Unified Correspondence

This workshop focuses on the meta-logical theory of unified correspondence and its applications to linguistics and management science.

Since the 1970s, correspondence theory has been one of the most important items in the toolkit of modal logicians. Unified correspondence [3] is a very recent approach, which has imported techniques from duality, algebra and formal topology [6] and exported the state of the art of correspondence theory well beyond normal modal logic, to a wide range of logics including, among others, intuitionistic and distributive lattice-based (normal modal) logics [4], non-normal (regular) modal logics [14], substructural logics [5], hybrid logics [8], and mu-calculus [1, 2].

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 [13], or of the phenomenon of pseudo-correspondence [7]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [9], the identification of the syntactic shape of axioms which can be translated into analytic rules of a proper display calculus [10], and the design of display-type calculi for the logics of capabilities and resources, and their applications to the logical modelling of business organizations [11].

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 defined uniformly in each mentioned signature, and which properly and significantly extends the Sahlqvist class.

This wealth of new techniques, results and insights is now ready to be put to use in the mathematical environments (both semantic and proof-theoretic) of logical systems which are suitable to address formalization problems in the target application areas of linguistics and management science. In fact, the logical work in both these fields already displays correspondence phenomena, albeit in an embryonic form, see e.g. [15] and [12].The aim of this workshop is therefore to foster new scientific collaborations among mathematical logicians using correspondence theoretic tools on the one hand and, on the other, researchers in linguistics and management science interested in applying logical methods.

References

[1] W. Conradie and A. Craig. Canonicity results for mu-calculi: an algorithmic approach.

Journal of Logic and Computation, forthcoming.

[2] W. Conradie, Y. Fomatati, A. Palmigiano, and S. Sourabh. Algorithmic correspondence

for intuitionistic modal mu-calculus. Theoretical Computer Science, 564:30-62, 2015.

[3] W. Conradie, S. Ghilardi, and A. Palmigiano. Unified Correspondence. In A. Baltag and

S. Smets, editors, Johan van Benthem on Logic and Information Dynamics, volume 5

of Outstanding Contributions to Logic, pages 933-975. Springer International Publishing,

2014.

[4] W. Conradie and A. Palmigiano. Algorithmic correspondence and canonicity for distributive modal logic. Annals of Pure and Applied Logic, 163(3):338-376, 2012.

[5] W. Conradie and A. Palmigiano. Algorithmic correspondence and canonicity for non-

distributive logics. Journal of Logic and Computation, forthcoming.

[6] W. Conradie, A. Palmigiano, and S. Sourabh. Algebraic modal correspondence: Sahlqvist

and beyond. Submitted, 2014.

[7] W. Conradie, A. Palmigiano, S. Sourabh, and Z. Zhao. Canonicity and relativized canonicity via pseudo-correspondence: an application of ALBA. Submitted, 2014.

[8] W. Conradie and C. Robinson. On Sahlqvist theory for hybrid logic. Journal of Logic

and Computation, forthcoming.

[9] S. Frittella, A. Palmigiano, and L. Santocanale. Dual characterizations for finite lattices

via correspondence theory for monotone modal logic. Journal of Logic and Computation,

forthcoming.

[10] G. Greco, M. Ma, A. Palmigiano, A. Tzimoulis, and Z. Zhao. Unified correspondence as

a proof-theoretic tool. Submitted, 2015.

[11] G. Greco, A. Palmigiano, and A. Tzimoulis. Algebraic proof theory for the logics of

organizations: a display-type calculus for capabilities and resources. In preparation, 2015.

[12] N. Kurtonina. Categorical Inference and Modal Logic. Journal of Logic, Language, and

Information, 7:399-411, 1998.

[13] A. Palmigiano, S. Sourabh, and Z. Zhao. Jonsson-style canonicity for ALBA-inequalities.

Journal of Logic and Computation, forthcoming.

[14] A. Palmigiano, S. Sourabh, and Z. Zhao. Sahlqvist theory for impossible worlds. Journal

of Logic and Computation, forthcoming.

[15] L. Polos, M. Hannan, and G. Hsu. Modalities in sociological arguments. Journal of

mathematical sociology, 34(3):201-238, July 2010.