Some Notes on Proofs with Alpha Graphs

It is well-known that Peirce’s Alpha graphs correspond to propositional logic (PL). Nonetheless, Peirce’s calculus for Alpha graphs differs to a large extent to the common calculi for PL. In this paper, some aspects of Peirce’s calculus are exploited. First of all, it is shown that the erasure-rule of Peirce’s calculus, which is the only rule which does not enjoy the finite choice property, is admissible. Then it is shown that this calculus is faster than the common cut-free calculi for propositional logic by providing formal derivations with polynomial lengths of Statman’s formulas. Finally a natural generalization of Peirce’s calculus (including the erasure-rule) is provided such that we can find proofs linear in the number of propositional variables used in the formular, depending on the number of propositional variables in the formula.