## Friday, November 05, 2004

### The Mathematics (or Logic) of Evidence in Law: What Is It For?

Mathematical analyses of evidence take a variety of forms and can serve a variety of purposes. Hence, one cannot identify just a single possible valid legal application of mathematical analyses of evidence. But if one wishes to gain some insight into factual, or evidential, inference (and to devise procedures to facilitate inference from evidence) in and for legal proceedings, of what use (if any) is mathematics?

Some people seem to think that legal researchers who fiddle with matters such as Bayes' Theorem are attempting to construct algorithms or some such things that describe how factual inference in trials or other legal proceedings works. It is possible that this is the aim of some legal researchers, but this is generally not what I am after when I try my hand at mathematical analysis of evidence: when I fiddle with Bayes' Theorem, fuzzy logic, or whatnot, I am not attempting to depict how the institution or practice of factual proof in legal proceedings such as trials actually works. I am trying to understand inference, but this is not the same thing as trying to describe how the legal system manages evidential inference.

What might a mathematical (or logical) theory of evidence in or for legal proceedings do? If one's aim in studying the mathematics of evidence is not to describe the legal management of evidence in law, why study the mathematics of evidence (and inference)?

I have asked myself this question before. I now ask the question again because it may be particularly important if one suspects that fuzzy logic says something interesting and important about factual inference in law. A fair question is, "What possible good would a fuzzy explanation of factual proof in law do?"

One answer (one that I have sometimes also given) is that a mathematical account of inference is a valuable heuristic procedure: it is a procedure that reveals to us the implications of our own (logical?) thinking.

This answer has some force, at least sometimes. It has particular force when the heuristic procedure conforms to our untutored intuitions about the workings of sound inference: the answer -- it's all about heuristics, stupid! -- has force, for example, if we already believe that our good thinking takes a Bayesian form and a mathematical account -- in this instance, a Bayesian account -- spells out for us clearly what we roughly but imperfectly already think. This kind of use of mathematics takes us, so to speak, where we already want to go. But this heuristic justification for mathematical analysis of evidence and inference sometimes runs into trouble ...

First, there are those pesky people who refuse to concede that the basic structure of their sound thinking is Bayesian. Well, let's put those silly people to one side for now. But there are other problems ...

Second, sometimes the mathematical calculations seem to run on, so to speak, by themselves -- to such an extent that even a person who thinks that the right logical procedure is being used might feel compelled to say, "I can't honestly say that those calculations represent what I think. I think the conclusion is correct -- I have to say this because I think the method of argument used here was correct and the premises, I think, were correct -- but I can't honestly say that the calculations here portray what I already, if only faintly, thought."

The difficulty with fuzzy logic may be related to this second difficulty: the procedure does not merely elucidate what is already in someone's head. Even if one can do the calculations and personally does the calculations, the procedures and calculations do not seem to be the calculator's. So in what sense (if any) is the mathematical (or logical) procedure "heuristic"?

The law is reluctant to allow legal reasoners -- e.g., jurors, judges -- to surrender their reasoning processes to other agents or mechanisms. The law permits this to happen sometimes, but not often. This is probably one reason why the law is particularly uneasy about analytical procedures that outrun the intuitions of its authorized reasoners.

But there may be a deeper reason why the law is uneasy about -- or uninterested in -- fuzzy logic. The law may take the view that (i) certain things -- e.g., ancillary generalizations, evidential hypotheses, warrants -- must be part of any good reasoning from evidence to possible facts and (ii) fuzzy logic does not make a place, at least not in any obvious way, for such essential features of evidential inference.

"So what?," a logician might say. You are confusing logic and psycho-logic, the logician might say. For example, the logician might add, contentiously, "Just because the law has the deluded notion that ancillary generalizations -- those things you call "evidential hypotheses" -- are necessary to sound argument and inference doesn't make it so. The job of logic is not to make the human psyche comfortable.The inestimable job of logic is to devise procedures that lead to correct answers!"

It has often been noted that people have a tendency to believe the things that please them and to disbelieve matters that make them unhappy. Is it so with law as well? Does the law like to think that its logic is a good logic because thinking so makes the law and lawyers comfortable with what they do and the (deluded) way they think? Is the apparent demand of law, lawyers, and (some) legal theorists that outsiders give lawyers a transparent logic attributable to this?

Another possible answer, of course, is that the intuitions of legal folk have been right all along and that good logic must have the characteristics that legal people have thought all along that good argument must have, and logicians only now (from a longer perspective) are beginning to appreciate this. Hence, on this view, the development of theories of argumentation, the elaboration of Toulmin's theory of argument and logic, and the like are welcome and long overdue developments and are the direction that future studies of the mathematics and logic of evidence in law should take.

But this kind of justification for mathematical or logical analysis of evidence and inference seems to leave fuzzy logic out in the cold. I am not yet prepared to do that. Question: Is there not a place for a mathematical account of evidence and inference that, even though not transparent (even mildly so) to most legal professionals, sheds light on some key features of legal argument, evidential inference, and human knowledge in general? For example, even if the mathematics of fuzzy logic escapes most of us -- and will continue to escape most of us -- is it possible that it is worth studying (but why?) because it grapples with a central(?) feature of at least some legal problems -- matters such as "partial existence"? And is it possible that even if fuzzy logic and its offshoots could and would never be used by professional protagonists in courtrooms, it might nevertheless say something very important about, e.g., factual inference and the way that forensic proof ought to be conducted or managed.(We have our theories about the behavior of plants and we think that some of those theories are correct even though most of us think that the plants themselves do not have in their heads -- do plants have heads? -- the theories that we think describe how they grow etc. Perhaps legal actors such as trial lawyers are sometimes the equivalent of headless vegetables.)

### Probability and Precision; Forms of Probability and Uncertainty

Probabilities are not necessarily precise. For example, we can say, "The probability of rain is between one-quarter and one-third." See generally International Society for Imprecise Probability Theory and Applications

Probabilities do not necessarily designate variability or indeterminacy in nature. Probabilities may instead represent ignorance. The former type of uncertainty may be called aleatory. (It goes by other names -- for example, chance.) The latter type of probability -- the one that represents degrees and forms of ignorance -- is often called epistemic probability or uncertainty. (This type of probability is also called different things. For example, it is sometimes called credal probability.) See Brian Weatherson, Keynes, Uncertainty and Interest Rates

The source or cause of uncertainty is important. It is important to know if our uncertainty about an event or hypothesis is attributable to the way the world works, to the chance elements in the world in which we live; or whether our uncertainty is attributable to the lack of information or our uncertainty about methods of assessing the information or evidence we have.

I have often fumbled (largely by remaining silent) in explaining the difference between my interest and the focus of people who are mainly interested in matters such as random variables and causality. The difference is that the people who are interested in the latter are generally interested in patterns of random or chance behavior in nature whereas I am more interested in incomplete evidence and inconclusive argument.

• The confusion between aleatory and epistemic probability or uncertainty is perhaps partly attributable to the fact that the aleatory properties of nature often shed light on the appropriate treatment of incomplete information and on appropriate argument from and about evidence and information. In addition, it is very often the case that we have uncertainty compounded, that (some amount of) chance is wrapped in (some degree of) ignorance.
• ## Thursday, November 04, 2004

### Fuzzy (Legal) Thinking -- or Precise Thinking about Fuzzy (Legal) Matters

Google "fuzzy sets" and you get 129,000 hits.

Google "fuzzy logic" and you get 600,000 hits.

Hey, jurisprudes and legal theorists! Do you think there may be a there there?

Why the general (albeit not universal) silence about fuzzy logic in the legal academy?

Don't lawyers believe in the importance of precise thinking about fuzzy and rough concepts? So why aren't they attracted (generally speaking) to a serious attempt to talk precisely about ambiguity, fuzziness, roughness, and such things? Is it because they see a basic flaw in the theoretical foundations of fuzzy logic? (This I truly doubt! They haven't gotten close enough to the theory to even begin to think about foundational issues.) Is it just because they can't get an intuitive handle on fuzzy sets, fuzzy probabilities, fuzzy measures of uncertainty, and all that? (Perhaps.) Is it because they think probability theory does a better job of describing the properties of imprecise language and imprecise concepts? (This I also doubt.)

Zadeh's more advanced work edges toward a [nominalist(?); semantic(?)] neo-Platonic [or, perhaps better described, "neo-Aristotelian"] notion of partial existence. Do legal theorists shy away from Zadeh because they cannot get a handle on the notion of a thing having some of the properties of some concept to some degree? I doubt this too!

• Whether legitimately or not, legal professionals think this way all the time. What did American lawyers do when confronted with institutions that are not quite banks but are very much like banks? They called such hybrid institutions "non-bank banks." Talk about putting aside the principle of non-contradiction! Talk about partiality of existence! Talk about penumbral concepts!
• Oops! This last item snuck in here via con law -- Griswold, J. Douglas, privacy, and all that. But there is an affinity here, no?
• To get back to the heart of the matter: What's the story here? Fear of fuzziness, is it?

• The Japanese will probably have to lead the way -- again.
• ## Sunday, October 31, 2004

### Local News: Plaintiff Fraud

Headline in Sunday Star-Ledger p. 31 (County News, October 31, 2004):
State fraud files suit against carpeting chain