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Admissible rules for logics containing S4.3. (English. Russian original) Zbl 0582.03009
Sib. Math. J. 25, 795-798 (1984); translation from Sib. Mat. Zh. 25, No. 5(147), 141-145 (1984).
The aim of this article is the investigation of rules of inference in logics containing S4.3 modal propositional logic. We use an algebraic approach to admissibility: the rule A/B is admissible in logic \(\lambda\) iff the quasi-identity \(A=1\Rightarrow B=1\) is true on the free algebra \(F_{\omega}(\lambda)\) of denumerable rank from the variety of modal algebras corresponding to \(\lambda\).
Theorem 5. The free algebra \(F_{\omega}(\lambda)\) has a finite basis of quasi-identities which is constructed by adding \(\diamond x\wedge \diamond \neg x=1\Rightarrow y=1\) to basis identities of \(F_{\omega}(\lambda)\) when \(\lambda\) \(\supseteq S4.3.\)
From this theorem we obtain that there exists an algorithm for recognizing admissibility rules in a modal logic \(\lambda\) by its axiom set when \(\lambda\) \(\supseteq S4.3.\)
Proposition 7. Let f/g be an admissible but not derivable rule in a logic \(\lambda\) \(\supseteq S4.3\). Then for every \(t_ i:\) \(f(t_ i)\not\in \lambda\). That is, each admissible, non derivable rule in \(\lambda\) \(\supseteq S4.3\) has no antecedent feasible in \(\lambda\).
Proposition 8. For all \(\lambda\) \(\supseteq S4.3\), the logic \(\lambda\) with additional rule of inference \(\diamond x\wedge \diamond \neg x/y\) is structurally complete.
In particular, when \(\lambda =S5\), these propositions give an affirmative solution to the questions of J. Porte [J. Philos. Logic 10, 409-422 (1981; Zbl 0475.03005)].

MSC:
03B45 Modal logic (including the logic of norms)
03G25 Other algebras related to logic
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[1] J. Porte, ?The deducibilities of S5,? J. Philos. Logic,10, 409-422 (1981). · Zbl 0475.03005 · doi:10.1007/BF00248735
[2] V. V. Rybakov, ?Admissible rules of pretable modal logics,? Algebra Logika,20, No. 4, 440-464 (1981). · Zbl 0489.03005
[3] L. L. Maksimova, ?Pretable extensions of the Lewis logic S4,? Algebra Logika,14, No. 1, 28-55 (1975).
[4] K. Fine, ?The logics containing S4.3,? Z. Math. Logic Grundl. Math.,17, 371-376 (1971). · Zbl 0228.02011 · doi:10.1002/malq.19710170141
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