Pseudo-Canonical Formulae are Classical

An original result about Hilbert Positive Propositional Calculus introduced in [11] is proven. That is, it is shown that the pseudo-canonical formulae of that calculus (and hence also the canonical ones, see [17]) are a subset of the classical tautologies.My work has been partially supported by EPSRC grant EP/J007498/1 and an LMS Computer Science Small GrantCaminati Marco B. - School of Computer Science University of Birmingham Birmingham, B15 2TT United KingdomKorniłowicz Artur - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155–167, 2011. doi:10.2478/v10037-011-0025-2.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1): 69–72, 1999.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133–137, 1999.Andrzej Trybulec. The canonical formulae. Formalized Mathematics, 9(3):441–447, 2001.Andrzej Trybulec. Classes of independent partitions. Formalized Mathematics, 9(3): 623–625, 2001.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990