Abstract
In a recent paper, Alexander argues that relaxing the requirement that sound knowers know their own soundness might provide a solution to Fitch’s paradox and introduces a suitable axiomatic system where the paradox is avoided. In this paper an analysis of this solution is proposed according to which the effective move for solving the paradox depends on the axiomatic treatment of the ontic modality rather than the limitations imposed on the epistemic one. It is then shown that, once the ontic modality is standardly introduced, the paradox still follows and, in addition, some puzzling consequences arise.
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Notes
In order to obtain the conclusion, K is to be interpreted as: it is known by someone at some time that.
To be more accurate, it is perfectly intuitive from a classical point of view, where negation is not conceived of as entailment of inconsistency. Indeed, every time we know that we wonder about p, i.e. every time when it is true that K(¬Kp ∧ ¬K¬p), we are in a condition where ¬Kφ for some true φ. This is why no principle like φ → ¬K¬Kφ is assumed in a classical framework. In addition, Alexander (2012) was able to show the interesting result that φ → ¬K¬Kφ implies φ → KKφ even without assuming (II).
This strategy was adopted by Edgington (1985), and developed in Edgington (2010). According to Edgington, what the principle of knowability actually states is that all actual truths are knowable. A logical characterization of Edgington’s interpretation of the paradox is given in Rabinowicz and Segerberg (1994), where the costs of the original proposal are highlighted and an improved solution is given in terms of a novel semantic characterization of the actuality operator. Edgington’s interpretation has been strongly criticized by Williamson (1987) and (2000, 12.5), but see Burgess (2009) and Kvanvig (2006) for a different implementation of the basic idea. Recently, a new interesting solution of the paradox, based on a definition of knowability in terms of existence of a proof, has been advanced in Dean and Kurokawa (2010).
This strategy was adopted by Tennant (1997) and originated a decennial debate with Williamson. See Tennant (2009, 2010), and Williamson (2000, 2009). The idea proposed by Tennant, i.e., to limit the scope of the knowability principle to propositions whose knowledge does not give rise to contradiction, is still a promising way out of the paradox. In the same vein, in Artemov and Protopopescu (2013) it is proposed to limit the principle of knowability with respect to stable propositions, i.e., propositions that, once true, cannot change their truth value. This is an elegant and intuitive solution, whose only problem seems to be that it prevents the general possibility of knowing that we don’t know some true proposition.
This is the strategy preferred by constructivists. The central intuition is that the last step in the argument is only classically valid and that the conclusion that can be achieved by using intuitionistic rather than classical logic, i.e., φ → ¬¬Kφ, can be accepted by a constructivist once the connectives are construed in the right way. A standard way in which the paradox can be interpreted from a constructivist point of view is proposed by Dummett (2009). See Fagin et al. (1995), Williamson (1982, 1988) for a discussion of the problems that constructivists have to take into account. See also the general proof-theoretic approach developed in Maffezioli et al. (2013), where it is shown that the adoption of intuitionistic logic allows us to block all the possible derivations of the paradox.
See Blackburn et al. (2001), ch 4, for a general introduction to modal systems. Note that instances of classical tautologies are allowed to contain modal operators. Hence, e.g., □p → □p is to be considered a classical tautology.
Since it is a straightforward consequence of the definition of general modal logic that the intersection of any set of general modal logics is in turn a general modal logic, S turns out to be the intersection of all the general modal logics over \( {\mathcal{L}}\) containing K3.
See Fagin et al. (1995, ch. 9) for an overall introduction to this kind of semantics.
See Blackburn et al. (2001, ch. 4) for an introduction to this procedure and proofs of the standard steps.
More precisely the schema K N(Kφ → φ), where K N is a string of N > 0 epistemic operators.
Alexander attributes K□1 to Salerno. However, as far as I can see, no such axiom is proposed. Actually, what we find there the assumption that what is necessarily false is impossible: □¬φ → ¬◇φ.
See Artemov and Protopopescu (2013) for an analysis of this position. Limiting knowledge to necessary propositions is a consequence of Edgington’s approach in Edgington (1985). Williamson (2000), criticizes this move by highlighting that Fitch’s paradox displays a limit to possible knowledge of contingent truths, so that it cannot be solved by limiting knowledge to necessary propositions.
To see that, consider a system where both □ and K are KT5 modalities and the following axiom is added: Kφ ↔ □φ. In this system □ and K are in fact the same modality, so that □φ → ¬□¬Kφ turns out to coincide with axiom 5: □φ → ¬□¬□φ, whereas the triviality axiom φ ↔ □φ is not derivable.
Let us remind that the axiomatic systems that are usually considered appropriate for capturing the notion of necessity are KT, KT4, and KT5, defined according to the following axioms:
K: □(φ → ψ) → (□φ → □ψ)
T: □φ → φ
4: □φ → □□φ
5: ¬□φ → □¬□φ
Rule of necessitation: \( {\vdash} \) φ ⇒ \( {\vdash} \) □φ
KT5 is commonly considered the right system to capture both the notion of logical necessity and the notion of metaphysical necessity. As far as I can see, no scholar has questioned the interpretation of the necessity operator, and so I will assume that one of the abovementioned notions is at work in the formalization of Fitch’s paradox. Nevertheless, as we will see, the full strength of KT5 is not needed in order to obtain the paradox.
This interesting variant and its fundamental problem has been suggested by an anonymous referee.
References
Alexander, S. (2012). A purely epistemological version of Fitch’s paradox. The Reasoner, 6, 59–60.
Alexander, S. (2013). An axiomatic version of Fitch’s paradox. Synthese, 190, 2015–2020.
Artemov, S., & Protopopescu, T. (2013). Discovering knowability: A semantic analysis. Synthese, 190(16), 3349–3376.
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. Cambridge: Cambridge University Press.
Burgess, J. (2009). Can Truth Out? In Salerno, J. (2009), pp. 147–162.
Dean, W., & Kurokawa, H. (2010). From the knowability paradox to the existence of proofs. Synthese, 176, 177–225.
Dummett, M. (2009). Fitch’s paradox of knowability. In Salerno, J. (2009), pp. 51–52.
Edgington, D. (1985). The paradox of knowability. Mind, 94, 557–568.
Edgington, D. (2010). Possible knowledge of unknown truth. Synthese, 173, 41–52.
Fagin, R., Halpern, Y., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge: MIT Press.
Hand, M. (2010). Antirealism and universal knowability. Synthese, 173, 25–39.
Kvanvig, J. L. (2006). The knowability paradox. Oxford: Oxford University Press.
Maffezioli, P., Naibo, A., & Negri, S. (2013). The Church–Fitch knowability paradox in the light of structural proof theory. Synthese, 190, 2677–2716.
Rabinowicz, W., & Segerberg, K. (1994). Actual truth, possible knowledge. Topoi, 13, 101–115.
Salerno, J. (2009). New essays on the knowability paradox. Oxford: Oxford University Press.
Salerno, J. (2010). Introduction to knowability and beyond. Synthese, 173, 1–8.
Tennant, N. (1997). The taming of the true, Chapter 8. Oxford: Oxford University Press.
Tennant, N. (2009). Revamping the restriction strategy. In Salerno, J. (2009), pp. 223–238.
Tennant, N. (2010). Williamson’s woes. Synthese, 173, 9–23.
Williamson, T. (1982). Intuitionism disproved? Analysis, 42, 203–207.
Williamson, T. (1987). On the paradox of knowability. Mind, 96, 256–261.
Williamson, T. (1988). Knowability and constructivism. Philosophical Quarterly, 38, 422–432.
Williamson, T. (2000a). Knowledge and its limits. Oxford: Oxford University Press.
Williamson, T. (2000b). Tennant on knowable truth. Ratio, 13, 99–114.
Williamson, T. (2009). Tennant’s troubles. In Salerno, J. (2009), pp. 183–204.
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Giordani, A. On a New Tentative Solution to Fitch’s Paradox. Erkenn 81, 597–611 (2016). https://doi.org/10.1007/s10670-015-9757-7
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DOI: https://doi.org/10.1007/s10670-015-9757-7