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Thursday, September 20, 2007

A Limitation on Dialetheism

First, I should acknowledge my indebtedness to Greg Littmann and Keith Simmons, whose essay “A Critique of Dialetheism” was the inspiration for this post.[1]

Dialetheism, for those not in the know, is the thesis that that some contradictions are true. It is platitudinous that some of the things people say are true and others false—and some, dialetheists add, are both true and false. At first sight, this is an odd yet intriguing view. But why believe it?

Dialetheism is usually motivated by considerations involving logical and semantic paradoxes, most famously by the Liar paradox. One of the most basic versions of the Liar is (1):

(1) This statement is false.

Is (1) true? If it is, then what it says is the case, and since what (1) says that it’s false, it’s false. But if (1) is false, then what it says is not the case, and since (1) says that it is false, it is false that it is false, and hence (1) is true. So if (1) is true, it’s false, and if it’s false, it’s true. Contradiction.

There are various things one could say at this point, but the important thing is what the dialetheist says, and what the dialetheist says is that (1) is both true and false. There are many things which can be said in favor of this view, some of them very compelling. There are also many things which can be said against it that are equally compelling. My aim, however, is not to argue either that dialetheism must be accepted or rejected as a matter of principle, but rather to show that the dialetheic treatment of logical and semantic paradoxes cannot be extended to all versions of the Liar. Consider for a moment (2):

(2) This statement has the same truth value as “0 = 1”.

Assume (2) is false. If so, it must have a different truth value than “0 = 1”, for what (2) says is that they have the same value. Since “0 = 1” is false, (2), if it has a different value, must be true. But if (2) is true, it has the same truth value as “0 = 1”, for that they have the same truth value is precisely what (2) says. Now if (2) is true, and it has the same truth value as “0 = 1”, then “0 = 1” must also be true, and hence we can conclude that 0 = 1 ![2]

We cannot give (2) a dialetheic treatment—holding that it is both true and false— for we can substitute any falsehood we like for “0 = 1” and use the paradox to show it must be true as well as false. We would then end up with trivialism—the view everything is both true and false! Since (2) cannot be solved by dialetheic means, it must have a different, consistent solution. There are many avenues we could pursue, such as tweaking the T-schema, holding that (2) expresses no proposition, adopting some form of the theory of types, etc. , but the point is that at least one of them must be successful. Granting this, why can’t we solve more traditional variants of the Liar in the same way? Dialetheism might still be true—in some attenuated epistemic sense of “might”—but even so it is not a perfectly general solution to all Liar-like paradoxes. If other kinds of Liar statements can be given the same treatment as (2), whatever that may be, dialetheism loses much of its motivation. If other reasons can be found for believing in true contradictions, well and good—but so long as consistent solutions are on the table, I think they ought to be preferred.


[1] “A Critique of Dialetheism”, in The Law of Non-Contradiction: New Philosophical Essays, Oxford University Press 2006. In particular I was inspired by their sentence (Z):

(Z) has the same complete and correct evaluation as the sentence ‘1+1=3’.

(Z) can be found in footnote 26 on page 333 (Paperback version).


[2]If we had started out by assuming (2) is true, we could have reached the same conclusion in half the time.

10 comments:

Rich said...

Interesting. So how does deduction work in a dialethic system? Is true-and-false treated as a third value in a Kleene-like logic?

Even simpler than that, how do the connectives work? For instance, let L be the Liar proposition, and let F be a false proposition. If L->F true, false or both?

(Apologies for being too lazy to read the papers you referenced to get my answers... feel free to send me scurrying off to the library)

Quirinius_Quine said...

Hi Rich,

Thanks for your comment. As far as formal logic is concerned, I've been trying to teach myself, and honestly I'm not very far along yet. However, from what I understand there are different ways of "going dialetheic": The most famous, Graham Priest's logic LP (for "Logic of Paradox"), does treat 'true-and false' as a third value. Others give up the truth-functionality of negation, so that the truth-table for negation looks just like that for an arbitrary p and q: both true, both false, p true and not-p false, p false and not-p true.

As for a true-and-false proposition implying a false one, I think that too depends on the system, specifically on whether true-and-false is designated or not.


I think this SEP article on paraconsistent logics explains things far better than I could:

http://plato.stanford.edu/entries/logic-paraconsistent/

(Don't worry, it's very short.)

Hope this helps answer your questions.

rich said...

Aha, that's interesting, I never actually knew what paraconsistent logic was :-)

Thanks a lot for the reading matter, will definitely take a look.

Anonymous said...

Hi, I just happened across your post whilst googling :-) and I have a question:

Regarding (2), couldn't one simply say that this statement has no truth value? Perhaps that's just another way of saying (2) isn't really a proposition, but to my mind, we could easily read the statement as follows:

0=1 has no truth value (unless we're speaking some tech language?).

So, the statement really says, "This statement has the same truth value as no truth value." Which seems to solve the problem... unless I'm missing something. Just wondering about that. Thanks.

Geoff

Quirinius_Quine said...

Hi anonymous,

That is a good point, and while one might take that view, I'm not sure whether it is viable or not. (If you're interested, you might want to check out my latest post "Does every proposition have a negation?".) In any case, if you think statements like "0=1" have no truth value, you can substitute any (contingent) falsehood you like and the argument will still go through. So let's run the argument with "Every human is shorter than six feet" in place of "0=1":

"(2) This statement has the same truth value as “Every human is shorter than six feet”.

"Assume (2) is false. If so, it must have a different truth value than “Every human is shorter than six feet”, for what (2) says is that they have the same value. Since “Every human is shorter than six feet” is false, (2), if it has a different value, must be true. But if (2) is true, it has the same truth value as “Every human is shorter than six feet”, for that they have the same truth value is precisely what (2) says. Now if (2) is true, and it has the same truth value as “Every human is shorter than six feet”, then “Every human is shorter than six feet” must also be true, and hence we can conclude that every human is shorter than six feet!"

Since there are some people who are six feet or taller, the conclusion of the above argument cannot be correct. Nevertheless, "Every human is shorter than six feet" has a truth value: It's (contingently) false. The problem with the argument is that you can take *any* false proposition you please and use the argument to show that it's true.

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