Dialetheists think that some contradictions are true.
Trivialists think that everything is true (and false).
Here's a trivialism problem that everyone has:
You can't express your rejection of trivialism in a way that isn't compatible with trivialism by the trivialist's lights. For example, if you say "trivialism is false," the trivialist will agree with you.
Here's a trivialism problem that only the dialetheist has:
They can't express their rejection of trivialism in a way that isn't compatible with trivialism by *their own* lights.
Here's what I mean:
If an ordinary nondialetheist says "trivialism is false," they've just said something that they think rules out the possibility that trivialism is true.
In a dialetheist says "trivialism is false," they've said something they *don't* think rules out the possibility that trivialism is true.
If they say "trivialism is (just) false" or "trivialism fails to be true in any sense" or "trivialism has an alethetic status that rules out the possibility that it could be true," they still haven't said anything that can be (by their lights) incompatible with the truth of trivialism.
...at least assuming that they take Liar sentences to be both true and false.*
After all, we can always construct Liars like:
"This sentence is (just) false."
"This sentence fails to be true in any sense."
"This sentence as an alethic status that rules out the possibility that it could be true."
....etc., etc., etc. This is, I take it, one of the reasons why the dialetheist's difficulties with making sense of claiming that some sentences are "just true" or "just false" are philosophically significant, a subject we'll be returning to next week. So that's dialetheism's trivialism problem (1).
While we're at it, though, here are two other special trivialism problems that the dialetheist has that no one else has:
(2) The Curry Paradox bears exactly the same relation to trivialism that the Liar Paradox bears to dialetheism. In both cases, we have paradoxical sentences involving self-referential truth talk, such that if we take such sentences to be meaningful, truth-evaluable, etc., the philosophical position in question simply follows. When it comes to the Liar, the dialetheist can accept all of those things, accuse anyone who tries to tamper with otherwise intuitive inferential rules involved in the paradox in order to avoid the paradoxical result of begging the question, etc. When it comes to Curry, they're forced into an awkward double standard. If they want to avoid trivialism, they have to start making exactly the sorts of moves they rail against when it comes to the Liar.
(3) A classical proof in classical logic shows that we can infer anything and everything from any contradiction.
It's easy to show that the dialetheist has a good, principled response to (3).
But, given (1) and (2), this shouldn't be taken as meaning that dialetheism doesn't have any trivialism problems that the rest of us don't.
*It would, technically, be possible to be a dialetheist without taking Liars to be both true and false--one could, for example, be convinced by Graham Priest's arguments about "the paradoxes of motion and change" but not by any of his other arguments--but I know of no dialetheist who doesn't analyze Liars dialethically. In every actual case of people who argue for dialetheism, the argument from the Liar and related semantic paradoxes is their *central* argument.