Ken Aizawa Martin Davies's Theory of Rule Following Martin Davies (1989, 1991) has developed a notion of what he calls a "causally systematic process." He uses this notion as a necessary condition on a system's being governed by a rule. In my talk today, I have two objectives. First, I will explain Davies's necessary condition on a system's following a rule, then provide two reasons for thinking that the condition is too strong. Second, I will examine one possible motivation for adopting Davies' condition, namely, that Davies's condition allows one to specify which of two competing mechanisms is responsible for computing a given semantic-level input-output function. I will then argue that the rationale does not in fact warrant the adoption of Davies's condition. Davies's Condition Consider the input-output mapping: (IO1) Input token Output red square coffee with milk blue square coffee without milk red round tea with milk blue round tea without milk What conditions determine whether a device that conforms to (IO1) is governed by the rules, (R1) If square, then coffee. If round, then tea. If red, then milk. rather, than by the rules, (R2) If red square, then coffee with milk. If blue square, then coffee without milk. If red round, tea with milk. If blue round, tea without milk. Davies suggests that the following condition provides an answer (DC) A system is governed by a particular rule r, only if there is a single causal mechanism dedicated exclusively to the processing of r. Applied to a set of rules, Davies's idea is that a system is governed by a set of rules, only if there is a dedicated mechanism for each rule in the set. Davies's condition is too strong: Round 1 I now want to provide a first argument that Davies's condition is too strong. The idea here is that Turing machines that we intuitively think should be said to be governed by rules, do not count as being governed by rules. We will begin with two simple illustrations, then explain the general nature of the problem. The general diagnosis will provide a rough indication of how widespread is the failure of Davies's condition. Consider a Turing machine that computes over the following alphabet {RS, BS, RR, BR, CM, C, TM, T}, where the symbols have the obvious semantic interpretations. Let this Turing machine have the program (TM1) S0 RS CM S1 S0 BS C S1 S0 RR TM S1 S0 BR T S1 The instructions in this Turing machine program have a straightforward interpretation. The first says that if the Turing machine is in state S0 scanning the symbol RS, then it should overwrite this with the symbol CM and go into state S1. The second says that if the Turing machine is in state S0 scanning the symbol BS, then it should overwrite this with the symbol C and go into state S1. Intuitively speaking, this program follows the look-up table rules of R2. Yet, according to Davies's condition it does not. The state S0 is a causal factor in all of the state transitions caused by the instructions in (TM1). State S0 is part of the mechanism for all of the state transitions brought about by the instructions in (TM1). So, not only does this Turing machine fail to by governed by the rules in R2, it fails to be rule governed at all. At this point, Davies's might contend that his theory was not meant to cover the case in which a system is governed by look-up table instructions. In fact, this example might be taken to show that treating Davies's theory as a theory of rule governance, rather than as simply a theory of causal systematicity, is a mistake. After all, look-up tables are, intuitively speaking, not causally systematic, and the foregoing accords with that intuition. Whatever attraction this line of reasoning may have, the following example will show that the problem extends beyond mere look-up tables. Consider a Turing machine that computes over the alphabet {red, blue, round, square, coffee, tea, milk, } and has the program, (TM2) S0 red milk S0 S0 blue S0 S0 milk R S1 S0 R S1 S1 square coffee S2 S1 round tea S2. This Turing machine will begin in state S0 with the read-write head scanning a square that has either a "red" or a "blue" in it, while the adjacent square on the right will contain either a "square" or a "round." If the first two squares contain "red" and "square", this Turing machine will replace these markers with a " " and a "coffee". This output will be understood to mean coffee without milk. If the first two squares contain "red" and "round", this Turing machine will replace these markers with a " " and a "tea". This output will be understood to mean tea without milk. Intuitively speaking, we might think that this machine has the semantic level rules of R1. In this scheme, Instructions 1 & 3 are the mechanism for "If red, then milk" Instruction 5 is the mechanism for "If square, then coffee" Instruction 6 is the mechanism for "If round, then tea." Yet, we find that this Turing machine also fails to meet Davies's criterion for following a rule. As we saw in our previous example, the state S0 plays a causal role all the computations. Moreover, S1 plays a causal role in the output of both "coffee" and "tea". So, it appears that a Turing machine using program (TM2) does not follow R1. In fact, it appears not to follow any rules by Davies's standards. Clearly, the general problem for Davies's theory is that it founders on the states that serve as certain kinds of "decision branch points" in Turing machine computations. Whenever a Turing machine must "decide" to carry out one course of computation, rather than another, the Turing machine will fail to meet Davies's condition. Davies's condition is too strong: Round 2 The foregoing argument relied upon a basic fact about the operation of Turing machines, namely, that at computational branch points, one state can be pressed into the service of more than one task. It also relied upon a bit of intuition mongering concerning what rules a system should be said to follow. This next argument that Davies's condition is too strong relies upon the same basic fact about the operation of Turing machines, but tries to sell a different intuition. Here is a meta-condition on a theory of the realization of computer programs: (MC1) A theory of what it takes to realize a computer program should allow for the conceptual possibility of realizing any Turing-equivalent computer program. This seems to me to be a relatively plausible meta-condition, but for present purposes of challenging (DC), I can do with a weaker meta-condition. (MC2) A theory of what it takes to realize a computer program should allow for the conceptual possibility of realizing the Turing machine program í = {, }. In condition (MC2), í is a Turing machine program that computes the successor function. The first instruction of í may be read as saying that, if the read-write head is in state S0 and scanning a 1, then the head should move right one square and stay in state S0. The second instruction may be read as saying that, if the read-write head is in state S0 and scanning a 0, then the head should write a 1 in its place and go into state S1. With these instructions, starting in state S0, and using some standard computational conventions, a Turing machine will take as input a string of 1's representing a number, scroll right across that string until it finds a 0 at the end, convert that 0 to a 1, then halt. Clearly, (DC) conflicts with our metatheoretic condition (MC2). According to (DC), no machine could realize the program í, since the state S0 is involved in more than one state transition. So, again, a technical feature of Turing machines, in conjunction with an intuitively plausible meta-condition on a theory of the realization of computer programs, suggests that (DC) is too strong. Undercutting the motivation for Davies's condition At the outset we asked what conditions determine whether a device that conforms to (IO1) is governed by the rules (R1), rather than (R2). Davies' condition provided us with an answer, hence some motivation for adopting his condition. I want to undercut this motivation by providing an alternative way to answer our original question. The alternative approach introduced here might be termed an "Instantaneous Description" approach, since it proposes to view rules as mechanisms that mediate transitions between successive instantaneous descriptions of the total state of a device. The notion of an instantaneous description is familiar in computation theory and formal language theory. For present purposes, I'll describe the idea with a Turing machine that computes over the alphabet {0, 1} and represents numbers as finite sequences of contiguous 1's occuring within some finite segment of the tape. The contents of the tape can be represented by writing down a 0, followed the contents of the finite segment of the tape containing all the 1's, followed by a final 0. Thus, if the Turing machine tape were to contain only three 1's on three successive squares, the contents of the Turing machine tape could be represented by "01110". This much satisfies to specify the contents of the tape, but a description of the total state of a Turing machine must also specify the state of the read-write head and the square of the tape it is scanning. To this end, we write the state of the read-write head, say, S0 or S192, to the immediate left of the symbol the read-write head is scanning. Thus, if a Turing machine were to begin in state S0 scanning the leftmost 1 in a string of three 1's, the instantaneous description of the Turing machine would be "0 S0 1 1 1 0". The alternative to Davies's condition on rule realization is the idea that rules specify possible transitions among instantaneous descriptions. So, we know what rules a given device is governed by once we know its possible transitions among instantaneous state descriptions. Further, we know what semantic level rules a system uses, once we know its possible transitions among semantically-evaluable instantaneous state descriptions. Insofar as this theory works, and clearly there will be objections to be addressed, there is no need to invoke Davies's condition to do this work. References Davies, M. (1989). Tacit knowledge and subdoxastic states. In George, A. (Ed.), Reflections on Chomsky. Oxford: Blackwell. (pp. 131-152). Davies, M. (1991). Concepts, Connectionism, and the Language of Thought. In Ramsey, W., Stich, S., and Rumelhart, D. (Eds.). Philosophy and Connectionist Theory. Hillsdale, NJ: Lawrence Erlbaum Associates.