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The paradoxes of material implication

Material implication is a concept used in formal logic to articulate conditional ("if...then") statements. For example, if I say "if I withhold food from the cur, then the cur will be hungry," let H symbolize "withhold food from the cur" and let D symbolize "the cur will be hungry." We can symbolize this statement in the following ways:

H ⇒ D

H → D

H ⊃ D

Here is the truth-table for material implication:

p q p q

We can discern the two "paradoxes of material implication" from this table. Rows 3 and 4 exhibit the paradox that "Whenever the antecedent is false, the whole condition is true" and rows 1 and 3 exhibit the paradox "Whenever the consequent is true, the conditional is true" (Suber).

For example, according to the first paradox, it is formally valid to say "If all trees are blue, then I am currently typing a sentence." I can also say "If all trees are blue, then the sky is green." In the first example, not only is it not true that trees are blue, but the color of trees seems to be totally irrelevant to the consequent, according to ordinary, commonsense reasoning. In the second example, this same apparent non-sequitur is operative, except the antecedent and consequent are both false. As for the second paradox, the material equivalent statement "If my name is Daniel, then the sky is blue" and "If the sky is green, then my name is Daniel" are both formally valid. In the first case, the antecedent and conseequent are both true but the consequent is semantically irrelevant to the antecedent, and in the second, the antecedent is false and irrelevant to the true consequent. As in the first paradox, there is an apparent non-sequitur, as the antecedent is irrelevant to the consequent, yet the material equivalent is formally valid.

What's up with the paradoxes of material implication? If they seem so counterintuitive, why do we want anything to do with them? Why not just get rid of them? The answer has to do with the importance of truth-functionality in connectives. We refer to a connective as truth-functional "if we can figure out the truth-value of the statement solely on the basis of the truth-values of its components"(Suber). Indeed, that's why truth-tables are so handy! They are perfectly suited for the use of truth-functional connectives. Determination of the truth-value of variables linked by a connective becomes a matter of mere input and output.

Suber summarizes the point that in logic, but you don't get something for nothing:

But to get truth-functionality we have to pay a big price. If the truth-value of a conditional compound is to be a function of the truth-values of its antecedent and consequent alone, then it will look only to the truth-values, not to the content, of the antecedent and consequent. So far this is consistent with our general desire to disregard content and represent only the logical form of statements and arguments. But if the content of the antecedent and consequent is irrelevant, then they may be utterly unrelated to one another. We have abandoned the requirement of ordinary implication that antecedent and consequent be mutually relevant or somehow connected. Truth-functionality requires the loss of relevancy (Suber)

Not all logicians are content to exchange relevance for truth-functionality. Some would rather forego truth-functionality and keep their relevance. An entire class of logics, called "relevance logics", have been developed in order to formalize conditionals whose antecedents and consequents must be semantically connected to one another, with the latter following semantically from the former. Perhaps the most ambitious work dedicated to articulating how this would work is the two-volume work The Logic of Relevance and Necessity, by Alan Ross Anderson and Nuel D. Belnap Jr.

Suber, Peter. "Paradoxes of Material Implication." Web. 1997. Retrieved from:

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