On reasoning expressions and the nature of deduction rules (Q1262861)

Reasoning expressions are those which express the reasoning procedure only by means of deduction rules and initial formulas (axioms or assumptions) without the help of any intermediate results in successive steps. They express the procedure systematically and concisely just like an algebraic form expressing a computing procedure. The deduction rules are mappings from formulas (premises) to a formula (conclusion). The elementary deduction rules are certain propositional connectives (but not necessarily truth functions), while the higher rules are certain binding operators (quantifiers etc.). Besides, the detachment rule is an inverse of the connective ``implication'', and is itself the kernel of the deduction method; while the suggestion rule (another inverse of ``implication'') is the kernel of induction method.