I am not $100\%$ clear as to what constitutes the difference between a rule of inference and the material conditional, at least in classical logic. I am using the truth-functional definition of the material conditional, commonly visualised through its truth table, but I'm not entirely sure what the formal definition of a rule of inference is. The wikipedia article defines it to be a particular kind of logical form, which seems to be a term from philosophical logic that I'm not familiar with, but reading that article didn't really answer my question. It pertains more to the mathematical side of things, and I am specifically interested in the interplay between the concepts on the syntactic and semantic level. As far as I can tell, any rule of inference can be 'captured' by a corresponding material conditional: if we take modus ponens as a well-known example, what is the difference between $$(a\land (a\to b))\to b$$ and $${a\to b,\text{ } a \over b}?$$ On a functional level, both statements seem to be expressing the same thing. What determines the need to use two separate terms and notations, and what, if anything, separates them?
Formal relationship between rules of inference and the material conditional
$\begingroup$
logic
soft-question
formal-systems