A Taxonomy of C-systems
TL;DR: An enormous variety of paraconsistent logics in the literature is shown to constitute C- System, and a novel notion of consistency is introduced.
Abstract: A thorough investigation of the foundations of paraconsistent logics. Relations between logical principles are formally studied, a novel notion of consistency is introduced, the logics of formal inconsistency, and the subclasses of C-systems and dC-systems are defined and studied. An enormous variety of paraconsistent logics in the literature is shown to constitute C-systems.
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01 Jan 2007
TL;DR: The Logicas da Inconsistencia Formal (LIFs) as mentioned in this paper form a classe of logicas paraconsistentes particularmente expressivas, in which a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to.
Abstract: Segundo a pressuposicao de consistencia classica, as contradicoes tem um cara[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocinio sensato pode entao ter lugar. Uma logica e paraconsistente se ela rejeita uma tal pressuposicao, e aceita ao inves que algumas teorias inconsistentes conquanto nao-triviais facam perfeito sentido. A? Logicas da Inconsistencia Formal, LIFs, formam uma classe de logicas paraconsistentes particularmente expressivas nas quais a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to. Como consequencia, as LIFs sao capazes de recapturar o raciocinio consistente pelo acrescimo de assuncoes de consistencia apropriadas. Assim, por exemplo, enquanto regras classicas tais como o silogismo disjuntivo (de A e {nao-,4)-ou-13, infira B) estao fadadas a falhar numa logica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presuncao de que estamos raciocinando em um ambiente consistente (neste caso, pelo acrescimo de (consistente-.A) como uma hipotese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustracoes destas logicas e de suas propriedades, mostrando que tais logicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocinio consistente dentro de tais sistemas inconsistentes sao tambem ilustradas Em cada caso, interpretacoes em termos de semânticas polivalentes, de traducoes possiveis ou modais sao fornecidas, e os problemas relacionados a provisao de contrapartidas algebricas para tais logicas sao examinados. Uma abordagem formal abstra[cjta e proposta para todas as definicoes relacionadas e uma extensa investigacao e feita sobre os principios logicos e as propriedades positivas e negativas da negacao
Abstract
348 citations
01 Sep 2006
TL;DR: This work provides a measure to quantify the inconsistency of a knowledgebase, thereby allowing for the comparison of the consistency of various knowledgebases, represented as first-order logic formulas, using quasi-classical (QC) logic.
Abstract: It is well-known that knowledgebases may contain inconsistencies. We provide a measure to quantify the inconsistency of a knowledgebase, thereby allowing for the comparison of the inconsistency of various knowledgebases, represented as first-order logic formulas. We use quasi-classical (QC) logic for this purpose. QC logic is a formalism for reasoning and analysing inconsistent information. It has been used as the basis of a framework for measuring inconsistency in propositional theories. Here we extend this framework, by using a first-order logic version of QC logic for measuring inconsistency in first-order theories. We motivate the QC logic approach by considering some formulae as database or knowledgebase integrity constraints. We then define a measure of extrinsic inconsistency that can be used to compare the inconsistency of different knowledgebases. This measure takes into account both the language used and the underlying domain. We show why this definition also captures the intrinsic inconsistency of a knowledgebase. We also provide a formalization of paraconsistent equality, called quasi-equality, and we use this in an extended example of an application for measuring inconsistency between heterogeneous sources of information and integrity constraints prior to merging.
217 citations
Cites background from "A Taxonomy of C-systems"
...Paraconsistent reasoning is important in handling inconsistent information, and there have been a number of proposals for paraconsistent logics (for reviews see [Hun98, CM02 , Pri02])....
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01 Jan 2004
TL;DR: The main aim of this paper is to review the measures of information and contradiction, and to study some potential practical applications, which have significant potential in developing intelligent systems that can be tolerant to inconsistencies when reasoning with real-world knowledge.
Abstract: Measures of quantity of information have been studied extensively for more than fifty years. The seminal work on information theory is by Shannon [67]. This work, based on probability theory, can be used in a logical setting when the worlds are the possible events. This work is also the basis of Lozinskii's work [48] for defining the quantity of information of a formula (or knowledgebase) in propositional logic. But this definition is not suitable when the knowledgebase is inconsistent. In this case, it has no classical model, so we have no “event” to count. This is a shortcoming since in practical applications (e.g. databases) it often happens that the knowledgebase is not consistent. And it is definitely not true that all inconsistent knowledgebases contain the same (null) amount of information, as given by the “classical information theory”. As explored for several years in the paraconsistent logic community, two inconsistent knowledgebases can lead to very different conclusions, showing that they do not convey the same information. There has been some recent interest in this issue, with some interesting proposals. Though a general approach for information theory in (possibly inconsistent) logical knowledgebases is missing. Another related measure is the measure of contradiction. It is usual in classical logic to use a binary measure of contradiction: a knowledgebase is either consistent or inconsistent. This dichotomy is obvious when the only deductive tool is classical inference, since inconsistent knowledgebases are of no use. But there are now a number of logics developed to draw non-trivial conclusions from an inconsistent knowledgebase. So this dichotomy is not sufficient to describe the amount of contradiction of a knowledgebase, one needs more fine-grained measures. Some interesting proposals have been made for this. The main aim of this paper is to review the measures of information and contradiction, and to study some potential practical applications. This has significant potential in developing intelligent systems that can be tolerant to inconsistencies when reasoning with real-world knowledge.
138 citations
Cites background from "A Taxonomy of C-systems"
...[11, 7]); Paraconsistent logics for which there is an object operator denoting “acceptable” inconsistency that can be used to differentiate acceptable and unacceptable inconsistencies [ 13 ]; Approximate entailment for which two sequences of entailment relation are defined (the first is sound but not complete, and the second is complete but not sound) which converge to classical entailment [65]; and Partial consistency checking for which ......
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...[28, 13 , 60]), so choosing a particular logic is already a real, non-trivial commitment....
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...A number of useful proposals have been made in the field of paraconsistent logics (see for example [28, 13 ])....
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TL;DR: New logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic are introduced and form part of a much larger family of similar logics.
Abstract: This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning.
116 citations
Cites background from "A Taxonomy of C-systems"
...Equivalently, if we call explosive the logics in which the Principle of Explosion: A, ¬A S B holds, for any A and B, we can assert then that a logical system is paraconsistent if, and only if, it is non-explosive (the equivalence of the two formulations above holds at least for logics having a m onotonic consequence relation —see [ CM01 ]—, like the ones we study here)....
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...Remember from last section that a logic with a consequence relation is called paraconsistent if it is non-explosive, that is, if there are formulas A and B such that A, ¬A / B. Following [CM99], we say that a logic has a strong negation if it has an operator ~ such that A, ~A B, for any formulas A and B. A paraconsistent logic in which all positive inferences hold and a strong negation is present is said to constitute a C-system (see ......
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...The reader should be aware that the concepts of contradict ion and inconsistency are not necessarily to be identified, as argued in [ CM01 ], and that distinct philosophical positions can be taken according to whether or not one adopts such an identification....
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...[ CM01 ] for a thorough discussion about such logics)....
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01 Jan 2007
TL;DR: Paraconsistent logics (PL) as discussed by the authors are logics of inconsistent but nontrivial theories, i.e., theories in which there is a formula (a grammatically well-formed expression of its language) such that the formula and its negation are both theorems of the theory; otherwise, the theory is called consistent.
Abstract: Publisher Summary This chapter discusses paraconsistent logics (PL) and paraconsistency. PL are the logics of inconsistent but nontrivial theories. A deductive theory is paraconsistent if its underlying logic is paraconsistent. A theory is inconsistent if there is a formula (a grammatically well-formed expression of its language) such that the formula and its negation are both theorems of the theory; otherwise, the theory is called consistent. A theory is trivial if all formulas of its language are theorems. In a trivial theory “everything” (expressed in its language) can be proved. If the underlying logic of a theory is classical logic, or even any of the standard logical systems such as intuitionistic logic, inconsistency entails triviality, and conversely. This chapter discusses da Costa's C-logics. This chapter elaborates on paraconsistent set theories, and shows, in particular, how they accommodate inconsistent objects, such as the Russell set. Ja´skowski's discussive logic is examined, and it is showed how it can be used in the formulation of the concept of partial truth. The chapter also examines annotated logic, and some of its applications.
104 citations
References
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Book•
01 May 2014
TL;DR: A collection of classic essays written throughout Popper's illustrious career, expounding and defending his 'fallibilist' theory of knowledge and scientific discovery.
Abstract: A collection of classic essays written throughout Popper's illustrious career, expounding and defending his 'fallibilist' theory of knowledge and scientific discovery. He applies his thinking not only to the philosophy of science, but also to a range of other concerns, from political theory to the mind-body problem.
4,621 citations
"A Taxonomy of C-systems" refers methods in this paper
...nd this was, in fact, a rediscovery of an argument used by the Pseudo-Scotus, much before.14 The use of contraposition (THEOREM 3.20) to the same purpose was pointed out in an argument by Popper (cf. [89], pp.320ff). Of course, in a logic where both the disjunctive syllogism and contraposition are invalid derivations, these arguments do not apply as such. 14 See Duns Scotus’s Opera Omnia, pp.288ff. Cf...
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TL;DR: In this paper, a process for sizing cellulose fibers or cellulose fiber containing materials and a composition for carrying out the process are described, and a method for sizing according to the general formula of R1 is presented.
Abstract: The present invention relates to a process for sizing cellulose fibers or cellulose fiber containing materials and to a composition for carrying out the process. More particularly the invention relates to a process for sizing according to which cellulose fibers or cellulose fiber containing materials in a manner known per se are brought into contact with compounds having the general formula WHEREIN R1 is an organic, hydrophobic group having 8 to 40 carbon atoms and R2 is an alkyl group having 1 to 7 carbon atoms or has the same meaning as R1.
1,915 citations
TL;DR: In this article, the notion of material adequacy of a definition of truth is defined as follows: if all these equivalences follow from it, then the definition is true if, and only if, p.X is true.
Abstract: X is true if, and only if, p. [...] we shall call a definition of truth “adequate” if all these equivalences follow from it. [...] The definition of truth which was outlined above [...] implies all equivalences of the form (T). In this connnection it is important to notice that the conditions for the material adequacy of the definition determine uniquely the extension of the term “true.” Therefore, every definition of truth which is materially adequate would necessarily be equivalent to that actually constructed. The semantic conception of truth gives us, so to speak, no possibility of choice between variaous non-equivalent definitions of this notion..
940 citations
Book•
01 Jun 1937
TL;DR: Carnap's entire theory of language structure appeared in The Logical Syntax of Language (1934) as mentioned in this paper, which led to his famous "Principle of tolerance" by which everyone is free to mix and match the rules of his logic in any way he wishes.
Abstract: Rudolf Carnap's entire theory of Language structure "came to me," he reports, "like a vision during a sleepless night in January 1931, when I was ill." This theory appeared in The Logical Syntax of Language (1934). Carnap argued that many philosophical controversies really depend upon whether a particular language form should be used. This leads him to his famous "Principle of tolerance" by which everyone is free to mix and match the rules of his language and therefore his logic in any way he wishes. In this way, philosophical issues become reduced to a discussion of syntactical properties, plus reasons of practical convenience for preferring one form of language to another. In a tour de force of precise reasoning, Carnap also indicated how two model languages could be constructed. This is one of three books which Open Court is making available in paperback reprint in its Open Court Classics series. The other two are Carnap's The Logical Structure of the World and Schlick's General Theory of Knowledge.
894 citations