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Hereditarily structurally complete modal logics. (English) Zbl 0836.03014
An inference rule is an expression $$r$$ of the form ${A_1 (x_1, \dots, x_n), \dots, A_m (x_1, \dots, x_n) \over B(x_1, \dots, x_n)},$ where $$A_j (x_1, \dots, x_n)$$, $$B(x_1, \dots, x_n)$$ are formulas built up from the variables $$x_1, \dots, x_n$$.
A rule $$r$$ is called admissible in a logic $$\lambda$$ iff for every collection of formulas $$C_1, \dots, C_n$$, if $$\forall j\;A_j (C_1, \dots, C_n) \in \lambda$$ then $$B(C_1, \dots, C_n) \in \lambda$$. A rule $$r$$ is called derivable in a logic $$\lambda$$ if $$B$$ is derivable from $$A_1, \dots, A_m$$ by using theorems and postulated inference rules of $$\lambda$$. A logic $$\lambda$$ is called structurally complete iff every admissible $$\lambda$$ rule is derivable in $$\lambda$$. A logic $$\lambda$$ is called hereditarily structurally complete if any logic extending $$\lambda$$ is structurally complete.
The main result of the paper is a necessary and sufficient condition for modal logics over $$K4$$ to be hereditarily structurally complete: a modal logic $$\lambda$$ is hereditarily structurally complete iff $$\lambda$$ is not included in any logic from a list of twenty special tabular logics.

##### MSC:
 03B45 Modal logic (including the logic of norms)
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##### References:
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