The soundness of modus ponens (?)

Question

I've been told that a proof can be provided to the effect that modus ponens is a sound logical principle. I know that modus ponens is a valid principle in classical logical but I can't see what it could mean for a logical principle to be sound. As a result, I can't see what a proof of modus ponens' soundness may look like. Can you help me?

Answer 1

Most likely it means the same as you're use to calling "valid".

The word "sound" is most often applied to entire logics:

• A logic is "unsound" (with respect to a particular semantics) iff there exists statements $A, B_1, B_2, \ldots, B_n$ such that the logic can derive $A$ as a consequence of the $B_i$s, and yet the semantics allows a situation where all the $B_i$s hold but $A$ doesn't.

• A logic is "sound" if it is not unsound.

The most likely interpretation of "modus ponens is sound" would be that if we take a logic where the only way to conclude anything is by modus ponens (that is, no axioms, no other rules of inference, no nothing), then that logic is sound.

And that is basically the same as what is also commonly expressed as "modus ponens is a valid inference rule".

---

In philosophy (and the early pages of some textbooks of mathematical logic) one can also call an argument "sound" if it is a valid deduction from premises that happen to be true. But there doesn't seem to be any reasonable way to extend that meaning to apply to a rule of inference in isolation.

Answer 2

According to Wikipedia " For modus ponens to be a sound argument, the premises must be true for any true instances of the conclusion. An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound."

Answer 3

I agree, it is hard to proof universal correctness, validity or soundness of a principle rigorously, mathematically due to lack of principle definition.

Yet frequent usage of Model Ponens in mathematics, and more recently, in AI, proves (albeit, in colloquial sense) principle relevance and utility.

Unlike Philosophy, in Formal Logic and AI proof systems Modus Ponens is most typically just a proof or inference rule, rather than some law or principle. Thus, soundness of Modus Ponens, can be defined as possibility to add Modus Ponens rule to an existing logic resulting in another.

I guess this is what hmakholm suggested (though do not take my word for granted, I am not logician).

So is Modus Ponens rule sound/valid universally ? Or at the least with well-known logics?

The answer is no. As you guessed, Modus Ponens fails in some well-respected non-classical logics https://math.stackexchange.com/a/2705757/466899 .

Perhaps, it is still possible to somehow adapt Modus Ponens to said logics.

Are those logic somehow deficient compared to the classical logic and should be dismissed? Quite possible, yet seems like a matter of taste. So, what concerns of Modus Ponens as a universal principle the question should be re-directed back to philosophers and hear out their arguments.

Well, opinion differ there. And some like Carol Lewis or Gilbert Harman would say that Modus Ponens Principle is utter nonsense, other refer to McGee conditional counterexamples https://math.stackexchange.com/a/143093/466899, others would mention contradictions to deontic and decision logic. I bet the most would say Modus Ponens is universally valid and great principle, yet should be careful where and how to use it.

I'd say the truth is out there.