KRIPKE MODELS FOR BADIOU'S ONTOLOGY AND PHENOMENOLOGY

In Badiou's philosophy two approaches can be recognized. The first and initial one is an ontology based on the set theory and Cohen's forcing method. It is presented in "Being and event". The second and late one is phenomenology or the theory of object, to which Badiou's second major book "Logics of worlds" is devoted. Both approaches admit various interpretations but perhaps one of the most natural of them is that suggested by Kripke models. The latter not only provide us with a proper semantics for Badiou's theory of object but also is instrumental in tracing a connection between his two approaches. The article deals with building Kripke models for early and late Badiou's philosophy, for his ontology and phenomenology. It first demonstrates two ways of constructing Kripke models. The first way follows Kripke's own presentation and happens to be useful in building a model for the initial Badiou's approach since it is based on the procedure of forcing. Thus Badiou's subjective procedure from "Being and event" can be depicted as a movement along the tree of possible worlds in a Kripke model directed to a final world determined by the generic set, this set itself being a collection of possible paths leading to this world. Moreover, subjective judgment (based on forcing) corresponds to subject's positioning in a particular world with particular formulas satisfied. The second way to construct Kripke models makes use of Goldblatt's reformulation of them in terms of hereditary sets and Heyting algebras. As demonstrated, this makes possible providing a semantics for Badiou's theory of object. Since these two formulations of Kripke models are equivalent, a ties between two Badiou's approaches thus established. In particular the subjective procedure from the early "Being and Event" can be considered as a special case of object from Badiou's late phenomenology. In addition, this allows us to demonstrate that the logic of both Badiou's approaches is intuitionistic. The article also considers few instructive examples.