From: Drew McDermott <drew.mcdermott@yale.edu>

Date: Mon, 1 Dec 2003 15:58:16 -0500 (EST)

Message-Id: <200312012058.hB1KwG413227@pantheon-po01.its.yale.edu>

To: www-rdf-rules@w3.org

Date: Mon, 1 Dec 2003 15:58:16 -0500 (EST)

Message-Id: <200312012058.hB1KwG413227@pantheon-po01.its.yale.edu>

To: www-rdf-rules@w3.org

[Graham Klyne] Can you please point me at a resource that explains the precise distinction between "deduction" and other forms of inference? Consulting my agent undergraduate logic textbook (by Angelo Margaris, published 1967), under "deduction" in the index we find a definition of "a" deduction, namely, a series of formulas that are either axioms or result from application of an inference rule from previous formulas. Then one could say that "deduction" (the technique) is whatever comes at the end of a "deduction" (the series of formulas). But that's not terribly enlightening. A better definition comes by taking into account the semantics of logical languages (found in another chapter). Anything that can be deduced is true in all models of a theory (and, if the theory is complete, vice versa). This is the reason that deduction is conservative: if you can think of any interpretation of the given facts, no matter how wild, in which the statements you start with are true, then if P is false in that interpretation it cannot be deduced. (Unless the statements you start with are inconsistent, in which case there _are_ no interpretations that make them all true.) When one philosopher says "P is possible," and the other retorts that it's "only logically possible," it's exactly this sense of possibility they have in mind. Those who expect great things from deduction hope to make many commonsense inferences logically necessary by supplying the appropriate axioms. For instance, we'd like to infer that you know your name. It may be physically impossible, or incredibly unlikely, that you have forgotten your name, but it's not logically impossible unless we supply an axiom that says "Everybody knows their own name." Then we think of the possibility of Alzheimer's, and realize that this is trickier than we thought. Techniques like probabilistic reasoning with Bayes nets can be thought of as deductive or nondeductive, and it is easy to slip from one mode to the other without realizing it. Let's assume that there is a deductive theory in which a Bayes net and its boundary conditions can be described, and the conclusions you arrive at are precisely those licensed by the usual algorithms. (Actually expressing this theory is probably harder than you think, but let that pass.) Now we will have a theorem such as P("Klyne knows his name", 0.9999976). So far, deduction. But if we slip to "Therefore, Klyne knows his name," we have interpreted the conclusion nondeductively. Decision theorists can postpone the inevitable one step further by having all _behavior_ depend only on expected utilities rather than beliefs. I don't need to actually _believe_ that Klyne knows his name; I just have to realize that if I want to answer the question "Does Klyne have a middle name?" the action with the highest expected utility is to send him an e-mail message with the question. One problem is that to prove that an action has the highest expected utility I have to be able to reason about all possible actions, not by running through an explicit list, but somehow. Another problem is that it is much more efficient to reason in terms of possibly wrong beliefs than in terms of certain probabilities. In the present example, I'd like to believe that after asking Klyne the question and getting the answer I will then know whether he has a middle name. But all I can conclude is that the conditional probability of "Klyne has a middle name" given that he replies "No" is 0.001495. (It's much higher than you'd expect because of the chance that he may conceal the truth, not out of malice, but in order to spoil the example.) -- Drew P.S. One might object that I can't really be certain about the probabilities, not to very many significant digits. No, but you'll almost certainly never be contradicted if you act as though these numbers really are completely accurate. -- -- Drew McDermott Yale University CS Dept.Received on Monday, 1 December 2003 15:58:17 GMT

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