Is that correct to say that semantic consequence is equivalent to the concept of sound argument in classical propositional logic? If it is the case, arguments or theories with contradictory premises are of no interest of study. If it is not the case, what is the rationale to lead with arguments and theories with contradictory premises if everything can be proved either true or false?
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A valid argument corresponds to semantical cons. A sound arg is a valid one with true premises.– Mauro ALLEGRANZACommented Apr 7, 2022 at 15:32
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Thus, an argument with inconsistent premises and a conclusion whatever is classically valid. See Ex Falso.– Mauro ALLEGRANZACommented Apr 7, 2022 at 15:43
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Any argument with a contradiction included will turn out to be valid under Mathematical Logic. Propositional logic falls under Mathematical logic. Sound arguments are not the concern in Mathematical logic---Validity is. Soundness means that all premises must be true in the real world & if so then the conclusion must necessarily follow to be true also. That is, the conclusion is impossible to be false while the premises are true. In this way all SOUND arguments must be valid. There is no such thing as a sound but invalid argument. Semantics indicates that is how a certain language works.– LogikalCommented Apr 7, 2022 at 16:03
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Everything cannot be proven true or false. You have some concepts wrong. If you are thinking PROOF must be available for something to be true you are mistaken. Many things are true without human beings being aware of them. For example there are other galaxies many light years away from us and likely have existed before our galaxy. There is no proof though because we haven't found all galaxies. Because we have no proof doesn't mean much of anything. We would be committing a fallacy to say "x must be true because we have no proof otherwise." Don't get too caught up with requiring PROOF.– LogikalCommented Apr 7, 2022 at 16:10
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Thanks for your comments. When I say proof, I'm referring to some deductive method for propositional classical logic. But, of course, I don’t take it as being the single method of reasoning.– Cesar TaclaCommented Apr 11, 2022 at 11:15
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In the way the terms are standardly used, logical or semantic consequence, validity, and entailment are interdefinable. An argument is (deductively) valid if and only if its conclusion is the consequence of its premises and if and only if the premises entail the conclusion. An argument is sound if it is valid and has true premises. So semantic consequence corresponds to validity, not soundness.
As you say, in classical propositional logic an argument with contradictory premises is of no interest, since it entails everything and can never be sound. There are non-classical logics that do not have this property: the term for these is paraconsistent.
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This refers to the principle of explosion ("from falsehood, anything follows") as pointed out above by Mauro Allegranza. And paraconsistency is the way to lead with contradictory propositions and yet do sound reasoning. Commented Apr 11, 2022 at 11:26