Existentially Quantified Statements

NOTE 1: An existentially quantified statement is a statement of the skeletal form:

(*) There exists an element, x, of a given set, S, such that a given statement, A, about this element, x, is true.

NOTE 2: An existentially quantified statement, (*), tells us something about the quantity of elements in the given set, S, for which the given statement, A, about these elements is true. More precisely, such a statement tells us that the number of elements of the given set, S, for which the given statement, A, about these elements is true is at least one.

QUESTION 3: How might one go about proving an existentially quantified statement, (*)?

ANSWER 4: One way to prove an existentially quantified statement, (*), is the so-called DIRECT METHOD OF PROOF OF AN EXISTENTIALLY QUANTIFIED STATEMENT.

NOTE 5: The basic skeleton of a DIRECT METHOD OF PROOF OF AN EXISTENTIALLY QUANTIFIED STATEMENT, *, can be summarized as follows:

NOTE 6: A PROOF OF (*) BY THE DIRECT METHOD OF PROOF OF AN EXISTENTIALLY QUANTIFIED STATEMENT:

	STEP 1: "Introduce a candidate." - In this step, one explicitly introduces an intelligently chosen specific element, x, of the given set, S, one for which one is able to complete the proof as follows.
	STEP 2: "Prove that the candidate which was introduced in STEP 1 is qualified." - In this step, one proves that the intelligently chosen specific element, x, of the given set, S, is one for which the given statement, A, about this element is true.
	STEP 3: Conclude from STEPS 1 and 2 that there exists an element of the given set for which the given statement about this element is true. After all, the candidate introduced in STEP 1 is, by the result of STEP 2, such an element of the given set (i.e. an element of the given set for which the given statement about this element is true).

NOTE 7: Here is an example of the DIRECT METHOD OF PROOF OF AN EXISTENTIALLY QUANTIFIED STATEMENT.

	PROPOSITION: There exists an integer x such that x + x = 2.
	PROOF OF PROPOSITION BY THE DIRECT METHOD OF PROOF OF AN EXISTENTIALLY QUANTIFIED STATEMENT:
	STEP 1 ("Introduce a candidate.") - Let x be the integer, 1.
	STEP 2 ("Prove that the candidate introduced in STEP 1 is qualified.") - Since x is the integer 1, x is an integer and x + x = 1 + 1 = 2.
	STEP 3 - Hence, there exists an integer x with the property that x + x = 2. (Indeed, the integer 1 is such an integer.)
	THIS COMPLETES THE PROOF OF PROPOSITION BY THE DIRECT METHOD OF PROOF OF AN EXISTENTIALLY QUANTIFIED STATEMENT:

NOTE 8: You may ask, "How did we 'find' our candidate"? This, while a reasonable question to ask, is not relevant to the issue of whether or not the above argument is a proof of the PROPOSITION. The argument is clearly a proof of the PROPOSITION. Indeed, it is about as "direct" a proof of this PROPOSITION as one could imagine. It proves directly that there exists a real number, x, with the property that x + x = 1 by exhibiting an intelligently chosen specific real number, x, with the property that x + x = 1. This, in a "nutshell", is the direct method of proof of an existentially quantified statement.

NOTE 9: Of course, we may have great difficulty in finding a candidate which we are convinced is one which we should introduce in STEP 1 (i.e. one for which we will be able to complete STEP 2). But this is irrelevant to the proof per se. Regardless of the method used to find our candidate, the proof begins with introducing our candidate (STEP 1). The proof continues with proving that our candidate is qualified (STEP 2). The proof ends with observing that we have just shown that there exists a qualified candidate (STEP 3). (Indeed, we have just shown that the candidate we introduced in STEP 1 is, by STEP 2, such a candidate.)

QUESTION 10: You may well ask, if the method used to find our candidate is irrelevant to the proof of the PROPOSITION, then why do we bother to employ any method to find our candidate? The answer to this question is, of course, obvious, and rather rhetorical. How else are we going to be able to start our proof that our candidate is qualified, if we haven't found a candidate which we can prove is qualified?