Paraconsistency is the study of logical systems with a non-explosive negation such that a pair of contradictory formulas (with respect to such negation) does not necessarily imply triviality, discordant to what would be expected by contemporary logical orthodoxy. From a purely logical point of view, the significance of paraconsistency relies on the meticulous distinction between the general notions of contradictoriness and triviality of a theory—respectively, the fact that a given theory proves a proposition and its negation, and the fact that (...) a given theory proves any proposition (in the language of its underlying logic). Aside from this simple rationale, the formal techniques and approaches that meet the latter definitional requirement are manifold. Furthermore, it is not solely the logical-mathematical properties of such systems that are open to debate. Rather, there are several foundational and philosophical questions worth studying, including the very question about the nature of the contradictions allowed by paraconsistentists. This entry aims to advance a brief account of some distinct approaches to paraconsistency, providing a panorama on the development of paraconsistent logic. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent (...) logical systems to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

It has been an open question whether or not we can define a belief revision operation that is distinct from simple belief expansion using paraconsistent logic. In this paper, we investigate the possibility of meeting the challenge of defining a belief revision operation using the resources made available by the study of dynamic epistemic logic in the presence of paraconsistent logic. We will show that it is possible to define dynamic operations of belief revision in a paraconsistent (...) setting. (shrink)

In this paper two systems of AGM-like Paraconsistent Belief Revision are overviewed, both defined over Logics of Formal Inconsistency (LFIs) due to the possibility of defining a formal consistency operator within these logics. The AGM° system is strongly based on this operator and internalize the notion of formal consistency in the explicit constructions and postulates. Alternatively, the AGMp system uses the AGM-compliance of LFIs and thus assumes a wider notion of paraconsistency - not necessarily related to the notion of (...) formal consistency. (shrink)

In this paper, the notion of degree of inconsistency is introduced as a tool to evaluate the sensitivity of the Full Bayesian Significance Test (FBST) value of evidence with respect to changes in the prior or reference density. For that, both the definition of the FBST, a possibilistic approach to hypothesis testing based on Bayesian probability procedures, and the use of bilattice structures, as introduced by Ginsberg and Fitting, in paraconsistent logics, are reviewed. The computational and theoretical advantages of (...) using the proposed degree of inconsistency based sensitivity evaluation as an alternative to traditional statistical power analysis is also discussed. (shrink)

This article is a preliminary presentation of conjunctive paraconsistency, the claim that there might be non-explosive true contradictions, but contradictory propositions cannot be considered separately true. In case of true ‘p and not p’, the conjuncts must be held untrue, Simplification fails. The conjunctive approach is dual to non-adjunctive conceptions of inconsistency, informed by the idea that there might be cases in which a proposition is true and its negation is true too, but the conjunction is untrue, Adjunction fails. While (...) non-adjunctivism is a well-known option, the other view is not so much studied nowadays, but it was not unknown in the tradition, and there are some positive suggestions, in recent literature, that the position is plausible and deserves to be developed. The article compares conjunctivism, non-adjunctivism and dialetheism, then focuses on some possible justifications, costs and benefits of the conjunctive view. (shrink)

The terms “model” and “model-building” have been used to characterize the field of formal philosophy, to evaluate philosophy’s and philosophical logic’s progress and to define philosophical logic itself. A model is an idealization, in the sense of being a deliberate simplification of something relatively complex in which several important aspects are left aside, but also in the sense of being a view too perfect or excellent, not found in reality, of this thing. Paraconsistent logic is a branch of philosophical (...) logic. It is however not clear how paraconsistent logic can be seen as model-building. What exactly is modeled? In this paper I adopt the perspective of looking at a particular instance of paraconsistent logic—paranormal modal logic—which might be seen as a model of a specific kind of agent: inductive agents. After ntroducing what I call the highlevel and low-level models of inductive agents, I analyze the extent to which the above-mentioned idealizing features of model-building appear in paranormal modal logic and how they affect its philosophical significance. (shrink)

In this paper, we present a non-trivial and expressively complete paraconsistent naïve theory of truth, as a step in the route towards semantic closure. We achieve this goal by expressing self-reference with a weak procedure, that uses equivalences between expressions of the language, as opposed to a strong procedure, that uses identities. Finally, we make some remarks regarding the sense in which the theory of truth discussed has a property closely related to functional completeness, and we present a sound (...) and complete three-sided sequent calculus for this expressively rich theory. (shrink)

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic (...) is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate and invalidate both versions of Explosion, such as classical logic and Asenjo–Priest’s 3-valued logic LP. On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics TS and ST, introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski’s and Frankowski’s q- and p-matrices, respectively. (shrink)

In this paper paraconsistent first-order logic LP^{#} with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^{#} is discussed.Axiomatical system HST^{#}as paraconsistent generalization of Hrbacek set theory HST is considered.

We introduce a modal expansion of paraconsistent Nelson logic that is also as a generalization of the Belnapian modal logic recently introduced by Odintsov and Wansing. We prove algebraic completeness theorems for both logics, defining and axiomatizing the corresponding algebraic semantics. We provide a representation for these algebras in terms of twiststructures, generalizing a known result on the representation of the algebraic counterpart of paraconsistent Nelson logic.

Two systems of belief change based on paraconsistent logics are introduced in this article by means of AGM-like postulates. The first one, AGMp, is defined over any paraconsistent logic which extends classical logic such that the law of excluded middle holds w.r.t. the paraconsistent negation. The second one, AGMo , is specifically designed for paraconsistent logics known as Logics of Formal Inconsistency (LFIs), which have a formal consistency operator that allows to recover all the classical inferences. (...) Besides the three usual operations over belief sets, namely expansion, contraction and revision (which is obtained from contraction by the Levi identity), the underlying paraconsistent logic allows us to define additional operations involving (non-explosive) contradictions. Thus, it is defined external revision (which is obtained from contraction by the reverse Levi identity), consolidation and semi-revision, all of them over belief sets. It is worth noting that the latter operations, introduced by S. Hansson, involve the temporary acceptance of contradictory beliefs, and so they were originally defined only for belief bases. Unlike to previous proposals in the literature, only defined for specific paraconsistent logics, the present approach can be applied to a general class of paraconsistent logics which are supraclassical, thus preserving the spirit of AGM. Moreover, representation theorems w.r.t. constructions based on selection functions are obtained for all the operations. (shrink)

This paper briefly outlines some advancements in paraconsistent logics for modelling knowledge representation and reasoning. Emphasis is given on the so-called Logics of Formal Inconsistency (LFIs), a class of paraconsistent logics that formally internalize the very concept(s) of consistency and inconsistency. A couple of specialized systems based on the LFIs will be reviewed, including belief revision and probabilistic reasoning. Potential applications of those systems in the AI area of KRR are tackled by illustrating some examples that emphasizes the (...) importance of a fine-tuned treatment of consistency in modelling reputation systems, preferences, argumentation, and evidence. (shrink)

There was a time when 'logic' just meant classical logic. The climate is slowly changing and non-classical logic cannot be dismissed off-hand. However, a metatheory used to study the properties of non-classical logic is often classical. In this paper, we will argue that this practice of relying on classical metatheories is problematic. In particular, we will show that it is a bad practice because the metatheory that is used to study a non-classical logic often rules out the very logic it (...) is designed to study. (shrink)

In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new (...) proof of S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency. (shrink)

This paper extends Fitting's epistemic interpretation of some Kleene logics, to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting's "cut-down" operator is discussed, rendering a "track-down" operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, (...) further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations. (shrink)

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once (...) the right set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants. (shrink)

Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic has not been made sense of, despite its well-developed mathematics. In this paper, I examine the nature of logic in order to understand what it means to make sense of logic. I then show that, just as one can make sense of non-normal modal logics (as Cresswell demonstrates), we can make `sense' of paraconsistent logic. Finally, I turn the tables on (...) classical logicians and ask what sense can be made of explosive reasoning. While I acknowledge a bias on this issue, it is not clear that even classical logicians can answer this question. (shrink)

In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non- contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to (...) philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we argue that an intuitive reading of the bivalued semantics for the logic mbC, a logic of formal inconsistency based on classical logic, fits in well with the basic ideas of an intuitive interpretation of contradictions. On this interpretation, the acceptance of a pair of propositions A and ¬A does not mean that A is simultaneously true and false, but rather that there is conflicting evidence about the truth value of A. (shrink)

'गोडेल के रास्ते' में तीन प्रख्यात वैज्ञानिकों ने अनिर्णय, अपूर्णता, यादृच्छिकता, गणनाऔरता और परासंगति जैसे मुद्दों पर चर्चा की। मैं Wittgensteinian दृष्टिकोण से इन मुद्दों दृष्टिकोण है कि वहाँ दो बुनियादी मुद्दों जो पूरी तरह से अलग समाधान है. वहाँ वैज्ञानिक या अनुभवजन्य मुद्दों, जो दुनिया के बारे में तथ्य है कि अवलोकन और दार्शनिक मुद्दों की जांच की जरूरत है के रूप में कैसे भाषा intelligibly इस्तेमाल किया जा सकता है (जो गणित और तर्क में कुछ सवाल शामिल हैं), (...) जो की जरूरत है एकटी कैसे हम वास्तव में विशेष संदर्भों में शब्दों का उपयोग देख कर फैसला किया. जब हम जो भाषा खेल हम खेल रहे हैं के बारे में स्पष्ट हो, इन विषयों को किसी भी अन्य की तरह साधारण वैज्ञानिक और गणितीय सवाल देखा जाता है. है Wittgenstein अंतर्दृष्टि शायद ही कभी बराबर किया गया है और कभी नहीं पार कर रहे हैं और के रूप में आज के रूप में प्रासंगिक हैं के रूप में वे 80 साल पहले थे जब वह ब्लू और ब्राउन पुस्तकें हुक्म दिया. अपनी असफलताओं के बावजूद-वास्तव में एक समाप्त पुस्तक के बजाय नोटों की एक श्रृंखला-यह इन तीन प्रसिद्ध विद्वानों के काम का एक अनूठा स्रोत है जो आधे से अधिक सदी से भौतिकी, गणित और दर्शन के खून बह रहा किनारों पर काम कर रहे हैं। दा कोस्टा और डोरिया Wolpert द्वारा उद्धृत कर रहे हैं (नीचे देखें या Wolpert पर मेरे लेख और Yanofsky 'कारण की बाहरी सीमा' की मेरी समीक्षा) के बाद से वे सार्वभौमिक गणना पर लिखा था, और उनके कई उपलब्धियों के बीच, दा कोस्टा में अग्रणी है paraconsistency. आधुनिक दो systems दृश्यसे मानव व्यवहार के लिए एक व्यापक अप करने के लिए तारीख रूपरेखा इच्छुक लोगों को मेरी पुस्तक 'दर्शन, मनोविज्ञान, मिनडी और लुडविगमें भाषा की तार्किक संरचना से परामर्श कर सकते हैं Wittgenstein और जॉन Searle '2 एड (2019). मेरे लेखन के अधिक में रुचि रखने वालों को देख सकते हैं 'बात कर रहेबंदर- दर्शन, मनोविज्ञान, विज्ञान, धर्म और राजनीति पर एक बर्बाद ग्रह --लेख और समीक्षा 2006-2019 3 एड (2019) और आत्मघाती यूटोपियान भ्रम 21st मेंसदी 4वें एड (2019) . (shrink)

Since its first appearance in 1966, the notion of a supervaluation has been regarded by many as a powerful tool for dealing with semantic gaps. Only recently, however, applications to semantic gluts have also been considered. In previous work I proposed a general framework exploiting the intrinsic gap/glut duality. Here I also examine an alternative account where gaps and gluts are treated on a par: although they reflect opposite situations, the semantic upshot is the same in both cases--the value of (...) some expressions is not uniquely defined. Other strategies for generalizing supervaluations are considered and some comparative facts are discussed. (shrink)

Принято считать, что невозможность, неполнота, Парапоследовательность, Несоответствие, Случайность, вычислительность, парадокс, неопределенность и пределы разума являются разрозненными научными физическими или математическими вопросами, имеющими мало или ничего общего. Я полагаю, что они в значительной степени стандартные философские проблемы (т.е. языковые игры), которые были в основном решены Витгенштейном более 80 лет назад. -/- Я предоставляю краткое резюме некоторых из основных выводов двух из самых выдающихся студентов поведения о Fсовременности, Людвиг Витгенштейн и Джон Сирл, на логическую структуру преднамеренности (ум, язык, поведение), принимая в качестве (...) отправной точки фундаментальное открытие Витгенштейна, что все действительно "философские" проблемы одинаковы-путаницы AB,как использовать язык в частности контекст, и поэтому все решения одинаковы, глядя на то, как язык может быть использован в рассматриваемом контексте, так что его условия истины (Условия удовлетворенности или COS) ясны. Основная проблема заключается в том, что можно сказать что угодно, но нельзя означать (государство ясно COS для) любое произвольное высказывание и смысл возможен только в очень специфическом контексте. -/- Я вскрыть некоторые писания некоторых из основных комментаторов по этим вопросам с точки зрения Витгенштейна в framework современной точки зрения двух систем мысли (популяризировал как "мышление быстро, думая медленно"), используя новую таблицу преднамеренности и новых двойных систем номенклатуры. Я показываю, что это мощная эвристическая для описания истинной природы этих предположенных научных, физических или математических вопросов, которые действительно лучше всего подходить как стандартные философские проблемы использования языка (языковые игры в терминологии Витгенштейна). (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs (...) recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices. (shrink)

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau (...) system with labels, signs, and suffixes is defined, extending the basic language $\mathscr{L}_{\mathbf{QB}}$ by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since $\phi \wedge \neg\phi$ does not ``explode'' with arbitrary syntactic consequences. (shrink)

Logic systems that can handle contradictions were being used for some time without having a general technical name. One of the main proposers of these systems, Newton da Costa, asked Francisco Miró Quesada to suggest him a name for those systems. In the historical letter that here we translate into English for the first time, Miró Quesada suggests three names to da Costa for this purpose: ‘ultraconsistent’, ‘metaconsistent’, and ‘paraconsistent’; explaining their pros and cons. -/- Paper based on a (...) letter by Francisco Miró Quesada Cantuarias to Newton da Costa, edited, translated, and annotated by Luis Felipe Bartolo Alegre. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...) ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at (...) how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. In spite of its failings—really a series of notes rather than a finished book—this is a unique source of the work of these three famous scholars who have been working at the bleeding edges of physics, math and philosophy for over half a century. Da Costa and Doria are cited by Wolpert (see below or my articles on Wolpert and my review of Yanofsky’s ‘The Outer Limits of Reason’) since they wrote on universal computation, and among his many accomplishments, Da Costa is a pioneer in paraconsistency. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for (...) both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed. (shrink)

Hal ini sering berpikir bahwa kemustahilan, ketidaklengkapan, Paraconsistency, Undecidability, Randomness, komputasi, Paradox, ketidakpastian dan batas alasan yang berbeda ilmiah fisik atau matematika masalah memiliki sedikit atau tidak ada dalam Umum. Saya menyarankan bahwa mereka sebagian besar masalah filosofis standar (yaitu, Permainan bahasa) yang sebagian besar diselesaikan oleh Wittgenstein lebih dari 80years yang lalu. -/- "Apa yang kita ' tergoda untuk mengatakan ' dalam kasus seperti ini, tentu saja, bukan filsafat, tetapi bahan baku. Jadi, misalnya, apa yang seorang matematikawan cenderung mengatakan (...) tentang objektivitas dan realitas fakta matematika, bukan filsafat matematika, tetapi sesuatu untuk pengobatan filosofis. " Wittgenstein PI 234 -/- "Filsuf terus melihat metode ilmu di depan mata mereka dan tak tertahankan tergoda untuk bertanya dan menjawab pertanyaan dalam cara ilmu tidak. Kecenderungan ini adalah sumber nyata metafisika dan memimpin filsuf menjadi gelap gulita. " Wittgenstein -/- Aku memberikan ringkasan singkat dari beberapa temuan utama dari dua siswa yang paling terkemuka perilaku zaman modern, Ludwig Wittgenstein dan John Searle, pada struktur Logis intensionality (pikiran, bahasa, perilaku), mengambil sebagai titik awal Penemuan fundamental Wittgenstein – bahwa semua masalah ' filosofis ' adalah sama — kebingungan tentang bagaimana menggunakan bahasa dalam konteks tertentu, sehingga semua solusi sama — melihat bagaimana bahasa dapat digunakan dalam konteks yang menjadi masalah sehingga kebenaranNya kondisi (kondisi kepuasan atau COS) jelas. Masalah dasar adalah bahwa seseorang dapat mengatakan apa-apa, tetapi orang tidak dapat berarti (negara yang jelas cos untuk) sembarang ucapan dan makna hanya mungkin dalam konteks yang sangat spesifik. -/- Saya membedah beberapa tulisan dari beberapa komentator utama pada isu ini dari sudut pandang Wittgensteinian dalam kerangka perspektif modern dari dua sistem pemikiran (Dipopulerkan sebagai ' berpikir cepat, berpikir lambat '), mempekerjakan meja baru intensionality dan baru sistem ganda nomenklatur. Saya menunjukkan bahwa ini adalah heuristik yang kuat untuk menggambarkan sifat sebenarnya dari hal ini ilmiah, fisik atau matematika masalah yang benar-benar terbaik didekati sebagai masalah filosofis standar bagaimana bahasa yang akan digunakan (permainan bahasa di Wittgenstein's terminologi). -/- Ini adalah pendapat saya bahwa tabel intensionality (rasionalitas, pikiran, pikiran, bahasa, kepribadian dll) yang fitur mencolok di sini menggambarkan lebih atau kurang akurat, atau setidaknya berfungsi sebagai heuristic untuk, bagaimana kita berpikir dan berperilaku, dan sehingga mencakup tidak hanya filsafat dan psikologi, tetapi segala sesuatu yang lain (sejarah, sastra, matematika, politik dll). Perhatikan terutama bahwa intensionalitas dan rasionalitas sebagai I (bersama dengan Searle, Wittgenstein dan lain-lain) melihatnya, mencakup baik sistem linguistik pertimbangan sadar 2 dan tidak disadari otomatis sistem prelinguistik 1 tindakan atau refleks. (shrink)

The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the (...) context of a subobject classifier, there should be a neutral label for the generic subobject, or if the label "true" appears instead, it carries no additional properties or assumptions. It has been shown that a natural isomorphism occurring in one of the definitions of topos need not be an isomorphism of the corresponding algebraic structures on the sets, so that a co-Heyting algebraic structure can be defined on the set of characteristic morphisms, which moreover will also be natural. The analyzes suggest that the notion of co-topos may be considered correct. However, it should be strongly emphasized that although the possibility of defining the notion of co-topos has been shown, its further full consequences should be very carefully examined, which can severely limit its logical applications. (shrink)

Consequence rleations over sets of "judgments" are defined by using "overdetermined" as well as "underdetermined" valuations. Some of these relations are shown to be categorical. And generalized soundness and completeness results are given for both multiple and single conclusion consequence relations.

In a recent work, Walter Carnielli and Abilio Rodrigues present an epistemically motivated interpretation of paraconsistent logic. In their view, when there is conflicting evidence with regard to a proposition A (i.e. when there is both evidence in favor of A and evidence in favor of ￢A) both A and ￢A should be accepted without thereby accepting any proposition B whatsoever. Hence, reasoning within their system intends to mirror, and thus, should be constrained by, the way in which we (...) reason about evidence. In this article we will thoroughly discuss their position and suggest some ways in which this project can be further developed. The aim of the paper is twofold. On the one hand, we will present some philosophical critiques to the specific epistemic interpretation of paraconsistent logic proposed by Carnielli & Rodrigues. First, we will contend that Carnielli & Rodrigues’s interpretation implies a thesis about what evidence rationally justifies to accept or believe, called Extreme Permissivism, which is controversial among epistemologists. Second, we will argue that what agents should do, from an epistemic point of view, when faced with conflicting evidence, is to suspend judgment. On the other hand, despite these criticisms we do not believe that the epistemological motivation put forward by Carnielli & Rodrigues is entirely wrong. In the last section, we offer an alternative way in which one might account for the epistemic rationality of accepting contradictions and, thus, for an epistemic understanding of paraconsistency, which leads us to discuss the notion of diachronic epistemic rationality. (shrink)

Dans 'Godel’s Way', trois éminents scientifiques discutent de questions telles que l’indécidabilité, l’incomplétude, le hasard, la calculabilité et la paraconsistence. J’aborde ces questions du point de vue de Wittgensteinian selon lesquelles il y a deux questions fondamentales qui ont des solutions complètement différentes. Il y a les questions scientifiques ou empiriques, qui sont des faits sur le monde qui doivent être étudiés de manière observationnelle et philosophique quant à la façon dont le langage peut être utilisé intelligiblement (qui incluent certaines (...) questions en mathématiques et en logique), qui doivent être décidés en regardant un comment nous utilisons réellement des mots dans des contextes particuliers. Lorsque nous obtenons clair sur le jeu de langue que nous jouons, ces sujets sont considérés comme des questions scientifiques et mathématiques ordinaires comme les autres. Les idées de Wittgenstein ont rarement été égalées et jamais dépassées et sont aussi pertinentes aujourd’hui qu’elles l’étaient il y a 80 ans lorsqu’il a dicté les Livres Bleus et Brown. Malgré ses défauts, vraiment une série de notes plutôt qu’un livre fini, c’est une source unique du travail de ces trois savants célèbres qui travaillent aux confins de la physique, des mathématiques et de la philosophie depuis plus d’un demi-siècle. Da Costa et Doria sont cités par Wolpert (voir ci-dessous ou mes articles sur Wolpert et mon examen de Yanofsky 'The Outer Limits of Reason') depuis qu’ils ont écrit sur le calcul universel, et parmi ses nombreuses réalisations, Da Costa est un pionnier dans la paraconsistence. -/- Ceux qui souhaitent un cadre complet à jour pour le comportement humain de la vue moderne de deux système peuvent consulter mon livre 'The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019). Ceux qui s’intéressent à plus de mes écrits peuvent voir «Talking Monkeys --Philosophie, Psychologie, Science, Religion et Politique sur une planète condamnée --Articles et revues 2006-2019 » 3e ed (2019) et Suicidal Utopian Delusions in the 21st Century 4th ed (2019) et autres. (shrink)

In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at (...) how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)

Priest holds anti-exceptionalism about logic. That is, he holds that logic, as a theory, does not have any exceptional status in relation to the theories of empirical sciences. Crucial to Priest’s anti-exceptionalism is the existence of ‘data’ that can force the revision of logical theory. He claims that classical logic is inadequate to the available data and, thus, needs to be revised. But what kind of data can overturn classical logic? Priest claims that the data is our intuitions about the (...) validity of inferences. In order to make sense of this claim, I will appeal to the Madhyamaka Buddhist philosopher Candrakīrti. I will then pose a problem for Priest’s anti-exceptionalism. Finally, I will suggest a way out of the problem for Priest. Whether or not he accepts my solution, I will let him decide. (shrink)

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to (...) be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences. (shrink)

在《哥德尔之路》中,三位杰出的科学家讨论了不可解性、不完整性、随机性、可估计性和副一致性等问题。我从维特根斯坦的观点出发来处理这些问题,即有两个基本问题有着完全不同的解决方案。有科学或经验问题,这是关 于世界的事实,需要研究观察和哲学问题,如何使用语言可理解(其中包括数学和逻辑中的某些问题),需要通过查看我们在特定上下文中实际使用单词的方式来决定。当我们清楚要玩哪种语言游戏时,这些话题就像其他话题一 样被视为普通的科学和数学问题。维特根斯坦的见解很少被平等,也从未被超越,今天和80年前他口述《蓝书》和《棕色书》时一样具有现实意义。尽管它的失败——实际上是一系列笔记,而不是一本已完成的书——这是这三 位著名学者作品的独特来源,他们半个多世纪以来一直在物理学、数学和哲学的流血边缘工作。达科斯塔和多里亚被沃尔珀特引用(见下文或我的文章沃尔珀特和我对亚诺夫斯基的"理性的外在极限"的评 论),因为他们写了通用计算,在他的许多成就中,达科斯塔是先驱参数一致性。 那些希望从现代两个系统的观点来看为人类行为建立一个全面的最新框架的人,可以查阅我的书《路德维希的哲学、心理学、Mind 和语言的逻辑结构》维特根斯坦和约翰·西尔的《第二部》(2019年)。那些对我更多的作品感兴趣的人可能会看到《会说话的猴子——一个末日星球上的哲学、心理学、科学、宗教和政治——文章和评论2006-201 9年第3次(2019年)和自杀乌托邦幻想21篇世纪4日 (2019) .

By Belief Revision it is understood a system that logically explains the rational process of changing beliefs by taking into account a new piece of information. The most influential approach in this field of study, the AGM system, proposed by Alchourrón, Gärdenfors, and Makinson, postulates rationality criteria for different types of belief change. In this paper I shall assess the relationship between those criteria and argue for an opposition between the principles of Information Economy and Consistency. Furthermore, I shall argue (...) that Paraconsistent Belief Revision manages to minimise this friction in the best possible way. (shrink)

Revision operation is the consistent expansion of a theory by a new belief-representing sentence. We consider that in a paraconsistent setting this desideratum can be accomplished in at least three distinct ways: the output of a revision operation should be either non-trivial or non-contradictory (in general or relative to the new belief). In this paper those distinctions will be explored in the constructive level by showing how the remainder sets could be refined, capturing the key concepts of paraconsistency in (...) a dynamical scenario. These are preliminaries results of a wider project on Paraconsistent Belief Change conduced by the authors. (shrink)

In the present paper, our objective is to examine the application of belief revision models to scientific rationality. We begin by considering the standard model AGM, and along the way a number of problems surface that make it seem inadequate for this specific application. After considering three different heuristics of informational economy that seem fit for science, we consider some possible adaptations for it and argue informally that, overall, some paraconsistent models seem to better satisfy these principles, following Testa (...) (2015). These models have been worked out in formal detail by Testa, Cogniglio, & Ribeiro (2015, 2017). (shrink)

“Trends in Logic XVI: Consistency, Contradiction, Paraconsistency, and Reasoning - 40 years of CLE” is being organized by the Centre for Logic, Epistemology and the History of Science at the State University of Campinas (CLEUnicamp) from September 12th to 15th, 2016, with the auspices of the Brazilian Logic Society, Studia Logica and the Polish Academy of Sciences. The conference is intended to celebrate the 40th anniversary of CLE, and is centered around the areas of logic, epistemology, philosophy and history of (...) science, while bringing together scholars in the fields of philosophy, logic, mathematics, computer science and other disciplines who have contributed significantly to what Studia Logica is today and to what CLE has achieved in its four decades of existence. It intends to celebrate CLE’s strong influence in Brazil and Latin America and the tradition of investigating formal methods inspired by, and devoted to, philosophical views, as well as philosophical problems approached by means of formal methods. The title of the event commemorates one of the three main areas of CLE, what has been called the “Brazilian school of paraconsistency”, combining such a pluralist view about logic and reasoning. (shrink)

In 2016 Beziau, introduce a more restricted concept of paraconsistency, namely the genuine paraconsistency. He calls genuine paraconsistent logic those logic rejecting φ, ¬φ |- ψ and |- ¬(φ ∧ ¬φ). In that paper the author analyzes, among the three-valued logics, which of these logics satisfy this property. If we consider multiple-conclusion consequence relations, the dual properties of those above mentioned are: |- φ, ¬φ, and ¬(ψ ∨ ¬ψ) |- . We call genuine paracomplete logics those rejecting the mentioned (...) properties. We present here an analysis of the three-valued genuine paracomplete logics. (shrink)

Dalam ' Godel ' s Way ' tiga ilmuwan terkemuka membahas isu seperti undecidability, ketidaklengkapan, kekasaran, komputasi dan paraconsistency. Saya mendekati masalah ini dari sudut pandang Wittgensteinian bahwa ada dua masalah dasar yang memiliki solusi yang sama sekali berbeda. Ada masalah ilmiah atau empiris, yang merupakan fakta tentang dunia yang perlu diselidiki masalah observationally dan filosofis mengenai bagaimana bahasa dapat digunakan secara jelas (yang mencakup pertanyaan tertentu dalam matematika dan logika), yang perlu diputuskan dengan mencarit bagaimana kita benar-benar menggunakan kata (...) dalam konteks tertentu. Ketika kita mendapatkan jelas tentang mana permainan bahasa yang kita bermain, topik ini dipandang sebagai pertanyaan ilmiah dan matematika biasa seperti orang lain. Wawasan Wittgenstein jarang sama dan tidak pernah melampaui dan seperti yang berkaitan dengan hari ini karena mereka 80 tahun yang lalu ketika dia mendikte buku Blue and Brown. Terlepas dari kegagalan-benar serangkaian catatan daripada buku selesai-ini adalah sumber yang unik dari pekerjaan tiga sarjana terkenal yang telah bekerja di tepi berdarah fisika, matematika dan filsafat selama lebih dari setengah abad. Da Costa dan Doria dikutip oleh Wolpert (Lihat di bawah atau artikel saya di Wolpert dan saya review yanofsky's ' The Outer batas dari alasan ') karena mereka menulis di Universal komputasi, dan di antara banyak prestasi, da Costa adalah pelopor dalam paraconsistency. -/- Mereka yang ingin komprehensif up to date kerangka perilaku manusia dari dua systEMS tampilan modern dapat berkonsultasi buku saya 'struktur Logis filsafat, psikologi, mind dan bahasa dalam Ludwig wittgenstein dan John Searle ' 2nd Ed (2019). Mereka yang tertarik pada tulisan saya lebih mungkin melihat 'berbicara monyet--filsafat, psikologi, ilmu, agama dan politik di planet yang ditakdirkan--artikel dan review 2006-2019 3rd ed (2019) dan bunuh diri utopian delusi di 21st Century 4th Ed (2019) . (shrink)

In celebration of Newton da Costa’s place in the history of paraconsistency, this paper considers the use and abuse of deliberate self-contradiction. Beginning with Parmenides, developed by Plato, and continued by Cicero, an ancient philosophical tradition used deliberately paraconsistent discourses to reveal the truth. In modern times, decisionism has used deliberate self-contradiction against Judeo-Christian revelation. • DOI:10.5007/1808-1711.2010v14n1p1.

Review of the book "Paraconsistent Logic: Consistency, Contradiction, and Negation by Water Carnielli and Marcelo Coniglio.

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...) equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

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. (shrink)

We present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language. We shall defend the view according to which logics of formal inconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically acceptable account of paraconsistency.

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann constant. (...) An overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)