For Monday, read Chapter 4, Sections 1 and 2. Nongraded homework: Problems on pages Graded HW #4 is due on Friday, Feb. 11, at the beginning of class.
A B GA → (B & G), ~ B A ~ B T T T T T T T T T T T T T T T T T T Answer: Invalid, proven by line seven
Is there an easier way to test for validity? Try going straight to an interpretation that makes all of the premises true and the conclusion false: A B C(A v ~ B) ↔ C,~ C A → C T The only possible way to make the conclusion false and the second premise true automatically makes the first premise false. So, the argument is valid.
Here’s another A B C(C v B) → ~ A, ~ C A → C T Invalid --Make A true and C false: False conclusion --C’s being false automatically makes the second premise true. --A’s being true makes the consequent of the first premise false; so we have to assign to the B to make the antecedent false (and the entire premise true).
Summary of the Procedure Assign truth-values to statement letters so as to make the conclusion false. If it’s possible to assign truth-values to the remaining statement letters in a way that makes all of the premises true, then the argument is invalid. If this cannot be done, the argument is valid.
Complications If there’s more than one way to make the conclusion false, then, before you conclude the argument is valid, be sure to check all of the possibilities (reaching a dead end in each case). Partly for this reason, when your answer is ‘valid’, we want you to explain in English the process by which you arrived at your answer.
A B DA → B, D → B, ~ B A ↔ D If A is true and D false, the conclusion is false. Then, because A is true, B has to be true (to make the first premise true). That, however, makes the third premise false. BUT, we have to check the other way of making the conclusion false before we answer ‘valid’. If D is true and A false, then B has to be true to preserve the truth of the second premise. But this makes the third premise false. So, the argument is valid.
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