Paradoxes of material implication

As the best known of the paradoxes, and most formally simple, the paradox of entailment makes the best introduction.

In natural language, an instance of the paradox of entailment arises:

It is raining

And

It is not raining

Therefore

George Washington is made of rakes.

This arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true).

The classical paradox formulae are closely tied to conjunction elimination,

• ${\displaystyle (p\land q)\to p}$

which can be derived from the paradox formulae, for example from (1) by importation. In addition, there are serious problems with trying to use material implication as representing the English "if ... then ...". For example, the following are valid inferences:

1. ${\displaystyle (p\to q)\land (r\to s)\ \vdash \ (p\to s)\lor (r\to q)}$
2. ${\displaystyle (p\land q)\to r\ \vdash \ (p\to r)\lor (q\to r)}$

but mapping these back to English sentences using "if" gives paradoxes.

The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is true that either (a) if John is in London then he is in France, or (b) if he is in Paris then he is in England." Using material implication, if John is not in London then (a) is true; whereas if he is in London then, because he is not in Paris, (b) is true. Either way, the conclusion that at least one of (a) or (b) is true is valid.

But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one does not know where John is, but nonetheless knows that if he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.

The second example can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or that if switch B is closed, the light is on." Here, the most likely natural-language interpretation of the "if ... then ..." statements would be "whenever switch A is closed, the light is on," and "whenever switch B is closed, the light is on." Again, under this interpretation both clauses of the conclusion may be false (for instance in a series circuit, with a light that comes on only when both switches are closed).

• Connexive logics were designed to exclude the paradoxes of material implication
• Correlation does not imply causation
• Counterfactuals
• False dilemma
• Import-Export
• List of paradoxes
• Modus ponens
• The Moon is made of green cheese
• Relevance logic arose out of attempts to avoid these paradoxes
• Vacuous truth

• Bennett, J. A Philosophical Guide to Conditionals. Oxford: Clarendon Press. 2003.
• Conditionals, ed. Frank Jackson. Oxford: Oxford University Press. 1991.
• Etchemendy, J. The Concept of Logical Consequence. Cambridge: Harvard University Press. 1990.
• "Strict implication calculus", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Sanford, D. If P, Then Q: Conditionals and the Foundations of Reasoning. New York: Routledge. 1989.
• Priest, G. An Introduction to Non-Classical Logic, Cambridge University Press. 2001.