CSE 472/572, Spring 2002

 Last Update: 1 December 2003 Note: or material is highlighted

For those of you still unconvinced of the truth table for →, the material conditional, you are not alone.

The main problem is that → does not exactly capture ordinary English "if-then".

There are notorious "paradoxes of the material conditional"; here's one:

P, ¬P /∴ Q
I.e., from a contradiction, any proposition whatsoever can be inferred!

To show this, all I have to do is show that there is no row of a truth table for that inference in which both P and ¬P are true but Q is false:

 P ¬P Q false true true false true false true false true true false false

Since there is no row of this truth table in which both P and ¬P are true, there certainly can't be a row in which not only are P and ¬P true but Q is false. So, the inference

P, ¬P /∴ Q
is "vacuously" truth-preserving.

Here is a syntactic proof of P, ¬P /∴ Q:

 1. P // assumption 2. ¬P // assumption *3. ¬Q // temporary assumption for ¬Elim *4. P // send, 1 *5. ¬P // send, 2 *6. Q // 4,5,3, ¬Elim 7. Q // return, 6

Or, more directly:

 1. P : assumption 2. ¬P : assumption 3. (P v Q) : 1, vIntro 4. Q : 2,3, vElim