Multi-valued
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
truth-value assignments.

Multi-valued (many-valued) logics were 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 Mordechaj Wajsberg who in 1935 proved the completeness conjecture presented by Lukasiewicz. 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 MV-center MV-center 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. Here are some of our related publications:

Multi-valued (many-valued) logics were 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 Mordechaj Wajsberg who in 1935 proved the completeness conjecture presented by Lukasiewicz. 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 MV-center MV-center 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. Here 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*.

- 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, Austria.P11U-04.

- 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.