|14.00 - 14.45||Matthias Baaz - Vienna University of Technology||Prenexation in Gödel Logics|
|15.00 - 15.45||Petr Cintula - Czech Academy of Sciences||General Neighborhood and Kripke Semantics for Substructural Modal Logics|
|16.00 - 16.45||George Metcalfe - University of Bern||Ordered Algebras and Model Completions|
The meeting will take place at
Exakte Wissenschaft building
In this lecture we characterize the First Order Gödel Logics where every formula is equivalent to a prenex normal form. (The prenex normal form might have any form.)
Frame semantics, given by Kripke or neighborhood frames, do not give completeness theorems for all modal logics extending, respectively, K
and E. Such shortcoming can be overcome by means of general frames, i.e. frames equipped with a collection of admissible sets of worlds (which is the range of possible valuations over such frame).
We export this approach from the classical paradigm to the substructural one by defining general A-frames over a given residuated lattice A (i.e., the usual frames with a collection of admissible A-valued sets).
We describe in details the relation between general Kripke and neighborhood A-frames and prove that, if the logic of A is finitary, all extensions of the corresponding logic E of A are complete w.r.t. general neighborhood frames.
A well-known theorem of classical model theory due to Robinson states that the (first-order) theory of totally ordered abelian groups has a model completion. However, it was proved by Glass and Pierce that this is not the case for the theory of lattice-ordered abelian groups. Similarly, it was proved by Lacava and Saeli that the theory of totally ordered MV-algebras has a model completion, but not the theory of all MV-algebras.
In this talk, we will generalize these negative results for lattice-ordered abelian groups and MV-algebras to see that the theory of a variety of commutative pointed residuated lattices has a model completion if and only if the variety admits uniform deductive interpolation and has equationally definable principal congruences. We will also see how positive model completion results can be rescued for certain varieties by adding an additional binary operator, demonstrating in particular that the theory of MV-Delta-algebras has a model completion.