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Precis of saving truth from paradox

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Notes

  1. And if one takes ‘False’ as a primitive, False(<W>) ↔ \(\neg\) W has as much claim to be part of the naive theory of falsehood as True(<W>) ↔ W has to be part of the naive theory of truth.

  2. Conversely, quite different model theories can lead to very similar or even identical solutions.

  3. There are related principles involving rejection in place of acceptance of negations: e.g. an analog of the Second is

    It is incoherent to accept True(<A>) while rejecting A.

    Standard dialetheist views (e.g. Priest 1987) accept this analog of the Second, but not the Second as given in the text. However, the current discussion is a lead-in to classical theories in which rejection just is acceptance of the negation, so I may as well stick with acceptance.

  4. For instance, consider theories that hold that Q is false and not true: they must hold

    [\(\neg\)True(<Q>), but Q] and also [True(<\(\neg\) Q>), but \(\neg\) \(\neg\) Q],

    thereby violating the First as regards Q and the Second as regards \(\neg\) Q. The dual theory, which holds that Q is true and not false, is similar.

  5. Like classical glut theories, these accept that Q is both true and not true. But classical glut theories do this while rejecting Q (and thus rejecting the inference from True(<Q>) to Q); whereas paraconsistent dialetheic theories accept both Q and \(\neg\) Q, and reject the classical principles that allow us to infer arbitrary conclusions from such contradictory pairs.

  6. Since writing the book, Anil Gupta has pointed out to me that the main theorem in the Appendix to Chapter 16 is an easy corollary of a result of his: the Reflection Theorem of Gupta and Belnap 1993. The book should also have said more about Brady 1989, which added a conditional not to Kleene logic but to a weaker relevance logic, and by a construction with some similarities to the one I used.

  7. It would be interesting to provide a similar generalization of the ordinary indicative conditional in English, which presumably is not the material conditional, to handle semantic paradoxes involving it. But there’s no hope of doing this until we have a good theory of how the ordinary indicative conditional behaves in absence of the semantic paradoxes.

References

  • Brady, R. (1989). The non-triviality of dialectical set theory. In G. Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: Essays on the inconsistent. Germany: Philosophia Verlag.

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  • Gupta, A., & Belnap, N. (1993). The revision theory of truth. USA: MIT Press.

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  • Priest, G. (1987). In contradiction. Holland: Martinus Nijhoff.

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  1. Philosophy Department, New York University, 5 Washington Place, New York, NY, 10003, USA

    Hartry Field

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Field, H. Precis of saving truth from paradox . Philos Stud 147, 415–420 (2010). https://doi.org/10.1007/s11098-009-9466-7

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