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Conference Paper (international conference)

Lindenbaum and Pair Extension Lemma in Infinitary Logics

Bílková Marta, Cintula Petr, Lávička Tomáš

: Logic, Language, Information and Computation, p. 130-144 , Eds: Moss L. S., de Queiroz R., Martinez M.

: WoLLIC 2018. International Workshop on Logic, Language, Information and Computation /25./, (Bogotá, CO, 20180724)

: GA17-04630S, GA ČR, GC16-07954J, GA ČR, JSPS-16-08, AV ČR

: Lindenbaum lemma, Pair extension lemma, Infinitary logic, Infinitary deduction rule, Strong disjunction, Prime theory

: 10.1007/978-3-662-57669-4_7

(eng): The abstract Lindenbaum lemma is a crucial result in algebraic logic saying that the prime theories form a basis of the closure systems of all theories of an arbitrary given logic. Its usual formulation is however limited to finitary logics, i.e., logics with Hilbert-style axiomatization using finitary rules only. In this contribution, we extend its scope to all logics with a countable axiomatization and a well-behaved disjunction connective. We also relate Lindenbaum lemma to the Pair extension lemma, other well-known result with many applications mainly in the theory of non-classical modal logics. While a restricted form of this lemma (to pairs with finite right-hand side) is, in our context, equivalent to Lindenbaum lemma, we show a perhaps surprising result that in full strength it holds for finitary logics only. Finally we provide examples demonstrating both limitations and applications of our results.

: BA

: 10201