lec0120 - 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) All years divisible by 400 are leap years 2) 2000 is divisible by 400 Therefore, 2000 is a leap year. Symbolically, consider setting up these three statements: p(x): x is divisible by 400 q(x): x is a leap year Now, the given information is 2200 x [p(x) q(x)] If we wish to determine the status of the current year, we add into our premise the statement p(2000) as being true. Using the Rule of Universal Specification, and Rule of Detachment, we can conclude that q(2000) is true; that is 2000 is a leap year.
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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”)
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lec0120 - Rule of Universal Specification This is a fairly...

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