In order to reason in a non-trivializing way with contradictions, para- consistent logics reject some classically valid inferences. As a way of re- covering some of these inferences, Graham Priest ([Priest, 1991]) proposed to nonmonotonically strengthen the Logic of Paradox by allowing the se- lection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory (...) assumptions. In Priest’s approach the inconsistent parts of models are compared in an extensional way by considering their inconsis- tent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known problem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In this paper, we demonstrate that Strong Reassurance holds for the extensional approach under a cardinality-based comparison of the incon- sistent parts of models. Furthermore, we introduce and investigate the metatheory of the class of first-order nonmonotonic inconsistency-tolerant construct over the extensional or quantitative comparisons of their respec- tive models. Core model-theoretic properties for these logics, such as the Löwenheim-Skolem theorems, along with other nonmonotonic properties, are further studied. (shrink)

It has recently been argued by Davey (2014) that inconsistency is never tolerated in science, but only discretely isolated. But when talking about inconsistencies in science, not much attention has been paid to the inconsistencies between theory and observation. Here I will argue that inconsistency toleration actually takes place in science, and that when we examine actual inconsistent theories, inconsistencies between theory and observation look anything but homogeneous. I will argue, appealing to certain properties of empirical theories, (...) especially holism, that at least two sub-types of inconsistencies regarding theory and observation can be distinguished: those that satisfy a criterion of observational independence and those that do not. (shrink)

Reasoning and change over inconsistent knowledge bases is of utmost relevance in areas like medicine and law. Argumentation may bring the possibility to cope with both problems. Firstly, by constructing an argumentation framework from the inconsistent KB, we can decide whether to accept or reject a certain claim through the interplay among arguments and counterarguments. Secondly, by handling dynamics of arguments of the AF, we might deal with the dynamics of knowledge of the underlying inconsistent KB. Dynamics of arguments has (...) recently attracted attention and although some approaches have been proposed, a full axiomatization within the theory of belief revision was still missing. A revision arises when we want the argumentation semantics to accept an argument. Argument Theory Change encloses the revision operators that modify the AF by analyzing dialectical trees—arguments as nodes and attacks as edges—as the adopted argumentation semantics. In this article, we present a simple approach to ATC based on propositional KBs. This allows to manage change of inconsistent KBs by relying upon classical belief revision, although contrary to it, consistency restoration of the KB is avoided. Subsequently, a set of rationality postulates adapted to argumentation is given, and finally, the proposed model of change is related to the postulates through the corresponding representation theorem. Though we focus on propositional logic, the results can be easily extended to more expressive formalisms such as first-order logic and description logics, to handle evolution of ontologies. (shrink)

In order to reason in a non-trivializing way with contradictions, paraconsistent logics reject some classically valid inferences. As a way to recover some of these inferences, Graham Priest proposed to nonmonotonically strengthen the Logic of Paradox by allowing the selection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory assumptions. In Priest’s approach the (...) inconsistent parts of models are compared in an extensional way by consid- ering their inconsistent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known prob- lem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In the following paper, we show that Strong Reassurance holds for the extensional ap- proach under a cardinality-based comparison of the inconsistent parts of models. We develop and study the meta-theory of a class of nonmonotonic inconsistency-tolerant logics based on the extensional and the quantitative comparisons of their respective models, including important model-theoretic properties, such as the Löwenheim-Skolem theorems, as well as principles of nonmonotonic inference. (shrink)

Inconsistency toleration is the phenomenon of working with inconsistent information without threatening one’s rationality. Here I address the role that ignorance plays for the tolerance of contradictions in the empirical sciences. In particular, I contend that there are two types of ignorance that, when present, can make epistemic agents to be rationally inclined to tolerate a contradiction. The first is factual ignorance, understood as temporary undecidability of the truth values of the conflicting propositions. The second is what I (...) call “ignorance of theoretical structure”, which is lack of knowledge of relevant inference patterns within a specific theory. I argue that these two types of ignorance can be explanatory of the scientists’ rational disposition to be tolerant towards contradictions, and I illustrate this with a case study from neutrino physics. (shrink)

In this paper I examine the problem of inconsistency toleration in the context of scientific pluralism. I argue that, first of all, the notion of inconsistency toleration has to be qualified with respect to the evaluative attitude that one takes towards a given scientific theory or theories. Second, I show which types of inconsistency toleration are compatible with two major approaches to scientific pluralism, the so-called modest and the radical one. In view of this (...) I suggest some points of demarcation between these two approaches. (shrink)

This thesis proposes and presents two new models for belief representation and belief revision. The first model is the B-structures model which relies on a notion of partial language splitting and tolerates some amount of inconsistency while retaining classical logic. The model preserves an agent's ability to answer queries in a coherent way using Belnap's four-valued logic. Axioms analogous to the AGM axioms hold for this new model. The distinction between implicit and explicit beliefs is represented and psychologically plausible, (...) computationally tractable procedures for query answering and belief base revision are obtained. ;The second model presents a method for relevance sensitive non-monotonic inference from belief sequences which incorporates insights pertaining to prioritized inference and relevance sensitive, inconsistency tolerant belief revision. Our model uses a finite, logically open sequence of propositional formulas as a representation for beliefs and defines a notion of inference from maxiconsistent subsets of formulas guided by two orderings: a temporal sequencing and an ordering based on relevance relations between the conclusion and formulas in the sequence. The relevance relations are ternary as opposed to standard binary axiomatizations. The inference operation thus defined easily handles iterated revision by maintaining a revision history, blocks the derivation of inconsistent answers from a possibly inconsistent sequence and maintains the distinction between explicit and implicit beliefs. In doing so, it provides a finitely representable formalism and a plausible model of reasoning for automated agents. (shrink)

Scientific pluralism is the view according to which there is a plurality of scientific domains and of scientific theories, and these theories are empirically adequate relative to their own respective domains. Scientific monism is the view according to which there is a single domain to which all scientific theories apply. How are these views impacted by the presence of inconsistent scientific theories? There are consistency-preservation strategies and inconsistency-toleration strategies. Among the former, two prominent strategies can be articulated: Compartmentalization (...) and Information restriction. Among the inconsistency-toleration strategies, we have: Paraconsistent compartmentalization and Dialetheism. In this paper, I critically assess the adequacy of each of these four views. (shrink)

Nowadays there is a growing tendency in the philosophy of science to think that some phenomena cannot be exhaustively explained, or even described, by a single theory or a particular approach. Thus, we are occasionally required to use various approaches in order to give account of the phenomenon we are analyzing. And sometimes, we can appreciate this as an invitation to be pluralist in certain respects about our understanding of a particular aspect in science. -/- During the last decade applications (...) of pluralism have increased and led to several other relevant debates in the philosophy of science. By the strengthening of pluralism, new questions have emerged, and more importantly, new alternatives have been offered in dealing with problems about how to interpret the ontology and the ontological commitments of particular theories, or how to apply strategies to analyze and understand specific episodes from the history of science, or how to work within different, and sometimes incompatible, explanations about a particular phenomenon, among others. In general, pluralism seems to be a very rich and yet not enough explored path for the philosophy of science in general. -/- Today, it is recognized that, at least for methodological purposes, the application of pluralism to the study of science can offer a great number of benefits. One of them would be the opportunity of analyzing the role that some epistemic virtues -such as scope, fruitfulness, consistency, and simplicity, to name just a few- play in the scientific activity. From the different pluralist positions, a lot has been said about empirical adequacy, refutability and explanatory power yet consistency power, yet, consistency has not been equally dealt with. -/- As a matter of fact, the lack of consistency and its philosophical implications have been studied from an angle that does not necessarily involve a pluralism of any kind. At the moment, it is commonly accepted that inconsistencies are more frequent in scientific development that the traditional philosophy of science could have expected and the idea that inconsistency is not always a synonym of logical anarchy, as it was suggested in the classical literature of logic and the philosophy of science, has been gaining support. All this has been possible, mostly, thanks to the emergence of paraconsistent logics and the availability of case studies that show how inconsistency is not an uncommon phenomenon in science. -/- But pluralism does not necessarily entail inconsistency toleration nor vice versa. Accordingly the main motivation for this volume is to explore the links between pluralism and inconsistency toleration in science, in order to connect the reflections on inconsistency toleration with broader and major issues in philosophy of science. In order to do so, we will suggest two different lines of investigation: first, to focus on the implication of some pluralistic accounts in the philosophy of science, regarding inconsistency; and second to analyze the implications of some paraconsistent approaches regarding pluralism in science. (shrink)

Moral conflicts are the situations which emerge as a response to deal with conflicting obligations or duties. An interesting case arises when an agent thinks that two obligations A and B are equally important, but yet fails to choose one obligation over the other. Despite the fact that the systematic study and the resolution of moral conflicts finds prominence in our linguistic discourse, standard deontic logic when used to represent moral conflicts, implies the impossibility of moral conflicts. This presents a (...) conundrum for appropriate logic to address these moral conflicts. We frequently believe that there is a close connection between tolerating inconsistencies and conflicting moral obligations. In paraconsistent logics, we tolerate inconsistencies by treating them to be both true and false. In this paper, we analyze Graham Priest’s paraconsistent logic LP, and extend our examination to the deontic extension of LP known as DLP. We illustrate our work, with a classic example from the famous Indian epic Mahabharata, where the protagonist Arjuna faces a moral conflict in the battlefield of Kurukshetra. The paper aims to avoid deontic explosion and allows to accommodate Arjuna’s moral conflict in paraconsistent deontic logics. Our analysis is expected to provide novel tools towards the logical representation of moral conflicts and to shed some light on the context-sensitive paraconsistent deontic logic. (shrink)

SummaryIt is not the inconsistency in the sense of classical logic that we have to tolerate. The dialectical reasoning, described by N. Rescher, is outside the domain where CI is defined. The apparent contradiction between CI and paraconsistent logic can be removed by realizing that PL is a widening of the conceptual framework of classical logic. In this new framework the meaning of some words was changed similarly as, according to N. Bohr, in quantum mechanics the words “particle” and (...) “wave” have changed their meaning. (shrink)

Toleration is one of the fundamental principles that inform the design of a democratic and liberal society. Unfortunately, its adoption seems inconsistent with the adoption of paternalistically benevolent policies, which represent a valuable mechanism to improve individuals’ well-being. In this paper, I refer to this tension as the dilemma of toleration. The dilemma is not new. It arises when an agent A would like to be tolerant and respectful towards another agent B’s choices but, at the same time, (...) A is altruistically concerned that a particular course of action would harm, or at least not improve, B’s well-being, so A would also like to be helpful and seeks to ensure that B does not pursue such course of action, for B’s sake and even against B’s consent. In the article, I clarify the specific nature of the dilemma and show that several forms of paternalism, including those based on ethics by design and structural nudging, may not be suitable to resolve it. I then argue that one form of paternalism, based on pro-ethical design, can be compatible with toleration and hence with the respect for B’s choices, by operating only at the informational and not at the structural level of a choice architecture. This provides a successful resolution of the dilemma, showing that tolerant paternalism is not an oxymoron but a viable approach to the design of a democratic and liberal society. (shrink)

We present the inconsistency-adaptive deontic logic DP r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A ∧ O ∼A, O A ∧ P ∼A or even O A ∧ ∼O A. On the other hand, DP r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP r interprets a given premise set ‘as (...) normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP r. (shrink)

I argue that a general logic of definitions must tolerate ω‐inconsistency. I present a semantical scheme, S, under which some definitions imply ω‐inconsistent sets of sentences. I draw attention to attractive features of this scheme, and I argue that S yields the minimal general logic of definitions. I conclude that any acceptable general logic should permit definitions that generate ω‐inconsistency. This conclusion gains support from the application of S to the theory of truth.

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)

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism proposed by Brown and Priest in, 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings of this application (...) of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from. (shrink)

This thesis analyzes from a philosophical perspective different models for nonmonotonic inference, belief revision and the handling of inconsistencies. ;The first chapter serves as an introduction to the subject, giving examples and analyzing the main concepts. As a result of these discussions, this thesis tries to: produce a refined map of the main notions related to this subject, maintain that there can be a fault tolerant logic that stands in support of fault tolerant reasoning, and defend the use of deductive (...) databases as models for changes of belief. ;The second chapter explores monotonic proposals to deal with uncertain reasoning. These include logics based on a classical notion of consequence, interpretations of defeasible reasoning as containing implicit premises, and systems in which temporal change is an integral part of the representation of states of affairs, via temporal indices or more elaborate devices. ;The third chapter explores quantitative approaches to nonmonotonic reasoning like dispositional logic, and theories of probabilistic reasoning including Bayesian, certainty factors, and Dempster-Shafer. ;The fourth chapter explores qualitative approaches. Here we mention conditional, default, and nonmonotonic logics, and the circumscription and autoepistemic formalisms. ;The fifth chapter studies different proposals to deal with contradictions. It starts with a review of the types of belief inconsistencies, and then it presents mechanisms to prevent, control or correct them. I explore paraconsistent logics, specially relevance logics, to "bracket" the area of conflict and allow inferences about non-related topics to continue unaffected. Then I examine different methods to get rid of the inconsistencies, including the use of dependency-directed backtracking and of relevance logic. ;Finally, I present my conclusions, and mention some of the open problems. (shrink)

Nicholas Rescher has argued we must tolerate inconsistency because of our cognitive limitations. He has also produced, together with R. Brandom, a serious attempt at exploring the logic of inconsistency. Inconsistency tolerance calls for a systematic rewriting of our logical doctrines: it requires a paraconsistent logic. However, having given up all aggregation of premises, Rescher's proposal for a paraconsistenl logic fails to account for the reductive reasoning Rescher appeals to in his account of inconsistency tolerance. A (...) non-adjunctive logic developed by P.K. Schotch and RE. Jennings provides just what Rescher's logic is lacking: It allows a tolerant attitude toward inconsistency while giving an account of reductive reasoning. (shrink)

I attempt to show that toleranceis part of the idea of American law: for any legalsystem must incorporate the capacity toaccommodate differences in order to meet theminimal standards necessary to apply a rule. There are multiple forms of tolerance, however, some ofwhich are inconsistent with liberal principles.By examining several lines of jurisprudencerelating to speech and privacy, I show thatAmerican law reflects elements of bothliberalism and conservative communitarianism. I attempt to reconcile these by suggesting they actuallyreflect a perfectionist foundation of liberalautonomy. (...) That is to say, American law doesnot value moral autonomy and reasoned discoursebecause they protect neutrality betweendifferent ideas of the good life: rather, thelaw reflects an idea of the good life that seesmoral autonomy as advancing well being.This perfectionist liberal foundation oftolerance reflects the evolution of Americanlaw. Through slavery, sexism and the controlof erotic speech we see how it expanded theideas of who is capable of rationaldiscourse and what activities incorporatethe exercise of reasoned moral autonomy;and how the law imposes this autonomouscapacity on individuals as the price ofcitizenship, even if they belong to groups whodeny the value of reason or autonomy. (shrink)

In this paper, I compare how it is that inconsistencies are handled in mathematics to how they are handled in chemistry. In mathematics, they are very precisely formulated and identified, unlike in chemistry. So the chemists can learn from the precision and the very well-worked out strategies developed by logicians and deployed by mathematicians to cope with inconsistency. Some lessons can also be learned by the mathematicians from the chemists. Mathematicians tend to be intolerant towards inconsistencies. There are some (...) philosophers of chemistry who attribute to chemists, collectively, a more tolerant attitude towards inconsistency, and so, mathematicians can learn from the chemists’ attitude to inform their choice of strategies. (shrink)

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6]. In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a (...) model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3, which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli. We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete. In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. (shrink)

This paper presents and discusses several methods for reasoning from inconsistent knowledge bases. A so-called argued consequence relation, taking into account the existence of consistent arguments in favour of a conclusion and the absence of consistent arguments in favour of its contrary, is particularly investigated. Flat knowledge bases, i.e., without any priority between their elements, are studied under different inconsistency-tolerant consequence relations, namely the so-called argumentative, free, universal, existential, cardinality-based, and paraconsistent consequence relations. The syntax-sensitivity of these consequence relations (...) is studied. A companion paper is devoted to the case where priorities exist between the pieces of information in the knowledge base. (shrink)

Increasingly, physicians are being asked to provide technical services that many believe are morally wrong or inconsistent with their beliefs about the meaning and purposes of medicine. This controversy has sparked persistent debate over whether practitioners should be permitted to decline participation in a variety of legal practices, most notably physician-assisted suicide and abortion. These debates have become heavily politicized, and some of the key words and phrases are being used without a clear understanding of their meaning. In this essay, (...) I endeavor, firstly, to clarify the meaning of some of these terms: conscience, conscientious action, professional judgment, conscientious objection, conscience clauses, civil disobedience, and tolerance. I argue that use of the term conscientious objection to describe these refusals by health care professionals is mistaken and confusing. Secondly, relying on a proper understanding of the moral and technical character of medical judgment, the optimal deference that the state and markets ought to have toward professions, and general principles of Lockean tolerance for a diversity of practices and persons in a flourishing, pluralistic, democratic society, I offer a defense of tolerance with respect to the deeply held convictions of physicians and other health care professionals who hold minority views on contested but legal medical practices. (shrink)

Is logical consistency required for a set of beliefs or propositions to be categorically coherent? An affirmative answer is often assumed by mainstream epistemologists, and yet it is unclear why. Cases like the lottery and the preface call into question the assumption that beliefs must be consistent in order to be epistemically rational. And thus it is natural to wonder why all inconsistent sets of propositions are incoherent. On the other hand, Easwaran and Fitelson have shown that particular kinds of (...) inconsistency entail the epistemically ‘irrationality’ of holding certain sets of beliefs. In cases of the latter kind of inconsistency, it seems more reasonable to insist that such sets of beliefs or propositions are categorically incoherent. What the precise relationship is between coherence and consistency depends on the nature of the coherence relation. We shall examine recent attempts to explicate the coherence relation in terms of probabilistic measures of confirmation or agreement to see what they can teach us about the relationship between coherence and consistency. We shall show that some probabilistic measures of coherence allow for inconsistent sets to be categorically coherent, while satisfying plausible epistemic rationality constraints. Other probabilistic measures of coherence impose very strong logical consistency requirements, and some measures are tolerant of most forms of inconsistency. As we try to understand what distinguishes coherence measures in this respect, we will also draw some important lessons about Bayesian confirmation measures and differences in the way that they treat contradictory propositions. (shrink)

Generally speaking, just war theory (JWT) holds that there are two just causes for war: self-defence and ‘other-defence’. The most common type of the latter is popularly known as ‘humanitarian intervention’. There is debate, however, as to whether these can serve as just causes for preventive war. Those who subscribe to JWT tend to be unified in treating so-called preventive war with a high degree of suspicion on the grounds that it fails to satisfy conventional criteria for jus ad bello; (...) – particularly the just cause and last resort criteria. Francisco di Vitoria held that the only just cause for war was ‘a wrong received’, which renders impossible any justification for preventive war. There are assumptions implicit in recent military practice, however – most notably, the US-led invasion of Iraq in 2003 – that challenge this ban on preventive war. Interestingly, both supporters and critics attempt to justify their views through the broader logic of JWT; viz., through a conception of what is good for both political communities and individuals, and through a legitimate defence of these goods. Supporters point to situations where so-called rogue states represent ‘grave and imminent risk’ of committing acts of aggression as grounds that justify preventive war; critics argue that to attack another political community on the basis of crimes not yet committed is a breach of the very rights JWT was created to defend. The advocate of preventive war does not appreciate important aspects concerning the morality of war. In the ongoing tension between Iran and The United States and her allies – if the rhetoric is to be believed – I am asked to tolerate a threat to my security and liberty, and to risk suffering aggression in defence of the rights of the antagonistic, but not yet aggressive, state. The crucial question is how such tolerance and risk fit in with the logic of just war: at what point, if any, does the risk of being attacked become great enough to justify declaring war in anticipation? In this paper I highlight some of the theoretical and practical difficulties in determining what counts as a grave and imminent threat, focusing especially on the complicated case of ‘imminence’ in the face of so-called ‘Weapons of Mass Destruction’. Secondly, I will argue that not only is the notion of preventive war inconsistent with the defence of the rights of political communities that JWT requires; it is also forbidden by the proportionality requirement of jus ad bellum. A risk of being subjected to aggression is the price for global peace. Whilst political communities can do much to prevent aggression and prepare themselves in case it occurs, the conditions for just war require that this prevention and preparation stop short of declaring war. We must live with a certain degree of risk in this area. (shrink)

In this paper I apply the concept of _inter-Model Inconsistency in Set Theory_ (MIST), introduced by Carolin Antos (this volume), to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s _Ultimate L_, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these (...) positions on the spectrum from inconsistency avoidance to inconsistency toleration. By doing so, I connect a debate in philosophy of set theory with a debate in philosophy of science about the role of inconsistencies in the natural sciences. While there are important differences, like the lack of threatening explosive inferences, I show how specific philosophical positions in the philosophy of set theory can be interpreted as reactions to a state of inconsistency similar to analogous reactions studied in the philosophy of science literature. My hope is that this transfer operation from philosophy of science to mathematics sheds a new light on the current discussion in philosophy of set theory; and that it can help to bring philosophy of mathematics and philosophy of science closer together. (shrink)

Most scientific theories are globally inconsistent. Chunk and Permeate is a method of rational reconstruction that can be used to separate, and identify, locally consistent chunks of reasoning or explanation. This then allows us to justify reasoning in a globally inconsistent theory. We extend chunk and permeate by adding a visually transparent way of guiding the individuation of chunks and deciding on what information permeates from one chunk to the next. The visual representation is in the form of bundle diagrams. (...) We then extend the bundle diagrams to include not only reasoning in the presence of inconsistent information or reasoning in the logical sense of deriving a conclusion from premises, but more generally reasoning in the sense of trying to understand a phenomenon in science. This extends the use of the bundle diagrams in terms of the base space and the fibres. We then apply this to a case in physics, that of understanding binding energies in the nucleus of an atom using together inconsistent models: the liquid drop model and the shell model. We draw some philosophical conclusions concerning scientific reasoning, paraconsistent reasoning, the role of logic in science and the unity of science. (shrink)

According to the naive theory of vagueness, the vagueness of an expression consists in the existence of both positive and negative cases of application of the expression and in the non- existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. The paper explores the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the transitivity (...) of the consequence relation. A lattice-theoretical framework for a whole family of “tolerant logics” is proposed and developed. Particular care is devoted to the relation between the salient features of the formal apparatus and the informal logical and semantic notions they are supposed to model. A suitable non-transitive counterpart to classical logic is defined. Some of its properties are studied, and it is eventually shown how an appropriate regimentation of the naive theory of vagueness is consistent in such a logic. (shrink)

Weber, Colyvan, and Priest have advanced glutty approaches to the sorites, on which the truth about the penumbral region of a soritical series is inconsistent. The major benefit of a glut-based approach is maintaining the truth of all sorites premisses while none the less avoiding, in a principled fashion, the absurdity of the sorites conclusion. I agree that this is a major virtue of the target glutty approach; however, I think that it can be had without gluts. If correct, this (...) result weighs heavily against the proposed glutty approach, at least given the default-consistency principle that all target glutty philosophers accept: posit gluts only if there is no consistent theory that enjoys the same virtues as the would-be glutty solution. (shrink)

It is widely believed that inconsistency is one of the greatest sins a scholar can commit. This issue is especially relevant in linguistics due to the rich diversity of data types, exceptions to the rules, counterexamples to the hypotheses, and background assumptions which constantly come into conflict with methodological principles. Bringing together ideas from linguistics and philosophy of science, this groundbreaking book seeks to answer the following questions: which kinds of inconsistency arise in linguistic theorising? Under which conditions (...) can inconsistencies be tolerated? And how can inconsistencies be resolved? It is the first study to develop a novel metatheoretical framework that accounts for the emergence and the resolution of inconsistency in linguistic theorising, and to reveal the strategies of inconsistency resolution in theoretical linguistics. Supported by detailed case studies, the findings of this metatheoretical analysis can be applied to improve the effectiveness of the working linguist's problem-solving activity. (shrink)

Are there independent standards of justice by which we are to measure our activities, or is justice itself to be understood in relativistic terms that vary with locality or historical period? I wish to examine briefly how far two inconsistent positions can both be accepted. I suggest that perhaps our ordinary understanding of reality itself—and in particular political reality—is essentially the outcome of a time of contest, and that there are areas of political reality where matters may be best seen (...) as still being contested. I thus question the need for a single internally consistent point of view, as if it alone were the answer to any particular political problem, and propose that a shared belief that reality is inconsistent may be a viable solution. Using the political scenario of Northern Ireland, I argue that justice requires the deliberate and institutionalised toleration of inconsistent views of the world. (shrink)

I elaborate and defend the inconsistency view on vagueness I have earlier argued for in my (2002) and (forthcoming). In rough outline, the view is that the sorites paradox arises because tolerance principles, despite their inconsistency, are meaning-constitutive for vague expressions. Toward the end of the paper I discuss other inconsistency views on vagueness that have been proposed, and compare them to the view I favor.

We put forward the hypothesis that there exist three basic attitudes towards inconsistencies within world views: (1) The inconsistency is tolerated temporarily and is viewed as an expression of a temporary lack of knowledge due to an incomplete or wrong theory. The resolution of the inconsistency is believed to be inherent to the improvement of the theory. This improvement ultimately resolves the contradiction and therefore we call this attitude the ‘regularising’ attitude; (2) The inconsistency is tolerated and (...) both contradicting elements in the theory are retained. This attitude integrates the inconsistency and leads to a paraconsistent calculus; therefore we will call it the paraconsistent attitude. (3) In the third attitude, both elements of inconsistency are considered to be false and the ‘real situation’ is considered something different that can not be described by the theory constructively. This indicates the incompleteness of the theory, and leads us to a paracomplete calculus; therefore we call it the paracomplete attitude. We illustrate these three attitudes by means of two ‘paradoxical’ situations in quantum mechanics, the wave-particle duality and the situation of non locality. (shrink)

Pierre Bayle’s theory of religious toleration has received much attention over the past three centuries, yet there is still little consensus surrounding the precise logic of Bayle’s argument, and even less consensus concerning whether that argument is successful or perhaps utterly inconsistent. One of the central themes in the literature concerns the role of skepticism in Bayle’s argument for toleration. Some argue that Baylean toleration is based entirely in a non-skeptical morality that is in turn based in (...) conscience, while others argue that Baylean toleration requires a foundation of skeptical doubt. In this paper I will offer an interpretation of Bayle’s argument for toleration as it is offered in the Ninth Letter of the Nouvelles lettres critiques (1685), and I will show that skepticism plays an essential role in the argument. (shrink)

Chomsky’s principle of epistemological tolerance says that in theoretical linguistics contradictions between the data and the hypotheses may be temporarily tolerated in order to protect the explanatory power of the theory. The paper raises the following problem: What kinds of contradictions may be tolerated between the data and the hypotheses in theoretical linguistics? First a model of paraconsistent logic is introduced which differentiates between week and strong contradiction. As a second step, a case study is carried out which exemplifies that (...) the principle of epistemological tolerance may be interpreted as the tolerance of week contradiction. The third step of the argumentation focuses on another case study which exemplifies that the principle of epistemological tolerance must not be interpreted as the tolerance of strong contradiction. The reason for the latter insight is the unreliability and the uncertainty of introspective data. From this finding the author draws the conclusion that it is the integration of different data types that may lead to the improvement of current theoretical linguistics and that the integration of different data types requires a novel methodology which, for the time being, is not available. (shrink)

Paraconsistent logics are characterized by rejection of ex falso quodlibet, the principle of explosion, which states that from a contradiction, anything can be derived. Strikingly these logics have found a wide range of application, despite the misgivings of philosophers as prominent as Lewis and Putnam. Such applications, I will argue, are of significant philosophical interest. They suggest ways to employ these logics in philosophical and scientific theories. To this end I will sketch out a ‘naturalized semantic dialetheism’ following Priest’s early (...) suggestion that the principles governing human natural language may well be inconsistent. There will be a significant deviation from Priest’s work, namely, the assumption of a broadly Chomskyan picture of semantics. This allows us to explain natural language inconsistency tolerance without commitment to contentious views in formal logic. (shrink)

Dirac’s treatment of his well known Delta function was apparently inconsistent. We show how to reconstruct his reasoning using the inconsistency-tolerant technique of Chunk and Permeate. In passing we take note of limitations and developments of that technique.

It is known that the full computation-tree logic CTL * is an important base logic for model checking. The bisimulation theorem for CTL* is known to be useful for abstraction in model checking. In this paper, the bisimulation theorems for two paraconsistent four-valued extensions 4CTL* and 4LCTL* of CTL* are shown, and a translation from 4CTL* into CTL* is presented. By using 4CTL* and 4LCTL*, inconsistency-tolerant and spatiotemporal reasoning can be expressed as a model checking framework.

We present a method for relevance sensitive non-monotonic inference from belief sequences which incorporates insights pertaining to prioritized inference and relevance sensitive, inconsistency tolerant belief revision. Our model uses a finite, logically open sequence of propositional formulas as a representation for beliefs and defines a notion of inference from maxiconsistent subsets of formulas guided by two orderings: a temporal sequencing and an ordering based on relevance relations between the putative conclusion and formulas in the sequence. The relevance relations are (...) ternary (using context as a parameter) as opposed to standard binary axiomatizations. The inference operation thus defined easily handles iterated revision by maintaining a revision history, blocks the derivation of inconsistent answers from a possibly inconsistent sequence and maintains the distinction between explicit and implicit beliefs. In doing so, it provides a finitely presented formalism and a plausible model of reasoning for automated agents. (shrink)

This edited volume collects essays on the four-valued logic known as Belnap-Dunn logic, or first-degree entailment logic. It also looks at various formal systems closely related to it. These include the strong Kleene logic and the Logic of Paradox. Inside, readers will find reprints of seminal papers written by the fathers of the field: Nuel Belnap and Michael Dunn. In addition, the collection also features a well-known but previously unpublished manuscript of Dunn, an interview with Belnap, and a new essay (...) by Dunn. Besides the original, monumental papers, the book also includes research by leading scholars. They consider the extraordinary importance of Belnap-Dunn logic from several perspectives. They look at how, philosophically, it has served as a basic system of inconsistency-tolerant reasoning, as the core of underlying logics for theories based on dialetheism, and, more recently, for theories based on Buddhist philosophy. Coverage also explores its contributions to computer science, such as knowledge representation and information processing. This mix of seminal papers and insightful analysis by top scholars offers readers a comprehensive outlook on Belnap-Dunn logic and its related expansions, which have been agenda setting for the debate on philosophical logic as well as philosophy of logic. The book will also enhance further discussion on the philosophical issues related to nonclassical logics in general. (shrink)

ABSTRACT The major approaches to belief revision and non monotonic reasoning proposed in the literature differ along a number of dimensions, including whether they are ?syntax- based? or ?semantic-based?, ?foundational? or ?coherentist?, ?consistence-restoring? or ?inconsistency-tolerant?. Our contribution towards clarifying the connections between these various approaches is threefold: ?We show that the two main approaches to belief revision, the foundations and coherence theories, are mathematically equivalent, thus answering a question left open in [Gar90, Doy92], The distinction between syntax-based approaches to (...) revision and approaches based on (semantic) preferential structures falls along similar lines, and their expressive equivalence is a consequence of this result. ?We formally clarify the connection between belief revision and non monotonic reasoning, in a particularly simple way which also throws light on the connection between consistence- restoring and reasoning-from-inconsistency approaches [BDP95]. ?As a direct application of the above, we show that Poole's (syntax-based) system of default reasoning and Shoham's preferential semantic for non monotonic reasoning are also expressively equivalent, in that they can represent the same set of non monotonic consequence relations. (shrink)

We present a model for first-order belief revision that is characterized by an underlying relevance-like relation and a background proof system. The model is extremely general in order to allow for a wide variety in these characterizing parameters. It allows some weakenings of beliefs which were initially implicit to become explicit and survive the revision process. The effects of revision are localized to the part of the theory that is influenced by the the new information. Iterated revision in this model (...) is handled trivially since the revision operator is constructive by definition. The usage of deductively limited proof systems permits an inconsistency tolerant model. The notion of a part of a theory capable of being influenced by new information (designed to accomodate the specific character of first-order languages) is shown to satisfy some intuitive and desirable properties. We show that for particular parametrizations, standard revision schemes can be embedded into our paradigm. (shrink)

It seems that Polish logic has always been open to considerationsconcerning the use of methods and results of formal logic within disciplines.We overview a couple of such Polish contributions to what may be called therealm of applied logic. We take a closer look at the formalization of naturalreasoning, inconsistency-tolerant logic, and at the formal analysis of causalnexus.

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

Many have contended that non-classical logicians have failed at providing evidence of paraconsistent logics being applicable in cases of inconsistency toleration in the sciences. With this in mind, my main concern here is methodological. I aim at addressing the question of how should we study and explain cases of inconsistent science, using paraconsistent tools, without ruining into the most common methodological mistakes. My response is divided into two main parts: first, I provide some methodological guidance on how to (...) approach cases of inconsistent science; and second, I focus on a peculiar type of formal methodologies for the scrutiny of inconsistent reasoning, the _Paraconsistent Alternative Approach_ (henceforth, PAA) and argue that PAA can enhance a more accurate understanding of sensible reasoning in inconsistent contexts. (shrink)

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)

Democratic societies tolerate intolerance, but that obligation finds its limit when the security of its citizens is jeopardized or its institutions of liberty are imperiled. Similarly, universities tolerate intolerance, but that obligation finds its limit when threatened by weaponized intolerance advocates who disenfranchise and denigrate community members and imperil academic norms and professional standards of conduct. Then, just as democratic societies must protect their threatened citizens and safeguard their imperiled institutions of liberty, so universities must protect their threatened community members (...) and safeguard their imperiled norms and standards. I argue for these conclusions by establishing a conflict between what the First Amendment legally permits university community members to express and what the norms of the university and the professional standards that structure academic freedom require from university community members. I argue that given the First Amendment, universities are legally obliged to tolerate even weaponized intolerance in campus public forums even if not in the classroom. I then recommend three responses to weaponized intolerance on campus that are consistent with the First Amendment: denunciation and protest, provision of safe space, and affirmation of academic values, norms, and standards. I reject three frequently encountered responses to weaponized intolerance as inconsistent with the First Amendment: heckler’s vetoes, student speech codes, and speaker bans. And I argue that one response—disruptive protest that falls short of a heckler’s veto—is legally permissible for students and faculty members but is ruled out for faculty members by academic norms and professional standards. (shrink)