- New
-
Topics
- All Categories
- Metaphysics and Epistemology
- Value Theory
- Science, Logic, and Mathematics
- Science, Logic, and Mathematics
- Logic and Philosophy of Logic
- Philosophy of Biology
- Philosophy of Cognitive Science
- Philosophy of Computing and Information
- Philosophy of Mathematics
- Philosophy of Physical Science
- Philosophy of Social Science
- Philosophy of Probability
- General Philosophy of Science
- Philosophy of Science, Misc
- History of Western Philosophy
- Philosophical Traditions
- Philosophy, Misc
- Other Academic Areas
- Journals
- Submit material
- More
Switch to: References
Citations of:
Adding involution to residuated structures
Studia Logica 77 (2):181 - 207 (2004)
Add citations
You must login to add citations.
|
|
|
We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice. |
|
|
|
|
|
|
|
|
Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions. |
|
|
Besides the better-known Nelson logic and paraconsistent Nelson logic, in 1959 David Nelson introduced, with motivations of realizability and constructibility, a logic called $\mathcal{S}$. The logic $\mathcal{S}$ was originally presented by means of a calculus with infinitely many rule schemata and no semantics. We look here at the propositional fragment of $\mathcal{S}$, showing that it is algebraizable, in the sense of Blok and Pigozzi, with respect to a variety of three-potent involutive residuated lattices. We thus introduce the first known algebraic (...) |
|
|
We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus ( DFL ) whose algebraic semantics is the class of distributive residuated lattices ( DRL ). The problem was left open in [8, 5]. We use the method of nuclei and quasi–embedding in the style of [10, 1]. |
|
|
This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which (...) |
|
|
We extend the lattice embedding of the axiomatic extensions of the positive fragment of intuitionistic logic into the axiomatic extensions of intuitionistic logic to the setting of substructural logics. Our approach is algebraic and uses residuated lattices, the algebraic models for substructural logics. We generalize the notion of the ordinal sum of two residuated lattices and use it to obtain embeddings between subvariety lattices of certain residuated lattice varieties. As a special case we obtain the above mentioned embedding of the (...) |
|
|
Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices. |
|
|
Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider (...) |
|
|
In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz (...) |
|
|
|
|
|
Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study (...) |
|
|
Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson conucleus algebras unify these examples and further extend them to the non-commutative setting. We study their structure, establish a representation theorem for them in terms of twist structures and conuclei that results in a categorical adjunction, and explore situations where the representation is actually an isomorphism. In the latter case, the adjunction is elevated to a categorical equivalence. By (...) |
|
|
In the paper Busaniche and Cignoli we presented a quasivariety of commutative residuated lattices, called NPc-lattices, that serves as an algebraic semantics for paraconsistent Nelson’s logic. In the present paper we show that NPc-lattices form a subvariety of the variety of commutative residuated lattices, we study congruences of NPc-lattices and some subvarieties of NPc-lattices. |
|
|
The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) |
|
|
We introduce a new product bilattice con- struction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lat- tice factors. Finally, we show how (...) No categories |
|
|
|
|
|
No categories  |
PhilPapers logo by Andrea Andrews and Meghan Driscoll.
This site uses cookies and Google Analytics (see our terms & conditions for details regarding the privacy implications). Use of this site is subject to terms & conditions.
All rights reserved by The PhilPapers Foundation
Server: philpapers-web-7c76586df8-pp8v9 uwo