Probabilistic representation of fuzzy logics Ondrej Majer Institute of Philosophy Academy of Sciences of the Czech Republic Prague majer@site.cas.cz Libor Behounek Institute of Computer Science Academy of Sciences of the Czech Republic Prague behounek@cs.cas.cz The idea of probabilistic interpretation of fuzzy logics dates back to the work of Robin Giles, (Giles 1974, 1977). He showed, using a game-theoretic framework, that a (fuzzy) value of a formula of Lukasiewicz logic can be represented in terms of probabilities of its subformulas obtained via a Lorenzen-style dialogue game. Recently Christian Fermueller extended Giles' result and proposed a representation of two other principal fuzzy logics - Goedel and Product (Fermueller, 2007). The connection between fuzzy logic and probability theory in both Giles' and Fermueller's work is rather loose - it remains on the level of isolated atomic events which are not assumed to be a part of a single probabilistic space. The main goal of this article is to make the connection more straightforward and to represent a formula of fuzzy logic as a pair of events in a probabilistic space of an appropriate kind. Our second goal is to extend the representation result to a wider class of fuzzy logics - in particular to those obtained as an ordinal sum of the Lukasiewicz, Goedel and Product logics. This gives us a probabilistic representation of an important class of fuzzy logics, namely those which correspond to continuous t-norms (see Hajek, 1998). - Christian G. Fermueller (2007), Revisiting Giles Game, to appear in: Logic, Games Philosophy, Majer, O., Pietarinen, A. Tulenheimo, T., (eds.), Springer 2007 - Robin Giles (1974), A non-classical logic for physics. Studia Logica 33, vol. 4, (1974), 399-417. - Robin Giles (1977), A non-classical logic for physics. In: R. Wojcicki, G. Malinkowski (Eds.) Selected Papers on Lukasiewicz Sentential Calculi. Polish Academy of Sciences, 1977, 13-51. - Hajek, Petr (1998), Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht.