Reliable Computing 10: 299–334, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Characterization of Interval Fuzzy Logic Systems
of Connectives by Group Transformations
LADISLAV J. KOHOUT
Department of Computer Science, Florida State University, Tallahassee FL 32306–4530, USA,
Department of Computer Science, John D. Odegard School of Aerospace Science, University of
North Dakota, Grand Forks, ND 58202–9015, USA, e-mail: email@example.com
(Received: 15 October 2002; accepted: 15 April 2003)
Abstract. The global structure of various systems of logic connectives is investigated by looking at
abstract group properties of the group of transformations of these. Such characterizations of fuzzy
interval logics are examined in Sections 4–9. The paper starts by introducing readers to the Checklist
Paradigm semantics of fuzzy interval logics (Sections 2 and 3). In the Appendix we present some
basic notions of fuzzy logics, sets and many-valued logics in order to make the paper accessible to
readers not familiar with fuzzy sets.
Many-valued logic interval-based reasoning plays increasingly important role in
fuzzy and other many-valued extensions of crisp logic. It is often overlooked that
many-valued logic reasoning has richer inference rule base than the crisp (i.e.
classical two-valued) logic. Indeed some rules that are not possible in the crisp
logic (which is point-based) do come to existence when we accept intervals as the
basic elements of the semantic space ([0, 1] or a more general lattice) from which
the logic expressions take their valuation.
To be of use in a diversity of application domains, the interval-valued inference
systems require formal semantics that are not derived on an ad hoc basis. The
formal semantics that is derived by means of an exact mathematical method, and
which also has a sound ontological and epistemological base is provided by the so
called checklist paradigm –. A number of distinct interval systems of fuzzy
logics arise when the fuzzy membership function
(S) of a fuzzy proposition S is
interpreted as a summarization of a two-valued (also called crisp) logical n-tuple
that represents a checklist that records yes-no answers to n questions concerning
the truth status of a logic statement. Semantics of fuzzy logics provided by this
interpretation of the fuzzy membership function has been called checklist paradigm