The logic of proofs LP was introduced in  and thoroughly studied in . LP\ud is a natural extension of the propositional calculus in the language representing\ud proofs as formal objects. Proof expressing terms are constructed using constants,\ud variables, and signs of natural operations on derivations. Then formula\ud t:F has the intended interpretation "t is a proof of F". LP is complete with\ud respect to the interpretation of t:F in the Peano arithmetic by an arithmetical\ud formula "t is a PA-derivation of F"
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