On the logic foundation of fuzzy reasoning
Information Sciences: an International Journal
The Paradoxical Success of Fuzzy Logic
IEEE Expert: Intelligent Systems and Their Applications
A triangular-norm-based propositional fuzzy logic
Fuzzy Sets and Systems - Logic and algebra
On equivalent forms of fuzzy logic systems NM and IMTL
Fuzzy Sets and Systems
Formalized theory of general fuzzy reasoning
Information Sciences—Informatics and Computer Science: An International Journal
Unified forms of fully implicational restriction methods for fuzzy reasoning
Information Sciences: an International Journal
Normal forms and free algebras for some extensions of MTL
Fuzzy Sets and Systems
Unified forms of Triple I method
Computers & Mathematics with Applications
A new theory consistency index based on deduction theorems in several logic systems
Fuzzy Sets and Systems
Undefinability of min-conjunction in MTL
Fuzzy Sets and Systems
Truth degree analysis of fuzzy reasoning
Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology
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In order to offer a logic foundation for fuzzy reasoning and to reduce the gap between fuzzy reasoning and artificial intelligence, a new formal deductive system L^* was introduced in 1996. It was proved that the prominent nilpotent minimum logic is a logic system equivalent to L^*. As is well known, in classical propositional logic, there is a one-to-one correspondence between deductively closed theories and topologically closed subsets of the Stone space of all maximally consistent theories. Recently, by means of the strong completeness of L^*, we obtained the structural and topological characterizations of maximally consistent theories in this logic. It was proved that the set of all maximally consistent theories over L^* with an induced topology is a Cantor space. In order to establish a similar connection between theories over L^* and closed subsets of this topological space to the classical case, we introduce in this paper the notion of three-valued Lukasiewicz theory and the notion of two-valued Lukasiewicz (Boolean) theory. We prove that there is also a one-to-one correspondence between deductively closed three-valued Lukasiewicz theories and closed subsets of the space, and Boolean theories correspond to the closed subsets of the subspace of all maximally consistent Boolean theories. Lastly, we give an alternative proof for the structural characterization of the maximality of consistent theories over L^* by induction on the complexity of formulas, which is independent of the strong completeness of L^* and also independent of the perfectness of the standard R"0-algebra.