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## 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. [Back] |