In a previous post I tried to undercut one of the main motivations for believing in dialetheism by giving the following argument:
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!
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.
I now think this argument doesn’t work. A dialetheist can simply say that (2) is both true and false while “0 = 1” is false only, because it is the statements’ conjoint falsity which accounts for the truth of (2), and not their conjoint truth. After all, it would appear that any statement of the form “p has the same truth value as q”, where q is both true and false, is itself both true and false, but surely this does not entail that p is both true and false. For example, one can easily generate statements of this form by substituting a Liar statement for q and an arbitrary statement for p, but as long as one rejects explosion—the principle that contradictions entail everything—this gives us no reason at all to think that p must also be both true and false. It may have taken me a year, but at least I caught my own mistake. :-P