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15.5 The Argument from Error A more important argument of Kripke’s is his Argument from Error. It begins with the observation that most any person’s dispositions may contain dispositions to make mistakes.5 For example, someone might systematically forget to ‘carry’ in certain circumstances. If we tried to read off the function that’s meant from the totality of a person’s dispositions, we will undoubtedly end up assigning the wrong meaning, intuitively speaking. To get around this, it seems, we have to specify, in non-intentional, non-semantic terms, a set of ideal conditions, C, under which one’s dispositions will be in perfect conformity with the meaning of one’s term, thus: (Refined Dispositional Theory): Necessarily, ‘S means plus by “+” ’ is true iff: For any two numerals ‘m’ and ‘n’ denoting particular numbers m and n, S is disposed, if conditions are C, and if queried about ‘m + n’, to reply ‘p’ where ‘p’ is a numeral denoting plus (m,n). Kripke’s way with this proposal is brief. He says: A little experimentation will reveal the futility of such an effort. Recall that the subject has a systematic disposition to forget to carry in certain circumstances: he tends to give a uniformly erroneous answer when well rested, in a pleasant environment free of clutter, etc. (1982: 32) (p.339) I think that Kripke is ultimately right that there are no such non-intentionally, non-semantically specifiable conditions under which it would be impossible for us to make a mistake in the application of ‘+’. But I think he doesn’t give us as much of an argument for this claim as we would like. What more can we say on its behalf?Well, it is certainly hard to see that there are a priori specifiable conditions, C, under which it would be impossible for us to make mistakes in the applications of our terms. It’s certainly not built into our concept of addition that there are such conditions C. Nor do I know of any other a priori source of insight into the nature of computation that indicates the existence of such conditions. But what’s to say that there aren’t such conditions that could be discoverable a posteriori? Indeed, how could Kripke claim to know that there aren’t such conditions? Has he done the relevant science? As a matter of fact, I think we can make a pretty strong case that such conditions can’t be discoverable a posteriori either. Recall, we are looking for a dispositional property that we can identify with the property of meaning addition. That means we are looking for conditions under which it would be, as a matter of metaphysical necessity, impossible for S to make mistakes in his use of ‘+’. Surely, though, the conditions under which S’s dispositions would be guaranteed to coincide with what he refers to by ‘+’, assuming that there are such conditions, would have to be contingent on the sort of creature S is, on what his actual cognitive apparatus is like. With creatures like us, being well rested, in a quiet environment, free of clutter, and so forth seems ideal for minimizing errors in arithmetic. But there are no doubt possible creatures out there (Jabba the Hutt, perhaps) who can grasp addition, but for whom a noisy environment, harsh lighting, and lack of sleep are better for doing arithmetic. But if we identified meaning addition with having such and so disposition under conditions-ideal-forus, we would have to say that Jabba the Hutt couldn’t mean addition. And that is implausible. Surely, all kinds of creatures with all sorts of cognitive profiles could instantiate the property of meaning addition. The upshot is that it is hard to believe that there are conditions C under which it would be metaphysically impossible for an adder to make a mistake in adding, whether these conditions are thought of as a priori or only a posteriori discoverable.6 15.6 The Argument from Finitude I believe that, between them, the Argument from Explanation and the Argument from Error provide a strong case against the reduction of meaning to dispositions, whether this reduction is thought to be of the a priori or the a posteriori type. (p.340) But I don’t think these arguments extend as far as threatening the Supervenience claim. Mere supervenience of meaning on the dispositional does not require us to be able to replicate the explanatory work of meaning in terms of dispositions. Nor does it require that meaning properties always be realized by the same sorts of dispositional fact. All it requires is that fixing the dispositional facts fixes the meaning facts, but not necessarily the other way around. This explains why I attach great importance to the last of the arguments that Kripke presents, the Argument from Finitude: I think that this argument does have the potential to undermine not only Reduction but Supervenience as well. Kripke states the argument thus: … not only my actual performance, but also the totality of my dispositions, is finite. It is not true, for example, that if queried about the sum of any two numbers, no matter how large, I will reply with their actual sum, for some pairs of numbers are too large for my mind—or my brain—to grasp. When given such sums, I may shrug my shoulders for lack of comprehension; I may even, if the numbers involved are large enough, die of old age before the questioner completes his question. (1982: 27) If my use of ‘+’ determinately refers to the plus function, then it correctly applies to a particular infinite set of triples (the one that corresponds to the plus function) and not to any of these other ones, the ones that correspond to the quus-like functions. Kripke poses a challenge to the dispositionalist: How could such a fact be captured dispositionally when we cannot be said to have dispositions to answer with particular numbers with respect to addition problems involving inaccessibly large numbers? It’s important to see that this isn’t a problem that arises merely from the ‘infinitary’ character of the dispositions in question, as some early commentators were inclined to think. Blackburn (1984) , for example, thought that there had to be something fishy about Kripke’s argument. In what sense are our dispositions finite, he asked? And how does it follow from our finitude that our dispositions are finite? Take a humble ordinary object such as this glass. It can have infinitary dispositions. It can be disposed to break when struck here, or when struck there, when struck at this angle or at that one, when struck at this location, or at that one. And so on, ad infinitum. Isn’t this the possession of an infinitary disposition? And if a mere glass can have infinitary dispositions, why couldn’t a human being? This reply is correct as far as it goes: it doesn’t follow from a thing’s being finite (whatever exactly that means) that it has only finite dispositions (whatever exactly those are). But just as it doesn’t follow from a thing’s being finite that it can’t have some particular infinitary disposition, it also doesn’t follow that it can have that disposition. Each attribution must be examined on its merits. The problem Kripke is pressing for the dispositional account depends on there being a difference between a human adder and the glass. The glass is disposed to break under a potential infinity of circumstances, but the human adder isn’t disposed to respond (p.341) with an answer to addition problems involving certain very large numbers. How does Kripke show this?7 Kripke’s argument seems to proceed as follows. First, he assumes a Counterfactual Conditional Analysis of Disposition talk: (CCAD) ‘S is disposed to do A if C’ is true iff if C, S would do A. And then he assumes a Possible Worlds Analysis of Counterfactual Conditionals: (PWACC) ‘If C, S would do A’ is true iff in the nearby worlds in which C, S does A. Neither of these assumptions is explicitly spelled out in Kripke’s argument; but they capture well the way in which his argument unfolds. With these assumptions in place, we can now articulate his argument as follows: in the case of the glass, it seems correct to say that its disposition to break when struck is infinitary because it’s clear that there is an infinite set of nearby worlds in each of which the glass is struck with a slightly stronger force than in the preceding world, and in each of which it breaks. But it looks to be not similarly plausible to say that a human being has a disposition, when asked about the sum of an arbitrary pair of numbers, no matter how large, to respond with their sum. Consider an arithmetical question involving a number that is bigger than I can properly take in, perhaps even bigger than I can take in within my lifetime. What happens in the nearest world in which I am asked this question (obviously by a being whose life span is longer than mine and who has nothing better to do with his time)? In the nearest such world, I would die before I could grasp which numbers are at issue. In the case of the glass, the existence of the infinite number of inputs or manifestation conditions—the different levels of force, angles, locations, and so forth—just follows from the nature of the glass qua physical object. No idealization is required. But to be able to respond to an arbitrary arithmetical problem I have to be able to grasp the numbers in question. And a capacity to grasp arbitrarily big numbers—the inputs in the arithmetical case—does not follow from our nature as thinking beings, and certainly not from our nature as physical beings. Indeed, at least at first blush, it seems that what does follow from our nature as finite biological beings that live for a finite period of time, is that we do not have that capacity: there are limits to the size of the numbers that we are able to grasp or process. Now, the functions plus and quus differ with respect to numbers that are, by assumption, inaccessible to a thinker. If a thinker, S, determinately refers to plus by his symbol ‘+’, then, with respect to a question Q involving an inaccessible number, it would be (p.342) correct for him to respond with a given number, say, p, and if he determinately refers to quus, then it would be correct for him to respond with a different number, say, p. The dispositionalist’s idea is that whether it would be correct for S to respond to Q with p or p is determined—a posteriori, admittedly, but nonetheless determined—by S’s dispositions. But, as we have just argued, S’s dispositions do not extend as far as the relevant correctness facts. He has no dispositions with respect to the inaccessible number that distinguishes between the p answer and the p* answer. How, then, could his dispositions determine that p is the correct answer as opposed to p*? 15.7 Dispositions and Conditionals This, then, is an articulated version of the argument that we may find in Kripke. The question is whether it works. One line of resistance to it has targeted its reliance on the conditional analysis of dispositions. It is widely agreed among philosophers that (CCAD) was decisively refuted by Martin (1994) . Martin presented counterexamples to both directions of the analysis, though I shall concentrate on the left to right direction. Our glass may be fragile; but it is not inconceivable, to use an example of David Lewis’s, that a sorcerer should lurk in the wings, watching and waiting, so that if ever the glass is dropped, then, practically instantaneously, he casts a spell that renders it unbreakable. Under this scenario, the glass’s fragility is ‘finked’: even though the glass has the disposition to break, a factor ensures that if ever it were to try to manifest itself the causal basis for the disposition would disappear. Johnston (1992) has also called attention to the way in which a disposition can be ‘masked’. Our glass is fragile, but it may not break if it is given the right sort of internal packing to fortify it against hard knocks. Our glass retains the disposition to break, but some factor in its environment blocks its manifestation, even as it retains that disposition. Given these problems for the conditional analysis, it looks possible to respond to Kripke’s argument in the way that Martin and Heil (1998) did: they concede that there are no nearby worlds in which, if I am queried about the sum of ‘m + n’ for arbitrary m, n, I will respond with their sum, but deny that it follows from this concession that I am not disposed to give an answer. That inference requires (CCAD), which has been rejected. Martin and Heil thus conclude: The infinity discoverable in P (or in any other disposition, mental or physical) will seem mysterious only so long as one fails to distinguish P as a disposition from its manifestations. (1998: 303) However, as Handfield and Bird (2008) have pointed out (in an excellent discussion of Martin and Heil), this conclusion seems hasty. For the connection between (p.343) dispositions and conditionals has been shown to be broken only in the cases of finks and masks. And the paradigm cases of finks and masks involve factors that are extrinsic to the object that has the relevant disposition (the sorcerer and the packaging are both extrinsic to the glass). However, the factors in virtue of which an agent is unable to compute the sums of inaccessible numbers look to be highly intrinsic to him. They are intrinsic to his cognitive powers. Hence, it’s hard to see that reliance on the existence of finks and masks can provide a good defense against the Argument from Finitude. 15.8 Idealized Dispositions For all that finks and masks show us, then, it looks as though we could continue to assume that our having dispositions vis-à-vis inaccessible numbers does amount to the truth of certain counterfactual conditionals. And, therefore, that we may continue to take the falsity of those conditionals to imply that we do not have the relevant dispositions. Although in what follows I will occasionally fall in with this way of talking, it is not essential to anything I will want to be claiming. Even if we assumed that disposition talk cannot be analyzed at all, let alone in terms of conditionals, I believe that the arguments presented below would still go through.8 We cannot defend dispositionalism against the Argument from Finitude simply by rejecting his implicit appeal to (CCAD). It might be thought, however, that there is a better way for the dispositionalist to go, one that would appeal to the notion of an ‘ideal disposition’. The thought is that although it isn’t true that For any two numerals ‘m’ and ‘n’ denoting particular numbers m and n, S is disposed, if queried about ‘m + n’, to reply ‘p’ where ‘p’ is a numeral denoting plus (m, n) it might be true to say that For any two numerals ‘m’ and ‘n’ denoting particular numbers m and n, S is disposed, if conditions are ideal, and if queried about ‘m + n’, to reply ‘p’ where ‘p’ is a numeral denoting plus (m, n). We originally introduced the idea of an ‘ideal disposition’ in order to deal with the fact that we all have dispositions to make mistakes. Even with respect to the accessible numbers, we are disposed, at least if conditions are unfavorable, to give answers that deviate from the answers that it would be correct for us to give. If dispositionalism is to (p.344) avoid assigning the wrong meaning, it has to assume that there is a set of non-trivially specifiable ideal conditions, under which we would be disposed to give answers that are in perfect conformity with the function we mean, at least with respect to the accessible numbers. In connection with the problem of Error, this notion of idealization attempts to idealize away from sources of error in our processing of numbers that we can grasp, rather than idealize away from our inability to grasp or process numbers that are above a certain size. Even with respect to this much more modest task, we have seen that it is very far from clear that there are such conditions. However, let us set those objections aside for the moment. Suppose that there are conditions, C, under which a subject is guaranteed to give the correct answer with respect to the accessible numbers. Might a notion of ideal conditions help not only with the error, but with the finitude problem as well? (Plausibly, such a solution would involve a different, or expanded, set of ideal conditions.) Kripke considers such a defense of dispositionalism: I have heard it suggested that the trouble arises solely from too crude a notion of disposition: ceteris paribus, I surely will respond with the sum of any two numbers when queried. And ceteris paribus notions of dispositions, not crude and literal notions, are the ones standardly used in philosophy and in science. Perhaps, but how should we flesh out the ceteris paribus clause? Perhaps as something like: if my brain had been stuffed with sufficient extra matter to grasp large enough numbers, and if it were given enough capacity to perform such a large addition, and if my life (in a healthy state) were prolonged enough, then given an addition problem involving two large numbers m and n, I would respond with their sum, and not with the result according to some quus-like rule. (1982: 27) Kripke’s response is dismissive: But how can we have any confidence in this? How in the world can I tell what would happen to me if my brain were stuffed with extra brain matter, or if my life were prolonged by some magic elixir? Surely such speculation should be left to science writers and futurologists. We have no idea what the results of such experiments would be. They might lead me to go insane, even to behave according to a quus-like rule … But of course what the ceteris paribus clause really means is something like this: If I somehow were to be given the means to carry out my intentions with respect to numbers that are presently too long for me to add (or to grasp), and if I were to carry out those intentions, then if queried about ‘m+n’ for some big m and n, I would respond with their sum (and not their quum). (1982: 27) Now, Kripke is surely right to say that we currently have no idea what the truth-value of the following counterfactual is: (Enhanced): If my brain were enhanced in certain specified ways, and my life were prolonged, I would answer with the sum to the question ‘m + n = ?’ for any two m and n, no matter how large. (p.345) We certainly do not know the truth of such a counterfactual a priori. Nor do we currently have any empirical evidence to support belief in (Enhanced) (pretending for the moment that it is determinate enough to be empirically tested). So, if the truth of a dispositional reduction of meaning turns on the truth of (Enhanced), we are certainly not now in a position to assert that such a dispositional reduction is true. However, and obviously, it doesn’t follow from this concession that dispositionalism is not true. All that follows is that it is not currently assertible. For all we currently know, there might be empirically discoverable conditions, C, specifiable non-intentionally and non-semantically, which are such that, if they obtain, then I will respond to arbitrarily large addition problems with their sum. And if there are such conditions, then it will be true to say that I am disposed to respond with the sum to addition problems involving inaccessible numbers.Given only Kripke’s arguments, then, we can say only that dispositionalism is not now assertible; we cannot say, what he seems to want to say, that it has been shown to be false. Once again, attention to the distinction between a priori and a posteriori forms of reduction seems to expose a gap in Kripke’s argument. Are there are any considerations that show that not only is dispositionalism not currently assertible, but that it is not true? I think there are. First of all, we can repeat the point made earlier, that conditions that are ideal for creatures like us may well not be ideal for other possible creatures who may also be able to grasp addition. However, I think there is a more fundamental worry with this particular deployment of ideal dispositions to deal with the problem of finitude. We can get at it by noting that there is something fishy about the dispositionalist helping herself to the dispositions she would have if her brain or cognitive powers were enhanced in various ways. After all, what she is trying to explain is how I, with my current cognitive powers and brain capacity, mean addition by ‘+’. If this is to be identical to my having a certain disposition, it should be identical to a disposition that I have more or less as I actually am, not to the dispositions that I would have if I were much more powerful than I actually am. In Boghossian (1989 : 31), I put the point as follows (relying explicitly on (CCAD), but the underlying lesson is independent of it): … not every true counterfactual form of the form If conditions were ideal, then, if C, S would do A can be used to attribute to S the disposition to do A in C. For example, one can hardly credit a tortoise with the ability to overtake a hare by pointing out that if conditions were ideal for the tortoise—if, for example, it were much bigger and faster—then it would overtake it. Obviously, only certain idealizations are permissible … To vary the example somewhat, it seems right to say that a humble Volkswagen Golf is unable, and so not disposed, to overtake the twelve-cylinder Bentley, when (p.346) they are both going at full tilt. But if, in judging this question, we were allowed to look at the capacities the Volkswagen would have if it were much faster and more powerful than it actually is, say as fast and powerful as the Ferrari 458, then we would be able to say that it is now able and disposed to overtake the Bentley. As a way of gauging the Volkswagen’s current dispositions, that would clearly be absurd.9 Similarly, it looks absurd to determine what dispositions I have in respect of ‘+’ by looking at the dispositions that I would have if I were much more cognitively powerful than I actually am. The dispositions relevant to a dispositional account of meaning are the dispositions I have, pretty much as I actually am, and not the dispositions that I would have if I were much more cognitively powerful than I actually am. Of course, among the dispositions I actually have are dispositions to respond in certain ways, when conditions are ideal—for example, when I am sober, well rested, in a quiet environment, and so forth. But that is different from the dispositions that I would have, if I were much more cognitively powerful than I actually am. In figuring out the first sort of disposition, we keep my cognitive powers more or less fixed, and simply vary the circumstances under which they are exercised. In figuring out the latter sort of disposition, we are allowed to look at situations under which my cognitive powers are far greater than they actually are.We could put the point this way. Let’s stipulate that there is a being, say God, that has dispositions to respond with the sum for arbitrarily large numbers m and n, and so is able to mean addition by ‘+’ on a dispositional view. How can that fact help me have the disposition so to respond, and so to mean addition on a dispositional view? I suppose that we could cook up an externalist view according to which I could get to mean addition by deferring to God. This would have the peculiar consequence that only those practicing the correct religion could mean addition rather than quaddition! And it would not help any of us finite beings at all, if there were no God. Seriously, though, if a dispositional view is to explain how it is that I mean addition by ‘+’, it had better be that it can do so in terms of my actual dispositions, rather than the dispositions that would be had by some superhuman version of me. These dispositions can be ones that I have in (certain kinds of) idealized circumstances, but they must be dispositions that I actually have, with my cognitive powers kept fixed.10 (p.347)