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
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 Fermüller extended Giles`
result and proposed a representation of two other principal fuzzy logics -
Gödel and Product (Fermüller 2007).
The connection between fuzzy logic and probability theory in both Giles' and
Fermüller'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, Gödel 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 Hájek, 1998).
Christian G. Fermüller (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.
Hájek, Petr (1998), Metamathematics of Fuzzy Logic, volume 4 of Trends in
Logic. Kluwer, Dordrecht.