Lattice-based logics form a wide family of very well known and intensely investigated logics which include classical and intuitionistic (modal) logics, many-valued logics, and substructural logics. In recent years, the mathematical theory of this class of logics has been systematically studied as a whole, yielding a rich semantic environment consisting of several types of relational structures, and a smooth proof-theoretic environment in which powerful results are driven by algebraic insights, which in turn can be systematically connected to and understood in terms of semantic properties.
While these results are very satisfactory from the viewpoint of mathematical logic, and while the analytic power and significance of the best known logics in this class is very well understood and exploited, most logics in this class remain very mysterious as to their conceptual significance. A case in point revolves around the distributivity axioms and which are valid e.g. in classical and intuitionistic (modal) logics but are not valid in most substructural logics. These axioms (among others) allow to interpret the logical operations of and as set-theoretic intersection and union respectively. This, in turn, makes it possible to understand and as natural language 'conjunction' (AND) and 'disjunction' (OR), respectively. Since the distributivity of AND and OR is hardwired in the meaning of these connectives, the failure of the distributive axioms immediately blocks the possibility of understanding and as natural language AND and OR.
In this tutorial, I will discuss a proposal for a possible solution. Namely, formulas of a lattice-based logic without distributivity should not be understood as sentences, but as terms denoting entities of different kinds, such as categories, concepts, theories (understood as sets of formulas closed under some broadly conceived forms of inference), and interrogative agendas. I will discuss alternative meanings of and under each interpretation, why the failure of distributivity becomes a desirable feature, and I will argue that these interpretations provide conceptually independent frameworks of reference for understanding the properties of some types of relational structures. Time permitting, I will argue that lattice-based logics without distributivity, in synergy with other logical frameworks, can provide very powerful formal tools to analyse issues which include the dynamics of decision-making via deliberation, market-dynamics, and competing theories in science.