Quantifiers

When we say that something is "always true" or "true in all cases" it is clear that the "something" is not a proposition. Why this "something" is not a proposition? Because, a proposition is just true or false, and there is an end of the matter. For example, there are no instances or cases of the proposition:

Socrates is a man.

It would be meaningless to speak of its being true "in all cases." From the discussion of propositional function we can see that the phrase like "in all cases" is only applicable to propositional function. We usually speak of such phrase when causation is being discussed. Now let us consider the example from the previous section:

If x is a flash of lightning, x is followed by thunder.

Now if you look closely, we are told, in general, that A is, in every instance, followed by B. Now if there are "instances" of A, we say that A is in every instance followed by B. that is, whatever x may be, if x is in A, it is followed by a B; that is, we are asserting that certain propositional function is always true. Sentences (expressions) involving such words as "all," "every," "a," "the," "some" require propositional functions for their interpretation. The way in which propositional functions occur can be explained by means of two of the above words, namely, "all" and "some."

There are two things that can be done with propositional function:

All other uses of propositional function can be reduced to these two.

Suppose we are given some propositional function and our job is to change into proposition. There are two ways to change propositional function into proposition.

1. To assign specific values to all propositional variables. (We have already seen this in the last section.)

2. To add quantifiers. (That is, to add ∀ and ∃ quantifiers.)

Quantifiers are words that refer to quantities such as "some" or "all" and tell how many elements a given predicate is true.

Consider the proposition:

This complex proposition cannot be broken down in the manner used in the propositional logic. We can perceive conjunctions and conditionals within the proposition but they are cemented together by the phrases "for all" and "there exists." If the given proposition occurred as component of a still more complex one, the given proposition would have to be treated as a unit in any truth-table analysis.

The modifying phrases "for all x" (or any linguistic equivalent like for each x) is called a universal quantifier; the modifying phrase "there exists y" (or any linguistic equivalent like for some y) is called an existential quantifier.