Friday, January 2, 2015

Modus Tollens

The companion to modus ponens is modus tollens, the way of denying.

If I say something like, "if P is true, Q is true," and add, "however, Q is not true," I can conclude that P is not true, either.  In short: "If P, then Q. Q is not true, so neither is P."  I know this is so because if P were true, Q would be true.  But Q isn't true, so P can't be.

Here is an example: If it is raining outside, then there are clouds in the sky.  There are no clouds in the sky, so it is not raining.  The first sentence says there must be clouds for it to rain.  The second sentence tells me that there are no clouds, so how could it be raining?  It couldn't be raining, and it isn't raining!

Another example: In order for me to have stabbed the man, I must have been next to him.  However, I wasn't even in the same room.  Therefore, it couldn't have been me!  This example deviates from the grammatical syntax of the other, but follows the same logical form: If P (I stabbed the man), then Q (I was next to him).  Q isn't true (I wasn't in the room) therefore P isn't true (I didn't stab him).

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