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Semantic solutions to the paradox of knowability
Bachelor Thesis
Bas Kortenbach 11041803
Department of Philosophy
Faculty of Humanities
University of Amsterdam
3 Contents - Abstract - Introduction - The paradox - Edgington’s solution - Objections to Edgington - Costs of Edgington - Kvanvig’s solution - Objections to Kvanvig - Costs of Kvanvig
- Comparison and conclusion
Abstract
This thesis discusses Fitch’s paradox of knowability; the derivation of Ɐp (p → Kp) from Ɐp (p → ◊Kp). Two related proposed solutions are examined and evaluated, starting with Edgington’s suggestion to replace Ɐp (p → ◊Kp) with Ɐp (Ap → ◊KAp), which proved effective and appealing. Kvanvig’s diagnosis of a fallacy in the paradox’s proof turned out to be defensible but controversial. The thesis’ last section compares the solutions and concludes they utilize the same weakness of Fitch’s argument, despite implementing different strategies to do so. The verdict on which strategy is preferable is left to the reader.
Introduction
The paradox of knowability is a logical problem, first introduced by Fitch (1963). It concerns the knowability principle: the claim that for any truth p it is possible, in principle, that somebody at some time comes to know p. The principle can be formalized in modal logic:
(1) Ɐp (p → ◊Kp)
Here, ‘Kp’ means that is known by someone at some time that p. The apparent paradox is that from this principle, which doesn’t seem to be obviously false, an entirely implausible conclusion can be derived with the help of modal logic and a few modest epistemic assumptions. Specifically, the knowability principle appears to entail that any truth not only
can be known, but that at some time it is in fact known:
(2) Ɐp (p → Kp)
This result is especially problematic for anti-realists, who tend to hold that there are no truths whose truth-conditions transcend our potential evidence, and that every truth is thus in principle knowable. Yet even realists and those undecided on the debate might be puzzled that a principle which is not prima facie false can be reduced to absurdity in a few simple logical steps.1
As a result of this, the paradox has been discussed extensively in the philosophical literature since Fitch’s original paper. Many different solutions have been proposed, utilizing
1 Williamson (2017, p. 14 - 15) amongst others disagrees, argueing that those unsympathetic towards
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an array of strategies. As with any paradox, the approaches to solve it can be sorted roughly into three categories: those that argue that one (or more) of the assumptions are false, those that show a fallacy in the derivation and those that claim the conclusion is actually tenable. Two closely related solutions, proposed by Dorothy Edgington and Jonathan Kvanvig, turn on semantic reasons for restricting the range of universal quantifier in the knowability principle. Interestingly enough, despite their similarities these approaches don’t fall into the same category. Edgington thinks that (1) is not the right formulation of the knowability principle, placing herself in the first class, whilst Kvanvig aims to reveal a fallacy in the proof to (2). These two are the proposed solutions I find most interesting, most intuitive and most appealing, and they will therefore be the main focus of this thesis.2 The central question can be formulated
as follows: how successful are Edgington’s and Kvanvig’s proposed solutions to the paradox of knowability, when considered separately and when compared to one another?
To answer this question, we will begin by discussing the actual paradox itself, walking through the derivation steps that lead us from (1) to (2). Next we will examine the answers that Edgington and Kvanvig have brought forward, in turn, including some objections and shortcomings. This will be followed by a comparison of the two solutions: how similar are they and where do they differ? Finally, I will offer an analysis of what I believe are the relative merits and defects of the two options.
The paradox
Recall that the paradox of knowability arises when we start with the seemingly plausible statement that all truths can be known:
(1) Ɐp (p → ◊Kp)
From (1), we can apparently derive the obviously false statement that all truths are at some time known:
(2) Ɐp (p → Kp)
As mentioned above, the derivation from (1) to (2) makes use of some assumptions about knowledge. Firstly it is assumed that knowledge distributes over conjunctions, so that knowledge of a conjunction implies knowledge of both conjuncts. In addition to that, it is presumed that knowledge implies truth; something which is known must be true. With these in mind, the argument starts from the reductio assumption that there is a truth p which is not known: p ^ ¬Kp. If this is the case, by (1) we can infer that it can also be known that it is the case, resulting in (3):
(3) ◊K(p ^ ¬Kp).
Since knowledge distributes over conjunctions, (3) implies that ◊(Kp ^ K¬Kp). However, because knowledge entails truth, K¬Kp implies that ¬Kp. Thus from ◊(Kp ^ K¬Kp) we obtain (4):
(4) ◊(Kp ^ ¬Kp)
But Kp ^ ¬Kp is a contradiction and therefore cannot possibly be true, so (4) is false. We’ve reached this point: the knowability principle conjoined with the existence of an unknown truth leads to an absurd conclusion. Thus in order to maintain (1), the reductio assumption that an unknown truth exists must be negated. So, given (1), there are no unknown truths:
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(5) ¬ꓱp (p ^ ¬Kp)
But (5) is (classically) equivalent to (2); if there are no truths which are never known, then all truths are at some time known.3 So those who claim (1), as anti-realists typically do, are also
committed to (2).
Edgington’s solution
Before diving into Edington’s proposal, one matter requires our attention. The method that Edgington suggests to resolve the paradox makes use of the concept of situation. It will be useful to clarify this notion before continuing. A situation or possibility is like a possible world, only less specific. If we flip a coin, for example, there are infinite possible worlds resulting from that action: the coin can land heads or tails, in one spot or another etc. But even given that the coin falls a certain way, it can do so whilst any unrelated event occurs or whilst it doesn’t. Hence the infinite possible worlds. Of course, we might only be interested in one aspect of all these possible worlds, like the side of the coin that lands up. In that case, we can divide the possible worlds into two categories: heads and tails. So even though there might be infinite possible worlds within one of these categories, we can choose to ignore the differences between them and consider only what they have in common. To do so, Edgington uses the term situation. So for example we can indicate the situation where the coin lands heads. Furthermore, we can think and reason about that possibility in a way that we can’t about specific possible worlds. For a possible world has an infinite amount of detail, so our finite minds are incapable of thinking about them specifically.
Our ability to think and reason about possibilities is crucial for Edgington’s argument. It requires that we are able to reason counterfactually, about situations which did not obtain. Specifically the proposal is based on the idea that, in principle, people in non-actual situations might reason about what is the case in the actual situation. This might solve the paradox. For suppose that p is actually an unknown truth. Would it be contradictory that somebody in a non-actual situation knows this? The answer seems to be no, because the supposition states merely that nobody in the actual situation knows that p, which does not mean that nobody at
all (meaning in any situation) knows that p.
Formally then, Edgington’s proposal is to replace (1) as the formulation of the knowability principle with the following:
(6) Ɐp (Ap → ◊KAp)
‘A’ stands for the actuality operator, which rigidly designates the actual situation. So (6) states that for every p which is true in the actual situation, it is possible that somebody in some situation knows that p is true in the actual situation.4 Let’s attempt to reconstruct the
derivation of the paradox, but this time starting with (6). Suppose there is actually an unknown truth p: A(p ^ ¬KAp). This means that, by (6), it is possible to know this: ◊KA(p ^ ¬KAp).
3 Williamson (1982) argues that anti-realist might avoid the paradox by adopting intuitionist logic, in which the
equivalence of (5) and (2) fails. See Percival (1990) for a critique of this suggestion.
4 Edgington’s principle need not be limited to truths regarding the actual situation. Although (6) only ranges
over truths of the form ‘Ap’, Edgington (1985, p. 567) provides a more general form of her principle which quantifies over all situations: Ɐs1Ɐp ((p is true in s1) → ꓱs2 (it is known in s2 that (p is true in s1))). This says that
for any situation, if something is true in it, this is known in some (potentially different) situation. Applying this to the actual situation results in (6).
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knowledge and the actuality operator distribute over conjunctions, this implies that ◊(KAp ^ KA¬KAp). Since knowledge implies truth, the latter conjunct entails that A¬KAp. So we obtain that ◊(KAp ^ A¬KAp). However, unlike (4), this result should not be absurd. It states that there is some situation s in which two facts obtain: (i) it is known that Ap and (ii) in the actual situation it is not known that Ap. As long as s is not the actual situation, this is not contradictory. So (6) is consistent with the possibility of unknown truths, and proponents of the former are not committed to rejection of the latter.
There is still a faulty argument which might lead some to think that ◊(KAp ^ A¬KAp) is nonetheless problematic, and it is important to understand this argument and how to block it. It relies on two generally accepted modal principles. For one, it presumes the rule of necessitation, which states that the necessity of a theorem is also a theorem: Ⱶp → Ⱶ□p. Furthermore it is presumed that the modal operators are interdefinable in this manner: □¬p ←→ ¬◊p. Given these principles the argument runs as follows: suppose for reductio that actually KAp ^ KA¬KAp. Since knowledge implies truth, the latter conjunct entails that A¬KAp. So we have both KAp and A¬KAp. However, for all p’s, it is actually the case that p ←→ Ap. Thus, A¬KAp actually implies that ¬KAp. This means we have a contradiction: KAp and ¬KAp. So we have to negate the reductio assumption and assert that ¬(KAp ^ A¬KAp). But if this is a theorem, so is its necessity: □¬(KAp ^ A¬KAp). Yet this means that ¬◊(KAp ^ A¬KAp). Thus the earlier conclusion that ◊(KAp ^ A¬KAp) would be false.
The reason this argument ultimately doesn’t succeed is that, in systems containing the actuality operator, the rule of necessitation must be restricted. For theorems that feature A, the rule fails. Take for example the formula p ←→ Ap. This is actually a theorem for every p. However, □(p ←→ Ap) fails, for in a non-actual situation, p might be false even though Ap is true (or vice-versa). Similarly, ¬(KAp ^ A¬KAp) is actually a theorem, but its necessity is not. Therefore ◊(KAp ^ A¬KAp) remains plausible.
Objections to Edgington
The big assumption of Edgington’s argument is that non-actual knowledge of the actual situation is plausible. For how could those in a counterfactual situation identify and refer specifically to the actual one? Certainly not by using the actuality operator, because in their situation it would not refer to the actual situation but to theirs. Edgington (1985, 2010) suggests that reference can be achieved by counterfactual descriptions: i.e. ‘the situation where x (instead of y)’. The situation thus referred to is the one identical to the actual one in all relevant respects, except (i) instead of y, x is the case, and (ii) whatever follows from (i) is the case. Similarly, in that counterfactual situation we might refer to the actual situation with the converse: ‘the situation where y (instead of x)’.
In support of this method of reference, Edgington has offered some examples: suppose that we flip a coin and bet on it landing heads, which it actually turns out to do. We might reason counterfactually and say that in the situation where it landed tails, we would have been sad about it. This would constitute knowledge about the non-actual. Similarly, in the situation where the coin had landed tails, we might realize that in the situation where it landed heads (which is the actual situation) we would have been happy. This appears to constitute non-actual knowledge of the actual situation.
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Yet Timothy Williamson (1987, 2017) argues against the coherence of this model. The problem he poses for it is that if a situation can be specified in this manner, knowledge of what is the case in that situation requires nothing more than knowledge of trivial logical truths. His argument is party motivated by the causal isolation of different possible worlds, which he thinks carries over to situations. Williamson notes that Edgington compares knowledge of non-actual situations to knowledge of non-current times, but that there’s an important asymmetry: there are causal chains from the past to the present, so that we may have knowledge of and effectively refer to past times. By contrast, there can be no causal chain from a counterfactual situation to us, since counterfactual situations are spatiotemporally isolated from the actual one. Therefore we shouldn’t be able to know anything about such a situation that isn’t trivial. Conversely those in counterfactual situations can’t have non-trivial knowledge of the actual.
One potential reply on Edgington behalf might be simply to point out that future situations are (for now) non-actual. Non-actual situations need not take place in non-actual worlds; they might take place in the actual world but at non-current times. Future knowledge of the current is as likely as current knowledge of the past is (perhaps in virtue of causal chains). It follows that non-actual knowledge of the actual situation is equally uncontroversial. One might think that those in future situations won’t refer to the present by counterfactual conditionals, so this response requires a different model of non-actual reference to the actual. But, unnatural as it may be, it seems entirely coherent to express knowledge of the past by saying ‘in the situation where it was (still) then, instead of now, p’ (provided that it is clear from context which time is meant by ‘then’). However, this response only validates knowledge of non-actual situations which take place in the past of the same world and timeline. If it were the only reply available, (6) would imply that for every p, if Ap, then there is a situation in the
present or future of the actual world in which it is known that Ap. This implies (Ap → KAp), a
conclusion about as strong and unlikely as (2). It is clear that a response to Williamson is required that also validates actual knowledge of non-actual situations in non-actual timelines, and its converse.
Edgington’s own answer is that if a situation s is defined as identical to a situation s+ except for the truth of one proposition, then s is never causally isolated from s+ (2010, p. 46). Most of what is true in s+ is also true in s, so much of our (non-trivial) knowledge of s+ carries over. Although there may not be a causal chain from s to s+, the two share a large part of their causal set-up. The best temporal analogy, therefore, does not lie in reference to past times but rather to future ones. There is no causal chain from the future to the present, but we can have knowledge (or at least reasonable belief) of the future because it shares a causal set-up with the present. When we plug a kettle in, we know the water will heat up in the future. Symmetrically, we know that in the situation where we had already plugged it in, the water would have started heating.
This appears to effectively block the initial objection. However, Williamson goes on to support his triviality allegation with a more formal argument (2017, p. 11). It’s designed to proof that any knowledge we might have of a counterfactual situation can be represented by a piece of trivial knowledge. Assume for example that we know that in the counterfactual situation that A, C obtains: A □→ C. The problem Williamson raises is that this very knowledge
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is then equally constituted by the following triviality: (A ^ C) □→ C. The proof lies in the following theorem of systems with counterfactual implication:
(7) (A □→ C) → ((A □→ P) ←→ ((A ^ C) □→ P))
It states that if A counterfactually implies C, then any P is counterfactually implied by A iff P is counterfactually implied by A ^ C. The theorem is quite intuitive: if A □→ C, then the counterfactual situation that A ^ C is the counterfactual situation that A. Thus A counterfactually implies the exact same P’s as A ^ C does. Williamson’s problem is made apparent when we fill in C for the place of P:
(8) (A □→ C) → ((A □→ C) ←→ ((A ^ C) □→ C))
Formula (8) is also a theorem, and it tells us that if A □→ C is the case, then A □→ C iff (A ^ C) □→ C. Given that by assumption we know that A □→ C, we obtain that (A ^ C) □→ C is equivalent to A □→ C. Thus our knowledge that (A ^ C) □→ C is equivalent to knowledge that A □→ C. But the former knowledge is trivial! Another way to put the problem is this: given that A □→ C, the counterfactual situation that A ^ C is the counterfactual situation that A. Thus, our knowledge that C obtains in the counterfactual situation that A ^ C constitutes knowledge that C obtains in the counterfactual situation that A. But the knowledge that C obtains in the situation that A ^ C is trivial.
It’s important to realize exactly what has and hasn’t been proven here. Specifically, it’s crucial that the preceding argument doesn’t establish that the mere trivial knowledge that (A ^ C) □→ C counts as or implies knowledge that A □→ C. Knowledge that (A ^ C) □→ C doesn’t in itself imply knowledge that A □→ C.5 The argument starts on the assumption that we know
that A □→ C. What it proves is that given that we have that knowledge, trivial knowledge that (A ^ C) □→ C represents knowledge that A □→ C.
The next question is whether this result is either surprising or problematic. I take it the answer is no. The reason is that if we assume that we know a certain fact p, we can always prove from this that knowledge of a tautology suffices for knowledge of p, in the spirit of Williamson’s argument. This is not just a characteristic of counterfactual conditionals. Because if we know that p is the case, then we also know that when q implies q, p is the case. Thus knowledge that q implies q suffices for knowledge that p is the case. But knowledge that q implies q is trivial! Thus we’ve proven that, given that we have knowledge that p, trivial knowledge that q implies q represents knowledge that p is the case.
Let’s consider another example, more similar to Williamson’s argument. Suppose that we know that the shortest spy is male. Given this fact, whatever is true of the shortest male spy is also true of the shortest spy and vice-versa, for they are the same person. Thus the shortest male spy is male iff the shortest spy is male. Therefore, knowledge that the shortest male spy is male constitutes knowledge that the shortest spy is male. But knowledge that the shortest male spy is male is trivial! Thus we’ve proven that, given that we know that the
shortest spy is male, knowledge of a triviality constitutes knowledge that the shortest spy is
male.
5 To see that it doesn’t, imagine that we only know (A ^ C) □→ C and attempt to derive A □→ C from this. We
can’t, because it is perfectly consistent with (A ^ C) □→ C to suppose that ¬(A □→ C): the fact that C obtains in the counterfactual situation where A ^ C obtains doesn’t mean that C obtains in the counterfactual situation where A obtains.
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Williamson’s objection comes down to this: Edgington argues that we can represent knowledge of other situations with statements like ‘in the counterfactual situation where p, q obtains’, formally p □→ q. However, if we allow such counterfactual statements to represent knowledge, we also allow trivialities like (p ^ q) □→ q to represent the same knowledge (assuming, like Williamson’s argument, that we already knew that p □→ q). Williamson concludes that there needs to be some constraint on which representations of such knowledge are permissible, so as to exclude the non-trivial representations. However, no constraint is available which doesn’t simply ad-hoc state that the representations must be non-trivial.
I think the examples I offered show that no such constraint is needed. If we allow statements like ‘the shortest spy is male’ to represent knowledge, we also allow trivialities like ‘the shortest male spy is male’ to represent that same knowledge (assuming that we already knew that the shortest spy is male). Yet this doesn’t prompt us to device some constraint on how knowledge of this type is represented. For as far as the problem Williamson has raised for Edgington’s account is problematic, it applies to all sorts of knowledge. It doesn’t arise because of anything specific related to counterfactual conditionals or to the method of expressing knowledge of non-actual situations with such conditionals.
Williamson’s objection, therefore, is not successful. In absence of a more effective objection, and considering the plausibility of the coin and kettle examples, it seems there’s no reason to doubt that non-actual knowledge of the actual situation is a reasonable idea and even a common phenomenon.
Costs of Edgington
One major limitation of Edgington’s solution to the paradox is that her version of the knowability principle doesn’t apply to all truths, but only certain types. As we metioned earlier, in footnote 3, the principle’s applicability is not limited to truths about the actual situation. However, even Edgington’s general principle only asserts the knowability of necessary truths. This is because propositions of the form ‘p is true in situation s’ are always necessarily true, if true at all. If it is true in situation s that p obtains, then it is true in every situation that in s, p obtains.
So both (6) and the more general principle only establish the possibility of knowing necessary truths. This might not be sufficiently satisfying to the realist. Recall that for anti-realists, the knowability principle is supposed to express a fundamental and therefore universal property of truth. The principle is meant to reflect that it is inherent in the nature of truth that it cannot transcend our cognitive limits. Thus, the anti-realist would want the knowability principle to apply to contingent truths as much as necessary ones.6
One comforting factor for the anti-realist might be that for every contingency p, there is at least a closely related necessary truth which is knowable, namely that p is true in the relevant situation. In the actual situation, for example, If p is a contingent truth, Edgington’s
6 To those who merely aim to solve the paradox for the sake of solving it, rather than for the sake of vindicating
anti-realism, the limited range of Edgington’s knowability principle need not be a problem at all. They can take Fitch’s proof as showing that (1) is untenable, and that knowability must be appropriately restricted to avoid contradiction.
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knowability principle doesn’t ensure us that p is knowable, but it does tell us that the related truth Ap is. Because of this, Edgington’s knowability principle still reflects a universal property of truth. Namely it states that for every truth, whether it’s necessary or contingent, its truth in the situations in which it is true is knowable. For necessary truths, this amounts to knowing that they are necessary truths. In the case of contingent truths it means that it’s knowable exactly in which situations they do and do not obtain.
I think many anti-realists, perhaps depending on their specific position, could accept this as an adequate representation of their original conception of the nature of truth. If every truth’s truth in the situations in which it is true is knowable, this means that truth is limited to our potential evidence in a significant and meaningful way. It would mean that there’s no proposition which is true in a certain situation without the possibility that somebody at some point knows this. We might reasonably conclude from this that reality does not exist independently of our cognitive capabilities, which is what anti-realists typically aim to assert by positing the knowability principle. The only concession is that some contingent truths are indeed unknowable. However, given that it can nonetheless be known that they are true in all the situations in which they are, the unknowability of the truths themselves doesn’t reflect any significant disparity between reality and our capacity for knowledge.
In conclusion, taking Edgington’s proposal as the correct solution of the paradox comes as the cost of limiting knowability to necessary truths. At first sight, paying this price seems to defeat the purpose for anti-realists, because it would mean admitting that apparently not all truths are knowable. However, once we realize that the unknowability of a contingent truth does not signify a genuine transcendence of our cognitive capacities, the result seems acceptable.
Kvanvig’s solution
We now turn to the resolution of the paradox proposed by Jonathan Kvanvig (1995). In short, Kvanvig claims that the derivation of the paradox contains a logically fallacious step. To see where, he requires that the paradox and its assumptions be formalized in more detail. He argues that the knowledge predicate ‘Kp’, read as ‘it is known by someone at some time that p’, is implicitly quantified. An explicit formulation of ‘Kp’ would be ‘ꓱxꓱt(Kxtp)’, where ‘Kxtp’ is a three place predicate read as ‘at time t, it is known by x that p’. The paradox is most accurately represented by substituting this quantified formula for every instance of Kp. So instead of (1), the knowability principle is formulated as (9):
(9) Ɐp (p → ◊ ꓱxꓱt(Kxtp)) From which we derive (10):
(10) Ɐp (p → ꓱxꓱt(Kxtp))
To do so, we assume for reductio that (11): (11) (p ^ ¬ꓱxꓱt(Kxtp))
The argument to contradiction proceeds in the familiar manner. This explicit reformulation itself does not solve the paradox; it merely enhances our understanding of its logic. In particular, it serves to reveal that when we substitute the reductio assumption for p in the knowability principle, we are really substituting a formula containing existential quantifiers from a non-modal context into a modal one. Kvanvig argues that this move is illegitimate,
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because it changes what the formula expresses. Fitch’s argument thereby commits a logical fallacy.
The reason that this type of substitution is inappropriate, according to Kvanvig, is that the domain of the quantifiers is restricted. What the domain is restricted to exactly is dependent on the context. This makes quantified sentences indexical, since the content of the proposition they express varies with varying contexts. Consider for example the statement ‘there are no more beers’, which could be formalized as ¬ꓱx(Bx), where Bx means x is a beer. In most cases, this statement is not meant to express the proposition that there are no more beers anywhere. Instead, the domain of the quantifier is implicitly restricted to a certain area, and it is usually determined by context which area that is. If I’m at home and say ‘there are no more beers’, this expresses that there are no more beers in my house. When I utter the same sentence at the supermarket, it expresses the entirely different proposition that there are no more beers in stock there. One of these two propositions may be true while the other isn’t, and carelessly substituting the sentence between different contexts leads to errors.
Similarly, Kvanvig proceeds, substituting quantified sentences into modal contexts isn’t freely permissible. Consider the statement that ‘there are no Belgian Ajax players’, formally ¬ꓱxAx(Bx) if Ax and Bx mean x is an Ajax player and Belgian, respectively. The domain of the existential quantifier in this formula is ‘modally restricted’ in the sense that it only ranges over Ajax players in the actual world, rendering the statement true. In a different modal context, the domain may include individuals in alternative possible worlds. In that case the same sentence expresses a different proposition, possibly a false one. Substituting a quantified sentence from a non-modal to a modal context may change the proposition it expresses. Thus the move made by Fitch’s argument is illegitimate.
In the specific case of the paradox, Kvanvig doesn’t explicate how the domain precisely changes (i.e. what is the domain before and after substitution), but it can be made out. Assumption (11) is meant to say that there is a truth p which is never known, but not that p cannot possibly be known, so the quantifiers are clearly restricted to the actual world. When we substitute (11) for p in (9) and apply modus ponens, we obtain (12):
(12) ◊ ꓱxꓱt (Kxt(p ^ ¬ꓱxꓱt(Kxtp)))
When we apply to (12) the assumptions that knowledge distributes over conjunctions and implies truth, we get (13):
(13) ◊ (ꓱxꓱt(Kxtp) ^ ¬ꓱxꓱt(Kxtp))
This is of course an impossible result. But the present question is what the domain of the second pair of existential quantifiers in (13) is. What (13) states is that there is a possibility or possible world w in which two things obtain: (i) p is known by somebody at some time, and (ii)
p is never known by anybody. But this means that the second set of quantifiers no longer
range over the actual world, but over w (which need not at all be the actual world). Somewhere along the way the domain changed.
As Kvanvig predicts, we can trace the change of domain back to the substitution of (11) in (9). To do so we look at the second pair of quantifiers in (12). (12) reads that there is a possibility or possible world w in which somebody at some time knows both (i) that p is true, and (ii) that nobody ever knows p. What does ‘nobody ever’ quantify over here? Clearly, it no longer quantifies over the actual world but over that possible world w in which the conjunction is known. Otherwise, the factivity of knowledge couldn’t be used to derive that it is true that
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nobody in w ever knows p, as is done to obtain (13). So the formula (p ^ ¬ꓱxꓱt(Kxtp)) expresses a completely different proposition after being substituted.
In response to these problems, Kvanvig notes, opponents of anti-realism may choose to introduce modally unrestricted quantifiers and attempt to reconstruct the paradox in terms of those. Of course, it won’t do to simply replace all the quantifiers in the assumptions with unrestricted versions, since that could alter the expressed content. At certain points it will therefore be necessary to add the intended restrictions in other ways. Suppose we let the cursive Ɐ and ꓱ denote the modally unrestricted versions of the universal and existential quantifiers respectively. The knowability principle can then be formalized as (14):
(14) Ɐp (p → ꓱxꓱt(Kxtp))
The principle is meant to represent a necessary property of true statements, so its universal quantifier may simply be replaced by the unrestricted variant. The existential quantifiers can be likewise replaced and the possibility operator left out, since ‘◊ ꓱxꓱt(Kxtp)’ is equivalent to ‘ꓱxꓱt(Kxtp)’: saying that it’s possible that there exists a certain y is equivalent to saying that there is, in the modally unrestricted sense, such a y. In the case of the reductio assumption, it won’t do to just replace the quantifiers and write (p ^ ¬ꓱxꓱt(Kxtp)), for this formula says that
p is an unknowable truth rather than just a never known one. Instead we can introduce a
predicate @x, meaning that x exists in the actual world, and formulate the following: (15) p ^ ¬ꓱxꓱt(@x ^ @t ^ Kxtp)
We can choose to define the actuality predicate in two ways, namely as designating the actual world rigidly or as being indexical. In the second case, what world it refers to depends on the context in which it is uttered. This clearly isn’t an option for those who wish to reinstate the paradox, because it would mean committing the same fallacy as before: to substitute (15) for
p in the knowability principle would be to substitute a sentence with an indexical nature into
a modal context. The alternative is to let @ rigidly designate the actual world, to avoid committing such a fallacy. But if we do that, substituting (15) for p in (14) won’t yield a contradiction at all. Applying the familiar derivation steps of Fitch’s argument, the result would be (16):
(16) ꓱxꓱt(Kxtp) ^ ¬ꓱxꓱt(@x ^ @t ^ Kxtp)
But this isn’t problematic. In order not to be contradictory, (16) merely requires that the x and
t whose existence is predicated by the first pair of quantifiers don’t exist in the actual world
but in some other possibility. So the success of Kvanvig’s argument is dependent on non-actual knowledge of the actual.7 Assuming that such knowledge is possible, there seems to be no
logically valid way to reconstruct the paradox.
Thus, Kvanvig concludes, when the logic of the paradox is properly understood it is revealed that its derivation turns on a logical fallacy. There appears to be no way to construct the paradox without committing an illegitimate substitution. Therefore, the knowability principle is perfectly consistent with the existence of unknown truths.
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Objections to Kvanvig
Timothy Williamson (2002, p. 285 - 289) has some fundamental disagreements with Kvanvig’s analysis. He doesn’t concur that the proposition expressed by quantified sentences can vary with varying modal contexts. Consider again the non-existence of Belgian Ajax players: ¬ꓱxAx(Bx). According to Williamson, in the actual world this formula doesn’t express the proposition that there are no Belgian Ajax players in the actual world. Nor does it express in any other possible world that there are no Belgian Ajax players in that world. Rather, in all possible worlds it expresses the same proposition: that there are no Belgian Ajax players. The fact that some part of a sentence refers to different objects when uttered in different worlds, as the term “Ajax players” does in this case, doesn’t mean that the sentence then expresses different propositions. In a different modal context the constituent term still contributes the same property to the proposition, namely that of being an Ajax player, so the proposition expressed remains the same. The term only refers to different objects because there are different objects carrying the property it contributes. So on Williamson’s account, if a quantified formula refers to varying objects with varying modal contexts, this doesn’t mean that the formula then also expresses varying propositions.
Williamson argues that Kvanvig’s position on this matter is counterintuitive (2002, p. 288). Consider the sentence “Dolberg is an Ajax player”. We are inclined to say that this expresses the same proposition across all modal contexts, namely that Dolberg is an Ajax player. What is and isn’t an Ajax player varies with varying modal contexts, but we don’t expect this to cause variance in the contribution that the term “Ajax player” makes to the proposition. The question Williamson asks is: why would this be any different for a quantified sentence like “There are no Belgian Ajax players”?
Kvanvig has roughly two options: either deny that there is a difference - by admitting that subject-predicate statements like “Dolberg is an Ajax player” can also express varying propositions - or explain why there is one. The former seems especially unattractive. Kvanvig wants to restrict substitution into modal contexts to sentences which invariably express the same proposition, so this reply would mean that such substation is illegitimate for simple subject-predicate statements as well as quantified sentences. This is presumably not how far we’re willing to go.
The second option looks more promising. Kvanvig could argue that a predicate’s role in contributing to the proposition is different when it appears in the scope of a quantifier, and that the nature of this distinct role allows the contribution to vary with varying modal contexts. However, a better response is available: in the case of a quantified sentence, it isn’t the predicate whose contribution to the proposition varies, but the quantifier itself. In the example of the sentence “There are no Belgian Ajax players”, the contribution of the term “Ajax players” remains the same across all possible worlds. However, the contribution to the proposition made by the words “there are” varies. With respect to any world w it signifies that “there are in world w”. This is in line with Kvanvig’s original argumentation, which claimed that it is the range (and thereby the propositional contribution) of the quantifiers that changed after modal substitution. Kvanvig can readily grant that the contribution made by a predicate like “Ajax player” remains unvaried, and that an unquantified sentence like “Dolberg is an Ajax player” thus always expresses the same proposition.
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Williamson elaborates his rejection of Kvanvig’s account with a further argument (2002, p. 288 - 289). It revolves around two central theses: (i) Kvanvig is committed to allowing the use of modally unrestricted quantifiers, and (ii) contrary to what Kvanvig claimed, allowing such quantifiers actually does revalidate the paradox (for as far as it was ever invalidated). The motivation for the first claim is that Kvanvig’s argumentation assumes that it’s contingent what things exist. Otherwise, the domain of the quantifiers wouldn’t change with respect to different possible worlds. But without unrestricted quantifiers, this very assumption would become impossible to express. Suppose all the F’s that actually exist are a1, …, an. We can
formalize this as follows:
(17) ꓱx1Fx1...ꓱxnFxn ((x1 ≠ x2) ^ … ^ (xn-1 ≠ xn) ^ (x1 = a1) ^ … ^ (xn = an)) ^ ⱯyFy ((y = a1)
v … v (y = an)).
This formula says there are at least n distinct F’s, names them as a1 to an, and states that there
are no more F’s than that. Williamson argues that on Kvanvig’s account, the proposition thus expressed is (partially) determined by the domain that the quantifiers range over, and that (17) therefore expresses a necessary truth. I take it his motivation for this conclusion is something like this: since a1, …, an are all the F’s that actually exist, they constitute the exact
domain of the quantifiers. Therefore the proposition (17) expresses is just that the sequence a1, …, an contains exactly n members, and that they are a1, …, an. This is necessarily true. But
that means that the necessitation of (17) is true.8 In that case, (18) is false:
(18) ◊¬(ꓱx1Fx1...ꓱxnFxn ((x1 ≠ x2) ^ … ^ (xn-1 ≠ xn) ^ (x1 = a1) ^ … ^ (xn = an)) ^ ⱯyFy ((y =
a1) v … v (y = an))).
Yet (18) is how we would normally express that there could have been different F’s. So on Kvanvig’s account it’s contingent what F’s there are, but the formula we’d typically use to express this, (18), is false. We are faced with two questions. First of all, what does (18) express in Kvanvig’s system? The answer can’t be what it usually expresses, because what it usually expresses is true and (18) is false. Furthermore we should ask how to express that it is contingent which F’s exist, now that (18) apparently cannot express this.
Typically, we’d expect (18) to express that it is contingent what F’s exist, but on Kvanvig’s view this is not the case. The reason that (18) has a different meaning in Kvanvig’s system is that (17) does, and (18) (on both accounts) expresses the contingency of (17). It says that in some possible world it is not the case that (17). Normally, (17) would express that the existing F’s are a1, …, an. Then (18) expresses that in some possible worlds, the existing F’s are
not precisely a1, …, an. However, as Williamson discusses, on Kvanvig’s account (17) expresses
the necessary truth that a1, …, an is the sequence a1, …, an. So (18) in this case expresses the
false proposition that the sequence a1, …, an might not have been itself.
How then might we express that there could have been different F’s? Williamson argues that Kvanvig must allow another method of quantifying for this purpose: modally unrestricted quantifiers. I’m not sure that this conclusion is warranted. There might yet be
8 The success of this particular move, from the necessity of the proposition expressed by (17) to □(17), depends
on a certain interpretation of □p. Namely □p is taken as the claim that the proposition expressed by p is necessarily true. If we were to interpret □p as the claim that the sentence/formula p itself was necessarily true, the move would fail. In that case, the necessity of the proposition expressed by (17) wouldn’t imply □(17), because on Kvanvig’s account the proposition expressed by (17) might be necessarily true without (17) itself being necessarily true.
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other ways to express the contingency of the existing F’s, for example by quantifying over possible worlds or situations:
(19) ꓱs ¬(ꓱx1Fx1...ꓱxnFxn ((x1 ≠ x2) ^ … ^ (xn-1 ≠ xn) ^ (x1 = a1) ^ … ^ (xn = an)) ^ ⱯyFy ((y
= a1) v … v (y = an))).
This reads that there exists a situation s such that there are different F’s than a1, …, an. Since
the quantifiers in (19) that range over the F’s appear within the scope of an existential quantifier ranging over situations, their domain is not the F’s in the actual situation. Instead they range over the F’s in that situation s. So (19) states that there is a situation s such that the F’s in that situation are not exactly a1, …, an. In other words: there might have been
different F’s. So I don’t think Kvanvig is compelled by this particular argument to allow modally unrestricted quantifiers. Nevertheless, he allows them. Therefore Williamson’s further argumentation that the paradox can be reconstructed in terms of such quantifiers requires our attention.
Recall that Kvanvig discussed and ultimately rejected the possibility of recreating the paradox in terms of modally unrestricted quantifiers. He argued that, if ꓱ denotes the unrestricted existential quantifier, ‘ꓱxꓱt(Kxtp)’ is equivalent to ‘◊ꓱxꓱt(Kxtp)’. Williamson wants to deny this very claim, instead arguing that ‘ꓱxꓱt(Kxtp)’ is equivalent to just ‘ꓱxꓱt(Kxtp)’. One consequence is that the seemingly innocent conclusion (16), which Kvanvig reached by applying Fitch’s argument to his unrestricted versions of the assumptions, is actually contradictory: ‘ꓱxꓱt(Kxtp) ^ ¬ꓱxꓱt(@x ^ @t ^ Kxtp)’. If we interpret ‘ꓱxꓱt(Kxtp)’ as Kvanvig does, it merely states that there is in some possible world a t and x such that x knows p at t. This does not contradict the second conjunct, because said possible world does not need to be actual. On Williamson’s interpretation however, it reads that there actually exists such knowledge, so that it does contradict the second conjunct.
So why does Williamson insist we interpret ‘ꓱxꓱt(Kxtp)’ in that way? The unrestricted quantifiers allow that the possible world where x and t exist is non-actual, so what ensures us that it isn’t? The reason is that the knowledge itself is not predicated to exist in the unrestricted sense, but predicated to exist, period. Williamson argues that ‘ꓱxꓱt(Kxtp)’ should be read as saying there exists knowledge that p, by some possible x at some possible t. So the knowledge exists in the ‘regular’, modally restricted sense that it actually exists. But if the knowledge actually exists, those who possess it must also actually exist, as must the time at which it is possessed.
I see that this is one way of interpreting the formula, but I think the alternative interpretation that fits Kvanvig’s argument is at least as intuitive and appealing. This alternative is simply to read it as saying it is in actuality the case, that some possible x and
possible t have knowledge of p. Since the x and t need not be actual, neither need the
knowledge itself. It seems a matter of what has priority: is it the asserted modal status of the
x and t that determines the modal status of the knowledge, or is it the asserted modal status
of the knowledge that determines the modal status of the x and t? This question strikes me as not easily answered. It certainly isn’t obvious that Williamson’s interpretation is the correct one. The following example might be one argument to the effect that it is not: consider the formula ‘ꓱxRx’, where Rx reads that x is red. Would we be more inclined to interpret this as saying ‘there exists some possible x, which is red’, or as saying ‘there exists redness, which is the property of some possible x’. The latter would mean that the redness is asserted to actually
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exist, and that therefore the x which possesses it must also actually exist. I think this is easily the less intuitive way to read that formula. The reason being that objects are in some sense prior to any properties they carry. The redness of an object does not and could not exist independently of the object, whereas the object could exist without its redness. Analogously, a time t, person x and truth p can exist without knowledge of p by x at t existing. The knowledge of p by x at t could not coherently be said to exist if it weren’t for the existence of
t, x and p. The knowledge is merely a result of the relative state of t, x and p. For this reason
it seems more natural to take the objects that something is predicated of as prior to the property that is predicated of them.
So it seems to me that the paradox doesn’t follow from versions of the assumptions with modally unrestricted quantifiers. If it did, perhaps this would be one reason to restrict use of such quantifiers, although independent reasons for doing so should ideally be offered to avoid ad hoc argumentation. All in all, I take it Kvanvig’s proposal is not affected by this related set of objections from Williamson.
Costs of Kvanvig
Adopting Kvanvig’s conception of quantified sentences and the propositions they express has some interesting, possibly unwanted consequences. One such result is discussed by Kvanvig, namely that the account requires a restriction on the rule of necessitation (1995, p. 494). The rule of states that if p is a theorem, so is its necessity: Ⱶp → Ⱶ□p. In other words, if a formula can be proven in any context, without assumptions, then so can its necessity. Kvanvig notes that this is to move from the necessity of a sentence p to the necessity of the proposition expressed by p.9 But for quantified sentences, the expressed proposition can vary modally.
Such a sentence may be true in every possible world, whilst the proposition it expresses in the current world comes out false in some other worlds. Then the proposition it currently expresses isn’t necessarily true, even though the sentence is. So the move from the necessity of a quantified sentence to the necessity of the proposition it expresses isn’t valid. Kvanvig concludes that the rule of necessitation should only apply to formula’s that designate the same proposition with respect to every world.
Kvanvig argues that this cost is not fatal. He points out that there are already other cases where the rule fails, such as (the formalization of) the sentence “I exist”. This sentence is true in every context of utterance, because whoever utters or thinks it exists. Yet the proposition it expresses when any person utters it is not necessarily true, for that person might not have existed. Once we accept that the rule of necessitation fails for some cases, adopting Kvanvig’s view only requires us to add another type of sentence to that list. Moreover, the reason that rule fails for the sentence “I exist” is similar to the reason it fails for quantified sentences, namely that the sentences express varying propositions. In light of this, Kvanvig’s account merely tells us that there are more such cases than we might have thought. I agree that this cost seems quite acceptable.
9 Just like Williamson’s contingency argument (see footnote 8), this point by Kvanvig presumes that □p is to be
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Comparison and Conclusion
In the introduction I mentioned that the two proposals were related, despite using different strategies and different formal solutions. Having reviewed both accounts in detail, it’ll be interesting to discuss what the similarity and dissimilarity between them consists in.
Philosophically, the two proposals share a common idea: that the paradox can be avoided if we consider non-actual knowledge of the actual. The idea is most obviously present in Edgington’s solution. If we combine her knowability principle (6) with the existence of an unknown truth and apply Fitch’s argument we eventually obtain ◊(KAp ^ A¬KAp). In order not to be contradictory, this result requires that the situation in which it’s known that Ap is non-actual. In the case of Kvanvig, non-actual knowledge of the actual was needed to avoid reconstruction of the paradox in terms of unrestricted quantifiers.
This all points towards a related but deeper similarity between the two solutions: they both diagnose and address the same problem in Fitch’s reasoning. That problem is that after the reductio assumption is substituted into the knowability principle, the assumption no longer seems to express merely that p is an unknown truth in the actual world, but that it is an unknown truth across all worlds. Given this assumption it is no surprise that the knowability of said p results in contradiction. This diagnosis of the argument is made explicit by Kvanvig, who points out the varying range of the quantifiers. It is clear that Edgington’s solution is designed to fix the same issue, for example when she speaks of “[…] the need for the operator 'actually', to preserve talk about the way things actually are, inside the scope of a modal operator […]” (1985, p. 562). She agrees with Kvanvig that the meaning of “it is not known that p” changes after it’s placed in a modal context, comparing it to the change that “it is not currently known” goes through when placed in the context “at some time it will be the case that it is not currently known” (1985, p. 559 – 562).
Where Kvanvig and Edgington differ is their approach to fix this issue. Kvanvig’s solution is crudely to put a restriction on what formulas can be substituted, so that simply only those formulas which never change their expression are substituted into the principle. Edgington in some sense restricts the principle even further, by only applying it to formulas of a certain form: ‘p is true in s’. This move then ensures that the entered formulas never change what they express, simply because formulas of this form never do.
The final question then is: which approach is preferable? At first glance, we might think that Kvanvig’s solution is better, because he allows the principle to be applied to all formula’s that don’t change what they express after modal substitution, whereas Edgington’s principle is limited to that distinct type of truth.10 As we discussed, this means that anti-realists siding
with Edgington have to slightly adapt their conception of what it means for truth to be limited to our potential evidence. However, Kvanvig avoids this only at a cost. In order to make sense of the restriction that only formulas which don’t change their expression can be substituted into modal contexts, we need a detailed account of which formulas do and do not change their expression like that. Moreover, this account needs to explain why quantified formulas like (p ^ ¬ꓱxꓱt(Kxtp) do. This required a novel conception of the relation between quantified sentences and the propositions they express, which proved controversial. In the section
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‘Objections to Kvanvig’, we managed to retain some credibility for it under heavy fire from Williamson. However, given its radical differences from the conventional way of understanding quantified sentences and their propositions, we might suspect that Kvanvig’s account is open to more potentially fatal charges from that direction. Edgington’s account is more conservative, and therefore significantly safer and less likely to fail in light of future evidence.
In conclusion, both proposed solutions come with their own pros and cons. Because of this, I remain impartial between them. I leave the final verdict for the reader to decide, so that they may opt for the method that best suits their philosophical preferences.
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