According to classical logic, from contradictory premises, anything can be inferred
(ex contradictione quodlibet).
The major motivation behind paraconsistent logic
is to challenge this stand point: paraconsistent logic is not explosive. This means that even if we are in certain circumstances,
e.g. doing a medical diagnosis, where the available information is inconsistent, the inference relation does not explode into triviality. Thus,
paraconsistent logic accommodates inconsistency in a sensible manner that treats inconsistent information as informative.

Belnap's first order paraconsistent logic, four possible values associated with a formula A are true, false, contradictory and unknown: if there is evidence for A and no evidence against A then A obtains the value true and if there is no evidence for A and evidence against A then A obtains the value false. A value contradictory corresponds to a situation where there is simultaneously evidence for A and against A and, finally, A is labeled by value unknown if there is no evidence for A nor evidence against A. A fundamental feature of paraconsistent logic is that truth and falsehood are not each others complements.

Öztürk and Tsoukiŕs gave 2007 a continuous valued extension of Belnap's logic and shaw how the graded values T(A), K(A), U(A) and F(A), are to be computed when an ordered couple (a,b), called evidence couple, is given. The intuitive meaning of a and b is the degree of evidence of a statement A and against A respectively. The values T(A), K(A), U(A) and F(A) are values on the real unit interval [0,1]; their intuitive meaning is the degree of truth, contradictory, unknown and falsehood, respectively, of the statement A. Moreover, Öztürk and Tsoukiŕs also demonstrated how such a fuzzy version of Belnap's logic can be applied in preference modeling. However, due to a lack of a clear algebraic framework, several open problems remained, including complete truth calculus, syntactical notions as proof and completeness, etc. Our group solved these problems in 2009 by showing how continuous valued paraconsistent logic can be equipped with an injective MV--algebr} structure, thus with Pavelka's fuzzy logic. As a consequence, complete truth calculus and a rich semantics and syntax are available. Moreover, we demonstrated how paraconsistency is related to logic with generalized quantifiers and GUHA logic.

Here are some of our recent papers

Belnap's first order paraconsistent logic, four possible values associated with a formula A are true, false, contradictory and unknown: if there is evidence for A and no evidence against A then A obtains the value true and if there is no evidence for A and evidence against A then A obtains the value false. A value contradictory corresponds to a situation where there is simultaneously evidence for A and against A and, finally, A is labeled by value unknown if there is no evidence for A nor evidence against A. A fundamental feature of paraconsistent logic is that truth and falsehood are not each others complements.

Öztürk and Tsoukiŕs gave 2007 a continuous valued extension of Belnap's logic and shaw how the graded values T(A), K(A), U(A) and F(A), are to be computed when an ordered couple (a,b), called evidence couple, is given. The intuitive meaning of a and b is the degree of evidence of a statement A and against A respectively. The values T(A), K(A), U(A) and F(A) are values on the real unit interval [0,1]; their intuitive meaning is the degree of truth, contradictory, unknown and falsehood, respectively, of the statement A. Moreover, Öztürk and Tsoukiŕs also demonstrated how such a fuzzy version of Belnap's logic can be applied in preference modeling. However, due to a lack of a clear algebraic framework, several open problems remained, including complete truth calculus, syntactical notions as proof and completeness, etc. Our group solved these problems in 2009 by showing how continuous valued paraconsistent logic can be equipped with an injective MV--algebr} structure, thus with Pavelka's fuzzy logic. As a consequence, complete truth calculus and a rich semantics and syntax are available. Moreover, we demonstrated how paraconsistency is related to logic with generalized quantifiers and GUHA logic.

Here are some of our recent papers

- Turunen, E.; Öztürk, M.and Tsoukiŕs, A.:
*Paraconsistent Semantics for Pavelka Style Fuzzy Sentential Logic*(submitted).

- Turunen, E.:
*Interpreting GUHA Data Mining Logic in Paraconsistent Fuzzy Logic Framework*. In A. Tsoukias (Ed.): ADT09, LNAI, (2009)1-10.

- Turunen, E.:
*A Paraconsistent Fuzzy Logic*.In R. Ramanujam and S. Sarukkai (Eds.): ICLA 2009, LNAI, (2009)77-88.

- Öztürk, M., Tsoukiŕs, A. and Turunen, E.;
*Continuous extension of 4 Belnap values*. Proceedings of LFA 2008: Rencontres francophones sur la Logique Floue et ses Applications. 16.-17. October, 2008. Lens, France.