Paradoxes of material implication

Implication, in logic, describes conditional if-then statements, e.g., "if it is raining, then I will bring an umbrella." There are many ways to formalise implication, of which material implication is one of the simplest. It equates the statement "if p then q" with the statement "either not p, or q". If one takes a truth-functional approach to implication, asking whether "if p then q" is true or false based only on the values of p and q, it is hard to see any other reasonable interpretation. However, accepting this causes problems.

The paradoxes of material implication are a group of formulas recognized as logical truths in classical logical theory, but which strike common intuition as somewhat questionable or even downright wrong. The classical formulas usually cited are:

# $p\; o\; (q\; o\; p)$
# $eg\; p\; o\; (p\; o\; q)$
# $(\; eg\; p\; land\; p)\; o\; q$
# $p\; o\; (q\; lor\; eg\; q)$

The paradoxes of material implication arise because of the truth-functional definition of material conditional – i.e., if/then – statements under which a conditional is said to be true merely because the antecedent is false or the consequent is true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logics must, by definition, reject (3) as false.) On the other hand, "If the White Sox win the World Series next year, then the Red Sox won it in 2004," is true simply because the Red Sox "did" win the World Series in 2004. By extension, any tautology is implied by anything whatsoever, since a tautology is always true.

The classical paradox formulas are closely tied to the formula,

* $(p\; land\; q)\; o\; p$

the principle of Simplification, which can be derived from the paradox formulas rather easily (e.g. from (1) by Importation). Since the principle of Simplification is generally considered to be a fundamental principle of logic, finding a system of logic that excludes the paradox principles, while still managing to be satisfactory in other respects, has been one of the primary agendas of modern logical theory.

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:

# $(p\; o\; q)\; land\; (r\; o\; s)\; vdash\; (p\; o\; s)\; lor\; (r\; o\; q)$
# $(p\; land\; q)\; o\; r\; vdash\; (p\; o\; r)\; lor\; (q\; o\; r)$

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 either true that if John is in London then he is in France, or that if he is in Paris then he is in England.", while the second can be read "If I close switch A and switch B, the light will go on. Therefore, it is either true that if I close switch A the light will go on, or that if I close switch B the light will go on." If the two switches are in series, then the premise is true but the conclusion is false. Thus, using classical logic and taking material implication to mean if-then is an unsafe method of reasoning which can give erroneous results.

References

*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.
*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.