Abstract
Graham Priest (In contradiction) introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Gödel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist.
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Notes
- 1.
For a gap theorist—one who says that certain sentences are neither true nor false—super-soundness is weaker than completeness. Presumably, a gap theorist does not want a complete theory.
- 2.
A referee pointed out that the new mathematical concepts introduced in a given era are typically explained in terms of the language of the preceeding era. Suppose that this goes on for several iterations. We explain the mathematics of, say, four or five generations of new concepts in a single language. I would think that we could not count on consensus as to what is correct in the original language.
- 3.
A referee suggested that a dialetheist might resist the connection between the formalized proof predicate, which goes via the coding, and actual proof. This possibility was raised in Shapiro (2002). Priest seemed to reject this, accepting the usual tight connection between statements of what is provable and their coded arithmetic counterparts.
- 4.
I don’t claim that these remarks constitute a formal proof of C. An attempt to formalize the informal argument presented here would presumably be a conditional proof, and it is open to the dialetheist to reject that for the conditional in C. Other attempts to reconstruct Curry-type reasoning in the formal system invoke contraction, which is rejected for the detachable conditional. Thanks to a referee for pointing this out.
- 5.
To follow the previous note, it is perhaps open to a dialetheist to claim that C is not true, in which case \(\lnot C\) is true. So if PA* is super-sound, there should be a proof, in PA*, of \(\lnot C\).
- 6.
A referee suggested that it is open for someone to claim that true contradictions arise only in semantic contexts, but that such contexts do (or at least might) result in true contradictions in the language of arithmetic. But this is not Beall’s view. As noted, he holds that there are no true contradictions in the language of arithmetic. As Beall (2009, p. 16) puts it, ‘The spandrels of ttruth bring gluts into our language, but they do not “spill” gluts back into our otherwise classical base language. In this way, the resulting dialetheism is very limited’. So for Beall, it is acceptable to reason with classical logic in arithmetic theories and in theories of syntax.
- 7.
This claim does not run afoul of the conclusions in Field (2006). The indicated argument takes place in a meta-language (one whose logic is classical).
References
Beall, J. (2009). Spandrels of truth. Oxford: Oxford University Press.
Field, H. (2006). Truth and the unprovability of consistency. Mind, 115, 567–605.
Kalmár, L. (1959). An argument against the plausibility of Church’s thesis. In Constructivity in mathematics (pp. 72–80). Amsterdam: North Holland.
Mendelson, E. (1963). On some recent criticisms of Church’s thesis. Notre Dame Journal of Formal Logic, 4, 201–205.
Priest, G. (2006). In contradiction: A study of the transconsistent (2nd, revised ed.). Oxford: Oxford University Press (1st ed., Dordrecht: Martinus Nijhoff Publishers, 1987).
Shapiro, S. (2002). Incompleteness and inconsistency. Mind, 111, 817–832.
Shapiro, S. (2009). We hold these truths to be self evident: But what do we mean by that? Review of Symbolic Logic, 175–207.
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Shapiro, S. (2019). Inconsistency and Incompleteness, Revisited. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_22
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