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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?

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  • $\begingroup$ With some more context, like what logic you are studying, we might be able to provide some more direct answers. $\endgroup$ – DanielV Jun 28 at 0:11
  • $\begingroup$ See Soundness of a proof system : "In an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). If the system allows Hilbert-style deduction, it requires only verifying the validity of the axioms and one rule of inference, namely modus ponens." $\endgroup$ – Mauro ALLEGRANZA Jun 28 at 12:47
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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.

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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."

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  • $\begingroup$ This meaning of the word "sound" is completely different from its meaning in mathematical logic. The last paragraph of Henning's answer indicates that philosophers may use the word "sound" in this way. But I think this passage on Wikipedia is potentially misleading. $\endgroup$ – Alex Kruckman Jun 28 at 14:05

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