Residuated Basic Logic I
نویسندگان
چکیده
We study the residuated basic logic (RBL) of residuated basic algebra in which the basic implication of Visser’s basic propositional logic (BPL) is interpreted as the right residual of a non-associative binary operator · (product). We develop an algebraic system SRBL of residuated basic algebra by which we show that RBL is a conservative extension of BPL. We present the sequent formalization LRBL of SRBL which is an extension of distributive full non-associative Lambek calculus (DFNL), and show that the cut elimination and subformula property hold for it.
منابع مشابه
Nilpotent Elements of Residuated Lattices
Ward and Dilworth 1 introduced the concept of residuated lattices as generalization of ideal lattices of rings. The residuated lattice plays the role of semantics for a multiple-valued logic called residuated logic. Residuated logic is a generalization of intuitionistic logic. Therefore it is weaker than classical logic. Important examples of residuated lattices related to logic are Boolean alg...
متن کاملResiduated Basic Logic II. Interpolation, Decidability and Embedding
We prove that the sequent calculus LRBL for residuated basic logic RBL has strong finite model property, and that intuitionistic logic can be embedded into basic propositional logic BPL. Thus RBL is decidable. Moreover, it follows that the class of residuated basic algebras has the finite embeddability property, and that BPL is PSPACE-complete, and that intuitionistic logic can be embedded into...
متن کاملFilter Theory of Bounded Residuated Lattice Ordered Monoids
Bounded residuated lattice ordered monoids (R -monoids) are a common generalization of pseudo-BL-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. In the paper we introduce and study classes of filters of bou...
متن کاملNon-Commutative EQ-Logics and Their Extensions
We discuss a formal many-valued logic called EQlogic which is based on a recently introduced special class of algebras called EQ-algebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated lattices. The goal of EQ-logics is to present a possible direction in the development of mathematical logics...
متن کاملA Survey of Generalized Basic Logic Al- gebras
Petr Hájek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval, and defined the corresponding algebraic models, BL-algebras, in the context of residuated lattices. The defining characteristics of BL-algebras are representability and divisibility. In this short note we survey recent developments in the study of divisible residuated lattices an...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- CoRR
دوره abs/1403.3354 شماره
صفحات -
تاریخ انتشار 2014