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15.9 Dispositions and Rules A number of writers sympathetic to dispositionalism are prepared to concede that we do need to work with the dispositions that we actually have, as opposed to those that we would have under enhanced conditions. But they maintain that, in the relevant sense, we do have the requisite dispositions. Martin and Heil describe this idea, somewhat abstractly, as follows: [Suppose] that an agent S possesses a disposition P, constituting S’s mastery of R, the plus rule … Now, imagine that at t1, S has P. At t2, S may acquire the capacity to form a range, L, of very large numbers. Call this capacity C. Note that the addition of C does not require the further addition of P; P is present already. Even at t1, P is ‘ready to go’ for adding numbers in L, numbers that, at t1, S lacks the capacity to consider or manipulate. It is the being-there-ready, without need of supplementation or alteration for any such numbers— whether S gains the capacity to manipulate them or not—that constitutes P’s infinity. S’s finitude with respect to addition results from limits on numbers S can consider or manipulate at any given time. This is a limitation not on P but on P’s manifestations owing to limitations in C, one of P’s reciprocal disposition partners. This is a limit not on magnitudes of numbers S (in virtue of possession of P) is prepared to add; the dispositional readiness encompassed by P is for any magnitude, and is in that sense infinite. (1998: 302) Here is a way of thinking about what Martin and Heil are suggesting here. Let’s concede that I don’t have a disposition to respond with the sum with respect to arbitrarily large numbers m and n. However, we can still make it plausible that I am now disposed to process any two numbers in the same way, according to the same rule, if only I could grasp those numbers. That is consistent with conceding that there are many numbers I can’t grasp. But it is also enough to credit me with computing addition rather than quaddition. We can motivate how Martin and Heil are thinking here—and they are certainly not alone in finding this an appealing line of thought—by thinking about the architecture of a Turing machine. Such a ‘machine’ has three distinct components. There is the ‘read/write head’, the tape, divided into cells, and the table of transition rules that specify on the basis of what it ‘sees’ in the current cell, what action it is to perform, and what state it is to transition to. For any Turing machine, it is the table of transition rules that determines the function that it computes, not any of the other components. Now, suppose that when we look at our own mental architecture, we see that we can isolate distinct components that are responsible for our capacity to add, in parallel to the way in which we can isolate such components in a Turing machine. There is, on the one hand, a component that corresponds to our capacity to retain in memory (p.348) the summands (the tape), a component that serves as the memory bus (the read/write head), and, finally, a component that embodies the algorithm or rule that we follow when we try to add numbers. Perhaps these components could be thought of as housed in distinct parts of a subject’s brain, in such a way that it can seem obvious that, no matter what numbers we are given to add, we would always subject them to the same procedure, no matter how large or small those numbers may be. That could just be a feature of the architecture. Then, it looks as though we could think of our meaning addition by ‘+’ as consisting just in the fact that we would subject any two summands to a particular procedure (adding them), even as we concede that there is a real limit to the size of the summands that we can grasp or consider. The thought is that we can isolate the fact that we are following a certain rule, from the fact that we are unable to grasp all the inputs over which that rule is defined.11 This idea is an appealing one in the case of addition and various other mathematical notions. It is not clear whether it could be made to apply across the entire range of concepts. What would correspond to the two factors in the case of the concept water or the concept ought? Let us stick to the case of plus, though, as we have been doing, and let’s see whether this factorization idea can help out a dispositional view, at least in this importantly representative case. Now, Wittgenstein’s great insight, of course, was that there is just as much of a problem saying what rule or algorithm a particular concrete mechanism is following as there is saying what concept it is deploying. So the appeal to rules or algorithms, housed in an isolable component of our mental architecture, can’t help all by itself. We still need to see how we are going to give a dispositional account of following a particular rule or algorithm, even if we are allowed to factorize meaning addition into a component that houses the rule, so to say, and others that house the inputs and outputs to that rule. And now, it would seem, we are right back where we started. We still face the problem that our brains will last for only a finite amount of time and have limited processing capacity. After a while, that bit of the brain that is said to house the rule is disposed to sputter rather than give an answer. And nothing we have been provided so far helps us get around this problem. 15.10 Computation and Physical Devices But can’t it be entirely determinate what rule or set of instructions a bit of our brain is employing? (p.349) After all, we each have in our offices a machine that is, in part, an adder. If there are such determinate facts about something as basic as a desktop computer, how could there fail to be such determinate facts about us and our brains?12 Stabler (1987) has provided an illuminating discussion of this question. His strategy relies on looking at an extremely simply device, much simpler than an adder, but similar to it in that it realizes an infinite function on the natural numbers. His example concerns an electrical circuit that computes the identity function on the natural numbers under an interpretation that maps a sequence of i voltage pulses at a specified input point into the number i, and similarly maps any sequence of i pulses at the output point into the number i. The circuit for the device may be represented by the following simple circuit diagram: input ————— outputAnd it looks as though all that would be needed in order to physically realize such a device is a simple wire connecting the input to the output. Now, of course, any real wire will transmit pulses for a while and then break down. Suppose it ceases to transmit after the fifty-seventh pulse. Why should we say that this wire is the realization of the infinite identity function? Why shouldn’t we say that it is a realization of the function that maps numbers less than fifty-eight into themselves and thereafter maps every number on to five? Stabler says: There is a natural response to this problem. It is just to point out that if the system had continued to satisfy conditions of normal operation for long enough, it would have composed arbitrary values of the identity function. In the case of the wire, the identity function is distinguished from other functions which agree only on actually performed computations; the physics of simple circuits tells us this. For example, we do not want to say that the wire realized a function F’ such that F’(58)=5, because we know from the simple physics of the device that if it had continued to satisfy the background conditions (of being a simple conductor), it would not have computed this value. So this is the requirement on the realization of any function: the system must be such that it would compute arbitrary values of the function if it lasted long enough. (1987: 9–10) The important point to note is that the claim that some particular physical system computes some particular function clearly rests on a particular choice of background or normal conditions, and that these conditions are not given just by looking at the device itself. Of course, describing our device as a conductor analytically implies certain conditions of normal operation. But we can imagine alternatives. We could suppose that our condition of normal operation did not involve the presence of a conductor but rather the presence of a conductor for fifty-seven pulses and the absence of any electrical connection thereafter. Then we should regard the wire as having realized not the identity function, but rather the function (p.350) Bent(x) = x if x < 58 = 0 otherwise. If the device were to continue to conduct voltage pulses past the fifty-seven mark, we would have to regard the device as malfunctioning. It would still be a device for computing the function Bent(x), although it would be giving incorrect answers. As Stabler remarks: … fixing the conditions of normal operation is crucial for making determinate claims about what function a system is computing. … However, as in the case of the wire, usually there is no problem seeing what the natural background condition is intended to be, even if it is not stated explicitly. (1987: 10) So: determinate facts about what function a system is computing rests on a choice of normal conditions for the operation of that system. And, such conditions are determined, at least in part, by the designer’s intentions. Interestingly, although Stabler’s piece is written as a criticism of Kripke’s finitude argument (at least in a version published before the book), he ends up in a position that is in perfect agreement with what Kripke has to say about the matter: Actual machines can malfunction: through melting wires or slipping gears they may give the wrong answer. How is it determined when a malfunction occurs? By reference to the program of the machine as intended by its designer, not simply by reference to the machine itself. Depending on the intent of the designer, any particular phenomenon may or may not count as a machine ‘malfunction’. A programmer with suitable intentions might even have intended to make use of the fact that wires melt or gears slip, so that a machine that is ‘malfunctioning’ for me is behaving perfectly for him. Whether a machine ever malfunctions and, if so, when, is not a property of the machine itself but is well defined only in terms of its program, as stipulated by the designer. (1987: 34–5) Now, if all of this is right, we can see why talk of machines determinately computing particular functions won’t help us with our problem. A machine may be said to determinately compute addition relative to the intentions of its designer and that designer’s selection of particular conditions of normal operation. But that would be a case of derived intentionality, not the sort of original intentionality that we are after in our own case. How is it determined in our own case, in the case of our own brains and cognitive systems, what the conditions of normal operation are? Not by further intentions, on pain of vicious regress; and not merely by our dispositions to respond in certain ways to certain stimuli. It is at this point that it is tempting to think that biology will help us with our problem (this is the idea behind ‘teleosemantics’). Couldn’t we make a case for saying that the biologically determined conditions for the normal operation of our cognitive mechanisms determine plus rather than quus as the function that we mean? (p.351) There are any number of problems with this line of thought, the most important of which, for present purposes, is that it’s hard to see that evolution would care about what we are disposed to do in respect of numbers that are nomologically inaccessible to creatures like ourselves. So, at least in the particular case we are looking at in detail, it’s hard to see how facts about natural selection are going to remove the indeterminacy that dispositionalism about meaning seems to entrain.13 15.11 Reduction versus Supervenience If the Argument from Finitude is correct, it doesn’t merely make trouble for dispositional analyses of meaning, or the identification of meaning properties with dispositional properties; it seems to make trouble even for such a weak thesis as that of the supervenience of determinate meaning facts on the dispositional facts. For the argument takes the form of showing that there is not enough in the supervenience base to sufficiently constrain determinate facts about meaning. The base seems too impoverished to do the job. If it is precisely plus rather than quus that we refer to, it looks as though there has to be a further fact doing the determining. Now, I can imagine someone trying to resist this conclusion by deploying the following line of thought:14 Look, what you are doing is covertly sneaking in an explanatory requirement: not only must the dispositional facts determine the reference facts; it must somehow be explainable how they do so. Then, since no explanation looks forthcoming you are concluding that they couldn’t determine it. But this is to impose an illegitimate explanatory requirement. Explanations come to an end somewhere. What makes you think they don’t come to an end precisely here, where one set of dispositional facts determines one determinate meaning fact as opposed to another? I agree that explanations come to an end somewhere and that it is not out of the question that they cometo an end precisely in principles connecting dispositional facts with determinate meaning facts. But that concession by itself does not absolve us from evaluating any particular determination claim for plausibility. Sometimes it can just be clear that a particular determination claim is wrong. Let me give what I take to be an especially compelling example that arises in a distinct although highly related context. It is provided by Stephen Schiffer’s objection to Timothy Williamson’s epistemicist view of vagueness.15 According to Williamson, a vague predicate, such as ‘bald’, has a perfectly determinate extension: some determinate number of hairs is required in (p.352) order for someone to count as not bald, it’s just that we don’t know what that number is. Furthermore, Williamson accepts the view that such determinate reference is not metaphysically primitive: it supervenes, broadly speaking, on the thinker’s psychology, including his referential intentions, and perhaps on his relations to his environment. Schiffer objects to this view by asking how there could be anything in our psychologies, or in their relations to their surroundings, or in any other remotely relevant location, that could determine such an exquisitely precise boundary. Consider the application of Williamson’s view to the command that I might give to someone in the course of photographing him: ‘Stand over there!’ On Williamson’s view, in saying ‘Stand over there!’ I must have referred to some very specific boundary within which the person being photographed must stand if he is to count as complying with my request, even if I don’t know precisely which boundary this is. It’s a legitimate criticism of this view to say that there doesn’t seem to be enough in any available fact about my psychology to attribute to me determinate reference to one exquisitely determinate boundary as opposed to another. It’s not just that I don’t know what it is. It is implausible to claim that there was such an exquisitely precise boundary that I must have been referring to. A similar worry arises in the arithmetical case. If we insist that if there is to be determinate reference at all, then it must be determined by one’s use-dispositions, then it seems to me that we must conclude that there couldn’t be determinate reference to plus as opposed to quus. If meaning supervenes on the dispositional facts, then meaning is not determinate. 15.12 Why Plus and Not Quus? There is another way of bringing out the implausibility of saying that meaning is both determinate and supervenes on the dispositional. It would follow from the combination of claims in question that, if I do determinately mean plus rather than quus, then that would not only be true, but would have to be true by metaphysical necessity: it would be metaphysically impossible for a thinker to mean quus by a primitive symbol of theirs. Here is the argument for this claim: (1) Let’s suppose that my dispositions over the accessible numbers are perfectly in conformity with Addition. Then, by definition, they are also perfectly in conformity with Quaddition as well. And suppose that, nevertheless, we insist that I determinately refer to Addition and not to Quaddition. (2) Any human being would be subject to the same finitude limitation as I am—he couldn’t have more dispositions than I have: for both of us there will be some inaccessible numbers.(3) Since anyone who is perfectly in conformity with Addition is said to refer to Addition, someone who referred to Quaddition would have to not be in conformity (p.353) with Addition. But the only respects in which his dispositions could be different from mine would be in respect of the accessible numbers. (4) But that means that in order to refer to Quaddition someone would have to have dispositions that are not in conformity with Quaddition with respect to the accessible numbers. (5) But it is absurd to claim that what it takes to refer to Quaddition is to deviate from what Quaddition requires with respect to the accessible numbers. (6) So it is not, after all, possible for a human being to refer to quus by a primitive (undefined) symbol of theirs. But isn’t this a peculiar conclusion? Surely, it needs explaining why it is metaphysically impossible for a human being to refer to this other function rather than to plus. What makes plus so special? Why, given that everything about my dispositions is compatible both with plus and all of these other functions, do my thoughts nevertheless gravitate inexorably to the plus function, ignoring every one of these other functions? 15.13 Further Naturalistic Facts Where have we got to so far? If the preceding anti-dispositional arguments are correct, and if we are to continue to believe both that meaning is determinate and that it is not primitive, then we must think that there are further facts that help determine meaning, over and above one’s dispositions to answer to certain arithmetical questions with certain answers. Soames, with whom we started, would not disagree. He is open to the idea that his dispositions alone don’t determine that he means plus. It’s just that he can’t see how anything could persuade him either of the claim that he doesn’t mean plus after all, or that his meaning plus is not determined by the various possible non-intentional facts about him. Let us turn, then, to these further naturalistic facts to see how they might help us restore determinacy. As Soames notes, four classes of fact spring to mind: (i) in addition to the disposition to answer in particular ways to particular arithmetical questions, there are further dispositional facts—for example, dispositions to ‘check and revise’ my work, dispositions to insist on one and only one ‘answer’ for any given question, dispositions to strive for agreement between my own answers and those of others, and so on; (ii) facts about the internal physical states of my brain; (iii) facts about my causal and historical relationships to things in my environment; (iv) facts about my relationship to my linguistic community and their dispositions to verbal behavior. Keeping our eye fixed on the finitude problem, let us go through each one of these further suggestions and ask how it might help make it determinate that I mean plus rather than quus. (p.354) Clearly, the further dispositional facts cannot help us. If we were primarily worried about the problem of error—weeding out the dispositions that are erroneous—then, perhaps it would be important to take into account such second-order dispositions. But it is hard to see how they could help with the problem of finitude. If there aren’t enough first-order dispositions, that problem will carry over to the secondorder dispositions: just as we don’t have first-order dispositions with respect to the inaccessible numbers, so we don’t have second-order dispositions with respect to those numbers.Furthermore, I don’t see how facts about the internal physical states of my brain are going to help, in addition to whatever role they may play in implementing my use-dispositions, a suggestion that we have examined in Section 15.9. My historical and causal relationships to my environment have, of course, been thought by some theorists to be highly relevant to the question of what I mean. But it is very hard to see how they could be relevant in this particular case, where we are dealing with an arithmetical notion like plus. Indeed, it hard to see how they could be relevant to any notion that purports to refer to an abstract object. Finally, what about relations that I may have to my linguistic community and to their verbal dispositions? Once again, of course, such relations have been thought to be crucial to determining what I mean (see Burge (1979) , for example). But the way in which they have been thought relevant cannot help us with the problem of finitude. For the kinds of cases with which their relevance is established concern only the parasitic case in which an ordinary person is said to mean what the experts mean because he defers to them. The problem of finitude, though, is a problem with determinacy that applies even to the experts. Recently, it has become popular, following a suggestion of Lewis (1983) , to claim that play with the notion of ‘naturalness’ can help us with this problem. The idea is that it is an important part of the theory of the meaning relation that we should attribute to our words natural properties, functions, or entities, other things being equal (once all other appropriate constraints, like charity, have been satisfied). To put the matter metaphorically, natural properties and functions serve as ‘reference magnets’, drawing our thoughts to them by default, without our having to do anything of a positive nature to ensure that that is what we end up referring to. Since, it is further alleged, plus is more natural than quus, it is determinately true that we refer to plus rather than quus. There are two ways to apply Lewis’s idea to the plus/quus problem. The first is to take the notion of ‘natural’ to be implicitly defined as ‘whatever it is that resolves the plus/quus problem (and all similar problems) in favor of plus’. The other is to take it to be implicitly defined by the solution it provides to some other philosophical problem (or small range of problems)—for example, that of objectively true similarity judgments, or lawlikeness, and then to show that whatever it is that is implicitly defined by that stipulation extends smoothly to the plus/quus problem and resolves it clearly in favor of plus. I think the first approach is unsatisfactory. (p.355) Consider, by way of analogy, the Gettier problem. No one would be satisfied by a solution to the Gettier problem that says there is something that makes for the difference between knowledge and justified true belief, and we are henceforth to call it ‘q’. Why would that be unsatisfactory? It’s hard to say precisely. But we have the sense that whatever it is that accounts for the difference between knowledge and justified true belief, it is not something primitive, something that can’t be explained in terms of something more fundamental. Something similar is true in the plus/quus case. If this problem has a solution along such Lewisian lines, it should be on the basis of something more fundamental. It’s not as antecedently intuitive, for example, that there are ‘natural’ functions, as it is that there are natural properties. As a result, it’s not as antecedently clear that an implicit definition of ‘natural’ as ‘whatever it is that resolves the plus/quus problem (and all similar problems) in favor of plus’ would pick out anything real. If there is to be a solution to the plus/quus problem along Lewisian lines, it will have to be because we have managed to introduce the notion of naturalness in connection with some other philosophical problem, and are then able to show that it extends in a satisfactory way to the plus/quus case and delivers a determinate verdict in favor of plus. But I don’t see much prospect of that. I see no obvious notion of naturalness that will cover both the notion of a natural property, as it might figure in an account of similarity or lawlikeness, and that of a natural function. This is obviously a large topic, and it deserves a more extended treatment than I am able to give it here. These brief remarks will have to suffice for now.