Order of Quantifiers

The order in which quantifiers appear is very important; it must be maintained when converting a formula to prenex normal form.

Consider the ambiguous sentence, ``Every man loves some woman.'' This sentence could be interpreted as:

  1. For every man, there is some woman (depending on who the man is) whom the man loves. This would be written:

    &forall x [Man(x) &rarr &exist y [Woman(y) &and Loves(x,y)]]

    and Skolemized:

    Man(x) &rarr
    [Woman(lover(x)) &and Loves(x,lover(x))]

  2. There is some woman (perhaps Marilyn Monroe) who is loved by every man. This would be written:

    &exist y &forall x [Man(x) &rarr [Woman(y) &and Loves(x,y)]]

    and Skolemized:

    Man(x) &rarr [Woman(a) &and Loves(x,a)]

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