In his Lectures on the History of Philosophy, Hegel develops a subtle analysis of Megarian paradoxes: the Liar, the Veiled Man and the Sorites. In this paper, we focus on Hegel's interpretation of...

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 mathematical concept of pragmatic truth, first introduced in Mikenberg, da Costa and Chuaqui (1986), has received in the last few years several applications in logic and the philosophy of science. In this paper, we study the logic of pragmatic truth, and show that there are important connections between this logic, modal logic and, in particular, Jaskowski's discussive logic. In order to do so, two systems are put forward so that the notions of pragmatic validity and pragmatic truth can be (...) accommodated. One of the main results of this paper is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is also discussed. (shrink)

We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

The article focuses on representing different forms of non-adjunctive inference as sub-Kripkean systems of classical modal logic, where the inference from □A and □B to □A ∧ B fails. In particular we prove a completeness result showing that the modal system that Schotch and Jennings derive from a form of non-adjunctive inference in (Schotch and Jennings, 1980) is a classical system strictly stronger than EMN and weaker than K (following the notation for classical modalities presented in Chellas, 1980). The unified (...) semantical characterization in terms of neighborhoods permits comparisons between different forms of non-adjunctive inference. For example, we show that the non-adjunctive logic proposed in (Schotch and Jennings, 1980) is not adequate in general for representing the logic of high probability operators. An alternative interpretation of the forcing relation of Schotch and Jennings is derived from the proposed unified semantics and utilized in order to propose a more fine-grained measure of epistemic coherence than the one presented in (Schotch and Jennings, 1980). Finally we propose a syntactic translation of the purely implicative part of Jaśkowski's system D₂ into a classical system preserving all the theorems (and non-theorems) explicilty mentioned in (Jaśkowski, 1969). The translation method can be used in order to develop epistemic semantics for a larger class of non-adjunctive (discursive) logics than the ones historically investigated by Jaśkowski. (shrink)

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and Cila. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa's logic C_1) but the whole hierarchy of da Costa's calculi (...) C_n. This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each C_n, which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa's hierarchy of C-systems as a family of (non deterministically) (n+2)-valued logics, where n is the number of "inconsistently true" truth-values and 2 is the number of "classical" or "consistent" truth-values, for every C_n. (shrink)

Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short (...) to get rapidly into the logical issues. This paper is relatively self-contained. A modest background on propositional modal logic is, though not strictly necessary, advisable. The reader might find useful the Compass papers Kracht (2011) and Negri (2011) (though these papers cover issues of more complexity than what is demanded to follow this paper). (shrink)

The aim of this paper is to present an algebraic approach to Jaśkowski’s paraconsistent logic D2. We present: a D2-discursive algebra, Lindenbaum- Tarski algebra for D2 and D2-matrices. The analysis is mainly based on the results obtained by Jerzy Kotas in the 70s.

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range (...) of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (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 metalogical 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 and by the logics of formal undeterminedness. 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. 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 LFI and undeterminedness, 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 one at a time. 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 nondeterministic matrices. (shrink)

From the technical point of view, philosophically neutral, the duality between a paraconsistent and a paracomplete logic lies in the fact that explosion does not hold in the former and excluded middle does not hold in the latter. From the point of view of the motivations for rejecting explosion and excluded middle, this duality can be interpreted either ontologically or epistemically. An ontological interpretation of intuitionistic logic is Brouwer’s idealism; of paraconsistency is dialetheism. The epistemic interpretation of intuitionistic logic is (...) in terms of preservation of constructive proof; of paraconsistency is in terms of preservation of evidence. In this paper, we explain and defend the epistemic approach to paraconsistency. We argue that it is more plausible than dialetheism and allows a peaceful and fruitful coexistence with classical logic. (shrink)

Chakraborty and Banerjee have introduced a rough consequence logic based on the modal logic S5. This paper shows that rough consequence logics, with many of the same properties, can be based on modal logics as weak as K, with a simpler formulation than that of Chakraborty and Banerjee. Also provided are decision procedures for the rough consequence logics and equivalences and independence relations between various systems S and the rough consequence logics, based on them. It also shows that each logic, (...) based on such an S, is theorem equivalent, but not necessarily equivalent, to the modal logic M-S. The paper also shows that rough consequence logic, which was designed to handle rough equality, is somewhat limited for that purpose. (shrink)

Though the social world is real and objective, the way that social facts arise out of other facts is in an important way shaped by human thought, talk and behaviour. Building on recent work in social ontology, I describe a mechanism whereby this distinctive malleability of social facts, combined with the possibility of basic human error, makes it possible for a consistent physical reality to ground an inconsistent social reality. I explore various ways of resisting the prima facie case for (...) social inconsistency. I conclude, however, that the prima facie case survives scrutiny, and draw out some of the ramifications. (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

Even when inconsistencies are present in our premise set, we can sensibly distinguish between good and bad arguments relying on these premises. In making this distinction, the inconsistency-adaptive approach of Batens strikes a particularly nice balance between inconsistency-tolerance and inferential strength. In this paper, we use the machinery of Batens’ approach to extend the paraconsistent approach to dialogical logic as developed by Rahman and Carnielli. In bringing these frameworks closer together, we obtain a dynamic mechanism for the systematic study of (...) dialogues in which two parties exchange arguments over a central claim, in the possible presence of inconsistencies. (shrink)

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A very common twofold view in contemporary philosophy is that classical logic is the correct view of logical consequence and that possibility conforms to classical logic in the sense that ‘possible worlds’ — whatever else they may be — are closed under classical logic. These two views are assumed in this paper. My aim in this paper is to show that a very natural ‘paraconsistent’ consequence relation is involved in the given view of possible worlds and logical consequence.

A common and much-explored thought is Łukasiewicz's idea that the future is ‘indeterminate’—i.e., ‘gappy’ with respect to some claims—and that such indeterminacy bleeds back into the present in the form of gappy ‘future contingent’ claims. What is uncommon, and to my knowledge unexplored, is the dual idea of an overdeterminate future—one which is ‘glutty’ with respect to some claims. While the direct dual, with future gluts bleeding back into the present, is worth noting, my central aim is simply to sketch (...) and briefly explore an alternative glutty-future view, one that is conservative—indeed, entirely classical—with respect to the present. The structure of the paper runs as follows. §1 briefly sketches the target gap picture of an indeterminate future yielding gappy claims at the present. §2 presents the direct dual idea—a glut picture of an overdeterminate future yielding glutty claims at present. §3 sketches the central idea, a more interesting glut picture in which the future contains contradictory states but the present remains entirely classical. §4 contains a general defence of the idea, leaving it open as to whether the gappy-future view enjoys substantive virtues over the proposed glutty-future view of §3. (shrink)

. This paper proposes a generalization of the theory of the process of explanation to include consistent as well as inconsistent situations. The generalization is strong, for example in the sense that, if the background theory and the initial conditions are consistent, it leads to precisely the same results as the theory from the lead paper (Halonen and Hintikka 2004). The paper presupposes (and refers to arguments for the view that) inconsistencies constitute problems and that scientists try to resolve them.

In an interesting experimental study, Bonini et al. (1999) present partial support for truth-gap theories of vagueness. We say this despite their claim to find theoretical and empirical reasons to dismiss gap theories and despite the fact that they favor an alternative, epistemic account, which they call ‘vagueness as ignorance’. We present yet more experimental evidence that supports gap theories, and argue for a semantic/pragmatic alternative that unifies the gappy supervaluationary approach together with its glutty relative, the subvaluationary approach.

The problem of future contingents is regarded as an important philosophical problem in connection with determinism and it should be treated by tense logic. Prior’s early work focused on the problem, and later Prior studied branching-time tense logic which was invented by Kripke. However, Prior’s idea to use three-valued logic for the problem seems to be still alive. In this paper, we consider partial and paraconsistent approaches to the problem of future contingents. These approaches theoretically meet Aristotle’s interpretation of future (...) contingents. (shrink)

Jaśkowski's discussive (discursive) logic D2 is historically one of the first paraconsistent logics, i.e., logics which 'tolerate' contradictions. Following Jaśkowski's idea to define his discussive logic by means of the modal logic S5 via special translation functions between discussive and modal languages, and supporting at the same time the tradition of paracomplete logics being the counterpart of paraconsistent ones, we present a paracomplete discussive logic D2p.

Henry Kyburg’s lottery paradox (1961, p. 197) arises from considering a fair 1000 ticket lottery that has exactly one winning ticket. If this much is known about the execution of the lottery it is therefore rational to accept that one ticket will win. Suppose that an event is very likely if the probability of its occurring is greater than 0.99. On these grounds it is presumed rational to accept the proposition that ticket 1 of the lottery will not win. Since (...) the lottery is fair, it is rational to accept that ticket 2 won’t win either—indeed, it is rational to accept for any individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 won’t win, accepting that ticket 2 won’t win, . . . , and accepting that ticket 1000 won’t win entails that it is rational to accept that no ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no ticket wins. (shrink)

Objects often manifest themselves in incompatible ways across perspectives that are epistemically on a par. The standard response to such cases is to deny that the properties that things appear to have from different perspectives are properties that things really have out there. This type of response seems worrying: too many properties admit of perspectival variance and there are good theoretical reasons to think that such properties are genuinely instantiated. So, we have reason to explore views on which things can (...) have the incompatible properties they appear to have across perspectives. This paper explores the view that things can have incompatible properties if the world is not a metaphysically unified place but is instead fragmented. There is a sensible notion of co-obtainment, on which two facts can each obtain without co-obtaining. Using this notion, we can step back from our embedded perspectives on the world without deeming the contents of those perspectives to be mere appearances. This renders th.. (shrink)

In 1948 Jaśkowski introduced the first discussive logic. The main technical idea was to take what holds to be what is true at some possible world. Some 2,000 years earlier, Jain philosophers had advocated a similar idea, in their doctrine of _syādvāda_. Of course, these philosophers had no knowledge of contemporary logical notions; but the techniques pioneered by Jaśkowski can be deployed to make the Jain ideas mathematically precise. Moreover, Jain ideas suggest a new family of many-valued discussive logics. In (...) this paper, I will explain all these matters. (shrink)

This book, provides a critical approach to all major logical paradoxes: from ancient to contemporary ones. There are four key aims of the book: 1. Providing systematic and historical survey of different approaches – solutions of the most prominent paradoxes discussed in the logical and philosophical literature. 2. Introducing original solutions of major paradoxes like: Liar paradox, Protagoras paradox, an unexpected examination paradox, stone paradox, crocodile, Newcomb paradox. 3. Explaining the far-reaching significance of paradoxes of vagueness and change for philosophy (...) and ontology. 4. Proposing a novel, well justified and, as it seems, natural classification of paradoxes. (shrink)

This volume examines the concept of falsification as a central notion of semantic theories and its effects on logical laws. The point of departure is the general constructivist line of argument that Michael Dummett has offered over the last decades. From there, the author examines the ways in which falsifications can enter into a constructivist semantics, displays the full spectrum of options, and discusses the logical systems most suitable to each one of them. While the idea of introducing falsifications into (...) the semantic account is Dummett's own, the many ways in which falsificationism departs quite radically from verificationism are here spelled out in detail for the first time. The volume is divided into three large parts. The first part provides important background information about Dummett’s program, intuitionism and logics with gaps and gluts. The second part is devoted to the introduction of falsifications into the constructive account and shows that there is more than one way in which one can do this. The third part details the logical effects of these various moves. In the end, the book shows that the constructive path may branch in different directions: towards intuitionistic logic, dual intuitionistic logic and several variations of Nelson logics. The author argues that, on balance, the latter are the more promising routes to take. "Kapsner’s book is the first detailed investigation of how to incorporate the notion of falsification into formal logic. This is a fascinating logico-philosophical investigation, which will interest non-classical logicians of all stripes." Graham Priest, Graduate Center, City University of New York and University of Melbourne. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

The AGM theory of belief revision provides a formal framework to represent the dynamics of epistemic states. In this framework, the beliefs of the agent are usually represented as logical formulas while the change operations are constrained by rationality postulates. In the original proposal, the logic underlying the reasoning was supposed to be supraclassical, among other properties. In this paper, we present some of the existing work in adapting the AGM theory for non-classical logics and discuss their interconnections and what (...) is still missing for each approach. (shrink)

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$$\scriptstyle \omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is (...) to define the deductive limit to this hierarchy, that is, its greatest lower deductive bound. The calculus $ \mathcal {C}$min, stronger than $ \mathcal {C}$$\scriptstyle \omega$, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of $ \mathcal {C}$min presented thereupon, possible-translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible-translations semantics are then used to provide both a semantics and a decision procedure for $ \mathcal {C}$Lim, the real deductive limit of da Costa's hierarchy. Possible-translations semantics also make it possible to characterize a precise sense of duality: as an example, $ \mathcal {D}$min is proposed as the dual to $ \mathcal {C}$min. (shrink)

David Lewis has argued that impossible worlds are nonsense: if there were such worlds, one would have to distinguish between the truths about their contradictory goings-on and contradictory falsehoods about them; and this--Lewis argues--is preposterous. In this paper I examine a way of resisting this argument by giving up the assumption that ‘in so-and-so world’ is a restricting modifier which passes through the truth-functional connectives The outcome is a sort of subvaluational semantics which makes a contradiction ‘A & ~A’ false (...) even when both ‘A’ and ‘~A’ are true, just as supervaluational semantics makes a tautology ‘A v ~A’ true even when neither ‘A’ nor ‘~A’ are. Connections with discussive logics and complications of the account are discussed, and some general morals are drawn. (shrink)

Scientific explanations arc subject to the occurrence of inconsistencies. To rule them out in many cases demands the construction of new theories. As the examples of complementary explanations show, that may take a while. Furthermore, even if possible in principle, it is not always reasonable to eliminate inconsistencies immediately, e.g., by bringing in a more sophisticated formal language. After all, under some circumstances a provisional, not fully coherent explanation may be better than none. In any case, we need a logically (...) controlled approach to such inconsistencies. Modern logic provides the tools which are necessary to solve this task. We will mention two alternative approaches. (shrink)

Considering the growing need for formal counterparts of causal nexus (AI is desperately looking for a good one!) and thus trying to construct appropriate relations within a formal framework one faces the problem that the notion of “causal connection” is by no means explained with sufficient precision. How to overcome the resulting difficulties?

Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.

In this paper, we consider the class of four-valued literal-paraconsistent-paracomplete logics constructed by combination of isomorphs of classical logic CPC. These logics form a 10-element upper semi-lattice with respect to the functional embeddinig one logic into another. The mechanism of variation of paraconsistency and paracompleteness properties in logics is demonstrated on the example of two four-element lattices included in the upper semi-lattice. Functional properties and sets of tautologies of corresponding literal-paraconsistent-paracomplete matrices are investigated. Among the considered matrices there are the (...) matrix of Puga and da Costa's logic V and the matrix of paranormal logic P1I1, which is the part of a sequence of paranormal matrices proposed by V. Fernández. (shrink)

The rule of adjunction is intuitively appealing and uncontroversial for deductive inference, but in situations where information can be uncertain, the rule is neither needed nor wanted for rational acceptance, as illustrated by the lottery paradox. Practical certainty is the acceptance of statements whose chances of error are smaller than a prescribed threshold parameter, when evaluated against an evidential corpus. We examine the failure of adjunction in relation to the threshold parameter for practical certainty, with an eye towards reinstating the (...) rule of adjunction in some restricted forms, by observing the conditions under which the overall chance of error of the joint statements can be variously bounded. (shrink)

Adopting a non-classical logic may not imply resigning the classical theories that have proven their worth. Nevertheless, the project of classical recapture poses some challenges, some of them specific to paraconsistent approaches. In this article, we analyse the consequences of introducing a recovery operator to subvaluationist logic. We argue that the classical recovery can indeed be carried out in a subvaluationist setting, but that doing so amounts to committing to a hierarchy of recaptures.

In this two-parts paper we present paranormal modal logic: a modal logic which is both paraconsistent and paracomplete. Besides using a general framework in which a wide range of logics  including normal modal logics, paranormal modal logics and classical logic can be defined and proving some key theorems about paranormal modal logic (including that it is inferentially equivalent to classical normal modal logic), we also provide a philosophical justification for the view that paranormal modal logic is a formalization of (...) the notions of skeptical and credulous plausibility. (shrink)

Pavel Florensky (1882–1937), a Russian theologian, philosopher, and mathematician, argued that the religious discourse is essentially contradictory and put forward the idea of the logical theory of antinomies. Recently his views raised interesting discussions among logicians who consider him a forerunner of many non-classical logics. In this paper I discuss four interpretations of Florensky’s views: paraconsistent, L-contradictory, non-monotonic and rhetorical. In conclusion I argue for the integral interpretation which unites these four approaches.

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.

We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova.

Being a pragmatic and not a referential approach tosemantics, the dialogical formulation ofparaconsistency allows the following semantic idea tobe expressed within a semi-formal system: In anargumentation it sometimes makes sense to distinguishbetween the contradiction of one of the argumentationpartners with himself (internal contradiction) and thecontradiction between the partners (externalcontradiction). The idea is that externalcontradiction may involve different semantic contextsin which, say A and ¬A have been asserted.The dialogical approach suggests a way of studying thedynamic process of contradictions through which thetwo (...) contexts evolve for the sake of argumentation intoone system containing both contexts.More technically, we show a new, dialogical, way tobuild paraconsistent systems for propositional andfirst-order logic with classical and intuitionisticfeatures (i.e. paraconsistency both with and withouttertium non-datur) and present theircorresponding tableaux. (shrink)

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical inference from (...) a probabilistic set of premisses. We will show how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency. (shrink)

We study logics determined by matrices consisting of a De Morgan lattice with an upward closed set of designated values, such as the logic of non‐falsity preservation in a given finite Boolean algebra and Shramko's logic of non‐falsity preservation in the four‐element subdirectly irreducible De Morgan lattice. The key tool in the study of these logics is the lattice‐theoretic notion of an n‐filter. We study the logics of all (complete, consistent, and classical) n‐filters on De Morgan lattices, which are non‐adjunctive (...) generalizations of the four‐valued logic of Belnap and Dunn (of the three‐valued logics of Priest and Kleene, and of classical logic). We then show how to find a finite Hilbert‐style axiomatization of any logic determined by a finite family of prime upsets of finite De Morgan lattices and a finite Gentzen‐style axiomatization of any logic determined by a finite family of filters on finite De Morgan lattices. As an application, we axiomatize Shramko's logic of anything but falsehood. (shrink)

We outline the rather complicated history of attempts at axiomatizing Jaśkowski’s discussive logic $$\mathbf {D_2}$$ D2 and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing $$\mathbf {D_2}$$ D2 in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of $$\mathbf {D_2}$$ D2 in the language Jaśkowski suggested in his second paper on discussive logic, by (...) following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of $$\mathbf {D_2}$$ D2, introduced by Ciuciura, in which negation is also taken to be discussive. (shrink)

This paper is an attempt to show that the subvaluation theory isnot a good theory of vagueness. It begins with a short review of supervaluation and subvaluation theories and procedes to evaluate the subvaluation theory. Subvaluationism shares all the main short-comings of supervaluationism.Moreover, the solution to the sorites paradox proposed by subvaluationists isnot satisfactory. There is another solution which subvaluationists could availthemselves of, but it destroys the whole motivation for using a paraconsistentlogic and is not different from the one offered (...) by supervaluationism. (shrink)

In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. We also give syntactic characterizations (...) of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems. (shrink)