7

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?

  • You'd still need semantics in addition to syntax. Let's say that removal of contraction blocks all the paradoxes (I doubt it), what is the Liar's truth value? Dialetheic answer is that it is true and false. More generally, it allows for complete theories with the truth predicate definable within them, and therefore removes the Godelian problems and the Tarski hierarchy. But that does require admitting some true contradictions. – Conifold May 10 '16 at 19:02
  • 2
    @Conifold but if we discard contraction, then the usual argument for the truth and falsity of the Liar fails, so it seems that it would be just as natural to consider it to be neither true nor false; so dialetheism would not be necessary. (I am not asking about other motivations for dialetheism; as I said, I am only asking about motivation coming from the logical paradoxes.) And if you can find a particular paradox that still goes through in the absence of contraction, that would be a great answer to the question! – Mike Shulman May 10 '16 at 20:36
  • I am not sure why truth value gaps are any better than dialetheias, they are the reason the intuitionist logic has no extensional semantics, and why the attitude should be anything but dialetheism, unless it is forced on us unavoidably. I see it more pragmatically, dialetheic logics have some technical upsides, including dissolution of semantic paradoxes, gapped non-extensional logics have others, it's a decision between calculi and should be made on balance with specific purpose in mind. Unfortunately, I do not have a specific paradox in mind that goes through without contraction though :( – Conifold May 10 '16 at 22:13
  • 2
    @Conifold I don't think this is an appropriate place for a general argument about the merits and demerits of dialetheism; I only asked one specific question about it. I did not mean to denigrate other motivations for dialetheism; I only wanted to know whether this particular motivation survives the denial of contraction. It sounds like you do not have an answer to my question. – Mike Shulman May 10 '16 at 22:31
  • @MikeShulman I'm not sure if it's what you're looking for, but dialetheism can also deal with the sorites paradox, where avoiding contraction doesn't help (e.g. the conditional formulation of the paradox requires only modus ponens to derive a contradiction). – Eliran May 11 '16 at 7:23
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.

0

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.

  • Actually, if I am understanding the reasoning on that Stanford page, such a position would lead to triviality, even if both contraction is abandoned and dialetheism is accepted; and surely triviality should be unacceptable to any serious thinker. But in my opinion, dialetheism is just as unacceptable. So I suppose he who accepted dialetheism would also accept reasoning in trivial systems. Which makes it all merely silly. – AnotherSmellyGeek Mar 12 '18 at 11:29
  • I'm not sure what the author of that particular page had in mind, but Russell's paradox (and Curry's paradox) does in fact depend on contraction. It's been shown rigorously that in affine logic one can have a consistent set theory with unrestricted comprehension. – Mike Shulman Mar 12 '18 at 14:07
  • For paradoxes of self-reference used as a motivation for dialetheism, see plato.stanford.edu/entries/dialetheism/#3.1 . – Mike Shulman Mar 12 '18 at 14:08
  • @Mike I'd love an explanation of how Russell's paradox depends on contraction (it's obvious to me how Curry's does). But I doubt it would fit in a comment here. – AnotherSmellyGeek Apr 4 '18 at 8:12
  • Russell's paradox is just the special case of Curry's paradox where the statement being proven is the logical absurdity $\bot$. – Mike Shulman Apr 4 '18 at 13:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.