# zbMATH — the first resource for mathematics

Kripke completeness of some intermediate predicate logics with the axiom of constant domain and a variant of canonical formulas. (English) Zbl 0772.03008
The paper gives the first examples of rather general completeness results in Kripke semantics for intermediate predicate logics. For an intermediate propositional logic $$J$$, $$J_ *$$ denotes the smallest predicate extension of $$J$$. Let $$K=\neg\neg\forall x\bigl(p(x)\lor\neg p(x)\bigr)$$, $$D=\forall x\bigl(p(x)\lor q\bigr)\supset_ .\forall x p(x)\lor q$$. The subsequent two theorems generalize some earlier partial results of Ono and Suzuki. Theorem 3.7. If $$J$$ is tabular, $$J_ *+D$$ is strongly Kripke complete. Theorem 3.9. If $$J$$ is a subframe logic, then $$J_ *+D$$, $$J_ *+K+D$$ are strongly Kripke complete.
The latter result is transferred to some kinds of confinal subframe logics (this notion was introduced by M. Zakhar’yashchev [Algebra Logika 28, No. 4, 402-429 (1989; Zbl 0708.03011)]): Theorem 3.11. If $$J$$ is a confinal subframe logic determined by a class of finite frames $$\{M_ i| i\in I\}$$ such that (for any $$i$$) every subset of $$M_ i$$ having an upper bound contains a maximal element of $$M_ i$$, then $$J_ *+D+K$$ is strongly Kripke complete. (As the author observes, D. Skvortsov has recently proved that 3.11 fails for arbitrary confinal subframe logics).

##### MSC:
 03B55 Intermediate logics
Full Text:
##### References:
 [1] K. Fine,Logics containing K4 part II Journal of Symbolic Logic 50 (1985), pp. 619-651. · Zbl 0574.03008 · doi:10.2307/2274318 [2] D. M. Gabbay andD. H. de Jongh,Sequences of decidable and finitely axiomatizable intermediate logics with the disjunction property,Journal of Symbolic Logic 39 (1974), pp. 67-79. · Zbl 0289.02032 · doi:10.2307/2272344 [3] T. Hosoi,Non-separable intermediate logics,Journal of Tsuda College 8 (1976), pp. 13-18. [4] V. A. Jankov,Conjunctively indecomposable formulas in propositional calculi,Izvestiya Akademii Nauk SSSR 33 (1969), pp. 13-38. (English translation)Mathematics of USSR, Izvestiya 3 (1969), pp. 17 ?35. · Zbl 0181.00404 [5] C. G. McKay,The decidability of certain intermediate logics,Journal of Symbolic Logic 33 (1968), pp. 258-264. · Zbl 0175.27103 · doi:10.2307/2269871 [6] H. Ono,Model extension theorem and Craig’s interpolation theorem for intermediate predicate logics,Reports on Mathematical Logic 15 (1983), pp. 41-58. · Zbl 0519.03016 [7] H. Ono,some problems on intermediate predicate logics,Reports on Mathematical Logic 21 (1987), pp. 55-67. · Zbl 0676.03016 [8] V. B. Shehtman,On incomplete propositional logics,Doklady Akademii Nauk SSSR 235 (1977), pp. 542-545. (English translation)Soviet Mathematics, Doklady 18 (1977), 985 ? 989. [9] N.-Y. Suzuki,An extension of Ono’s completeness result,Zeitschrift f?r Mathematische Logik und Grundlagen der Mathematik 36 (1990), pp. 365-366. · Zbl 0691.03014 · doi:10.1002/malq.19900360410 [10] M. V. Zakharyashchev,Syntax and semantics of superintuitionistic logics,Algebra i Logika 28 (1989), pp. 402-429. (English translation)Algebra and Logics 28 (1989), pp. 262 ? 282.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.