On Jaśkowski-type semantics for the intuitionistic propositional logic

One of remarkable results of Jaskowski is the construction of a se? quence of finite matrices adequate for the intuitionistic propositional logic (INT). This result was published in 1936 in the paper [2] where only a very condensed sketch of proof is to be found. It was 17 years before a more detailed proof was published in [4] by Bose who worked out some modification of the strategy suggested by Jaskowski for eluding the lemma which the was unable to prove. A detailed proof following closely the Jaskowski's strategy is presented in [7]. The sequence of finite matrices adequate for INT was obtained by Jaskowski as a result of alternate application of the operation of direct power and s.c. V ? operation to the two-element Boolean algebra. This T ? operation of Jaskowski can be considered as a special case of the sum operation for pseudo-Boolean algebras introduced later by Troelstra [8]. Suppose we are given the pseudo-Boolean algebras 31 and 93 with the universes A, B such that {!<%} = {Ojb} = Ar\B. Then the sum 3??23 is the pseudo-Boolean algebra with the universe AuB and the lattice-ordering <5I@S = <^u <su (J. x B). The result of Jaskowski's T-operation performed on the pseudo-Boolean algebra 31 is isomorphic to 31 ?A where R is the two-element Boolean algebra. Thus, the diagram of the lattice-ordering of T(3I) can be obtained from that of 31 by adding the new greatest element. The sequence {^n\n = 1,2,...} constructed in [2] is given by the conditions: 3i = ?, 3n+i = r((3?)w). Denoting the content of 3n by E(%n) and the set {1,2, ...} of positive intigers by N one can express the main result of Ja?kowski as follows: