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Reports on Mathematical Logic

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No. 33

Alexej P. PYNKO

IMPLICATION SYSTEMS FOR MANY-DIMENSIONAL LOGICS
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 [4], Theorem 13.2.

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