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.