Lesson PROOF OF THE FALLACY OF THE INVERSE
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This proof uses the law of equivalence. The law of equivalence states: If two statements have the same truth values for all possible conditions in the truth table, then the statements are equivalent and can be used inter-changeably. The fallacy of the inverse involves the use of the implies statement. The implies statement is symbolically shown as: p -> q This means that: If p is true, then q must be true in order for the implies statement to be true. This logic only applies if p is true. If p is false, then q can be true or false without making the implies statement false because the necessary condition for proving the implies statement is false is that p is true. The truth table for the implies statement is shown below: <pre> p q (p -> q) - - -------- T T T T F F F T T F F T </pre> If p is true and q is true, then the implies statement is true. If p is true and q is false, then the implies statement is false. If p is false, then the implies statement is true regardless of whether q is true or false. The inverse of the implies statement is symbolically shown as: ~p -> ~q The fallacy of the inverse states that: THE ASSUMPTION THAT THE INVERSE OF THE IMPLIES STATEMENT IS EQUIVALENT TO THE IMPLIES STATEMENT IS FALSE. In order to prove the fallacy of the inverse, we assume that the inverse of the implies statement is equivalent to the implies statement and then show that it can't be because it violates the law of equivalence. We start with the assumption that: ~p -> ~q is equivalent to p -> q What this means is: If p implies q is true, then ~p implies ~q is also true. If p implies q is false, then ~p implies ~q is also false. The truth table for the inverse of the implies statement is shown below: <pre> p q (p -> q) ~p ~q (~p -> ~q) - - -------- -- -- ---------- T T T F F T T F F F T T F T T T F F F F T T T T </pre> If (p -> q) is to be equivalent to (~p -> ~q), then they both must have the same truth values for all possible conditions in the truth table. Equivalence means that their truth values agree for all possible conditions in the truth table. If both are true or if both are false, then the equivalence statement is true. If one is true and the other is false, then the equivalence statement is false. We can immediately see that they do not have the same truth table values for all possible conditions in the truth table, but we will shown that through the use of the equivalency statement. The equivalency statement is symbolically shown as: (p -> q) <-> (~p -> ~q) The truth table for (p -> q) <-> (~p -> ~q) is shown below: <-> translates to IS EQUIVALENT TO <pre> p q ~p ~q (p -> q) (~p -> ~q) (p -> q) <-> (~p -> ~q) - - -- -- -------- ---------- ----------------------- T T F F T T T T F F T F T F ***** F T T F T F F ***** F F T T T T T </pre> Each row is a condition of the truth table. There are four possible conditions to be tested. This is because we are dealing with two variables with two possible truth values each. The variables are p and q. The possible conditions are: p is true and q is true. p is true and q is false. p is false and q is true. p is false and q is false. There are no other possible conditions involving these two variables. The truth table for the equivalency statement is not true for all conditions. Look at the second and third rows in the truth table. The truth values for (p -> q) and (~p -> ~q) do not agree. This violates the requirements of the law of equivalency. This proves that the two statements of (p -> q) and (~p -> ~q) are not equivalent. This is called the fallacy of the inverse. It means that any assumption that the statement p -> q is equivalent to ~p -> ~q is false. If you are given an argument such as: p -> q ~P therefore ~q then you can immediately reject it by reason of the fallacy of the inverse.
