Monday, January 15, 2007
Real-world semantics is in the eye of the beholder
The notion of semantics in logic can be confusing. I found some good discussions of possible world semantics as based on interpretations, in a book I have on Reserve for my AI courses:
Brachman, R.J. and Levesque, H.J., Knowledge representation and reasoning. Elsevier/Morgan Kaufmann, San Francisco, CA, 2004.
I'll paraphrase their comments on page 20.
The meaning of "hammer" in some interpretation is no more or less than those objects that we consider to be hammers. As far as FOL is concerned, we do not try to say what a hammer is in the way a dictionary might, describing its shape, materials, size, or weight. All we need to say is which objects are and which are not hammers.
On pages 24 and 25, there are further statements to the effect that we can build a system that can be told that Hammer(h1) is true in some user-intended interpretation, so that the system can later come to believe other sentences that are true in that interpretation. "Hammer" is a predicate, and "h1" a term that we create; they have no other existence beyond our creation of them in our system.
I quote their fundamental tenet of knowledge representation:
“Reasoning based on logical consequence is weak, and only allows safe, logically guaranteed conclusions to be drawn. However, by starting with a rich collection of sentences as given premises, including not only facts about particulars of the intended application, but also those expressing connections among the non-logical symbols involved, the set of entailed conclusions becomes a much richer set, closer to the set of sentences true in the intended interpretation. Calculating these entailments thus becomes more like the form of reasoning we would expect of someone who understood the meaning of the terms involved."
Figure 7.6 in the AIMA textbook for the course illustrates some of this from a slightly different perspective.
Brachman, R.J. and Levesque, H.J., Knowledge representation and reasoning. Elsevier/Morgan Kaufmann, San Francisco, CA, 2004.
I'll paraphrase their comments on page 20.
The meaning of "hammer" in some interpretation is no more or less than those objects that we consider to be hammers. As far as FOL is concerned, we do not try to say what a hammer is in the way a dictionary might, describing its shape, materials, size, or weight. All we need to say is which objects are and which are not hammers.
On pages 24 and 25, there are further statements to the effect that we can build a system that can be told that Hammer(h1) is true in some user-intended interpretation, so that the system can later come to believe other sentences that are true in that interpretation. "Hammer" is a predicate, and "h1" a term that we create; they have no other existence beyond our creation of them in our system.
I quote their fundamental tenet of knowledge representation:
“Reasoning based on logical consequence is weak, and only allows safe, logically guaranteed conclusions to be drawn. However, by starting with a rich collection of sentences as given premises, including not only facts about particulars of the intended application, but also those expressing connections among the non-logical symbols involved, the set of entailed conclusions becomes a much richer set, closer to the set of sentences true in the intended interpretation. Calculating these entailments thus becomes more like the form of reasoning we would expect of someone who understood the meaning of the terms involved."
Figure 7.6 in the AIMA textbook for the course illustrates some of this from a slightly different perspective.
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