pre-theoretic perspective to a pre-theoretical perspective. What do our best theories of vagueness tell us about these issues? Do they tilt the scale in favour of Kagan’s position or indicate that his conclusion is mistaken?
According to this view, there is a unique and precise point at which a growing person suddenly becomes tall, at which the addition of a grain of sand turns a non-heap to a heap, and at which the loss of a single hair makes a person bald. Likewise, in
HARMLESS TORTURERS, there is a unique and precise threshold beyond which the victim is in pain. We just do not know exactly where this threshold is, and neither does the victim. On this view, we can reject the inductive premise in sorites cases, and this will leave us with room to agree that Kagan is correct.
The epistemic view might seem incredible in its claim that there are sharp boundaries in all sorites cases. Still, it has the theoretical advantage of preserving classical logic – this is why its leading proponents typically put considerable effort into explaining why various nonclassical logics fail. However, instead of retaining classical logic and accepting the epistemic view, we could accept a non-classical logic such as three-valued logic.
THREE-VALUED LOGIC: For borderline cases, it is neither true nor false that Fx.
Instead, Fx takes the intermediate third truth-value indefinite.
(See Halldén 1949; Łukasiewicz 1970/1920; Tye 1990, 1994)9 According to this view, it is neither true nor false, but indefinite, whether a person who is (say) 178 cm is tall or not. Likewise, when (say) 120 switches are flipped in
HARMLESS TORTURERS, it is indefinite whether the victim is in pain. Typically, proponents of three-valued logic think that vagueness is due primarily, not to ignorance, but semantic imprecision (see e.g. Tye 1990).
Interestingly, according to three-valued logic, there will be a sharp boundary at which a growing person becomes tall, and at which flipping a single switch could make the sentence “the victim is in pain” true. Why? Since there are only three possible truth values, there must be some place in a sorites series when the truth value changes from indefinite to true. It cannot be indefinite whether the sentence
“the victim is in severe pain” is true or indefinite.
In fact, according to three-valued logic, there are two sharp boundaries in a sorites case: one between the false and indefinite cases, and one between the indefinite and true cases. For instance, there is a flipping of a switch such that the statement “the victim is in pain” becomes indefinite, not false, and there is a different flipping of a switch such that the statement “the victim is in pain” ceases to be indefinite and becomes true.
9 There are some different suggestions for how to denote and understand the third truth-value. For instance, Tye (1994) denotes it “indefinite”, Halldén (1949) calls it “meaningless” and Łukasiewicz (1970/1920) labels it “possible”.
Adherents of three-valued logic will reject the inductive premise of the sorites argument. It is not true for all n that if xn is F, then xn+1 is F. So, three-valued logic implies that Kagan is correct. It is a conceptual truth that the flipping of a switch might make the statement “the victim is in pain” true.10
Finally, supervaluationism might turn out to be our best theory of vagueness. Like most proponents of three-valued logic, supervaluationists like Kit Fine (1975), David Lewis (1986c) and Rosanna Keefe (2000) take vagueness to stem from our indecisiveness about how to use our concepts, not our ignorance. In a passage that is often quoted, Lewis writes:
The only intelligible account of vagueness locates it in our thought and language. The reason it’s vague where the outback begins is not that there’s this thing, the outback, with imprecise borders; rather there are many things, with different borders, and nobody has been fool enough to try to enforce a choice of one of them as the official referent of the word “outback”.
(Lewis 1986c: 213)
Presumably, we can understand the vagueness of the statement “the victim is in pain” in a similar way. It is not that there is this thing, the victim’s being in pain, imprecisely described; rather there are many things, each precisely described, and nobody has been fool enough to try to enforce a choice of one of them as the official referent of the “the victim’s being in pain”. As Fine (1975) puts it: “vagueness is ambiguity on a grand and systematic scale” (282).
Supervaluationists then claim that a vague statement like “the victim’s being in pain” is true (in the sense they explain) just when it is true on each acceptable description; that is, on each acceptable way of making it perfectly precise. This idea is commonly phrased in the following way:
SUPERVALUATIONISM: A statement is supertrue and therefore true if and only if it is true on all its acceptable completely sharp sharpenings.
(See Fine 1975; Keefe 2000; Broome 2004)
10 Notice that, in general, many-valued logics (four-, five-valued logics, and so on) will entail that a single flipping of a switch may make the statement “the victim is in pain” true.
For example, the statement “Tek is tall”, said about Tek, who is 178 cm, is not supertrue (and hence not true), because there are completely sharp sharpenings of
“tall” according to which 178 cm does not count as tall: “180 cm or taller” could be an acceptable and completely sharp sharpening of “tall”, for instance.11 Similarly, assuming that it is vague whether the victim is in pain when 120 switches are flipped in HARMLESS TORTURERS, the supervaluationist can hold that it is not true (because not supertrue) that the victim is in pain. There are acceptable sharpenings of “being in pain” according to which the victim is in pain, and acceptable sharpenings of
“being in pain” according to which he is not in pain. Things would have been different if Tek had been 198 cm, or if 700 switches had been flipped. There is arguably no acceptable sharpening of “tall” according to which a height of 198 cm is not tall, and no acceptable sharpening of “being in pain” according to which the affective state a torture victim is in when 700 switches are flipped is not pain.
Instead of introducing an additional truth-value (as three-valued logic does), supervaluationism accommodates truth-value gaps. For instance, if Tek has a height of 178 cm, the statement “Tek is tall” is neither true nor false. However, this does not mean that this statement assumes some further, third truth value. Likewise, when 120 switches are flipped in HARMLESS TORTURERS, it is neither true nor false (nor indefinite) that the victim is in pain. For this reason (and a few others), supervaluationism is substantially compatible with classical logic.
According to supervaluationism, then, (2b) is false. On each completely sharp sharpening of “being in pain”, there is a sharp boundary between the pain states where the victim is in pain and the pain states where he is not. This may seem strange. The following examples may help to explain how it can be so. Consider the statement:
(A) There is a sharp boundary between persons who are tall and persons who are not.
However we sharpen “tall”, (A) will be true. This means that (A) is supertrue – and therefore true. If, for instance, we sharpen “tall” so that a person is tall if and only if they are taller than exactly 178 cm, there will be a sharp boundary between people who are tall and people who are not. Everyone will either belong to the set of those who are tall or to the set of those who are not tall. Similarly, if we sharpen “tall” so that a human being is tall if and only if they are taller than exactly 180 cm, there will be a sharp boundary between tall and non-tall people, although of course slightly fewer individuals will now belong to the set of those who are tall.
The same goes for the following statement:
11 In what follows, when I talk about sharpenings, I mean completely sharp sharpenings.
(B) There is a sharp boundary between the states in which the victim is in pain and the states in which he is not in pain.
This statement will be true on all sharpenings of “in pain”, and therefore supertrue, and therefore true. Generally, statements of the form “There is a sharp boundary between the states in which x is F and the states in which x is not F” come out true on all sharpenings of F (see Fine 1975; Keefe 2000). Putting the point differently, we can say that supervaluationism entails that the inductive premise (For all n, if xn
is F then xn+1 is F) is false. On each sharpening of F, there will be some n for which xn is F and xn+1 is not F.12
So, the epistemic view, three-valued logic and supervaluationism all entail that (2b) is false. A sharp boundary separates cases where the victim is in pain and cases where he is not. This opens up the possibility that Kagan’s conclusion is correct after all. It might be a conceptual truth that there are no imperceptible difference cases. Our pre-theoretic verdict about the inductive premise is most likely mistaken.13
Another Way to Reach the Same Conclusion
We could reach the same conclusion without the apparatus of the epistemic view, or three-valued logic or supervaluationism. Argument #1 prompts us to reject one of the following:
• (1) when no switches are flipped, the victim is not in pain,
• (2b) for all n, if the victim is in no pain in sn, he is also in no pain in sn+1,
• (¬3) the victim is in pain when 1,000 switches are flipped, or
• the validity of the argument.
Kagan thinks we have to reject (2b), and by extension (2a). Nefsky argues that we are not forced to do so, as we might instead reject (1), (¬3) or the validity of the argument. However, Nefsky’s argument is not fully convincing. She is right that we
12 Note, however, that it is indeterminate where the sharp boundary is located. This is indeterminate because the location of the boundary differs from sharpening to sharpening. The range of admissible sharpenings is also indeterminate.
13 We cannot definitely conclude that our pre-theoretic verdict is mistaken. Other theories of vagueness (e.g. the already mentioned accounts by Unger, Russell and Dummett) locate the problem elsewhere. These accounts, however, have problems of their own (see e.g. Keefe 2000:
18-26). So, we will still do best to rely on the verdicts of the epistemic view, three-valued logic and supervaluationism.
are not forced to reject (2b). However, on closer reflection, it would seem that doing so is likely the best option. It seems clearly wrong to deny (1). Likewise, the idea that the victim is not in excruciating pain when all the switches are flipped is farfetched, to put it mildly, so (¬3) is safe. Moreover, the argument is almost certainly valid. As Dummett (1975) argues, to reject sorites arguments as invalid we must give up some fundamental rules of inference, and doing so would be a high price to pay. Sorites arguments rely merely on the inference rules of modus ponens and universal instantiation. The rejection of these rules would have far-reaching effects.14 This leaves us with (2b). This premise might appear correct at first sight, but it is not as obviously correct as the other premises, or as difficult to challenge as the validity of the argument. So, it appears that were we forced to identify a problem with this sorites argument, we should point to (2b). Given this, we have independent reasons to think that the verdict on this issue given by the epistemic view, three-valued logic and supervaluationism is correct.
Conceptual Sharp Boundaries Do Not Have to Be Perceptual
While each of the epistemic view, three-valued logic and supervaluationism implies that (2b) is false, none of these accounts implies that (2a) is false. This indicates that there is a problem in the implication from (2a) to (2b). To explain how this can be the case, we have to go back and consider the different ways of handling vagueness more closely.
On the epistemic view, there is always an unknown sharp boundary in sorites cases.
However, this boundary does not have to correspond to a sharp boundary in the world. It may instead correspond to a sharp boundary in our language. According to Timothy Williamson (1994), for instance, the location of the sharp boundary is determined by the meaning of the vague predicate in question, and meaning, in turn, supervenes on use. If we are to determine whether the victim is in pain when 120 switches are flipped, we have to consider the meaning of the predicate “is in pain”.
This in turn requires us to consider how this predicate is used in ordinary language.
In many cases, the meaning of a predicate is stabilised by natural divisions, making small changes in use irrelevant. With this in mind, Williamson says, a “slightly increased propensity to mistake fool’s gold for gold would not change the meaning or the extension of the word ‘gold’” (1994: 231). However, he warns that this is not true of vague predicates, since where these are concerned there are no natural divisions that might help stabilise the boundary. When it comes to vague predicates, meaning supervenes wholly on use. So, even if, on the epistemic view, there is a sharp boundary in HARMLESS TORTURERS, this boundary does not exist because the
14 If you think that the universal quantifier might give rise to validity problems, argument #1 could be restated without it, using only modus ponens.
victim perceives it. Its existence supervenes on the way we use the predicate “is in pain” in ordinary language.
Elaborating this point, we can note that Williamson (1994: 180-84) also argues that even if an object is red, an observer will not necessarily know that it is red. The observer might be mistaken. This, he continues, goes for any observational property, including “being square” and (presumably) “being in pain”. Consider again
HARMLESS TORTURERS, and imagine that we flip the switches one at a time, and that after flipping each switch we ask the victim, “Are you in pain?” It seems likely that at some interval in this series of switch-flipping the victim would be unsure how to answer. Perhaps when the 120th switch is flipped, he will answer: “Well, I feel something, but I am unsure of whether I should call it pain.” As more and more switches are flipped, he might become increasingly confident that he has passed the threshold for being in pain, and if we require him to give a simple “Yes” or “No”
answer, he will eventually stop answering “No” and say “Yes”. There is nothing in this story to indicate that the victim would necessarily perceive a difference in pain between two adjacent states.
The upshot is that the epistemic view does not ensure that there is a perceptual difference of the kind Kagan needs. The victim might be in pain without perceiving pain (i.e. the statement “the victim is in pain” could be true even where the victim does not recognise what he is feeling as pain), and there might be a difference in pain even though the victim does not perceive any difference in the pain.
Three-valued logic runs into a similar problem, and for similar reasons. While it implies that a sharp conceptual boundary marks the point at which the statement
“the victim is in pain” becomes true rather than indefinite (and another where this statement becomes indefinite and no longer false), three-valued logic fails to imply that there is a sharp perceptual boundary where the victim starts being in pain. And so, again, it will not deliver Kagan the result he needs.
Likewise for Supervaluationism, but for a different reason. One might assume that if a statement about the world is true, this truth will correspond to something in the world. However, if we take supervaluationism at face value, this is not always the case. As Bertil Rolf (1981) points out, a statement like “there exists a sharp boundary between red and pink” comes out as true even though there is no such boundary. It does so because on each completely sharp sharpening of “red” and
“pink” there is a sharp boundary between these colours. Rolf questions supervaluationism as such on these grounds.
Faced with Rolf’s objection, supervaluationists have two options: they can bite the bullet and agree that supervaluationism separates statements of existence from actual existence, or they can restrict the range of statements to which supervaluationism applies. Keefe (2000) and Lewis (1993) take the first option.
Broome (2004) takes the second. Either way, supervaluationism does not entail that the victim can perceive a difference in pain between a certain pair of adjacent
affective states. If we bite the bullet, the fact that supervaluationism entails that (B) is true will tell us nothing about what the victim perceives. And if we restrict the range of statements to which supervaluationism applies in order to avoid biting the bullet, supervaluationism will not be applicable to (B) (my earlier discussion implicitly assumed that it was). In what follows, for the sake of simplicity I will assume that supervaluationism is applicable to statements like (B).
According to the supervaluationist, then, there is sharp boundary between the states in which the victim is in pain and the states in which he is not (i.e. (B) is true), even if there is no sharp boundary separating the states in which the victim perceives pain and the states in which he does not.
To sum up, the epistemic view, three-valued logic and supervaluationism all entail that there are sharp conceptual boundaries in imperceptible difference cases like
HARMLESS TORTURERS. However, they do not entail that these boundaries are perceptible. Where argument #1 is concerned, they entail that (2b) is false even if (2a) is true. This means that the implication from (2a) to (2b) is invalid. In turn, this means that argument #1 does not show what Kagan wants it to show: that, necessarily, there is a perceptible threshold in all collective harm cases. This should not surprise us. Even if it is a conceptual truth that there is a sharp border between the states where the victim is in pain and those where he is not, this tells us nothing about whether the victim can perceive this difference. The first point is about language, and the second is about perception.
Moreover, since argument #1 fails to show that the flipping of a switch makes pain perceptibly worse, it also fails to show that the expected utility approach gives the right verdict in all collective harm cases. Only perceptible differences are morally relevant according to THE PERCEPTIBILITY PRINCIPLE, and because there is still a live possibility that no flipping of a switch makes a perceptible difference in pain, it may still be the case, for all that has been said, that no flipping of a switch makes a morally relevant difference.