# Why does the Curry paradox require a separate solution in dialetheism?

On the Dialetheism entry on SEP, it is stated that, although dialetheism can offer a solution to the Liar Paradox (by accepting the Liar sentence as a true dialetheia), dialetheists need a separate treatment for the Curry paradox. However, one could see a relation between Curry and Liar as follows.

The Liar has form:

L: ~L.

Where the tilde is the logical negation. Curry has form:

C: C -> P.

Where -> is an implication and P is an arbitrary proposition (including, possibly, ⊥). Let us rewrite L (resp. C) in the following logically equivalent form:

L: (~L) or ⊥.

C: (~C) or P.

Then, it is clear that Liar is a particular case of Curry where P = ⊥. Now, a dialetheist way of dealing with Liar is to accept L as both true and false. Let us now look at the following derivation for Curry:

By LEM, either C or ~C holds.

Assume C. Then, by definition of C, we have (~C) or P. The first disjunct is false; therefore, the second disjunct must hold. Hence P.

Next, assume ~C. This implies (~C) or P. But this is the definition C, hence C holds. Contradiction. Hence P.

In both cases, P holds.

It seems to me that, here, we are using the fact that both C and ~C cannot hold together. However, if we assume that (in a dialetheist way) that C and ~C can hold together without rendering our theory trivial, we get:

By LEM, either C or ~C holds.

Assume C. Then, by definition of C, we have (~C) or P. Now, we can have either ~C or P.

Next, assume ~C. This implies (~C) or P. But this is the definition C, hence C holds. Hence, ~C or P.

In other words, by accepting the contradiction we cannot prove P via Curry.

In short, my question is:

Why is dialetheism enough to deal with Liar, but not with Curry, given my conceptualization of the two paradoxes?

• Based on my reading of that SEP entry, I'd say: because the Curry process allows trivialization even without a contradiction. So it is informative to combine the problems, though, as you appear to have done, so this line of reasoning may prove more fruitful than that discrepancy would suggest. A sustained answer to your question would hopefully include references to analysis of conditionals generally, but then for now, this isn't a topic I know enough about to respond to more confidently. Commented Jul 15 at 12:34

Simplifying the material conditional into a disjunction is a classical logic move. You can't do it in arbitrary logics, because it might not properly capture the essence of the material conditional, if that logic does not allow the same rules of inference as classical logic (and in particular, if it does not allow disjunctive syllogism, which is commonly banned by paraconsistent logics).

What is the essence of the material conditional? If we have P → Q, and we have P, then we should be able to conclude Q (modus ponens). There are a few logics which break this rule, most of them involving defeasible reasoning, but most logics take this to be the defining property of the material conditional, and go out of their way to ensure it remains valid, no matter what happens to disjunction and negation.

So let's look at this again:

1. We define P: P → Q, and ignore the issue of self-reference for the sake of argument.
2. If we do not have LEM, then there's the option of declaring P neither true nor false. This is a valid resolution to the paradox, but it is probably unsatisfying under dialetheism, which usually does include LEM.
3. If LEM is in play, then P may be true. If P is true, then we can immediately expand it into P → Q, and then use the above rule of inference to conclude Q. Dialetheism can't reasonably object to this case of the argument.
4. Alternatively, P may be false. Under classical logic, this immediately allows us to arrive at P → Q, because a false statement implies "everything." Since P → Q is P, that is a contradiction, so classical logic takes this as proof-by-contradiction that P is true. This gives us at least two obvious points of departure from classical logic - firstly, we may decide that you can't infer P → Q from ~P (which makes sense, because that's the syntactic equivalent of the principle of explosion). Secondly, we may decide that you can't reject the case just because it leads to P ∧ ~P (which makes sense, because dialetheism says that contradictions are not "problems" to be rejected) - but that's not actually a solution, because if we have P ∧ ~P, then we have P, and that puts us back in the P case again.

So, if you want a logic that allows the self reference in Curry's paradox and doesn't devolve into trivialism, you need it to conclude that the implication is false, and you need it to avoid concluding that this makes the implication true. That suggests some kind of relevance logic, although I suspect that other paraconsistent logics are also up to the task.

Why is dialetheism enough to deal with Liar, but not with Curry, given my conceptualization of the two paradoxes?

Dialetheism, by itself, is not a formal logical system. It is simply a general principle that statements may be both true and false simultaneously. It does not describe how the material conditional should work, so a paradox that contains the material conditional may not be possible to properly address without defining how your logic interprets the conditional.

By contrast, the Liar is simply P: ~P. Any logic that allows P ∧ ~P necessarily resolves the Liar, and it is obvious that any logic implementing dialetheism must necessarily allow P ∧ ~P.

Actually, setting aside the full question of how conditionals "should" work in logic, I think I do know the answer to this... If I remember correctly, Graham Priest promotes something called "the uniformity principle" (or, it's called by a name like that one...). This is that there are different logical paradoxes but which have structurally-similar solutions; at its most extreme (and I don't know that Priest is an extremist, although I've heard tell he's not quite a logical pluralist, either...), the idea would be that all logical paradoxes have similar-structure solutions.

So Priest's solution to the liar paradox is to recalibrate the restrictions on contradictions in his system(s), but by itself, this doesn't seem as if it would be "relevant to"(!) the generalized Curry problem. The "big thing" about the Curry problem is that it's paradoxical without being essentially negation-based. So it's like a "dual of" the Yablo problem, because that's negation-based but isn't supposed to be self-referential, whereas Curry sentences do self-refer; and logical paradoxes often turn on these two parameters (negation and self-reference). If Priest's theory is, at its heart, a theory of negation, then it seems as if its solutions to paradoxes will depend on dealing with the vagaries of this negativity.

But then, other things being equal, Priest faces a little "dilemma":

1. According to the uniformity principle, the structure of the solution to the liar and Curry paradoxes "should be similar to an important extent."
2. For that to be so, however, would require that both be structurally grounded in problems of negation theory. It seems that Curry paradoxes don't "need" negation on that level.
3. "In practice," if we can show that a structural similarity between solutions-to-paradoxes obtains (or: there is a tendency to this uniformization), this will be in showing how one specific solution "leads to" another. For example, it would be nice if solving the liar paradox meant solving at least one variant of the knower paradox, which would hopefully lead to solving another variant, and then maybe the Yablo problem besides, and so on. But so we can't use Priest's theory of negation to extrapolate a solution to the Curry problem, as a problem of conditional logic per se, from his solution to the liar paradox.

But this would be, such as it is, more specific to Priest's individual theory/theories, and/or other such theories. As far as the general question of combining paradoxes goes, this could be taken for a moment in combining logics (here, the word "logics" is not quite being used as when we speak of "many logics," although on a deeper level that is still exactly what we are very well speaking of). And so like, in logics where conditionals can be "converted" into sequences of other operations, incl. maybe negation, why couldn't we finagle a translation of a Curry sentence into a sentence essentially involving negation? I think that's what you mean to be trying to show, so maybe I'm just reading it too slowly, though, then.