Dialetheias in the absence of contraction/absorption?

Question

One of the arguments given for dialetheism is that the paradoxes of self-reference, such as the Liar paradox and Russell's paradox, are most naturally regarded as dialetheias (both true and false). To avoid triviality, dialetheists must use a paraconsistent logic in which ex contradictione quodlibet fails.

However, in a sufficiently expressive system, such as natural language or set theory with naive comprehension, paraconsistency is not enough to guarantee nontriviality, because of Curry's paradox. The standard response seems to be to use a logic that also lacks the contraction/absorption law A → (A → B) ⊢ A → B, thereby blocking the derivation of Curry's paradox.

It seems to me, though, that discarding contraction also blocks the other paradoxes of self-reference. So once we are willing to do without contraction, is there anything remaining of the motivation for dialetheism coming from these paradoxes? Are there any semantic or set-theoretic paradoxes whose derivation does not require contraction and hence can still be argued to be dialetheias?

Answer 1

Since asking this question I have realized that even if contraction-free logic does not reproduce any literal dialetheias of the form "P and not-P", it gets pretty close. For instance, in naive set theory phrased in classical linear logic, there is a statement P such that we can prove

(P ⅋ P) ∧ (¬P ⅋ ¬P)

where ⅋ is the multiplicative/intensional disjunction, aka "fission". Namely, let P be the statement R∈R, where R = {x|x∉x} is the Russell set. Then "P ⅋ P" is equivalent to ¬P → P, while ¬P ⅋ ¬P is equivalent to P → ¬P, both of which follow directly from the definition of R.

Answer 2

Russell's paradox (set theory) does not depend on contraction. However whether it can be used as an argument for dialetheism is doubtful. The standard interpretation is that it is a demonstration that unrestricted set comprehension cannot be permitted in a system that is intended to be useful or interesting.

I suppose it is possible that one could continue to permit unrestricted set comprehension, and interpret the paradox as showing that one should adopt dialetheism, and hence conclude that x both is and is not a subset of itself, the barber both does and does not shave himself, the library catalogue both does and does not list itself, etc etc. But I am not aware of any philosophers adopting this interpretation.

Particularly relevant part of the linked-to article above:

It seems, therefore, that proponents of non-classical logics [including dialetheism] cannot claim to have preserved NC [naive set comprehension] in any significant sense, other than preserving the purely syntactical form of the principle, and neither intuitionism nor paraconsistency plus the abandonment of Contraction will offer an advantage over the untyped solutions of Zermelo, von Neumann, or Quine.