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.