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.
