Reference constructions in the single-conclusion proof logic
We propose an extension of the propositional proof logic language by the second-order variables denoting the reference constructors (like 'the formula which is proven by x'). The proof logic in this weak second-order language is axiomatized via the calculus FLPref, the (Functional) Logic of Proofs with References. It is supplied with the formal arithmetical semantics: we prove that FLPref is sound with respect to arithmetical interpretations and is a conservative extension of propositional single-conclusion proof logic FLP. Finally, we demonstrate how the language of FLPref can be used as a scheme language for arithmetic and provide the corresponding proof conversion method.
reference proof theory explicit modal logic single conclusion logic of proofs proof term unification