Unified correspondence as a proof-theoretic tool

Abstract

The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap.These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into 'analytic' structural rules of display calculi. In this context, a rule is 'analytic' if adding it to a display calculus preserves Belnap's cut-elimination theorem.In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of nonclassical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvisttype classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices).We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present paper characterize the space of properly displayable DLE-logics. Keywords: Display calculi, unified correspondence, distributive lattice expansions, properly displayable logics. Math. Subject Class. 03B45, 06D50, 06D10, 03G10, 06E15. , and mu-calculus [11,12]. The breadth of this work has stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [40,15], or of the phenomenon of pseudocorrespondence [19]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [27]. Finally, the insights of unified correspondence theory have made it possible to determine the extent to which the Sahlqvist theory of classes of normal DLEs can be reduced to the Sahlqvist theory of normal Boolean expansions, by means of Gödel-type translations [20]. These and other results have given rise to a theory called unified correspondence [13]. Contents Tools of unified correspondence theory.The most important technical tools in unified correspondence are: (a) a very general syntactic definition of the class of Sahlqvist formulas, which applies uniformly to each logical signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives; (b) the algorithm ALBA, which effectively computes first-order correspondents of input term-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which, like the Sahlqvist class, can be defi...

From Display to Labelled Proofs for Tense Logics

“…A general solution which establishes the conservativity of display calculi for tense logics over their modal fragments, by making use of algebraic semantics, has been presented in [17]. Our work here obtains this result syntactically in the context of tense logics with modal general path axioms by exploiting the translations developed in the previous sections (i.e., corollary 4.5): As shown in [15], this result can also by proved syntactically for the more restricted set of modal path axioms by leveraging the separation property.…”

Logics for Rough Concept Analysis

“…Definition 6. An abstract Kent algebra (aKa) is a structure A = (L, s , s , ℓ , ℓ ) such that L is a lattice, and s , s , ℓ , ℓ are unary operations on L validating (11), (12) and (13). We let KA denote the class of abstract Kent algebras.…”

“…The inequality corresponding to the lower variant of IA3, which was analytic in the presence of distributivity, is not analytic inductive in the absence of distributivity (cf. [12,Definition 55]). However, the inequality corresponding to the upper variant of IA3 is analytic inductive, and hence can be captured in terms of an analytic structural rule.…”

Lattice Logic Properly Displayed

“…To conclude, a proof theory for axiomatic extensions and expansions of general lattice logic is comparably not as modular as that of the axiomatic extensions and expansions of the logic of distributive lattices, which can rely on the theory of proper display calculi [50,38]. The idea guiding the approach of the present paper, which we will elaborate upon in the next sections, is that, rather than trying to work our way up starting from a calculus for lattice logic, we will obtain a calculus for lattice logic from the standard proper display calculus for the logic of distributive lattices, by endowing it with a suitable mechanism to block the derivation of distributivity.…”

“…To argue that the calculus D.LL introduced in Section 5 adequately captures lattice logic, we follow the standard proof strategy discussed in [38,37]. Let | = HA denote the semantic consequence relation arising from the heterogeneous algebras introduced in Section 3.…”