Abstract
Reformulation strategies for solving Fitch’s paradox of knowability date back to Edgington (Mind 94:557–568, 1985). Their core assumption is that the formula \(p\rightarrow \Diamond Kp\), from which the paradox originates, does not correctly express the intended meaning of the verification thesis (VT), which should concern possible knowledge of actual truths, and therefore the contradiction does not represent a logical refutation of verificationism. Supporters of these solutions claim that (VT) can be reformulated in a way that blocks the derivation of the paradox. Unfortunately, these reformulation proposals come with other problems, on both the logical and the philosophical side (see Percival in Aust J Philos 69:82–97, 1991; Williamson in Knowledge and its limits, Oxford University Press, Oxford, 2000; Wright in Realism, meaning and truth, Blackwell, Oxford, 1987). We claim that in order to make the reformulation idea consistent and adequate one should analyze the paradox from the point of view of a quantified modal language. An approach in this line was proposed by, among others, Kvanvig (Nous 29:481–499, 1995; The knowability paradox, Oxford University Press, Oxford, 2006) but was not fully developed in its technical details. Here we approach the paradox by means of a first order hybrid modal logic (FHL), a tool that strikes us as more adequate to express transworld reference and the rigidification needed to consistently express this idea. The outcome of our analysis is ambivalent. Given a first order formula we are able to express the fact that it is knowable in a way which is both consistent and adequate. However, one must give up the possibility of formulating (VT) as a substitution free schema of the kind \(p\rightarrow \Diamond Kp\). We propose that one may instead formulate (VT) by means of a recursive translation of the initial formula, being aware that many alternative translations are possible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
See Brogaard and Salerno (2013) for a complete overview of the solution strategies.
This point can be made clearer by a formal consideration. In any possible worlds semantics for modal propositional logics with an actuality operator @, the formula \(@p \leftrightarrow \Box @p\) is valid. Not only that: if we allow accessibility relations to be serial (i.e. there are no dead ends for accessibility) then for any sequence Seq of modal operators we have that \(@p \leftrightarrow Seq @p\) is valid. Since knowledge is factive, accessibility for K is reflexive and a fortiori serial. The same goes for the metaphysical possibility operator \(\Diamond \), which is often read as an \(S4\) operator (at least). Therefore, we obtain that \(@p \leftrightarrow \Diamond K @p\) is already valid and that \(@p \leftrightarrow K @p\) is too, viz. modal collapse strikes back.
A response to this objection and a further elaboration of the original proposal may be found in Edgington (2010).
See Mendelsohn and Fitting (1999), particularly Chap. 8, for a detailed discussion of actualist and possibilist quantification and their formal rendering in first order modal logic.
One further consideration is in order here. As our sloppy formalization \(\forall {\varPhi }\) suggests, quantification over propositions is also an issue. This would bring in an additional level of complexity and possibly entail a consequence that few would sustain: even truths that are not expressible in our language must be knowable. As argued by Burgess (2008), a simple cardinality argument can convince us that such truths are legitimate, and therefore (VT) should in any case be restricted to propositions that can be expressed within the language. However, full quantification over propositions will not be taken into account here as in most discussions on the Fitch paradox. We indeed read the \({\varPhi }\)s and \(\Psi \)s as placeholders for sentences of our language.
Hybrid logic originated with Arthur Prior’s work on tense logics Prior (1967) and Prior (1968). Prior’s philosophical aim was to extend the language of tense logics (the “A-series” language) in order to capture the expressive power of a first order language with an earlier-later relation (the “B-series” language), the latter being, according to him, an inadequate language for talking about time. Long after Prior’s death hybrid logic was reinvented by Passy and Tinchev (1985) and Passy and Tinchev (1991). Since the late 1990s interest in this area has significantly grown and hybrid logic is by now a well-established and active research area within modal logics. For a general overview see Chap. 7 of Blackburn et al. (2001) or Areces and ten Cate (2006).
On the intuitive level varying domain semantics correspond to actualist quantification while constant domain semantics is the natural way of expressing possibilist quantification. However, our choice does not mean embracing an actualist metaphysics. Insofar as the latter is, technically speaking, a special case of the former, the formalism can be easily adapted to the needs of a possibilist view.
The introduction of the downarrow binder in hybrid logics is due to Goranko (1994) and Goranko (1996). Neither of these papers uses the now standard syntax or semantics of hybrid logic. Goranko’s early formulation is more closely related to the Vlach-style syntax of retrieval operators (see Vlach 1973). The now standard syntax and semantics—used in this paper—and its relation to Prior’s idea of quantifying over nominals, was introduced by Blackburn and Serligman (1995). The more recent Yanovich (2015) shows how Vlach-Cresswell systems can be related to the standard apparatus of downarrow in hybrid logic.
For simplicity in this example we are assuming non-reflexivity for \(R_{\Diamond }\) but there would no problem in making our point, with a slightly different formula, for a reflexive \(R_{\Diamond }\).
An additional problem is how to express (VT) for the full language of FHL, and this is an even more complex issue. A possible option is represented by the following completion of the translation above:
$$\begin{aligned} \sigma ^{v}_{u}(i) {::}= & {} i \\ \sigma ^{v}_{u}(z'){::}= & {} z' \\ \sigma ^{v}_{u}(@_{i} \phi ) {::}= & {} @_{i}\sigma ^{i}_{i}(\phi ) \\ \sigma ^{v}_{u}(@_{z} \phi ) {::}= & {} @_{z}\sigma ^{z}_{z}(\phi ) \\ \sigma ^{v}_{u}(\downarrow z'. \phi ) {::}= & {} \downarrow z'. \sigma ^{v}_{z'}(\phi ) \\ \end{aligned}$$The above translation has the positive characteristic of keeping the de re/de dicto distinction in the consequent of (VT). For example, \(P(a) \wedge \lnot \exists x K_{x}{\downarrow }z'. P(z':a)\) leads to \({\downarrow }v. \Diamond {\downarrow }u. \exists y K_{y} (P(v:a) \wedge \lnot @_{v} \exists x@_{u} K_{v:x} {\downarrow }z. {\downarrow }z'. P(v:z':a))\). Since \(P(v:z':a)\) is equivalent with \(P(z':a)\), this atomic formula refers to the object denoted by \(a\) at the final world we arrive at. In contrast, \(P(a) \wedge {\downarrow }z'. \lnot \exists x K_{x} P(z':a)\) generates \({\downarrow }v. \Diamond {\downarrow }u. \exists y K_{y} (P(v:a) \wedge {\downarrow }z'. \lnot @_{v} \exists x@_{u} K_{v:x} {\downarrow }z. P(v:z':a))\). Here \(P(z':a)\) refers instead to the object denoted by \(a\) at the world we start from.
References
Areces, C., & ten Cate, B. (2006). Hybrid logics. In P. Blackburn, F. Wolter, & J. van Benthem (Eds.), Handbook of modal logics. Amsterdam: Elsevier.
Blackburn, P., & Marx, M. (2002). Tableaux for quantified hybrid logics. In U. Egly & C. Fernmüller (Eds.), Automated reasoning with analytic tableaux and related methods, international conference, TABLEAUX 2002, Copenhagen, pp 38–52.
Blackburn, P., & Serligman, J. (1995). Hybrid languages. Journal of Logic, Language and Information, 4, 251–273.
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. Cambridge tracts in theoretical computer science (Vol. 53). Cambridge: Cambridge University Press.
Braüner, T. (2005). Natural deduction for first-order hybrid logic. Journal of Logic, Language and Information, 14, 173–198.
Braüner, T. (2011). Hybrid logic and its proof theory. New York: Springer.
Brogaard, B., & Salerno, J. (2008). Knowability, possibility and paradox. In V. Hendricks & D. Pritchard (Eds.), New waves in epistemology. New York: Palgrave Macmillan.
Brogaard, B., & Salerno, J. (2013). Fitch’s paradox of knowability. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy.
Burgess, J. (2008). Can truth out? In J. Salerno (Ed.), New essays on the knowability paradox (paraître). Oxford: Oxford University Press.
Edgington, D. (1985). The paradox of knowability. Mind, 94, 557–568.
Edgington, D. (2010). Possible knowledge and unknown truth. Synthese, 173(1), 41–52.
Fitch, F. B. (1963). A logical analysis of some value concepts. Journal of Symbolic Logic, 28, 135–142.
Goranko, V. (1994). Temporal logic with reference pointers. In Proceedings of the 1st international conference on temporal logic, LNAI, Vol. 827, Springer, pp 133–148.
Goranko, V. (1996). Hierarchies of modal and temporal logics with reference pointers. Journal of Logic, Language and Information, 5, 1–24.
Hansen, J. U. (2007). Hybrid logic with applications. Master’s thesis, University of Copenhagen.
Hart, W. D. (1979). The epistemology of abstract objects. Proceedings of the Aristotelian Society, 53, 153–165.
Hart, W. D., & McGinn, C. (1976). Knowledge and necessity. Journal of Philosophical Logic, 5, 205–208.
Kennedy, N. (2014). Defending the possibility of knowledge. Journal of Philosophical Logic, 43, 579–601.
Kvanvig, J. (1995). The knowability paradox and the prospects for anti-realism. Nous, 29, 481–499.
Kvanvig, J. (2006). The knowability paradox. Oxford: Oxford University Press.
Mendelsohn, R. L., & Fitting, M. (1999). First-order modal logic. Dordrecht: Kluwer.
Passy, S., & Tinchev, T. (1985) Quantifiers in combinatory pdl: Completeness, definability, incompleteness. In FCT, pp. 512–519.
Passy, S., & Tinchev, T. (1991). An essay in combinatory dynamic logic. Inf Comput, 93(2), 263–332.
Percival, P. (1991). Knowability, actuality and the metaphysics of context-dependence. Australasian Journal of Philosophy, 69, 82–97.
Prior, A. N. (1967). Past, present and future. Oxford: Oxford University Press.
Prior, A. N. (1968). Papers on time and tense. Oxford: Clarendon Press.
Quine, W. V. (1953). Reference and modality. In From a logical point of view. Harvard University Press.
Quine, W. V. (1956). Quantifiers and propositional attitudes. Journal of Philosophy, 53(5), 177–187.
Rabinowicz, W., & Segerberg, K. (1994, March). Actual truth, possible knowledge. In R. Fagin (Ed.), Proceedings of the 5th conference on theoretical aspects of reasoning about knowledge, Pacific Grove, CA, USA. Morgan Kaufmann, pp. 122–137.
Vlach, F. (1973). ‘now’ and ‘then’: A formal study in the logic of tense anaphora. PhD thesis, University of California Los Angeles.
Wehmeier, K. F. (2003). World travelling and mood swings. In B. Löwe, W. Malzornm, & T. Räsch (Eds.), Foundations of the formal sciences (pp. 257–260). Dordrecht: Kluwer.
Wehmeier, K. F. (2004). In the mood. Journal of Philosophical Logic, 33, 607–640.
Williamson, T. (1987a). On the paradox of knowability. Mind, 96, 256–261.
Williamson, T. (1987b). On knowledge and the unknowable. Analysis, 47, 154–158.
Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press.
Wright, C. (1987). Realism, meaning and truth. Oxford: Blackwell.
Yanovich, I. (2015). Expressive power of ’now’ and ’then’ operators. Journal of Logic, Language and Information, 24, 65–93.
Acknowledgments
Special thanks to my anonymous reviewer 2 for many insightful comments and suggestions and the tone of his or her remarks. I am also particularly indebted to Jens Ulrik Hansen and Patrick Blackburn for discussion, inspiration and comments on previous versions of this paper. Thanks also to Sonja Smets, Timothy Williamson, Jeremy Seligman, Jeroen Smid and all the participants of the Modality and Modalities workshop (Lund, Mai 2014). Last but not least, to my friend Neil Kennedy, whose paper on Defending the Possibility of Knowledge pushed me to pursue a direction of inquiry we started scrutinizing together, some years ago, in Paris. My research is funded by the Swedish Research Council (VR) through the project “Logical modelling of collective attitudes and their dynamics”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Proietti, C. The Fitch-Church Paradox and First Order Modal Logic. Erkenn 81, 87–104 (2016). https://doi.org/10.1007/s10670-015-9730-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-015-9730-5