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)

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)

. 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.

Scientific inquiry is typically focused on particular questions about particular objects and properties. This leads to a multiplicity of models which, even when they draw on a single, consistent body of concepts and principles, often employ different methods and assumptions to model different systems. Pluralists have remarked on how scientists draw on different assumptions to model different systems, different aspects of systems and systems under different conditions and defended the value of distinct, incompatible models within science at any given time. (...) Paraconsistentists have proposed logical strategies to avoid trivialization when inconsistencies arise by a variety of means. Here we examine how chunk and permeate, a simple approach to paraconsistent reasoning which avoids heterodox logic by confining commitments to separate contexts in which reasoning with them is taken to be reliable while allowing ‘permeation’ of some conclusions into other contexts, can help to systematize pluralistic reasoning across the boundaries of plural contexts, using regional climate models as an example. The result is a kind of unity for science—but a unity achieved by the constrained exchange of specified information between different contexts, rather than the closure of all commitments under some paraconsistent consequence relation. (shrink)

This paper is a contribution to the program of constructing formal representations of pragmatic aspects of human reasoning. We propose a formalization within the framework of Adaptive Logics of the exclusivity implicature governing the connective ‘or’.Keywords: exclusivity implicature, Adaptive Logics.

We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...) can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus. (shrink)

In this paper, I present the modal adaptive logic $AJ^{r}$ (based on S5) as well as the discussive logic $D_{2}^{r}$ that is defined from it. $D_{2}^{r}$ is a (nonmonotonic) alternative for Jaśkowski's paraconsistent system D₂. Like D₂, $D_{2}^{r}$ validates all single-premise rules of Classical Logic. However, for formulas that behave consistently, $D_{2}^{r}$ moreover validates all multiple-premise rules of Classical Logic. Importantly, and unlike in the case of D₂, this does not require the introduction of discussive connectives. It is argued that (...) this has clear advantages with respect to one of the main application contexts of discussive logics, namely the interpretation of discussions. (shrink)

In this paper Grice's requirements for assertability are imposed on the disjunction of Classical Logic. Defining material implication in terms of negation and disjunction supplemented by assertability conditions, results in the disappearance of the most important paradoxes of material implication. The resulting consequence relation displays a very strong resemblance to Schurz's conclusion-relevant consequence relation.

While the analytical philosophy of science regards inconsistent theories as disastrous, Chomsky allows for the temporary tolerance of inconsistency between the hypotheses and the data. However, in linguistics there seem to be several types of inconsistency. The present paper aims at the development of a novel metatheoretical framework which provides tools for the representation and evaluation of inconsistencies in linguistic theories. The metatheoretical model relies on a system of paraconsistent logic and distinguishes between strong and weak inconsistency. Strong inconsistency is (...) destructive in that it leads to logical chaos. In contrast, weak inconsistency may be constructive, because it is capable of accounting for the simultaneous presence of seemingly incompatible structures. However, paraconsistent logic cannot grasp the dynamism of the emergence and resolution of weak inconsistencies. Therefore, the metatheoretical approach is extended to plausible argumentation. The workability of this metatheoretical model is tested with the help of a detailed case study on an analysis of discontinuous constituents in Government-Binding Theory. (shrink)

Analyzing situations where information is partial, incomplete or contradictory has created a demand for quantitative belief measures that are weaker than classic probability theory. In this paper, we compare two frameworks that have been proposed for this task, Dempster-Shafer theory and non-standard probability theory based on Belnap-Dunn logic. We show the two frameworks to assume orthogonal perspectives on informational shortcomings, but also provide a partial correspondence result. Lastly, we also compare various dynamical rules of the two frameworks, all seen as (...) generalizations of classic Bayes’ conditioning. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

This paper presents and illustrates a formal logic for the abduction of singular hypotheses. The logic has a semantics and a dynamic proof theory that is sound and complete with respect to the semantics. The logic presupposes that, with respect to a specific application, the set of explananda and the set of possible explanantia are disjoint . Where an explanandum can be explained by different explanantia, the logic allows only for the abduction of their disjunction.

The present paper introduces a belief merging procedure by majority using the standard format of Adaptive Logics. The core structure of the logic ADM c (Adaptive Doxastic Merging by Counting) consists in the formulation of the conflicts arising from the belief bases of the agents involved in the procedure. A strategy is then defined both semantically and proof-theoretically which selects the consistent contents answering to a majority principle. The results obtained are proven to be equivalent to a standard majority operator (...) for bases with partial support. (shrink)

In this paper, I will characterize a new class of inconsistency-adaptive logics, namely inconsistency-adaptive modal logics. These logics cope with inconsistencies in a modal context. More specifically, when faced with inconsistencies, inconsistency-adaptive modal logics avoid explosion, but still allow the derivation of sufficient consequences to adequately explicate the part of human reasoning they are intended for.

In this paper, I want to substantiate three related claims regarding causal discovery from non-experimental data. Firstly, in scientific practice, the problem of ignorance is ubiquitous, persistent, and far-reaching. Intuitively, the problem of ignorance bears upon the following situation. A set of random variables V is studied but only partly tested for (conditional) independencies; i.e. for some variables A and B it is not known whether they are (conditionally) independent. Secondly, Judea Pearl’s most meritorious and influential algorithm for causal discovery (...) (the IC algorithm) cannot be applied in cases of ignorance. It presupposes that a full list of (conditional) independence relations is on hand and it would lead to unsatisfactory results when applied to partial lists. Finally, the problem of ignorance is successfully treated by means of ALIC, the adaptive logic for causal discovery presented in this paper. (shrink)

This book contains a selection of the papers presented at the Logic, Reasoning and Rationality 2010 conference in Ghent. The conference aimed at stimulating the use of formal frameworks to explicate concrete cases of human reasoning, and conversely, to challenge scholars in formal studies by presenting them with interesting new cases of actual reasoning. According to the members of the Wiener Kreis, there was a strong connection between logic, reasoning, and rationality and that human reasoning is rational in so far (...) as it is based on logic. Later, this belief came under attack and logic was deemed inadequate to explicate actual cases of human reasoning. Today, there is a growing interest in reconnecting logic, reasoning and rationality. A central motor for this change was the development of non-classical logics and non-classical formal frameworks. The book contains contributions in various non-classical formal frameworks, case studies that enhance our apprehension of concrete reasoning patterns, and studies of the philosophical implications for our understanding of the notions of rationality. (shrink)

We present a logic, \, for the discovery of deterministic causal regularities starting from empirical data. Our approach is inspired by Mackie’s theory of causes as INUS-conditions, and implements a more recent adjustment to Mackie’s theory according to which the left-hand side of causal regularities is required to be a minimal disjunction of minimal conjunctions. To derive such regularities from a given set of data, we make use of the adaptive logics framework. Our knowledge of deterministic causal regularities is, as (...) Mackie noted, most often gappy or elliptical. The adaptive logics framework is well-suited to explicate both the internal and the external dynamics of the discovery of such gappy regularities. After presenting \, we first discuss these forms of dynamics in more detail. Next, we consider some criticisms of the INUS-account and show how our approach avoids them, and we compare \ with the CNA algorithm that was recently proposed by Michael Baumgartner. (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)

It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the (...) rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results are inspired by and based on Mycielski (1981). (shrink)

In this paper, I will argue that Logics of Formal Inconsistency $$ can be used as very sophisticated and powerful methods of classical recapture. I will compare $LFIs$ with the well-known non-monotonic logics by Batens and Priest and the ‘shrieking’ rules of Beall. I will show that these proposals can be represented in $LFIs$ and that $LFIs$ give room to more complex and varied recapturing strategies.

In a rather general setting, we prove a number of basic theorems concerning computational complexity of derivability in adaptive logics. For that setting, the so-called standard format of adaptive logics is suitably adopted, and the corresponding completeness results are established in a very uniform way.

This paper answers the philosophical contentions defended in Horsten and Welch . It contains a description of the standard format of adaptive logics, analyses the notion of dynamic proof required by those logics, discusses the means to turn such proofs into demonstrations, and argues that, notwithstanding their formal complexity, adaptive logics are important because they explicate an abundance of reasoning forms that occur frequently, both in scientific contexts and in common sense contexts.

For Tarski logics, there are simple criteria that enable one to conclude that two premise sets are equivalent. We shall show that the very same criteria hold for adaptive logics, which is a major advantage in comparison to other approaches to defeasible reasoning forms. A related property of Tarski logics is that the extensions of equivalent premise sets with the same set of formulas are equivalent premise sets. This does not hold for adaptive logics. However a very similar criterion does. (...) We also shall show that every monotonic logic weaker than an adaptive logic is weaker than the lower limit logic of the adaptive logic or identical to it. This highlights the role of the lower limit for settling the adaptive equivalence of extensions of equivalent premise sets. (shrink)

. In this paper, adaptive logics are studied from the viewpoint of universal logic (in the sense of the study of common structures of logics). The common structure of a large set of adaptive logics is described. It is shown that this structure determines the proof theory as well as the semantics of the adaptive logics, and moreover that most properties of the logics can be proved by relying solely on the structure, viz. without invoking any specific properties of the (...) logics themselves. (shrink)

The present paper is devoted to computational aspects of propositional inconsistency-adaptive logics. In particular, we prove (relativized versions of) some principal results on computational complexity of derivability in such logics, namely in cases of CLuN r and CLuN m , i.e., CLuN supplied with the reliability strategy and the minimal abnormality strategy, respectively.

Belnap-Dunn logic, sometimes also known as First Degree Entailment, is a four-valued propositional logic that complements the classical truth values of True and False with two non-classical truth values Neither and Both. The latter two are to account for the possibility of the available information being incomplete or providing contradictory evidence. In this paper, we present a probabilistic extension of BD that permits agents to have probabilistic beliefs about the truth and falsity of a proposition. We provide a sound and (...) complete axiomatization for the framework defined and also identify policies for conditionalization and aggregation. Concretely, we introduce four-valued equivalents of Bayes’ and Jeffrey updating and also suggest mechanisms for aggregating information from different sources. (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)

The paper contains a survey of (mainly unpublished) adaptive logics of inductive generalization. These defeasible logics are precise formulations of certain methods. Some attention is also paid to ways of handling background knowledge, introducing mere conjectures, and the research guiding capabilities of the logics.

This paper contains different approaches to solve the problem how to construct the ultimate position out of one’s interventions in a discussion after possibly one or more position changes. In all approaches it is the aim to come as close as possible to human reasoning. Therefore all logics are adaptive logics. The first logic is an extension of an adaptive translation into S5 of the Rescher-Manor mechanisms. The second one is a dynamic proof theory based on a technique using indices. (...) In the end a satisfactory solution is given by a dynamic proof theory expressing the idea of prioritized compatibility, i.e. compatibility step by step. (shrink)

Inconsistency-adaptive logics isolate the inconsistencies that are derivable from a premise set, and restrict the rules of Classical Logic only where inconsistencies are involved. From many inconsistent premise sets, disjunctions of contradictions are derivable no disjunct of which is itself derivable. Given such a disjunction, it is often justified to introduce new premises that state, with a certain degree of confidence, that some of the disjuncts are false. This is an important first step on the road to consistency: it narrows (...) down suspicion in inconsistent premise sets and hence locates the real problems among the possible ones. In this paper I present two approaches for handling such new premises in the context of the original premises. The first approach may apparently be combined with all paraconsistent logics. The second approach does not have the same generality, but is decidedly more elegant. (shrink)