That is, some things both are and are not true, some things both are and are not answers to that question, some things both are and are not fulfillments of a promise, etc. This is are all extensions of a paraconsistent response to the liar.

But, notice that the solution to the liar involves focusing on "truth" and noting that some things are both true and not true. So, it is not a red herring to focus on truth. It just turns out that we need to realize that whatever we say about truth should be plausibly extended to cover these other paradoxes.

Maybe, though, I've misunderstood what you meant when you said that focusing on "truth" is a red herring.

Lewis Powell said...

I don't mean to suggest that a proper resolution to the Liar does not have consequences for the truth-predicate. I think what I meant was something more like this:

If these other paradoxes are of a kind with the Liar, then a proper resolution will involve re-describing the paradoxes at a higher level of abstraction, and giving a response that addresses them jointly.

So, for instance, at the level of language, the paraconsistent response you mention involves maintaining that a class of predicates will be such that for some objects, they both apply and fail to apply to those objects.

One way to think about what I am suggesting is that proposed resolutions of the Liar should be evaluated (in part) on their extensibility to these related paradoxes. Prima facie, it looks like a paraconsistent view would be easily "portable" to the related paradoxes. Some responses, however, might not be. So, to determine whether or not something is a solution to the liar might involve seeing whether the solution can answer questions that are not about the truth predicate at all.

I did not mean to suggest that there would be nothing to say about the truth predicate in responding to the Liar.

Joshua said...

That all sounds right to me. However, are there any solutions to the liar paradox that can't obviously be adapted for the other paradoxes?