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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?

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