Order-processing vs. Quus
Sunday, April 24, 2005
Kripke's interpretation of Wittgenstein's rule-following paradox is certainly one of the most influential to date but also one of the most controversial. Its starting point is an example based on an imaginary function named quus and defined as such:
Let Å be the symbol of this function: for two integers a and b,
a Å b = a+b, if a and b < 57
a Å b = 5, otherwise.
In other words, quus coincides with addition for numbers below 57 but yields a constant, 5, for all numbers above that threshold. When I use the symbol '+', do I mean plus or quus? How can I be sure that it is the one rather than the other? This is the example Kripke uses to illustrate the rule-following paradox.
In many respects, the order-processing example is similar to the quus function. For instance, both are set in a formal context; arithmetic, for quus, and accounting, for the order-processing example. This is important, as it is explicitly stated by Kripke and Wittgenstein, in order to avoid any confusion between rule-following and vagueness related problems. The rule following paradox does not arise because we allow the meaning of certain terms to be vague. We see it appearing even in contexts where meaning is perfectly clear-cut.
Another similarity is the reliance of both examples on a function with two possible outputs, depending on how the rules governing its use are interpreted. In the case of the order-processing example, this function is the system's sub-routine which retrieves the unit rate / price for each row of an invoice. Either the values returned by this sub-routine come from a single table or from two tables, one for internal invoices and the other for ordinary ones.
But there are also a number of aspects on which the quus function and the order-processing example are in stark contrast. We will examine three of them in detail below:
Craziness: In the case of quus, there is a clear dissymmetry between the protagonists in the debate. One of them, "the skeptic" is said to be "bizarre" or even "crazy". Furthermore, factors like "bouts of frenzy" or "LSD" have to be invoked to explain why me might change our interpretation. The whole enterprise of challenging the usual addition procedure seems rather strange and far fetched. By contrast, no such surreal atmosphere is present in the order-processing example. The debate which ultimately leads to the infinite regress occurs between sober-minded professionals in a corporate conference room. In the course of the argument we may imagine them splitting into several groups, each supporting a different interpretation. But there is no reason to suppose that any of these groups should appear to be "bizarre" or "crazy". Some may be suspected of being dishonest, but this is another matter. Contrary to Kripke's scenario, which could hardly take place in practice, meetings like the one described in the order-processing example, are routine happenings. Any MIS professional with a minimum of experience can recall at least one or two of them. This is, I believe, more than a matter of taste. While the atmosphere of strangeness surrounding the quus example gives the impression that the rule-following paradox is an arcane problem solely of interest to philosophers, the order-processing example shows that it is a concern to all users of language. More specifically, in the case of quus, the infinite regress takes place during the philosophical analysis of the example. It appears to be a part of philosophy, not of ordinary language. By contrast, the order-processing example shows that such regresses may occur in the midst of perfectly ordinary language games, independent of any philosophical inquiry.
Past vs. present: One of the central themes of Kripke's text is the notion that comparisons in time are an essential feature of the rule-following paradox. For example, the skeptic is represented as questioning "whether I am presently conforming to my previous linguistic intentions" (p. 12, emphasis Kripke's). In another passage, it is said that if I gave 5 as the result of 68 plus 57 "I would have made a mistake not in mathematics, but in the supposition that I had accorded with my previous linguistic intentions." (p. 9, emphasis mine). And so on. Further in the text, this theme is used as one of the main arguments to support the parallel drawn between Wittgenstein and Hume : "Both develop a skeptical paradox, based on questioning a certain nexus from past to future" (p. 62). The parallel with Hume is, in turn, one of the main reasons which lead Kripke to picture Wittgenstein's remarks as a skeptical solution to a skeptical paradox.
The order-processing example, however, does not involve any such comparison in time. The debate leading to the infinite regress described in the "four possible answers" section, arises within the time of a meeting. The question which triggers it is concerned with the comparison of the present state of the system with the present content of the specification. Of course, this content is supposed to have remained constant for a certain period, and this is no doubt important. But the comparison itself occurs in the present. Indeed, how could we compare a thing of the past with one of the present if the thing of the past did not persist into the present? It should also be noted that at no point were the past intentions of the people involved considered to be relevant. People may have been honest or dishonest when they wrote the spec. They may or may not have had certain intentions regarding the correct way to process invoices. But it is not what matters now. What is at stake, in the meeting, is the relationship between the specification and the system, as it can be observed in the present.
Mental facts. Another characteristic trait of Kripke's interpretation is its focus on facts about the mind. This focus is, for example, quite clearly expressed in the already mentioned passage about "my previous linguistic intentions". Besides this one, the text contains numerous instances of such expressions as "facts about me", "facts about my mind", "facts about my mental state" and so on. They are used each time the consequences of the paradox need to be spelled out, such as, for example, in "This, then, is the skeptical paradox [...] there is no fact about me that distinguishes between my meaning plus [...] and my meaning nothing at all" (p. 21). Admittedly, there is some fluctuation in this usage. Sometimes, phrases such as "nothing in my internal mental history or my external behavior ..." (p. 21) are used, suggesting a wider scope. Sometimes the word "fact" alone is used, without any additional qualifier. Nonetheless, it is quite clear that the emphasis is put on "facts about my mind" throughout most of the text, and in particular in the part of it which deals with the "solution" of the paradox (chapter 3).
In this respect also, quus function and order-processing example differ markedly. Indeed, the latter does not support any particular emphasis on mental aspects. It is, as we have seen, quite the contrary. The expert is explicitly asked to give an answer about the compliance of the system with the specification regardless of the causes which may account for the content of this document. This excludes, among other things, the intentions of the people who wrote it or reviewed it. However, despite this restriction, rule-following seems to be just as paradoxical in this case as in any other. We hit upon an infinite regress and are thus led to the usual conclusion: words appear to lose their meaning and, as Kripke puts it, "the entire idea of meaning vanishes into thin air".
quus as a special case
As we have seen, Kripke's quus function and the order processing example lead us to view the rule-following paradox very differently. If we follow Kripke, we are led to see it as a kind of philosophical oddity about how past mental facts may or may not justify our present behavior. If we take the order-processing example as a paradigm, we rather come to see it as a fairly widespread problem appearing every time we try to compare an act with a "specification". Do we have to choose between "philosophical oddity" and "widespread problem"? I think not, as these two positions are not contradictory, or, at least, not in an irreducible way.
In a more precise form, the "widespread problem" position says that infinite regress lurks whenever we attempt to test the compliance of a particular course of action by comparing it with a particular rule expression; i.e. a piece of text or discourse, which we take to be the formulation of a rule, and which is what Wittgenstein calls an "interpretation of a rule". In the question (Qb) mentioned in the previous section ("is the system in accord with the specification?"), the word "specification" refers to such a piece of written text and this is why we came to the conclusion that "depth grammar" did not allow such sentences. This is, in my view, the most complete and general way to see the rule-following paradox and I believe it is the way Wittgenstein saw it too.
Kripke's method is an analysis of the possible reactions of an individual confronted with the challenge of the quus function. Frightened into believing that he may be crazy or under the influence of a mind-altering drug, this individual is tempted to reach for the apparent security of what Kripke calls a "fact": "...the proposal that I always meant quus is absolutely wild. Wild it indubitably is, no doubt it is false; but if it is false, there must be some fact about my past usage that can be cited to refute it." (p. 9, emphasis mine) What this fact consists in is, at first, not stated. But it is said that it must "justify" the answers the individual is tempted to give (p. 11) and it is implied that it may consist in a set of instructions that the individual gives himself (p. 13) and which "compel" him to give the right answer. At last (p. 15) a candidate appears, in the form of an algorithm for performing additions. Once this procedure is fully spelled out, written down so to speak, the infinite regress starts. The word "count", used in spelling out the procedure, is challenged as possibly meaning "quount". Then, after an additional sentence is added to the procedure description in an attempt to plug this leak, the word "independent", which appeared in this last sentence, is challenged as possibly meaning "quindependent", and so on. Therefore it is only after the procedure has been spelled out explicitly that the infinite regress appears. If it had been said that the individual had an addition procedure in his head without spelling it out, it could not have taken place. It is only once a definite piece of text describing this procedure is exhibited that it becomes possible to "work on it" in order to produce the regress. This concept of a definite piece of text is what we referred to earlier as a "specification", a "rule expression" or "an interpretation of a rule". It therefore appears that what Kripke calls a "fact" turns out to be very close to what we called as such. Obviously, this is quite probably not the way Kripke intended it. What we claim is that, as soon as Kripke tries to spell out in greater detail what the "fact" might be, the thing which materializes under his pen is a piece of text which is, in all but name, a "specification". If this is granted, the parallel with the order-processing example becomes quite clear.
In both cases, someone is concerned to make sure that the correctness of his behavior cannot be challenged. In both cases the solution chosen is to commit oneself to a piece of normative text against which the behavior is to be checked. In the order-processing example, this text is actually written and is called a "specification". In the quus example, this text is not written by the imaginary subject but, as we have seen, by Kripke himself, when he is driven to propose an "algorithm" as a possible candidate for the "fact" he is challenging himself to find. This is why, in this case, the infinite regress appears to be part only of the philosophical discourse.
There is a preoccupation with time in both examples, but the role it plays is not, in my opinion, as central as Kripke takes it to be. We have seen that a comparison between a piece of text and a course of action can only take place in the present. Even in the case of the quus function this comparison takes place in the present of Kripke's discourse. Time plays a role in both examples but only as a motivation for writing the text; not as an essential factor contributing to its ultimate failure in guaranteeing the behavior's correctness. As we have seen, a piece of text (even one not actually written but learned by heart, for example) looks like an appropriate way to guarantee against future challenges because it persists through time without changes. But one could also imagine other motivations for writing a normative text; like, for example, to make private intentions manifest and, therefore, "objective" (a piece of text is an object). Even if the time motivation is present in both the quus function and the order- processing examples, and is probably present in most cases where a piece of normative text actually gets written in practice, it does not play a role in the paradox itself, which could, conceivably occur even in a case in which no such motivation was present.
The role of mental states or intentions appear to be similar to that of time. They are present in the background of the order-processing example as aspects that the specification should precisely enable us to ignore. In the quus function they are put much more into focus but the goal is ultimately the same. If a "fact" was found which justified the future behavior of the subject, it would make it possible for us to ignore those past intentions it is supposed to faithfully represent. Besides the fact that intentions are an important philosophical topic, their prominence in Kripke's text is also a consequence of his method. Challenging the imaginary subject's certainty about his own past intentions looks like the only tactic powerful enough in order to instill a sufficient amount of doubt into his mind about the correctness of his own additions.
In the light of what has just been said, we may now rephrase Kripke's interpretation of the rule-following paradox as such : there is no representation of my past mental intentions which can fulfill the normative goal of compelling or justifying my present use of words and symbols such as '+'. This, indeed, appears as a special case of the wider interpretation, stated above, which says that no representation whatsoever can fulfill that goal.
If one is mainly concerned with the relationship between intentions and meaning, Kripke's version may seem to be both satisfactory and sufficient. However, its excessively narrow focus has, in my view, one important undesirable effect. In one noteworthy passage of Wittgenstein on Rules and Private Language (pp. 74 to 78) Kripke develops the idea that while the rule-following paradox makes it impossible to explain meaning in terms of truth conditions, its "solution" licenses a switch to an interpretation in terms of assertability conditions. What this type of conditions consist in is outlined in the following passage: "All that is needed to legitimize assertions that someone means something is that there be roughly specifiable circumstances under which they are legitimately assertable" (p. 78). In my view, the danger with this view lies in the fact that the phrase "roughly specifiable circumstances" looks ominously like the "specification" of the order-processing example. The risk is real to interpret assertability conditions as a warrant for a concept of meaning as something which can be captured into pieces of texts with a normative purpose. That this is impossible, as another consequence of the rule-following paradox, may be overlooked if one sticks to Kripke's narrow interpretation.