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Admissible and derivable rules in intuitionistic logic. (English) Zbl 0797.03001
The paper considers admissible and derivable inference rules in intuitionistic propositional logic. A finite set $$\Gamma$$ of formulas is said to have the same admissible and derivable consequences if, for every formula $$C$$, intuitionistic logic is closed under the rule $$\Gamma/C$$ iff $$\Gamma\lvdash C$$. $$\Gamma$$ has the disjunction property for admissibility of admissibility of the rule $$\Gamma/C\lor D$$ implies $$\Gamma\lvdash C$$ or $$\Gamma\lvdash D$$. It is proved that Harrop formulas and anti-Harrop formulas (which are equivalent to conjunctions of formulas $$p\to A$$, where $$p$$ is a propositional variable) have the same admissible and derivable consequences and the disjunction property for admissibility.

##### MSC:
 03B20 Subsystems of classical logic (including intuitionistic logic)
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##### References:
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