Abstract

Let L = be an absolutely free algebra generated by the nite set of generators fp1; p2; : : :g , where Con is a nite sequence of operations denoted by sentential con- nectives. Lk = is the subalgebra of L generated by variables p1; : : : ; pk. The rules are dened in the usual man- ner. Sb is the consequence determined by the rule of substitution. If C is a consequence operation of L, R a set of rules, M a generalized matrix , then C, CR, CnR, CnM, M denote respectively: C = Sb, the consequence obtained by adding the set of rules R, to the rules of C, the consequence determined by the rules of R, the consequence determined by the matrix M, the relation such that for all formulas a; b, a M b if and only if for every valuation in M va = vb. A substitution e : L ! Lk is called a k-substitution. We often write M instead of CnM. Some of the notions not dened in this paper can be found in [7,6]