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It is introduced an immune dynamic n-valued Łukasiewicz logic I D Ł n on the base of n-valued Łukasiewicz logic Ł n and corresponding to it immune dynamic M V n -algebra ( I D L n -algebra), 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras (M , R , ◊) that combine the varieties of M V n -algebras M = (M , ⊕ , ⊙ , ∼ , 0 , 1) and regular algebras R = (R , ∪ , ; , ∗) into a single finitely axiomatized variety resembling R-module with "scalar" multiplication ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic I D Ł n with application in immune system. [ABSTRACT FROM AUTHOR]

This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logicIŁL —a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009). to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that w ⊩ ψ —which for ILis a relation between worlds w and formulas ψ , and can be seen as a function taking values in the booleans (w ⊩ ψ) ∈ B —becomes a function taking values in the unit interval (w ⊩ ψ) ∈ [ 0 , 1 ] . An appropriate monotonicity restriction (which we call sloping functions) needs to be put on such functions in order to ensure soundness and completeness of the semantics. [ABSTRACT FROM AUTHOR]

We study two notions of definability for classes of relational structures based on modal extensions of Ł ukasiewicz finitely-valued logics. The main results of the paper are the equivalent of the G oldblatt-T homason theorem for these notions of definability. [ABSTRACT FROM AUTHOR]