Abstract
The thesis that every truth is knowable is usually glossed by decomposing knowability into possibility and knowledge. Under elementary assumptions about possibility and knowledge, considered as modal operators, the thesis collapses the distinction between truth and knowledge (as shown by the so-called Fitch-argument). We show that there is a more plausible interpretation of knowability—one that does not decompose the notion in the usual way—to which the Fitch-argument does not apply. We call this the potential knowledge-interpretation of knowability. We compare our interpretation with the rephrasal of knowability proposed by Edgington and Rabinowicz and Segerberg, inserting an actuality-operator. This proposal shares some key features with ours but suffers from requiring specific transworld-knowledge. We observe that potential knowledge involves no transworld-knowledge. We describe the logic of potential knowledge by providing models for interpreting the new operator. Finally we show that the knowability thesis can be added to elementary conditions on potential knowledge without collapsing modal distinctions.
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Notes
From the chapter “A Dialogue”, James (2002, p. 295).
In a technical sense, our approach may also be seen as decomposing. But we do not decompose into a quantification over worlds and the assessment of a knowledge state therein.
Truth in all models at the 0-point is what is called weak validity in Rabinowicz and Segerberg (1994). Strong validity would be truth in all models at all points. Both notions of validity have their use and there is little point in discussing which is the right one in the present context.
In these examples we understand possibility in the widest, i.e. logically weakest sense, sometimes called metaphysical possibility. This is the sense that is at issue in discussing the merits of the PossK-version of the knowability principle vis-à-vis the Fitch-argument. Fara (2010) argues to essentially the same conclusion as to the relation between knowability and the possibility that one knows. But he does so on the basis of a stronger intermediate step: “Sometimes one might have the capacity to do so-and-so even though it is not possible that one does so-and-so in any sense of possible obtained by restriction of metaphysical possibility” (p. 68). Although this thesis is certainly interesting, it goes well beyond what is required for decoupling the knowability thesis Ver from its interpretation PossK in terms of possible knowledge.
Based on the observation that Ver does not entail PossK, Fara points out in the final part of Fara (2010) that knowability may be interpreted as a capacity to know without inviting the Fitch-argument nor the transworld-knowledge challenge discussed in the last section. The general strategy is thus the same as in the present paper. Fara does not wish to commit himself on the basis and nature of that capacity. In particular, there is no attempt at a semantical analysis of knowability claims under the capacity-reading. Sometimes such neutrality can be prudent. But in this case it remains unclear in which sense capacity can be construed as a sentential operator as suggested in the final pages of Fara (2010). Moreover, we do not know enough about the aimed at notion of a capacity-to-know to confidently judge that it is immune to fitching and to transworld-embarrassments. What seems plausible enough though, is the thesis that some way of cashing out that notion is safe in this sense. One such way is explained here, though it is one that—contrary to Fara’s suggestion—does not decompose knowability into a knowledge- and a capacity-operator.
I.e. where I believed that I finished the event in less than 8 h because I shaped up to perform the feat or because I was taken in by some illusion to the effect.
The minimal principles are those one finds in the logic KD4, a common—though not uncontroversial—starter for doxastic logics. It extends the set of tautologies by the 4-schema \(\Box A \rightarrow \Box \Box A\), the D-schema \(\Box A \rightarrow \diamond A\), and closes under Modus Ponens and Necessitation, \(A / \Box A\).
Moore-sentences are of the form “\(P\) and I do not believe that \(P\).” They may well be true but cannot be properly asserted. A standard way of accounting for this, at first sight paradoxical characteristic is by observing that assertion must be taken to express belief, whence the assertion of a Moore-sentence expresses contradictory content. Note that we may substitute knowledge for belief in a Moore-sentence, thus obtaining a Fitch-sentence. Just like Moore-sentences, asserting a Fitch-sentence sounds odd, a fact that can be explained by the thesis that knowledge is the norm of assertion: someone who asserts that \(P\) signals that he knows that \(P\). On this explanation, the assertion of a Fitch-sentences would likewise have contradictory content. See Moore (1944), where Moore’s paradox first appeared in print and Williamson (2000, ch. 11) for a statement of the knowledge account of assertion.
The caveat is necessary because the agent may come to have contradictory beliefs—perhaps only for a short moment—by a mechanism beyond doxastic control, such as perception. The picture we draw here includes the option that the evidence may be rejected (choice (a) in the menu). This option is not considered in classical belief revision theory (“AGM”). But AGM may be extended to cover the case, as in the theory of non-prioritised belief revision; see Hansson (1999).
Essentially the same idea gave rise to the notion of a hyperproposition, as introduced in Fuhrmann (1999). Hyperpropositions represent belief states together with their fallback positions in response to recalcitrant evidence. The fallbacks are determined by the choice that needs to be exercised in retracting beliefs.
The credibility of the picture drawn here depends on resolving in one way or another an important issue concerning belief revision. We said that a belief state is a potential state for the agent (“within reach”), if it can be reached from the agent’s state by a series of legitimate belief changes. Thus rational change functions have not only to determine a successor set of of beliefs but also a successor choice function. Only thus shall we get a belief state that can again respond to evidence in a non-trivial way—Rott (2001) calls this “the problem of categorical matching” between the input and the output of a belief change function. The problem of finding rationality constraints for generating choice function that rule changes after changes is the topic of theories of iterated belief change; see Darwiche and Pearl (1997) and also Spohn (2012).
Our use of the notion of a potential belief state owes much to the work of Isaac Levi; see in particular Levi (1991). Levi, however, requires strong structural constraints on the family of potential belief states generated by a given state. (Such families should, according to Levi, form complete atomic Boolean algebras, so as to serve as adequate tools in inquiry.) Levi’s constraints are well worth considering, given the ways in which he makes use of potential belief states. Our purposes here are more modest and thus we can afford to remain non-committal with respect to conditions that turn no wheel in the present context.
Lest not forget that the conditions on knowledge, assumed in the argument, have a normative character too.
For more on modal modifiers see the companion paper (Fuhrmann forthcoming).
This way of generating legitimate transitions is not represented in our formal models, since the language to be modelled is not expressive enough to refer to such apparatus.
In fact, as pointed out above, a little less suffices for the Fitch-argument.
Correspondence is here meant in the sense that a schema is valid in a class of structures just in case the structures satisfy the condition in question. Thus, as to RNp, clearly the condition validates the rule. For the converse (the necessity of the condition for the rule) assume that if \([\![A]\!]=W\), then \(\exists S': S\le S'\) and \((\forall a)\ S'_a\models \langle \mathbf{K \rangle }A\). Let \(A = \top \). Then \(\exists S': S\le S'\); so the condition holds. — Likewise, the validity of Mp follows immediately from the condition. For the converse consider a model with relations \(S, S'\) and \(S''\) such that \(S\le S'\) and \(S\le S''\) and choose atoms \(P\) and \(Q\) such that \(S'_a \subseteq [\![P]\!]\) and \(S''_a \subseteq [\![Q]\!]\). Then \(a\models \langle \mathbf{K \rangle }P\) and \(a\models \langle \mathbf{K \rangle }Q\). Now the condition fails if and only if there is no \(R\) with \(S\le R\) and \(R_a\subseteq [\![P \wedge Q]\!]\). But this is just the condition under which \(a\not \models \langle \mathbf{K \rangle }(P\wedge Q)\). So, if the condition fails, so does the schema.
The further correspondences claimed below follow similarly. Here is, to illustrate once more, the argument for the necessity of Knowability (see the next footnote) for PotK: Consider a model with (1) \(a\models P\), for some \(P\), and (2) for all \(S'\), if \(S\le S'\), then \(S'_a\not \models P\). So the condition fails. But from (2) we have \(a\not \models \langle \mathbf{K \rangle }P\), whence, given (1), \(a\not \models P\rightarrow \langle \mathbf{K \rangle }P\).
The condition of perfectibility is stronger than what is needed for the validity of PotK. We adopt it here for the sake of simplicity. For a sufficient and necessary condition we need to switch the order of the quantifiers: Instead of there being a single state in which every truth is known, it suffices (and is also necessary for PotK) that for every truth there is a state in which it is known, i.e.
$$ \begin{aligned} a\in X \Rightarrow \exists S': S\le S' \,{ \& }\, S'_a \subseteq X. \end{aligned}$$(Knowability)
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Acknowledgments
I am grateful to those who have commented on various occasions on earlier versions of this paper, in particular, Johan van Benthem, Ali Esmi, Dominik Kauß, Manfred Kupffer, Hannes Leitgeb, Isaac Levi, Oliver Petersen, Hans Rott, Sonja Smets, and Roy Sorensen. Two anonymous referees have commented in uncommon detail and have given me excellent advice which, I hope, I have succeeded in turning into an improved version of the paper.
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Fuhrmann, A. Knowability as potential knowledge. Synthese 191, 1627–1648 (2014). https://doi.org/10.1007/s11229-013-0340-y
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DOI: https://doi.org/10.1007/s11229-013-0340-y