Seminar series by Bruno Poizat
An introduction to Positive Logic
Bruno Poizat
University of Lyon
Date: Monday-Thursday, October 15-18, 2012
Time: 5:30PM
Venue: CS103.Abstract:
An introduction to Positive Logic
Positive Logic is the logic without negation, that is, where only conjunctions, disjunctions and existential quantifications are used in the formation of the formulae. The sentences that are considered in PL are a special kind of universal and of inductive statements, which have the property to cross the inductive limits of homomorphisms.
The Model Theory of positive logic has been revitalized a few years ago by Itai Ben Yaacov, out of a kind of internal model-theoric necessity, to treat situations where the logic with negation has nothing to say. The main feature of the model theory of PL is that we restrict our attention to existentially closed models, and doing so PL appears as a generalisation of the Logic with negation (which paradoxically is interpretable in PL !), which corresponds to the case where all the structures under consideration are EC.
This serie of lectures will expose the basis of the theory, following the paper I. Ben Yaacov and BP, Fondements de la Logique Positive, which has been published in the Journal of Symbolic Logic some 3 (or 4 ?) years ago. It should attract the attention not only of people interested in Logic, but also of mathematicians of the algebraic persuation.
Topics to be covered: Ch. 0. Prelude to Positive Logic : Algebraically Closed Fields and Existentially Closed Groups
Ch. 1. Homomorphisms and Positive Formulae
Ch. 2. Inductive Limits and Inductive Classes
Ch. 3. Inductive and Universal Sentences
Ch. 4. Compactness of First Order Logic
Ch. 5. Companion Theories ; Model-complete Theories
Ch. 6. Spaces of Types