Some Locally Tabular Logics with Contraction and Mingle

Ai-ni Hsieh


Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. An algebra in IP is called semiconic if it decomposes subdirectly (in IP) into algebras where the identity element t is order-comparable with all other elements. The semiconic algebras in IP are locally finite. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x ≈ (x → t) → x. It follows that if an axiomatic extension of RMO has ((p → t) → p) → p among its theorems then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized.

Słowa kluczowe: Residuation, mingle, semiconic, locally tabular, quasivariety. Mathematics Subject Classification (2000): 03B47, 03G25, 06D99, 06F05, 08C15
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