Dynamic Łukasiewicz logic and its application to immune system

AbstractIt is introduced an immune dynamicn-valued Łukasiewicz logicID{\L }_nIDŁnon the base ofn-valued Łukasiewicz logic{\L }_nŁnand corresponding to it immune dynamic$$MV_n$$MVn-algebra ($$IDL_n$$IDLn-algebra),$$1< n < \omega$$1<n<ω, which are algebraic counterparts of the logic, that in turn represent two-sorted algebras$$(\mathcal {M}, \mathcal {R}, \Diamond )$$(M,R,◊)that combine the varieties of$$MV_n$$MVn-algebras$$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$M=(M,⊕,⊙,∼,0,1)and regular algebras$$\mathcal {R} = (R,\cup , ;, ^*)$$R=(R,∪,;,∗)into a single finitely axiomatized variety resemblingR-module with "scalar" multiplication$$\Diamond$$◊. Kripke semantics is developed for immune dynamic Łukasiewicz logicID{\L }_nIDŁnwith application in immune system

Algebraic semantics for one-variable lattice-valued logics

The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setting of first-order lattice-valued logics, where formulas are interpreted in algebraic structures with a lattice reduct. In particular, axiomatizations are obtained for modal counterparts of one-variable fragments of a broad family of these logics by generalizing a functional representation theorem of Bezhanishvili and Harding for monadic Heyting algebras. An alternative proof-theoretic proof is also provided for one-variable fragments of first-order substructural logics that have a cut-free sequent calculus and admit a certain bounded interpolation property

One-variable fragments of first-order logics

The one-variable fragment of a first-order logic may be viewed as an "S5-like" modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases -- notably, the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic -- but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically-defined first-order logic -- spanning families of intermediate, substructural, many-valued, and modal logics -- to admit a natural axiomatization. More precisely, such an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, building on a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.Comment: arXiv admin note: text overlap with arXiv:2209.0856

Monadic Wajsberg hoops

Wajsberg hoops are the { , →, 1}-subreducts (hoop-subreducts) of Wajsberg algebras, which are term equivalent to MV-algebras and are the algebraic models of Lukasiewicz infinite-valued logic. Monadic MV-algebras were introduced by Rutledge [Ph.D. thesis, Cornell University, 1959] as an algebraic model for the monadic predicate calculus of Lukasiewicz infinitevalued logic, in which only a single individual variable occurs. In this paper we study the class of { , →, ∀, 1}-subreducts (monadic hoop-subreducts) of monadic MV-algebras. We prove that this class, denoted by MWH, is an equational class and we give the identities that define it. An algebra in MWH is called a monadic Wajsberg hoop. We characterize the subdirectly irreducible members in MWH and the congruences by monadic filters. We prove that MWH is generated by its finite members. Then, we introduce the notion of width of a monadic Wajsberg hoop and study some of the subvarieties of monadic Wajsberg hoops of finite width k. Finally, we describe a monadic Wajsberg hoop as a monadic maximal filter within a certain monadic MValgebra such that the quotient is the two element chain.Fil: Díaz Varela, José Patricio. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Cimadamore, Cecilia Rossana. Universidad Nacional del Sur. Departamento de Matemática; Argentin

Finite axiomatizability in Łukasiewicz logic

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On a Definition of a Variety of Monadic ℓ-Groups

In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL 0 of integral residuated lattices with bottom, which generalize MV-algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor K∙, motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV-algebras and the corresponding category MV∙ of monadic MV-algebras induced by “Kalman’s functor” K∙. Moreover, we extend the construction to ℓ-groups introducing the new category of monadic ℓ-groups together with a functor Γ♯, that is “parallel” to the well known functor Γ between ℓ and MV-algebras.Facultad de Ciencias Exacta