We prove that if GCH holds and τ = 〈κα : α < η 〉 is a sequence of infinite cardinals such that κα ≥ |η | for each α < η, then there is a cardinal-preserving partial order that forces the existence of a scattered Boolean space whose cardinal sequence is τ.

Niko Kolodny It is often said that there is a special class of norms, ‘rational requirements’, that demand that our attitudes be related one another in certain ways, whatever else may be the case.1 In recent work, a special class of these rational requirements has attracted particular attention: what I will call ‘requirements of formal coherence as such’, which require just that our attitudes be formally coherent.2 For example, we are rationally required, if we believe something, to believe what it (...) entails. And we are rationally required, if we intend an end, to intend what we take to be necessary means to it. The intuitive idea is that formally incoherent attitudes give rise to a certain normative tension, or exert a kind of rational pressure on each another, and this tension, or pressure, is relieved just when one of the attitudes is revised. As John Broome observes, these requirements are, by their nature, ‘wide scope’, which is to say that there is no particular attitude that one must have or lack in order to satisfy them. This is because they require just formal coherence, and there is no particular attitude that one must have or lack in order to be formally coherent. (shrink)

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...) is ramifiable” implies the existence of a proper class of inaccessible cardinals. (shrink)

It might be thought that we could argue for the consistency of a mathematical theory T within T, by giving an inductive argument that all theorems of T are true and inferring consistency. By Gödel's second incompleteness theorem any such argument must break down, but just how it breaks down depends on the kind of theory of truth that is built into T. The paper surveys the possibilities, and suggests that some theories of truth give far more intuitive (...) diagnoses of the breakdown than do others. The paper concludes with some morals about the nature of validity and about a possible alternative to the idea that mathematical theories are indefinitely extensible. (shrink)

Alan Gewirth's Reason and Morality , in which he set forth the Principle of Generic Consistency, is a major work of modern ethical theory that, though much debated and highly respected, has yet to gain full acceptance. Deryck Beyleveld contends that this resistance stems from misunderstanding of the method and logical operations of Gewirth's central argument. In this book Beyleveld seeks to remedy this deficiency. His rigorous reconstruction of Gewirth's argument gives its various parts their most compelling formulation and (...) clarifies its essential logical structure. Beyleveld then classifies all the criticisms that Gewirth's argument has received and measures them against his reconstruction of the argument. The overall result is an immensely rich picture of the argument, in which all of its complex issues and key moves are clearly displayed and its validity can finally be discerned. The comprehensiveness of Beyleveld's treatment provides ready access to the entire debate surrounding the foundational argument of Reason and Morality . It will be required reading for all who are interested in Gewirth's theory and deontological ethics and will be of central importance to moral and legal theorists. (shrink)

Peter Vanderschraaf’s Strategic Justice provides a defense of the egalitarian bargaining solution. Vanderschraaf’s discussion of the egalitarian solution invokes three arguments typically given to support the Nash bargaining solution. Overall, we reinforce Vanderschraaf’s criticism of arguments in favor of the Nash solution and point to potential weaknesses in Vanderschraaf’s positive case for the egalitarian solution.

In *Cognition and Commitment in Hume's Philosophy*, Don Garrett argues for the coherence of Hume's philosophy when it is viewed as work in cognitive psychology. Without denying this, I argue that there is more to Hume's standpoint than cognitive psychology. Specifically, Hume's standpoint shifts as the level of inquiry changes. A descriptive cognitive psychology is one standpoint that he occupies. However, he occupies other standpoints as well: the commonsense standpoint of the vulgar is one; the radical doubt of the skeptic (...) is another. Such a radical perspectivism is central to Hume's writings. (shrink)

The principle of maximum entropy is a general method to assign values to probability distributions on the basis of partial information. This principle, introduced by Jaynes in 1957, forms an extension of the classical principle of insufficient reason. It has been further generalized, both in mathematical formulation and in intended scope, into the principle of maximum relative entropy or of minimum information. It has been claimed that these principles are singled out as unique methods of statistical inference that agree with (...) certain compelling consistency requirements. This paper reviews these consistency arguments and the surrounding controversy. It is shown that the uniqueness proofs are flawed, or rest on unreasonably strong assumptions. A more general class of inference rules, maximizing the so-called Re[acute ]nyi entropies, is exhibited which also fulfill the reasonable part of the consistency assumptions. (shrink)

I argue that the medieval form of dialectical disputation known as obligationes can be viewed as a logical game of consistency maintenance. The game has two participants, Opponent and Respondent. Opponent puts forward a proposition P; Respondent must concede, deny or doubt, on the basis of inferential relations between P and previously accepted or denied propositions, or, in case there is none, on the basis of the common set of beliefs. Respondent loses the game if he concedes a contradictory (...) set of propositions. Opponent loses the game if Respondent is able to maintain consistency during the stipulated period of time. The obligational rules are here formalised by means of familiar notational devices, and the application of some game-theoretical concepts, such as (winning) strategy, moves, motivation, allows for an analysis of some crucial properties of the game. In particular, the primacy of inferential (syntactic) relations over semantic aspects and the dynamic character of obligations are outlined. (shrink)

In this paper, consistency is understood as absolute consistency (i.e. non-triviality). The basic constructive logic BKc6, which is adequate to this sense of consistency in the ternary relational semantics without a set of designated points, is defined. Then, it is shown how to define a series of logics by extending BKc6 up to contractionless intuitionistic logic. All logics defined in this paper are paraconsistent logics.

Any inconsistent theory whose underlying logic is classical encompasses all the sentences of its own language. As it denies everything it asserts, it is useless for explaining or predicting anything. Nevertheless, paraconsistent logic has shown that it is possible to live with contradictions and still avoid the collapse of the theory. The main point of this paper is to show that even if it is formally possible to isolate the contradictions and to live with them, this cohabitation is neither desired (...) by working scientists not desirable for the progress of science. Several cases from the recent history of physics and cosmology are analyzed. (shrink)

We have seen that despite Feferman's results Gödel's second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a (...) system as we are willing to entertain seriously then a consistency proof for it will yield no epistemological gain. But in other cases philosophers might argue that the proof uses methods which are merely different rather than stronger than those available in the system in question. This claim has been made, for example, in the case of the constructive consistency proofs for elementary number theory. Similar philosophical investigations can be made on relative consistency proofs, since these differ from each other in the principles they employ. For example, most relative consistency proofs can be carried out within elementary number theory, but without using the theory of real numbers, no one has been able to prove the consistency of Quine's ML relative to that of his NF.What about the consistency of all mathematics or of some strong system for set theory? How do we answer the skeptic? Since here a convincing proof is not possible, we have established that the skeptic demands too much. We cannot be certain that our axioms are free from contradiction and must treat them as hypotheses which may be abandoned or modified in the face of further mathematical experience. This attitude is taken by many foundational workers who also go on to voice opinions about thelikelihood that various systems are consistent. Since these opinions are variously supported by appeals to the clarity of the mathematical concept formalized, the existence or non-existence of ‘weird’ models for the system and actual empirical experience with the system, this is surely a fruitful area for philosophical research. (shrink)

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question of (...) whether we have a Dutch Book Argument for Conditionalization is left open. (shrink)

In this paper, we show that certain choiceless models of ZF originally constructed using an almost huge cardinal can be constructed using cardinals strictly weaker in consistency strength.

Thought: A Journal of Philosophy, EarlyView.

Rule-consequentialism has been accused of either collapsing into act-consequentialism or being internally inconsistent. I have tried to develop a form of rule-consequentialism without these flaws. In this June's issue of Utilitas, Robert Card argued that I have failed. Here I assess his arguments.

To consciously bolster behaviour that is disapproved by others (i.e., stigmatised behaviour) people may hold and report a favourable attitude toward the behaviour. However, achieving such bolstering outside awareness may be more difficult. Explicit attitudes were measured with self-report measures, and the Implicit Association Test was used to assess implicit attitudes toward behaviour held by stigmatised actors (smokers) and nonstigmatised actors (vegetarians and omnivores). Smokers' showed greater attitude-behaviour consistency in their explicit attitudes toward smoking that in their implicit attitudes. (...) By contrast, vegetarians and omnivores showed attitude-behaviour-consistency at both implicit and explicit levels. Smokers' implicit negative attitudes toward smoking may reflect its status as a stigmatised behaviour, or its addictive nature. (shrink)

Tragic choices arise during the COVID-19 pandemic when the limited resources made available in acute medical settings cannot be accessed by all patients who need them. In these circumstances, healthcare rationing is unavoidable. It is important in any healthcare rationing process that the interests of the community are recognised, and that decision-making upholds these interests through a fair and consistent process of decision-making. Responding to recent calls to safeguard individuals’ legal rights in decision-making in intensive care, and for new authoritative (...) national guidance for decision-making, this paper seeks to clarify what consistency and fairness demand in healthcare rationing during the COVID-19 pandemic, from both a legal and ethical standpoint. The paper begins with a brief review of UK law concerning healthcare resource allocation, considering how community interests and individual rights have been marshalled in judicial deliberation about the use of limited health resources within the National Health Service. It is then argued that an important distinction needs to be drawn between procedural and outcome consistency, and that a procedurally consistent decision-making process ought to be favoured. Congruent with the position that UK courts have adopted for resource allocation decision-making in the NHS more generally, specific requirements for a procedural framework and substantive triage criteria to be applied within that framework during the COVID-19 pandemic are considered in detail. (shrink)

The most immediately appealing model for formal constraints on degrees of belief is provided by probability theory, which tells us, for instance, that the probability of P can never be greater than that of (P v Q). But while this model has much intuitive appeal, many have been concerned to provide arguments showing that ideally rational degrees of belief would conform to the calculus of probabilities. The arguments most frequently used to make this claim plausible are the so-called "Dutch Book" (...) arguments. (shrink)

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π (...) 1 0 T ω α · , α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T , T β stands for β times iterated consistency, and ≡ Π 1 0 denotes mutual Π 1 0 -conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal. (shrink)

Stephen Yablo [23,24] introduces a new informal paradox, constituted by an infinite list of semi-formalized sentences. It has been shown that, formalized in a first-order language, Yablo’s piece of reasoning is invalid, for it is impossible to derive falsum from the sequence, due mainly to the Compactness Theorem. This result casts doubts on the paradoxical character of the list of sentences. After identifying two usual senses in which an expression or set of expressions is said to be paradoxical, since second-order (...) languages are not compact, I study the paradoxicality of Yablo’s list within these languages. While non-paradoxical in the first sense, the second-order version of the list is a paradox in our second sense. I conclude that this suffices for regarding Yablo’s original list as paradoxical and his informal argument as valid. (shrink)

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

The assumption of rational expectations (RE) plays two roles in economic models: it imposes restrictions on behaviour of agents, and it ensures model consistency. Dissatisfaction with RE on behavioural grounds has, in a variety of models, led to its replacement by more behaviourally plausible postulates. However, replacing RE by ad hoc behavioural postulates may result in internally inconsistent models. This work introduces a conceptual framework within which the nature of the issue can be described, and points to potential problems (...) that the abandonment of RE entails. We argue that the RE-based notion of consistency is model-specific rather than general, and introduce a weaker consistency condition that is relevant for non-RE models. To assess the consistency of these models, we propose a test and illustrate its use taking an example from the recent literature. Broader implications of the findings are discussed. (shrink)