Abstract
If one, in order to evaluate ¬Kp, follows the BHK condition for negated formulas, and takes Kp to be untrue in all possible worlds, Fitch’s paradox is no threat to the antirealist. However, the semantics become intolerably inexpressive. On the other hand, if one interprets ¬Kp as saying that Kp is untrue in the actual world, another way-out of the paradox presents itself. I sketch which one this is, and I describe the intuitionistic models, in which it can be applied. I show that, within these models, one can built the knowability principle, while, at the same time, not everything is known in every world. Moreover, by applying the Beth condition for existential quantification, I show that there are worlds of these models, where a sentence saying that we will come to know something we now ignore is true/established. So, in these models, where the knowability principle holds good, not only are there worlds, in which not everything is known, but these worlds can prove that much as well.
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Notes
- 1.
A moment’s thought will convince you that the crucial question here is whether the intuitionist can recognize something as truth-valueless. By the moment that she does, one has positive evidence that she can impose decidability on Kp as well.
- 2.
Thanks to an anonymous referee for this point.
- 3.
We skip the predicate logic part here, since it is inessential for the argument.
- 4.
This formulation belongs to our external, classical, metalanguage, but concerns eternal sentences of the Signature. This is important to notice, since it might be the case that one such sentence is true sub specie of our external metalanguage, while, at the same time, it might be untrue in some worlds of the model. I do this for simplicity. The principle can be built with equivalence classes of “being now true” sentences.
- 5.
Notice, again, that this sentence makes part of our external metalanguage. This is important, because p is not true in the worlds, where it is unknown, as neither is p&¬K(p).
- 6.
Notice that the more general schema: “w forces (∃x)f(x) if and only if there is a bar B for w, such that for every world w′ ∈ Β, (i) w′ > w, and (ii) there is a y, such that f(y) is forced in w′ ”, should not be valid. For example consider the case where sometime in every possible future there lives a person more than 200 years old. The non validity of the more general schema is the reason why ¬(∃p)(p&¬Kp) is valid in the models. It does not depend on future bars but is evaluated in situ.
- 7.
If you do not feel at ease with both q and ¬q being possible, when q is mathematical, consider the example in the following note.
- 8.
A more everyday life candidate for q is the following: Think of any decision you have not made as yet, that you can, in principle, make, and which is still within your powers either to make or not to make; then, construct an eternalized sentence for today’s “I will have taken this decision by the end of the day”.
References
Beth, E. (1956). Semantic construction of intuitionistic logic. Kon. Nederlandse Ac. Wetenschappen afd. Letteren: Mededelingen, 19(11), 357–388.
Brogaard, B., & Salerno, J. (2007). Knowability, possibility and paradox. In D. Pritchard & V. Hendricks (Eds.), New waves in epistemology (pp. 270–299). New York: Palgrave Macmillan.
Brouwer, L. E. J. (1981). In D. van Dalen (Ed.), Cambridge lectures on intuitionism (Appendix). Cambridge: Cambridge University Press.
Dummett, M. (1977). Elements of intuitionism. Oxford: Oxford University Press.
Edgington, D. (1985). The paradox of knowability. Mind, 94, 557–568.
Edgington, D. (2010). Possible knowledge of unknown truth. Synthese, 173(1), 41–52.
Kapantaïs, D. (2009). The sea-battle and intuitionism. Revised and augmented version of my Ph.d. Thesis, University of Bern (2007).
Kapantaïs, D. (2011). Intuitionistic formal semantics for future contingents. In P. Nomikos (Ed.), Proceedings of the eighth panhellenic logic colloquium, Ioannina (pp. 68–72). University of Ioannina.
Kripke, S. (1965) Semantical analysis of intuitionistic logic I. In M. Dummett & J. Crossley (Eds.) Formal systems and recursive functions. Proceedings of the Eighth Logic Colloquium, Oxford (pp. 92–130). Amsterdam: North-Holland.
Kvanvig, J. (1995). The knowability paradox and the prospects of anti-realism. Noûs, 29, 481–499.
Kvanvig, J. (2006). The knowability paradox. Oxford: OUP.
Van Dalen, D. (1986). Intuitionistic logic. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. III, pp. 225–339). Dordrecht: Reidel.
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Kapantaïs, D. (2013). Intuitionistic Semantics for Fitch’s Paradox. In: Karakostas, V., Dieks, D. (eds) EPSA11 Perspectives and Foundational Problems in Philosophy of Science. The European Philosophy of Science Association Proceedings, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-01306-0_3
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