Quasi-Nelson algebras and fragments

2021 ◽  
pp. 1-29
Author(s):  
Umberto Rivieccio ◽  
Ramon Jansana
Keyword(s):  
Residuated Lattices ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Nelson Algebras ◽  
Algebraic Language ◽  
Nelson Logic ◽  
Algebraic Counterpart

Abstract The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒ ew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

Relational semantics and a relational proof system for full Lambek calculus

1998 ◽  
Vol 63 (2) ◽  
pp. 623-637 ◽  
Author(s):  
Wendy MacCaull
Keyword(s):  
Proof Theory ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Complete Proof ◽  
Proof Search ◽  
Proof Tree ◽  
Relational Logic ◽  
Completeness Theorems

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

scholarly journals Glivenko theorems for substructural logics over FL

2006 ◽  
Vol 71 (4) ◽  
pp. 1353-1384 ◽  
Author(s):  
Nikolaos Galatos ◽  
Hiroakira Ono
Keyword(s):  
Propositional Logic ◽  
Substructural Logics ◽  
Residuated Lattices ◽  
Substructural Logic ◽  
Classical Propositional Logic ◽  
Double Negation ◽  
Glivenko’S Theorem

AbstractIt is well known that classical propositional logic can be interpreted in intuitionistic prepositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part of the paper, we also discuss some extended forms of the Koltnogorov translation and we compare it to the Glivenko translation.

scholarly journals NONASSOCIATIVE SUBSTRUCTURAL LOGICS AND THEIR SEMILINEAR EXTENSIONS: AXIOMATIZATION AND COMPLETENESS PROPERTIES

2013 ◽  
Vol 6 (3) ◽  
pp. 394-423 ◽  
Author(s):  
PETR CINTULA ◽  
ROSTISLAV HORČÍK ◽  
CARLES NOGUERA
Keyword(s):  
Algebraic Approach ◽  
Substructural Logics ◽  
Unit Interval ◽  
Residuated Lattices ◽  
Lambek Calculus ◽  
Deduction Theorem ◽  
The Real ◽  
Special Stress ◽  
Ordered Algebras

AbstractSubstructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and nonassociative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e., logics complete with respect to linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.

scholarly journals Quasi-N4-Lattices

2021 ◽  
Author(s):  
Umberto Rivieccio
Keyword(s):  
Paraconsistent Logic ◽  
General Setting ◽  
Extension Property ◽  
Double Negation ◽  
Structure Representation ◽  
Nelson Algebras ◽  
Nelson Logic ◽  
Double Negation Law ◽  
Logical Calculi ◽  
Twist Structure

Abstract Within the Nelson family, two mutually incomparable generalizations of Nelson constructive logic with strong negation have been proposed so far. The first and more well-known, Nelson paraconsistent logic , results from dropping the explosion axiom of Nelson logic; a more recent series of papers considers the logic (dubbed quasi-Nelson logic ) obtained by rejecting the double negation law, which is thus also weaker than intuitionistic logic. The algebraic counterparts of these logical calculi are the varieties of N4-lattices and quasi-Nelson algebras . In the present paper we propose the class of quasi- N4-lattices as a common generalization of both. We show that a number of key results, including the twist-structure representation of N4-lattices and quasi-Nelson algebras, can be uniformly established in this more general setting; our new representation employs twist-structures defined over Brouwerian algebras enriched with a nucleus operator. We further show that quasi-N4-lattices form a variety that is arithmetical, possesses a ternary as well as a quaternary deductive term, and enjoys EDPC and the strong congruence extension property.

scholarly journals Negation and Implication in Quasi-Nelson Logic

2021 ◽  
Vol 27 (1) ◽  
pp. 107-123
Author(s):  
Thiago Nascimento ◽  
Umberto Rivieccio
Keyword(s):  
Intuitionistic Logic ◽  
Constructive Logic ◽  
Strong Negation ◽  
Completeness Result ◽  
Nelson Algebras ◽  
Nelson Logic

Quasi-Nelson logic is a recently-introduced generalization of Nelson’s constructive logic with strong negation to a non-involutive setting. In the present paper we axiomatize the negation-implication fragment of quasi-Nelson logic (QNI-logic), which constitutes in a sense the algebraizable core of quasi-Nelson logic. We introduce a finite Hilbert-style calculus for QNI-logic, showing completeness and algebraizability with respect to the variety of QNI-algebras. Members of the latter class, also introduced and investigated in a recent paper, are precisely the negation-implication subreducts of quasi-Nelson algebras. Relying on our completeness result, we also show how the negation-implication fragments of intuitionistic logic and Nelson’s constructive logic may both be obtained as schematic extensions of QNI-logic.

Nelson’s logic 𝒮

2020 ◽  
Author(s):  
Thiago Nascimento ◽  
Umberto Rivieccio ◽  
João Marcos ◽  
Matthew Spinks
Keyword(s):  
Residuated Lattices ◽  
The Other ◽  
Algebraic Semantics ◽  
Intended Interpretation ◽  
Nelson Logic

Abstract Besides the better-known Nelson logic ($\mathcal{N}3$) and paraconsistent Nelson logic ($\mathcal{N}4$), 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 (crucially lacking the contraction rule) with infinitely many rule schemata and no semantics (other than the intended interpretation into Arithmetic). We look here at the propositional fragment of $\mathcal{S}$, showing that it is algebraizable (in fact, implicative), 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 semantics for $\mathcal{S}$ as well as a finite Hilbert-style calculus equivalent to Nelson’s presentation; this also allows us to clarify the relation between $\mathcal{S}$ and the other two Nelson logics $\mathcal{N}3$ and $\mathcal{N}4$.

Which structural rules admit cut elimination? An algebraic criterion

2007 ◽  
Vol 72 (3) ◽  
pp. 738-754 ◽  
Author(s):  
Kazushige Terui
Keyword(s):  
Intuitionistic Logic ◽  
General Class ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Inference Rules ◽  
Lambek Calculus ◽  
Free Logic ◽  
Algebraic Semantics ◽  
Propagation Property ◽  
Structural Rules

AbstractConsider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structure-free logic. The aim of this work is to give such a condition by using algebraic semantics.We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set ℛ of structural rules, the cut elimination theorem holds for FL enriched with ℛ if and only if ℛ satisfies the propagation property.As an application, we show that any set ℛ of structural rules can be “completed” into another set ℛ*, so that the cut elimination theorem holds for FL enriched with ℛ*. while the provability remains the same.

scholarly journals Priestley duality for (modal) N4-lattices

10.29007/p4ch ◽  
2018 ◽  
Author(s):  
Ramon Jansana ◽  
Umberto Rivieccio
Keyword(s):  
Recent Work ◽  
Substructural Logics ◽  
Priestley Duality ◽  
Algebraic Semantics ◽  
Nelson Logic ◽  
Wide Family ◽  
Algebraic Counterpart

N4-lattices are the algebraic semantics of paraconsistent Nelson logic, which was introduced as an inconsistency-tolerant counterpart of the better-known logic of Nelson. Paraconsistent Nelson logic combines interesting features of intuitionistic, classical and many-valued logics (e.g., Belnap-Dunn four-valued logic); recent work has shown that it can also be seen as one member of the wide family of substructural logics.The work we present here is a contribution towards a better topological understanding of the algebraic counterpart of paraconsistent Nelson logic, namely a variety of involutive lattices called N4-lattices.

A Minimal Substructural Logic in Temporal Database

2014 ◽  
Vol 36 (8) ◽  
pp. 1592-1601 ◽  
Author(s):  
Dong-Ning LIU ◽  
Yong TANG ◽  
Shao-Hua TENG ◽  
Zhe LIN
Keyword(s):  
Substructural Logic ◽  
Temporal Database
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