Abstract
The paper discusses some ways in which the phenomenon of borderline cases may be thought to bear on the traditional philosophical idea that certain domains of facts are fully open to our view. The discussion focusses on a very influential argument (due to Tim Williamson) to the effect that, roughly, no such domains of luminous facts exist. Many commentators have felt that the vagueness unavoidably inherent in the description of the facts that are best candidates for being luminous plays an illicit role in the argument. The paper investigates this issue by centring around the idea that vagueness brings with itself borderline cases, and that these in turn generate absence of a fact of the matter and hence epistemically benign lack of knowledge. It is argued that, given the possibility of absence of a fact of the matter, the idea of luminosity should be reformulated using the notion of determinacy, and that the resulting reformulation is not immediately subject to the original anti-luminosity argument. However, it is shown that the specific understanding of determinacy required by this strategy validates a new argument against the reformulated version of luminosity. Moreover, reflection on the connection between mistake and absence of a fact of the matter offers another argument against such version, with the surprising upshot that, granting the soundness of the original anti-luminosity argument, not even the determinacy of a certain fact would guarantee its knowability.
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Notes
The question of what confidence requirements (if any) there are on knowledge is arguably of great momentum for some of the issues touched on by this paper (see Zardini (2012c) for an extensive discussion). However, given that the question is peripheral to the specific arguments developed in the paper, sometimes we will, for simplicity, acquiesce in the traditional view that what is required for knowledge at the level of confidence is belief.
This first conjunct of the definition is needed in order to guarantee the factivity of being in a position to know (i.e. if one is in a position to know that P, then P), which it is not clear is already enforced by the second conjunct. That depends on how the notion of a ground’s being already available is further explicated, an issue on which I shall remain silent in this paper. To give a flavour of at least one worry here, suppose that one knows that beliefs whose content is expressed by a sentence of the form ‘I am entertaining the thought that P’ are self-verifying and typically knowledgeable. The notion of a ground’s being already available may well be explicated in such a way that, although one is not actually entertaining the thought that snow is white, it is true that, in order for one to come to know that one is entertaining the thought that snow is white, one only needs to come to believe that one is entertaining the thought that snow is white on grounds one has already available. Under these assumptions, leaving out the first conjunct of the definition would result in a notion of being in a position to know such that, although one is not actually entertaining the thought that snow is white, one may nevertheless be in a position to know that one is entertaining the thought that snow is white.
Throughout, I use square brackets to disambiguate constituent structure in English.
What is the relation between being in a position to be known and being knowable? The modality involved in the notion of being knowable is of course in just as much need of explication and investigation as the modality involved in the notion of being in a position to be known. However, this much can be said: on at least one usual philosophical understanding of the notion (prominent for example in the tradition of semantic anti-realism, see e.g. Tennant (2000)), being knowable amounts in effect to some fairly strict possibility of being in a position to be known—a possibility that is at least no weaker than a nomological one relative to the state of the world at which the claim of knowability is to be evaluated. Hence, being in a position to be known implies being knowable but not vice versa. For example, I might not know whether there are dogs, the only dog might be next room, in excellent perceptual conditions for all and only those entering the room, and there might be no way of coming to know that there are dogs other than by perceiving the only existing dog. In such a situation, although I am not (yet) in a position to know that there are dogs, it is knowable for me that there are dogs, since I can very easily enter next room and perceive the dog, thereby coming to be in a position to know that there are dogs (thus, if I also form the belief that there are dogs on those grounds, I will know that there are dogs). Having drawn this distinction, I will move freely from being knowable to being in a position to be known when that is warranted by the context.
I don’t claim a perfect match between the visual metaphor of being fully open to our view and its proposed epistemic cash value in terms of being in a position to be known. An interesting case of potential divergence is familiar from the epistemological literature: in barn-façade country (see Goldman (1976)), the fact that that structure is a barn is intuitively fully open to Henry’s view, although, at least according to many epistemologists, Henry is not in a position to know that that structure is a barn. This being noted, since I don’t think that the potential divergence makes any difference to the specific arguments I develop in this paper, I’ll henceforth assume that there is in this sense no gap between a fact’s being fully open to our view and its being fully open to our knowledge. This is not to say that the potential divergence is in no way relevant to the general topic I’ll be addressing: since the argument against the traditional philosophical idea which is the starting point of my discussion basically consists in the mere claim that certain beliefs are unreliable, one can reasonably wonder—on the strength of apparently similar examples such as Henry’s—how that is supposed to show that every domain of facts is not even fully open to our view (over and above its merely not being fully open to our knowledge).
For simplicity, I’m following very closely the set-up of Williamson (2000), pp. 93–113. Barring problems of apparent hyperintensionality (which we’ll set aside here) and assuming (as we’ll do) that there is a set of all cases and that all conditions are bivalent at all cases (i.e. at every case α, either true at α or false at α), a condition can be modelled by a set of cases (that is, most naturally, by the set of all and only those cases at which the condition is true). Arguably, the possibly most distorting feature of such a set-up is that it models the objects of knowledge, belief and confidence with entities whose truth and falsity are relative to subjects and times, while there are serious, if not necessarily compelling, reasons to think that the truth and falsity of what one knows, believes and is confident about are not relative to subjects or times (see e.g. Stalnaker (1981), pp. 145–147 and Richard (1981) respectively). Nevertheless, for uniformity, I’m henceforth going to assume conditions to be the objects of knowledge, belief and confidence, adapting traditional claims and arguments to this assumption. See fn 11 for a brief indication of how to recast the discussion to follow if this rather peculiar feature of the set-up is abandoned.
Throughout, I’ll use ‘\(\langle \varphi \rangle\)’ as a singular term referring to the condition expressed by \(\varphi. \)
Only ‘seems’. A central point of this paper is that, on many theories of vagueness, a condition might be concerned with a domain of facts fully open to our view without being luminous in the sense of Definition 3 (see Sect. 3 and fn 32).
Sometimes (as here) I will implicitly assume that the range of ‘n’ and of its like is restricted to the set {x: 1 ≤ x ≤ 1, 000, 000, 000} if that’s clear enough from the context.
In my view, it is precisely this inference in the derivation of (KMAR) that is fallacious and irremediably so, since, because of considerations I cannot go into here, one’s confidence in α n that one feels cold can arguably be reliable even if one’s confidence in α n+1 that one feels cold is mistaken. I develop and defend this view at length in Zardini (2012c).
Indeed, in the very same condition (i.e. \(\langle\)One feels cold\(\rangle\)). In the text, I use a more general formulation since that is the formulation one would need to use if one wanted to run the AL-argument under the alternative assumption that the objects of knowledge, belief and confidence are insensitive to (subjects and) times, so that they are either (objectively and) eternally true or (objectively and) eternally false (see fn 6). On this assumption, the relevant object of a subject s’s confidence in α n is, roughly, the (subject- and) time-insensitive proposition that s feels cold at t n , while the relevant object of s’s confidence in α n+1 is, roughly, the different and yet very similar (subject- and) time-insensitive proposition that s feels cold at t n+1.
In this paper, I will neither contest the general principle nor the interpretation offered in the text of what the principle requires in our framework. I think both issues are worthy of further investigation (Zardini (2012c) tries to make some progress on the latter issue).
The reasoning is valid in most logics of some philosophical interest. However, it is not valid in the family of “tolerant logics” developed by Zardini (2008a), (2008b), pp. 93–173, (2012b), logics which have been designed exactly for accommodating a view on vagueness which accepts the relevant sorites premises as correctly expressing what the vagueness of an expression consists in. On such a view, (SOR) is true (indeed, it is true for reasons independent of the AL-argument, as it correctly expresses what the vagueness of ‘feels cold’ consists in), but, because of the shift to a tolerant logic, it is now harmless and consistent with (POSNEG). As a consequence, on this view, the AL-argument has no bite, even if, up to (SOR), it may well be sound. To give the AL-argument its best chance, I’ll henceforth ignore the view, and assume with most theorists that (SOR) does contradict (POSNEG).
As has just been made explicit in the text, this specific result depends on the assumption that the negation of a condition concerned with a domain of facts fully open to our view is itself concerned with a domain of facts fully open to our view. I find the assumption very plausible for many conditions that are good candidates for being concerned with a domain of facts fully open to our view. In any event, even dropping that assumption we can still conclude that [it is not the case that C although there is no fact of the matter as to whether C] (and so although it is not a fact of the matter that it is not the case that C). In spite of its no longer being a truth-functional (or quantificational) contradiction, such a conclusion is typically considered inconsistent by NFMU-theorists (and absence-of-a-fact-of-the-matter theorists more generally), and rightly so in my view (notice that the conclusion is indeed inconsistent given the rule of inference (D-INTRO) introduced and defended in Sect. 5). Hence, absence of a fact of the matter is still inconsistent with the luminosity of the relevant condition in the sense of definition 3, whether or not the condition’s negation is also concerned with a domain of facts fully open to our view. Thanks to Paul Égré for raising this issue.
Appealing as such a stance with respect to absence of a fact of the matter might seem, it must be mentioned that it sits very uncomfortably with classical patterns of reasoning. For take again a condition \(\langle C\rangle\) such that there is no fact of the matter as to whether C. Then, since absence of a fact of the matter is incompatible with knowledge, it is neither knowable that C nor knowable that it is not the case that C. However, by the law of excluded middle, either C or it is not the case that C, wherefore, reasoning by cases with side premises, either C and it is unknowable that C or [it is not the case [that C] and it is unknowable that it is not the case that C]. Few would staunchly hold that such a result is unproblematically consistent with the spirit of the claim that there is no fact concerning \(\langle C\rangle\) about which we are ignorant (see Field (2003), pp. 458–459 for a forceful presentation of this problem; Field’s own conclusion is that we should give up the law of excluded middle for the relevant claims).
Notice that (FACT) makes sure that (DLUM) respects the factivity of being in a position to know (see fn 2).
In a classical setting, (DLUM) has the following consequence that may be worth mentioning. By familiar classical reasoning, there is a last n such that in α n one determinately feels cold, and so, by (DMAR), in α n+1 [one still feels cold even if one does no longer determinately feel cold]. Thus, there is a case in which [one feels cold but one does not determinately feel cold]. That is certainly a problematic claim, but it is only very marginally more problematic than, for example, the claim that either there is case in which [one feels cold but it is not determinately the case that one feels cold] or there is a case in which [one does not feel cold but it is not determinately the case that one does not feel cold]. And, reasoning by cases with side premises, the latter claim follows independently of (DLUM) merely from the law of excluded middle together with the side premise that, in some case, it is neither determinately the case that one feels cold nor determinately the case that one does not feel cold. In fact, by familiar classical reasoning, the former claim itself follows independently of (DLUM) once it is further assumed (extremely plausibly on an NFMU-theory of vagueness) that the last case in which one feels cold is a case in which it is neither determinately the case that one feels cold nor determinately the case that one does not feel cold. Whatever problem emerges with these and similar claims should thus be imputed to classical logic rather than to (DLUM) (see also fn 15). Thanks to Aidan McGlynn for discussions of this issue.
Having made this qualification, in the following I will ignore it, given that the latter kind of theory is in this dialectic considerably less interesting in view of its inability to provide an independent reason for preferring (DLUM) over (LUM).
Here and in the following, I’m assuming that the situation described by the AL-argument is such that borderline cases are the only possible source of absence of a fact of the matter.
I mean talk of assertoric content not to be restricted to sentences expressing ordinary, subject-insensitive propositions (see Definition 2 and fn 6). In particular, I’m going to presuppose that it applies to (open) sentences expressing less ordinary, subject-sensitive conditions.
Here and in the following, ‘fact’ is used to cover both actually obtaining and merely possibly obtaining facts. Moreover, first-order talk of actually obtaining and merely possibly obtaining facts is simply adopted for convenience and could in principle be eliminated in favour of higher-order locutions. What is not so eliminable is some version or other of talk of truth-making. This is not the place to elaborate on this issue: an intuitive understanding of truth-making will suffice for our purposes. Finally, as for truth itself, I’m assuming a notion of truth that validates the relevant instances of the schema ‘ ‘P’ is true iff P’.
In the following, I will use ‘strong’ and its relatives to denote the ordering of strength of assertoric contents. It is crucial to see that equivalence in this ordering between the assertoric contents of \(\varphi\) and ψ does not guarantee that \(\varphi\) is unrestrictedly inter-substitutable with ψ even outside of (direct- or indirect-) quotation environments. For example, supposing that the reference of ‘Antonius’ is fixed as the inventor of the telephone, the assertoric content of ‘Antonius liked spaghetti’ is at least as strong and at least as weak as (and so equivalent with) the assertoric content of ‘The inventor of the telephone liked spaghetti’: in the intended sense, whatever fact whose obtaining would make ‘Antonius liked spaghetti’ true is also a fact whose obtaining would make ‘The inventor of the telephone liked spaghetti’ true and vice versa. (Indeed, given that any ordering ≤ requires the anti-symmetry property that, if x ≤ y and y ≤ x, then x = y, equivalence in the ordering of strength of assertoric contents implies identity.) However, while ‘It is possibly the case that [Antonius liked spaghetti and it is not the case that the inventor of the telephone liked spaghetti]’ is true, the result of substituting ‘Antonius liked spaghetti’ for ‘The inventor of the telephone liked spaghetti’ would yield a falsity (see fn 27 for another example of failure of inter-substitutability of sentences having equivalent assertoric contents). We thus need also another domain of entities (“ingredient senses”, following the terminology of Dummett (1991), p. 48) and an ordering of strength on them such that equivalence in that ordering between the ingredient senses of \(\varphi\) and ψ does guarantee that \(\varphi\) is unrestrictedly inter-substitutable with ψ at least outside of (direct- or indirect-) quotation environments. I will use ‘strong*’ and its relatives to denote such an ordering of strength of ingredient senses. Notice that I will stick to ordinary English and use ‘stronger’ (‘stronger*’) and its relatives to mean what would be meant by ‘strictly stronger’ (‘strictly stronger*’) and its relatives in mathematical English, using ‘at least as strong as’ (‘at least as strong* as’) and its relatives to mean what would be meant by ‘stronger’ (‘stronger*’) and its relatives in mathematical English.
Here is the implicit reasoning a bit more in detail. If there obtains a fact f whose obtaining makes ‘One feels cold’ true, then, by our assumption about truth in fn 21, one feels cold and, by a natural assumption, there is a fact of the matter as to whether one feels cold. Hence, by Definition 5, it is determinately the case that one feels cold. But, by our assumption about truth in fn 21, that suffices for ‘It is determinately the case that one feels cold’ to be true. Since, given natural assumptions, this conclusion has been secured by f’s obtaining, f’s obtaining makes ‘It is determinately the case that one feels cold’ true.
I’m here adapting a bit Wright’s argument to fit our setting in ways that seem to me innocuous.
True, ‘at least 50 ft tall’ is much less vague than ‘tall’, yet it also presents borderline cases. To get an idea of how this can be so, suppose that there is some wood on the top of a tree such that it would make a difference as to whether the tree is at least 50 ft tall or not, but also such that it is borderline whether it belongs to the tree or not.
For example, not only is the ingredient sense of ‘It is determinately the case that \(\varphi\)’ stronger* than the ingredient sense of ‘It is possibly the case that \(\varphi\)’, it is even stronger* than the ingredient sense of \(\varphi\) itself (see fn 27).
Let’s briefly and informally mention the most characteristic feature of such a behaviour. This consists basically in the fact that (D-INTRO) cannot be unrestrictedly employed in an argument some of whose premises are assumptions to be discharged (hence, (D-INTRO) cannot be unrestrictedly employed in a sub-argument for conditional proof, reductio, reasoning by cases etc.). To give an example, while in many standard logics (like classical, intuitionist, or minimal) whenever \(\varphi\) entails ψ the conditional ‘If \(\varphi, \) then ψ’ is a logical truth, this is not so once a rule like (D-INTRO) is added to one’s logic (more generally, letting \(\Upgamma\) be an arbitrary set of premises, what fails is the principle of conditional proof that, whenever \(\Upgamma, \varphi\) entails ψ, \(\Upgamma\) entails ‘If \(\varphi, \) then ψ’). For, while (D-INTRO) makes \(\varphi\) entail ‘It is determinately the case that \(\varphi\)’, the conditional ‘If \(\varphi, \) then it is determinately the case that \(\varphi\)’ is not a logical truth—indeed, some of its instances are arguably untrue. (To see this, let \(\varphi\) be ‘One feels cold’ and suppose that there is no fact of the matter as to whether one feels cold. Then, by contraposition, ‘If one feels cold, then it is determinately the case that one feels cold’ yields ‘If it is not the case that it is determinately the case that one feels cold, then it is not the case that one feels cold’, which is arguably untrue in this situation, as it has a true (indeed, determinate) antecedent and an indeterminate consequent. Notice that, since the untrue (in the situation just described) ‘If one feels cold, then it is determinately the case that one feels cold’ can be obtained from the true ‘If one feels cold, then one feels cold’ by substituting sentences that have equivalent assertoric contents, we have here another example of failure of inter-substitutability of sentences having equivalent assertoric contents (see also fn 22): indeed, quite generally, the ingredient sense of ‘It is determinately the case that \(\varphi\)’ is stronger* than the ingredient sense of \(\varphi. \))
Theorem 2 is a close kin of a result offered by Fara (2003), pp. 196–205. The important difference consists in the fact that Fara’s result uses the principle obtained from (DMAR) by substituting ‘in α n+1 one does not determinately not feel cold’ for ‘in α n+1 feels cold’. The result is more significant than Theorem 2 in that the new principle is weaker than (DMAR); it is less significant in that it also requires higher-order versions of the new principle, whereas theorem 2 does not require higher-order versions of either (DMAR) or the new principle (although, of course, (CD) and (D999,999,998DMAR) jointly entail the relevant higher-order versions of both (DMAR) and the new principle). Theorem 4 is a close kin of a result offered by Gómez-Torrente (1997), pp. 243–245. The important difference consists in the fact that Gómez-Torrente’s result uses higher-order versions of (DMAR). The result is more significant than Theorem 4 in that it requires neither (CD) nor (D999,999,998DMAR); it is less significant in that it also needs to appeal to higher-order versions of (DMAR), whereas Theorem 4 does not need to appeal to them (although, of course, (CD) and (D999,999,998DMAR) jointly entail the relevant higher-order versions of (DMAR), and hence the sense in which Gómez-Torrente’s result is less significant than Theorem 4 is only the sense in which Theorem 4 shows how what in that result is merely a not very compelling assumption actually follows from jointly stronger premises each of which is very compelling). (I note in passing that the (CD)-based reasoning of Theorem 4 takes its inspiration from a related argument developed in Zardini (2012a).) Neither Fara nor Gómez-Torrente relate their discussion to the AL-argument, let alone to (DLUM). Thanks to Paul Égré for raising the issue of the relationships between Theorems 2 and 4 and these two important results in the literature.
I wish to emphasise that, in this discussion, the operative sense of ‘mistake’ is that of objective mistake, a failure located in the brute relation between a subject’s belief (and confidence) and reality rather than in the formation of the belief (and confidence) in the light of the subject’s epistemic position. I don’t contest that there is a sense of ‘mistake’ and its like which targets the epistemic pedigree of a subject’s belief (and confidence), and which is thus such that e.g. a belief formed on the basis of tossing a coin to the effect that the number of stars in the universe is even would count as mistaken even if the number of stars in the universe were in fact even. This epistemically loaded sense of ‘mistake’ and its like would be of dubious relevance for the discussion connecting mistake and reliability. For example, it might be that a (epistemic) norm of belief (and confidence) is knowledge: in such case, the general principle that knowledge requires reliability would have extremely problematic consequences. For recall that, in our framework, we interpret the principle as requiring that, for every n, in α n one knows that one feels cold only if in α n one’s confidence that one feels cold is reliable, and that we take this in turn to require that a very similar confidence in a very similar condition formed on a very similar basis in a very similar case—e.g. the confidence one has in α n+1 in \(\langle\)One feels cold\(\rangle\)—is not mistaken. The epistemically loaded sense of ‘mistake’ would then require that the latter confidence is also knowledgeable, thus generating a sorites premise which, in conjunction with (POSNEG) and the factivity of knowledge, would entail the sceptical conclusion that in α1 one does not know (and, by (FORM), is not in a position to know) that one feels cold.
Somehow related to the point made in fn 29, I wish to emphasise that, in this discussion, the operative sense of ‘mistake’ is that of representational mistake, failure to represent reality (the totality of obtaining facts) as it is. It may be thought that there is a sense of ‘mistake’ and its like which, while still purely objective, permits as not mistaken a positive degree of confidence (or even belief) in at least some conditions representing no fact of the matter, and which is thus such that e.g. today a belief (or at least a confidence) that there will be a sea-battle tomorrow would not count as mistaken even if today there were no fact of the matter as to whether there will be a sea-battle tomorrow. Such view has a dubious plausibility even with respect to the domains for which it is not clearly a non-starter (possibly among others, the future, vagueness and conditionals): the very real intuitions on which it relies concerning the warrantedness of belief (or at least confidence) are more plausibly taken to tell against the assumption that the relevant condition does not represent any fact of the matter. Be that as it may, belief (and even confidence) that is in this sense objectively not mistaken albeit representationally mistaken is very plausibly all the same sufficient for giving rise to a charge of unreliability against a very similar confidence in a very similar condition formed on a very similar basis in a very similar case. For example, suppose that a respected general said that there was a sea-battle yesterday and also that there will be a sea-battle tomorrow. Suppose also that, while it is a fact of the matter that there was a sea-battle yesterday, there is no fact of the matter as to whether there will be a sea-battle tomorrow. Suppose finally that, purely on the basis of this testimony and without knowing that there is no fact of the matter as to whether there will be a sea-battle tomorrow, one becomes confident both that there was a sea-battle yesterday and that there will be a sea-battle tomorrow. Then, very plausibly, the fact that the latter confidence is placed in a condition representing no fact of the matter (and so is representationally mistaken) does give rise to a charge of unreliability against the former confidence.
I won’t go into the details of this, but I note that the particular application of reductio to follow is unobjectionable even in view of the remarks in fn 27.
Having argued this much against (DLUM), I hasten to add that in Zardini (2012c) I try to rescue (LUM) by focussing instead on that phenomenon of vagueness consisting in the apparent absence of sharp boundaries, and that the considerations offered in that paper can easily be converted into a rescue of (DLUM) itself from the two arguments against it that I have put forth in this paper (given that these arguments ultimately rely on a crucial kind of inference at work both in the derivation of (KMAR) and in the derivation of (DSOR), see fn 10). Such a rescue is particularly important if an NFMU-theory of vagueness is correct. For in that case, because of the reasons pointed out in Sect. 3, (LUM) has anyways to be abandoned in favour of the more guarded (DLUM), without this constituting in any way a retreat from the idea that some domains of facts are fully open to our view.
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Acknowledgements
Earlier versions of the material in this paper have been presented in 2008 at the Arché Audit and at the Arché Basic Knowledge Seminar (University of St Andrews); in 2009, at the Formal Epistemology Project Research Seminar (University of Leuven), at a research seminar at the University of Aarhus, at the 5th SOPHA Congress (University of Geneva), at the COGITO Epistemology Seminar (University of Bologna) and at the NIP Formal Epistemology Seminar (University of Aberdeen); in 2010, at the 2nd Workshop on Vagueness and Physics, Metaphysics, and Metametaphysics (University of Barcelona). I’d like to thank all these audiences for very stimulating comments and discussions. Special thanks go to Derek Ball, Annalisa Coliva, Laura Delgado, Dylan Dodd, Julien Dutant, Paul Égré, Gabriel Gomez, Chris Kelp, Hannes Leitgeb, Dan López de Sa, Aidan McGlynn, Sebastiano Moruzzi, Sven Rosenkranz, Ian Rumfitt, Johanna Seibt, Jason Stanley, Brian Weatherson, Robbie Williams and Crispin Wright. I’m particularly grateful to Stew Cohen for his support, feed-back and advice on my luminosity-related work throughout the years. But my greatest intellectual debt here is obviously to Tim Williamson, whose seminal articles on the topic provide much of the foundation for the present investigations. In writing this paper, I have benefitted, at different stages, from an AHRC Postdoctoral Research Fellowship and a UNAM Postdoctoral Research Fellowship, as well as from partial funds from the project FFI2008-06153 of the Spanish Ministry of Science and Innovation on Vagueness and Physics, Metaphysics, and Metametaphysics, from the project CONSOLIDER-INGENIO 2010 CSD2009-00056 of the Spanish Ministry of Science and Innovation on Philosophy of Perspectival Thoughts and Facts (PERSP) and from the European Commission’s 7th Framework Programme FP7/2007–2013 under Grant FP7-238128 for the European Philosophy Network on Perspectival Thoughts and Facts (PETAF).
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Zardini, E. Luminosity and determinacy. Philos Stud 165, 765–786 (2013). https://doi.org/10.1007/s11098-012-9973-9
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DOI: https://doi.org/10.1007/s11098-012-9973-9