Universal Quantifier

When we say that a propositional function is true "in all cases," or "always", we mean that all its values are true. If f(x) is the function, and "a" is the right sort of object to be an argument to f(x), then f(a) is to be true. Note that "a" can be anything i.e. no matter how "a" may have been chosen. For example,

If 'a' is human, 'a' is mortal.

is true and it is true whether 'a' is human or not. (In fact, every proposition of this form is true.) Therefore, it is easy to see that the proposition function:

If x is human, x is mortal

is "always true," i.e., "true in all cases."

And again, 

The proposition stand for "There are no unicorns."

is the same as:

The proposition function f(x) stands for "x is not a unicorn."

And it is true in all cases.

Note that when we assert something about propositions, we are really asserting that certain propositional functions are true in all cases. For example, consider the following assertion:

(p or q) implies (q or p)

We do not assert the above as being true only for particular p or q, but as being true of any p or q. (Note that here we are talking about the principle of deduction. See the principle of deduction.)

Not only the principle of deduction, but all the fundamental proposition of logic, consist of assertions that certain propositional function are always true. It is part of the definition of logic that all its propositions are completely general, i.e. they all consist of the assertion that some propositional function containing no constant terms is always true.

Enough with the philosophy. Now let us talk about the above mentioned concept from the computational point of view. Apart from the replacement of variables by constants there is still another way way in which propositions can be obtained from propositional functions. Let us consider the following formula known as commutative law:

x + y = y + x

From the discussion of propositional function, we know it is a propositional function containing the two variables x and y that is satisfy by any arbitrary pair of numbers. In other words, whatever be the numerical constants may be, we always obtain a true formula. We express this fact in the following manner.

for all numbers x and y, x + y = y + x

The expression just obtained is a true proposition. 

The most important theorem of mathematics are formulated in a similar manner and they are so-called universal proposition, which asserts that arbitrary "things" of certain category (the word here is "class" - see theory of classes) have such and such a property.

Consider an sentence:

All human beings are mortal.

Another way to write the above sentence is:

∀ human beings x, x is mortal.

or more formally,

∀ x ∈ S, x is mortal

where S denotes the set of all human beings and the symbol ∀ denotes "for all" and is called the universal quantifier.

Note that the domain of the predicate variable is generally indicated:

Universal Proposition

Example: Truth of Universal Proposition.

Let domain D = {1, 2, 3, 4, 5}, and consider the universal proposition:

∀ x ∈ D, x² ≥ x.

Our problem is to show that this proposition is true.

Lets apply the method of exhaustion to show the truth of the universal proposition. That is, check that "x² ≥ x" is true for each element x in D.

Hence, the given universal proposition, ∀ x ∈ D, x² ≥ x, is true.

Example: Falsity of Universal Proposition

Consider the universal proposition:

∀ x ∈ D, x² ≥ x.

But this time, the domain of a predicate variable is the set of all real number, R.

Now, our problem is to show that this universal proposition is false.

Here, we try to find a counterexample to show that the given proposition is false.

lets take x = 1/2

Then, clearly x is in R. Since 1/2 is a real number but

(1/2)² = 1/4 (not ≥ ) 1/2

Hence, the given universal proposition "∀ x ∈ D, x² ≥ x" is false.