Abstract
Predicates like knowable, believable or evincible each are associated with Fitchlike paradoxes. Given some plausible assumptions, the prima facie reasonable hypotheses that what is true is knowable/believable/evincible entail, respectively, the decidedly unreasonable conclusions that what is true is known/believed/evinced. I argue that all Fitchlike paradoxes admit of a common diagnosis and give a uniform semantics for predicates like knowable that avoids the paradoxes while accounting for the intuitive meaning of these predicates. Moreover, I argue that a semantics of the same shape is to be given to similar predicates like erasable or legible, whose simple analyses likewise face broadly Fitchlike problems. This semantics also highlights and explains the contextsensitive nature of such predicates.
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Notes
In fact, San’s theorem is more general still, dealing not merely in 1bridging principles, but in nbridging principles. The case where \(n=1\) suffices for present purposes and everything said in this paper applies, mutatis mutandis to the general result as well.
A clear case can, for instance, be made for knowledge under deontic modality. If it is permitted to know that the museum was robbed factivity under deontic modality would seem to entail that it is permitted that the museum was robbed, which is a mistake. So there is at least some prima facie reason to doubt the factivity of knowledge under modal operators.
In the literature on the truth norm for belief, a Fitchlike paradox is sometimes discussed as the problem of blindspot propositions, propositions that are not truly believable, e.g. It is raining and I do not believe that it is (Sorensen, 1988). McHugh (2012) uses Fitchlike reasoning about blindspots to argue against any kind of prescriptive truth norm (i.e. norms linking the truth of a proposition to deontic believability) and in favor of evaluative truth norms (i.e. norms whereby one assesses the correctness of a given belief). See Sect. 4 for further discussion.
Yale Weiss (2019) presents a Fitchlike paradox for what is true is believable The argument is a special case of San’s theorem and hence depends on on Axiom D. Weiss recommends to give up this axiom and accept that contradictions can be believed.
Schloeder (2018) and McIntosh (2020) discuss an approximately Fitchlike paradox for the knowledge norm, what is known is assertible The argument takes a different shape than the various corollaries of San’s theorem, so it is tangential here. Adopting the semantics for assertible I develop later will however also avoid this problem with claiming what is known is assertible.
Exceptions might include words like considerable or responsible that aren’t in any obvious sense about what one can apprehend about or do to some thing.
As is usual in the debate surrounding Fitch’s paradox, I will assume that it is known that p means that at some time, some agent knows p (see, e.g. (Fuhrmann, 2014)).
The account also faces a technical objection related to actuality. On the standard semantics for the operator A, it is the case that when Ap, then also \(\Box Ap\). So if all epistemically accessible worlds are possible worlds, then Ap entails KAp and we again get the paradoxical conclusion that all actual truths are known to be actually true. By adjusting the semantics of A one can resolve this formal issue (Rabinowicz & Segerberg, 1994) but this does not answer the conceptual questions of how nonactual knowers know what is actual (Williamson, 1987).
Cases like PRESSURE complicate the matter, since in such cases there is no action that, when pursued, imparts the knowledge that p. To appreciate Heylen’s point, we can restrict our attention to cases that are not like PRESSURE, i.e. to p that can be learned by pursuing an action.
It follows that in worlds that are not the outcome of performing a, all sentences of the form \([a]^ p\) are vacuously true and all those of the form \(\langle a\rangle ^\) are trivially false. This may be counterintuitive (as vacuous truths often are), but need not concern us for anything that is to follow. In the principles formulated here, inverse operators for an action a will only be evaluated at worlds that are the result of performing a.
I thank a referee for pressing me on this point.
I thank a referee for discussion on this point.
It is however a little more than the knowability principle since there may be more sets of worlds than there are propositions. It coincides with the knowability principle only on finite frames.
One may also consider the possibility of there being infinitely many actions that, when successful, impart the knowledge that p. On the formal side, it is straightforward to accommodate infinite choice. On the conceptual side, a potential issue is that one cannot have de re knowledge of all these actions. Since the problem of de re knowledge already appears in the finite case, I will put infinite choice to the side.
The intuition seems clear enough, but it takes some additional work on the formal side to bring it out. Under the received definitions of choice actions and inverses, \(w\Vdash [a\cup b]^p\) if and only if for all v with \(vR^aw\) or \(vR^bw\) it is the case that \(v\Vdash p\). This does not capture the intuition that \([a\cup b]^\) means that before any of a and b, it was the case that p. This is because there could be a world v with \(vR^aw\) that is itself the outcome of performing b in another world. So \(w\models [a\cup b]^ p\) does not mean that p is the case before any of a and b were performed because p might be witnessed by a world that is the outcome of b.
It is, hence, more apt to define inverses compositionally, leaving the received definition intact for atomic actions, and letting \([a\cup b]^p\) decompose into \([a]^((p \wedge [b]^\bot )\vee [b]^p) \wedge [b]^((p\wedge [a]^\bot )\vee [a]^p)\). That is, the claim that before any of a and b it was the case that p decomposes into the claim that: before a it was either the case that p and b has not happened, or that before b it was the case that p; and before b it was either the case that p and a has not happened, or that before a it was the case that p. This captures the intuition that p should be witnessed by a world that is the outcome of neither a nor b.
Again, the dynamic view is advantageous. The counterfactual view can help itself to the algebra of actions as well. But to spell out such compositional definitions in the language of counterfactuals would again require additional formal bulk to ensure that a counterfactual with a particular description in its antecedent accesses the right worlds.
As beliefs are not always formed by conscious action, the set of beliefforming actions should be thought of as including any action whose outcome may be the formation of a belief, e.g. actions related to perception, reflection or reasoning, among others.
McHugh (2012) calls norms stating which beliefs are obligatory to acquire prescriptive norms and likewise contrasts them with evaluative norms.
As with belief, it may not be immediately obvious what adoringforming, desireforming, detestforming or forgetting actions are. Presumably, those include all actions that can result in one being in the corresponding mental state. An adoringforming action may be simple perception of something whose appearance can result in one forming adoration towards it.
Possibly, predicates like sustainable or livable can be included in the present account by associating with them a certain set of temporally extended actions so that, e.g., a process being sustainable means that there is an extended action such that after any part of that action, the process is again as it was before the action, i.e. it is sustained.
Despite the similarities, it should be stressed that ability predicates are different from dispositional predicates. An ability predicate is used to describe something in virtue of what can be done with it; a dispositional predicate describes an intrinsic property. A salient question is whether, like the counterfactual analysis of knowability was helped by going dynamic, the counterfactual analysis of dispositions can be improved by similarly going dynamic. I plan to explore this elsewhere.
A third challenge to the counterfactual analysis of dispositions is mimicking. A paper cup is not fragile, but if someone is nearby who will immediately step on the cup if it is dropped, then the counterfactual were it dropped, it would break is true. Here the problem is not that a fragile object does not satisfy the counterfactual, but that a nonfragile object does satisfy it. It is unclear how a similar challenge could be levelled against the dynamic analysis of ability predicates. An object is properly called breakable if there is an (easy) action one could do that results in the object being broken. It is immaterial whether the action does so in a somewhat roundabout way.
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Acknowledgements
I am grateful to Jan Heylen, Lorenz Demey and the audience of the workshop on the concept and scope of knowability, held at KU Leuven in 2023, for comments on earlier versions of this material.
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Appendix
Appendix
Here is why why the dynamic definitions of knowability, believability etc are immune to San’s theorem.
Theorem 6.1
Let \(\Box\) be a normal modal satisfying axiom D (i.e. \(\Box p\rightarrow \Diamond p\)) and any 1bridging principle. Let \(\blacklozenge\) be any normal modal. If \(p\rightarrow \blacklozenge \Box p\) is valid, then \(p\leftrightarrow \Box p\) is valid.
The central step in San’s proof is to show that with the assumptions of the theorem, one can show that \(\vdash \Box p\rightarrow p\). This plus the assumptions of the theorem are as required for Fitch’s original proof of his paradox. Since Fitch’s original derivation is blocked by the dynamic conception, this does not work here.
However, it would already be a bad result if a principle like what is true is \(\Box\)able were to entail that if \(\Box\) satisfies any 1bridging principle, it is factive (e.g. this would appear to rule out such principles where \(\Box\) is belief or evidence). So it is useful to see that already the central step fails when ‘\(\Box\)able’ is understood dynamically. Here is how this step goes.

1.
It is easy to show that Axiom D and the 1bridging principle jointly entail that there is a natural number j such that \(\Box ^j p \rightarrow \Diamond ^{j+1}p\) is valid, where \(\Box ^j\) is a sequence of j many \(\Box\)’es and \(\Diamond ^{j+1}\) is a sequence of \(j+1\) many \(\Diamond\)’s. This entails that \(\Box (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\) is a contradiction. To see this, note that \(\Box (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\) entails \(\Box ^jp\wedge \Box \lnot \Diamond ^jp\), which entails \(\Box ^jp\wedge \Box ^{j+1}\lnot p\). This contradicts \(\Box ^jp\rightarrow \Diamond ^{j+1}p\). Because \(\blacklozenge\) is a normal modal, it follows that \(\lnot \blacklozenge \Box (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\).

2.
Now, assume towards a reductio that \(\Box ^{j1}p\wedge \lnot \Diamond ^jp\). This entails that \(\blacklozenge \Box (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\) by the assumption that \(p\rightarrow \blacklozenge \Box p\) is valid. Contradiction to the result of step 1. Thus, \(\Box ^{j1}p\rightarrow \Diamond ^jp\) by reductio. Note that we derived \(\Box ^{j1}p\rightarrow \Diamond ^jp\) from \(\Box ^{j}p\rightarrow \Diamond ^{j+1}p\). Iterate this step to derive \(\vdash p\rightarrow \Diamond p\). It follows that \(\vdash \Box p\rightarrow p\), as \(\Box\) is normal.
What happens when one attempts to apply the same reasoning to the dynamic conception? The principle what is true is \(\Box\)able is now not expressed as \(p\rightarrow \blacklozenge \Box p\), but as \(p\rightarrow \exists a \in \mathcal {A}.\langle a\rangle \Box [a]^ p\). Note that in step 1, we can still follow the argument to conclude that \(\Box (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\) is a contradiction. And since ‘\(\exists a \in \mathcal {A}.\langle a\rangle\)’ is a normal possibility modal, it follows that \(\lnot \exists a \in \mathcal {A}.\langle a\rangle \Box (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\). But, now, in step 2, the assumption \(\Box ^{j1}p\wedge \lnot \Diamond ^jp\) entails \(\exists a \in \mathcal {A}.\langle a\rangle \Box [a]^ (\Box ^{j1}p\wedge \lnot \Diamond ^jp)\), which does not contradict the result of the first step, due to the intervening modal operator \([a]^\).
Thus, San’s proof strategy cannot at all be applied to the dynamic conception of ability predicates, for the same reason that Dynamic Knowability is immune to the original Fitch paradox. Roughly put, both results follow from the observation that a particular claim is contradictory, so it is contradictory under any normal modal while a ‘what is true is \(\Box\)able’ principle appears to entail that just this claim occurs under a normal modal. In the dynamic conception, this is not so, as what follows from ‘what is true is \(\Box\)able’ is that another claim, one featuring the \([a]^\) modality, occurs under the normal modal operator.
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Schloeder, J.J. Ability predicates, or there and back again. Philos Stud 181, 1877–1902 (2024). https://doi.org/10.1007/s11098024021788
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DOI: https://doi.org/10.1007/s11098024021788