It is widely accepted that a theory of truth for arithmetic should be consistent, but -consistency is a highly desirable feature for such theories. The point has already been made for first-order languages, though the evidence is not entirely conclusive. We show that in the second-order case the consequence of adopting -inconsistent theories of truth are considered: the revision theory of nearly stable truth T # and the classical theory of symmetric truth FS. Briefly, we present some conceptual (...) problems with ω-inconsistent theories, and demonstrate some technical results that support our criticisms of such theories. (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.

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...) not jointly, lack the problematic feature. (shrink)

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I (...) argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. -/- (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, is unjustified, in general. This means my position is stronger against this objection than the paper suggests, since omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences.). (shrink)

En este artículo, tengo dos objetivos distintos. En primer lugar, mostrar que no es una buena idea tener una teoría de la verdad que, aunque consistente, sea omega-inconsistente. Para discutir este punto, considero un caso particular: la teoría de Friedman-Sheard FS. Argumento que en los lenguajes de primer orden omega inconsistencia implica que la teoría de la verdad no tiene modelo estándar. Esto es, no hay un modelo cuyo dominio sea el conjunto de los números naturales en el cual esta (...) teoría de la verdad pueda tener una interpretación consistente. En ese sentido, la introducción del predicado veritativo no mantiene la ontología estándar. Además, cuando se considera un lenguaje de orden superior, la situación es aun peor. En teorías de segundo orden con semántica estándar, la misma introducción produce una teoría que no tiene modelo. Por eso, si la omega-inconsistencia es un mal síntoma, la insatisfacibilidad de una teoría es aun peor. En segundo lugar, propongo abandonar el principio de unión de teorías FSn y aceptar una extensibilidad indefinida de teorías FS0, FS1, FS2, FS3,.... De acuerdo a mi punto de vista, la secuencia de teorías tiene las mismas virtudes que FS sin sus desagradables consecuencias. In this paper, I have two different purposes. Firstly, I want to show that it's not a good idea to have a theory of truth that is consistent but omega-inconsistent. In order to bring out this point, it is useful to consider a particular case: FS. I argue that in First-order languages omega-inconsistency implies that a theory of truth has not standard model. Then, there is no model whose domain is the set of natural numbers in which this theory of truth could acquire a consistent interpretation. So, in theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. I add that in Higher-order languages the situation is even worst. In second order theories with standard semantic the same introduction produces a theory that doesn't have a model. So, if an omega-inconsistent theory of truth is bad, an unsatisfiable theory is really bad. Secondly, I propose to give up the union principle of theories FSn and accept an indefinite extensibility of theories FS0, FS1, FS2, FS3,... According to my view, the sequence of theories has the same virtues of FS without its disgusting consequences. (shrink)

This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency (...) and shows thus, pace Cobreros et al., that the result in McGee does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic. (shrink)

Theories where truth is a naive concept fall under the following dilemma: either the theory is subject to Curry’s Paradox, which engenders triviality, or the theory is not trivial but the resulting conditional is too weak. In this paper we explore a number of theories which arguably do not fall under this dilemma. In these theories the conditional is characterized in terms of non-deterministic matrices. These non-deterministic theories are similar to infinitely-valued Łukasiewicz logic in that they (...) are consistent and their conditionals are quite strong. The difference is the following: while Łukasiewicz logic is \-inconsistent, the non-deterministic theories might turn out to be \-consistent. (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)

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)

Classical logic is explosive in the face of contradiction, yet we find ourselves using inconsistent theories. Mark Colyvan, one of the prominent advocates of the indispensability argument for realism about mathematical objects, suggests that such use can be garnered to develop an argument for commitment to inconsistent objects and, because of that, a paraconsistent underlying logic. I argue to the contrary that it is open to a classical logician to make distinctions, also needed by the paraconsistent logician, which allow (...) a more nuanced ranking of theories in which inconsistent theories can have different degrees of usefulness and productivity. Facing inconsistency does not force us to adopt an underlying paraconsistent logic. Moreover we will see that the argument to best explanation deployed by Colyvan in this context is unsuccessful. I suggest that Quinean approach which Colyvan champions will not lead to the revolutionary doctrines Colyvan endorses. (shrink)

Douglas Patterson argues that the best way to respond to the semantic paradoxes that arise in natural language is to take natural language semantics to be (explosively) inconsistent. According to Patterson, to understand a natural language is to share with others cognition of a false semantic theory. Patterson’s main argument runs as follows. English is expressively rich. So, the first sentence occurring in this review could be.

Theories are conceptualized as predictors of phenomena using computable functions acting on prior world information. Formally, the concept of bounded prior world recursive function is defined and used as a candidate for a potential theory. An artificial world of fact is then constructed for which there exist two inconsistent best theories, in that they cannot be improved upon, and these theories are maximally inconsistent in that every best theory is a compromise. It is argued that in such (...) a world the scientific objective would be to find these maximally inconsistent theories. The construction is motivated as an attempt to illustrate the status of theory in the social sciences and in economics, in particular. (shrink)

An Omega Point Theory says that reality is making progress from some initial state to some final state. It moves from some Alpha Point (the initial state) to some Omega Point (the final state). The progress is an increase in some quality. For example, reality is making progress from the chaotic to the orderly; or it is making progress from the simple to the complex; or from the mindless to the mental; or from evil to good. Here we focus on (...) the Omega Point theory of Peirce. An Omega Point Theory is an evolutionary cosmology – it says that the universe is evolving from the Alpha Point to the Omega Point. Peirce refers to his evolutionary cosmology as a Cosmogonic Philosophy would suppose that in the beginning – infinitely remote – there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence. This feeling, sporting here and there in pure arbitrariness, would have started the germ of a generalizing tendency. Its other sportings would be evanescent, but this would have a growing virtue. Thus the tendency to habit would be started; and from this, with the other principles of evolution, all the regularities of the universe would be evolved. At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future. (Peirce, Collected Papers , 6.33). (shrink)

In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive (...) answer to the last question: there are times when it is legitimate to believe in inconsistent objects. (shrink)

Canadian Journal of Philosophy, 37: 263–282. [preprint] This paper is a critical discussion of Mathias Frisch’s book Inconsistency, Asymmetry, and Nonlocality.

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)

We study groups definable in tame expansions of ω-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair, where K is an algebraically closed field and G is a multiplicative subgroup of K× with the Mann property, we show that the pair satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in in terms of algebraic groups (...) in K and interpretable groups in G. Furthermore, we compute the Morley rank and the U-rank in and both ranks agree. (shrink)

In this paper I critically evaluate a number of current "consistent inconsistency theories" and then briefly motivate a rival position. The rival position challenges a consistent inconsistency theory, by sharing many of its basic commitments without suffering the problems that such a theory appears to face.

An attempt to discover what can be learned from the recent work of Frank Tipler on the Omega Point theory requires an analysis of his framework of understanding from scientific, philosophical, and theological perspectives. A critique of his crucial ideas, and of the salient points raised by some of his critics, can then be undertaken within the compass of his strengths. A critique of the critiques of Tipler's work allows one to evaluate the extent and limitations of his contributions.

We study the class of all strongly constructivizable models having \(\omega \) -stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean \(\Sigma ^1_1\) -algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of all Boolean \(\Sigma ^1_1\) -algebras. This gives a characterization to the (...) Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with \(\omega \) -stable theories. (shrink)

In this article I will show how the goal-directed proof procedure for the propositional fragment of the 'inconsistency-adaptive' logic CLuN(r)from Batens (2005, J. Appl. Logic, 3, 221-250) called pCLuN(r) allows for generating sensible abductive explanations from finite inconsistent theories. Before this is shown, I give a more general account on how goal-directed proof procedures contribute to the study of abduction as backward reasoning. Between these two parts, I describe the goal-directed proof procedure for the propositional fragment of the 'inconsistency-adaptive' (...) logic CLuNr from Batens (2005, J. Appl. Logic, 3, 221-250). (shrink)

The ubiquitous assertion that the early calculus of Newton and Leibniz was an inconsistent theory is examined. Two different objects of a possible inconsistency claim are distinguished: (i) the calculus as an algorithm; (ii) proposed explanations of the moves made within the algorithm. In the first case the calculus can be interpreted as a theory in something like the logician’s sense, whereas in the second case it acts more like a scientific theory. I find no inconsistency in the first case, (...) and an inconsistency in the second case which can only be imputed to a small minority of the relevant community. (shrink)

Kirchhoff’s 1882 theory of optical diffraction forms the centerpiece in the long-term development of wave optics, one that commenced in the 1820s when Fresnel produced an empirically successful theory based on a reinterpretation of Huygens’ principle, but without working from a wave equation. Then, in 1856, Stokes demonstrated that the principle was derivable from such an equation albeit without consideration of boundary conditions. Kirchhoff’s work a quarter century later marked a crucial, and widely influential, point for he produced Fresnel’s results (...) by means of Green’s theorem and function under specific boundary conditions. In the late 1880s, Poincaré uncovered an inconsistency between Kirchhoff’s conditions and his solution, one that seemed to imply that waves should not exist at all. Researchers nevertheless continued to use Kirchhoff’s theory—even though Rayleigh, and much later Sommerfeld, developed a different and mathematically consistent formulation that, however, did not match experimental data better than Kirchhoff’s theory. After all, Kirchhoff’s formula worked quite well in a specific approximation regime. Finally, in 1964, Marchand and Wolf employed the transformation of Kirchhoff’s surface integral that had been developed by Maggi and Rubinowicz for other purposes. The result yielded a consistent boundary condition that, while introducing a species of discontinuity, nevertheless rescued the essential structure of Kirchhoff’s original formulation from Poincaré’s paradox. (shrink)

This paper accomplishes three goals. First, the essay demonstrates that Edmund Husserl’s theory of meaning consciousness from his 1901 Logical Investigations is internally inconsistent and falls apart upon closer inspection. I show that Husserl, in 1901, describes non-intuitive meaning consciousness as a direct parallel or as a ‘mirror’ of intuitive consciousness. He claims that non-intuitive meaning acts, like intuitions, have substance and represent their objects. I reveal that, by defining meaning acts in this way, Husserl cannot account for our experiences (...) of countersensical, absurd, or impossible meanings. Second, I examine how Husserl came to recognize this 1901 mistake in his 1913/14 Revisions to the Sixth Logical Investigation. I discuss how he accordingly reformulates his understanding of non-intuitive meaning acts from the ground up in those Revisions, where this also allows for him to properly account for the experience of impossible meanings. Instead of describing them as mirrors of intuitions, Husserl takes non-intuitive meaning acts to be modifications of intuitions, where they have no substance and do not represent their objects. Finally, in the conclusion to this essay, I demonstrate how this fundamental change to his understanding of meaning consciousness forced Husserl to revise other central tenets of his philosophy, such that the trajectory of his thought can only be properly understood in light of these revisions to his theory of non-intuitive meaning consciousness. (shrink)

For several decades now philosophers have discussed apparent examples of internally inconsistent scientific theories. However, there is still much controversy over how exactly we should conceive of scientific theories in the first place. Here I argue for a new approach, whereby all of the truly important questions about inconsistency in science can be asked and answered without disagreements about theories and theory-content getting in the way. Three examples commonly described as ‘internally inconsistent theories’ are analysed in (...) the light of this approach. In the process, the question ‘Is the theory inconsistent or not?’ is identified as a bad, or at least unimportant, question. (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)

The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We prove this (...) in a general setting for a family of relevant connexive logics, extract the essential features of the proof in order to obtain a sufficient condition for the consistency of a theory in arbitrary logics, and finally consider some concrete examples of consistent mathematical theories in Abelian logic. The upshot is that on this way of understanding deductive closure, common to relevant logics, there is a rich and interesting kind of interaction between inconsistent logics and their theories. We argue that this suggests an important avenue for investigation of inconsistent logics, from both a technical and a philosophical perspective. (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)

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)

This article focuses on the Omega Point, an essential concept in Teilhard de Chardin’s evolutionary metaphysics. In certain passages about the Omega Point, Teilhard mentions the primeval atom hypothesis, a theory about the beginning of the universe proposed by Georges Lemaître, another contemporary Jesuit priest who was also a scientist. Although Teilhard and Lemaître are essentially evolutionists, besides being Jesuit priests, their evolutionary metaphysics and their philosophies of science are radically divergent, and two important differences are presented here – about (...) the eventual cosmic determinism and about the relationship between scientific knowledge and religious beliefs. This article also shows how Teilhard’s original concept was projected into two cosmological theories in which it reveals an essential transfiguration by losing its initial anthropic and religious matrix – the cosmological anthropic principle, proposed by physicists John Barrow and Frank Tipler, and the Omega Point Theory proposed later by Frank Tipler. (shrink)

We present the axioms of Alternative Set Theory in the language of second-order arithmetic and study its ω- and β-models. These are expansions of the form , M ⊆ P, of nonstandard models M of Peano arithmetic such that ⊩ AST and ω ϵ M. Our main results are: A countable M ⊩ PA is β-expandable iff there is a regular well-ordering for M. Every countable β-model can be elementarily extended to an ω-model which is not a β-model. The Ω-orderings (...) of an ω-model are absolute well-orderings iff the standard system SS of M is a β-model of A−2. There are ω-expandable models M such that no ω-expansion of M contains absolute Ω-orderings. There are s-expandable models which are not β-expandable. For every countable β-expansion M of M, there is a generic extension M[G] which is also a β-expansion of M. If M is countable and β-expandable, then there are regular orderings <1, <2 such that neither <1 belongs to the ramified analytical hierarchy of the structure , nor <2 to that of . The result can be improved as follows: A countable M ⊩ PA is β-expandable iff there is a semi-regular well-ordering for M. (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)

Summary It is here shown that the relativistic doctrine of the relativity of simultaneity is untenable and that both the special and general theories of relativity are inconsistent. It is also shown that the theories can perhaps be made consistent, but excessively weak, through the reintroduction of absolute space and a weakening of the Lorentz transformations. Non-relativistic hypotheses for some events thought to require relativity are suggested. Finally, some conjectures are made on how so wrong a theory could (...) have been accepted by so many for so long. (shrink)

Kirchhoff’s diffraction theory is introduced as a new case study in the realism debate. The theory is extremely successful despite being both inconsistent and not even approximately true. Some habitual realist proclamations simply cannot be maintained in the face of Kirchhoff’s theory, as the realist is forced to acknowledge that theoretical success can in some circumstances be explained in terms other than truth. The idiosyncrasy (or otherwise) of Kirchhoff’s case is considered.

In this paper, the problem of purifying an assumption-based theory KB, i.e., identifying the right extension of KB using knowledge-gathering actions (tests), is addressed. Assumptions are just normal defaults without prerequisite. Each assumption represents all the information conveyed by an agent, and every agent is associated with a (possibly empty) set of tests. Through the execution of tests, the epistemic status of assumptions can change from "plausible" to "certainly true", "certainly false" or "irrelevant", and the KB must be revised so (...) as to incorporate such a change. Because removing all the extensions of an assumption-based theory except one enables both identifying a larger set of plausible pieces of information and renders inference computationally easier, we are specifically interested in finding out sets of tests allowing to purify a KB (whatever their outcomes). We address this problem especially from the point of view of computational complexity. (shrink)

Direct reference theory faces serious prima facie counterexamples which must be explained away (e.g., that it is possible to know a priori that Hesperus = Phosphorus). This is done by means of various forms of pragmatic explanation. But when those explanations that provisionally succeed are generalized to deal with analogous prima facie counterexamples concerning the identity of propositions, a fatal dilemma results. Either identity must be treated as a four-place relation (contradicting what just about everyone, including direct reference theorists, takes (...) to be essential to identity). Or direct reference theorists must incorporate a view that was rejected in pretty much our first lesson about identity—namely, that Hesperus at twilight is not identical to Hesperus at dawn. One way of the other, the direct reference theory is thus inconsistent with basic principles concerning the logic of identity, which nearly everyone, including direct reference theorists, take as starting points. (shrink)

The problem of how to accommodate inconsistencies has attracted quite a number of researchers, in particular, in the area of database theory. The problem is also of concern in the study of belief change. For inconsistent beliefs are ubiquitous. However, comparatively little work has been devoted to discussing the problem in the literature of belief change. In this paper, I examine how adequate the AGM theory is as a logical framework for belief change involving inconsistencies. The technique is to apply (...) to Grove’s sphere system, a semantical representation of the AGM theory, logics that do not infer everything from contradictory premises, viz., paraconsistent logics. I use three paraconsistent logics and discuss three sphere systems that are based on them. I then examine the completeness of the postulates of the AGM theory with respect to the systems. At the end, I discuss some philosophical implications of the examination. (shrink)

Martin-Löf's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom - treating the category of propositions as a set and thereby enabling higher order quantification - leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of all well-founded sets). The occurrence (...) of the contradiction is explained in set theoretical terms. Crucial here is the way a proof-object of an existential proposition is understood. It is shown that also Russell's paradox can be translated into type theory. The type theory extended with the axiom mentioned above contains constructive higher order logic, but even if one only adds constructive second order logic to type theory the contradictions arise. (shrink)

The old quantum theory of black body radiation was manifestly logically inconsistent. It required the energies of electric resonators to be both quantized and continuous. To show that this manifest inconsistency was inessential to the theory's recovery of the Planck distribution law, I extract a subtheory free of this manifest inconsistency but from which Planck's law still follows.

This article argues that the reduction of the morality of killing in war to the morality of killing in self-defense by ‘reductive-individualist’ revisionist just war theories is inconsistent, because when those theories apply the moral notion of self-defense to the morality of killing in war, they do not preserve the two conceptions of the “individual” inherent in this notion. The article demonstrates this inconsistency in two steps: First, it disentangles the two conceptions of the individual inherent to the (...) notion of self-defense, namely (1) that the individual is an “entity” potentially bearing a right to self-defense (unlike, e.g., groups) and (2) that the individual is a “particular,” where “particular” signifies that every human is different from every other human. The conception of the individual as a “particular” is tied to the idea that a justification grounded in a rule of self-defense is necessarily “concrete,” in the sense of referring to individually given and specific perceptions or cases, as opposed to “abstract,” in the sense of being detached from specific perceptions or cases. The article then demonstrates that reductive individualism reflects the first notion of the individual, but not the second. Due to the “loss” of the individual as a “particular”, the reductive-individualist reduction of the morality of killing in war to the morality of killing in self-defense is inconsistent, and hence its justification of killing in war grounded in self-defense is not concrete. Since such a justification must be concrete, reductive individualism cannot offer a justification for belligerent killing. (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)