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?


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.

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  • I agree with the general idea, but it is not that mathematicians are not interested in those other things, they just have different ways of going about it than packing complications into the base language. Statistics deals with uncertainty, mathematical models in science incorporate causality, there are formalizations of natural languages and modalities, etc. – Conifold Jun 24 at 5:02
  • I agree, but you still need conditionals other than MI to express more complex relationships. For example, conditional probability expresses the degree of uncertainty of a conditional, but it is not reducible to MI. It is neither A → P(B) nor P(A → B). Conditional obligations are not expressible using only MI and a unary obligation operator. Also, there are non-classical logics and they have their own conditionals: intuitionistic implication is not a truth-function, nor is relevant implication, etc. – Bumble Jun 24 at 5:55
  • Logical implication is a notion that's very important to mathematicians and which may be expressed verbally in terms of conditionals, but it isn't the same as the material conditional since the material conditional can be used to express relationships which just happen to be true in one domain of discourse but aren't necessarily true in all possible worlds. – Hypnosifl Jun 24 at 6:23
  • But most of this is expressible in ZFC or its segment, which is the base language, and reasoning there uses plain MI. Everything else is emulated at a higher level, including even non-classical logics oftentimes. – Conifold Jun 24 at 7:32

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