Making Sense of Paraconsistent Logic: The Nature of Logic, Classical Logic and Paraconsistent Logic

2012 ◽  
pp. 15-25 ◽  
Author(s):  
Koji Tanaka
Keyword(s):  
Classical Logic ◽  
Paraconsistent Logic ◽  
Making Sense

scholarly journals Paraconsistent Annotated Logic Algorithms Applied in Management and Control of Communication Network Routes

Sensors ◽  
2021 ◽  
Vol 21 (12) ◽  
pp. 4219
Author(s):  
João Inácio Da Silva Filho ◽  
Jair Minoro Abe ◽  
Alessandro de Lima Marreiro ◽  
Angel Antonio Gonzalez Martinez ◽  
Cláudio Rodrigo Torres ◽  
...  
Keyword(s):  
Communication Network ◽  
Communication Networks ◽  
Classical Logic ◽  
Paraconsistent Logic ◽  
Logic Gates ◽  
Main Property ◽  
Computational Method ◽  
Stream Control ◽  
Routing Management ◽  
And Control

This paper presents a computational method based on non-classical logic dedicated to routing management and information stream control in communication networks. Paraconsistent logic (PL) was used to create an algorithmic structure whose main property is to accept contradiction. Moreover, a computational structure, the denominated paraconsistent data analyzer (PDAPAL2v), was constructed to perform routing management in communication networks. Direct comparisons of PDAPAL2v with a classical logic system that simulates routing conditions were made in the laboratory. In the conventional system, the paraconsistent algorithms were considered as binary logic gates, and in the tests, the same adjustment limits of PDAPAL2v were applied. Using a database with controlled insertion of noise, we obtained an efficacy of 97% in the detection of deteriorated packets with PDAPAL2v and 72% with the conventional simulation system. Functional tests were carried out, showing that PDAPAL2v is able to assess the conditions and degradation of links and perform the analysis and correlation of various inputs and variables, even if the signals have contradictory values. From practical tests in the laboratory, the proposed method represents a new way of managing and controlling communication network routes with good performance.

FACING INCONSISTENCY: THEORIES AND OUR RELATIONS TO THEM

Episteme ◽  
2013 ◽  
Vol 10 (4) ◽  
pp. 351-367 ◽  
Author(s):  
Michaelis Michael
Keyword(s):  
Classical Logic ◽  
Paraconsistent Logic ◽  
Indispensability Argument ◽  
Mathematical Objects ◽  
The Face ◽  
Inconsistent Objects ◽  
Inconsistent Theories

AbstractClassical logic is explosive in the face of contradiction, yet we find ourselves using inconsistent theories. Mark Colyvan, one of the prominent advocates of the indispensability argument for realism about mathematical objects, suggests that such use can be garnered to develop an argument for commitment to inconsistent objects and, because of that, a paraconsistent underlying logic. I argue to the contrary that it is open to a classical logician to make distinctions, also needed by the paraconsistent logician, which allow a more nuanced ranking of theories in which inconsistent theories can have different degrees of usefulness and productivity. Facing inconsistency does not force us to adopt an underlying paraconsistent logic. Moreover we will see that the argument to best explanation deployed by Colyvan in this context is unsuccessful. I suggest that Quinean approach which Colyvan champions will not lead to the revolutionary doctrines Colyvan endorses.

scholarly journals Rewriting the History of Connexive Logic

2022 ◽  
Author(s):  
Wolfgang Lenzen
Keyword(s):  
Twentieth Century ◽  
Classical Logic ◽  
Paraconsistent Logic ◽  
Historical Analysis ◽  
Relevance Logic ◽  
Overwhelming Majority ◽  
Lewis Carroll ◽  
Self Consistent ◽  
History Of ◽  
Connexive Logic

AbstractThe “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties (like the condition that no proposition implies its own negation) only apply to “normal” (e.g., self-consistent) antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning (“humbly”) connexive implication don’t give rise, however, to anything like a non-classical logic.

The logical inconsistency in making sense of an ineffable God of Islam

Philotheos ◽  
2020 ◽  
Vol 20 (1) ◽  
pp. 68-116
Author(s):  
Abbas Ahsan ◽  
Keyword(s):  
Classical Logic ◽  
Driving Force ◽  
Analytic Philosophy ◽  
Logical Inconsistency ◽  
Making Sense ◽  
The Law ◽  
Made In

With the advent of classical logic we are continuing to observe an adherence to the laws of logic. Moreover, the system of classical logic exhibits a prominent role within analytic philosophy. Given that the laws of logic have persistently endured in actively defining classical logic and its preceding system of logic, it begs the question as to whether it actually proves to be consistent with Islam. To consider this inquiry in a broader manner; it would be an investigation into the consistency between Islam and the logic which has been the predominant driving force of analytic philosophy. Despite the well documented engagement and novel contributions made in the field of logic by Arab and Islamic theologians/logicians, I think this question deserves examination not just in terms of classical logic but also from perspectives which go beyond classical logic, namely, non-classical logic. Doing so, would I believe, retain this inquiry within the purview of analytic philosophy despite the reference to non-classical logic. To be more specific, this question would be directed toward the Islamic theologian who espouses the system of classical logic in attempting to make sense of an absolute ineffable God of Islam. The inquiry would seek to determine if classical logic is consistent (amenable) in making sense of an absolute ineffable God of Islam. This would principally involve an analysis which determines whether the metaphysical assumptions of the laws of logic (more specifically the law of non-contradiction) are consistent in making sense of an absolute ineffable God of Islam. I shall argue that it is inconsistent. I shall establish my position on this matter by demonstrating why classical logic is inconsistent (not amenable) with an absolute ineffable God of Islam. Although, I am principally concerned with classical logic, my argument is as applicable to all earlier systems of logic as much as it is to classical logic. This is on the basis that both systems of logic, namely, all preceding systems and classical logic, consider the laws of logic as defining features.

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.

PARACONSISTENT CIRCUMSCRIPTION: PRELIMINARY REPORT

1996 ◽  
Vol 10 (06) ◽  
pp. 679-686 ◽  
Author(s):  
ZUOQUAN LIN
Keyword(s):  
Classical Logic ◽  
Preliminary Report ◽  
Paraconsistent Logic ◽  
Nonmonotonic Logic ◽  
Logic Of Paradox

In this paper we describe the paraconsistent circumscription by the application of predicate circumscription in a paraconsistent logic, the logic of paradox LP. In addition to circumscribing the predicates, we also circumscribe the inconsistency. The paraconsistent circumscription can be well characterized by the minimal semantics which is both nonmonotonic and paraconsistent. It brings us advantages in two respects: nonmonotonic logic would be nontrivial while there was a contradiction, and paraconsistent logic would be equivalent to classical logic while there was no effect of a contradiction.

scholarly journals Implication, Equivalence, and Negation

2021 ◽  
Vol 27 (1) ◽  
pp. 31-45
Author(s):  
Avron Arnon
Keyword(s):  
Classical Logic ◽  
Linear Logic ◽  
Paraconsistent Logic ◽  
Type System ◽  
Relevant Logic ◽  
Axiom Schema ◽  
Propositional Constants ◽  
Hilbert Type ◽  
Axiomatic Extension

A system $HCL_{\overset{\neg}{\leftrightarrow}}$ in the language of {$ \neg, \leftrightarrow $} is obtained by adding a single negation-less axiom schema to $HLL_{\overset{\neg}{\leftrightarrow}}$ (the standard Hilbert-type system for multiplicative linear logic without propositional constants), and changing $ \rightarrow $ to $\leftrightarrow$. $HCL_{\overset{\neg}{\leftrightarrow}}$ is weakly, but not strongly, sound and complete for ${\bf  CL}_{\overset{\neg}{\leftrightarrow}}$ (the {$ \neg,\leftrightarrow$} – fragment of classical logic). By adding the Ex Falso rule to $HCL_{\overset{\neg}{\leftrightarrow}}$ we get a system with is strongly sound and complete for ${\bf CL}_ {\overset{\neg}{\leftrightarrow}}$ . It is shown that the use of a new rule cannot be replaced by the addition of axiom schemas. A simple semantics for which $HCL_{\overset{\neg}{\leftrightarrow}}$ itself is strongly sound and complete is given. It is also shown that  $L_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , the logic induced by $HCL_{\overset{\neg}{\leftrightarrow}}$ , has a single non-trivial proper axiomatic extension, that this extension and ${\bf  CL}_{\overset{\neg}{\leftrightarrow}}$ are the only proper extensions in the language of { $\neg$, $\leftrightarrow$ } of $ {\bf  L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , and that $ {\bf  L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ and its single axiomatic extension are the only logics in {$ \neg, \leftrightarrow$ } which have a connective with the relevant deduction property, but are not equivalent $\neg$ to an axiomatic extension of ${\bf R}_{\overset{\neg}{\leftrightarrow}}$ (the intensional fragment of the relevant logic ${\bf R}$). Finally, we discuss the question whether $ {\bf  L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ can be taken as a paraconsistent logic.

scholarly journals What trends in non-classical logic were anticipated by Nikolai Vasiliev?

2013 ◽  
Vol 19 ◽  
pp. 122-135 ◽  
Author(s):  
В.И. Маркин
Keyword(s):  
Classical Logic ◽  
Paraconsistent Logic ◽  
Original System ◽  
Intensional Logic ◽  
Temporal Logics ◽  
Modal And Temporal Logics ◽  
Imaginary Logic

In this paper we discuss a question about the trends in non-classical logic that were exactly anticipated by Niko- lai Vasiliev. We show the influence of Vasiliev’s Imaginary logic on paraconsistent logic. Metatheoretical relations between Vasiliev’s logical systems and many-valued predicate logics are established. We also make clear that Vasiliev has developed a sketch of original system of intensional logic and expressed certain ideas of modal and temporal logics.

In Pursuit of the Non-Trivial

Episteme ◽  
2019 ◽  
pp. 1-16
Author(s):  
Colin R. Caret
Keyword(s):  
Classical Logic ◽  
Fundamental Problem ◽  
Paraconsistent Logic ◽  
Paraconsistent Logics ◽  
Scientific Theories ◽  
Preferential Treatment ◽  
Logical Reconstruction ◽  
Inconsistent Theories

AbstractThis paper is about the underlying logical principles of scientific theories. In particular, it concerns ex contradictione quodlibet (ECQ) the principle that anything follows from a contradiction. ECQ is valid according to classical logic, but invalid according to paraconsistent logics. Some advocates of paraconsistency claim that there are ‘real’ inconsistent theories that do not erupt with completely indiscriminate, absurd commitments. They take this as evidence in favor of paraconsistency. Michael (2016) calls this the non-triviality strategy (NTS). He argues that this strategy fails in its purpose. I will show that Michael's criticism significantly over-reaches. The fundamental problem is that he places more of a burden on the advocate of paraconsistency than on the advocate of classical logic. The weaknesses in Michael's argument are symptomatic of this preferential treatment of one viewpoint in the debate over another. He does, however, make important observations that allow us to clarify some of the complexities involved in giving a logical reconstruction of a theory. I will argue that there are abductive arguments deserving of further consideration for the claim that paraconsistent logic offers the best explanation of the practice of inconsistent science. In this sense, the debate is still very much open.

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