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On the admissible rules of intuitionistic propositional logic. (English) Zbl 0986.03013
For $$A$$, $$B$$ propositional formulas, $$A|\mskip-6mu\sim B$$ means that for each substitution $$\sigma$$, $$\sigma(A)$$ intuitionistically valid implies $$\sigma(B)$$ intuitionistically valid. The author introduces the expression $$A\vartriangleright B$$ for propositional formulas $$A$$, $$B$$, and defines the following system of derivations for such expressions:
Axiom schemes:
$$(A\to r\vee s)\vee t\vartriangleright(A\to r)\vee (A\to s)\vee (A\to p_1)\vee\cdots\vee$$
$$\vee(A\to p_n)\vee t$$ for $$A= \bigwedge^n_{i=1} (p_i\to q_i)$$,
$$A\vartriangleright B\quad$$ where $$(A\to B)$$ is intuitionistically vaild.
Rules:
$${C\vartriangleright A, C\vartriangleright B\over C\vartriangleright A\wedge B},\quad {A\vartriangleright B, B\vartriangleright C\over A\vartriangleright C}$$.
A Kripke model is called AR-model, if for every finite set $$\{u_1,\dots, u_n\}$$ of nodes, there is a node $$u$$ such that $u\preceq u_1,\dots, u_n\wedge\forall u'\succ u\;(u_i\preceq u'\text{ for some }i\in \{1,\dots, n\}).$ It is shown that the following are equivalent:
a) $$A|\mskip-6mu\sim B$$.
b) $$A\vartriangleright B$$ is derivable.
c) In every AR-model, $$A$$ valid implies $$B$$ valid.
Reviewer: A.Tauts (Tallinn)

##### MSC:
 03B20 Subsystems of classical logic (including intuitionistic logic)
##### Keywords:
intuitionistic logic; admissible rules; Kripke models
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##### References:
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