Discrete Structures

Lecture Notes Supplement, 22 Sep 2010

Last Update: 23 September 2010

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§1.3: Predicates & Quantifiers (cont'd)

Knowledge Representation & Translation:

  1. How to Represent "Toy Gun", etc.:

    1. In class today, I represented "Mark Twain was a wise man" in FOL as "Wise(mark-twain) ∧ Man(mark-twain)".

    2. But I pointed out that this doesn't work for all Adjective+Noun combinations.

      • E.g., "Dumbo is a small elephant" shouldn't be represented as "Small(dumbo) ∧ Elephant(dumbo)",
        because it would logically follow that Dumbo is small
        (but we assume that all elephants are large).

      • Similarly, "x is a toy gun" shouldn't be represented as "Toy(x) ∧ Gun(x)",
        because—although toy guns are toys—they aren't guns.

      • Similarly, "Fred is an alleged murderer" shouldn't be represented as "Alleged(fred) ∧ Murderer(fred)",
        because—although Fred may be alleged to be something—he might not be a murderer.

      • And how should we represent "Ann swam quickly"?

  2. It's not that we can't figure out some way to represent these.

    1. Rather, the problem is that our language for FOL is not rich enough to represent these in such a way that we can reason correctly about them.

    2. We need to extend FOL, just as FOL was itself an extension of propositional logic.

      • In an earlier lecture, I described some of those extensions.

        One of them, "modal" logic, can be used to handle "swimming quickly".

        Other extensions can handle the other cases.

    3. To read more about this, take a look at:

      • Parsons, Terence (1970), "Some Problems concerning the Logic of Grammatical Modifiers", Synthese 21: 320–334.

      • NEW "The Meaning of Noun Phrases"

        • Part of a longer document called "Common Sense Problem Page": "Commonsense reasoning is a central part of intelligent behavior. In contrast to expert knowledge, which is usually explicit, most commonsense knowledge is implicit. One of the prerequisites to developing commonsense systems is making this knowledge explicit. The goal of the formal commonsense reasoning community is to encode this knowledge using formal logic."

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