lec0907 - Rule of Universal Specification This is a fairly...

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Rule of Universal Specification This is a fairly obvious rule, but one that is important: If an open statement is true for all possible replacements in the designated universe, then that open statement is true for each specific individual member in that universe. Symbolically speaking, we have: If 2200 x p(x) is true, then we know that p(a) is true, for each a in the universe for x. Here is a simple example using this rule. Consider the following premises: 1) Each actor/actress on the TV show Friends is a millionaire. 2) Jennifer Aniston is an actress on the TV show Friends. Therefore, Jennifer Aniston is a millionaire. Symbolically, consider setting up these three statements: p(x): x is an actor/actress on Friends. q(x): x is a millionaire. Now, the given information is 2200 x [p(x) q(x)] If we wish to determine the financial status of Jennifer Aniston, we add into our premise the statement p(Jennifer Aniston) as being true. Using the Rule of Universal Specification, and Rule of Detachment, we can conclude that q(Jennifer Aniston) is true; that is Jennifer Aniston is a millionaire.
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Let’s go ahead and look at another example in greater detail. Consider each of these open statements for the next example: p(x): x is a show on prime-time TV q(x): x is a show on late-night TV r(x): x is a soap opera Now, consider the following argument: No soap opera is on prime-time TV or late-night TV All My Children is a soap opera. Therefore, All My Children is not on prime-time TV. (Note: Let A stand for “All My Children”) 1) 2200 x [p(x) q(x) ¬ r(x)] Premise 2) r(A) Premise 3) p(A) q(A) ¬ r(A) 4) ¬¬ r(A) ¬ (p(A) q(A)) Contrapositive 5) r(A) ¬ (p(A) q(A)) Law of Double Negation 6) r(A) ( ¬ p(A) ¬ 7) ¬ p(A) ¬ q(A) Steps 2 & 6 & Rule of Detachment (Modus Ponens) 8) ∴¬ p(A) Simplification
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The Rule of Universal Generalization If an open statement p(x) is proved to be true when x is replaced by any arbitrarily chosen element c from our universe, then the universally qualified statement 2200 x p(x) is true. (This rule also extends beyond one variable.) We can use this to formally show that IF
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lec0907 - Rule of Universal Specification This is a fairly...

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