Re: plain rules, please [was: Semantic Web Rule Language (SWRL) 0.5 released]

My question about the distinction between "deduction" and other forms of 
inference was posed to help me better understand the points you have been 
making about the utility of non-deductive inferences.

 From your response, I think that "deduction" is the process of finding a 
proof in some theory.  Thus, "deductions" (deduced results) are precisely 
those that are proven to be true in some (accepted) proof theory.  (And, 
maybe, for a proof theory to be acceptable, it must be sound with respect 
to some accepted model theory.)

On this basis, I understand the further points you are making to be that 
there may be useful results (inferences) that cannot be proven.  Which, I 
guess, takes us into issues of how dependable one needs results to be in 
order for them to be useful.

Am I following your key points?

(This leaves me wondering if it is not generally possible to turn any 
non-deduction into a deduction by strengthening the accompanying proof 
theory.  Picking an example from another thread here:  based on a given 
knowledge of airports, I might usefully infer, via NAF, that the closest to 
my current location is LHR, because I don't know of a closer one (and 
there's a general presumption that I know about airports close to my 
current location).  This is not a provable deduction, but maybe it is made 
so by adding to the proof theory concerned an axiom to the effect that a 
given list of airports is complete.)

#g
--

At 15:58 01/12/03 -0500, Drew McDermott wrote:
>    [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.

------------
Graham Klyne
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