A b s t r a c t.
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 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 parameterized 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 metaequivalence 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 , Theorem 13.2.