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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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##### References:
 [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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