scholarly journals On the strongest three-valued paraconsistent logic contained in classical logic and its dual

2020 ◽  
Author(s):  
C A Middelburg
Keyword(s):  
Equivalence Relation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Paraconsistent Logic ◽  
The Other ◽  
Paper Properties ◽  
Logical Equivalence

Abstract $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ is a three-valued paraconsistent propositional logic that is essentially the same as J3. It has the most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic. However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned. In this paper, properties concerning the logical equivalence relation of a logic are used to distinguish $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ from the others. As one of the bonuses of focusing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent and commutative laws for conjunction and disjunction. For most properties of $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ that have been proposed as desirable properties of a reasonable paraconsistent propositional logic, its paracomplete analogue has a comparable property. In this paper, properties concerning the logical equivalence relation of a logic are also used to distinguish the paracomplete analogue of $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ from the other three-valued paracomplete propositional logics with those comparable properties.

scholarly journals Behavioural reasoning for conditional equations

2007 ◽  
Vol 17 (5) ◽  
pp. 1075-1113 ◽  
Author(s):  
MANUEL A. MARTINS ◽  
DON PIGOZZI
Keyword(s):  
Equivalence Relation ◽  
Wide Class ◽  
Propositional Logic ◽  
Object Oriented ◽  
Equational Logic ◽  
Classical Propositional Logic ◽  
Logical Equivalence ◽  
Programming Techniques ◽  
Hidden Data ◽  
Partial Specification

Object-oriented (OO) programming techniques can be applied to equational specification logics by distinguishing visible data from hidden data (that is, by distinguishing the output of methods from the objects to which the methods apply), and then focusing on the behavioural equivalence of hidden data in the sense introduced by H. Reichel in 1984. Equational specification logics structured in this way are called hidden equational logics, HELs. The central problem is how to extend the specification of a given HEL to a specification of behavioural equivalence in a computationally effective way. S. Buss and G. Roşu showed in 2000 that this is not possible in general, but much work has been done on the partial specification of behavioural equivalence for a wide class of HELs. The OO connection suggests the use of coalgebraic methods, and J. Goguen and his collaborators have developed coinductive processes that depend on an appropriate choice of a cobasis, which is a special set of contexts that generates a subset of the behavioural equivalence relation. In this paper the theoretical aspects of coinduction are investigated, specifically its role as a supplement to standard equational logic for determining behavioural equivalence. Various forms of coinduction are explored. A simple characterisation is given of those HELs that are behaviourally specifiable. Those sets of conditional equations that constitute a complete, finite cobasis for a HEL are characterised in terms of the HEL's specification. Behavioural equivalence, in the form of logical equivalence, is also an important concept for single-sorted logics, for example, sentential logics such as the classical propositional logic. The paper is an application of the methods developed through the extensive work that has been done in this area on HELs, and to a broader class of logics that encompasses both sentential logics and HELs.

scholarly journals Control effects as a modality

2009 ◽  
Vol 19 (1) ◽  
pp. 17-26 ◽  
Author(s):  
HAYO THIELECKE
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
The Other ◽  
Classical Propositional Logic ◽  
Double Negation ◽  
Other Hand

AbstractWe combine ideas from types for continuations, effect systems and monads in a very simple setting by defining a version of classical propositional logic in which double-negation elimination is combined with a modality. The modality corresponds to control effects, and it includes a form of effect masking. Erasing the modality from formulas gives classical logic. On the other hand, the logic is conservative over intuitionistic logic.

scholarly journals On density of truth of the intuitionistic logic in one variable

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Zofia Kostrzycka
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
Propositional Variable ◽  
Intuitionistic Propositional Logic ◽  
International Audience

International audience In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment $\{ \to ,\vee ,\bot \}$ of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case.

scholarly journals ON ISOTOPISMS OF COMMUTATIVE PRESEMIFIELDS AND CCZ-EQUIVALENCE OF FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1243-1258 ◽  
Author(s):  
LILYA BUDAGHYAN ◽  
TOR HELLESETH
Keyword(s):  
Equivalence Relation ◽  
The Other ◽  
Ccz Equivalence ◽  
Positive Divisor ◽  
Positive Integers

A function F from Fpnto itself is planar if for any [Formula: see text] the function F(x+a)-F(x) is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime p and any positive integers n and m, the indicators of the graphs of functions F and F' from Fpnto Fpmare CCZ-equivalent if and only if F and F′ are CCZ-equivalent.We also prove that, for any odd prime p, CCZ-equivalence of functions from Fpnto Fpm, is strictly more general than EA-equivalence when n ≥ 3 and m is greater or equal to the smallest positive divisor of n different from 1.

Generic substitutions

2005 ◽  
Vol 70 (1) ◽  
pp. 61-83 ◽  
Author(s):  
Giovanni Panti
Keyword(s):  
Continuous Function ◽  
Free Algebra ◽  
Classical Logic ◽  
Propositional Logic ◽  
Dual Space ◽  
Extensive Literature ◽  
Cantor Space ◽  
Natural Measure ◽  
The Subject ◽  
Stochastic Properties

AbstractUp to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these.

scholarly journals THE UBIQUITY OF CONSERVATIVE TRANSLATIONS

2012 ◽  
Vol 5 (4) ◽  
pp. 666-678 ◽  
Author(s):  
EMIL JEŘÁBEK
Keyword(s):  
Propositional Logic ◽  
Broad Class ◽  
Paraconsistent Logic ◽  
Lambek Calculus ◽  
Classical Propositional Logic ◽  
Consequence Relation ◽  
Deductive Systems ◽  
Nonclassical Logics ◽  
Conservative Translations

AbstractWe study the notion of conservative translation between logics introduced by (Feitosa & D’Ottaviano2001). We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.

Principal type-schemes and condensed detachment

1990 ◽  
Vol 55 (1) ◽  
pp. 90-105 ◽  
Author(s):  
J. Roger Hindley ◽  
David Meredith
Keyword(s):  
Type Theory ◽  
Propositional Logic ◽  
Modus Ponens ◽  
The Other ◽  
Combinatory Logic ◽  
Minimal Amount ◽  
Principal Type ◽  
Condensed Detachment

The condensed detachment rule, or ruleD, was first proposed by Carew Meredith in the 1950's for propositional logic based on implication. It is a combination of modus ponens with a “minimal” amount of substitution. We shall give a precise detailed statement of rule D. (Some attempts in the published literature to do this have been inaccurate.)The D-completeness question for a given set of logical axioms is whether every formula deducible from the axioms by modus ponens and substitution can be deduced instead by rule D alone. Under the well-known formulae-as-types correspondence between propositional logic and combinator-based type-theory, rule D turns out to correspond exactly to an algorithm for computing principal type-schemes in combinatory logic. Using this fact, we shall show that D is complete for intuitionistic and classical implicational logic. We shall also show that D is incomplete for two weaker systems, BCK- and BCI-logic.In describing the formulae-as-types correspondence it is common to say that combinators correspond to proofs in implicational logic. But if “proofs” means “proofs by the usual rules of modus ponens and substitution”, then this is not true. It only becomes true when we say “proofs by rule D”; we shall describe the precise correspondence in Corollary 6.7.1 below.This paper is written for readers in propositional logic and in combinatory logic. Since workers in one field may not feel totally happy in the other, we include short introductions to both fields.We wish to record thanks to Martin Bunder, Adrian Rezus and the referee for helpful comments and advice.

scholarly journals Here and There with Arithmetic

2021 ◽  
pp. 1-15
Author(s):  
VLADIMIR LIFSCHITZ
Keyword(s):  
Programming Language ◽  
Propositional Logic ◽  
Answer Set Programming ◽  
The Other ◽  
Arithmetic Operations ◽  
First Order ◽  
The Relationship ◽  
Answer Set

Abstarct In the theory of answer set programming, two groups of rules are called strongly equivalent if, informally speaking, they have the same meaning in any context. The relationship between strong equivalence and the propositional logic of here-and-there allows us to establish strong equivalence by deriving rules of each group from rules of the other. In the process, rules are rewritten as propositional formulas. We extend this method of proving strong equivalence to an answer set programming language that includes operations on integers. The formula representing a rule in this language is a first-order formula that may contain comparison symbols among its predicate constants, and symbols for arithmetic operations among its function constants. The paper is under consideration for acceptance in TPLP.

scholarly journals Forward and Backward Chaining with P Systems

2014 ◽  
pp. 149-158
Author(s):  
Sergiu Ivanov ◽  
Artiom Alhazov ◽  
Vladimir Rogojin ◽  
Miguel A. Gutiérrez-Naranjo
Keyword(s):  
Computer Science ◽  
Propositional Logic ◽  
Membrane Computing ◽  
Modus Ponens ◽  
The Other ◽  
P Systems ◽  
Backward Chaining ◽  
Other Hand

One of the concepts that lie at the basis of membrane computing is the multiset rewriting rule. On the other hand, the paradigm of rules is profusely used in computer science for representing and dealing with knowledge. Therefore, establishing a “bridge” between these domains is important, for instance, by designing P systems reproducing the modus ponens-based forward and backward chaining that can be used as tools for reasoning in propositional logic. In this paper, the authors show how powerful and intuitive the formalism of membrane computing is and how it can be used to represent concepts and notions from unrelated areas.

scholarly journals An Holistic Extension for Classical Logic via Quantum Fredkin Gate

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1178
Author(s):  
Hector Freytes ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Computation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Classical Propositional Logic ◽  
Mixed States ◽  
Fredkin Gate ◽  
Bipartite States ◽  
Extended Notion

An holistic extension for classical propositional logic is introduced in the framework of quantum computation with mixed states. The mentioned extension is obtained by applying the quantum Fredkin gate to non-factorizable bipartite states. In particular, an extended notion of classical contradiction is studied in this holistic framework.

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