Paradoxes of Material Conditional

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

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

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:

• Lepore, Ernest (2000), Meaning and Argument: An Introduction to Logic through Language (Malden, MA: Blackwell), §A3 ("Conditionals"), esp. §A3.1.1 ("Paradox of Implication), p. 317, and §A3.1.3 ("Paradox of Implication Revisited"), pp. 319-320.

• Anderson, Alan Ross, & Belnap, Nuel D., Jr. (1975), Entailment: The Logic of Relevance and Necessity (Princeton, NJ: Princeton University Press).