Quantifiers

Quantifiers are modifiers which give information regarding "howmany" objects there are of a given type. In other words, they giveinformation about the quantity of such objects.

For example, consider objects of the following type: real numbers whosesquare is equal to 2.

The statement:

There exists a real number x such that x * x = 2.

asserts something about how many objects there are of this type by using thequantifier "there exists".

In this particular case, more precisely, this statement asserts thatthere is at least one object of this type.

Of course, "we all know" that there are exactly two objects of this type. Indeed, we not only know that there are exactly two objects of this type. We know exactly which two objects these are. They are "-square root of 2" and "square root of 2".

This statement doesn't tell us such precise information about objects ofthis type. This statement simply asserts that there is at least one such object.

Since there is indeed at least one such object, this assertion (i.e.this statement) is true. Otherwise, this statement would have beenfalse.

Such a statement, a statement that asserts that there is at least oneobject of a given type, is called an existence statement. Such astatement asserts (rightly or wrongly) that we are not talking about"nothing" when we are talking about objects of that type. This isprecisely what such an assertion states, because such a statementasserts that there is at least one object of that type. Such a statementtells us nothing about whether or not there is more than one object of that type. It simply addresses the question of whether or not there isat least one object of that type.

As one might expect from this discussion, the modifier "there exists" is called anexistential quantifier.

Existential quantifiers occur quite often in discussions regardingobjects. When they occur, they often occur out of necessity. The removalof their presence at one or more places in such a discussion can renderpreviously meaningful statements in the discussion meaningless.

For instance, consider the affect of removing the existential quantifierfrom the above statement. The resulting statement is:

x * x = 2.

Unless this statement was made in some context in which priorinformation about "x" had been given, this statement, unlike theexistential statement from which it was extracted, is meaningless. The meaningless nature of this statement can be seen, in particular, byconsidering the fact that we have no way of knowing, in the absence ofsuch a context, whether this statement is true or false, even if we knew that "x" was a real number.

The verity of these observations can be further demonstrated byconsidering what we would have to say if someone came to us and said "x* x = 2." and then asked us whether we believed what they just asserted.What would we say to such a person? We might be inclined to respond to themwith something along the lines of "I don't know. Could you tell mesomething about "x" that I might be able to use to figure out whether ornot what you just told me about "x" is true?".

On the other hand, if someone came to us and said "There exists a realnumber x such that x * x = 2." and then asked us whether we believedwhat they just stated, we could say to them, "Yes. Indeed, "square rootof 2" is such a real number."

Another type of quantifier which occurs quite often in discussionsregarding objects is the so-called universal quantifier "for every".

For example, consider the following universally quantified statement:

For every real number x, x * x = 2.

This statement also asserts something about how many objects there are of the type in question (i.e. real numbers whose square is equal to 2).Namely, this assertion states that all real numbers are of this type.

Of course, "we all know" that this universally quantified statement,unlike the previously discussed existentially quantified statement, isfalse. After all, 5 is a real number whose square is not equal to 2.

Consider the three statements which we have considered in this noteabout quantifiers:

	(1) There exists a real number x such that x * x = 2.
	(2) x * x = 2.
	(3) For every real number x, x * x = 2.

Observe how the particular quantifiers occurring in a statementsignificantly affect the meaning of a statement.

Statement (1), the one with the existential quantifier is not onlymeaningful, it is also true.

Statement (2), the one without any quantifier, is meaningless, apartfrom some context in which further information about "x" is present; information which, hopefully, we could use to make sense of this statement.

Statement (3), the one with the universal quantifier, is not onlymeaningful, it is also false.

Other illustrative examples, showing the affect of removal or insertion of one or more quantifiers on the meaning of a discussion (i.e. on the meaning of a sequence of one or more statements), such as, for example, a proof, can easily beconstructed.

Let this suffice, for now, to demonstrate the wisdom of the following advice:

Pay attention to quantifiers.