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 ${\mathrm {FLP}_{ref}}$, the (Functional) Logic of Proofs with References. It is supplied with the formal arithmetical semantics: we prove that ${\mathrm {FLP}_{ref}}$ is sound with respect to arithmetical interpretations and is a conservative extension of propositional single-conclusion proof logic ${\mathrm {FLP}}$. Finally, we demonstrate how the language of ${\mathrm {FLP}_{ref}}$ can be used as a scheme language for arithmetic and provide the corresponding proof conversion method.