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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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