# Can a valid argument be said to be unsound if the set of premises is unsatisfiable (inconsistent)?

I'm asking in a strict propositional logic sense. Suppose that I have a set of premises that is logically unsatisfiable (or inconsistent), i.e. they can not be all True simultaneously, that argument must be valid since there are no cases where all premises are True and the conclusion is False.

But then can it automatically follow that the argument is unsound, because soundness is defined as validity + trueness of all premises and the given premises can not be all True.

I'm confused about this since normally when we examine the soundness of an argument, we need to know the particular truth of the premises in the world.

• Yes; see Validity and Soundness. Commented Sep 18, 2023 at 7:06
• An argument, valid or otherwise, is unsound even when some of its premises happen to be false, they do not need to be unsatisfiable. By the way, such an argument is not necessarily valid because it is unclear what "unsatisfiable" means. If they are "unsatisfiable" merely for some physical reasons it may yet be invalid because logic does not care about physics. It is only if they are logically "unsatisfiable" (inconsistent) that it must be valid. Commented Sep 18, 2023 at 7:11
• @Conifold Yes, I'm talking about inconsistent, the text I'm using refers to such a set of propositions as unsatisfiable. Commented Sep 18, 2023 at 7:43
• You have it right. An argument with inconsistent premises is valid, since there is no way for the premises to be true and the conclusion false. Such an argument can never be sound, since there is no way for all the premises to be true, and a sound argument is a valid argument with true premises. All of this relates to classical logic, and some others. It does not apply to paraconsistent logics. Commented Sep 18, 2023 at 10:16
• I was taught anything can be proven true from a false premise. I believe this includes unsatifiable or inconsistent premises. Commented Sep 19, 2023 at 4:26

I'm asking in a strict propositional logic sense.

Ok, I would offer an answer if the question concerned logic, but "propositional logic" is not logic.

Mauro Allegranza's comment:

"Propositional logic is the study of the meanings of, and the inferential relationships that hold among, sentences based on the role that a specific class of logical operators called the propositional connectives have in determining those sentences' truth or assertability conditions. As early as Aristotle it was observed that propositional connectives have a logical significance, and over many centuries piecemeal observations about some of their properties were made."

First, there is just one logic, the logic of the human mind, so there aren't different sorts of logics, one propositional, the other not. So, the the qualification "propositional" just disqualifies proposition logic as logic.

Second, logic is a cognitive capacity, not an academic discipline, nor a formal system etc. This also disqualifies propositional logic.

Third, we already have an expression, i.e. "formal logic", to refer to the academic discipline where logicians try to understand logic and produce a formal model of it. So, we don't need a new name.

Fourth, the horseshoe is the only model of the implication which was ever tested by the people who developed propositional logic, so propositional logic is not the result of any scientific investigation. It is just a mathematical theory which has no proven relation to human logic, and can be, and has been, comprehensively, falsified as a potential model of the implication.

The inferential relationships... in the plural?! There is just one inferential relationship, namely, the implication.

Further, Bertrand Russell already decided, in 1906 I think, what was the model of the implication to be used in propositional logic. In fact he just redefined the notion of implication! This is not science, this is make-believe. So, the study, if there ever was one, all but stopped at that point.

The one logical relationship that hold between any two sentences cannot be determined by mathematicians. You need to start from natural languages and from actual occurrences thereof, and you need to take into account the pragmatic of verbal interactions, in their actual context. This is something mathematicians have never been interested doing. Bertrand Russell never really tried. Frege didn't try. Both Russel and Frege just dismissed natural language as illogical! These people never had a chance.

Aristotle's syllogistic shows that he understood that natural language can be used to produce logical statements. Propositional logic is a regression.

[propositional logic] Obviously IS logic.

Just as much as the horseshoe is the implication.

That is, not at all.

There is no decree in science. We have to study nature to find out how it works, and academics have never even tried to do that in the case of logic. One model, and they decreed it was the implication, even though it has been comprehensively falsified! This is not science, this is make-believe.

• See Propositional Logic: "Propositional logic is the study of the meanings of, and the inferential relationships that hold among, sentences based on the role that a specific class of logical operators called the propositional connectives have in determining those sentences’ truth or assertability conditions. As early as Aristotle it was observed that propositional connectives have a logical significance, and over many centuries piecemeal observations about some of their properties were made. " Obviously IS logic. Commented Sep 21, 2023 at 13:11