Posted by: hrummel | December 1, 2008

X implies Y is true when X is false, Y is true.

The logic statement X implies Y is true when X is false and Y is true has always confused me. When spoken it seems strange that “the Germans won WW2 implies that rabbits are cute” is a true statement. It seems perfectly logical X implies Y is false when X is true and Y is false. “rabbits are cute implies that the Germans won WW2” is a false statement. However when X is false, we are not given any information about how it relates to Y. Thus mathematicians have chosen to assign truth to this statement when X is false. The result is, when looking at a Venn diagram, that we have a truth value everywhere except when A is true and B is false.

P implies Q

As to why mathematicians like having X implies Y to be true when X is false, we have simple representations for all other causes. We definitely wish it to be false when X is true and Y is false. This means that a true statement would imply a false statement, which should not happen. 1+1=2 should not imply 1+1=3. When looking at a Venn diagram, the statements that make sense are when X is true, and Y is false, the statement is false, when X is true and Y is true, the statement is true. Suppose we were to change the truth table such that X false, Y false, X->Y false then we would arrive at the representation “Y is true”.

Hypothetical Venn diagram

Suppose we were to change the truth table such that X false, Y true, X->Y false then we would arrive at the representation “X if and only if Y”

Hypothetical Venn diagram

Suppose we were to change the truth table such that anytime X is false, X->Y is false, we would arrive at the representation for “X and Y”.

Hypothetical Venn

Thus choosing “X implies Y” to be true anytime X is false we arrive at a new truth table that is not constructed with another easy expression, and we avoid redundancy. This is most likely why mathematicians have chosen the truth values that exist.

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Responses

  1. aww… you think that rabbits are cute!

    pretty Venn diagrams, too.

  2. I like this explanation. It is probably one of the reasons, I agree.

  3. Suppose “X implies Y is false when X is false and Y is true” is a theorem of our logic L. i.e. (X & ~Y) -> ~(X -> Y). Then (X -> Y) -> ~(X & ~Y), modus ponens. Using de Morgan’s laws, the RHS is equivalent to ~X & Y. Then both ((X -> Y) -> ~X) and ((X -> Y) -> Y)). But both of these statements run contrary to our intuitive understanding of implication! The first says that, “If Y follows from X, then X is untrue”. The second says that “If Y follows from X, then Y is true”.

    The basic problem, then with NOT assuming that “X implies Y is true when X is false and Y is true”, is that we are led to a contradiction. I have only sketched a proof of this assertion, but believe it can be made rigorous.

    Do you see any difficulties with this line of reasoning?

  4. So very true.

  5. Use CKod (http://ckod.sourceforge.net/_/) for creating your “truth table”.


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