What are the arguments of philosophers against the reasoning which justifies the horseshoe from truth-functionality?

Question

There is a reasoning in mathematical logic which is meant to prove that the horseshoe is the only logical operation which fits our notion of conditional.

The reasoning starts from the idea that the conditional is truth-functional, that there are therefore only 16 possibilities, and that we can dismiss all but the horseshoe as a candidate for expressing the logic of the conditional.

To clarify, the reasoning implies that either the conditional is truth-functional, and then the only possible model of it is the horseshoe, or the horseshoe is not the correct model and then the conditional is not truth-functional.

To clarify further, we can look at Bumble's reply to the comment I made on his answer:

If you assume a conditional is a bivalent truth function in two variables, and that it is neither gappy nor glutty, i.e. for any given values of its arguments it always has a unique truth value, then the material conditional is the only truth function that obeys modus ponens, modus tollens and does not obey affirming the consequent.

The way mathematicians conceive of truth-functionality, to say that the conditional is truth-functional implies that it is "a bivalent truth function in two variables, and that it is neither gappy nor glutty, i.e. for any given values of its arguments it always has a unique truth value".

Consequently, any proof that this implies that "the material conditional is the only truth function that obeys modus ponens" would be the same sort of proof as the one I am referring to above. The principle is the same: you find good reasons to eliminate possibilities until only the horseshoe remains.

My question is:

What are the arguments of philosophers against this reasoning?

Thank you for providing scholarly references.

Answer 1

There are only a few people who continue to try to argue that the material conditional is the only correct way to represent conditionals. The material conditional is useful in certain contexts, including within mathematics, but most conditionals that you come across in ordinary language do not fit the mould. The material conditional is useful in cases where you want to treat "if A then B" as a bivalent truth-function in two variables whose logic is neither gappy nor glutty. Many other formal accounts of conditionals have been proposed, but none have achieved any general consensus.

Here are some of the main reasons why the material conditional does not work as a general account of conditionals.

1. A ⊃ B is entailed by ¬A, but it is not always the case that "if A then B" follows from A being false. Frank Jackson has argued that you can get round this by appeal to the pragmatics of how conditionals are used, but IMHO this doesn't work and even if it did, it wouldn't explain away all the other differences.

2. ¬(A ⊃ B) entails A and ¬B, but "if A then B" being false does not in general entail that A is true and B is false. In typical cases, the truth of "if A then B" is independent of the truth of A.

3. (A ⊃ B) ∨ (B ⊃ A) is a tautology, but it is not the case in general that for any pair of propositions A, B one of "if A then B" or "if B then A" is true.

4. When a conditional "if A then B" is uncertain, it is better represented using a statement of conditional probability, i.e. that P(B|A) is equal to some value or obeys some inequality. The material conditional systematically gives the wrong value for the uncertainty of a conditional. The probability P(A ⊃ B) is equal to P(¬A ∨ B) and this only agrees with P(B|A) in the degenerate case when the conditional is either certainly true or certainly false. So, the material conditional does not cope with uncertain conditionals.

5. The material conditional does not sit correctly with the existential quantifier. "There is something such that if it is F then it is G" is not correctly represented as (∃x)(Fx ⊃ Gx). The latter would be true if there existed anything that is not F. There is a whole bunch of counterexamples along these lines.

6. The material conditional does not work with quantifiers other than the universal quantifier. You can correctly express "all Fs are Gs" as (∀x)(Fx ⊃ Gx). But in English we have many quantifiers, e.g. a lot, most, many, some, a few, hardly any, etc. If you try to use these with the material conditional you get the wrong result. For example, you can gloss, "most birds fly" as "for most things x, if x is a bird then x flies". But expressing this if/then as a material conditional gives the wrong truth conditions. "Most penguins fly" would come out true, because most things are not penguins.

7. In a similar vein, the material conditional does not sit correctly with adverbs of quantification. A sentence such as "usually, if a person buys a donkey they pay cash for it" does not give the correct truth conditions if formalised using a material conditional.

8. Conditionals are often used to express a dispositional property or a causal relation. These are not truth functional and so cannot be represented using the material conditional.

9. More generally, many conditionals have a modal force. Sometimes these can be expressed using strict implication, which is a material conditional within the scope of a modal box operator, but not always. And even when they can, strict implication still has different truth conditions from those of a material conditional.

10. Some conditionals are counterfactual, which is to say they have antecedents that are known to be false, but without being trivially true. Most treatments of conditionals consider these to be quite different from indicative conditionals, though the differences are exaggerated.

11. A ⊃ B entails (A ∧ D) ⊃ B for any arbitrary D, which is to say that material conditionals obey strengthening of the antecedent. In practice, many conditionals are nonmonotonic and do not have this property.

12. Conditionals are not just used to form conditional statements and can be used to conditionalise many speech acts. We have conditional questions, commands, offers, obligations, threats, promises, bets, etc. These conditionals are typically quite different from material conditionals are not truth functional.

There is a huge literature on conditionals and most of what we know about them has been learned in the last 50 to 60 years or so. A good start is to look at some of the articles in the Stanford Encyclopedia, particularly on Indicative Condiitonals, Counterfactuals Conditionals, and The Logic of Conditionals.

Some other references are:

• Jonathan Bennett, Conditionals: A Philosophical Guide (2003).
• David Sanford, If P then Q (2003).
• Ernest Adams, The Logic of Conditionals (1975).
• David Lewis, Counterfactuals (1973).
• Robert Stalnaker, several papers collected in Knowledge and Conditionals (2019).
• Dorothy Edgington, “On Conditionals”, Mind, Vol. 104, pp. 235–329, (1995).
• Angelika Kratzer, Modals and Conditionals (2012).
• Nicholas Rescher, Conditionals (2007).
• Michael Woods, Conditionals (1997).
• Igor Douven, The Epistemology of Indicative Conditionals (2015).