A Dilemma for Dispositional Answers to Kripkenstein’s Challenge

Kripkenstein’s challenge is usually described as being essentially about the use of a word in new kinds of cases ‒ the old kinds of cases being commonly considered as non-problematic. I show that this way of conceiving the challenge is neither true to Kripke’s intentions nor philosophically defensible: the Kripkean skeptic can question my answering “125” to the question “What is 68 plus 57?” even if that problem is one I have already encountered and answered. I then argue that once the real nature of Kripkenstein’s challenge is properly appreciated, one extremely popular strategy to try to meet it, what usually goes by the name of “semantic dispositionalism”, loses much of its appeal. Along the way, I also explain that Kripkenstein’s challenge is actually two distinct challenges ‒ one concerning the mental state of meaning, or intending, something by a sign and the other concerning the meaning (referentially conceived) of linguistic expressions.

"red" or not?), 1 which is not true to Kripke's intentions. 2 The goal of this paper is to show that this tendency not to fully appreciate, or at least not to properly stress, the generality of Kripkenstein's challenge has prevented philosophers from realizing that a very popular anti-skeptical strategy, namely semantic dispositionalism (in the mainstream version embraced by Heil andMartin 1998, Warren 2020, and many others), is much more problematic than it seems: even if it turned out that dispositionalism has the resources to handle the classic objections concerning mistakes, finitude, and normativity, there would still be another, much more basic, reason to think that this approach just cannot work.
The paper is structured as follows. The first part of Sect. 1 describes the formulation of the paradox Kripke gives at the start of the second chapter of his book, a formulation that, indeed, does (misleadingly) suggest that the argument is essentially about the use of a word in new kinds of cases. I then turn to introduce, in the second half of Sect. 1, the dispositional strategy. Sect. 2 calls attention to the fact that Kripkenstein's paradox is really two paradoxes -no doubt intertwined with each other, but in principle distinct. I would not say that anyone is especially confused about this point, but I call attention to it nonetheless because my anti-dispositionalist argument is easier to formulate if we focus on Kripkenstein's second paradox, which is sometimes neglected. Sect. 3 then explains why Kripkenstein's argument is not essentially about the use of a word in new kinds of cases. Finally, Sect. 4 argues that, once the full generality of Kripkenstein's challenge is properly appreciated, the dispositional strategy loses much of its appeal. More precisely, I argue that dispositionalists face a dilemma. Either they grant that our dispositions do not come with what we may call "a temporal qualification", or they do not. If they grant the assumption, however, they have to admit that our dispositions do not have enough structure to meet Kripkenstein's real challenge. And if they do not grant the assumption, their view turns out to be question-begging.
Instead of speaking of using "+" to denote addition or grasping (the rule for) addition, in the rest of the book Kripke usually speaks of meaning addition by "+" (see, e.g., p. 9). For reasons that will become clear in the next section, I prefer to avoid speaking of meaning in this context and therefore I will refer to the mental state in question as to the mental state of "intending" a certain thing by a certain word (or symbol). 3 Anyway, whatever we call it, there is a feature of this mental state that is key to Kripke's initial formulation of the paradox: One point is crucial to my "grasp" of this rule. Although I myself have computed only finitely many sums in the past, the rule determines my answer for indefinitely many new sums that I have never previously considered.
[…] my past intentions regarding addition determine a unique answer for indefinitely many new cases in the future (Kripke, 1981, p. 8).
The addition function determines my answer for indefinitely many -in fact, infinitely many -new sums in the sense that every time I am asked for the result of an addition problem I have never previously considered there will be only one answer consistent with that function. And this means that, insofar as by "+" I intend addition, every time I am asked for the result of a new "+" problem there will also be only one answer consistent with what I intend -only one "metalinguistically correct" answer, in Kripke's wording (p. 8).
It is with this background that Kripke introduces the skeptical challenge around which Kripkenstein's paradox revolves: Let me suppose […] that "68 + 57" is a computation that I have never performed before. Since I have performed […] only finitely many computations in the past, such an example surely exists. In fact, the same finitude guarantees that there is an example exceeding, in both its arguments, all previous computations. I shall assume in what follows that "68 + 57" serves for this purpose as well. I perform the computation, obtaining, of course, the answer "125". I am confident […] that "125" is the correct answer. It is correct both in the arithmetical sense that 125 is the sum of 68 and 57, and in the metalinguistic sense that "plus", as I intended to use that word in the past, denoted a function which, when applied to the numbers I called "68" and "57", yields the value 125. Now suppose I encounter a bizarre sceptic. This sceptic questions my certainty about my answer, in what I just called the "metalinguistic" sense. Perhaps, he suggests, as I used the term "plus" in the past, the answer I intended for "68 + 57" should have been "5"! […] perhaps in the past I used "plus" and "+" to denote a function which I will call "quus" and symbolize by "⊕". It is defined by: Who is to say that this is not the function I previously meant by "+"? […] Now if the sceptic proposes his hypothesis sincerely, he is crazy; such a bizarre hypothesis […] is absolutely wild. Wild it indubitably is, no doubt it is false; but if it is false, there must be some fact about my past usage that can be cited to refute it (1981, pp. 8-9).
There are several levels to this challenge. At the most superficial one, we are asked to show that "125", and not "5", is indeed the metalinguistically correct answer to the never previously considered problem "68 + 57". 4 But saying that "125" is the metalinguistically correct answer is saying that "125" is the answer consistent with what we have always intended by "+"; therefore, at a deeper level what the skeptic is asking us is to show him that the function we have always intended by "+" is addition, and not that hymn to gerrymandering that is quaddition. But, of course, it is not like the skeptic really believes someone may routinely intend quaddition by "+". By challenging us to show him that the function we have always intended by "+" is addition, not quaddition, the skeptic is trying to make us recognize that we cannot really make sense of the difference between intending addition and quaddition (Kripke, 1981, p. 11), which in turn would show that -even though it may look like something we are all familiar with -we cannot really make sense of the very notion of intending a certain thing by a certain word (p. 13).
Having introduced the skeptic's challenge, Kripke goes on to dismiss a number of attempts to meet it. He rejects the notion that the challenge can be met by insisting that I must have internalized directions about how to use "+" (1981, pp. 15-18), or by appealing to simplicity considerations (pp. 38-40), or to irreducible mental states (pp. 41-53), and so on. The arguments Kripke employs against these various strategies for trying to make sense of intending provide, when taken together, quite a powerful inductive argument to the conclusion that there is no such mental state (see Guardo 2020a, pp. 4049-4054). And it is this inductive argument that has come to be known as "Kripkenstein's paradox".
There is no questioning that, by setting things up the way he does, Kripke ends up suggesting that the reason why his Wittgenstein believes that we cannot make sense of intending is that there is no way to make sense of the "crucial point" he had called attention to at the start of his discussion, namely that "[…] my past intentions regarding addition determine a unique answer for indefinitely many new cases in the future" (1981, p. 8). 5 After all, at its most superficial level the skeptic's challenge is to show that "125", and not "5", is indeed the metalinguistically correct answer to "68 + 57", a problem that Kripke takes care to make clear I had never previously considered, thereby implicitly suggesting that if "68 + 57" were not a new problem the issue of showing what is the metalinguistically correct answer to it (the one answer consistent with what I have always intended by "+") would not even arise -the metalinguistically correct answer would just be the one I have always given! 6 It is likewise hard to deny that if the feature of my intending addition by "+" that needs explaining is its embracing also the addition problems I have never considered, the move of identifying that mental state with (some of) my dispositions concerning that symbol makes sense, at least prima facie. After all, even assuming, with Kripke, that "68 + 57" is a computation I have never performed before, it is clear that for most of my life I have anyway been disposed to answer "125" if queried about "68 + 57" (at least in certain conditions). Here is, then, a general formulation of the dispositional view of the mental state of intending addition by "+": One intends addition by "+" if and only if their dispositions concerning the use of "+" in conditions γ track that function, i.e. if and only if, for any ν 1 , ν 2 , ν 3 such that ν 3 is the value of the addition function for the arguments ν 1 and ν 2 , they are disposed to answer "ν 3 " if asked about "ν 1 + ν 2 " in conditions γ.
Of course, the notion that my dispositions embrace the whole addition function, and not just a finite segment of it, is controversial. But it can be defended, and it has been defended (see, e.g., Blackburn 1984, Shogenji 1993, Heil and Martin 1998, and Warren 2020; see also Guardo 2022, § 2 for some discussion). Anyway, my point here is not that going dispositional is the right way to respond to the challenge that Kripke (on behalf of Wittgenstein) seems to be issuing in the passage I quoted above. What I am saying is just that if the feature of my intending addition by "+" that needs explaining is its embracing also the addition problems I have never considered, then going dispositional is quite a natural, and indeed somewhat promising, move. This is, I think, really hard to deny. 7 In any case, the issue of whether my dispositions can, or cannot, embrace the whole addition function is a red herring, for -as I will show in Sect. 3 -the real challenge at the heart of Kripkenstein's paradox is not the one I introduced in this section. In particular, the old cases (the "+" problems I have already been asked about, the shades of color I have already bumped into) are just as problematic as the new ones. Before arguing for this point, however, I want to show that, even if we were to set to one side the considerations I will develop in Sect. 3, the argument I sketched in this section would only be one half of Kripkenstein's paradox.
Before moving on, I want to introduce a bit of terminology that I feel will come in handy in the rest of the paper.
First, from now on, I will refer to the argument of this section as "the Ersatz paradox about intending". "Ersatz" is there to signal that the challenge it revolves around is not the one at the heart of Kripkenstein's real paradox. As for the qualification "about intending", its job will be that of reminding the reader that even setting to one side the fact that Kripkenstein's paradox is not essentially about the use of a word in new types of cases, the argument of this section leaves out -or at least does not make explicit enough -a key dimension of Kripkenstein's problem, which regards the meaning of words as much as it regards the mental state of intending.
Second, the next section will begin to give the reader a more well-rounded picture of Kripkenstein's views by introducing a second paradox, which concerns word meaning and which parallels the Ersatz paradox about intending; since, however, this second paradox, too, is essentially about new kinds of cases, I will refer to it as to "the Ersatz paradox about word meaning".
So much for the terminology. Let us now turn to more substantive issues.

The Ersatz Paradox About Word Meaning
The source of the Ersatz paradox about intending is that this mental state's referring to its intentional object is supposed to generate correctness criteria, in the metalinguistic sense of "correctness" defined above, that outstrip the boundaries of actual usage (my intending, say, addition by "+" makes it so that every time I am asked for the result of a new "+" problem there will be only one metalinguistically correct answer, only one answer consistent with what I intend) and, at least prima facie, it is not clear how that should work. Sometimes, however, Kripke seems to have in mind a different, albeit related, problem, one concerning the meaning of words and symbols. The problem can be raised for any "referential" view of meaning; that is: for any view of meaning -such as Kaplan's (1989), Chalmers' (1996Chalmers' ( , 2006, or MacFarlane's (2014) -which has that the meaning of a linguistic type is a function which takes possible referents/extensions/denotations as values -or, less formally, a rule explaining how the referent, extension, or denotation of that type depends on the value of certain parameters. 8 Just as intending's referring to its intentional object should generate metalinguistic correctness criteria that outstrip the boundaries of actual usage, an expression's referring (relative to the value of the relevant parameters) to its referent is supposed to generate correctness criteria that, again, outstrip the boundaries of actual usage. If "+" refers to addition (and "68" to 68, etc.), then the correct answer to the problem "68 + 57", the one answer consistent with what "+" means, is "125" -and this even if nobody has ever encountered that problem before. And here one can raise a multilevel challenge that parallels the one we have seen in the case of intending: show me that "125", and not "5", is indeed the right answer (the one consistent with the meaning of "+"); show me that "+" really means plus, not quus; show me that there is indeed a difference between "+"'s meaning plus and its meaning quus; and, finally, show me how to make sense of the notion of a sign's meaning something. Now, Kripke's discussion of the first challenge -the one about intending -is intertwined with a parallel discussion of this second challenge, and so his inductive argument to the conclusion that there is no such thing as intending is intertwined with a parallel argument to the conclusion that there is no such thing as the meaning of a word or symbol (at least as conceived in the context of the referential framework). This second skeptical argument is the Ersatz paradox about word meaning. 9 A good way to see that sometimes the problem Kripke has in mind is the one about word meaning, not that about intending (and a good way to see that the two problems must be carefully distinguished), is to ask ourselves why he builds his plus/quus scenario the way he does. In particular, the key question is: why does Kripke describe the case in such a way that the answers I gave to the "+" problems I encountered in the past are as consistent with the quaddition function as they are with addition?
Here is what I am sure looks like quite a natural answer: if the scenario had not been built this way, the skeptic's challenge would have been completely devoid of interest -for the answer would have been obvious. Let us assume, for example, that I have already encountered the problem "68 + 57" in the past and each time I have given the answer "125". It is then clear that if now I am asked again about that problem, the metalinguistically correct answer, the one consistent with my past history, will be "125", not "5".
That is not, however, how we defined metalinguistic correctness! The metalinguistically correct answer is not the one consistent with my past history; it is the one consistent with a proper part of my past history -namely, with what I have always intended by "+". And even though my having always answered "125" when queried about "68 + 57" is part of my past history at large, it does not belong to the relevant region of that history, the one concerning that particular mental state.
Here it is useful to keep in mind that when the skeptic challenges us to show him that "125", and not "5", is indeed the metalinguistically correct answer to the never previously considered problem "68 + 57", that is just the most superficial level of his challenge. What he really wants from us is that we show him how to make sense of the notion of intending a certain thing by a certain word. It should be clear that trying to meet such a challenge by citing facts about overt behavior cannot help in any way -unless you believe we can go full Skinner and identify the mental state in question with such facts. 9 Two things. First, if one views the relation between word meaning and intending in a certain way, the two Ersatz paradoxes end up being more than just two somewhat analogous arguments. If you believe that the reason why, say, "+" means addition is that that function is what people usually intend by that symbol, then it is clear that the conclusion of the Ersatz paradox about intending (there is no such mental state) entails that of the Ersatz paradox about word meaning (there is no such thing as the meaning of words and symbols) -for the very problem of word meaning would reduce to that of the ontological status of intending. That being said, since, as I argue below, some things he says cannot be made sense of in terms of intending, I take it to be clear that Kripke believes that it is at least in some sense possible that facts about word meaning do not reduce to facts about intending.Second, the Ersatz paradox about word meaning is an argument to the conclusion that there is no such thing as reference (and hence meaning, referentially conceived), not an argument to the conclusion that the referents of our words do not exist. The idea is that "+" does not refer to the addition function -nor to anything else, for that matter -not that there is no such thing as addition. Of course, if there is no such thing as the fact that "+" refers to addition, it does not make much sense to call addition a "referent"; but the point still stands: the argument is not an argument to the conclusion that what we usually take to be the referents of our words do not exist (this is, I believe, what allows us to resist the argument of van Inwagen 1992; see esp. pp. 140-141).
But then why does Kripke construe his scenario the way he does? Here is what I think is going on. Already here, in the midst of his introduction of the Ersatz challenge about intending, Kripke inadvertently switches to the parallel, but distinct, problem about word meaning. Therefore, what he has in mind when he takes care to stipulate that my answers to past "+" problems do not discriminate between addition and quaddition is that if "68 + 57" had already been consistently answered with "125", it would be trivial that now the correct answer to that problem -"correct" in the sense that it is the one consistent with the meaning of "+" -is "125", and not "5". After all, it seems legitimate to assume that past usage determines meaning at least partially, with regard to the actually encountered cases 10 -whereas with intending it did not make much sense to go super-behaviorist and assume some kind of constitutive relation between past usage and that mental state. 11 Both the problem about word meaning and that about intending can be stated in terms of consistency between a present fact, my answering "125" to "68 + 57", and some past facts. Which past facts, however, depends on the problem. And in particular, it depends on the problem whether past facts concerning overt behavior are relevant or not. If the problem we are interested in is the one about word meaning, then past facts concerning overt behavior are definitely relevant. If the problem we have in mind is that about intending, then these facts are not relevant. But at least sometimes, Kripke makes clear that he takes past facts concerning overt behavior to be relevant, which means that, at least sometimes, the problem he has in mind is the one about word meaning, not that about intending.
Of course, there are also places in which it is eminently clear that the challenge Kripke has in mind is the one about intending -as when, in arguing against specific strategies to try to block the paradox, he assumes (1981, pp. 23 and 40) that we have some kind of non-inferential access to that mental state (see Guardo 2020b). But, as I think I have just shown, sometimes the problem at issue is the one about word meaning.
Let us take stock. In this section I have argued that, in his book, Kripke is interested in two parallel (and somewhat intertwined) but ultimately distinct challenges, one about the mental state of intending a certain thing by a given sign and the other about the meaning of words and symbols. Kripke introduces each challenge as a challenge concerning only new kinds of cases: how can my intending addition in some way also embrace problems I have never even considered? And how can the meaning of "+" encompass sums nobody has ever thought about? However, this is misleading, for the distinction between old and new kinds of cases is actually immaterial to 10 In the case of "+", you can think about it this way. The meaning of "+" may be identified with its referent (see footnote 8), which in turn we can conceive of as a set of ordered triples. Past usage would then partially determine the meaning of "+" in the sense that when a problem "ν 1 + ν 2 " is consistently answered with "ν 3 ", the triple < ν 1 , ν 2 , ν 3 > is immediately added to "+"'s meaning set. 11 Two caveats are in order. First, while Kripke focuses on my past linguistic behavior (after all, inadvertent switches of topic aside, his interest is in correctness as consistency with what I have always intended), it is clear that here the relevant linguistic behavior is that of the entire community. Second, remember that in this section I am bracketing the fact that the distinction between old and new kinds of cases does not really matter. Hence, when I say that it seems legitimate to assume that past usage determines meaning at least partially, well, that "seems" is really doing a lot of work. the problems Kripke really has in mind. This is what I will show in the next section. Finally, in Sect. 4 I will argue that, even though it does have a good deal of intuitive plausibility when we focus exclusively on his preliminary formulations, dispositionalism ends up losing much of its promise as soon as we turn to Kripke's real worries.

The Real Paradoxes
Lest it be thought that I am tilting at windmills here, let us start with a selection of excerpts from papers by various prominent commentators of Kripke's essay which should establish that Kripkenstein's challenge is indeed routinely portrayed as being essentially about new kinds of cases. Here is Hannah Ginsborg introducing the paradox: Suppose that all your previous uses of the word "plus" and of the "+" sign have involved numbers less than 57. You are now asked "What is 68 + 57?" and you answer "125". But a skeptic proposes the hypothesis that by the word "plus", or the "+" sign, you previously meant not addition, but quaddition, where x quus y is the sum of x and y if x and y are less than 57, and otherwise 5. If you are to use the word "plus" as you used it in the past, the skeptic says, then, on the hypothesis that you meant quaddition, you ought to answer "5" (2011, p. 227; both here and in the following quotes the italics are mine).
Note that Ginsborg never comes back to this characterization of the problem to warn us that the assumption that all our previous uses of "plus" have involved (only) numbers less than 57 is actually inessential; so either she takes Kripkenstein's challenge to be only about the new kinds of cases or she believes that depicting the challenge that way is a harmless simplification -because the fact that the challenge is more general than that does not have any momentous consequence. Here are analogous passages by other authors: Suppose one is asked to perform an addition other than any one has encountered before […]. In confidently giving a particular answer, one will naturally have a thought that is problematic: namely […] that in returning this answer one is keeping faith with one's understanding of the "plus" sign.
[…] what could constitute one's being in such a state? (McDowell, 1984, p. 226). When I come to do a calculation, which we suppose I have never done before, I certainly believe myself to follow a principle.
[…] I believe that if I am faithful to yesterday's principle, I should say "57 + 68 is 125". […] I most certainly should not say that 57 + 68 is 5. Nor of course should I say that there is more than one answer to that problem, or that the problem is indeterminate, so that there is no answer at all. […] The sceptic asks me to point to the fact that I am being faithful to yesterday's rule only by saying one thing, and not these others (Blackburn, 1984, p. 287). (Pettit, 1990, p. 2). 12 […] even if we grant that whatever I meant by "+" must conform to my past calculations using that symbol, these calculations are not sufficient to determine that I meant addition by it, since the values of the addition function far outstrip the limited number of calculations I have made (Soames, 1998, p. 314).

The first and main element in the definition of rules is the stipulation that rules are normative constraints […] which are relevant in an indefinitely large number of decision-types
Just like Ginsborg, McDowell, Blackburn, Pettit, and Soames never come back to these characterizations of the problem to warn us that they are making a rhetorically useful but ultimately dispensable assumption. So again: either they take Kripkenstein's challenge to be only about the new kinds of cases or they believe that depicting the challenge that way is a harmless simplification.
Let us say that I now bump into a shade of color I had never seen before and wonder how it should be categorized. This is what I call "a new kind of case" -here you may want to add that this kind of case is new to me (maybe you had already bumped into this particular shade), but of course there are also kinds of cases that are new in general (by which I mean: to the linguistic community in its entirety). If, on the other hand, the shade in question is one I had already seen, and categorized, before (instantiated either by another object or by the very same one I am gazing at right now) we have what we can call "a new instance of an old kind of case" -or just "a new instance", for short. 13 The distinction applies, of course, also to mathematical examples. In Kripke's plus/quus scenario, being asked for "68 + 57" is a new kind of case -new to me, of course; but it would be easy to modify the scenario so that it is new to the linguistic community in its entirety. Now, let us assume, with Kripke, that I unhesitatingly answer "125". And let us also assume that, at a later time, I am asked again for the same problem. This would be a new instance of an old kind of case.
With this distinction in hand, we can rephrase the thesis of this section thus: even though they are usually portrayed as being essentially about the use of a word in new kinds of cases (new to me in the case of intending, new in general in that of word meaning), Kripkenstein's challenges also concern the new instances of the old kinds of cases -the skeptic can question my answering "125" to "68 + 57" even if that is a problem the members of my linguistic community have already encountered countless times and consistently answered that way. Now the question is: what kind of evidence can be brought in support of this claim?
Well, for one, this is what Kripke himself (1981, p. 52, note 34) says explicitly: […] although it is useful […] to begin the presentation of the puzzle with the observation that I have thought of only finitely many cases, it appears that in principle this particular ladder can be kicked away. Suppose that I had explicitly thought of all cases of the addition table. How can this help me answer the question "68 + 57"? Well, looking back over my own mental records, I find that I gave myself explicit directions. "If you are ever asked about "68 + 57", reply "125"!". Can't the sceptic say that these directions, too, are to be interpreted in a non-standard way?
I know of only five works that at least mention this footnote, namely a two-and-ahalf page review of Kripke's book by Harry Deutsch (1986), Kathrin Glüer's book on the normativity of meaning (1999), Alex Miller's introduction to the collection Rule-Following and Meaning (2002), Romina Padro's PhD dissertation (2015), and James Shaw's recent book on Wittgenstein's metasemantics (2022). 14 This is quite an important passage, though. In it, Kripke does not just make clear that he takes Wittgenstein's challenge to concern the new instances of any old kind of case, too (my present response can be questioned even if "68 + 57" is a problem I have already encountered and answered; indeed, even if I gave myself explicit directions about it). He also explains why it is so: focusing on one of the skeptic's strategies to cast doubt on my response (that of suggesting non-standard interpretations for any set of directions I may have internalized), he notices that my answers to a new instance of an old kind of case can be questioned in roughly the same way the skeptic can question my answers to a new kind of case.
The best way to see that Kripkenstein's challenges concern also the new instances of the old kinds of cases, however, is to set quaddition to one side and consider, as Kripke himself sometimes does, an example à la Goodman (1954). 15 For simplicity's sake, I will focus on the issue of word meaning.
The first order of business is to give a definition for the term "grue". Kripke's (1981, p. 20, note 15) is that "[…] past objects were grue if and only if they were (then) green while present objects are grue if and only if they are (now) blue". Now, just as Kripke's original skeptic challenged us to show him that "+" means plus, and not quus, we can imagine a second skeptic who challenges us to show her that "green" means green, and not grue. Let us then imagine that we are dealing with such a skeptic, that she hands us a green object O, that she asks us what color O is, and that our answer is "Green". This is definitely a new instance of an old kind of case; and yet, our skeptic can question our answer pretty much in the same way in which Kripke's original skeptic questioned the answer "125". After all, if "green" means grue, to categorize O as "green" is just wrong -since O is a present object, it is (now) green, and present objects are grue if and only if they are (now) blue. The new instances of the old kinds of cases are just as problematic as the new kinds of cases. 14 Thanks to two exceptionally helpful reviewers, one for this and one for another journal.
15 That Goodman's grue and bleen scenarios differ in interesting respects from Kripke's quaddition case is widely recognized in the literature (see, e.g., Podlaskowski forthcoming, § 1); the difference I want to call attention to, though, has gone, to the best of my knowledge, largely unnoticed. It is also worth noting that sometimes people do focus on such scenarios (see, e.g., Miller 2011, pp. 459-461); my point is, in a certain sense, that we should always do that.
What characterizes an example à la Goodman is that the alternative referent is defined by means of a temporal switch: in the past, an object was grue if and only if it was green, but now an object is grue if and only if it is blue. Such examples can therefore be built for mathematical lingo, too. Let us say, for instance, that in the past x grus y equaled x plus y, but now x grus y always equals 5, and let us suppose that I meet a skeptic who challenges me to show them that "+" means plus and not grus. If I am asked for "56 + 56" and I answer "112", the skeptic can question my answer even if "56 + 56" has already been consistently answered with "112" -that is: even if I am dealing with a new instance of an old kind of case. After all, if "+" means grus, to answer "56 + 56" with "112" is just wrong -since now x grus y always equals 5.
In the next section I will explain how all this can be brought to bear on the issue of the viability of semantic dispositionalism as a straight solution to Kripkenstein's paradox. The argument I will put forward can be applied both to dispositionalism about word meaning and to dispositionalism about intending. However, my idea is easier to explain when the focus is on the issue of word meaning, and therefore that is the problem I will concentrate upon.
But before turning to that task, and following the suggestion of a reviewer for this journal, I will briefly sketch a conjecture as to why Kripke decided to introduce (what he took to be) Wittgenstein's challenge in terms of new kinds of cases, rather than in terms of new instances. To be clear, this is pure speculation, and other hypotheses are no doubt possible. That being said, this is what I suspect may have happened.
Kripke begins the preface to his book by noting that the main part of the work had been delivered at various places either as lectures, series of lectures, or seminars (Kripke, 1981, p. vii), which suggests that the book's footnotes were added only at a later stage, while the work was being prepared for publication. And since Kripke explicitly recognizes that the observation that I have thought of only finitely many cases is "a ladder that can be kicked away" only in a footnote, this in turn suggests that Kripke came to fully understand the nature of (what he took to be) Wittgenstein's challenge only while he was preparing his work for publication. At that point, of course, he could have come back to his original presentation of the problem to revise it accordingly; but, realizing that it was quite an intuitive and powerful way to introduce the challenge, he ended up deciding not to touch it. And that is why the version of the problem discussed in the main text assumes that I have thought of only finitely many cases and it is only in a footnote that we are told that the assumption is unnecessary.

A Dilemma for Dispositionalism
The general idea at the root of the argument I am going to put forward is that dispositionalists face a dilemma: either they accept a certain, I believe very plausible, assumption about the structure of our linguistic dispositions, or they do not: if they do accept the assumption, they have to admit that our dispositions lack the required structure to meet Kripkenstein's real challenge; and if they do not, their view ends up begging the question against the skeptic -either way, the dispositionalist project fails.
Since the assumption in question is, I think, quite plausible indeed, I believe it is rhetorically more effective to start by taking it for granted -thereby leaving it implicit. This means that I will start by simply arguing (subsection 4.1) that dispositionalism cannot meet Kripkenstein's real challenge. Having done that, I will call attention (subsection 4.2) to the assumption my argument relied on and, finally, I will turn to explain why rejecting it is of no help to the dispositionalist.

The First Horn
Let simple semantic behaviorism be the view that meaning facts, conceived along the lines of the referential picture, are just facts about the actual linguistic behavior of the relevant community. Simple semantic behaviorism enjoys, no doubt, some intuitive plausibility. However, it is clearly too simple a view to stand any chance of withstanding serious scrutiny. Take, for example, the following quaddition-like function ("Q", for short): where "ν" stands for a number so huge that none of us has ever even thought about it -nor about any number larger than it. Q mimics the more respectable addition function for quite some time, but the two functions are, of course, two distinct functions. Therefore, the fact that "+" means addition and the unactualized, and somewhat disturbing, possibility that "+" means quaddition are two distinct things. But if meaning facts were just facts about actual behavior there would be no difference whatsoever between "+"'s referring to addition and its referring to quaddition -since, ex hypothesi, our linguistic behavior does not discriminate between these two functions. 16 Hence, meaning facts are not just facts about actual behavior and simple semantic behaviorism is false.
Let us now consider a variant of simple semantic behaviorism -what I will call "enhanced semantic behaviorism". Enhanced semantic behaviorism is the result of appending to its simpler counterpart the assumption that the linguistic community has encountered every possible case: there are no new kinds of cases anymore; only 16 Here by "linguistic behavior" I mean things like our answering a given "+" problem in a certain way. Broadening the scope of the notion to encompass also things like definitions etc. would not help with the paradox for reasons relating to Wittgenstein's remarks about a rule for interpreting a rule (see Kripke 1981, pp. 15-20).In this connection, arguments such as those of Putnam 1980 andGauker 2003, Chap. 2 are also relevant. Take, for example, Gauker's argument (for its relation to Putnam's see Gauker 2003, pp. 43-44). Gauker argues that (1) "If a set of mental representations has one model, then […] it will have many, many, wildly divergent models" (p. 42) and that (2) " […] there is no good way to identify the correct interpretation from among the class of models" (p. 44) -it does not help, for instance, to assume that "[…] the interpretation of beliefs can assign to basic, noncompound mental predicates only properties that are in some sense privileged", such as properties that are directly perceptible or properties that in some sense carve nature at the joints (pp. 44-45). Now, Gauker's strategy to build divergent models (pp. 40-41) parallels Kripke's strategy to build quaddition-like functions, and his arguments for (2) can be adapted to show that assuming, say, that "+" can denote only functions that are sufficiently natural, or simple, will not be enough to turn simple semantic behaviorism into a viable view. new instances of old kinds of cases. If, once again, we focus on "+", this means that the linguistic community, as a whole, has encountered, and answered, every possible "+" problem. Enhanced semantic behaviorism is, as it were, simple semantic behaviorism in incredibly favorable (indeed, impossibly favorable, maybe even unintelligibly favorable!) circumstances.
Not much reflection is needed to see that quaddition scenarios like the one sketched above cannot be used to refute enhanced semantic behaviorism. After all, if the linguistic community has already encountered every possible "+" problem, there is just no number so huge that none of us has ever thought about it. We can, however, revert to the grus function that, at the end of last section, we defined by saying that in the past x grus y equaled x plus y, but now x grus y always equals 5. Once again, the fact that "+" means addition and the unactualized possibility that "+" means gruaddition are two distinct things; this becomes especially clear if you pay attention to the fact that if "+" refers to the addition function and now I am queried about "68 + 57", the correct answer (the one consistent with the meaning of "+") is "125", whereas if "+" means gruaddition the correct answer is "5". However, if meaning facts were just facts about our linguistic behavior there would be no difference between "+"'s meaning addition and its meaning gruaddition, and this even under the assumption that we have already encountered every possible "+" problem -because of the way we defined gruaddition, our past behavior just cannot discriminate between the two functions. Hence, meaning facts cannot be just facts about actual behavior, even under the assumption that the linguistic community has encountered every possible case, which means that semantic behaviorism fails even in the most favorable circumstances possible, which in turn means that enhanced semantic behaviorism fares no better than its simpler counterpart.
Enhanced semantic behaviorism is not, of course, a plausible metaphysics of meaning -even bracketing the problem I have just called attention to. After all, it is just simple semantic behaviorism, a very naïve view, plus an assumption which is clearly false. It is worth discussing, though, because its failure entails that of the dispositional view. When you ask me for the sum of 68 and 57 and I give you the answer "125", that response is the manifestation of a disposition of mine. 17 Therefore, when we assume that the linguistic community has encountered and answered every possible "+" problem, what we are assuming is that we had the relevant dispositions, and we manifested them. Therefore, when we notice that our having encountered and answered every possible "+" problem does not suffice to discriminate between addition and gruaddition, the point is that even a full manifestation of a dispositional array that in some way covers every possible "+" problem is not enough to discriminate between these two functions. But if the full manifestation of such a dispositional array cannot discriminate between addition and gruaddition, the dispositional array itself surely cannot do that either, which means that if meaning facts were just facts about our dispositions there would be no difference between "+"'s meaning addition and its meaning gruaddition -even under the (by the way, questionable) assump-tion that these dispositions cover every possible "+" problem. But the fact that "+" means addition and the unactualized possibility that "+" means gruaddition are, quite obviously, two distinct things. Hence, meaning facts cannot be just facts about the linguistic community's dispositions and dispositionalism is false.

The Second Horn
The point of the argument in the previous subsection is that our dispositions do not have enough structure. The plus/grus scenario shows that the fact that "plus" means addition must have enough structure not just to tell us that the correct response to "What is 68 plus 57?" is "125", but to prescribe answers to a series of questions, namely to every instance of the schema "What is 68 plus 57 at t i ?" -for the scenario shows that we should reason not in terms of kinds of cases, but in terms of their particular instances. This means that in order for the fact that "plus" means addition to be dispositional in nature, we should have a disposition for each question in the series: the disposition to answer "125" if asked "What is 68 plus 57?" at t 1 , the disposition to answer "125" if asked "What is 68 plus 57?" at t 2 , and so on (where the role of "at t 1 ", "at t 2 ", etc. is that of helping specify the relevant stimulus, not that of saying at what time one has the disposition in question). Now, here one could doubt that we have this many dispositions, but the real problem, at least according to the argument I sketched, is that our dispositions do not have the required kind of structure: they do not come with a temporal qualification. I have no doubts that on the day of my fiftieth birthday I will be disposed to answer "125" if asked "What is 68 plus 57?", but the notion that now I have the disposition to answer "125" if asked "What is 68 plus 57?" on the day of my fiftieth birthday (where the role of the clause "on the day of my fiftieth birthday" is that of helping specify the relevant stimulus) seems to me to be an overly intellectualistic view of our linguistic dispositions.
One could, however, disagree with my assessment. Whether the speakers' dispositions come with a temporal qualification or not is, after all, an empirical issue which only future advancements in neuroscience will be able to settle. 18 That it is not the case that now I have the disposition to answer "125" if asked "What is 68 plus 57?" on the day of my fiftieth birthday is, therefore, a substantive assumption of the argument in the previous subsection.
Rejecting this assumption, however, is not going to be of much help to the dispositionalist program. Saying that one may be disposed to answer "125" if asked about "68 + 57" on the day of their fiftieth birthday is ascribing to that person a contentful state -a state that, in some way, refers to that person's fiftieth birthday; a state involving some kind of language-of-thought representation of that day. But (at least since Boghossian 1989, § 3) it is generally agreed that Kripkenstein's challenge can be raised with regard to any contentful state. And our case seems to be no exception. Just as one may wonder what, if anything, makes it the case that "green" means green and not grue, and just as the challenge can be raised to make sense of the notion that 18 That our current understanding of the human brain is nowhere near the level required to address such problems is clear from even just a casual perusal of any neuroscience textbook, such as the standard Kandel, Schwartz, Jessell, Siegelbaum, and Hudspeth 2013. by "+" we all intend addition and not some other function, it is legitimate to ask what makes it the case that a certain dispositional state of mine refers to the day of my fiftieth birthday -and not, say, to the day of my wedding. Therefore, if a dispositionalist opts to try to rebut the argument I put forward by maintaining that the speakers' dispositions do come with a temporal qualification, the net result of their move is that they find themselves with a problem perfectly analogous to the one they started with, only located at another level. The problem moved, but the solution is no nearer for that. 19

Conclusion
Most of this paper focused on the task of trying to make clear the real point of Kripkenstein's challenge -that it does not concern only the new kinds of cases, but also all the new instances of the old kinds of cases. This means that most of the paper has been, as it were, stage setting. But all that stage setting enabled us to see, I hope, that any attempt to offer a dispositional answer to Kripkenstein's challenge, a straight solution to Kripkenstein's paradox in terms of speakers' dispositions, faces a very basic problem. If dispositionalists grant that our linguistic dispositions do not come with a temporal qualification, they have to admit that these dispositions lack the required structure to meet the challenge. And if they try to argue that our dispositions do come with a temporal qualification, their answer seems to be begging the question against the skeptic (since the skeptic's challenge can be raised with regard to any contentful state). This seems to me a dilemma very much worthy of the attention of anyone who believes that some version of semantic dispositionalism must be able to kill Kripkenstein's monster.
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