Paradoxes of the Material Conditional

We understand perfectly what the material conditional is. We define A⊃B truth-functionally as being equivalent to ¬(A&¬B), and there is nothing paradoxical in the material conditional itself. Paradoxes, however, arise when we construe our ordinary indicative conditionals as material conditionals. On this construal, the truth value of "If A, then B" (henceforth, A→B) is determined solely by the truth values of A and B. For example, A⊃B is true for every false A and for every true B. This generates the following paradoxical consequences.

Contents 1 Negated Antecedent to Conditional 2 Affirmed Consequent to Conditional 3 Other Paradoxes 3.1 A⊃¬A 3.2 (A⊃B)∨(B⊃A) 3.3 Strengthening of the Antecedent 3.4 Conditional Excluded Middle 3.5 Disjunctive Paradox of Material Conditional

Negated Antecedent to Conditional

If A→B and A⊃B are logically equivalent, the inference form of "¬A; therefore, A→B" is valid; in other words, the falsity of A is logically sufficient for the truth of A→B. However, it seems counterintuitive to say that the mere fact that it will not rain entails the truth of "If it rains, the lecture will be cancelled." Moreover, if the truth of ¬A implies the truth of A→B, we could equally infer from the mere fact that it will not rain, that "If it rains, the lecture will not be cancelled." But it seems absurd to maintain both that "If it rains, the lecture will be cancelled" and "If it rains, the lecture will not be cancelled."

Here is one possible response of the material conditional theorist of indicative conditionals. How do we test out intuitions about validity of an inference? The direct way is to imagine that we know for sure that the premise is true, and to consider what we would then think about the conclusion. Now when we know for sure that ¬A, we have no use for thoughts beginning "If A, ...." When you know for sure that Harry didn't do it, you don't go in for "If Harry did it, ...." In this circumstance conditionals have no role to play, and we have no practice in assessing them. The direct intuitive test is, therefore, silent on whether "If A, B" follows from ¬A. If our smoothest, simplest, generally satisfactory theory has the consequence that it does follow, perhaps we should learn to live with that consequence." (Dorothy Edgington, Conditionals).

Is then the material conditional account our smoothest, simplest, generally satisfactory theory of indicative conditionals? It seems controversial.

Affirmed Consequent to Conditional

Given the theory's assumption that A→B is equivalent to A⊃B, the inference form of "B; therefore, A⊃B" is valid; in other words, the truth of B is logically sufficient for the truth of A⊃B. However, it seems counterintuitive to say that the mere fact that the lecture will not be cancelled entails the truth of "If the lecturer dies on his way, the lecture will not be cancelled."

This is maybe less obviously unacceptable than Negated Antecedent to Conditional. The rationale is that if it is a fact that the lecture will not be cancelled, then it remains a fact even if the lecturer dies on his way. Christoper Gauker (2005, pp. 92-93) observes that when the example is reformulated in the past tense, there is less temptation to think it invalid. Here is his example:

In this case, Gauker holds, if it is a fact that I met you yesterday, then it remains a fact even if (per impossible) I died the night before. Moreover, we recognize a clear contrast between this inference in the past tense indicative and the following inference in the past tense subjuctive:

Cleary this is invalid. At least, the subjunctive inference strikes us as worse than the indicative one. Gauker holds that a simple explanation of this fact about how they strike us would be that the indicative version is valid and the subjunctive version is not.

Other Paradoxes

A→¬A

Indeed, we can see lots of conditional statements and inferences using conditionals whose meaning and validity are severely distorted when the involved conditionals are understood as material conditionals. For example, the statement of the form A→¬A surely seems self-contradictory, while it is consistent when the connective is a material conditional.

(A⊃B)∨(B⊃A)

Given the equivalence of A→B and A⊃B, the statement of the form A→B∨B→A is a tautology. Put another way, we could infer A→B from the mere falsity of B→A if they are understood as material conditionals. In this case, the inference seems to say that for any two statements there must be one which implies the other, which sounds unacceptable.

Strengthening of the Antecedent

If A→B and A⊃B are logically equivalent, the inference form of "A⊃B; therefore, (A&F)→B is valid. But it is usually supposed that Strengthening of the Antecedent is invalid for indicative conditionals as well as for subjunctive conditionals. Consider, for example, the following inference:

Strengthening of the Antecedent is closely related to the Affirmed Consequence to Conditional because the latter can be thought as the strengthening of a null antecedent.

Conditional Excluded Middle

Given the equivalence of A→B and A⊃B, the inference form of A→(B∨C); therefore, (A→B)∨(A→C) is valid. Against this form of argument, Lewis (1973, Counterfactuals, p. 80) gave the following counterexample, although he used subjunctive not indicative conditional:

While the fact that Bizet and Verdi are compatriots is sufficient to ensure that either Bizet is Italian or Verdi is French, that fact is sufficient to ensure neither that Bizet is Italian nor that Verdi is French.

Disjunctive Paradox of Material Conditional

Given Strengthening of the Antecedent and Conditional Excluded Middle, the inference form of P∨Q therefore, (A→P)∨(A→Q) is valid. Strengthening of the Antecedent allows the inference from P→Q to (A&P)→Q. Conditional Excluded Middle allows the inference from A→(P∨Q) to (A→P)∨(A→Q). Here is a counterexample due to Gauker (2005, p. 109).

From the fact that either you will draw a red card or you will draw a black card, it does not follow that either drawing the top card is sufficient to ensure that you draw a red card or drawing the top card is sufficient to ensure that you draw a black card.