Rewriting the History of Connexive Logic

The “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.

Paraconsistent annotated logic applied to industry assets condition monitoring and failure prevention based on vibration signatures

In this study, we introduced an expert system (ESvbrPAL2v), responsible for monitoring assets based on vibration signature analysis through a set of algorithms based on the Paraconsistent Annotated Logic – PAL. Being a non-classical logic, the main feature of the PAL is to support contradictory inputs in its foundation. It is therefore suitable for building algorithmic models capable of performing out appropriate treatment for complex signals, such as those coming from vibration. The ESvbrPAL2v was built on an ATMega2560 microcontroller, where vibration signals were captured from the mechanical structures of the machines by sensors and, after receiving special treatment through the Discrete Fourier Transform (DFT), then properly modeled to paraconsistent logic signals and vibration patterns. Using the PAL fundamentals, vibration signature patterns were built for possible and known vibration issues stored in ESvbrPAL2v and continuously compared through configurations composed by a network of paraconsistent algorithms that detects anomalies and generate signals that will report on the current risk status of the machine in real time. The tests to confirm the efficiency of ESvbrPAL2v were performed in analyses initially carried out on small prototypes and, after the initial adjustments, tests were carried out on bearings of a group of medium-power motor generators built specifically for this study. The results are shown at the end of this study and have a high index of signature identification and risk of failure detection. These results justifies the method used and future applications considering that ESvbrPAL2v is still in its first version.

THE SYSTEM OF PERSONAL PRONOUNS IN NATURAL LANGUAGE: A MEREOTOPOLOGICAL APPROACH

Проанализирована логическая структура системы подлежащих естественного языка, выраженных местоимениями или аналогичными с лингвистической точки зрения, объектами, а именно: личными местоимениями, как эксплицитными, так и имплицитными, подлежащими неопределенно-личных и безличных (pro) предложений, а также нефинитных клауз (PRO). В процессе анализа оценивалось содержание имплицитных пропозиций, соответствующих тому или иному выбору подлежащего (агента) из универсума агентов. Для процедуры логического анализа был разработан мереотопологический метод, позволяющий работать как с классической, так и неконсистентными логиками. Показано, что подлежащие (агенты) в естественном языке могут выбираться как по процедурам, соответствующим классической логике, так и по процедурам, соответствующим различным неконсистентным логикам. А именно, подлежащие безличных предложений (pro) выбираются соответственно паракомплектной логике (допускающей контрарное противоречие), подлежащие нефинитных клауз (PRO) выбираются соответственно параконсистентной логике (допускающей субконтрарное противоречие), а подлежащие как неопределенно-личных, так и автореферентных предложений выбираются соответственно неалетической логике (допускающей контрадикторные противоречия). The logical structure of the system of subjects in the natural language is examined. The subjects were limited to those expressed by either pronouns or other similar (from a linguistic viewpoint) phenomena, namely: personal pronouns (both explicit and implicit), subjects of indefinite sentences, subjects of impersonal sentences (pro), and subjects of non-finite clauses (PRO). The implicit propositions corresponding to different modes of choosing the subjects (agents) from the universe of agents are analysed. For this purpose, a mereotopological method has been developed which allows to deal with both consistent (classical) and inconsistent logics. It was demonstrated that, in the natural language, the subjects (agents) can be chosen using the procedures that are governed by either consistent or inconsistent logic. Namely, the subjects of impersonal sentences (pro) are to be chosen according to the paracomplete logic (allowing the contrary contradiction), and the subjects of non-finite clauses (PRO), according to the paraconsistent logic (allowing the subcontrary contradiction), whereas the subjects of both indefinite and self-referential sentences are to be chosen according to the non-alethic logic (allowing the contradictory contradiction).

Characterization and classification of numerical data patterns using Annotated Paraconsistent Logic and the effect of contradiction

This work describes the development of a computational mathematical model that uses Annotated Paraconsistent Logic - APL and a concept derived from it, the effect of contradiction, to identify patterns in numerical data for pattern classification purposes. The APL admits paraconsistent and paracomplete logical principles, which allow the manipulation of inconsistent and contradictory data, and its use allowed the identification and quantization of the attribute related to the contradiction. To validate the model, series of Raman spectroscopies obtained after exposure of proteins, lipids and nucleic acids, collected from cutaneous tissue cell samples previously examined for the detection of cancerous lesions, identified as basal carcinoma, melanoma and normal, were used. Initially, the attributes related to contradiction, derivative and median obtained from spectroscopies were identified and quantified. A machine learning process with approximately 31.6% of each type of samples detects a sequence of spectroscopies capable of characterizing and classifying the type of lesion through the chosen attributes. Approximately 68.4% of the samples are used for classification tests. The proposed model identified a segment of spectroscopies where the classification of test samples had a hit rate of 76.92%. As a differential and innovation of this work, the use of APL principles in a complete process of training, learning and classification of patterns for numerical data sets stands out, with flexibility to choose the attributes used for the characterization of patterns, and a quantity of samples of about one third of the total required for characterization.

Quasi-N4-Lattices

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.

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

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.

Model Theory in a Paraconsistent Environment

The purpose of this thesis is to develop a paraconsistent Model Theory. The basis for such a theory was launched by Walter Carnielli, Marcelo Esteban Coniglio, Rodrigo Podiack, and Tarcísio Rodrigues in the article ‘On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency’ [The Review of Symbolic Logic, vol. 7 (2014)].Naturally, a complete theory cannot be fully developed in a single work. Indeed, the goal of this work is to show that a paraconsistent Model Theory is a sound and worthy possibility. The pursuit of this goal is divided in three tasks: The first one is to give the theory a philosophical meaning. The second one is to transpose as many results from the classical theory to the new one as possible. The third one is to show an application of the theory to practical science.The response to the first task is a Paraconsistent Reasoning System. The start point is that paraconsistency is an epistemological concept. The pursuit of a deeper understanding of the phenomenon of paraconsistency from this point of view leads to a reasoning system based on the Logics of Formal Inconsistency. Models are regarded as states of knowledge and the concept of isomorphism is reformulated so as to give raise to a new concept that preserves a portion of the whole knowledge of each state. Based on this, a notion of refinement is created which may occur from inside or from outside the state.In order to respond to the second task, two important classical results, namely the Omitting Types Theorem and Craig’s Interpolation Theorem are shown to hold in the new system and it is also shown that, if classical results in general are to hold in a paraconsistent system, then such a system should be in essence how it was developed here.Finally, the response to the third task is a proposal of what a Paraconsistent Logic Programming may be. For that, the basis for a paraconsistent PROLOG is settled in the light of the ideas developed so far.Abstract prepared by Bruno Costa Coscarelli.E-mail: [email protected]: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

Implication, Equivalence, and Negation

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.