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Math Help - Is a Name or Variable alone a WFF?

  1. #1
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    Is a Name or Variable alone a WFF?

    So the formation rules of predicate logic sit pretty well with me...but one thing that we don't seem to have covered explicitly is precisely whether a name or variable by itself is a wff. It doesn't seem to me intuitively that say "Superman" by itself should be a well-formed formula, but unfortunately I can't explain why this would be the case :P

    I would have thought that for a name or variable to be part of a wff, it would have to be preceded by a predicate of some sort. So "Superhero(Superman)" is a wff, and "Superman" is not.

    Am I right? If not, why? Thanks in advance!
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  2. #2
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    A constant (a superman in your example) and a variable are terms and terms alone are not atomic sentences. An atomic sentence can have a form like "Predicate(term)" [superhero(superman) in your example] or a term = term. Atomic sentences alone can be WFFs in first order logic (see below syntax).

    I assume you can read a BNF form. Belows are the syntax of first order logic with equality. A "Sentence" below denotes a WFF in FOL.

    \text{Sentence} \rightarrow \text{AtomicSentence}
    | \text{(Sentence Connective Sentence)}
    | \text{Quantifier Variable, ... Sentence}
    | \neg Sentence

    \text{AtomicSentence} \rightarrow \text{Predicate(Term,...)} | \text{Term=Term}

     \text{Term} \rightarrow \text{Function(Term,..)} | \text{Constant} | \text{Variable}

    \text{Quantifier} \rightarrow \forall | \exists
    \text{Connective} \rightarrow \wedge | \lor | \Rightarrow | \Leftrightarrow
    \text{Constant} \rightarrow \text{Superman} |... //your example in Q
    \text{Predicate} \rightarrow \text{Superhero} |... //your exampe
    Last edited by aliceinwonderland; January 28th 2009 at 03:09 AM. Reason: Error correction
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  3. #3
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    Much thanks for the concise reply!
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