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Filtrations in Intermediate Logics via Locally Finite Reducts of Heyting Algebras
November 12, 2015 @ 12:00 pm - 2:00 pm
There are two standard model-theoretic methods for proving the finite model property for modal and intermediate logics, the standard filtration and selective filtration. The standard filtration for modal and intuitionistic logics was first defined algebraically by McKinsey and Tarski in the 1940’s. In the 1960’s Lemon and Segerberg defined filtrations model theoretically. Selective filtrations were introduced by Gabbay and further developed by Fine and Zakharyaschev (1970’s and 1980’s).
In this talk I will give an algebraic description of filtrations in intuitionistic logic via locally finite reducts of Heyting algebras. In particular, I will show that the algebraic description of the standard filtration is based on the implication-free reduct of Heyting algebras, while that of selective filtration on the disjunction-free reduct. In order to define standard filtrations algebraically we will need to work with free Boolean extensions of distributive lattices. This will enable us to define the least and greatest filtrations algebraically. I will conclude by mentioning several consequences of this approach for intermediate and modal logics.