Abstract
The concept of being in a position to know is an increasingly popular member of the epistemologist’s toolkit. Some have used it as a basis for an account of propositional justification. Others, following Timothy Williamson, have used it as a vehicle for articulating interesting luminosity and anti-luminosity theses. It is tempting to think that while knowledge itself does not obey any closure principles, being in a position to know does. For example, if one knows both p and ‘If p then q’, but one dies or gets distracted before being able to perform a modus ponens on these items of knowledge and for that reason one does not know q, one is still plausibly in a position to know q. It is also tempting to suppose that, while one does not know all logical truths, one is nevertheless in a position to know every logical truth. Putting these temptations together, we get the view that being in a position to know has a normal modal logic. A recent literature has begun to investigate whether it is a good idea to give in to these twin temptations—in particular the first one. That literature assumes very naturally that one is in a position to know everything one knows and that one is not in a position to know things that one cannot know. It has succeeded in showing that, given the modest closure condition that knowledge is closed under conjunction elimination (or ‘distributes over conjunction’), being a position to know cannot satisfy the so-called K axiom (closure of being in a position to know under modus ponens) of normal modal logics. In this paper, we explore the question of the normality of the logic of being in a position to know in a more far-reaching and systematic way. Assuming that being in a position to know entails the possibility of knowing and that knowing entails being in a position to know, we can demonstrate radical failures of normality without assuming any closure principles at all for knowledge. (However, as we will indicate, we get further problems if we assume that knowledge is closed under conjunction introduction.) Moreover, the failure of normality cannot be laid at the door of the K axiom for knowledge, since the standard principle NEC of necessitation also fails for being in a position to know. After laying out and explaining our results, we briefly survey the coherent options that remain.
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Notes
See Williamson (2000: §4.2) and (e.g.) Smithies (2019: Ch. 7). As far as we can tell, the expression ‘in a position to know’ came to be widely used in general epistemology due to the influence of Williamson’s discussion of luminosity, while earlier it (along with variations on ‘can know’ and ‘is able to know’) was sometimes used as glosses on the epistemic logician’s ‘K’ operator (see note 5).
See Williamson (2000: 282).
An anonymous referee pointed out (in effect) that such examples might well be used to argue that being in a position to know also isn’t closed under any inference rules: It’s natural to describe a sudden death scenario with a sentence along the lines of: ‘Since he died before he could perform the inference, he was not in a position to know its conclusion.’ Fair enough, but such examples don’t strike us as very decisive. The trouble is that there are many senses of ‘in a position to know’, and to do its job the proposed counterexamples would have to work not just for some sense (content) ‘in a position to know’ expresses in some context—that’s easy—but for all senses ‘in a position to know’ is capable of expressing that are plausibly the ones that philosophers who think ‘in a position to know’ has a normal modal logic have in mind. Certainly there is at least a sense of ‘in a position to know’ and a corresponding sense of ‘could have known’ in which its true that someone who is just about to perform a modus ponens but is struck dead before completing the inference was in a position to know and could have known its conclusion, and any philosopher who is dead set on having a notion of ‘in a position to know’ that obeys a normal modal logic can insist that it is that sense (or one of those senses) that he or she has in mind. Cf. Lewis (1976: 150): ‘Whenever the context leaves it open which facts are relevant, it is possible to equivocate about whether I can speak Finnish’—and likewise about whether the unfortunate subject was in a position to know or could have known the conclusion of the modus ponens. And if the philosopher proposing the counterexample fixes the context in a way that leaves no room for equivocation, then the philosopher who claims to have a normal modal logic-obeying notion of ‘in a position to know’ can fix the context another way and say: ‘We’re talking at cross purposes. You are right that, given his untimely death, the subject was not in a position to know the conclusion, but given his general cognitive capacities the subject was in a position to know the conclusion, and when I say ‘in a position to know’, I mean in a position to know given the subject’s general cognitive capacities’’.
By a ‘logical truth’ we mean simply any theorem of the logic characterized by the axioms and rules under consideration; the notion is syntactic, not semantic. For semantic notions of ‘logical truth’, the idea that one is in a position to know every logical truth is no more plausible than the idea that set of ‘logical truths’, in the relevant semantic sense, is axiomatizable (in some cases it is, in others it isn’t). Part of the appeal of the idea that one is in a position to know all logical truths, we take it, comes from the idea that one could in principle prove any of them in a finite number of steps.
For example, Berto and Hawke (2018: 6) use ‘in a position to know’ to gloss the epistemic operators when describing what they call ‘[t]he standard approach to (multi-agent) epistemic logic’. See also Hilpinen’s (1970) use of ‘in a position to know that’ and Williamson’s (1990: 5–10) use of ‘the subject is able to activate knowledge that’. Admittedly some characterizations of being in a position to know are less suggestive of a normal modal logic: notably Williamson’s (2000: 95) remark that what one is in a position to know is ‘open to one’s view, unhidden, even if one does not yet see it’ could easily be taken to suggest that a subject lacking in logical acumen is not able to know some logical consequence of what he knows due to its being hidden from his view. In what follows we hope to offer more decisive objections to the normality assumption. (Thanks to an anonymous referee for discussion here.).
We don’t want to be able to necessitate theorems of the logic of actuality, since some of those truths are contingent. Hence the proviso.
Here is a further axiom that may strike some readers as plausible, which says that being in a position to know is factive:
(TKP)KPφ → φ.
(TKP), however, plays no role in our results. We also lean towards the view that (TKP) has false instances (at least insofar as ‘in a position to know’ being used in an ordinary and not as a term of art), and that its apparent factivity is the result of something like a presupposition: see Sect. 4.
When we say that one ‘knows φ’, where φ is a sentence, we mean that one knows the proposition expressed by φ—that is, we mean what is formalized by Kφ. The sentence α, of course, would have expressed a truth as long as it had its actual character (in the sense of Kaplan 1989), but it would not have expressed the same truth as it actually does.
One should read our ‘only gets one shot’ in a modal rather than temporal way.
In order to smoothen the presentation, we will pretend that K∧ is an operator in the language. K∧ is an operator in the standard loose sense in which a formula like T(x) in one free variable is a predicate. If we had λ-expressions in the language we could avoid this loose talk by proving things about the operator λp.∃qK(q ∧ p), but that would require adding the axioms of the λ-calculus to the logic and applying them in the derivations in the "Appendix"—a significant increase in complexity with little payoff.
The formal result that underwrites these remarks is this:
(KKP) ⊢ KP(φ → (α ∧ φ)) → (KPφ → K(α ∧ φ)).
For the record, we realize that they are not ‘logical truths’ according to the Tarskian permutation-invariance criterion of logicality, but we deny that this has any bearing on the epistemological issues this paper is concerned with: see Goodsell and Yli-Vakkuri (2020) for discussion.
Lines 13–14 in the proof of (3) in the "Appendix". We maintain that the axioms of actuality derive their ultimate justification from the axiom of choice in higher-order logic. See Goodsell and Yli-Vakkuri (2020) for relevant results and Yli-Vakkuri and Hawthorne (forthcoming) for discussion of implications for epistemology.
There are precedents for giving up (KP/◇) in the work of David Chalmers and Declan Smithies.
We have in mind Chalmers’ proposal in Constructing the World regarding ‘ideal a priori warrant’: in some cases ‘there exists an (ideal a priori) warrant for believing p even though the warrant cannot be used to know p.’ Chalmers’ example, too, involves the actuality operator. However, he has in mind a sentence that expresses a proposition necessarily equivalent to the proposition that snow is white in any world (holding its character fixed) in which snow is white:
When p is the proposition expressed by the semantically fragile sentence S discussed at the end of the third excursus (‘Snow is white iff actually snow is white’), one can argue that there exists a proof of p even though it is impossible to use it to prove p. In particular, there exists an abstract proof of S using the logic of ‘actually’. S expresses p in the actual world, so this abstract proof of S is also an abstract proof of p. But if one were to use the proof to prove S, S would express p' rather than p, so one would not prove p (Chalmers 2012: 93).
Smithies’ idea is that ‘being in an epistemic position to know’ is can be a ‘finkish’ disposition to know under certain circumstances—a disposition that would disappear if those circumstances obtained:
As a rough heuristic, you’re in an epistemic position to know that p just in case you would know that p if your doxastic response to your epistemic position were sufficiently rational. More precisely, you would know that p if you were to properly base a doxastically justified belief that p on your propositional justification to believe that p. This is only a heuristic because there are finkish cases in which you cannot respond rationally to your epistemic position without thereby changing it (Smithies 2019: 349–350).
The paradigm cases of finkish dispositions to A in circumstances C discussed in the literature, however, are all ones in which it is metaphysically possible for the bearer of the disposition to A in C: see Lewis (1997) and Yli-Vakkuri (2010).
Note that Smithies’ picture seems to be one on which being in a position to know requires being in fact propositionally justified. This fits poorly with the kind of externalism we favor, according to which one might be disposed to form a safe belief in φ and thus be in a position to know but currently have no evidence for φ and, thus, arguably, no propositional justification.
Another way to develop the thought is to go for something along the following lines: Being in a position to know that φ does not entail possibly knowing φ but rather possibly knowing some proposition suitably similar to φ. But this strategy seems even less promising. It is very much out of step with how the concept of being in a position to know is used in the literature, and we have no good idea of how to develop this thought in a systematic and satisfying way.
Yet another strategy borrows an idea from the literature on Fitch’s paradox: being in a position to know φ entails possibly knowing @φ (see Edgington 1985 and Schlesinger 1985: 103–6). Our concerns about this idea very much mirror Williamson’s concerns about the Edgington/Schlesinger proposal (see Williamson 2000: 292–5). One concern is that, on a fine-grained conception of propositions it is extremely difficult to know @φ in a counterfactual situation, because, while @ is a convenient guise for singling out the actual world in the actual world, there is no convenient guise for singling out the actual world in counterfactual situations. Of course, on a coarse-grained conception of propositions (according to which necessarily equivalent propositions are identical) this problem doesn’t arise, but then @φ, if true, will be identical to every necessary truth—there being only one—and the new principle would be a terrible surrogate for the old one, since it will say that being in a position to know a fact entails knowing the necessary truth. There is a lot more to say about how coarse-grained conceptions of propositions interacts with the logic of @ and knowledge: see Yli-Vakkuri and Hawthorne (forthcoming) for more.
A third way to develop the thought is to say that being in a position to know φ entails the possibility of knowing a certain proposition under some guise one actually associates with the sentence φ. On one version of this proposal the guise could be just that sentence itself, on another it is the Kaplanian character of φ, on a third it is the primary intension of φ, on a fourth it is the language of thought sentence one correlates with φ.
Suppose K¬Kφ and ⊢ φ. By (K/KP), KP¬Kφ. Since ⊢ φ, by (NECKPK), KPKφ.
In a survey article, David Beaver and Bart Geurts begin their (non-alphabetic) list of ‘lexical classes widely agreed to be presupposition triggers’ with:
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factives (Kiparsky and Kiparsky, 1971).
Berlusconi knows that he is signing the end of Berlusconism.
→ Berlusconi is signing the end of Berlusconism (Beaver and Geurtz, 2011).
Here they are citing the article that initiated the study presupposition in linguistics, Paul Kiparsky and Carol Kiparsky’s ‘Fact’, which in turn begins:
The object of this paper is to explore the interrelationship of the English complement system. Our thesis is that the choice of complement type is in large measure predictable from a number of basic semantic factors. Among these we single out for special attention presupposition by the speaker that the complement of the sentence expresses a true proposition (Kiparsky and Kiparsky, 1971: 345, emphasis in the original).
Kiparsky and Kiparsky introduce the phenomenon using ‘Two syntactic paradigms’, two lists of ‘predicates which take sentences as their subjects’, labelled ‘Factive’ and ‘Non-factive’ (ibid.). Lauri Karttunen’s (1971) classic paper, in which he argues, contra Kiparsky and Kiparsky, that the choice of sentential complement-taking verb does not alone determine whether the speaker presupposes the truth of the complement, begins:
There is a class of verbs that are commonly called ‘factive’ verbs. …
There is a general agreement that factive verbs involve presuppositions, though it seems that nobody quite understands what we mean by the term ‘presupposition’ (Karttunen, 1971: 55).
And ‘the hallmark of presuppositions, as well as the most thoroughly studied presuppositional phenomenon, is projection’ (Beaver and Geurtz, 2011)—projection of the kind we saw examples of in the main text.
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Acknowledgements
We would like to thank David Chalmers, Cian Dorr, Kevin Dorst, Julien Dutant, Peter Fritz, Jeremy Goodman, Alex Kaiserman, Amir Kiani, Clayton Littlejohn, Beau Madison Mount, Chunxiao Qu, Alex Roberts, Sven Rosenkrantz, Declan Smithies, Ching Hei Yau, and audiences at the Dianoia Institute of Philosophy at Australian Catholic University, the 2020 Workshop on Philosophical Applications of Modal Logic at the University of Glasgow, and the Oxford Epistemology Reading Group for helpful comments and discussions.
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Appendix: Proofs of the main results
Appendix: Proofs of the main results
We begin with
(3) (KKP) + (NECKP) + (DISTK) ⊢ KPφ ↔ Kφ.
Proof
Below is an abbreviated derivation of KPφ ↔ Kφ in L + (KKP) + (NECKP) + (DISTK), where ‘K’ (boldface) indicates provability using classical logic and (NEC□) from (K□), and
α := ∀p(p ↔ @p)
α* := ¬K(α ∧ φ)
1. α* → □ @α* | (RIG@) |
2. α* → □ (∀p(p ↔ @p) → (α* ↔ @α*)) | (UI), (NEC□) |
3. α* → □((α* ↔ @α*) ↔ α*) | 1, K |
4. α* → □(∀p(p ↔ @p) → α*) | 2, 3, K |
5. α* → □(α → α*) | 4 abbreviated |
6. ¬K(α ∧ φ) → □(α → ¬K(α ∧ φ)), | 5, with α* unabbreviated |
7. ¬K(α ∧ φ) → □((α ∧ φ) → ¬K(α ∧ φ)), | 6, K |
8. □(K(α ∧ φ) → (α ∧ φ)) | (TK), (NEC□) |
9. ¬K(α ∧ φ) → □¬K(α ∧ φ) | 7, 8, K |
10. ¬K(α ∧ φ) → ¬◇K(α ∧ φ) | 9, K |
11. ◇K(α ∧ φ) → K(α ∧ φ) | 10 |
12. ◇K(α ∧ φ) → Kφ | 11, (DISTK) |
13. p ↔ @p | (T□), (T@) (RIG@) |
14. ∀p(p ↔ @p) | 13, (UG) |
15. KP(φ → (α ∧ φ)) | 14, (NECKP) |
16. KPφ → KP(α ∧ φ) | 15, (KKP) |
17. KP(α ∧ φ) → ◇K(α ∧ φ) | (KP/◇) |
18. KPφ → ◇K(α ∧ φ) | 16, 17 |
19. KPφ → Kφ | 12, 18 |
20. Kφ → KPφ | (K/KP) |
21. KPφ ↔ Kφ | 19, 20 |
(1) (KKP) + (NECKP) ⊢ KPφ ↔ K∧φ.
Proof
To show this, replace line 12 in the above with K(α ∧ φ) → ∃pK(p ∧ φ), where p is not free in φ (which is equivalent by contraposition and quantifier duality to an instance of (UI)), and proceed:
11. ◇K(α ∧ φ) → K(α ∧ φ) | |
12. K(α ∧ φ) → ∃pK(p ∧ φ) | UI |
13. p ↔ @p | (T□), (T@) (RIG@) |
14. ∀p(p ↔ @p) | 13, (UG) |
15. α | 14 abbreviated |
16. φ → (α ∧ φ)) | 15 |
17. KP(φ → (α ∧ φ))) | 16, (NECKP) |
18. KPφ → KP(φ ∧ α) | (KKP) |
19. KP(φ ∧ α) → ◇K(φ ∧ α) | (KP/◇) |
20. KPφ → ∃pK(p ∧ φ) | 11, 12, 18, 19 |
21. K(p ∧ φ) → KP(p ∧ φ) | (K/KP) |
22. KP((p ∧ φ) → φ) | (NECKP) |
23. KP((p ∧ φ) → φ) → (KP(p ∧ φ) → KPφ) | (KKP) |
24. KP(p ∧ φ) → KPφ | 22, 23 |
25. K(p ∧ φ) → KPφ | 21, 24 |
26. ¬KPφ → ¬K(p ∧ φ) | 25 |
27. ¬KPφ → ∀p¬K(p ∧ φ) | 26, (UG) |
28. ¬∀p¬K(p ∧ φ) → KPφ | 27 |
29. ∃pK(p ∧ φ) → KPφ | 28, definition of ∃p |
30.∃pK(p ∧ φ) ↔ KPφ | 29, 20 |
(2) The logic of K∧ in L + (KKP) + (NECKP) is normal.
Proof
Immediate from (1).
(4) The logic of K in L + (KKP) + (NECKP) + (DISTK) is normal.
Proof
Immediate from (3).
(5) L + (NECKP) is closed under (NECK∧).
Proof
To show this, let (NECKP) ⊢ φ and continue from line 11 in the proof of (3) as follows.
11. ◇K(α ∧ φ) → K(α ∧ φ) | |
12. p ↔ @p | (T□), (T@) (RIG@) |
13. ∀p(p ↔ @p) | 12, (UG) |
14. φ | By hypothesis |
15. φ ∧ α | 13, 14 |
16. KP(φ ∧ α) | 15, (NECKP) |
17. KP(φ ∧ α) → ◇K(φ ∧ α) | (KP/◇) |
18. K(α ∧ φ) | 11, 16, 17 |
19. K(α ∧ φ) → ∃pK(p ∧ φ) | UI and quantifier duality |
20. ∃pK(p ∧ φ) | 18, 19 |
(6) L + (NECKP) + (DISTK) is closed under (NECK).
Proof
Immediate from the proof of (5).
(7) (KKP) ⊢ KP(φ → ψ) → (KPφ → ◇Kψ).
Proof
Immediate from axiom (KP/◇).
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Yli-Vakkuri, J., Hawthorne, J. Being in a position to know. Philos Stud 179, 1323–1339 (2022). https://doi.org/10.1007/s11098-021-01709-x
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DOI: https://doi.org/10.1007/s11098-021-01709-x