There was presumably a reason the ordinance was enacted, for example it might have been thought that motor vehicles disturb the peace that a park is supposed to possess. In that case to determine whether or not roller skates are motor vehicles, we should determine whether or not skating disturbs the peace.
The difficulty in deciding the roller skate question presumably comes from the difficulty in deciding whether or not roller skating is a disturbing activity. One might imagine particular cases of particularly exuberant skating that would border on disturbing, but one can also imagine cases of people skating peacefully, in complete harmony with the park atmosphere. So some instances of roller skating would be considered "operating a motor vehicle" for the purposes of the ordinance, and other instances would not. But any given instance either qualifies or doesn't. Some may be difficult to decide, but ultimately a decision is made as to the tolerability. Probabilities enter because, assuming stationarity, there is some probability that an arbitrarily chosen instance will be disturbing. This probability would presumably be used in deciding whether or not roller skates should be explicitly included in the ordinance or not. Words certainly have fuzzy boundaries. Wittgenstein gave the example of "games". There is no definition of game that will suffice to decide all cases in agreement with actual usage. Rather games exhibit "family resemblances" among each other, so that any particular game is always similar in some way to other things that are called games, but need not have any particular feature in common with all other games. There are particular features that apply to large groups of games and account for their game-ness, but there is no one set of features that applies to all games. However, this does not mean that individual instances have different degrees of "game-ness". One can certainly talk that way, and assign say a fun value to each particular action in a sequence, and then combine the values in some way in order to decide whether the sequence as a whole is fun. But this is just a heuristic way of dealing with the combination of different factors to make a decision. In the same way, Bayesians assign belief values to events, and combine them according to the laws of probability to get a final "likelihood" that is used to make the final decision. That the Bayesian claims to be assigning probabilities, doesn't make this assignment any less arbitrary than the assignment of a set-membership function. The nice thing about the Bayesian framework is its relatively clear prescription for reasoning under uncertainty, once all the "probabilities" have been assigned, and the intuitive appeal of probability in decision making. In any case a decision is ultimately made. The need to make an ultimate decision seems to be particularly important in law. J.P. -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Behalf Of Peter Tillers Sent: Tuesday, December 30, 2003 4:01 PM To: [EMAIL PROTECTED] Subject: [UAI] causal_ vs_ functional models One or two members of the list have made points like the following one by David Larkin: >With standard Bayesian techniques it is possible to reason probabilistically about variables with non-binary domains. For example, in a standard belief network one might find a node labeled "Temperature" which can take on the values "Low", "Medium", and "High". A fuzzy-logic based interpretation of the variable would amount to a simple relabeling. The node could be renamed, "The object is hot", taking a truth value of "0", "0.5", or "1". Mathematically, there would be no difference. You would say, according to bivalent logic, "The object is hot" can only be true or false, and we can only reason probabilistically about the chance that it will be true or false, and therefore partial truth cannot be addressed. But standard belief networks are not bivalent (i. e., the variables are not binary), but in general multi-valent. Vague and perception-based degrees of truth, such as the temperature example given above, are commonly used to describe the states of variables.< I have a question (that I penned some time ago) for members of this list who share David Larkin's sentiment: How would I use the procedure of assessing the probabilities of multivalent variables to determine the probability (of the proposition or hypothesis) that "roller skates" are "motor vehicles" for purposes of an ordinance prohibiting the operation of a motor vehicle in a park? Would you propose that one of the hypotheses whose probability I am to assess is "roller skates have the property of being a 'motor vehicle' only to a small extent"? (I am taking "motor vehicle" as a primitive property or attribute, rather than as one that is decomposable into subordinate properties or attributes.) If this formalism were to show us anything, we would need to figure out what events or evidence or symptoms etc. we should use in expressions such as P(E|"'roller skates' have small amount of the property 'motor vehicle'") Most legal scholars (if they understood probability theory well, that is) and some probability theorists would be tempted to conclude that the concept of a conditional probability cannot readily be used to address uncertainties of the sort found immediately above. If so, can we use a different kind of "formalism"? Is it _conceivable_ that we could do so? If so, what would that way of thinking be like? Is it useful to think in terms of shifting sets (rough sets) or varying degrees of membership in a set (fuzzy sets)? If not, why not? *** In the world of accounting there are often alternative ways of describing the same financial state of affairs. The descriptions one gets often seem different; but they are often not inconsistent, and sometimes they amount to paraphrases, different but equivalent ways of saying the same thing. Sometimes (but not always) the differences between probability theorists and fuzzy set theorists strike me as being like that. In response to formulation x -- e.g., think of this uncertainty as "degree of truth" -- there is the response: e.g. #1: we can instead think of this problem, or we can instead describe this problem, as a situation in which we are trying to assess our uncertainty about a continuum of possible states, we can instead speak of the probabilities of continuous variables; e.g. #2: we can instead speak of the probabilities of non-continuous but multiple disjoint possible states. Speaking as a non-mathematician: sometimes such alternative ways of describing uncertainty seem to me to be, at bottom, "the same", and if I were to prefer one way of talking over another, I would want to have some _reasons_ for talking about a particular kind of uncertainty one way rather than another way. Although I think there are in fact some advantages (sometimes) to using the grammar of conditional probability to talk about hypotheses about legal norms -- in this regard I am in a small minority among legal scholars --, I also cannot escape the sense that the grammar of conditional probability just does not "get at" some of the types of uncertainty that are found in legal reasoning (and in other contexts as well). Think of the underlying intuition of rough sets, the notion that some words or classifications seem to flip and shift, that they have fluctuating perimeters: How does one get at this type of uncertainty, effectively capture it, by talking in the language of conditional probabilities? I do not deny that it could be "done" if one tried "hard enough" -- standard probability theory is protean: it can be interpreted in an immense variety of ways -- but in the end I think I would still wonder if deployment of notions such as conditional probabilities, likelihood ratios etc. would be the best way to think about words that seem to have "rough edges" or to about the appropriate categorization of things that seem to fall into two or more categories that to some extent seem disjoint to some degree but not strictly or completely disjoint. Since I am not a fuzzy or rough set theorist, I could not begin to prove that my doubts (such as those mentioned immediately above) about the relative fertility of the standard probability calculus in some contexts are justified or that fuzzy sets or rough sets make it possible to think in a more orderly and more productive manner about uncertainties such as those mentioned in the preceding paragraph. This is why I (desperately) need the help of logicians and mathematicians and probability theorists etc. But if you were to advise me, I would want you to keep this in mind: I would not think you have made a good defense of your preferred theory -- your preferred theoretical apparatus -- merely by showing that your theory or apparatus can "handle" some problem that a rival theory also claims to handle. The question, it seems to me, is which theory does a _better_ job of "handling" the problem at hand. Nothing I say here, of course, answers the last question. Here I am not even _attempting_ to answer the last question. Furthermore: I confess that I do not know the answer to the last question. I only know that conditional probability talk sometimes just doesn't seem to get at the kind of uncertainty that I sometimes see in law. (The reverse is also the case: talk about fuzzy and rough sets also sometimes just doesn't seem to fully get at the kind of uncertainty that I think I sometimes see in law [unless, of course, we are forced to agree with the thesis that the standard probability calculus is just a special case in fuzzy sets].) When considering the general question I pose in this message, please keep in mind that I am a legal professional and that lawyers and legal professionals are very interested in "word wars" -- in battles over the meaning of words and in _arguments_ about the meaning or scope of legal classifications. The way a word or classification is conventionally used is part of the kind of argument that a lawyer often makes. But only part. Lawyers appeal to reasons other conventional usage when they argue about the "proper" or "appropriate" or "correct" interpretation of some legal phrase or classification. (Much depends on the context in which a lawyer is operating. When lawyers draft contracts, they are more likely to emphasize conventional {legal} interpretation. When they litigate, they are likely to emphasize conventional usage considerably less.) Peter T P.S. I suppose that a discussion list -- even one as distinguished as this -- is not the appropriate venue for a discussion of the implication of the principle of non-contradiction for the question of the (logical) coherence of fuzzy set-based arguments. (When I have time -- but I am running out of time! -- I will reconsider this basic(?) question in my own autodidactic way. [There is a crasser but more trenchant way to describe my efforts at self-education. The adjective I have in mind is a hyphenated expression that begins with "h" and ends with "d".]) **** Peter Tillers http://tillers.net Professor of Law Cardozo School of Law 55 Fifth Avenue New York, New York 10003 U.S. (212) 790-0334 [EMAIL PROTECTED] FAX (212) 790-0205
