Kripke’s Conjuring Trick
(University of East Anglia Philosophy Department)
(Manchester University Sociology Department)
Kripke’s Conjuring Trick
Yet another paper on Kripke’s Wittgenstein, another challenge to his sceptical considerations about rules, now fully twenty years since those considerations were first published? This might seem a delicate, even a perilous affair. But we believe it to be necessary. Why? Because we believe that even many of those who want to resist Kripke’s interpretations, arguments or conclusions have failed to place the problem with these far enough back. Even those, such as for instance McDowell or Boghossian, who have most effectively or brilliantly undercut Kripke’s Wittgenstein, have not, in our view, been sufficiently sceptical of Kripkean scepticism. Almost all of Kripke’s readers have allowed that the problem which Kripke tries to delineate can at least in some sense be delineated. It is that presupposition which we challenge. We believe that Kripke has taken in most of even his more ‘sceptical’ readers, at least to the extent of apparently presenting a genuine philosophical problem, a real and novel challenge. We believe that so many have been taken in at least to this extent because even when some incoherence has been detected in Kripke, it has not been noted quite how very early in Kripke’s presentation that incoherence begins.
It is this deficit in the large literature on Kripke’s Wittgenstein which we aim to correct. We will shortly explain how.
But first, a preliminary question or two. Why is it so important to get Wittgenstein right, here? Why is it so important to show that Kripke is wrong? Why care about the precise details of how to understand what is philosophically most apposite to say concerning perhaps recondite matters of rules, meaning, understanding and scepticism?
Because we think that these questions are not so recondite. Not only have they more or less dominated Wittgenstein scholarship over the past generation, but they are vital for a proper understanding of the relevance of Wittgenstein’s thought to the philosophy of education, among other areas. If one fails to understand the nature of our rule-folowing practices, then one risks failing to see how education is possible.
Kripke’s sceptic threatens our understanding of same. We think it vital, therefore, to show just how fundamentally his ‘challenge’ fails.
“The decisive movement in the conjuring trick has been made, and it was the very one we thought quite innocent.” (Wittgenstein, PI para.308)
In ‘On Certainty’, Wittgenstein advises questioning the intelligibility of philosophical scepticism, refusing ‘the sceptic’s’  attempt to impose an insatiable onus of justification upon us, and subjecting ‘the sceptic’s’ supposed doubts to searching interrogation. More precisely, Wittgenstein encourages us to wonder whether anyone need ever take the relevant forms of words we sometimes find ourselves inclined towards uttering to be genuinely and ultimately attractive, whether those forms of words can ever actually satisfy us; whether, in short, there can be, even temporarily, ‘positions’ worth calling ‘scepticism(s)’ which could satisfy our real needs: linguistic, epistemic, or pragmatic.
The prevalent style of response to ‘the sceptic’  tends to involve attempting to rebut the latter’s challenges, but such efforts are, in our view, both futile and unnecessary. The fact that we humans, and we philosophers, cannot satisfy ‘the sceptic’s’ apparent demand for justification or what-have-you does not put us at any epistemic disadvantage, and certainly does not mean that we are therefore bereft of justification for what we do. ‘The sceptic’s’ contention that we cannot justify X is correct only in a superficial way, and it is misguided to suppose that this point exposes the dubiety or insecurity of the relevant practices. Our practices do not -- not even ‘ultimately’ -- rest upon, are not founded in, an imaginary quite general justification (and justification tends, after all, to be usefully described (for prophylactic purposes) as something which takes place only within one practice or another.). ‘The sceptic’ offers only purported ‘doubts’, baseless pretences of disbelief, not authentic and substantiated queries, and does not, therefore, deserve to be offered justifications or anything remotely resembling them, in response. Or this, at any rate, is what we shall endeavour to substantiate in what follows; this is what we shall try to persuade those who (through feeling strongly the attractions of the words of scepticism, or of the dialogue or dialectic with and against it) imagine themselves to be imagining otherwise.
Thanks to Saul Kripke, scepticism in the philosophy of language (and certain associated or implied forms of ‘Anti-Realism’) has been given a renewed period of vitality, though Kripke’s attempt to extract a ‘sceptical paradox’ from the ‘Philosophical Investigations’  fundamentally does nothing to change the situation outlined above. Kripke himself is perfectly ready -- acting as the surrogate of an imagined sceptic -- to indulge ‘the sceptic’s’ purported challenges, to look for a general justification for our practices of rule-following and meaning-attribution, and, failing to find any, to conclude that ‘the sceptic’s’ challenges are unanswerable. We, for our part, cannot presuppose the intelligibility of ‘the sceptic’s’ arguments, do not accept that these arguments do in fact ever satisfy the person entertaining them or purportedly putting them forward as accurate representations of ‘what [they] want to say’, and regard Kripke’s patent difficulties in giving voice to ‘the sceptic’s’ purported challenges as more than trivial or provisional ones. This last point will turn out to be of fundamental importance, for it is our overriding contention that ‘the sceptic’s’ arguments are at very best ineffable. And this word -- “ineffable” -- is to be read à la Cora Diamond and company. That is, ‘the sceptic’s’ inability to say what s/he wants is not a trivial matter, for ‘the sceptic’ does not gesture at an ineffable truth -- rather, ‘the sceptic’s’ words come away from her intent in using those words, until eventually s/he is (we hope) persuaded that there is no point in uttering those words in the first place; or, at least, any more.
We will seek to demonstrate our contention by highlighting the clash between meaning scepticism and the scenarios that appear to give it form. In particular, we will analyse in detail the way in which this clash is absolutely inevitable in and from the very first moves in the dialectic of Kripke’s argument.
Kripke’s opening movements
“Now suppose I encounter a bizarre sceptic. This sceptic questions my certainty... . Perhaps, he suggests, as I used the term ‘plus’ in the past, the answer I intended for ‘68 + 57’ should have been ‘5’! Of course the sceptic’s suggestion is obviously insane. My initial response to such a suggestion might be that the challenger should go back to school and learn to add. Let the challenger, however, continue...” (Kripke, ‘Wittgenstein on rules...’ p.8).
Anyone who thoughtfully considers Kripke’s arguments ought to be able to see that there is something basically perverse about them. Kripke is, on the one hand, only too ready to own up to the ludicrous character of ‘the sceptic’s’ position, describing ‘the sceptic’ as “bizarre”, and ‘the sceptic’s’ suggestions as “obviously insane”. The response to such a one that they should “go back to school” can seem inappropriately naive -- especially when addressed to someone who is, after all, manifestly quite capable of doing addition. However, this is arguably by no means so inappropriate a response to someone who is either so deranged or jejune as to instead seriously propose to “Let the challenger...continue”. Returning the floor to ‘the sceptic’ merely permits further vacuities.
That this is so, however, must be shown to any person taking ‘the sceptic’s’ remarks, ‘the sceptic’s’ bizarre opening question and suggestion, not to be mere vacuities. Let us then show this.
Concretely, let us see how ‘the sceptic’s’ purported doubt that “as I have used the term “plus” in the past, the answer I intended for 68 plus 57 should have been 5” (which would be the result given, so Kripke tells us, if we were actually computing the ‘quus’ function rather than the ‘plus’ function) can be rightly be said to be without grounds:
Does ‘the sceptic’ adduce any evidence on behalf of this suggestion? ‘The sceptic’ is not, of course, suggesting that one some, any or all previous occasion, had I carried out this addition, I would have arrived at the result ‘5’; for, of course, ‘the sceptic’ has no specific grounds which suggest or support any such definite suggestion...
The imagined challenger’s suggestion is baseless because it can be shown to be entirely unevidenced and unexplained and thus empty. It does not involve the proposal that there is some system of arithmetical operation that we actually have in which 68 plus 57 does equal five. The addition of 68 and 57 is not being seriously proposed as a sum to which the imagined ‘quus’ function really applies, for ‘the challenger’ is making no definite assertion that the result in this case should actually be five, but is only offering the quite indefinite contention that (perhaps) at some point in the process of addition we should... what, exactly? 68 + 57 = 5 is not any kind of proposal, but a mere placeholder, an ‘as it were’ example only, to represent some numbers which, of course, the challenger cannot specify directly. Let the challenger continue???
The imagined challenge does not, then, involve any definite assertion such as might be specifically gainsaid; which means, of course, that there can be no specific remark which we can make to counter it; but this should be treated as a deficiency of the would-be challenge, not as manifesting any failure on our part. ‘The challenger’s’ challenge at this stage is, then, that we cannot rule out a certain imagined possibility as to what we might do or might have done if, in the past, we had done something that we did not in fact do. We cannot rule out the possibility that we might have ... what, exactly?
The example that we are given is this: that one might, at the moment, give the answer 125 to the addition 68 + 57, and be entirely confident that this was the right answer, wholly in accord with the rules of arithmetic. This, we may feel, is the answer one should always give to this sum. And yet, can one rule out the possibility that one might have, in the past (i.e. perhaps just a moment ago, or five minutes ago, or in one’s youth) actually and with equal confidence in the rightness of it, given some other answer?  Can we, ex hypothesi and absent evidence of what we did do, produce convincing evidence of what we would have done, to establish that what we would have done would have been consistent with that we now say is the right thing to do -- might we not have blithely done something quite other than what we now do and also insisted we would have done on the same prior occasion? Far be it from our imagined challenger to venture that we did something quite other than we now say we do by, for example, producing our old schoolbooks containing completed sums with totals quite at odds with those which we have just now done again, and without reference to our earlier efforts -- for, of course, such a test, besides requiring irksome effort, would (almost certainly) show consistency between the results then and the results now.
‘The sceptic’s’ need to argue about what we would have done is occasioned by his need to put his suggestions beyond the reach of any counter-evidence -- or, at least, of anything that ‘the sceptic’ is prepared to accept as counter-evidence. Hence ‘the sceptic’ raises the question not with respect to this or that sum, nor with respect to some large but doable collection of sums, but with respect to no sum in particular, with respect to some sum that we have not previously done. It is of course quite impossible to prove that one would have done a certain thing in a past situation if one assumes that the only reliable way of proving that one would do a certain thing just is to do it, for, of course, one cannot prove that one would have done a certain thing by adducing evidence that one did in fact do it when the issue as to whether one would have done it arises only from the fact that one did not do it... . Kripke’s ‘Wittgensteinian sceptic’ asks us to prove (other than via arithmetical manuals, other people’s calculations, etc.) that, for some unidentified sum, which we have not done before, we would not, had we done that sum in the past, have given a very different answer than the one we now so confidently treat as correct. What basis, he asks, excluding all arithmetical grounds for asserting that 125 is the correct answer, do we have for asserting that 125 is the correct answer?
It is part of Kripke’s strategy to further insulate the sceptical ‘challenges’ from all possible evidential grounds by treating ‘the sceptic’s’ questions as ‘metalinguistic’, thereby quite calculatingly excluding the arithmetical basis for asserting the correctness of a certain answer to a sum, or of invoking the stark difference which is apparent at the moment of reading between the ‘plus’ function and the ‘quus’ function, both in their formulation as abstract functions and in their illustration. The quus function, with its injunction to sum everything over a certain number to ‘5’ certainly does not at all closely resemble the operation of addition, and the equations 68+57=125 and 68+57=5 are certainly not equivalent. (Further, Kripke is, nominally at least, not challenging our current arithmetical practices and uses of words; and it actually cannot, therefore, currently be denied that these differences are apparent, actual, substantial differences.) In the face of these patent, glaring differences, Kripke wants to ask: What grounds do we have for thinking we are following a ‘plus’ and not a ‘quus’ function... despite the fact that the ‘quus’ function does not, as just explained, closely resemble the ‘plus’ function, with the latter being something entirely familiar to us, normally unproblematically presumed by us, while the former is something utterly alien?
You don’t always get what you want
“The more narrowly we examine actual language, the sharper becomes the conflict between it and our requirement.” (Wittgenstein, PI para.107).
Is this that we have been discussing in the above section, though, the ‘sceptical challenge’ that Kripke is attempting to articulate? Kripke is seeking to work his way toward the conclusion that the notion of consistency in addition (or in the use of words in general) is without content, that there is no contentful way of saying, still less of ‘guaranteeing’, in advance of doing some sum (or of using some word), that the way in which one does the sum or uses the word is consistent with the rule. Kripke, by formulating the issue with respect to what one might have done in the past, makes it very tempting to say that his scepticism is as to whether there is any content to the notion of consistency between past and present practice; but, of course, that cannot be allowed as the formulation, for, as we shall detail, ‘the challenger’s’ argument in practice evades comparison with past practice -- it is about ‘past practice’ only in a very ‘particular’ (and arguably purely notional) sense.
Does the notion of “quus” assist ‘the sceptical challenger’ in saying just what, exactly, the challenge amounts to? It is, we suggest, both an indispensable (rhetorically and ‘logically’  ), but also -- ironically -- a useless notion; for it appeals to, depends upon, the very notions of consistency and inconsistency that are up for challenge! Let us explain: 
The argument from ‘quus’ is that there is allegedly some arithmetical function which is (exactly like) addition up to a point, but which specifically differs from it; one which yields, for all the sums that someone has done, the results which (regular) addition yields, and which may, as we go on talking and acting, continue to yield the results of (regular) addition, but which might (just might) at some point come into play to yield results other than that which we would now, at this moment, regard as the correct ones. Again, of course, this is not a substantive proposal, to the effect that there is such a function, which has certain determinate resemblances to, and differences from, arithmetic as we, and ‘the sceptic’, do it; for, as before, the apparent viability of ‘the sceptic’s’ challenge requires that there be no firm proposal in place. The stating of particular arithmetical properties could only be on pain of exposing ‘the sceptic’s’ argument to the test of seeing what we actually did do at the particular juncture in the series of additions nominated as the point of departure between ‘plus’ and ‘quus’.
Thus it may look as though ‘the sceptic’s’ challenge asserts this: “We cannot rule out the possibility that though we think we are proceeding according to the function ‘plus’ we may nonetheless be proceeding according to some different function, ‘quus’. Since we cannot rule this out, we cannot produce decisive evidence that we are doing plus and not ‘quus’.” However, the challenge is actually this: “We cannot rule out the possibility that, however many sums we do, and however extensively they conform to the pattern of regular addition, nevertheless, for some sums, amongst those we have not yet done, we shall respond to them in a way very different from (regular) addition.” The actual challenge, then, is a challenge to provide a determinate way of countering an indeterminate suggestion, namely, that in some vague sense we might be following some other arithemtical function than addition. The only characteristics with which that alternative function is endowed are those of resembling (regular) addition for all cases of addition hitherto encountered, but differing in ways which no one can specify because:
((a)) nobody is seriously proposing that such a function, coherently definable, exists, and that ‘quus’ is it; and
((b)) its only characteristic is that if it did exist no one can say in just what respects it does differ from addition, save that the point of departure could be manifest only by sums which we have not yet done.
Kripke can’t example what he means (-- this is a grammatical remark). ‘68 + 57’ is, of course, an example of arithmetic; specifically, of ‘plus’ in operation. This is a sum which adds up to 125 and not to 5. If ‘quus’ did require that we answer ‘5’ and not ‘125’ to this addition sum then that would be evidence that we are following plus and not quus. Except, of course, that ‘quus’ is (practically) indiscernible from ‘plus’ for all the sums we have so far done  in that ‘plus’ and ‘quus’ coincide over vast tracts of addition. The example of having 68 + 57 add up to 125 and 5 is ‘only illustrative’: if there were some function ‘quus’ which related to ‘plus’ in the required way, then it would be entirely coincident with ‘plus’ over vast tracts of addition. When it did differ, the differences between it and (our regular) addition would be as sharp and stark as if it were the case that one were to answer ‘5’, not ‘125’, to the addition sum ‘68+57’.
Someone might object here that we are flatly contradicted by the evidence of Kripke’s text, in which the ‘quus’ function appears to be quite clearly stated/defined (x quus y = x plus y if x, y <57, and otherwise =5  ); but this is not so, for our concern is with the use which must be made by an expositor of rule-scepticism of such a statement. Our concern here is not with establishing precisely how little Kripke’s statement resembles a properly formed mathematical function but rather with the status of such a formulation as his. Our question might be put thus: does this formulation comprise a serious proposal or merely an item of would-be stage-setting, an arithmetical Potemkin village?
Let us continue the line of objection a moment longer, in order to clarify our point here: Couldn’t Kripke be saying that ‘quus’ is the function people apply? But: What basis could he have for asserting this? If he is asserting this, then it is easy enough to give the sum (e.g.) 68+57 to people who have not done it before and see whether they answer with 125 or 5. It is only the fact that the ‘quus’ function ‘sketched’ in the opening pages is not advanced as a serious proposal which makes Kripke’s a sceptical argument, for if it were advanced as a determinate suggestion about what people could do, then we could check it out. What is essential to the formulation and ‘employment’ by Kripke of ‘quus’ is, however, that it is not put forward as a serious proposal in its own right, but merely as a conceivable but imagined instance of rule-following (or otherwise) -- so, of course the sceptical challenge which it poses would not be answered by checking what people did, and finding that they answered 125 and not 5. For that would not rule out ‘the sceptical point’, which is not that there is a ‘quus’ function as ‘specified’, but that there might be some function from among any number of arbitrarily conceivable ones which would differ from our addition in the same perverse-seeming way that 68+57 equalling 5 does. Kripke himself can have no reason for thinking that the quus function, as specified, really is the function he or we are all following, rather than any one of the other, arbitrarily conceivable functions or quasi-functions that could have been conjured out of thin air for use in illustration.
It should be clear, then, that were Kripke to attach himself to the ‘quus’ function as specified in the early pages of ‘Wittgenstein on rules and private language’ as a serious proposal, he would thereby liquidate the ‘sceptical’ status of his discussion; for he would have to substantiate the falseness of our supposition by demonstrating the correctness of his proposal, rather than merely deploying it as a means of casting doubt on our suppositions without having to back up his own contentions. Kripke’s ‘sceptic’ is deeply traditional in the sense of purportedly needing to argue nothing stronger than ‘it is not logically impossible’ that (in this case) we are following a ‘quus’ not a ‘plus’ function, to achieve the objective of apparently shifting on to us the burden of proof. Arguing that there is a quus function and that we are following it would be an altogether different and vastly more problematic (because, we think, straightforwardly refutable) project.
The difficulties compound: the purported point about ‘quus’ just cannot be stably, coherently stated. One can perhaps say, if the argument for ‘quus’ were conceded, then addition would not exist. People were always and are only doing quaddition, in which case it is a nonsense to talk about a function ‘quus’ which differs from ‘plus’ since the two are names for one and the same function: plus and quus are identical. However, one can then ask what has been gained, for, it now appears, the only difference is in the change of name. (Perhaps it is emerging more clearly now why, at the start of this paper, we emphasized that there are no words, no sentences, which will end up satisfying the would-be entertainer of scepticism, that s/he will not find a way of expressing what s/he wants to, and will be left frustrated at the ‘constraints’ imposed upon us by our language -- as if ‘behind’ our language one could think the pure truth of scepticism.)
‘The sceptic’ cannot, we think, settle for that; because the idea of there being a difference, albeit indiscernible, between ‘plus’ and ‘quus’ is an essential to Kripke’s motivation for invoking the notion of ‘quus’ in the first place -- for the whole challenge which the introduction of ‘quus’ seeks to communicate is that we are unable to substantiate that we are doing ‘plus’ because we cannot rule out the possibility that we might equally well be doing ‘quus’. It is (purportedly) much less than certain, therefore, that we are following ‘plus’. We are not allowed to eliminate that alternative possibility through pointing to what we are doing, since everything that we can demonstrably identify counts just as much for the alternative possibility, since ‘quus’ can parallel, overlap with, be identical to, addition in every case we can adduce. Hence, our inability to eliminate the possibility that we are doing ‘quus’ could (only) seem significant if there were enough of a difference between ‘plus’ and ‘quus’ to matter. If plus were the same as quus in every case then there would be no sense in talking about ‘quus’ as an alternative possibility. If there were no purported difference between, at the least, what we think  ‘plus’ is and what ‘quus’ is (which is, presumably, a candidate for what we take ‘plus’ to be really is) then, of course, there is no other possibility (than that we are doing what we think we are doing -- namely, adding) to be ruled out, and the sense of a challenge evaporates.
‘The sceptic’, it now emerges, must maintain no less than three quite inconsistent things -- concealing their inconsistency to a degree by depending on them at different points in the argument:
((A)) that there is no difference between plus and quus;
((B)) that there is a difference between plus and quus, albeit an indiscernible one; ((C)) that there is a difference, and that it is discernible -- thus the identification of a ‘quus function’ and the 68+57=5 example is essential to conveying the idea of how great the difference between ‘plus’ and ‘quus’ could be, but (contra ((A))) that example draws upon the notion of a substantial, determinate difference between them -- between, presumably, what we think we would do if we did a certain sum (under the spell of ‘plus’ while reflecting on these matters) and what we would actually do when we came to that sum (for the latter would clearly have to be markedly incongruous with our before-the-fact intuitions for the ‘5’ vs. ‘125’ contrast, to have any philosophical significance). But actually it is quite reasonable to say, as we detail in the next section, that if plus and quus are identical, then one cannot argue that we might be doing ‘quus’ and not ‘plus’ but only that we are doing ‘plus’ (when we do arithmetic). What kind of sceptical challenge is it which in actuality asserts: When you think you are adding you are really doing ‘plus’?! (It sounds more like a sceptical challenge to say: when you think you are doing ‘plus’ you might really be doing ‘quus’; but, again, if one then says “But ‘quus’ is just what I call what you call ‘plus’” then ‘the sceptic’ and ourselves are, to borrow a phrase of Quine’s, only words apart. At most; for ‘the sceptic’ can in actuality find no words with which genuinely to part us, no words which genuinely separate us.)
One obvious way might be thought to remain in which ‘the sceptical challenger’ could parry: by holding that the above does not capture the force of ‘the sceptic’s’ challenge, which is to the very idea of consistency. Which is, of course, true, in intent at least, for the elaboration of Kripke’s argument holds that the idea of consistency (and therefore, conjointly, of inconsistency) in doing addition and in the use of words in general is ‘groundless’. Well indeed, in an important sense it is non-grounded; but that does not mean that it is, therefore, thrown into any doubt.
In any case, challenging the very idea of consistency yields the kind of massive and immediate incoherence that we have already noted (and will note in relation to further aspects of the sceptical case) -- for the way in which the ‘quus’ argument is brought in precisely presupposes the idea of consistency, presupposes that the pattern of ‘quus’ is in some important respects inconsistent with the pattern of ‘plus’, that (up to some point which we have not yet reached) we proceed according to both ‘plus’ and ‘quus’ because, to that point, they are consistent with each other, and that beyond that point, at least for some of the argument (!), they are different from each other, their answers are inconsistent, and would / could differ as drastically as ‘5’ and ‘125’ as answers to the same sum.
Leaving everything as it is
“With cheerful indifference [O.K.] Bouwsma would say to a Kripkean sceptic: forget it! Your doubt makes no difference.”
(Hacking, “On Kripke’s...”, p.292.)
Perhaps the root difficulty in attempting to state ‘the sceptic’s’ position is becoming more apparent. In order to state an argument worth calling ‘the sceptic’s’ argument, it is necessarily one which must be presented as confronting the way we think things are. Things are not, perhaps, the way we conceive them as being. They might be different than they seem. But how might they be different -- how would it be ‘if ‘the sceptic’s’ thesis were true? The only answer to this which can be given, if one must be given, is actually: just as they are... . If ‘the sceptic’s’ thesis that ‘there was/is no determinate sense to the rules of arithmetic or indeed of language generally’ were true, then it would always and ever have been the case that there was no determinate sense to rules or words, and that though the way in which we have been proceeding seems to proceed according to rules, with consistency between subsequent applications and uses, there would have been no such consistency between them. What, in that case, would be the difference between it seeming in every particular that there is consistency in our additions and in the way we use our words -- and that there is a contrast between following the rule and deviating from it -- and there being no such difference? There would, of course, be no tangible, manifest, detectable, discernible difference between them. Being generous to Kripke’s lexicon, one might say: The world in which there is consistency is (insofar as anything that anyone can point to is concerned) exactly the same as one in which there is none, differing in not one in-principle-identifiable feature.
It is, of course, necessary for the ‘quus’ argument that the only evidence for the ‘quus’ function is exactly the same as the evidence for the ‘plus’ function: the ‘quus’ function is something which, for actual instances, gives us the results which we call ‘results of addition’, which we say derive from the application of ‘plus’, making the notion of ‘quaddition’ derivative upon that of addition. ‘The sceptic’ can only propose the notion of ‘quaddition’ by borrowing our understanding of addition -- quaddition is an intelligible function only insofar as it is one which (in terms of the sums we have done) gives us the results of what we have hitherto termed ‘addition’. The specification could not, of course, be intelligibly accomplished the other way around -- for, as noted more than once above, ‘the sceptic’ is in practice incapable of saying what ‘quaddition’ is actually like in respect of its (actual, as opposed to notional) difference from addition (for, of course, ‘quaddition’ was never anything but a notional entity). In short, the only ‘evidence’ there is for ‘quaddition’ is the fact of addition.
However, and we have alluded to this already, matters can get -- are -- even more tortured; for ‘the sceptic’ might well reply that our arguments above are grist to the mill, that the attempt to say that ‘quaddition’ differs from ‘addition’ is misleading precisely because it suggests that they are two different things, when this is not at all the claim that ‘the sceptic’ wants to make. Nor is it; but the difficulty is, again, ‘the sceptic’s’, not ours. ‘The sceptic’ wants to show us that there is an intelligible difference between what we believe to be the case and what is (actually, only ‘what might be’ (in most cases??) the case) but is seeking, thereby, to advance the position that we are incapable of becoming aware of that difference:
We proceed (or, if forced to reflect, we claim to proceed) by adding, but we could be wrong about that (supposedly) -- we might be quadding. Given a sum, we add to a particular total -- presented with 68+75, we total to 125. We have followed the rules of arithmetic, the correctness of our addition justified by our conformity with those rules. But what if, instead of proceeding according to those rules -- something we could not do were there no rules to conform to, if there was no substance to the idea that a rule fixed any next step in a sequence -- we were simply ‘taking unjustified leaps in the dark’? Plainly there could be no experiential difference, for there is no feeling which is marked as literally a ‘following of the rules of addition’ feeling, so our feeling that we are justified in taking the steps that we do is nothing other than finding that these steps come to us naturally -- we obey the rules ‘blindly’, as Wittgenstein said. Further, there need be no experiential content to our inspection of the sum and the derived total: seeing whether the total is justified can only be carried out or ‘tested’ by doing the sum over again. The fact that our arithmetical operations are ‘leaps in the dark’ rather than justified applications of a rule or procedure is not something, then, that can be distinctively experienced -- making leaps in the dark would, in this connection, be experienced as a matter of following the dictates of a rule, just as our operations with arithmetic have been so experienced (even if, ‘if’ ‘the sceptic’ were ‘right’, they would be leaps in the dark), and the conviction that the achieved result of the addition follows from the proper application of the rule would be integral to the sense of the figure-in-total as ‘the right result’. Making the calculations was of course not experienced as ‘taking leaps in the the dark’, even if the ‘the sceptic’ were to insist that they really are just such leaps. Indeed, and ironically, Kripke can again be used against himself here, for he says himself that drawing ‘quus’ type conclusions would feel right to us. Consequently, the addition 68+57=125 should be a better example of ‘the sceptic’s’ case than 68+57=5! But plainly, it isn’t. The (former) sum exhibits what has come naturally to us, and is the one which provides what Kripke, and his readers, will designate the correct result (even though, allegedly -- for this is the core of ‘the sceptic’s’ case -- there is no rule to justify our assigning that total to the sum); whilst the other does not seem natural at all -- i.e. it looks wrong, is to us clearly wrong (and Kripke does not of course deny the fact that though there are no rules to justify one conclusion over another we nonetheless react toward one total very differently than toward another). However, whilst the better example, 68+57=125, cannot play an effective illustrative role in setting out ‘the sceptic’s’ case (for to try to make it play that role, the role that 68+57=5 plays, would highlight the absurdity of that case rather too drastically and instantly), it is no small feature of ‘sceptical’ arguments that they must recurrently resort to inadequate surrogates for what they are ‘trying to say’.
On conversing with ‘the sceptic’
“For the sceptic to converse with me at all, we must have a common language... . Of course ultimately if the sceptic is right the concepts of meaning and of intending one function rather than another will make no sense... . But before we pull the rug out from under our own feet, we begin by speaking as if the notion that at present we mean a certain function by ‘plus’ is unquestioned and unquestionable.” (Kripke, ‘Wittgenstein...’ pp.11-14)
What we have illustrated, then, is that ‘the sceptic’ is in a vicious double-(at least)-bind: there must be a difference, but there is no difference; s/he explains what the difference is, then denies that the explanation makes sense / means anything. This second point draws out that there is something extremely odd about debating with ‘the meaning sceptic’. Kripke writes:
“For the sceptic to converse with me at all we must have a common language. So I am supposing that ‘the sceptic’ provisionally is not questioning my present usage of the word ‘plus’. He agrees that, according to my present usage, 68+57 denotes 125... he merely questions whether my present usage agrees with my past usage, whether I am presently conforming to my previous linguistic intentions... He [‘the sceptic’] does not -- at least not initially -- deny or doubt that addition is a genuine function.” 
If ‘the sceptic’ denies the existence of a common language, and if a common language is the essential precondition of a conversation, then it is not possible to have a conversation with ‘the sceptic’. It is not possible for the following reason: If ‘‘the sceptic’ is right’ (in what s/he wants to mean  ), then we simply are not, however we proceed, whatever argumentative steps Kripke takes, having a conversation with ‘the sceptic’. Though Kripke’s ‘sceptic’ does not assert at the very outset the conclusion that we are not entitled to the ‘assumption’ that any of our words have meaning, this is plainly the terminus of the argument, one toward which s/he is always pretty rapidly manoeuvring, not one which is ‘incidentally’ its outcome. What ‘the sceptic’ is asserting, implicitly, is not that some of our words have no meaning, whilst others do. The differentiation between our past and present usages is gratuitous to ‘the sceptic’s’ arguments per se, and comprises only a façon de parler (It is misleading at best to present the matter as though doubt could be focussed on some particular expression uttered at some particular time when ‘the sceptic’s’ doubt wants to be about every word ever in his our mouths and his.). ‘The sceptic’ does not want ultimately to distinguish between our past and present usages, but is putting forward arguments which (‘ultimately’) purportedly apply to all of them: there is no fact which establishes what a word means, and hence no basis for saying it means one thing and not another, or assigning it any determinate meaning at all. Hence, if the conversation-facilitating gambit is adopted, it can only involve a misleading presentation of ‘the sceptic’s’ argument, setting this out as though it applied only to past usages, when, of course, it is actually applied to past and present alike.
This of course eventuates in ‘the sceptic’s’ direct self-contradiction (such that s/he is not actually succeeding in saying anything), for:
“How can one doubt whether one means plus (rather than, say, quus) in the present on grounds of doubt about whether “plus” meant plus in the past. For, if one’s present meanings are thrown into doubt, then the doubt one raises (in the present) about the past must also be thrown into doubt. One cannot entertain that one meant quus by “plus” on past occasions unless one knows (or, better, presumes) now the meanings of “plus’ and “quus”. So I see no way in which present use can be undercut without an undercutting of the very past use with which present use was supposed to be undercut.” 
It would be deeply misguided to treat ‘the sceptic’, as too many philosophers we think do, as being able and (‘generously’) willing to abstain for a while from the flat out statement of the sceptical ‘position’, of comprehensive doubt about the capacity of any word to mean. Does s/he make this concession to us? No; leaving our present usages unquestioned conveniences, aids and abets, rather, ‘the sceptic’. (After all, ‘if words do not have meaning, if they do not mean one thing rather than another’, then ‘sceptics’ cannot say what they (want to) mean, such as this, either.) And this dynamic is clearly at work in the situation as described above.
It is the ‘Tractatus Logico-Philosophicus’  which contains the best advice to ‘the sceptic’ -- that ‘whereof’ we cannot speak, ‘thereof’ we cannot speak, and we ought not to describe ourselves as trying to. ‘The sceptic’ supposes that s/he has something to say, which is that no-one can have anything to say -- including presumably ‘the sceptic’. Or are the words ‘Words have no determinate meanings’ somehow exempt from the general ban on meaning, the only words which can now legitimately be used?! If we do not suspend Kripke’s ‘sceptic’s’ assumption with respect to our discourse, our conversation with ‘the sceptic’, then, of course, it is ‘the sceptic’ who cannot continue, for it is s/he, not we, who must be the first to abandon dependence upon the ‘assumption’ of a common language. If we do not ‘assume’ that there is a common language, but rather, and unselfishly and generously, agree to attempt to adopt, from the beginning, Kripke’s ‘sceptic’s’ position on this matter for the sake of argument, then still ‘the sceptic’ is really no better off, for now we have no reason to suppose that ‘the sceptic’s’ words have any more meaning than ours ‘allegedly’ do, that ‘the sceptic’ is asserting or denying anything definite. To paraphrase Kripke: who is to say what the words “Words cannot have determinate meaning” might mean in ‘the sceptic’s’ mouth; or if they mean anything at all (Perhaps they mean (e.g.) what we (think we) mean by “68+57=125”?!)? Presumably, we do not ourselves know what we mean (or even mean anything) when we wonder idly (in) that way -- if that is what we are doing; if we are doing anything? In all of this, of course, the sceptic is undercutting, at least as much as our supposedly questionable presupposition, his own sceptical standpoint for framing any sceptical doubts. As one of us has argued on another occasion:
“[Our] point has been not that the conclusions of such arguments undercut the premises which undergird them, nor merely that the conclusions cannot be stated, but that the premises too, the arguments in toto, simply cannot be stated (they don’t get as far as actually having conclusions). For at every stage such arguments (such premises) presuppose what they deny.” 
Providing for oneself
“In giving explanations I already have to use language full-blown (not some sort of preparatory, provisional one).”
(Wittgenstein, PI para.120).
It may be that some readers are still inclined to think that much in the way of questions that make sense, problems to which philosophy needs must give constructive answers, remain to be addressed in and after Kripke’s text, and the ‘literature’ it has spawned. Let us then focus in on a further (related) specifically misleading element about Kripke’s ‘sceptic’s’ making of a ‘concession’ to us, one which is integral to the thought that there may be genuine puzzles here, even if they are ‘hard to express’; we refer to the presentation of ‘the sceptic’s’ ‘concession’ as a ‘provisional’ step; “I am supposing that the sceptic, provisionally, is not questioning my present usage of the word ‘plus’; he agrees that, according to my present usage, ‘68 plus 57’ denotes 125.”  This move is supposed merely to facilitate the setting out of the sceptical argument, enabling ‘the sceptic’  first to cast doubt on our past usages, and then, having obtained argumentative leverage with those, ingenuously to extend them to our present ones, as though this were the upshot of the first part of the argument, rather than simply its full (and, we have argued, incoherent) rather than its oblique statement. The truly misleading element is, however, the pretence that this concession is made “provisionally” to get the argument going, as though at some point in its course the ‘assumption’ that we have made is to be withdrawn. It is not. (Precisely as with the ‘assumption’ of a common language, it would not make sense to suppose that this ‘provision’ that Kripke makes for himself could intelligibly be withdrawn.)
And yet: at some point in the argument ‘the sceptic’ will reveal that all along the assumption that any of our words can have determinate meanings was what the sceptical argument has been directed against! An essential ‘assumption’ of Kripke’s argument was ... just what the argument was directed against! A deep internal tension indeed (The argument is just the same for present usage as for past usage, as discussed above.).
In sum: There is a difference between saying what ‘the sceptic’ says and suspending the ‘assumption’ that there is a common language for, of course, ‘the sceptic’ does not suspend -- cannot (grammatically) suspend -- this assumption at any point in the argument. And yet suspending this assumption is exactly what Kripke’s ‘sceptic’ is committed to doing. Things are just the same with Kripke’s ‘provisionally’ not problematising the present -- it is another aspect of just the same difficulty.
Further, to speak of ‘conceding’ present usage, to ‘concede’ the supposition of a common language, itself supposes that ‘the sceptic’ knows what the difference is between proceeding as if there were a common language, and proceeding minus that ‘assumption.’ It would therefore be perfectly reasonable --and most ‘interesting’ -- for us not to make the concession but simply but rather to invite a defender of scepticism to implement the program that s/he wishes to pursue, and to see how things would then proceed. The difficulty we have in relating to ‘the sceptic’ in the absence of a common language is not like that which we can encounter in a real situation where we lack a common language, as when someone who speaks only English meets someone who speaks only Chinese. In the case of ‘the sceptic’, s/he seemingly speaks exactly the same language that we do (One might say: Kripke’s book is written mostly in pretty plain English, after all!) -- save for the supposed fact that there is no such thing as ‘the same language’. Does suspending the ‘assumption’ that there is a common language make any difference to what ‘the sceptic’ says, to how s/he talks? Though we offered a moral of the ‘Tractatus’ earlier, in a sense we admit that it would not necessarily be incumbent of ‘the sceptic’ to remain silent. S/he might implement her program by going on garrulously; for since ‘the sceptic’s’ (and everyone else’s) words have never meant anything, and that has never stopped anyone talking in the past, why should it do so now? Why not just go on spouting nonsense, albeit nonsense that is indifferentiable from our ordinary use of language (i.e. from what is in fact the paradigm of sense!)?
Indeed, there is a further reason for supposing that the withdrawal of the ‘assumption’ of a shared language does not make any difference, apart from the fact that one cannot see what particular difference to anything one might say would follow from withdrawal of the assumption of a common language, in the absence of any identifiable step one could take to register the fact that one had given up that ‘assumption.’ The further reason is this: that one couldn’t say anything that would, without all the above difficulties of ostensibly denying what it is in practice affirming, register this fact.
It seems, then, that the actual force of ‘the sceptic’s’ argument, if any, is that there is no discernible difference between talking as if there were a shared language, and talking in a shared language simpliciter, that the capacity to carry on a conversation does not hinge upon the assertion or denial of the contention that a shared language is available -- for the simple truth is that there is no way to say what would be the difference between the case in which there was a shared language, and one in which there was/is not.
But this is a lesson one can learn without all the headaches and angst of scepticism; for we might usefully say, paraphrasing Wittgenstein, that if everything looks as if there is linguistic meaning then there is linguistic meaning. (An analogy with money might help here: It is only slightly tendentious to say that “If it’s used as money then it is money. It doesn’t matter whether its a cheque written on the side of a cow, or a piece of plastic, or what have you, or even forged; if it works, it’s money!” In particular, even if an individual knows that a note they own is forged or that a credit card they have is stolen generally doesn’t stop them being able to use it (If a forged note or coin fools everyone, and permanently enters the system of exchange, then ‘to all intents and purposes’ it is money!). That someone believes that they can only speak nonsense needn’t stop them being able to make perfect sense, to speak meaningfully.) As was mentioned in connection with Hacking’s take on meaning scepticism earlier, it seems the only answer we can give (/need to give) to the question, “Grant if you possibly can that the sceptic’ is right, how ought we to respond?” is “Carry on as usual!” 
“We get a more radical divergence from Kripke, however, if we suppose that the thrust of Wittgenstein’s reflections is to cast doubt on...the thesis that whatever a person has in her mind, it is only by virtue of being interpreted in one of various possible ways that it can impose a sorting of extra-mental items into those that accord with it and those that do not.” (John McDowell, “Wittgenstein on meaning and intentionality, p.45)
Kripke’s chief idea of how to search for relevant facts via which to found meaning involves an inventory of the mental realm, and in particular a search for some intention. What is really queer is not the absence of any requisite fact or super-fact, but Kripke’s idea of what it would be to follow a rule, and of what might justify my confidence that we are adding, not quadding. Kripke’s notion of learning a rule is that one’s past intentions regarding addition should determine a unique answer for indefinitely many new cases. This is, indeed, the basis upon which Kripke seeks to shift the issue from an arithmetical one to a ‘metalinguistic’ one. It is thus Kripke’s assumption, from the start, that the idea of ‘grasping a rule’ requires something over and above our being enabled to take the steps that the rule requires, that if one performs a computation to add 68 to 57 “obtaining, of course, the answer ‘125’ [then] I am [merely?] confident, perhaps after checking my work [how many times?], 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.”  It is Kripke’s way of formulating what is involved in adding, or indeed in learning to add, that exposes the matter to ‘the sceptic’s’ doubt. The admittedly bizarre sceptic challenges thus: that if one is “now so confident that, as I used the symbol ‘+’, my intention was that ‘68+57’ should turn out to denote 125, this cannot be because I explicitly gave myself instructions that 125 is the result of performing the addition in this particular instance. By hypothesis I did no such thing.” 
At this point we might reiterate that the notion of ‘addition’ is already simply being ‘reiterated’; though here it is not so much the word that is being reiterated as the idea; ‘addition’ is already being imagined in a doubled or homuncular fashion, it is being as it were ‘put back behind itself’. That is: To suppose that one could ‘give oneself instructions’ that, when one added 68 to 75 this would compute at 125 is an absurd caricature of what is involved in learning to add. It is certainly true that ‘by hypothesis’ one does no such thing -- insofar as ‘giving oneself instructions’ that certain computations one had not done should have a certain result is no part of the learning or practice of addition. Since the remark about the absence of such a ‘mental operation’ as giving oneself instructions is attributed to ‘the sceptic’, we are presumably meant to understand it not as a truism about addition, but as a reproachful comment -- the suggestion is, presumably, how could one learn addition if not by ‘giving oneself instructions’?
The absurdity of the notion -- especially in such a context as this -- of ‘giving oneself instructions’ is, we trust, patent. One learns, in learning addition, not the answers to particular sums, but a general procedure of figuring out, of working out, the answers of indefinitely many such sums.
In being taught addition we are being taught how to use a method, how to get the right answer, and not being taught a long list of right answers. The picture which apparently captivates Kripke is of having learned addition as being equivalent to having, somehow, learned all the right answers, having (somehow) learned them in advance of doing the sums (Would certain fantastically competent idiot savants then provide a ‘straight solution’ to Kripke’s worries, to his ‘sceptical problem’?!). If we have not, at some point, ‘given ourselves the instruction’ that an addition 68 + 57 should come out at 125, how can we be sure that this outcome is ‘what we had in mind’ (though Kripke would prefer no scare-quotes) when we grasped the rule?
However, the whole point about learning a rule or procedure is that it does not involve knowing how to go on in an occurrent sense, it does not involve having any (or, at least, many) outcomes in mind  -- and, if one does not have an outcome in mind for a particular sum at the time when one learns the rule, then the outcome that one gets cannot be either the same as or different from the outcome that one had in mind! Certainly, one’s confidence that one’s arithmetical practice is consistent cannot be assured by looking for mental occurrences, such as intending that a certain sum should issue in a certain result, and hence searching one’s memory for the occurrence of any such laying down of intentions can give no reassurance -- but does this mean that one has no basis for one’s confidence, that one should now worriedly wonder whether there is any consistency in one’s arithmetical practice? Hardly. And is that only because we have defeated ‘meaning-scepticism’? No; it is because there isn’t any such thing as meaning-scepticism to defeat. Our arithmetical practices literally are fine as they are; they need no philosophical shoring-up against Kripke.
Let us sum up. Many philosophers have felt that Kripke correctly identifies a major strand of thinking in Wittgenstein’s PI, a strand that purportedly finds alarming the absence of a certain kind of ‘grounds’ which one might have imagined that one needed, in order to be justified in using any given word in a certain determinate way, or in the way which comes most naturally to one. Kripke’s ‘sceptical challenge’ is purportedly one heuristically vivid way of making this point, of dramatizing this strand allegedly in Wittgenstein’s own thinking. We have questioned ‘from the ground up’ whether there is in fact any way whatsoever of making this point vivid. We have done so by means of challenging the notion that any coherent challenge whatsoever is laid down by Kripke’s ‘Wittgensteinian sceptic’.
Thus the mainstream debate around Kripke’s Wittgenstein has been over what the overall conclusion immanent in PI is, and in particular whether PI embraces (or, leaving Wittgenstein-exegesis aside, whether one ought really, philosophically, to embrace) ‘meaning-scepticism’, or whether rather the moral is only that a certain wrong view of what it is to understand a word, or to attach a meaning to it, leads to ‘meaning-scepticism’. By contrast, we have gone beyond the latter ‘moral’, by means of questioning whether meaning-scepticism has any intelligibility whatsoever. The mainstream debate over Kripke’s work finds interesting questions such as how one is to understand counterfactuals about meaning (e.g. “If she means addition by ‘plus’, then she would have given the answer ‘125’ if asked what ‘68 + 57’ makes”), given that PI 138-202 ‘appears’ to make it problematic what the truth of such counterfactuals consists in. By contrast, we have tried to bring out the bizarreness of the first clause in such counterfactuals, the emptiness of the shuffle involved in any notion of the meaning of terms ‘lying behind’ them, given that Kripke has no way of satisfying himself or us about how his ‘alternative meanings’ for such terms (‘quaddition’) are to be construed, are to be understood. In short, we argue, it is actually not that the premises of meaning scepticism -- which Kripke’s Wittgenstein suggests lead inexorably to meaning-scepticism -- need not be maintained, or are false. Rather the ‘premises’ of ‘meaning-scepticism’ turn out not to amount to anything whatsoever (e.g. ‘quaddition’ does not turn out to be a determinate mathematical function). Neither Kripke nor his supporters nor most of his opponents have succeeded in meaning anything at all by ‘meaning-scepticism’. They have not even succeeded in describing what ‘meaning-scepticism’ would be.
In the ‘Philosophical Investigations’, Wittgenstein tries mighty hard to persuade us away from the idea that understanding or meaning are some kind(s) of mental occurrence or mental state, and to recognize that the words “understand” and “mean” are not the names of episodes or processes, but rather, roughly, for capacities or abilities. Learning to add is a matter of acquiring an ability to do certain kinds of operations, and, as such, we do not acquire evidence of, or confidence in, our ability to do something by introspection, by locating some present or prior mental occurrence which evidenced the possession of that capacity. We acquire evidence of and confidence in our abilities by exercising them, by finding that we can do certain sorts of things, by a record of having successfully done those very things on previous occasions -- or, of course, by making determinations as to how the thing to be done stands to what one is confident one can do. We can be confident that we can lift twenty three and a half kilogram weights -- we have never lifted this weight before, but it is less than weights which we have quite easily lifted. Could we lift one hundred kilograms? We have not tried to lift this weight before either, but we do not know whether we could lift it or not -- not just because we have never tried to lift this particular weight before, but because this weight is much greater than any weight we have previously tried to lift. Of course, even our confidence can very definitely be misplaced: the fact, if it is one, that we have lifted one hundred kilograms before can be the basis for our confidence that we can lift this weight, here and now, but, of course, we may find that we cannot now do this, that we can no longer lift this weight -- age and lack of relevant exercise have now taken their toll. Kripke’s argument is correct, then, only in the extremely attenuated sense that it correctly supposes that the basis for our confidence that we know how to proceed with adding -- and that it is adding, not ‘quadding’ or whatever, that we are doing -- does not derive from having undertaken any self-conscious laying down of the intention that, when we come to add 68 to 57 it will come out as 125. But he fails to draw the proper implication, that this is because such a laying down is in general immaterial to the possession of such confidence, and that the absence of such a laying down does not count against the consistency of our practice. After all, to lay down such an intention would not be something we could do in advance of (doing) the sum, it would be a matter of doing the sum itself -- our confidence that 125 was the correct answer, entirely consistent with everything we have been taught, would normally come from doing the sum correctly (i.e. in just the way we were taught).
In the first instance, of course, our confidence that we can get the answers right comes from authority; our teacher simply tells us that our response is or is not the right answer. Our confidence about our capacity to get the sum “Take 68, then add 57 to it” right would not normally derive from the fact that we have done it before, but from the fact (if such it is) that we have been taught arithmetic, such that, until we reach the limits of our abilities, we can do such sums, without hesitation, with ease.
And, quite simply, no ‘meaning-scepticism’ with which we have been presented has anything whatsoever to say to the contrary of any of this.
“The ladder must be finally kicked away.” (Kripke, ‘Wittgenstein...’, p.21)
We have intended and endeavoured to test the arguments of ‘the sceptic’, as voiced by Kripke, tirelessly to complete (self-)destruction -- or (simultaneously) to what use they can be put, to what sense can be made out of them (Not quite zero -- though almost indiscernibly more than that...).
We might then sum up by saying that Kripke’s conjuring tricks (some of which one might easily miss, so very early do they enter into his text) -- his prestidigitation -- aim to create the illusion that the irrefutability of ‘the sceptic’s’ arguments is more than an artifact of their contrivance. But an illusion it is, from the very outset, albeit in certain respects and in certain contexts a very tempting one; until the ladder is, as it must be, kicked away from under it; until it comes crashing down like a house of cards; or until, to use a more gentle image, the rabbits are placed gently back into their hats, into their homes.
Are we spoilsports, spoiling the philosopher’s games? Unfortunately, in philosophy it is all important to unmask the conjuror’s tricks, rather than to allow ourselves to be delighted and taken in by them. We contend that even many of those who have correctly resisted both Kripke’s interpretation of Wittgenstein and his conclusions about meaning do not show clear evidence of having understood what we have tried to show clearly: that Kripke’s quus ‘thought-experiment’ is simply not well-defined in the first place. That Kripke’s dramas and sleights-of-text (and the urges and compulsions of those who write after him) cannot, consequently, alter in the slightest the humdrum facts about digits -- about numbers and our lives with them -- and about language-use and human life more generally, that they (merely) appear so radically to challenge.
“[Solipsism] can never be demonstrably refuted, yet in philosophy it has never been used otherwise than as a sceptical sophism, i.e. a pretence. As a serious conviction, on the other hand, it could only be found in a madhouse, and as such it stands in need of a cure rather than a refutation.” (Arthur Schopenhauer, ‘The world as will and representation’)
The last word on Kripke, however, needs perhaps to re-orient one toward what we have hinted at periodically in the body of our text, above, by means of referring to the futility of efforts to refute scepticism, to the attractiveness and temptingness of the forms of words used by sceptics, and to the importance of the Cavellian project of endeavouring to understand the deep cultural and human reasons for ‘our’ condition, our condition of being subject to influence sometimes from even the most evidently absurd sceptical doctrines. Rule-scepticism as a substantive philosophical doctrine is utterly absurd -- it fails even to get off the ground in the very slightest. But showing so, and even persuading someone rationally so, does not necessarily remove all its charms.
We have shown that there is nothing worth calling genuine conversation with ‘the sceptic’. But that does not mean that someone genuinely attracted by Kripke’s words will be effectively (and without residue) persuaded by our words.
Scepticism is in the final analysis a matter not of narrowly intellectual conviction but of mood, albeit often of mood consequent upon a particular mode(s) of intellectual comportment and of a certain ‘over-rational’ concentration upon (say) the contents of our minds rather than our embodiment and embeddedment in a world of ground, of fellow actors, and of practices (of getting things done). Our entire discussion can only hope to convince someone whose grip on and placement in practices of intersubjectively-comprehensible rule-following etc. etc. is pretty secure. If someone actually is in the grip of a frenzy, or is subject to LSD flashbacks (scenarios gestured at by Kripke), then even our efforts at cure -- cure of the intellectual disease stoked by Kripke -- are most unlikely to be effective.
For our own part, we are thankfully subject to such conditions only extremely rarely. Perhaps Kripke or some of those impressed by his thinking have the misfortune to be less ontologically and epistemologically secure. But what we find repugnant is those who purport to be taking up or at least considering a serious sceptical ‘position’, while actually they have no doubts, and no empathy for what it is like really to have terrifying thoughts along the lines of “Maybe I don’t even know how to use any words properly any more; maybe I’m losing my mind” cross one’s mind.
For those who are really in the grip of the kind of thoughts proposed by Kripke, weighty and genuine conversational interaction is going to be an extremely tricky and probably counter-productive enterprise, and so even our Wittgensteinian efforts at diagnosis and cure are unlikely to be efficacious. For those -- and we suspect that this class includes virtually all philosophers virtually all the time -- who do not know scepticism as other than a sophism, our occasionally polemical or short-tempered tone in this piece is perhaps appropriate. We have hoped in this paper to coax any readers genuinely tempted by Kripke’s writing (but yet far from the terror that is almost inevitably going to attach to really feeling an abyssal absence of meaning) away from his clever trickery and back toward the everyday, as present in and implicated in Wittgenstein’s own work. Such coaxing has in the main proceeded by attempting to take seriously Kripke’s words (and the temptations they express), and finding that there is just nothing that we can count as doing so. But for those readers and writers who pretend to be themselves impressed by Kripke’s (would-be) doubts, but are actually just using those ideas for the purposes of propounding abstract intellectual (e.g. semantic) doctrines or of intellectual gymnastic, we have little respect. Such philosophers need to understand not only that the conjuring trick does not get off the ground, but that -- to those who nevertheless actually feel as if it does -- the matter is too weighty (and too much a matter of mood rather than of purely rational conviction) to play philosophical games with.
We have not offered a refutation of scepticism, but rather a fragment of a cure of the impulse toward scepticism, an impulse which is a matter less of narrowly philosophical than of existential origination and significance. And one reason why this is only a fragment is: that a fully effective cure, insofar as there can be any such thing, likely involves changes in one’s life and perhaps the life of one’s entire society. So we do not have a very rosy picture of the likely effectiveness of our writing here. Cartesian and ‘Kantian’ etc. impulses  toward Kripkean scepticisms are likely in the end to diminish seriously only if the kinds of changes in ‘form of life’ which Wittgenstein urged parenthetically but powerfully upon his disciples and readers -- upon his (and our) culture -- were to be realized...
 We place ‘the sceptic’ in scare-quotes as a way of recurrently -- indeed, perhaps annoyingly (and thus effectively?) -- reminding the reader of a point which follows from our main point (that being that ‘the sceptic’ can have nothing intelligible to say, and cannot therefore be considered to have an authentic position to hold); namely, that in fact no-one exists (not even a philosopher) who could accurately be described as trying to be a rule-sceptic. (We would be inclined to go further: no-one actually exists, to our knowledge, who would even attempt to self-identify as ‘a rule-sceptic’ (or anything cognate), to describe themselves thus. This, if true, also buttresses our main point, for it points to the misleading philosophical rhetoric which often accumulates around philosophical scepticism and the figure (truly, only a textual/linguistic figure, even if in some contexts an important one) of ‘the sceptic’. (Cf. also n.11, below.))
 Though not the only style. More effective in our view, for instance, is that offered by Cora Diamond in her “Rules: Looking in the right place” (in Phillips and Winch (eds.), Wittgenstein: Attention to particulars (New York: St.Martin’s Press, 1989)), where the approach is by way of emphasizing the life, the contexts, in which rules live, and of stressing the distance from that life not only of Kripke’s approach but also of that of many of his would-be ‘constructive’ opponents. See also Diamond’s The Realistic Spirit (Cambridge, MA: MIT, 1991), and Peter Winch’s criticisms of Kripke in his Trying to make sense (Oxford: Blackwell, 1987). An approach which shares a great deal in common with our own is Sharrock and Button’s “Do the right thing: Rule-finitism, rule-scepticism and rule-following”, in Human Studies 22 (1999), 193-210.
 As will become clear, what we are in fact rejecting is not ‘the sceptic’s’ claim that we have no justification, but the suggestion that we need justification; that in the kind of cases ‘the sceptic’ raises, it is intelligible to ask for any justification where we do not normally give any, or for further justification than we could ordinarily give. (Though see the ‘Epilogue’, below, for a cautionary note.)
 Saul Kripke, Wittgenstein on rules and private language (Cambridge, MA: Harvard, 1982; henceforth “Kripke”). We will not spend time establishing the point, made extensively elsewhere, that Kripke’s reading has little to do with what is actually in Wittgenstein’s text, the purported paradox being attached to PI only by the most bizarrely selective reading (e.g. the complete ignoring by Kripke of all but the first three sentences of the crucial paragraph of PI, no. 201). See e.g. Baker and Hacker’s Rules, Scepticism and Language (Oxford: Blackwell, 1984); or A.Collins’s paper in Mid West Studies in Philosophy 17 (1992); or Read’s writings on Wittgenstein, Kripke and Goodman (from whom Kripke in the first instance got his central ideas), including (though somewhat tangentially) the paper “The riddle of the new riddle”, Journal of Thought 33:2 (1998), to which the present paper is a kind of sequel; and also n.3, above.
 See n.2, above; though we do not deny that the Kripkian challenges might well be (psychologically /rhetorically) unanswerable for someone in the grip -- in the play -- of their ‘logic’. I.e. We do not regard Realist / Anti-sceptical approaches as at all well-off in their ‘dialectic’ with Kripke’s ‘sceptic’. We will not claim that consistency in practice is metaphyiscally fixable -- only that consistency is consistency... . Consistency in practice(s) can be neither justified nor evacuated by means of philosophical analysis.
 There is no need to formulate
this in a cautionary way such as, “Doing addition as we understand it” since
there is no difference -- there can be
no difference -- between addition and ‘addition as we understand it’. It is not
as if there is some other kind of addition than that that we understand. The
phrase “as we understand it” brings in an unnecessary and misleading suggestion
of contrast, of a bare possibility of addition otherwise than we understand it.
(For amplification, see David Cerbone’s “How to do
things with wood”, in Crary and Read (eds.) The
New Wittgenstein (
 In the following section, “You don’t always get what you want”, we respond to the objection (to our argument here) that Kripke does in fact make a serious proposal for a modified arithmetical system.
 One might possibly, with equal confidence, have given a different answer, but one would, in that case, have been wrong. The fact that one might give a wrong answer does not, of course, count against the fact that some other number than the one given provides the right answer -- that, in this case, it is 125, not 5. The fact that one might have made a mistake does not contribute to the point Kripke aims for, that ‘(meta-)linguistically’ there is no such right answer.
 We are, therefore, provisionally rejecting the category of ‘contingent scepticism’ or ‘a posteori scepticism’ as nonsensical, as a locution that would best be withdrawn from circulation; for surely philosophical scepticism can hardly be a scientific matter -- it cannot hinge upon the availability of evidence with which to settle a matter. Contrast R.Fogelin’s attempted taxonomy of scepticisms in the opening pages of Hume’s Skepticism in the Treatise of Human Nature (London: Routledge, 1985); see also the closing page and notes of R.Read’s “In what sense is Kripke’s ‘Wittgensteinian paradox’ a scepticism?: On the essential opposition between epistemological and metaphysical aspects of rule-scepticism”, in De Philosophia 12 (1996),pp.117-132; and n.24, below, on Ian Hacking’s intriguing and problematic attempted ‘exemplification’ of rule-scepticism (To be fair to him, Hacking is explicit on p.114 of his paper in agreeing with us that “It is an essential feature of sceptical arguments that one can never give real examples”). Our point here ought not of course to be taken as implying that ‘philosophical scepticism’ does not need to be explained -- to be comprehensible -- if it is to be anything; nor as implying that its exponents do not often write as if there were or could be evidence that could support sceptical pronouncements.
 We will return to develop this point in later sections, where (to anticipate) we argue that this is an anomaly in Kripke’s would-be meaning-nihilistic position (though again, not an easily-avoidable anomaly, not something there by mere happenstance); for if that position were being seriously and coherently advanced then one could not adopt a ‘take it or leave it’, ‘on again / off again’ attitude to the denial that our practices of addition and our uses of words generally have the meaning we take them to. Ifone takes Kripke ‘at face value’ and applies his arguments ‘with full scope’, then it is inevitably the case that Kripke cannot, even ‘nominally’, exempt our current mathematical etc. practices from the purported general challenge to meaning which he mounts.
 The exposition of our argument is made more difficult by the unavoidable incoherence in Kripke’s own, for it is a part of that argument to deny the differences between ‘plus’ and ‘quus’ which are so patent. The intended (but ‘ineffable’) sense of the argument --or, more carefully, what Kripke wants to say -- is that ‘plus’ and ‘quus’ are not different because (for all we know) plus really is quus. Arthur Collins (op.cit.) has argued, in one way quite rightly, that Kripke has no need whatever of the notion of quus, for it is of ‘plus’ and that alone that Kripke is attempting, in his tormented would-be-sceptical way, to speak. However, we maintain that the notion of ‘quus’ is forced upon Kripke because of his rhetorical tactics (which appeal to patterns of inference well-established by virtue of the philosophical currency of Goodman’s “grue” etc.) and his argumentative strategy. This strategy, as explained below, is doomed to grapple unsuccessfully with the desire to convey the sense that the adoption of sceptical attitudes would make tangible and profound differences by giving some illustration of how they would do so, whilst purportedly confronting (but in fact, avoiding!) the fact that the sceptical arguments themselves hold that these differences are undetectable. Kripke needs both to give the impression that “quus’’ requires us to something different than “plus”does -- and not to mean that at all! (This is a version of the general diagnosis that we have already made: that there is an unavoidable clash between what Kripke wants to do with his words (i.e. produce an argument for ‘meaning-scpeticism’) and what he finds himself able to do with them (i.e. generate various inchoate and pathetic sketches of scenarios which cannot actually be filled out). The problem here is just the same as the problem facing Frege’s attempt to speak of ‘logical aliens’, a problem detailed for example in the papers by Cerbone, Crary and Conant in Crary and Read (eds.) The New Wittgenstein (London: Routledge, 2000). Kripke’s ‘quus’ idea is no better off than was Frege’s ‘logical aliens’ idea, a hundred years earlier. The vacuity and absurdity of Kripke’s project was thus already clearly laid out in advance by Wittgenstein’s own complete undermining of Frege in this regard.)
 See again p.8 of Kripke, also p.12f.; and the sections of our paper following this one.
 I.e. Given the ‘logic’ of the ‘Kripktic’ argument we are examining here...
 Our argument here has elective affinities with S.Mulhall’s, in his important paper “No smoke without fear: The meaning of ‘grue’” (Phil. Quarterly 39 (1989), 166-189). Unlike Mulhall, however, we do not court the hazard of appearing to use (i.e. to depend upon in the structure of our own argument) ‘gruespeak’ and ‘quuspeak’, the hazard of using the notion(s) of ‘quus’ (or ‘grue’) that we (and he) are challenging/critiquing. (In fairness to Mulhall, let us here quote Ian Hacking (on p.283 of “On Kripke’s and Goodman’s uses of ‘grue’ (Philosophy 68 (1993), pp.269-295)) on his piece; “Mulhall may be writing in an ironic vein, intending in fact to show that gruespeak is incoherent [by means of using it and reducing it to absurdity]”. Indeed, this is apparently so (Mulhall; personal communication); but we fear many of Mulhall’s readers will not realize this.)
 The notion of ‘quus’ is invoked to suggest the point that doing addition might not be what we think it is, that we are not able to prove that we are doing addition by pointing to what we have done or are doing (!), for we could, for all we know, be following a system which would suddenly reveal itself to have properties other than those we had hitherto taken it to have, which we -- presumably, though how is anyone to say whether we do or don’t? -- would not presently take it to have.
 Though not the ‘final’ challenge, for the arguments now being discussed pave the way for the challenge to identify the kind of fact about an individual’s mental state which would establish that one was following any rule at all, not just one rule rather than another. See Read’s “In what sense... ” (op. cit.) for discussion of whether this further manoeuvre helps Kripke’s strategy in the slightest.
 As so often in trying to take meaning-sceptical arguments seriously, there is a devastating irony available here: If we can understand ‘quus’, and if we can understand the notion of a continued ‘quusification’ of our language (Kripke sketches, in desperate endeavour to buttress his exposition, further terms such as “quum”, “quimilar”, “quame” etc. -- see e.g. p.23, pp.20-21), then what is to stop us from re-interpreting these terms or others used by Kripke as ‘plus’ etc. were re-interpreted, i.e. in a quus-like fashion? E.g. We could reinterpret the numeral “57” in the ‘definition’ of ‘quus’ to mean “57 in all contexts except definitions of ‘quus’, and otherwise minus 63” (‘Who is to say’ it doesn’t mean this?!). That way, (our definition of) quus would be functionally equivalent for all numbers to plus -- here is another way then, one which Kripke can hardly say we are not entitled to, of stopping the scepticism from even getting the slightest millimetre off the ground. [One could go on; e.g. Why not read “quusification” in a quus-like fasion, such that quusifying terms is actually by our lights doing nothing to them (except when applied to terms marked as quus-like)? We leave it to the reader to go on having some knockabout fun through what she can do, with these materials, for herself (cf. Wittgenstein’s Culture and Value p.77 (Chicago: U.Chicago, 1980 (Transl. Winch; posthumous))].
 Recalling here that, as argued earlier ‘quus’ as Kripke initially ‘defines’ it is merely a placeholder.
 It might be worth noting here that the way in which Kripke and many of his commentators talk of the ‘the function ‘plus’’ as though it were more ‘basic’ than arithmetic itself, or of ‘+’ as though it were more basic than ‘plus’ (or vice versa!) is most peculiar. It as if “each one contented us at least for a moment, until we thought of yet another standing behind it” (PI, para.201).
 See p.8 of Kripke: “Let us suppose, for example, that ‘68 + 57’ is a computation that I have never performed before... . In fact...finitude guarantees that there is an example exceeding, in both its arguments, all previous computations.”
 Kripke, p.9.
 Cf. Ian Hacking’s argument in his ingenious paper, “Rules, scepticism, proof, Wittgenstein” (in Exercises in Analysis (Cambridge: Cambridge U.Press, 1985)). Hacking sets out an actual historical example which could be described as an abyssal indeterminacy found in the rules of Chess; but the very fact that it is an actual example means that ((i)) it cannot be generalized across language/games, and ((ii)) it can be ‘checked out’, as it was, by what historicaly occured.
 Indeed, as was pointed out in n.19 above, it is only by virtue of (by his own lights, unjustifiably) presupposing that the terms he uses -- terms like “quus”, “quum”, “quusify” and “quimilar”, not to mention all the other more familiar terms (terms of the English language, indeed) featuring in his text -- are not themselves open to the threat of massive sudden disjunction in our use and understanding of them, that Kripke can even begin to engage us with his drama, with his arguments. We have to presuppose throughout that his own discourse is not exposed to the very difficulties that he dramatises. But why should we?
 Compare again Hacking’s “Rules...” (op.cit.); Hacking (p.124; see also p.120) describes the point that “any application of any rule could be unprecedented and distinguishing” as “the sceptical doctrine”. But this must be a milder scepticism than Kripke is implicitly interested in, for this “sceptical doctrine” suggested by Hacking certainly does not imply that most or all applications of a rule could be unprecedented or distinguishing. Moreover -- a devastating irony, given Kripke’s sponsorship of a ‘community solution’ -- it must be agreed implicitly or explicitly by participants in a social group that an application, an instance, is (e.g.) unprecedented. As ethnomethodology tells us, the actual practices of any group will make manifest and ordinary the work of dealing with (e.g.) peculiar examples, situations, or words that they are presented with. There is no such thing as a general proof that words could be being used bizarrely.
 Whence originate our doubts (and A.Collins’s) about introducing a named function at all. Kripke wants to say: perhaps plus is identical with quus. Kripke wants to open a space for the possibility of reversing the direction of derivation of ‘quus’ (which involves, in actuality, only the addition -- excuse the word -- of a weird wrinkle to the already-given friendly face of ‘plus’) by treating the notion of ‘quus’ as though it could be the primary one. “Plus is (really) quus” sounds like it is saying something, and certainly sounds like it is denying something: that plus is just plain old (mostly) unproblematical plus. Saying “Plus is (really) ‘plus’” lacks anything like that impact! It is to try to attain the psychological effect of the first of these two statements that Kripke is rhetorically compelled to expand his vocabulary.
 The sense of “think”, or of “imagine”, in contexts like this is, as we would hope is evident to the reader by now, somewhat peculiar. Imagining a concept in isolation is often tantamount to imagining merely ‘de dicto’, to imagining that we can imagine something (For exposition, see Cora Diamond’s “Ethics, Imagination and the method of the ‘Tractatus’” (Wiener Reihe 1991, pp.55-90), reprinted in The New Wittgenstein (op.cit.)). Nearly all of the present piece involves engaging the reader with sentences and ‘thoughts’ which we either simply cannot find ways of understanding consistently -- or which turn out to be mundane facts and truisms etc. disguised as something more.
 It may be worth pointing out here another irony of this initial consistency, and of the ‘definition’ Kripke gives of ‘quus’ (parasitically upon ‘plus’): that Kripke appears to think that arithmetical rules ought to be quite literally rails to infinity. He defines ‘quus’ as though the idea of ‘plus’ is completely unproblematic for us, reflectively, and to infinity (see also n.25, above). Hacking (on p.294 of “On Kripke’s...” (op.cit.)) argues that it is (t)his Metaphysical/Platonistical Realist presupposition that apparently sets one up for the scepticism. If one has already read and understood (e.g.) Wittgenstein or Dewey on rules and human practices, one will not be tempted in the first place by Kripke’s ‘Realist’ construal of rules. (And furthermore, how could one so much as understand (t)his idea of rules, as rails to infinity, if Kripkian scepticism were on ‘the right track’?)
 Hacking’s gloss on this point (see pp.289-291 of his “On Kripke’s...” (op.cit.)) is to distinguish between ‘fearful’ scepticism (e.g. serious inductive scepticism) and ‘existential’ scepticism (about whether one is being true to one’s (past) self -- e.g. in word-meanings). Kripke’s is then at best an existential scepticism -- one with no consequences whatsoever. (Though see the ‘Epilogue’, below, for clarification of just what we mean by this.)
 It might seem as though quus can be interdefined ‘symmetrically’ with plus, as in a certain sense grue genuinely can be interdefined ‘symmetrically’ with green and blue (see e.g. p.282f. of Hacking’s “On Kripke’s...” (op.cit.)). But in fact this is false; ‘quus’ is worse off than ‘grue’. Here is about as close as one can get to such a definition of ‘plus’: X plus Y means X quus Y for X,Y<57, and otherwise means what X quus Y would mean if one went on in the same way, rather than giving the answer ‘5’.
Clearly, this a ‘viciously’ circular and entirely useless definition, depending among other things on an anti-quus-like reading of ‘same way’.
 Kripke himself invokes Wittgenstein’s use of this particular turn of phrase (on p.17), but as though it conveyed a notion equivalent to that of “an unjustified or random stab in the dark”. Doing a piece of arithmetic, working out the answer to a sum, is not, however, at all what we should ordinarily consider an “unjustified stab in the dark” -- the latter would specifically contrast with such cases, as when, lacking the means to calculate an answer, we make a guess. It is much better, we think, to understand Wittgenstein’s use of “blindly” as referring only to the fact that we apply the rules of addition we have been taught without prior deliberation, following them without reflection and without question. Think of applying Kripke’s characterisation of following a rule as an unjustifed leap in the dark to a practical case such as that of one’s behaviour upon suddenly seeing a traffic light up ahead! (For discussions of ‘experiencing’ the difference between rule-following and other behaviour, see J.McDowell’s work; and also J.Guetti and R.Read, “Acting from rules: ‘Internal relations’ vs. ‘Logical existentialism’ (International Studies in PhilosophyXXVIII:2 (1996), pp.43-62).
 Kripke, pp.12-13.
 I.e. Our (scare-quoted) locution here does not amount to a retraction of our underlying argument that there is no such thing as ‘the sceptic’ being right here (unless the position is re-read so charitably as to amount to the kind of banalities and reminders we are all able to agree on -- see the section on “[Kripke’s] intentions”, below), because no sense can be extracted from persistent attempts to state ‘‘the sceptic’s’ position’, due to its irresolvable internal tensions. Thus ‘If ‘the sceptic’ were right...’ is only a pseudo-counterfactual.
 Though this is explicitly enunciated as early as p.13, after only 5 pages of the argument. One does not have to wait for the more famous announcements of meaning-nihilism on p.21 and p.55.
 See p.21 and p.13 of Kripke; also, p.55, p.62, p71.
“Is there a legitimate way to raise doubts about the immediate future ‘from the
perspective of’ a doubted immediate
past?”, Wittgenstein Jahrbuch (
 “The unstatability of Kripkean scepticisms” (Philos. Papers XXIV:1 (1995), pp.67-74), p.71. And, from the same page, a remark anticipating the detailed argument of the present paper: “When we ask whether by “+” in the past we meant addition, we have in particular to presuppose that “addition” means . . . addition. But...this last re-statement is simply a shuffle -- it does not explain anything; and so nothing can have been put into doubt when we supposedly doubted that “+” meant addition.”
 Kripke, p.12.
 Seemingly; for of course, as we have already made clear, exemption of our present use from those arguments can only be a pretence, for there is nothing relevant to those arguments about our past usages which differentiates them from our present ones.
 Because, again, the suspension of the ‘assumption’ of a common language would have retrospective force, and would, thus, deny that the prior course of the discussion had ever had any meaning, that it could ever have been a discussion. The sceptic’ must retain the ‘assumption’ of a common language even as s/he denies it, must adopt the assumption not just as a first step in the argument, but as an essential presumption throughout its course; and must assert the conclusion about the meaninglessness of words in the very words his conclusion denies have meaning.
 In sum: Kripkian ‘existential scepticism’, purporting to ‘build’ on the ‘fact’ that “There can bo no such thing as meaning anything by any word” (Kripke, p.55) has at best zero consequences!
 See Kripke p.8f. (And for an effective critique at a higher level of generality of this Kriptic notion, see McDowell’s paper, which is to be found in MidWest Studies in Philsophy XVII (1992), 40-52.)
 Kripke, p.8. [Parenthetical insertions ours...]
 Ibid. .
 As Cavell has long led us to expect (see e.g. his Conditions Handsome and Unhandsome, (Chicago: U.Chicago Pr., 1990)), the defender of scepticism will try to place a special aura around certain words, will try to put something forward, something for which, though, s/he ‘cannot find the words’; no words will satisfy ‘the sceptic’ as representing what s/he wants to say or propose, even at this initial stage in the presentation of the sceptical ‘viewpoint’, even in this, Kripke’s ‘elementary’ scene of instruction. That only a few major philosophers (Winch, Diamond (cf. n.3, above), to some extent McDowell) have understood this, have taken such a ‘Cavellian’ line on Kripke’s pseudo-scepticism as we have endeavoured to detail in this paper, has been one of our primary motivations in writing the paper. (See the ‘Epilogue’, below, for more on the consequences of our own ‘Cavellianism’.)
 There is nothing mysterious about the notion of managing to get even what are called ‘infinite’ results from ‘finite’ procedures and beings, contra Chomsky et al, for talk about infinity often means (in mathematical practice) simply that there is no ‘stopping’ injunction included in the rules that we learn. It is, of course, integral to the notion of a ‘rule’ or ‘procedure’ that it is not in principle a ‘one-off’ matter (see, e.g., PI 199) -- but this does not require anything over and above our setting down something as a rule.
 It seems that Kripke and those impressed at least with the questions he raises want a model for understanding rule-following which makes it into something essentially different -- much as philosophers have traditionally been anxious to ‘understand’ inductive methods in a manner which makes them into something quite different (usually, either into deduction or into pieces of scrap).
 One must be wary here, of course, not to give the impression that mathematics consists in some kind of authoritarianism, for one’s deep dependence on one’s teacher is only an essential to one’s getting started. The teaching, as it develops, gives me ways of checking and cross-checking mathematical results for myself, and, once I have developed proficiency, of checking up on the teacher too.
 In the Remarks on the Foundations of Mathematics (London: Blackwell, 1978 (posthumous; revised ed.); transl. Anscombe), Wittgenstein tells us that mathematics is a matter of transforming symbols according to paradigms, and that it is the correspondence between what I do and the paradigm that can be used to waylay doubts. My confidence that I can do addition is not based upon my possession of inner resolutions, but upon my capacity to consult paradigms of arithmetical practice -- either in the form of lists of answers (is this how Kripke wishes to imagine all of arithmetic -- only with the lists being mental?), or (more often) of perspicuous paradigms of calculative practice. (It is very easy to misunderstand Wittgenstein’s work on maths. A paper which attempts to head off misunderstanding of Wittgenstein (and Frege) hereabouts is Read’s “Logicism and Anti-Logicism are both equally bankrupt and unnecessary”, forthcoming in the Proceedings of the Austrian Ludwig Wittgenstien Society, 2001/2.)
 Or rather, unvoiced, or pseudo-voiced -- there actually isn’t any such thing as their being voiced.
 Quoted on p.xvi of Louis Sass’s The Paradoxes of Delusion: Wittgenstein, Schreber and the Schizophrenic Mind (Ithaca: Cornell, 1994). Sass’s book, and indeed his entire corpus, is strongly recommended to those interested in the issues we raise in this ‘Epilogue’.
 See James Conant’s “Varieties of scepticism” (forthcoming in his Scepticism and Interpretation) for fuller explication of these terms.