Notre Dame Journal of Formal Logic

Modal Consequence Relations Extending S4.3: An Application of Projective Unification

Wojciech Dzik and Piotr Wojtylak

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We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 (the lattice of quasivarieties of S4.3-algebras) is countable and distributive and it forms a Heyting algebra.

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Notre Dame J. Formal Logic, Volume 57, Number 4 (2016), 523-549.

Received: 30 November 2012
Accepted: 11 November 2013
First available in Project Euclid: 19 July 2016

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Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45} 08C15: Quasivarieties 68T15: Theorem proving (deduction, resolution, etc.) [See also 03B35]
Secondary: 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45] 03B35: Mechanization of proofs and logical operations [See also 68T15]

admissible rules consequence relations projective unification $\mathbf{S4.3}$ structural completeness quasivarieties


Dzik, Wojciech; Wojtylak, Piotr. Modal Consequence Relations Extending $\mathbf{S4.3}$ : An Application of Projective Unification. Notre Dame J. Formal Logic 57 (2016), no. 4, 523--549. doi:10.1215/00294527-3636512.

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