О прикладных теориях с суперинтуиционистскими логиками.

Abstract

There is a continuum of so called «superintuitionistic logic». Usual they are studied from a purely logical point of view. But many logically different calculi coincide when used in applied theories. Some results about «applied classification of logical calculi» are presented here. Arithmetic are considered here as semi-formal system with co-rule. Arithmetic with finitevalued superintuitionistic logics are classical. Arithmetic with a logic of linear chains in classical. Minimal arithmetic-based theories are definitionally equivalent to intuitionistic.

$\neg\neg\exists x A(x)\Rightarrow\exists x\neg\neg A$ implies classical logic in any theory with two different elements.

$\forall x (\neg A(x)\Rightarrow\exists yB(x,y))\Rightarrow\forall x\exists y(\neg A(x)\Rightarrow B(x,y))$ implies $(\neg A\Rightarrow B\vee C)\Rightarrow(\neg A\Rightarrow B)\vee(\neg A\Rightarrow C)$ in any theory with two different elements.