Dynamic Łukasiewicz Logic and Dynamic MV-algebras

Dynamic Łukasiewicz logic and its application to immune system

“…Notice that the axiom Ax0 is well known as axiom of basic modal logic K , the axioms Ax1-Ax6 have been proposed by Segerberg in 1977 (see also Kozen and Parikh 1981) for dynamic logic, but in a terms, and Ax7-Ax8 are specific axioms for modal Łukasiewicz logic (Di Nola and Grigolia 2004;Hansoul and Teheux 2006). Notice also the works (Di Nola et al 2020;Teheux 2014) on dynamic Łukasiewicz logic.…”

“…Similarly to dynamic propositional Łukasiewicz logic introduced in Di Nola et al (2020), we introduce immune dynamic n-valued Łukasiewicz logic I DL n , the main difference of which are in interpretations. Immune dynamic n-valued Łukasiewicz logic I DŁ n is designed for representing and reasoning about propositional Łukasiewicz logic expected 1 M V n -algebras where introduced by Grigolia in Grigolia (1977).…”

“…One of the semantics of I DL n is Kripke semantics that is similar to the ones that have been done in Di Nola et al (2020). In the presented case, the formula [a ∪b]ϕ means that whenever laboratory experiment a or b is successfully done, a state is reached where ϕ holds, whereas the formula (a; b) ϕ means that there is a sequence of consecutive laboratory experiments a and b such that a state is reached where ϕ holds.…”

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“…One of the semantics of IDL n are Kripke semantics [18] that is similar to the ones that have been done in [22]. In the presented case the formula [a ∪ b]φ means that whenever laboratory experiment a or b is successfully done, a state is reached where φ holds, whereas the formula ⟨(a; b)⟩φ means that there is a sequence of consecutive laboratory experiments a and b such that a state is reached where φ holds.…”

“…Following K. Segerberg [41] (1977), D. Kozen [31] (1979) and V. Pratt [38] (1980), who have been introduced dynamic (classical) propositional logic, that is a formal system for reasoning about programs, and dynamic algebras, dynamic propositional Lukasiewicz logic DP L (dynamic n-valued propositional Lukasiewicz logic DP L n ) and dynamic M V -algebras (dynamic M V n -algebras) are introduced and theories of the logic DP L (DP L n ) and dynamic M V -algebras (M V n -algebras) are developed [22] (2020). Dynamic M V -algebras (dynamic M V n -algebras) are algebraic counterparts of the logic DP L (DP L n ), that in turn represent two-sorted algebras that combine the varieties of M V -algebras (M V n -algebras) and regular algebras into a single finitely axiomatized variety resembling R-module with scalar multiplication.…”