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Implication systems for many-dimensional logics

Pynko A.

Reports on Mathematical Logic

1999

No. 33

11-27

The main result of the present paper is equivalence of the following conditions, for any k-dimensional logic L: (i) L has a full-replacement implication system, i.e., a finite set of k-dimensional formulas with 2k variables that in a natural way adopts the Identity axiom and and the Modus Ponens rule for the ordinary implication connective; (ii) L has an unary-replacement implication system, i.e., a finite set of k-dimensional formulas with k+1 variables that in a different way adopts the Identity axiom and the Modus Ponens rule for the ordinary implication connective; (iii) L has a parametrized local deduction theorem; (iv) L has the syntactic correspondence property that is essentially the restriction of the filter correspondence property to deductive L-filters over the formula algebra alone; (v) L is protoalgebraic in the sense that the Leibniz operator is monotonic on the set of deductive l-filters over every algebra; (vi) L has a system of equivalence formulas with parameters that defines the Leibniz operator on deductive L-filters over every algebra. We also present a family of specific examples which collectively show that the above mataequivalence doesn't remain true when in (i) "2k" (resp., in (ii) "k+1") is replaced by "2k-1" (resp., by "k"). This, in particular, disproves the statement of [4], Theorem 13.2.