Logics based on linear orders of contaminating values

2019 ◽  
Vol 29 (5) ◽  
pp. 631-663 ◽  
Author(s):  
Roberto Ciuni ◽  
Thomas Macaulay Ferguson ◽  
Damian Szmuc
Keyword(s):  
Classical Logic ◽  
Infinite Family ◽  
Linear Ordering ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Linear Orders ◽  
Wide Range ◽  
Countably Infinite ◽  
Wide Family ◽  
Classical Truth

AbstractA wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula $\varphi $, any complex formula in which $\varphi $ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behaviour is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper.

scholarly journals Logical consequence relations in logics of quasiary predicates

2016 ◽  
pp. 029-043 ◽  
Author(s):  
O.S. Shkilniak ◽  
Keyword(s):  
Classical Logic ◽  
Logical Consequence ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
The Difference

Logical consequence is one of the most fundamental concepts in logic. A wide use of partial (sometimes many-valued as well) mappings in programming makes important the investigation of logics of partial and many-valued predicates and logical consequence relations for them. Such relations are a semantic base for a corresponding sequent calculi construction. In this paper we consider logical consequence relations for composition nominative logics of total single-valued, partial single-valued, total many-valued and partial many-valued quasiary predicates. Properties of the relations can be different for different classes of predicates; they coincide in the case of classical logic. Relations of the types T, F, TF, IR and DI were in-vestigated in the earlier works. Here we propose relations of the types TvF and С for logics of quasiary predicates. The difference between these two relations manifests already on the propositional level. Properties of logical consequence relations are specified for formulas and sets of formulas. We consider partial cases when one of the sets of formulas is empty. It is shown that relations P|=TvF and R|=С are non-transitive, some properties of decomposition of formulas are not true for R|=С, but at the same time the latter can be modelled through R|=TF. A number of examples demonstrates particularities and distinctions of the defined relations. We also establish a relationship among various logical consequence relations.

Sequent Calculi for Global Modal Consequence Relations

Studia Logica ◽  
2018 ◽  
Vol 107 (4) ◽  
pp. 613-637
Author(s):  
Minghui Ma ◽  
Jinsheng Chen
Keyword(s):  
Consequence Relations ◽  
Sequent Calculi

The extensions of the modal logic K5

1985 ◽  
Vol 50 (1) ◽  
pp. 102-109 ◽  
Author(s):  
Michael C. Nagle ◽  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Abstract Characterization ◽  
Propositional Modal Logic ◽  
Normal Logic ◽  
Countably Infinite ◽  
Infinite Set

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

Non-Isomorphic Maximal Orders with Isomorphic Matrix Rings

2000 ◽  
Vol 43 (4) ◽  
pp. 413-417
Author(s):  
A. W. Chatters
Keyword(s):  
Infinite Family ◽  
Matrix Rings ◽  
Maximal Orders ◽  
Countably Infinite

AbstractWe construct a countably infinite family of pairwise non-isomorphic maximal ℚ[X]-orders such that the full 2 by 2 matrix rings over these orders are all isomorphic.

Yet another constructivization of classical logic

1998 ◽  
Author(s):  
Stefano Baratella ◽  
Stefano Berardi
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
A Posteriori ◽  
Classical Formula ◽  
Constructive Proofs ◽  
Classical Truth

The aim of this paper is to provide a way of extracting the constructive content of a certain family of classical proofs directly from the proofs themselves. The paper itself is written in a purely constructive style. Our work is inspired by the game interpretations of classical logic due to Novikov (1943) and Coquand (1995). These interpretations date back to Gentzen (1969) and Bernays (1970) and were recently studied by Coquand (1995) who made use of technical tools developed by Novikov (1943). We will introduce an interpretation which is a short and compact description of the meaning assigned to classical formulas by Coquand’s interpretation. Contrary to Coquand, we will completely avoid any game terminology, by making use of the intuitionistic notion of continuous computation. A posteriori, our interpretation turns out to be related to Kreisel’s no-counterexample interpretation (Kreisel 1957) but, compared with his, it provides simpler constructive proofs. The reader is referred to Baratella and Berardi (1997) for a number of examples of constructive proofs provided by our interpretation that can be used for a comparison. Indeed, our interpretation is a fragment of Coquand’s that can be easily expanded to a variant of his. However, we claim that our interpretation suffices as long as we are only interested in the constructive meaning of classical formulas (whilst we need Coquand’s if we are interested in computations lying behind the constructive meaning). We will support this claim by proving, as the main result, that our interpretation is intuitionistically complete, in the same way as Coquand’s (Herbelin 1995). That is, we will intuitionistically prove that a formula is derivable in infmitary classical logic if and only if its interpretation holds. Since infinitary classical logic is classically complete, loosely speaking we can restate our result as follows: the classical truth of a classical formula is intuitionistically equivalent to the intuitionistic truth of the constructive interpretation of the formula. We also recall that Godel’s Dialectica interpretation is not intuitionistically complete (see section 7). In this regard, see also Berardi (1997). In addition, we point out that, contrary to the game interpretations, our interpretation is not a sort of reformulation of what is going on in the sequent calculus.

Consistent disjunctive sequent calculi and Scott domains

2021 ◽  
pp. 1-24
Author(s):  
Longchun Wang ◽  
Qingguo Li
Keyword(s):  
Fixed Point ◽  
Propositional Logic ◽  
Continuous Functions ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Continuous Solutions ◽  
Syntactic Representation ◽  
Scott Continuous ◽  
Recursive Domain Equations ◽  
Standard Domain

Abstract Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

Hydrostatic compression model for sandy soils

2008 ◽  
Vol 45 (8) ◽  
pp. 1169-1179 ◽  
Author(s):  
Javier Vallejos
Keyword(s):  
Void Ratio ◽  
Sandy Soils ◽  
Infinite Family ◽  
Hydrostatic Compression ◽  
Stress Scale ◽  
Wide Range ◽  
Different Types ◽  
Loading And Unloading ◽  
Unique Framework ◽  
Mean Stresses

Based on elastic considerations, a new four parameter model of the hydrostatic compression curves of sandy soils is presented. Using a unique set of parameters, this model is capable of accurately representing the infinite family of compression curves that result when soil is subjected to stresses below particle breakage. The parameters are readily estimated from two hydrostatic tests conducted at different initial densities during loading and unloading stages. The resulting equations are able to predict the effect of progressive stiffness increase and compressibility reduction that occur as void ratio decreases during both loading and unloading conditions. The same normalized stress scale employed for the hydrostatic compression curves can also be used to describe the steady state line, which leads to a unique framework for representing the stress–void ratio curves of sandy soils. A comprehensive series of simulations on 13 different types of sandy soils in both loose and dense states was performed under a wide range of mean stresses during loading and unloading conditions. The simulations gave very satisfactory agreement with published experimental results.

scholarly journals Labeled Sequent Calculus for Orthologic

2018 ◽  
Vol 47 (4) ◽  
Author(s):  
Tomoaki Kawano
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Sequent Calculi

Orthologic (OL) is non-classical logic and has been studied as a part of quantumlogic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this paper, we introduce new labeled sequent calculus called LGOI, and show that this sequent calculus solve the above problems. It is alreadyknown that OL is decidable. We prove that decidability is preserved when theimplication connective is added to OL.

scholarly journals A novel family of 1-D robust chaotic maps

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dhrubajyoti Mandal
Keyword(s):  
Mathematical Models ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Infinite Family ◽  
Practical Applications ◽  
Smooth Maps ◽  
Wide Range ◽  
Chaotic Signals ◽  
Piecewise Smooth Maps ◽  
Parameter Values

AbstractChaotic dynamics of various continuous and discrete-time mathematical models are used frequently in many practical applications. Many of these applications demand the chaotic behavior of the model to be robust. Therefore, it has been always a challenge to find mathematical models which exhibit robust chaotic dynamics. In the existing literature there exist a very few studies of robust chaos generators based on simple 1-D mathematical models. In this paper, we have proposed an infinite family consisting of simple one-dimensional piecewise smooth maps which can be effectively used to generate robust chaotic signals over a wide range of the parameter values.

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