Equivalence Thesis

Question

What is, if any, the canonical justification accepted in mathematical logic for the Equivalence Thesis, asserting (1) that indicative conditionals are truth-functional logical expressions and (2) that their truth conditions are exactly those of the Material Implication, as defined by its truth table?

Answer 1

It is misleading to speak of an "equivalence thesis". Mathematicians typically use material implication (MI) as a conditional because it is useful to do so. In simple contexts it obeys the implicational rules that we expect a conditional to obey. It is possible to show that if you require a truth-functional dyadic connective for bivalent classical logic, then MI is the only such connective that obeys modus ponens and the rule of conditional proof. All of which is to say that if you assume classical, bivalent logic and you require a truth-functional conditional then MI is the appropriate connective to use. Mathematicians typically do use classical logic most of the time, and truth-functional connectives make life nice and simple, and hence the common use of MI in mathematics.

However, attempting to argue that all conditionals, or even all indicative conditionals, are MIs is tricky because it is obviously false. I have given some of the reasons why there are non-truth-functional conditionals in the answer to this question.

However, if you look at those examples, they are concerned with situations that mathematicians are not interested in. In the real world we sometimes use conditionals to express causal relationships, but there are no causal relationships in mathematics. In the real world we sometimes use conditionals to express dispositional properties, but mathematical objects do not have dispositional properties. In the real world, pretty much everything is uncertain, including conditionals, so we need conditionals that handle uncertainty, which MI does not. Mathematicians, for the most part, have the luxury of ignoring uncertainty: you prove P or you prove not-P, so there's no need to be uncertain about P. In the real world we conditionalise speech acts other than propositions - sentences that lack truth values - but mathematicians are not interested in such things.

The upshot is that MI is useful in mathematics just because of the limited interests of mathematicians. It is a special case of a conditional, but a special case that happens to be useful in the contexts that are relevant to mathematics.