English%20?%20FOL:%20universal%20quantifier

1
English ? FOL universal quantifier

• The structure of the universal quantifier is
• ?X(... ?...)
• Key words in an English sentence that indicates
  an universal quantifier
• all , every, each, any
• For every x, if x is a student and x likes
  mathematics, the x does not likes Oscar
• (1) ?X(S (X) ? M(X) ? ? L(X, o))

2
English ? FOL universal quantifier

• no (when negating an existential quantifier)
• No student who likes mathematics also likes
  Oscar.
• There does not exist an x such that x is a
  student and x likes mathematics and x likes
  Oscar.
• (2) ? ?X(S (X) ? M(X) ? L(X, o))
• (1) is equivalent to (2)
• ((1) ltgt (2))

3
English ? FOL existential quantifier

• The structure of the existential quantifier is
• ? X(... ?...)
• Key words in an English sentence that indicates
  an existential quantifier
• some, there is, there are, there exists, at least
  one
• There is an x such that x is a student pilot, and
  there is a y so that y is an instructor and x
  flies to chicago with y, but it is not the case
  that x attends a meeting and x is accompanied by
  y
• (3) ?X(SP(X) ? ?Y (I(Y) ? F(X, c, Y) ?
• ? (M(X) ? A(X, Y))))

4
English ? FOL existential quantifier

• not all (when negating an universal quantifier)
• Not all student pilots who fly a plane to Chicago
  with an instructor attend a meeting accompanied
  by the instructor.
• (4) ? ?X(SP(X) ? (?Y ((I(Y) ? F(X, c, Y)) ?
• (M(X) ? A(X, Y))))
• (3) is equivalent to (4)
• ((3) ltgt (4))

5
English ? FOL ambiguities

• There are no hard and fast rules for translating
  English into FOL.
• For example, the articles a or an in some
  cases imply an existential quantifier and in
  others an universal quantifier.
• Oscar flies a plane to Chicago with an
  instructor
• ? X(I(X) ? F(o, c, X)
• A student will do well at college if he/she
  studies
• ?X(S(X) ? D(X) ?W(x))

6
English ? FOL ambiguities

• Another ambiguity that occurs in English is that
  plural nouns not preceded by all or some are
  sometimes translated using a universal
  quantifier. At other times, these nouns are
  translated using an existential quantifier.
• Viruses cause colds (some viruses cause colds)
• Cats are animals (all cats are animals)
• The translations depends upon the meaning, and in
  some cases, the meaning may not be clear. The
  problem is that English is ambiguous, whereas FOL
  is not.