Quantifiers: General Statements

Suppose we want to make a general statement about a class of things rather than an individual object. Suppose we want to say that all cities have mayors.

We can accomplish this by introducing a quantifier, which tells us whether Mx is being asserted of all x (in this case, every city in the world) or of some x (at least one city in the world).

In the first case, we use a universal quantifier, symbolized by an x in parentheses at the front of the statement: (x)(Mx), which can be translated as, "For any x, x has a mayor."

In the second case, we use the existential quantifier, symbolized by a backward "E" and placed, once again, at the front of the statement: (x)Mx, which can be translated as, "For some x, x has a mayor."

An expression of the form Px is called an open sentence, and its variable is said to be free. When we preface it with a quantifier, the quantifier binds the variable by telling us how to interpret it, and the result is a closed sentence or statement.

Universal statements are treated as conditionals, particular statements as conjunctions. The four traditional standard forms of categorical statements are translated as follows:

A:  All S are P (x)(Sx Px)
E:  No S is P (x)(Sx ~ Px)
I:  Some S are P (x)(Sx Px)
O:  Some S are not P (x)(Sx ~ Px)

Singular statements | General statements


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