Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly (...) addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary. (shrink)

Even though metaphysical dependence has been a subject of a lively debate in contemporary metaphysics, it is rare in such a debate to seriously consider the possibility that the metaphysical dependence relations among the things in the reality is inconsistent. This paper focuses on two philosophers of the Kyoto School, Kitaro Nishida and Keiji Nishitani, who challenge the common supposition that the structure of reality is consistent. In this paper, we show that Nishida’s logic of place is a version of (...) inconsistent foundationalism, according to which absolute nothingness as a foundational element does not depend on anything but depends on itself, and that Nishtani’s theory of the field of emptiness is a version of inconsistent coherentism, according to which emptiness does not depend on anything but depends on everything else (and possibly on itself). (shrink)

The article is an extended critical discussion of Kevin Scharp’s Replacing Truth. Scharp’s case for the claim that the concept of truth is inconsistent is criticized, and so is his case for the claim that the concept of truth must be replaced because of its inconsistency.

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)

For centuries, inconsistencies were seen as a hindrance to good reasoning, and their role in the sciences was ignored. In recent years, however, logicians as well as philosophers and historians have showed a growing interest in the matter. Central to this change were the advent of paraconsistent logics, the shift in attention from finished theories to construction processes, and the recognition that most scientific theories were at some point either internally inconsistent or incompatible with other accepted findings. The new interest (...) gave rise to important questions. How is `logical anarchy' avoided? Is it ever rational to accept an inconsistent theory? In what sense, if any, can inconsistent theories be considered as true? The present collection of papers is the first to deal with this kind of questions. It contains case studies as well as philosophical analyses, and presents an excellent overview of the different approaches in the domain. (shrink)

Peter Vickers examines 'inconsistent theories' in the history of science--theories which, though contradictory, are held to be extremely useful. He argues that these 'theories' are actually significantly different entities, and warns that the traditional goal of philosophy to make substantial, general claims about how science works is misguided.

The main thesis of this paper is that we sometimes are disposed to accept false and even jointly inconsistent claims by virtue of our semantic competence, and that this comes to light in the sorites and liar paradoxes. Among the subsidiary theses are that this is an important source of indeterminacy in truth conditions, that we must revise basic assumptions about semantic competence, and that classical logic and bivalence can be upheld in the face of the sorites paradox.

Mathias Frisch provides the first sustained philosophical discussion of conceptual problems in classical particle-field theories. Part of the book focuses on the problem of a satisfactory equation of motion for charged particles interacting with electromagnetic fields. As Frisch shows, the standard equation of motion results in a mathematically inconsistent theory, yet there is no fully consistent and conceptually unproblematic alternative theory. Frisch describes in detail how the search for a fundamental equation of motion is partly driven by pragmatic considerations (like (...) simplicity and mathematical tractability) that can override the aim for full consistency. The book also offers a comprehensive review and criticism of both the physical and philosophical literature on the temporal asymmetry exhibited by electromagnetic radiation fields, including Einstein's discussion of the asymmetry and Wheeler and Feynman's influential absorber theory of radiation. Frisch argues that attempts to derive the asymmetry from thermodynamic or cosmological considerations fail and proposes that we should understand the asymmetry as due to a fundamental causal constraint. The book's overarching philosophical thesis is that standard philosophical accounts that strictly identify scientific theories with a mathematical formalism and a mapping function specifying the theory's ontology are inadequate, since they permit neither inconsistent yet genuinely successful theories nor thick causal notions to be part of fundamental physics. (shrink)

Several writers have argued that the state lacks the moral standing to hold socially deprived offenders responsible for their crimes because the state would be hypocritical in doing so. Yet the state is not disposed to make an unfair exception of itself for committing the same sorts of crimes as socially deprived offenders, so it is unclear that the state is truly hypocritical. Nevertheless, the state is disposed to inconsistently hold its citizens responsible, blaming or punishing socially deprived offenders more (...) often or more harshly than other offenders, even when the crimes are the same. The state’s stable disposition to inconsistently hold offenders responsible undermines its standing to hold offenders responsible for the same reasons that hypocrisy undermines standing; instead of making an unfair exception of itself, the state makes an unfair exception of others. Strikingly, this means that the state lacks the standing to hold anyone responsible for a crime for which it is unfairly disposed to hold citizens responsible inconsistently, not just socially deprived offenders. Thus, it is even more urgent that the state regain its moral standing by working toward a justice system that holds offenders responsible consistently. (shrink)

A popular and enduring approach to the liar paradox takes the concept of truth to be inconsistent. Very roughly, truth is an inconsistent concept if the central principles of this concept (taken together) entail a contradiction, where one of these central principles is Tarski's T-schema for truth: a sentence S is true if and only if p, (where S says that p). This article targets a version of Inconsistentism which: retains classical logic and bivalence; takes the truth-predicate “is true” to (...) pick out a property (and determine a non-empty extension relative to a given world); and holds that liar sentences exhibit a certain kind of indeterminacy in truth-value. Call such a view Modest Inconsistentism since it is somewhat more conservative in its outlook than various other forms of Inconsistentism. Such a modest view has its attractions: we retain the thesis that the liar sentence is meaningful; we get to respect the claims that there are truths and that there is a property of truth; we get to keep classical logic and bivalence; and, prima facie, no strengthened liar paradox is in the offing. The main aim in this paper is to show that Modest Inconsistentism, despite its initial attractions, is in deep trouble—because it does, after all, give rise to a strengthened liar paradox. We shall also see that there are related kinds of theory which are also subject to the same worry. (shrink)

I erect a framework within the semantic view of theories for explaining the empirical success of internally inconsistent models and theories, with scientific realism in mind. The framework is an instance of the ‘content-driven’ approach to inconsistency, advocated by both Norton (Philos Sci 54:327–350, 1987) and Smith (Stud Hist Philos Sci 19:429–445, 1988a, In: Fine A, Leplin J (eds) PSA1988, 1988b), whose ideas my analysis aims to clarify and substantiate.

Plato’s view on pleasure in the Republic emerges in the course of developing the third proof of his central thesis that the just man is happier than the unjust. Plato presents it as the “greatest and most decisive” proof of his central thesis, so one might expect to find an abundance of scholarly work on it. Paradoxically, however, this argument has received little attention from scholars, and what has been written on it has generally been harshly critical. I believe that (...) this treatment of the argument has been unfair and that the relevant passages deserve a more careful and charitable interpretation. In this paper, I take up two serious charges that scholars have leveled against this proof, that it is inconsistent and that it involves a “fatal ambiguity”. I show that these charges result from misinterpreting Plato’s text, and I offer an alternative interpretation of the relevant passages. In doing so, I hope to shed some light on the complexities involved in Plato’s unappreciated third proof. (shrink)

In a recent issue of this journal, M. Frisch claims to have proven that classical electrodynamics is an inconsistent physical theory. We argue that he has applied classical electrodynamics inconsistently. Frisch also claims that all other classical theories of electromagnetic phenomena, when consistent and in some sense an approximation of classical electrodynamics, are haunted by “serious conceptual problems” that defy resolution. We argue that this claim is based on a partisan if not misleading presentation of theoretical research in classical electrodynamics.

Opponents of abortion are commonly said to be inconsistent in their beliefs or actions, and to fail in their obligations to prevent the deaths of embryos and fetuses from causes other than induced...

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)

Most opposition to induced abortion turns on the belief that human fetuses are persons from conception. On this view, the moral status of the fetus alone requires those in a position to provide aid—gestational mothers—to make tremendous sacrifices to benefit the fetus. Recently, critics have argued that this pro-life position requires more than opposition to induced abortion. Pro-life theorists are relatively silent on the issues of spontaneous abortion, surplus in vitro fertilisation human embryos, and the suffering and death of born (...) persons due to lack of access to food, shelter and medical care. Colgrove et al call such arguments inconsistency arguments, arguing they ‘do not matter’ and mischaracterise them as ad hominem attacks. Here, I argue these are better understood as moral dilemmas. While some critics argue pro-life inaction is evidence that they do not really believe human fetuses are persons, I contend this inaction is likely the result of resolvable confusion rather than moral negligence. (shrink)

The concept of measuring inconsistency in information was developed by John Grant in a 1978 paper in the context of first-order logic. For more than 20 years very little was done in this area until in the early 2000s a number of AI researchers started to formulate new inconsistency measures primarily in the context of propositional logic knowledge bases. The aim of this volume is to survey what has been done so far, to expand inconsistency measurement to (...) other formalisms, to connect it with related topics, and to provide ideas for further research in a topic that is particularly relevant now in view of the many inconsistencies in the massive amount of information available. The book contains 11 chapters. The first chapter, by John Grant, gives his original motivation for starting this field, explains why it was formulated in a highly mathematical manner, presents important material that was omitted from the original paper, and provides ideas about the use of dimensions in measuring inconsistency. The second chapter, by Matthias Thimm, is a survey that covers most of the research on inconsistency measures up to 2017. The other 9 chapters, all by experts either in inconsistency measures, or in the topic under consideration, or both, connect inconsistency measures with argumentation, disjunctive logic programming, fuzzy logic systems, modal logics, multiset representation, paraconsistent consequence, probabilistic logic, relational databases, and spatio-temporal databases."-- back cover. (shrink)

Abstract In this paper I discuss one apparent counterexample to the rationality constraint on belief ascription. The fact that there are inconsistent believers does not seem compatible with the idea that only rational creatures can be ascribed beliefs. I consider Davidson's explanation of the possibility of inconsistent believers and claim that it involves a reformulation of the rationality constraint in terms of the believers' subscription to norms of rationality. I shall argue that Davidson's strategy is partially successful, but that the (...) emphasis on the notion of subscription badly fits with the motivations behind his theory of interpretation. (shrink)

This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal (...) class='Hi'>Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning. (shrink)

The Uniform Determination of Death Act (UDDA) provides that “an individual who has sustained either (1) irreversible cessation of circulatory and respiratory functions or (2) irreversible cessation of all functions of the entire brain, including the brain stem, is dead.” We show that the UDDA contains two conflicting interpretations of the phrase “cessation of functions.” By one interpretation, what matters for the determination of death is the cessation of spontaneous functions only, regardless of their generation by artificial means. By the (...) other, what matters is the cessation of both spontaneous and artificially supported functions. Because each UDDA criterion uses a different interpretation, the law is conceptually inconsistent. A single consistent interpretation would lead to the conclusion that conscious individuals whose respiratory and circulatory functions are artificially supported are actually dead, or that individuals whose brain is entirely and irreversibly destroyed may be alive. We explore solutions to mitigate the inconsistency. (shrink)

William Simkulet has recently criticised Colgrove et al ’s defence against what they have called inconsistency arguments—arguments that claim opponents of abortion act in ways inconsistent with their underlying beliefs about human fetuses. Colgrove et al presented three objections to inconsistency arguments, which Simkulet argues are unconvincing. Further, he maintains that OAs who hold that the fetus is a person at conception fail to act on important issues such as the plight of frozen embryos, poverty and spontaneous abortion. (...) Thus, they are morally negligent. In response, we argue that Simkulet has targeted a very narrow group of OAs, and so his criticisms are inapplicable to most OAs. We then explain why his responses to each of Colgrove et al’s objections do not succeed, even for this restricted group. Finally, we note that Simkulet fails to provide evidence for his claims regarding OAs’ supposed failures to act, and we show that OAs veritably do invest resources into these important issues. We conclude that Colgrove et al ’s reasons for rejecting inconsistency arguments remain intact. (shrink)

This paper presents a new argument against A-theories of time. A-theorists hold that there is an objective now (present moment) and an objective flow of time, the latter constituted by the movement of the objective now through time. A-theorists therefore want to draw different pictures of reality—showing the objective now in different positions—depending upon the time at which the picture is drawn. In this paper it is argued that the times at which the different pictures are drawn may be taken (...) to be normal times or hypertimes. If they are normal times then the A-theory is inconsistent, or else collapses to the B-theory—and appealing to primitive tense operators will not help A-theorists avoid this conclusion. If the times are hypertimes then the A-theory is consistent, but deeply problematic none the less. (shrink)

It is argued that a certain form of the view that the semantic paradoxes show that natural languages are "inconsistent" provides the best response to the semantic paradoxes. After extended discussions of the views of Kirk Ludwig and Matti Eklund, it is argued that in its strongest formulation the view maintains that understanding a natural language is sharing cognition of an inconsistent semantic theory for that language with other speakers. A number of aspects of this approach are discussed and a (...) few objections are entertained. (shrink)

It is widely recognized that scientific theories are often associated with strictly inconsistent models, but there is little agreement concerning the epistemic consequences. Some argue that model inconsistency supports a strong perspectivism, according to which claims serving as interpretations of models are inevitably and irreducibly perspectival. Others argue that in at least some cases, inconsistent models can be unified as approximations to a theory with which they are associated, thus undermining this kind of perspectivism. I examine the arguments for (...) perspectivism, and contend that its strong form is defeasible in principle, not merely in special cases. The argument rests on the plausibility of scientific knowledge concerning non-perspectival, dispositional facts about modelled systems. This forms the basis of a novel suggestion regarding how to understand the knowledge these models afford, in terms of a contrastive theory of what-questions. (shrink)

A critic may attack an arguer personally by pointing out that the arguer’s position is pragmatically inconsistent: the arguer does not practice what he preaches. A number of authors hold that such attacks can be part of a good argumentative discussion. However, there is a difficulty in accepting this kind of contribution as potentially legitimate, for the reason that there is nothing wrong for a protagonist to have an inconsistent position, in the sense of committing himself to mutually inconsistent propositions. (...) If so, any such charge seems to be irrelevant. The questions to be answered in this essay are: what, if any, is the dialectical rationale for this type of criticism, and in what situations, if any, is this kind of charge dialectically legitimate? It will be shown that these attacks can be dialectically legitimate, in special circumstances, and that they can be seen as strategic␣manoeuvres where a party attempts to reconcile his dialectical and his rhetorical objectives. (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)

An argument for Trivialism, the view that natural languages are logically inconsistent, is provided that does not rely on contentious empirical assumptions about natural language terms such as “and” or “or.” Further, the view is defended against an important objection recently mounted against it by Thomas Hofweber.

I identify an inconsistency in Jago’s theory of truth. I show that Jago is committed to the identity of the proposition that the proposition that A is true and the proposition that A. I show that Jago is committed to the proposition that A being true because A if the proposition that A is true. I show that these two commitments, given the rest of Jago’s theory, entail a contradiction. I show that while the latter commitment follows from Jago’s (...) theory of truth, even by Jago’s own lights he should not be committed to the identity of the proposition that the proposition that A is true and the proposition that A. (shrink)

The paper explains how a paraconsistent logician can appropriate all classical reasoning. This is to take consistency as a default assumption, and hence to work within those models of the theory at hand which are minimally inconsistent. The paper spells out the formal application of this strategy to one paraconsistent logic, first-order LP. (See, Ch. 5 of: G. Priest, In Contradiction, Nijhoff, 1987.) The result is a strong non-monotonic paraconsistent logic agreeing with classical logic in consistent situations. It is shown (...) that the logical closure of a theory under this logic is trivial only if its closure under LP is trivial. (shrink)

Inferentialists about scientific representation hold that an apparatus’s representing a target system consists in the apparatus allowing “surrogative inferences” about the target. I argue that a serious problem for inferentialism arises from the fact that many scientific theories and models contain internal inconsistencies. Inferentialism, left unamended, implies that inconsistent scientific models have unlimited representational power, since an inconsistency permits any conclusion to be inferred. I consider a number of ways that inferentialists can respond to this challenge before suggesting my (...) own solution. I develop an analogy to exploitable glitches in a game. Even though inconsistent representational apparatuses may in some sense allow for contradictions to be generated within them, doing so violates the intended function of the apparatus’s parts and hence violates representational “gameplay.”. (shrink)

William Simkulet has recently criticised Colgrove et al’s defence against what they have called inconsistency arguments—arguments that claim opponents of abortion (OAs) act in ways inconsistent with their underlying beliefs about human fetuses (eg, that human fetuses are persons at conception). Colgrove et al presented three objections to inconsistency arguments, which Simkulet argues are unconvincing. Further, he maintains that OAs who hold that the fetus is a person at conception fail to act on important issues such as the (...) plight of frozen embryos, poverty and spontaneous abortion. Thus, they are morally negligent. In response, we argue that Simkulet has targeted a very narrow group of OAs, and so his criticisms are inapplicable to most OAs. We then explain why his responses to each of Colgrove et al’s objections do not succeed, even for this restricted group. Finally, we note that Simkulet fails to provide evidence for his claims regarding OAs’ supposed failures to act, and we show that OAs veritably do invest resources into these important issues. We conclude that Colgrove et al’s reasons for rejecting inconsistency arguments (en masse) remain intact. (shrink)

This paper demonstrates that there is an inconsistency in functionalism in psychology and philosophy of mind. Analogous inconsistencies can be expected in functionalisms in biology and social theory. (edited).

Several authors maintain that we are capable of having inconsistent intentional states, either in cases of illusion, in certain cases of imagination, or because the observable world is (partly) inconsistent and we perceive it as such. These views are all premised on the assumption that inconsistent intentional states—even if acknowledged—are peculiar and have nothing essential to do with our perceptual capacities. In the present article, I would like to present, and argue for, a much stronger thesis: that inconsistent intentional states (...) underlie the possibility of having intentional content in mind. I argue for this thesis based on a Husserlian phenomenological analysis of our grasp of objects, which I formulate in terms of incompatibility semantics. (shrink)

In this work, the first of a series, we study the nature of informal inconsistency in physics, focusing mainly on the foundations of quantum theory, and appealing to the concept of quasi-truth. We defend a pluralistic view of the philosophy of science, grounded on the existence of inconsistencies and on quasi-truth. Here, we treat only the ‘classical aspects’ of the subject, leaving for a forthcoming paper the ‘non-classical’ part.

The paper examines philosophical issues that arise in contexts where one has many different models for treating the same system. I show why in some cases this appears relatively unproblematic (models of turbulence) while others represent genuine difficulties when attempting to interpret the information that models provide (nuclear models). What the examples show is that while complementary models needn’t be a hindrance to knowledge acquisition, the kind of inconsistency present in nuclear cases is, since it is indicative of a (...) lack of genuine theoretical understanding. It is important to note that the differences in modeling do not result directly from the status of our knowledge of turbulent flows as opposed to nuclear dynamics—both face fundamental theoretical problems in the construction and application of models. However, as we shall, the ‘problem context(s)’ in which the modeling takes plays a decisive role in evaluating the epistemic merit of the models themselves. Moreover, the theoretical difficulties that give rise to inconsistent as opposed to complementary models (in the cases I discuss) impose epistemic and methodological burdens that cannot be overcome by invoking philosophical strategies like perspectivism, paraconsistency or partial structures. (shrink)

Research Ethics Committees (RECs) are frequently a focus of complaints from researchers, but evidence about the operation and decisions of RECs tends to be anecdotal. We conducted a systematic study to identify and compare the ethical issues raised in 54 letters to researchers about the same 18 applications submitted to three RECs over one year. The most common type of ethical trouble identified in REC letters related to informed consent, followed by scientific design and conduct, care and protection of research (...) participants, confidentiality, recruitment and documentation. Community considerations were least frequently raised. There was evidence of variability in the ethical troubles identified and the remedies recommended. This analysis suggests that some principles may be more institutionalized than others, and offers some evidence of inconsistency between RECs. Inconsistency is often treated as evidence of incompetence and caprice, but a more sophisticated understanding of the role of RECs and their functioning is required. (shrink)

At various times, mathematicians have been forced to work with inconsistent mathematical theories. Sometimes the inconsistency of the theory in question was apparent (e.g. the early calculus), while at other times it was not (e.g. pre-paradox na¨ıve set theory). The way mathematicians confronted such difficulties is the subject of a great deal of interesting work in the history of mathematics but, apart from the crisis in set theory, there has been very little philosophical work on the topic of inconsistent (...) mathematics. In this paper I will address a couple of philosophical issues arising from the applications of inconsistent mathematics. The first is the issue of whether finding applications for inconsistent mathematics commits us to the existence of inconsistent objects. I then consider what we can learn about a general philosophical account of the applicability of mathematics from successful applications of inconsistent mathematics. (shrink)

This is a philosophical and historical investigation of the role of inconsistent representations of the same scientific phenomenon. The logical difficulties associated with the simultaneous application of inconsistent models are discussed. Internally inconsistent scientific proposals are characterized as structures whose application is necessarily tied to the confirming evidence that each of its components enjoys and to a vision of the general form of the theory that will resolve the inconsistency. Einstein's derivation of the black body radiation law is used (...) as an example application of such a strategy and is contrasted with planck's derivation. (shrink)

Buchanan highlighted the inconsistencies that arise when public services are financed by general taxation. Citizens increase their demand for services, even though citizens are reticent to increase taxation. Buchanan invited readers to explore the impact of different assumptions of politicians’ behaviour. In this chapter, attention focuses on the way that vote maximising governments are likely to respond to the divorce between receipt and payment for services. Buchanan illustrated his analysis with reference to the National Health Service in the UK. Predictions (...) are tested with reference to the pattern of excess demand in the National Health Service between 1970 and 2012. (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)

I revisit my earlier arguments for the (trivial) inconsistency of natural languages, and take up the objection that no such argument can be established on the basis of surface usage. I respond with the evidential centrality of surface usage: the ways it can and can't be undercut by linguistic science. Then some important ramifications of having an inconsistent natural language are explored: (1) the temptation to engage in illegitimate reductio reasoning, (2) the breakdown of the knowledge idiom (because its (...) facticity isn't comfortable with every sentence being true and false). Restoring the utility of the knowledge idiom motivates regimentation - the consistentist reinterpretation of natural language inferences (as they occur in real time). It's then sketched how to give the semantics of a natural language despite its inconsistency: Necessary and sufficient truth conditions are dropped in favor of necessary truth conditions and sufficient truth conditions. (shrink)

This is a discussion of different ways of working out the idea that the semantic paradoxes show that natural languages are somehow “inconsistent”. I take the workable form of the idea to be that there are expressions such that a necessary condition of understanding them is that one be inclined to accept inconsistent claims (an conception also suggested by Matti Eklund). I then distinguish “simple” from “complex” forms of such views. On a simple theory, such expressions are meaningless, while on (...) a complex theory they are not. I argue that complex theories are incompatible with truth conditional semantics and that simple theories are only coherent when the inconsistent claims are metalingusitic attributions of meaning. I close with a discussion of the version of the simple metalinguistic theory I have defended in “Understanding the Liar” and other papers. (shrink)

This paper discusses the problem of inconsistencies in the metaphorical conceptualizations of time that involve motion within the framework of conceptual metaphor theory (CMT). It demonstrates that the TIME AS A PURSUER metaphor contrasts with the reverse variant TIME AS AN OBJECT OF PURSUIT, just as the MOVING TIME metaphor contrasts with the MOVING OBSERVER variant. Such metaphorical conceptualizations of time functioning as pairs of minimally differing variants based on Figure-Ground reversal are, strictly speaking, inconsistent with one another. Looking at (...) these inconsistencies from a wider perspective suggests that time may belong to a separate category of conceptual phenomena. This paper puts forward a proposal to approach time from the perspective of “phenomena of the third kind”, which according to Keller’s thesis include conceptual establishments resulting from human cognition, but not of human design. (shrink)