logics are logical calculi in which there are more than two
possible truth values. Traditionally, logical calculi are bivalent—that
is, there are only two possible truth values for any proposition, true
and false (which generally correspond to our intuitive notions of truth
and falsity). But bivalence is only one possible range of truth values
that may be assigned, and other logical systems have been developed
with variations on bivalence, or with more than two possible
first studied by a Polish mathematicial Jan Lukasiewicz
in 1920's. Łukasiewicz worked on multi-valued logics, including his own
three-valued propositional calculus, the first non-classical logical
calculus. An early important contributor was a Polish Jew
Wajsberg who in 1935 proved the completeness conjecture presented
A third pioneer was C.C. Chang who in 1958 gave another proof for the
Completness Theorem of Lukasiewicz's infinite valued logic. In 1998 a
Czech mathematician Petr
Hajek generalized Lukasiewicz logic to what is
now called BL-logic.
We are studying Hajek's BL-logic mainly from an algebraic point of
view. Known by Lindenbaum-Tarski Theorem, for each logic there is a corresponding abstract algebra. For classical bivalent logic this
algebra in Boolean, for BL-logic it is BL-algebra,
which is a special residuated lattice .
In spite of multiplicity of nonclassical logics and various residuated lattices, Lukasiewicz infinite valued logic and
MV-algebras have a specific position
in the realm of many-valued and fuzzy reasoning; Mundici
is a central figure in establishing a relation between Lukasiewicz logic and other mathematical realms,
e.g. to abelian l-groups with strong unit, to AFC*-algebras and to De Finetti's subjective probability theory, just to give a few examples.
Also our experience consolidates the special position of Lukasiewicz infinite valued logic among nonclassical logics.
Firstly, in 2002 we introduced many-valued similarity algorithms to construct fuzzy inference systems on a firm mathematical basis.
In this study it was shown that Pavelka's approach to fuzzy logic offers a natural framework to model experts' vague knowledge in decision making and
control applications. Algebraically the approach is based on injective MV-algebras
The feasibility of these similarity algorithms was testified in several real life applications, see e.g. our papers from
2002 and 2003.
Secondly, we proved in 2001 that a subset MV(L) of complemented element, the
of a BL-algebra L, is the largest MV-subalgebra of L. From a logical point of view this has, among others, a consequence
that the negative part of Hajek's BL-logic reduces to Lukasiewicz logic. In our publications from
1999, 2001, 2005, 2007, and 2008, by utilizing the MV-center theory, we listed several such properties of a BL-algebra L that L has that property if, and only if
the corresponding MV(L) has the property.
are some of our related publications:
- Mertanen, J. and Turunen, E.: States on semi-divisible generalized residuated lattices reduce to states on
MV-algebra. Fuzzy Sets and Systems 159(2008), 3051-3064
- Turunen, E.: Deduction Theorem in Monoidal Logic.
Journal of the Calcutta Mathematical Society (2)2008, 1-4.
- Turunen, E., Mertanen, J.: States on semi-divisble residuated lattices.
Soft Computing Vol.12 (4)2008, 353-357.
- Turunen, E.: Hyper Arhimedean BL-algeras are MV-algebras.
Mathematical Logic Quaterly 53 No 2, 177-182 (2007).
- Turunen, E.: Semilocal BL-algeras . Proceedings of the IX
International IFSA Congress 2005. 28.-31. July 2005, Beiging, China. 252-256.
- Mänttäri, A., Luukka, P., Parkkari, J. and Turunen, E.: Predicting maximal heart rate from the UKK 2 km walk test results: soft
computing and linear regression methods. 8th annual congress of the EUROPEAN COLLEGE of SPORT SCIENCE. 9-12 July 2003 Salzburg,
- Niittymäki, J. and Turunen, E.: Traffic signal control on similarity logic reasoning. Fuzzy
Sets and Systems 133(2003) 109-131
- Dubrovin, T., Jolma, A. and Turunen, E.: Fuzzy model for real-time reservoir operation. Journal
of Water Resources Planning and Management 128(2002) 66-73.
- Kukkurainen, P. and Turunen, E.:Many-valued Similarity Reasoning. An Axiomatic Approach.
International Journal of Multiple Valued Logic 8(2002) 751-760.
- Turunen, E., Sessa, S.: Local BL-algebras. International Journal of Multiple Valued Logic 6(2001) 229-249.
- Turunen, E.: Boolean deductive systems of BL-algebras. Archive for Mathematical Logic 40(2001) 467-473.
- Turunen, E.: BL-algebras of Basic Fuzzy Logic. Mathware and Soft
Computing 6(1999) 49-61.