From WouterBeek.com

Contents 1 Introduction 2 Straight solutions & The problem of ethics 2.1 On the explicit treatment of a cognitive correlate to meaning 2.2 The restrictive conditions specifying Kripke’s problem 2.3 A straight solution to Kripke’s paradox 2.4 The problem of meaning and the problem of ethics 3 Introducing the cognitive contiguity as an alternative to Kripke’s skeptical solution 3.1 Kripke’s skeptical solution 3.2 Blind inclinations and other stabs in the dark 3.3 The gap between rule and action 3.4 The gap between different media 3.5 The cognitive contiguity 3.6 The linguisticity condition 3.7 Levels of explanation 4 Rule-following within the cognitive contiguity 4.1 The cognitive contiguity and personal meaning 4.2 Error-correction on a higher level; expanding the notion of normativity 4.3 The cognitive contiguity within the communal framework 4.4 Various conditionals 4.5 The interface between the individual and the community: static variant 4.6 The interface between the individual and the community: dynamic variant 4.7 Some closing remarks 5 Bibliography 6 References 7 Publication history

Introduction

With Wittgenstein on Rule-Following and Private Language^[1] Kripke gave a spur to the philosophical literature on rule-following and meaning. In this work he describes a so-called skeptical paradox that shows that there can be no fact about a person as to whether he or she means anything by any word. This is done by introducing a skeptic who claims that whenever I made use of the sign ‘+’ in the past, I did - contrary to my present intuition - not really mean the addition function, but actually the so-called quus-function. This function is defined to be entirely identical to the addition function, except for (at least) a single - as of yet unobserved - pair of numbers x and y, for which the outcome "x quus y" deviates from the outcome of "x + y". Because of the infinite character of the addition function, it is clear that such an alternative quus-function can always be defined since there will always be such an, as of yet unobserved, pair of input numbers. The same holds for the general case of meaning, since every term can be applied to 'the world' in a potentially infinite number of ways. E.g. the word 'red' can be applied to an indefinite number of entities. And even proper nouns (or names) which we generally think of as pertaining to a single entity, are - in reality - applied to the indefinite and unlimited set of possible appearances of the singular entity that they (are assumed to) represent. In short, the use of any sign whatsoever, in order for it to be a sign, discloses a (potential) infinity of applications that are normative, i.e. that have normative force.

But the skeptic is of course wrong in his supposition. I did, in fact, mean the addition function and not some strange quus-function. I do not have to prove my understanding of the concept of addition by first writing down all the possible inputs, paired to their respective outputs which, due to limitations of both space and time, is clearly impossible. I only have to know the formula, since the formula seems to be the shortcut representing an infinite number of input-output tuples. Thus in knowing the formula my mind has already captured the infinite extension of the applicability of the addition function.

It is therefore my task to come up with this fact about my mental state, representing the addition function and determining all future applications, in order to prove, once and for all, that with the '+'-sign I actually did mean the addition function all along. Such a fact would provide a legitimation for my intuition that all the future applications of the plus-function are – somehow, in some mysterious way – already inside of me (inside my head I suppose), prior to having performed this or that specific calculation.

Kripke, then, considers the various facts of my mental state that I could come up with as constituting the meaning of the ‘+’-sign. All these proposals, however, turn out to be deficient in their account of my meaning addition by ‘+’, since they do not seem to be able to explain the two principle properties of meaning, namely (1) that the extension of the meaning of a term is (potentially) infinite, whereas a mental state is finite, and (2) the normativity of meaning.

The latter requirement connects up with the notion of rule-following, since whenever I mean something (e.g. addition by ‘+’) I am, at the same time, following a rule (e.g. so that my answers should be in accordance with the rules of addition), with respect to which my behavior can be said to be justified or not. In other words, meaning is essentially a normative notion.^[2]

But how could a finite mental state, operating in an entirely causal fashion, ever lead to the grasp of a (potentially) infinitely applicable and normatively justified function?

Kripke's answer to this question is that finite mental states can simply not do this. He therefore concludes that there can be no fact whatsoever about a person that constitutes hos or her meaning anything by any word. The conclusion extends to the topic of rule-following, in that there are no facts about a person that could constitute his or her following a rule. Assuming one adheres to a realist notion of truth-ascription, it follows that our talk about meanings must be meaningless. Since this conclusion is taken to be unacceptable by Kripke himself, he seeks to provide an alternative, non-factual description of our use of meaning ascriptions in our language. According to this theory, the meanings of signs can be accounted for only in terms of justification conditions, with no further regard to (cognitive) facts. The justification conditions would arise out of the similarities and dissimilarities between irreducible and unexplainable 'blind inclinations'.

Knowing the addition function then, is no longer amounts to being in a certain mental state, applying my understanding of the regularity of addition to certain specific sample inputs so that by the soundness of my captured knowledge (in my head) I am , more or less automatically, lead to act in the, normatively speaking, right way, i.e. by writing down the right outcome. But instead, whenever I make an application of the addition function, e.g. '57 + 68?', I am simply having an irreducible inclination to answer '125'.

In the present article I will argue that Kripke’s explanation of meaning and rule-following in terms of such unexplainable blind inclinations is too radical. For I deem it perfectly possible to explain the precise way in which meaning functions at a factual level. This will be done by introducing meaning-constituting facts in combination with a community-based system of error-correction and conditioning.

Straight solutions & The problem of ethics

On the explicit treatment of a cognitive correlate to meaning

In this article it is my intention to seek to integrate the notion of a cognitive fact into a successful theory of meaning and rule-following. Now some may object that Kripke himself, nor any of his followers, has ever explicitly objected to such a cognitive correlate to meaning. It may even be said that what Kripke seeks to indicate by the term ‘blind inclination’ might, actually, be this very cognitive basis for meaning.^[3] But even though the presence of a cognitive correlate has never been thus explicitly refuted, neither has a positive account of its residual role within the framework of Kripke’s notion of meaning and rule-following - as of yet - been given. I therefore still take it to be a non-trivial task of contemporary philosophy to come up with an explicit specification of such a cognitive account, and to specify – again, in an explicit and lucid manner – the specific role of this correlate within the framework of meaning.^[4]

One might, on the other hand, object that various philosophers have in the past tried to put various mental facts at the basis of a theory of meaning. But, sadly, all these proposed solutions have by now been sufficiently refuted.^[5] It may thus be suggested that such a cognitive explanation cannot be provided with regard to the problem of meaning and rule-following at all. One would wonder why the account that is provided in the present article would succeed where all of its ideological predecessors so obviously failed.

It seems surprising that all of the cognitive accounts that have been proposed thus far have at the same time embraced the view of treating of the language-user in isolation. They all try to show that such a mental fact about meaning indeed exists – and in this they agree with the present treatment –, but they at the same time disregarded the communal use of meaning ascriptions that are needed in order to initialize and shape those individual cognitive facts – and in this second aspect they sharply differ from the present treatment. I will start out by illustrating the, as I will argue, insurmountable problem that one comes to face if one is trying to provide such an individualized 'straight solution'. However, before we can look at the problems that are to be faced when one seeks to provide a straight solution to Kripke’s paradox, we first have to make clear the constraints that Kripke poses on any such a candidate solution.

The restrictive conditions specifying Kripke’s problem

The first way in which a cognitive correlate could be integrated into a theory of meaning and rule-following, is by providing a straight solution to Kripke’s paradox.^[6] In order for a solution to be a straight solution, one must first seek to clearly establish the conditions under which the paradox was originally formed. The problem arises as soon as one is asked to provide the fact that constitutes one’s meaning a certain word or statement or (which is similar) one’s following a certain rule. One may come up with various facts, but not all of them will be accepted, for Kripke poses two restrictions that must be met in order for such a fact to be adequate. One of these restrictions is explicitly specified, the other is left implicit, but be unambiguously distilled from his text.^[7]

The first of these restrictions is called the normativity condition. It requires that the fact that one comes up with is able to account for the normativity that we intuitively observe in our everyday use of the notion of meaning (and rule-following). For example, if we mean the addition function by the ‘+’-sign, this indicates that, when the question "57 + 68?" is posed, we should answer ‘125’ and not, say, ‘5’. So by meaning something we at the same time seem to be following a rule, associated with the meaning of the term, according to which we can be said to act in a right or in a wrong way (i.e. to follow the rule or to deviate from it).

The second restriction that a fact that seeks to account for the meanings of words and other expressions should adhere to is called the reduction condition. Kripke does not state this condition explicitly, but it can be unambiguously derived from the section concerning the refutation of the idea of there being an irreducible mental state of meaning.^[8] This condition states that the fact we provide should not explain the normativity of meaning by making use of the notion of normativity itself. In other words the notion of normativity must be explained in non-normative terms. So what is required here is that normativity be defined in terms of notions that are purely descriptive.

A straight solution to Kripke’s paradox

In the vast literature on Kripke’s Wittgenstein on Rule-Following and Private Language various attempts have been made to provide a straight solution to the paradox. Since all of these proposed straight solutions have by now been refuted, one may come to doubt whether such a solution could ever be reasonably considered to exist. And indeed, it would seem to be an epitome of vanity to suppose that one could provide a successful solution where so many others before have failed. But when looking at the problem of identifying a fact constitutive of meaning, one could only wonder why so many ‘solutions’ have been proposed in the first place, while there is clearly only a single possible non-skeptical way out of the paradox. When one observes Kripke’s restrictive conditions for a moment, one must inevitably come to the conclusion that there is only one straight solution that could be given to counter the claims that the skeptic has put forward. When we look at the two restrictive conditions that were proposed in the previous section, we see that we have before us two separate systems of statements:

1. A system of factual (or assertive) statements. These are statements in which the word ‘is’ occurs and that describe the factual aspects of meaning.
2. A system of normative statements regarding our applications of the use of a certain word. These are statements in which the word ‘ought’ occurs and that do not necessarily describe what we factually do, but what we – in the light of the meaning of a word or sentence – should do.

The system of normative statements (2) must exist in order for the straight solution to adhere to the normativity condition (i.e. that there be a normative aspect to meaning and rule-following). The system of descriptive statements (1) is needed in order to settle the reduction condition (i.e. that there be a non-normative basis for meaning).^[9] Now the enterprise of providing a straight solution turns out to be this: to give an account of how the system of normative statements (2) could, in its entirety, ever be derived from the system of descriptive statements (1). Such a potentially successful straight solution to the paradox would, under the present restrictions, necessarily consist of an explication of the interface between systems (1) and (2), in such a way that a full deduction – of (2) out of (1) – may be established. This ‘interface’ could be represented – for simplicity’s sake – in some (finite) set of axioms, relating is- to ought-sentences. In addition to specifying these interfacing axioms, one would have to make it conceivable (or at least probable) that these axioms – or some extensional equivalent of them – are indeed present in our head, i.e. that the interface is a fact about the mental state of a person taken in isolation. It is obvious that the thus specified entity would then exactly be the cognitive correlate that we were after in the first place, i.e. the fact that would provide us with a straight solution to Kripke’s paradox. It is obvious that there is no other way of providing a potentially successful straight solution to Kripke’s problem, since the restrictive conditions that we were given in the previous section narrow the number of possibilities down to one.

The problem of meaning and the problem of ethics

The undertaking that we described in the previous section, i.e. that of deriving 'ought'-sentences from 'is'-sentences would, if successful, not only provide us with a straight solution to Kripke’s meaning-paradox, but would - at the same time - provide us with a solution to the principal problem of ethics, as was noted a very long time ago by David Hume.^[10] Because Kripke restricts the domain we are allowed to search for these is/ought-interfacing axioms to the mind of an individual person, this search consequently becomes a very deep and (semi-) religious undertaking. Since normativity is assumed to form itself within a person taken in isolation, there must be supposed to be some divine path of reasoning, located somewhere within me, with respect to which my causal operations are to be compared; it is from this very causal path, that determines what I actually do, that I am able to derive what path I should have taken, i.e. what I should have done. One would wonder why we hadn’t followed the error-ridden normative path in the first place, putting our causal alternative aside altogether. The only possible answer to this question would be that these normative notions are very deeply hidden within ourselves and are not very easily known. But even though such a deep personal basis could be suggested to exist in the case of ethics, when considering the case of meaning, we see that such (semi-) religious system could never really apply. The reason for this is that in the case of ethics we are concerned with what should be done according to God (and we might have some religious space residing in our souls that provides us with a system for generating ‘ought’-sentences), whereas in the case of meaning we are considered with what should be done according to a certain language-community (so that the ‘ought’ is grounded in communal practice).

The same is true for rule-following. There are two ways in which the utterance “I am following a rule.” can be used. In its 'ethical' sense, not referring to anything outside myself, I am following the rule 'within' (whatever that may be). And in the sense of following communally established rules, in the case of which the utterance is equivalent to “I am said to follow a rule according to some significant subset of the members of my language-community.” The first sentence is an ethical description of the self, whereas the second sentence is a description of a non-ethical, communal ascription.

The difference between the two forms of normativity becomes clear now: no normativity-generating system for meaning and rule-following could ever reside in my soul, for the community could change its opinion at the drop of a hat, and my personal system of axioms would have to mimic their changes consequently. Since we look at a person taken in isolation here, there is of course no way in which such changes could be explained, and the process of mimicking the whims of the community would be rendered entirely mysterious. If today the water is polluted with some strange substance causing everyone from now on to mean hammer by ‘nail’ and nail by ‘hammer’, and I am the only one to remain unaffected by this unfortunate event, then I will have to reestablish my meanings of the words ‘hammer’ and ‘nail’, while the community’s meanings aren’t said to have been altered in any way at all. This stands in sharp contrast to the problem of ethics, in which it doesn’t matter a bit what the community actually does. If, for example, today the water is polluted with some strange substance causing everyone from now on to find it perfectly normal that people whose length is between - let's say - 1.50 and 1.52 meter are to be brutally murdered by all means available to society, it is the community that will have to change its conduct in the light of some ethical law and surely not the sole individual who was lucky enough to remain unaffected by this unfortunate event of unmatched moral delusion.

We see now that the undertaking of finding an individual meaning-constituting fact causes us to conflate the – entirely different – notions of ethics and meaning. The undertaking of finding a straight solution therefore could never have been right in the first place. Since the normativity of meaning is established with respect to men and not with respect to God, we must necessarily involve the communal view into the total picture and discard the notion of meaning as obtaining to a person taken in isolation. This move, away from the individual and towards the community, is paralleled by Kripke but, as we shall see in the below, on entirely different grounds and for fundamentally different reasons.

Introducing the cognitive contiguity as an alternative to Kripke’s skeptical solution

Kripke’s skeptical solution

Kripke’s solution to the skeptical paradox consist of abandoning the project of seeking to provide an account of how meaning works from the ground up, i.e. from a description of the facts up to the normative meanings that arise out of them. There is no way in which we could come to understand such a factual basis of meaning:

There can be no such thing as meaning anything by any word. Each new application we make is a leap in the dark; any present intention could be interpreted so as to accord with anything we may choose to. So there can be neither accord, nor conflict. [Kripke1982, p. 55]

If we apply to this the Tractarian notion of truth, in which meaning consists of the truth conditions of a sentence, we derive at the unacceptable conclusion that a statement in which the word ‘meaning’ occurs must be meaningless too. But then all our talk about meaning would become entirely senseless! Kripke is reluctant to accept this, and points out that - luckily - the Philosophical Investigations^[11] define meaning in a quite different way. Here the meaning of a sentence consists of the conditions under which this sentence is asserted or denied, and of the role that such a sentence plays in our lives. Instead of searching for the meaning of a sentence in an absolute and human-independent way, we are now to search for a description of the way in which the word ‘meaning’ is used in a language community.

So Kripke’s theory of meaning consists of a description of how we come to ascribe rule-following behavior to others. We say that a person is following a rule if he or she reacts in exactly the same manner as I would, counterfactually, have done. Were I to be asked to provide an answer to the daunting question of "5 + 8?", I would have answered '13'. Now this person here in front of me is in fact asked this very same question, and her answers is exactly the same as my answer would have been had I, counterfactually, been asked the same question.

The thing that remains to be explained is how such a similarity relation between a person’s response – as I observe it at a certain moment in time –, and my own (counterfactual) response – as it appears to me on this very same moment in time – is to be established.

Blind inclinations and other stabs in the dark

Meaning is – according to Kripke – not grounded in any thing. Our behavior is the most basic of building blocks there is to a theory of meaning, beyond which there is really nothing at all:

It seems that my application of it [= a rule] is an unjustified stab in the dark. I apply the rule blindly. [Kripke1982, p. 17]

In his attempt to provide a behavioral basis for the practice of meaning-ascriptions in a linguistic community, Kripke proposes a similarity-relation to exists that is able to establish whether the blind inclination of someone else is similar to the blind inclination that I am having. Whenever one suggests a similarity-relation to exists, it seems only natural to ask: “similar to what?”. Such a basis, with respect to which the similarity of two inclinations might be established is, however, not given. And yet such a basis seems to be necessary in order to end the chain of explanations that otherwise would never have come to an end. We seem to be obliged to pose such a relation, establishing the similarity between our blind inclinations and those of others with respect to nothing at all, or else we end up in an infinite regress. This may at first seem quite hard to swallow, but in the end all explanations must come to an end somewhere, and it may just be the case that the end lies here and that indeed no further reduction is possible.

But we could still wonder whether this similarity-relation is really the end of our chain of explanations, or whether some further reduction might not still be possible. And then we find that the problem of regarding the similarity-relation between blind inclinations as basic is only deepened by Kripke’s assumption that this relation is much more complicated then an ordinary similarity-relation would be. It would need to be much more fine-grained then the plain boolean similar/dissimilar-distinction, for so much is hinted at in Kripke's text:

Suppose, however, the child gets almost all ‘small’ addition problems right. For larger computations, the child can make more mistakes than for ‘small’ problems, but it must get a certain number right and, when it is wrong, it must recognizably be ‘trying to follow’ the proper procedure, not a quus-like procedure, even though it makes mistakes. [Kripke1982, p. 90]

The problem is thus clearly intensified, for it now simply will no longer be enough to state that “the teacher judges that the child has given the same answer that he himself would give”. For in order to establish that there is a different degree of failure between a child answering ‘5’ when asked "57 + 68?", and a child answering - say - ‘124’ when posed with the same question, the teacher would need to have more that the mere knowledge of the number that he himself is inclined to give (which is probably ‘125’).

We could claim that the teacher, when observing the answer ‘5’ of the first student, is having a more uncanny feeling than he would have had when he would, at the same time, have observed the other students’ answer (i.e. ‘124’). Maybe the word ‘feeling’ is not in place here, for the notion of similarity is blind and does not consist in any thing, so might actually not even consist of a feeling which, after all, is a mental entity (or event).

But this very act of waiving aside the problem of accounting for this second student’s answer, that in reality may not be present at this particular moment at all, is clearly not in accord with our ordinary usage of the term ‘degree of error’. For we know there to be a very good reason why the first answer is plainly wrong, whereas the second one is (probably) a minor error that is still in (relative) accordance with the proper rule. This is not due to some completely unspecifyable and indeterminable ‘feeling’ that arises in us whenever we see the answers before us. Instead we observe a difference in degree of error because the number ‘124’ could arise from a minor alteration in the right algorithm that leads to the right answer ‘125’, whereas the answer ‘5’ can only be derived from a heavily altered form of the right algorithm, i.e there are more altering steps required in order to derive at such a faulty algorithm. Maybe under certain circumstances the answer ‘5’ to the question "57 + 68?" could be regarded as a minor error too, but it is clear that we can always come up with some second wrong answer that is clearly further removed from the right answer than the first wrong answer was. E.g. responding ‘shipwreck’ when posed with the question "57 + 68?".

Kripke, at least to a certain extent, recognizes the richness of the similarity relation, for example in:

Any individual who claims to have mastered the concept of addition will be judged by the community to have done so if his particular responses agree with those of the community in enough cases, especially the simple ones (and if his ‘wrong’ answers are not often bizarrely wrong, as in ‘5’ for ‘68 + 57’, but seem to agree with ours in procedure.) [Kripke1982, p. 91-92]

We are thus assumed to be able to make sense of a similarity between alleged procedures. The procedures themselves, since they do not consist of any fact, cannot really exist at all. We only observe the similarity and/or dissimilarity of our blind inclinations. It then remains altogether unclear how we are to base the notion of an unobserved procedure on the basis of these observed individual similarities. It would of course be possible to fit some function upon the perceived similarities, but multiple functions could be equally made to fit these, and the initial problems gets duplicated at a lower level.

The similarity-relation’s assumed richness seems thus very hard to understand. But also the notion of entirely blind inclinations, in its own respect, is not unproblematic. If my blind inclinations are primitive, then it is clear that I can know nothing about them in advance. They can only ‘occur’ to me when I am posed with a certain question. But surely there must be a relation between the question that is being posed, or the signs featuring in it, and the inclination I thereupon have to answer it. For otherwise I would only be able to give answers at random, in disregard of the question that was being posed. Yet my answers seem to be attached to the questions in some odd way that we feel to have some deeper basis.

Another problem that arises here was already apparent in the phrase ‘at the same time’ in the above mentioned case of the teacher judging one student’s blind inclination to be more dissimilar to his own than some other student’s answer that appeared to him at the same time. There must, in order for the system of blind inclinations to function at all, be a certain continuity in my inclinations with respect to the same question being posed at different occasions. If someone asks me ‘57 + 68?’ and I say ‘125’ today, but will answer ‘5’ tomorrow, how could that be? Well, my inclinations are primitive, so how would I know? Others could correct me in this, sure, but their inclinations might equally change on a day-to-day basis. We therefore have to assume that these inclinations – at least in the general case – cannot differ so much, for otherwise the game of meaning-ascriptions could not be played at all. It would destroy the point of the language game. But surely such a continuity must be accounted for in one way or another. It is not evident that the majority of people have approximately the same unreducible inclinations every day. We therewith seem to know something about these inclination's nature and it would seem that these blind inclinations weren’t that blind after all, but actually have a certain structure and certain properties.

A further problem with respect to this similarity-relation arises. If a blind inclination would consist of some thing, then it would be evident that the private language argument – which Kripke claims to follow from his skeptical solution – is applicable to my knowledge of this personal blind inclination too, and that such an inclination would thereby be rendered unintelligible. This is of course only natural, since the very idea of a blind inclination was that it would be able to stop the unsuccessful search for personal facts of meaning. But if a blind inclination is not a thing, we are not quite sure what it could be. We therefore have a similarity-relation that holds between entities that do not really exist. We would now seem not only to have to regard our blind inclinations as absolutely primitive, but the entire similarity relation itself. We do not know what is similar to what with respect to which. We only know that there is some kind of similarity (we cannot even talk of a ‘feeling’ of similarity here).

Having to accept the blind inclinations themselves as basic and unintelligible was already quite hard to swallow, and the added acceptance of a basic and unexplainable – but nevertheless very complex – similarity-relation, makes it even harder. Given all the above objections to accepting the blind inclinations as primitive, we come to wonder whether Kripke’s account can be accepted at all and whether a more satisfying account, that could explicate the same practice of ascribing meaning and rule-following on a deeper level, could not be given here.

The gap between rule and action

Before proceeding with our alternative model, we shall first seek to identity the principle problem of meaning and rule-following. For what was so ultimately attractive about the skeptical solution in the first place? The fundamental problem of the factual account was that we couldn’t, for various reasons, explain how a rule could ever be followed, or a meaning could ever be accounted for. As Wittgenstein already noted, seemingly insurmountable problems arise whenever we put forward a rule and expect someone to act in accordance with it.^[12] For what is the connection between a rule and a subsequent act? This problem is illustrated by an example in the Philosophical Investigations, sections 139 through 141. In these sections Wittgenstein describes his well-known cube-argument: when we use the word ‘cube’ it seems as if the meaning of the word consists of a mental image of a cube. We see the cube before our mental eye, as it were. We can turn it around, manipulate it, etc. Applying the word ‘cube’ would then be to apply it to newly observed entities that stand in a specific relation to this mental image of ours. But – Wittgenstein asks us – how do we know in which manner we are to relate this mental image to the newly observed entity? Let us assume the observed entity is indeed a real cube. We could imagine a projection method, relating the individual points (or lines, or surfaces) from the mental image to those of the newly observed cube. But since a similar projection method could be provided in which our mental image of the cube would be projected – in a likewise manner – onto a prism, it would seem to be necessary to also explicitly formulate the projection function that we use when applying the word ‘cube’. But now we have opened the way for an infinite regress, for this very projection-method could be interpreted in a non-standard way too. We would then have to provide an indefinite number of interpretation rules, the one regulating the other, without ever reaching the act of establishing an identity relation between our mental image and the object observed. We would thus never be able to apply the word ‘cube’ in a rightful way.

The same problem was already treated earlier of in the Investigations, where the gap between the rule itself and the behavior that is to be in accordance with this rule was filled with ‘doubt’:

A rule stands there like a sign-post. Does the sign-post leave no doubt open about the way I have to go? [Wittgenstein1953, § 85]

In the section following the road sign, a lookup table is considered.^[13] Such a lookup table could be used by a person to find the appropriate building materials that are designated by the letters of the alphabet. The first column of the table features the letters, the second column shows the images depicting the building blocks that are to be associated by these letters. The problem is now how the table is to be read. The standard use would of course be to pair the letter in cell (N, 1) to the image in cell (N, 2), for an arbitrary row number N. But there are also non-standard possibilities, e.g. the pairing of the letter from cell (N, 2) to the image of cell (N+1 modulus the number of rows in the table, 2). This example is similar to the application of the word ‘cube’, for we are again required to provide an indefinite number of interpretation rules in order to mitigate the gap between rule and rule-following behavior.^[14]

The gap between different media

The problem with the road sign was that it is (probably) made out of a piece of wood of a specific shape (maybe there were, in addition to this, some signs draw onto the wooden surface). The table was (probably) printed on a sheet of paper (but it could also be displayed on a computer screen, etc.). The mental image of the cube was, regardless of the word ‘mental’, represented as a picture. It was supposed to be ‘drawn’ on an imaginary ‘sheet of paper’ in my head. Now to bridge the rule-action gap – in these above mentioned examples – is really to bridge the gap between on the one hand a piece of wood, certain lines on a piece of (real or imaginary) paper, and on the other hand certain forms of rule-following behavior. This, being put thus bluntly, is of course impossible. The reason for this impossibility is that the medium of representation – shapes of wood, signs on paper – is fundamentally different from the medium in which my actions are represented. How could a piece of wood or a sheet of paper ever come to determine any one of my actions?

The cognitive contiguity

The problem in the above was that we were inclined to provide interpretations in the same medium as the rule that was to be explicated. The actual gap is not (only) the one between different formulations in the same medium, but is (also) a gap between the different media in which the rule and the action are embedded. No interpretation can rule out the possibility of a new non-standard interpretation of any rule (or could rule out the absence of an interpretation, for there was nothing forcing an interpretation upon us in the first place). The problem here is the word (the image, the sign). Its very character bars the way to a sensible account of action and application. The only way, that I know of, in which it is possible to bridge the gap between rule and rule-governed practice, is by bringing everything into the flesh (when an animal is concerned), or into the silicon (when a machine is concerned). There is a cognitive contiguity connecting the sensors to the actuators, with the processing unit of thought residing in between. This is the only way in which the rule and the rule-following behavior could be causally connected, while at the same time circumventing the, above observed, rule-behavior gap. The idea behind this is that whereas between two representations in different media there must always be an infinite reformulatory regress, as long as the representations reside in the same medium, there must always be a finite numbers of reformulation steps between the two.

But wasn’t this cognitive account already sufficiently refuted by Kripke? For how could, in the case of addition for example, an infinite number of applications flow out of a finite mental state? Surely a table, summing all of the possible applications, would need to be infinite. And surely a set of Peano axioms could always be interpreted in various ways. But this is just because Kripke was thinking of mental states as actually inscriptions on an imaginary piece of paper in the head (in a similar way as was the case in Wittgenstein’s cube-argument). But a cognitive account, ensuring the continuity between knowing a rule and behaving in accordance with that rule, can be provided that is entirely based on the neurological medium (and its convenient contiguity property).

A neural network (NN) consists of a number of neurological wires, all of which are initially connected to each other. At the one end of this collection of connections are the input neurons where, due to electrical stimulation, information enters the network; on the other end of the NN are the output neurons that, again due to electrical stimulation, provide the output of the system. The whole process of deriving one’s actions from one’s sensations is achieved by the intermediate neurons that connect the two ends of the total system. This intermediate section of the NN is not shaped according to some rule, table or any other linguistic model whatsoever; its entire functioning is derived at by providing a (sufficiently large) set of training instances, consisting of stimulus/response-pairs, and a computational model that alters the way in which the strengths of the interconnections among the neurons will have to change in the light of the agreement or disagreement that the present NN shows with respect to the individual training examples. Due to this process the NN is shaped in such a way that, when presented with a new input, the system is very likely to produce the right outcome. The right outcome here is that the NN, when presented with the stimulus from some stimulus/response-pair, is able to provide exactly the same response as was already present in this pair. Whenever this is not the case – and such a situation can surely never be entirely ruled out, since because of the nature of the NN we have no guarantee as to whether it will function correctly for some as of yet unobserved input – the computational model, again, shapes the weights among the various neurons so as to correct for the newly observed error.

The linguisticity condition

Kripke’s meaning-constituting fact has to adhere to the two restrictive conditions stated in section 1.2. But, as was alluded to in the previous sections, a third such restriction exists. It is the restriction that the fact that one comes up with needs to be linguistic. This third restriction is, just as the second one – namely the reduction condition – nowhere specified explicitly in the work of Kripke. It is – unlike the second restriction – not all that obvious to be extracted from the text, and therefore needs some exegetical work on the part of the reader (which is probably one of the reasons why it has not been identified before).

Kripke, on p. 15-16, is considering the possibility of someone meaning addition by ‘plus’ in terms of the instructions this person is giving him- or herself, or in terms of some sort of algorithm that this person is following. He alludes extensively to the linguistic aspects of these instructions, stating:

Despite the initial plausibility of this objection, the skeptic’s response is all too obvious. True, if ‘count’, as I used the word in the past, referred to the act of counting (and my other past words are correctly interpreted in the standard way), then ‘plus’ must have stood for addition. But I applied ‘count’, like ‘plus’, to only finitely many past cases. Thus the skeptic can question my interpretation of my past usage of ‘count’ as he did with ‘plus’. […] [Kripke1982, p. 16]

Both the following of an algorithm and the acting in accordance with a set of instructions are assumed to be dressed in language (i.e. to have a linguistic medium). The description of the meaning of the term ‘addition’ is believed to consist of other terms like ‘count’. But this is a somewhat strange example, since one ordinarily would like to define a term in more basic – preferably atomic – terms (instead of in more complex terms). We would normally want to define ‘addition’ in terms of the successor function for example. By defining a term in subsequently more basic terms we will eventually end up with a description that is entirely dressed in so-called basic instructions. But these basic instructions are themselves not linguistic entities at all, they are the actuators in a neurological system or the hardwired instructions on a computer chip. Kripke therefore misses the fundamental point, namely that the linguistic program or algorithm is actually just a convenient shorthand notation for a sequence of instructions that consists of the, quite unlinguistic, connections between tiny parts of silicon that reside inside a concrete machine or within neurological connections in the brain. Nowhere does Kripke consider this essentially nonlinguistic counterpart of a machine. But surely a machine cannot be build out of language alone.

We now move to page 33-34, where Kripke comes to speak about the notion of a machine more explicitly. On p. 33 he first introduces the notion of the machine program - which is of course linguistic, i.e. embedded in a linguistic medium - and thereafter the notion of the program as an abstract mathematical object - of which I have no idea what this should consist in, nor what medium it should have. Kripke thereafter considers a third possibility, i.e. “a concrete machine, made of metal and gears (or transistors and wires)”. The concrete machine + software instructions would then be the metaphor for the brain operating according to some linguistic instructions that can ultimately be reduced to some basic, non-linguistic or hardware-based instructions. We would then consider the concrete machine to exist of hardware only, with software as a convenient way for us to define the machine’s operations that might as well have been define into ultimately reduced hardware-level operations exclusively. But here, again, Kripke only considers the machine that is in need of software in order to operate, but forgets that any software instruction must ultimately be reduced to some sequence of hardware instructions, or else the program would not run on a machine at all.

Kripke is eventually lead to reduce the concrete machine to the machine program, which is the linguistic description of the machine, entirely. The role of the machine itself is trivialized and the program takes its place. We might now wonder how such a program, without the aid of the ‘transistors and wires’ could ever come to function. After this absurd replacement, of the machine by its program, it is only natural that the problem now seems to be insurmountable. How could the machine ever come to operate? Not by interpreting the program! Kripke states this as follows:

[…] the appeal to the designer’s program makes the physical machine superfluous: only the program is really relevant.” [Kripke1982, p. 34]

The third restrictive condition that a meaning-constituting fact, according to Kripke, should adhere to becomes clear now. He says:

The machine as physical object is of value only if the intended function can somehow be read off from the physical object alone. [Kripke1982, p. 34]

^[15]

I take this last quotation as to be extraordinarily revealing with respect to Kripke’s own understanding of the problem. For not only does Kripke require a fact about my mental state that constitutes my meaning something with some specific sign, he also requires that this fact be linguistic. The meaning must be read off from the fact. The cognitive account that I am proposing has no such a linguistic correlate. It is thus an apparent departure from Kripke’s treatment.

Levels of explanation

Now one might – when reading the previous section – object that this is all very true indeed, and that when one is to take Kripke’s words quite literally (and places them in a certain context) one indeed seems to have a point in this. But one might still wonder whether this ‘linguisticity condition’ is really all that important for the theory of meaning that Kripke sought to set out. It might be argued that it is only evident that there may be some non-linguistic methods that constitute the linguistic states of mind that Kripke describes (before he refutes then), but that this is all just the mere question of how things are implemented at a deeper level, whereas the fundamental question should be whether this deeper non-linguistic account has any bearing on the theory of meaning.^[16] In other words: does it really touch upon Kripke’s exposition at all? Let me be very clear about this: it does not. And the reason for this is easy enough to see. By taking the blind and complex similarity relation between inclinations as foundational for a theory of meaning, we have come to look at the way in which meanings are implemented in the mind as something quite unrelated to the theory of meaning.

It is not my intention to quarrel over the viability of this account of meaning at all. For if one chooses to accept the existence of blind inclinations, if one assumes their relative continuity both within a single person and within a community at large, if one takes the relative similarity between the blind inclinations of most of the people within a community for granted, if one accepts a similarity relation to exists of which it is altogether unclear between what entities it holds and with respect to what it could ever come to establish such a similarity, and if one does not demand that a theory of meaning has to account for the very specific discriminations between blunt and minor errors, for which we – intuitively – seem to have very good grounds, then Kripke’s theory of meaning must surely be correct and the implementation question has indeed become quite uninteresting with respect to meaning.

But one might have gone a little step further, not only leaving out the notion of a cognitive fact to meaning, but also the practice of communal agreement and disagreement, and claim that there is really no similarity between your inclination and mine, but that whenever we make a meaning ascription, this is just an irreducible act of God (She may even provide us with the impression that we nonetheless do have a feeling that we have certain reasons to think we mean something by our words, but in the end this is all merely imaginary and due to Her providence only). Such an account would probably – after some minor alterations maybe – fair quite well, explaining everything about meaning that there is to explain. Every investigation that goes beneath this level could then be considered to be about some, quite unrelated, implementational details.

Rule-following within the cognitive contiguity

The cognitive contiguity and personal meaning

Of course there exists such a thing as personal meaning. And of course it is ‘normative’ (in some way). Let us look at what we do when we make a specific computation in everyday life, let us look at what Kripke himself does when he makes a calculation in ordinary life:

I perform the computation [= ‘68 + 57’], obtaining, of course, the answer ‘125’. I am confident, perhaps after checking my work, than ‘125’ is the correct answer. [Kripke1982, p. 8]

The emphasis lies on the phrase ‘perhaps after checking’. For how could we ever be in a position to check our work privately, without submitting it to some external authority (a colleague or a teacher for example), while at the same time refuting the existence of any form of private meaning regarding the plus function? Surely we perform the calculation, but this is just the way in which the causal chain functions. How would recalculating (i.e. checking ourselves) be of any more use than buying a second morning paper in order to check whether what the first one said was true?^[17] But the morning paper example is not all that ridiculous, for if I buy a newspaper with a blank page in it, I am able to establish – with some measure of certitude – that the first version of the newspaper is faulty, by comparing it to several other copies of the very same newspaper. One could of course object that such a method cannot be used to investigate the truth of an expression that features in the newspaper (which Wittgenstein in his example seems to intend). But, even though such an example could – given some ingenuity – be come up with, we have multiple memories and a multitude of methods in which to establish the truth of a statement, the validity of a claim, or the time at which a train is leaving^[18], just as there are many different publications that can all be about the same topic. (We can establish the truth of a statement in the morning paper – again with some measure of certitude – by comparing it with other sources.) Since such correcting processes within the self clearly exists, we are interested in investigating their nature.

What is going on in Kripke’s head when he is checking his answer to an addition problem? Well, we of course cannot know that, but we could come up with a computational model that would simulate Kripke’s process of self-assessment (and, possibly, self-correction). For a computational model could come to recognize a faulty operation by performing the same task several times, and comparing the answers afterwards. If these answers are all similar, we have no knowledge of whether the computational process does represent the addition function or not. But if the answers are dissimilar, there must be a fault somewhere. If only one of the answers is out of line, the system could decide this particular outcome to deviate from the (apparently) standard solution and would identify it as a fault. Now I could of course never know whether this is indeed the process that occurred in Kripke’s head, but it sure seems to come very close to my introspection of how I calculate the outcomes to additions problems myself.

We are of course not always making the same calculation some specified number of times, and afterwards following a calculus in order to establish which of the answers is most probably the right one. Moreover we, most of the time, make every calculation only a single time. In the usual case we do not seem to hesitate, and we do not seem to reflect upon our answer at all. (The answer seems be a ‘stab in the dark’.) We are only considering the process of recalculation and self-correction, when we get a certain feeling that the answer we have derived at could be incorrect. Now this might lead some to suggest that our providing the answer to an addition problem is indeed a stab in the dark. And a lot can be said in favor of it. But it is not the only possible explanation here. We can imagine an alteration to our above proposed system that accounts for these self-correction evoking moments also.

We could imagine there to be multiple causal processes operating in parallel. We could imagine such a collection of processes to exist under the header of – what we normally call – the ‘addition function’. The most prominent of these processes, and the one we are generally inclined to signify when using the word ‘addition’ or ‘plus’, is the addition of numbers according to the following specific algorithm: we first add the two rightmost numbers of the given inputs, we carry a 1 in case these calculations exceed 9, then shift our focus one position to the left, etc., until there are no more numbers left. Let us call this addition1. It is easy to see that this process can really function within us in an entirely causal way, for computers can do it too. (And surely we can do what a computer could do.) But it is important to observe that it is not the only possible causal process that comes up with right answers. There are (many) other processes besides, providing similar answers. For example there could be a process, let us call it addition2, that is less thoroughgoing than the first one, providing answers that are much coarser, telling us in what range the answer to our addition problems will reside. One could call this an ‘estimate’, and it would then estimate the outcome of a given sum based on the number of digits that feature in the inputs. The rationale is that if one of the inputs consists of n digits, then the outcome couldn’t possibly consist of less than n digits. It is equally easy to see how this function could operate successfully in an entirely causal way (again we could write a program for a certain machine to carry it out). It is easy to come up with an arbitrary number of additional addition functions. We are thus able to specify a collection of addition functions addition1 through additionM. These functions could differ considerably in the representational form of their in- and outputs, the precision of their outcomes, and the space & time complexities that are required by their calculation methods. But they are all addition functions (in some way), for what they have in common is that they all calculate the sum of two inputs, even though the output of some functions may lie within a range of numbers instead of pinning the outcome down to one definitive answer. (We could call the addition functions that determine the outcome to be a single number ‘determinate addition functions’ and call the other ones ‘approximate addition functions’ or – as in the above – ‘estimates’).

Now when the system is solving the addition ‘67 + 58’ by the use of function addition1, and due to some disturbing external event its causal process is temporarily deformed, it could come up with the answer ‘5’. Well this answer is ‘clearly wrong’. And the reason why it is ‘clearly wrong’, and not just ‘wrong’, is that the parallel process addition2, which wasn’t affected by the temporal disturbance, comes up with the claim that the answer should consist of at least two digits. Provided there exists a meta-process that links all the inputs and outputs of all of the different processes to one another, and that establishes whether the derived outcomes are in conflict or not, we would be able to model the self-correcting behavior of Mr. Kripke to a considerable extent. This account was still couched in entirely causal terms. The so-called ‘meta-rule’ was not a normative rule at all, since the way in which it determined the functioning of the sub-rules that fell under it, could be specified in an entirely causal manner.

But one could still think of some more fundamental errors occurring in those very meta-rules, that would thereby introduce the same problems that were identified before, only now residing at a higher level. In order for our model to be successful we therefore have to provide a different explanation for the error-correcting process of these meta-rules. This shall be done in the below by embedding the individual into a community of other, similar individuals.

Error-correction on a higher level; expanding the notion of normativity

By adhering to the cognitive contiguity we were able to explain why we have school children and computer programs who can, in principle, perform an indefinite number of calculations involving the addition function. One could say that such a program or such a child is able to determine the outcome for an indefinite number of addition problems, and so a finite mental state could, as a matter of fact, come to embody an potentially infinite application of a certain meaning (e.g. addition). In the preceding section we however postponed the problem of potential errors that might occur during this process (and that cannot be corrected by a personal correction model). We will address these problems here.

For what if the program contains a bug, or the child is put on LSD? Well, in such cases this potentially infinite determining capacity would obviously be violated. That is exactly the reason why we say that there is a bug in the program right now (or that the child is behaving erratically). Both the program and the child’s behavior are now no longer in accordance with other instantiations of the addition functions (i.e. with other programs or with the behavior of other children). But do we not first need to have a criterion, a crystalline case that is, in order to define what it is to deviate from this exemplary case? Shouldn’t a rule have to be specified in order for us to be able to violate it? By pointing to the Philosophical Investigations, or some other recent book of philosophy, it may be illustrated that the existence of such common rules, with respect to which one is able to compare his or her conduct, is highly speculative. But this is only natural: meaning is not some thing, it is brought about in communication. Kripke’s major achievement was to show that the notion of meaning has to be described in terms of the meaning ascriptions that we make in ordinary, communal life. And this is very true indeed. The only problem with Kripke’s account is that there is now no medium at all in which the concept of meaning, as it operates in the individual, could be grounded. We were said to have certain blind inclinations, and by comparing these with one another we were considered to be able to establish whether we are to ascribe meaning to another person or not. By taking these inclinations as the atoms of our theory of meaning, and by obscuring the precise functioning of the similarity-relation that must hold between them, the whole process became needlessly mysterious.

The cognitive contiguity within the communal framework

We are able to account for communal error-corrections by providing a description of the way in which the mental correlates to meaning that reside in a person’s head are shaped and altered by the communal interplay of meaning ascriptions. The principle thing that is needed for such a system of meaning is that positive and negative replies of community members are recognized and understood. The affirmative or negative replies should be paired to the most recent action undertaken and should thus come to influence a person’s thoughts. Here we again need an obscure similarity relation of which it isn’t all that clear to make out with respect to what things should be similar. Let us illustrate this with a concrete example.

A rudimentary explanation of Wittgenstein’s cube-argument in the cognitive contiguity would be that the extensional equivalent of an image of a cube resides in a part of our brain, as modeled in a neural network. The word ‘cube’ is represented in another portion of the same network en is linked to the image in some way. The observed cube enters our cognitive system through the senses (how else would it be able to do so), and now the attribution of the word ‘cube’ to the newly observed cube (or more specifically: to its sensatory correlate in the brain) is established if and only if there is enough similarity between my prototypical ‘image’ of a cube and the sensation of the cube that is to be identified. Well, how is this similarity-relation established? What we are concerned with here is determining whether the distance, which can be measured in various ways, between the neural representations of two similar entities (e.g. two cubes) is indeed smaller than all of the distances between the sensory perception of the newly observed cube and any other prototypical representation in my brain, for otherwise it would be equally possible to attach a word wrongly (e.g. attaching the word ‘cube’ to a prism). But this similarity function isn’t nearly as obscure as the one that Kripke posed, for we could think of it as a continual reestablishment of the connections between certain neurons; a process that is guided by the continuous experiences that an individual has, i.e. the incessant flow of training data that shapes the neural network so as to make it converge to a system that – for a considerable number of cases – comes to rightly assign the word ‘cube’ to a cube.^[19] This process will eventually, in the general case, come to establish a similarity relation between cubes that is stronger than the similarity relation that holds between e.g. a cube and a prism.

The problem of establishing a model that accounts for these respective similarity relations to operate in the right way is a (maybe even ‘the’) central problem of the research area of machine learning today. And within this discipline it is well known that the thus established similarity relations that reside in a neural network can only come to operate in the desired manner, when their implementation is based on a vast assortment of representational assumptions. But since the medium in which these representations reside is contiguous – linking sensation, through cognition, to action – no fundamental problems arise. And though it might be argued that the various representational preconditions are chosen quite arbitrarily, so as to make the system ‘work’ in a particular situation, it might as well be supposed that human cognition is reliant on some set of representational methods too. This set of preconditions might be formed by both our cultural background and our biology.

Various conditionals

What we observe in our every-day use of the word ‘meaning’ are sentences like “If Jones means addition by the ‘+’-sign, then he should answer ‘125’ when asked ‘68 + 57?’.”, or in a slightly more general formulation:

NC If person P means M with sign S, then he or she should answer A_i when asked Q_i.

This is not the way in which Kripke defines meaning-ascribing conditionals, however. He uses the following alternative formulation:

NCK If person P means M with sign S, then he or she will answer A_i when asked Q_i.

The problem with both of these conditionals is that the concept of meaning that we sought to define, by specifying the conditions of its use, are already mentioned in the antecedents. Kripke therefore introduces the notion of the contraposition of a sentence, so that the non-normative consequence of sentence NCK becomes the antecedent of the new sentence:

NCKC If person P does not answer A_i when asked Q_i, then we cannot assert that he or she means M with sign S.^[20]

But not only have the antecedent and consequent changed places. A much more fundamental change in their natures occurred. In the original case we had a conditional between a fact and an expression regarding a specific addition problem: If someone means addition by ‘+’ (a fact in the head), then he or she should answer ‘125’ when asked ‘57 + 68?’ (an expression). But due to Kripke’s reversal we now have a conditional between an expression regarding a specific addition problem and an expression regarding a meaning ascription:

If someone does not answer ‘125’ when asked ‘57 + 68?’, then do not say that he or she means addition by ‘plus’.

This is the real difference between NC and NCKC. The problem we had with Kripke’s treatment in the above, was of course that linguistic expressions had eventually come to replace all facts. But now that we have reestablished the inclusion of facts in our theory of meaning, we will have to revert to a meaning-ascribing conditional that holds between a fact and an expression again. I shall therefore provide a new contraposed conditional in which the process of interaction within a language community is incorporated:

If someone does not answer ‘125’ when asked ‘57 + 68?’, then in my communal interaction with such a person I shall not take him or her to mean the same thing as I do. Either he or she is wrong, or I am. A process of conditioning of the one by the other – possibly by recourse to some other members of our language-community – will eventually result in some alteration in our factual correlates to meanings.

There is no problem in reverting this conditional, so that it ends up in its original for again:

If, within my language-community, I take someone else to mean addition by plus – because his answers have in enough cases been identical with mine –, then this person should answer ‘125’ when asked ‘57 + 68?’.

The interface between the individual and the community: static variant

We have still left unspecified how this process of communal agreement could operate. I shall set out a very simple model of how this can be done. For a more advanced version one would have to enrich the model with a computational method that accords for the relative weights of specific question-and-answer pairs, personal peculiarities and differences in acceptance and endurance between persons, etc.^[21] We start out by summing up a certain number of individual everyday practices, in which a certain person P believes another person R to answer the same questions q_i in the same way a_i:

QA_{1, …, m, P, Q}: Bel(P, Ans(R, q₁, a₁)) ^ … ^ Bel(P, Ans(R, q_m, a_m)) ^ Bel(P, Ans(P, q₁, a₁’)) ^ … ^ Bel(P, Ans(P, q_m, a_m’)) ^ Bel(P, Sim(a₁, a₁’)) ^ … ^ Bel(P, Sim(a_m, a_m’))
(For persons P and Q, questions q_i and answers a_i.)

The conjunction of a certain number of such individual examples from everyday practice will eventually bring about a belief within P, that R must mean the same thing with the sign S as he does. This could – due to the problem of induction – never be stated absolutely, and we are therefore considering a provisional attribution of meaning here (which is why the meaning-predicate operates in the belief space of person P only, and not in the world (which would be treating of meaning as if it were ethical)):

M: QA_{1, …, m, P, Q} → Bel(P, Means(Q, S, M))
(For sign S and meaning M, where S occurs in all q_i.)

When someone believes some other person to mean M by sign S, then this is the same thing as ascribing this meaning to this person. By including the person in whose belief state the provisional meaning ascription takes place as an argument, we can formulate our meaning-ascription predicate:

MA: Bel(P, M(Q, S, M)) == AM(P, Q, S, M)

Once established, a meaning-ascription could – under certain conditions – be retracted. This process is, naturally, quite similar to the before formulated positive case:

QA_{1, …, m, P, Q}: Bel(P, A(Q, q₁, a₁)) ^…^ Bel(P, A(Q, q_m, a_m)) ^ Bel(P, A(P, q₁, a₁’)) ^…^ Bel(P, A(P, q_m, a_m’)) ^ Bel(P, ¬ Sim(a₁, a₁’)) ^…^ Bel(P, ¬ Sim(a_m, a_m’))

And:

M: QA_{1, …, m, P, Q} → Bel(P, ¬ M(Q, S, M))

Linking this to our meaning-ascription predicate in the following way:

MA: Bel(P, ¬ M(Q, S, M)) == ¬ AM(P, Q, S, M)

The interface between the individual and the community: dynamic variant

So far we have only been concerned with what I would call a static factual communal model of meaning. The word ‘static’ here indicates that in the above model meanings can be ascribed to a person (or retracted), but the facts about meaning inside every person’s belief state remain perfectly fixed (i.e. it only accounts for the variety in meaning ascriptions and not for the variety in meanings themselves). The model should therefore be expanded by adding a conditioning component that causes the meaning facts of individuals to change in the light of the agreement/disagreement that a person has with his or her peers. Such a conditioning model has of course already been suggested in section 2.5, it is a neural network.

Such a network could be complemented by considering various variables that replicate certain personal peculiarities of the members of the language-community, such as the willingness to learn and the conditioning influence of a correction by others (e.g. when my professor corrects me, this is considered to have a more thoroughgoing influence on my psyche than when some madcap does), etc. We have then modeled a dynamic factual communal model of meaning that would provide tremendous insights in the way in which a language community could be explained to operate in a less mystical manner.

Some closing remarks

According to Stein, Kripke was wrong in formulating the contraposed form of the normative conditional as:

If Jones does not come out with ‘125’ when asked ‘68 + 57?’, then we cannot assert that he means addition by ‘+’.” [Stein1997, p. 37]

Stein’s problem with the form of this contraposition is that the word ‘should’, which is indicative of a system of normative statements, was here – by Kripke – replaced by the words ‘come out’, which are indicative of a system of descriptive statements. The proper contraposed conditional would, according to Stein, have been:

If it is not the case that Jones should reply ‘125’ when asked ‘68 + 57?’, then it is not the case that he means addition by ‘+’.

But this is a very strange statement that almost never occurs in every-day linguistic usage. The problem is that it assumes the normativity to be part of the statement, i.e. that the statement (or some part of it) could itself be normative. We would be inclined to ask: normative according to whom or what? What we are actually after is not a connection, through the use of a conditional, of a non-normative with an absolutely normative part (to bring up the principle problem of ethics once more), but a connection between the utterances of an individual and the meaning-ascribing utterances that are provided by other individuals with regard to the practices of the former individual. Beyond this system of communication between agents – the problems they solve, the utterances they make, the meaning-ascriptions and –retractions they make, the ways in which their mutual ascriptions come to shape their personal meaning-constituting facts – nothing else exists in the realm of meaning. (It is all quite ordinary actually.) Justification of one’s words should therefore not be sought to be established in any absolute manner whatsoever, but it is always to be established in the light of the reactions of one’s peers. This is the only thing that is needed for a successful system of meaning and rule-following to exists. It does not necessarily imply that the way in which language really functions could not (also) be guided by some divine intervening being, providing us with some absolute normativity once in a while, or with a mystique similarity relation, but when Occam’s razor cuts off these unnecessary embellishments, the bare theory that was outlined in the above remains.

Bibliography

Boghossian, P.A. 1989. “The Rule-Following Considerations.” In: Mind. No. 98. p. 507-549.

Goldfarb, W. 1985. “Kripke on Wittgenstein on Rules.” In: The Journal of Philosophy. No. 82. p. 471-488.

Hume, D. 1740. A Treatise of Human Nature.

Kripke, S. 1982. Wittgenstein on Rule-Following and Private Language. An Elementary Exposition. Cambridge: Harvard University Press.

McGinn, C. 1984. Wittgenstein on Meaning. Oxford: Basil Blackwell.

Stein, H.P. 1997. The Fiber and the Fabric. An Inquiry into Wittgenstein’s Views on Rule-Following and Linguistic Normativity. Amsterdam: Institute for Logic, Language and Computation.

Wittgenstein, L. 1953. Philosophical Investigations. Trans., G.E.M. Anscombe. Oxford: Basil Blackwell.

Wittgenstein, L. 1978. Remarks on the Foundations of Mathematics. Eds., G. H. von Wright, R. Rhees, and G. E. M. Anscombe. Trans., G. E. M. Anscombe. 3rd Ed. Oxford: Basil Blackwell.

References

1. ↑ Kripke1982
2. ↑ But even the notion of 'normativity', in the sense of saying of a person's uses of a term that it is 'good' or 'bad', can only be sensibly applied whenever a language game to this very specific purpose is presupposed. In this language game of spreading norms such apparent meta-utterances could, with respect to the allowances of the grammar, be expressed. The notion of normativity that we are concerned with here, however, is the one that is not determined by any such a specific language game, nor by some set of those. Rather, the notion of normativity that we are interested in is something that is a fundamental characteristic of the notion of a language game (any language game) as such, constituting its boundary conditions as it were.
3. ↑ Such might be concluded from Kripke1982, p. 100-101, ff. 81.
4. ↑ The lack of specification of the role of the cognitive correlate of meaning can be illustrated by calling to mind that Kripke’s account of meaning – regardless whether or not his blind inclinations are to be regarded as actually signifying certain cognitive facts – can be specified without recourse to any such a notion of a cognitive entity.
5. ↑ See Kripke1982, Ch. 2 and Stein1997, Ch. 1.III.
6. ↑ This was attempted by - among others - Goldfarb1985 and McGinn1984.
7. ↑ I follow Stein in the identification of these two conditions. I will also use his terminology in naming these condition, because the names he uses reflect the intended conceptions in a clear way.
8. ↑ The idea of an irreducible, sui generis, mental state of meaning that is (as of its very nature) normative, is introduced by Kripke on page 51. Kripke responds to this suggestion in the following way:

   Such a move may in a sense be irrefutable [...]. But it seems desperate: it leaves the nature of this postulated primitive state – the primitive state of meaning addition by ‘plus’ – completely mysterious. [Kripke1982, p. 51]

9. ↑ Observe that, strictly speaking, it is not necessary for system (1) to consist of assertive or factual statements. It might contain statements of some other modality (or of a mix of statements of various modalities). Just as long as none of these statements is normative, all is allowed here. Maybe some would object to the freedom of using statements of any imaginable modality, except for the normative one, here. For example those that seek to provide an entirely extensional account of meaning would surely object. Such persons could then impose additional restrictive conditions in order to exclude e.g. non-extensional statements from featuring in system (1). I will here make such a restriction and will thus talk about descriptive statements where one is allowed, according to Kripke’s restriction, to talk about non-normative statements (a set that is considerably wider).
10. ↑ Hume, only summarily but nevertheless notoriously, introduced the problem in Hume1740, `Book III: Of Morals, Part I: Of Virtue and Vice in General, Section i: Moral Distinctions Not Derived from Reason'.
11. ↑ Wittgenstein1952.
12. ↑ It is important to note here that Wittgenstein gives an entirely different solution to the problem that is introduced here. It might even be said that he doesn’t see a real problem here at all. The problem would then rather be the way in which an ordinary practice gets problematized. We are however only considering Kripkestein in this article, and we use the citations of Wittgenstein in this passage in the light of the problems that face Kripke (in the role of Kripkestein).
13. ↑ Wittgenstein1953, § 86.
14. ↑ The same problem, of the existence of an insurmountable gap between thinking a rule and acting in accordance with it, is again touched upon in the later sections 431 through 434 in the Investigations.
15. ↑ It must be noted that the passage this sentence was quoted from occurs in brackets, indicating that Kripke probably thought it less important or less central to his argument.
16. ↑ Kripke says the following about the cognitive underpinnings of a theory of meaning:

    The rough uniformities in our arithmetical behavior may or may not some day be given an explanation on the neurophysiological level, but such an explanation is not here in question. [Kripke1982, p. 97]

17. ↑ I refer here to the following citation:

    (As if someone were to buy several copies of the morning paper to assure himself that what it said was true.) [Wittgenstein1953, § 265]

18. ↑ This example is also drawn from Wittgenstein1953, § 265.
19. ↑ The instances in which a cube is not rightly identified (according to my language-community), then there is a reason for my being wrong in identifying the object unjustly (e.g. it is a boundary case of a cube). (The same holds for the unjust attribution of the word 'cube' to a non-cube (e.g. a prism).)
20. ↑ This sentence is a generalization of the specific instance that Kripke mentions:

    If someone means addition by ‘+’ then, if he remembers his past intention and wishes to conform to it, when he is queried about ’68 + 57’, he will answer ‘125’. [Kripke1982, p. 89]

21. ↑ An example of these further conditions is that for different people there could be different values for m in QA.

Publication history

This article was originally written sometime in July 2007, for a course on Wittgenstein's Philosophical Investigations, taught by Prof. Dr. M. Stokhof. It has been somewhat altered since.