Abstract
Some dialetheists have claimed that one of the central advantages of their approach to the Paradoxes of Self-Reference is that they are able to offer a unified solution to structurally similar paradoxes that arise in the semantic and set-theoretic realms (Priest in Mind 103:25–34, [12] and Beyond the limits of thought. Clarendon Press, Oxford, [16]). They argue that since the structures of all of these paradoxes conform with the Inclosure Schema (IS), the Principle of Uniform Solution (PUS) dictates that we should solve them all the same way. But the dialetheist’s approach to PUS collapses when it comes to the Curry Paradox, to which any solution based on dialetheism seems inapplicable. We show that a particular version of a ‘paracomplete’ theory (inspired by Field in Saving truth from paradox. Oxford University Press, Oxford, [8]) has available to it a way of avoiding these problems that the dialetheist cannot mirror without losing Modus Ponens. JC Beall has suggested a way of minimizing this loss, but Beall’s strategy runs up against a further difficulty about the PUS. This one involves the Irrationalist’s Paradox. We conclude with a dilemma: The dialetheist can either reject the principles that are used to accommodate ordinary reasoning when Modus Ponens or Disjunctive Syllogism fail, and thus live without the ability to mimic these inferences, or they can sacrifice the Principle of Uniform Solution by solving an extremely Liar-like semantic paradox in a way that has nothing to do with their solution to the Liar and Curry’s Paradox. In either case, the prospects for a plausible and truly uniform dialetheic solution to the paradoxes of self-reference are grim.
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Notes
- 1.
See Priest [16, Chaps. 9–11], for details. The crucial point brought out by Priest’s set-theoretic model of this structure is that in all Inclosure paradoxes, Closure and Transcendence are a ‘package deal’. Objects transcend the relevant categories because they belong to them, and they belong to them because they transcend them.
- 2.
Oddly, in Priest’s comparison of “Field’s approach” and dialetheism, he uses the PUS to critique Field’s views on set theory without acknowledging that Field’s views on the semantic paradoxes could easily be extended into the set-theoretic realm [17, pp. 48–51].
- 3.
Alan Musgrave reports being shocked that Priest told him that reductio is “not always” valid. “Logicians”, Musgrave complains, “used to think an argument form was valid, full-stop, or invalid, full-stop, not valid here and invalid there” [9, p. 338]. Given the assumption that there are true contradictions, though, it’s not obvious why it should be so absurd to say that argument forms that preserve truth in contexts where no contradictions are true should fail to do so in inconsistent domains.
- 4.
- 5.
Once this framework is established, Priest goes on to introduce various complexities to the scrambled egg argument, and to make other arguments against trivialism. Whether or not all of his arguments go through is beside the point for our current purposes. The important part of the discussion from Doubt Truth to Be a Liar for our purposes here is not Priest’s anti-trivialist argumentation, but his conceptual framework for how anti-trivialist argumentation should proceed.
- 6.
The ‘paradoxes of material implication’ play no role in motivating Field’s conditionals, which work just like classical conditionals in contexts where Excluded Middle holds. Crucially, though, they behave very differently in paracomplete contexts. This in term allows Field to generate a transfinite hierarchy of determinateness operators, such that he can say of any given paradoxical sentence that it is (in some sense or another) not determinately true. The hypothetical version of paracomplete theory being proposed would thus have costs in terms of expressive completeness.
- 7.
Can dialetheists simply reject Modus Ponens as a way out? We will consider this option, and note its significant costs, in the next sections.
- 8.
- 9.
A desperate way to avoid this might be to deny that there’s any feature that sentence can have that always make it rationally rejectable. The problem with this move is that it makes the whole notion of rational rejection rather mysterious. If there’s no feature of sentences that makes them rationally rejectable, can we say anything about them except that they’re rejectable because they’re rejectable?
- 10.
Actually, it’s not clear that Beall could endorse a hierarchical solution here without giving up on his transparent theory of truth. As he points out in his discussion of Field’s solution to the paradoxes in Spandrels of Truth [1] one of the things transparent truth predicates are good for is referring to every level of an infinite hierarchy at the same time. Thus, we could have sentence (10):
(10) It is irrational in at least one of the infinitely many senses of irrational to believe this sentence.
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Burgis, B., Bueno, O. (2019). Liars with Curry: Dialetheism and the Prospects for a Uniform Solution. In: Rieger, A., Young, G. (eds) Dialetheism and its Applications. Trends in Logic, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-30221-4_1
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