CSE 472/572, Spring 2002

Paradoxes of Material Conditional

Last Update: 1 December 2003

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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:

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:

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

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

For more information, see:

Copyright © 2002-2003 by William J. Rapaport (rapaport@cse.buffalo.edu)
file: 572/S02/paradoxes.of.mat.cond.2003.12.01.html