MACM 101 Lecture 10: Lecture 10 Part 2_ Theorems and Proofs
21 Dec 2018
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If an open statement becomes true for all values of the universe, then it is true for each specific individual value from that universe. Reason premise: p(socrates) q(socrates), rule of universal specification, p(socrates, q(socrates) premise modus ponens. Take any number c such that 2c 6 = 0. Then 2c = 6, and, finally c = 3. As c is an arbitrary number this proves the theorem. Look at the first and the last steps. In the first step instead of the variable we start to consider its generic value, that is a value that does not have any specific property that may not have any other value in the universe. In the last step having proved the statement for the generic value we conclude that the. If an open statement p(x) is proved to be true when x is assigned by any arbitrary chosen (generic) value from the universe, then the statement x p(x) is also true.
