6 Algebras and coalgebras

Abstract

This chapter1 sketches some of the mathematical surroundings of modal logic. First, we discuss the algebraic perspective on the field, showing how the theory of universal algebra, and more specifically, that of Boolean algebras with operators, can be used to prove significant results in modal logic. In the second and last part of the chapter we describe how modal logic, and its model theory, provides many natural manifestations of the more general theory of universal coalgebra.

Expressiveness Modulo Bisimilarity: A Coalgebraic Perspective

“…For the valuation W on S we take W := W f . It is immediate by (21) and the fact that W f = W • f , that the map f : S → S is in fact a T Q -coalgebra morphism. Finally, we need to come up with a designated point r of S := S , σ which is mapped to r by f .…”

“…¡ Remark 2.7 In many cases, including those of Kripke frames and models, behavioral equivalence is the same as bisimilarity, but in cases where the two notions diverge, behavioral equivalence is the more natural notion. For the purpose of this paper it suffices to work with behavorial equivalence, and we do not need to discuss generalisations of the notion of a bisimulation to coalgebras of arbitrary type, referring the reader to [21] for more information.…”

‘Rolling Down the River’: Saul Kripke and the Course of Modal Logic

“…One reason why the latter system has been so hard to define well, avoiding model-theoretic and proof-theoretic catastrophes, is the fact that is a system combination of two modal logics, 16 whose behaviour depends critically on the mode of combination. Even so, modal predicate logic is obviously important to old and new theory and applications -and in that sense, Kripke's paper is still highly relevant, decades later.…”

Bisimulation for Neighbourhood Structures

“…For further reading on coalgebra we refer to [23,26]. Extended discussions on neighbourhood semantics can be found in [7,12].…”