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English FOL: universal quantifier

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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))

- 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) <=> (2))

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))))

- 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) <=> (4))

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))

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.