Abstract
Marton (2019) argues that that it follows from the standard antirealist theory of truth, which states that truth and possible knowledge are equivalent, that knowing possibilities is equivalent to the possibility of knowing, whereas these notions should be distinct. Moreover, he argues that the usual strategies of dealing with the Church–Fitch paradox of knowability are either not able to deal with his modal-epistemic collapse result or they only do so at a high price. Against this, I argue that Marton’s paper does not present any seriously novel challenge to anti-realism not already found in the Church–Fitch result. Furthermore, Edgington (1985) reformulated antirealist theory of truth can deal with his modal-epistemic collapse argument at no cost.
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Notes
Marton calls \(\hbox {RM}_{\Diamond }\) ‘\(\Diamond \)-Elim’.
I am grateful to an anonymous reviewer who showed this.
As Mackie (1980) pointed out, it suffices to assume a special case of \(\hbox {T}_{K}\), namely: \(K\lnot K\phi \rightarrow \lnot K\phi \). Jago (2010) has shown that one can still derive epistemic collapse even if one replaces (1) with the following weaker principle: \(K\left( \phi \wedge \psi \right) \rightarrow \Diamond \left( K\phi \wedge K\psi \right) \).
San (2020) has proved a ‘general collapse’ theorem: a logic with (A-R), \(\hbox {D}_{K}\) and some ‘n-level bridging principle’ (e.g., \(\hbox {T}_{K}\) is a one-level bridging principle) with \(\hbox {RN}_{\Box }\) and \(\hbox {RM}_{K}\) as rules entails a ‘nth degree modal collapse’.
If the K-operator means that some human at some time knows, then Marton has to cite scenarios in which no individual human being at some point in time is able to foresee that in the future no human beings are alive anymore.
Marton does not use \(\hbox {RM}_{K}\) explicitly. He seems to tacitly appeal to \(\hbox {K}_{K}\) (i.e., \(K\left( \phi \rightarrow \psi \right) \rightarrow \left( K\phi \rightarrow K\psi \right) \)) or perhaps on a weakening of \(\hbox {K}_{K}\) and \(\hbox {RM}_{K}\), i.e., \(\vdash \phi \rightarrow \psi \Rightarrow \,\vdash K\left( \phi \rightarrow \psi \right) \rightarrow \left( K\phi \rightarrow K\psi \right) \).
See Horsten (1994) for the following inference rule for a system of modal-epistemic arithmetic: for every provable formula it is possible that there is a mathematician who has a proof of it.
Or weaker versions of those principles (cf. footnote 3).
I mention only the approaches explicitly discussed by Marton. There are other approaches, e.g., the dynamic-epistemic approach of van Benthem (2004) and Balbiani et al. (2008), which is a combination of the restriction and the reformulation strategy. Note that on this approach knowability (understood as known after an announcement) is not factive, which Marton presumably reckons to be pricey. See footnote 14 for another example of the reformulation strategy.
Alternatively, replace (VA) and \(4_{\Diamond }\) with \(\Diamond A\phi \rightarrow A\phi \), \(\hbox {T}_{\Diamond }\) and \(\hbox {RM}_{K}\).
The simplified versions can be found in: Heylen (2016, 2020a). One of the simplification is the following: Rabinowicz and Segerberg (1994) define V as a function from sentence letters and pair of worlds to truth values, whereas here V is defined as a function from sentence letters and worlds to truth values. The function V could instead have been defined in the same way as Rabinowicz and Segerberg (1994) do and then it could have been stipulated that V should agree on all pairs of worlds, if the first elements of the pairs of worlds are the same. In other words, the models used here are a special case of the models described by Rabinowicz and Segerberg (1994).
I am grateful to an anonymous reviewer for a suggestion that turned my original model in one based on a frame on which \(\mathbf{S5} _{K}\) is valid.
Williamson (1987, 2000) questions the existence of non-trivial non-actual knowledge about the actual world, while Edgington (2010) thinks that there is. Heylen (2020a) objects to Edgington’s theory, because it comes with possible omniscience: there is an accessible state at which all truths are known. Schlöder (2019) reformulates Edgington’s theory to address these concerns. Heylen (2020b) argues that Schlöder’s version of the knowability thesis overgenerates knowledge.
Another reformulation strategy has been pursued by Fuhrmann (2014), who uses the notion of ‘potential knowledge’. This notion is expressed with the help of a primitive, unanalyzed operator, \(\langle K \rangle \). With no modal (\(\Diamond \)) and no epistemic (K) operators as syntactical components of this operator, a modal-epistemic collapse result with that operator is not possible. One can also transform the model used in the proof of Theorem 5 into a model for potential knowledge, showing that Fuhrmann’s theory is also free of modal-epistemic collapse. In the light of footnote 13, it is noteworthy that potential knowledge is an ‘intra-world affair’.
I am leaving out the clauses for the quantifiers.
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Acknowledgements
I would like to thank two anonymous reviewers for their very helpful feedback—see especially footnotes 2 and 12 for two substantial contributions. Furthermore, I would like to thank Felipe Morales Carbonell, Harmen Ghijsen and Lars Arthur Tump for their comments on earlier versions of this paper. Finally, I would like to thank the audience of the CLPS Seminar (Leuven, 2 October 2020) to which I presented this paper.
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Heylen, J. Anti-Realism and Modal-Epistemic Collapse: Reply to Marton. Erkenn 88, 397–408 (2023). https://doi.org/10.1007/s10670-020-00353-x
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DOI: https://doi.org/10.1007/s10670-020-00353-x