The logic BKc1 is the basic constructive logic for weak consistency in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the contraposition axioms.

A relation is obtained between weak values of quantum observables and the consistency criterion for histories of quantum events. It is shown that “strange” weak values for projection operators always correspond to inconsistent families of histories. It is argued that using the ABL rule to obtain probabilities for counterfactual measurements corresponding to those strange weak values gives inconsistent results. This problem is shown to be remedied by using the conditional weight, or pseudo-probability, obtained from the multiple-time (...) application of Lüders’ Rule. It is argued that an assumption of reverse causality implies that weak values obtain, in a restricted sense, at the time of the weak measurement as well as at the time of post-selection. Finally, it is argued that weak values are more appropriately characterized as multiple-time amplitudes than expectation values, and as such can have little to say about counterfactual questions. (shrink)

A relation is obtained between weak values of quantum observables and the consistency criterion for histories of quantum events. It is shown that “strange” weak values for projection operators always correspond to inconsistent families of histories. It is argued that using the ABL rule to obtain probabilities for counterfactual measurements corresponding to those strange weak values gives inconsistent results. This problem is shown to be remedied by using the conditional weight, or pseudo-probability, obtained from the multiple-time (...) application of Lüders’ Rule. It is argued that an assumption of reverse causality implies that weak values obtain, in a restricted sense, at the time of the weak measurement as well as at the time of post-selection. Finally, it is argued that weak values are more appropriately characterized as multiple-time amplitudes than expectation values, and as such can have little to say about counterfactual questions. (shrink)

ABSTRACT: A relation is obtained between weak values of quantum observables and the consistency criterion for histories of quantum events. It is shown that ``strange'' weak values for projection operators always correspond to inconsistent families of histories. It is argued that using the ABL rule to obtain probabilities for counterfactual measurements corresponding to those strange weak values gives inconsistent results. This problem is shown to be remedied by using the conditional weight, or pseudo-probability, obtained from the (...) multiple-time application of Luders' Rule. It is argued that an assumption of reverse causality implies that weak values obtain, in a restricted sense, at the time of the weak measurement as well as at the time of post-selection. Finally, it is argued that weak values are more appropriately characterised as multiple-time amplitudes than expectation values, and as such can have little to say about counterfactual questions. (shrink)

We prove that the Gödel incompleteness theorem holds for a weak arithmetic T = IΔ0 + Ω2 in the form where Cons H (T) is an arithmetic formula expressing the consistency of T with respect to the Herbrand notion of provability.

Since Varian (Econometrica 50:945–973, 1982; Review of Economic Studies 50:90–110, 1983) made checking for consistency with revealed preference conditions more accessible to empirical researchers; researchers have often used revealed preference procedures to test their maintained hypotheses and narrow the scope of their demand studies. The tests developed by Varian are for the direct utility function, while researchers estimating demand systems often find it convenient to model consumer behavior with an indirect utility function. Unfortunately structure revealed in the direct utility (...) function does not necessarily follow for the indirect utility function. To remedy this problem, we set forth conditions for checking consumer data for consistency with revealed preference conditions for minimization and weak separability of the indirect utility function. (shrink)

I argue that a compulsive seeking for just one sense of consistency is hazardous to rationality, and that observing the subtle distinctions of reasonableness between individual and groups may suggest wider, structuralistic notions of consistency, even relevant to re-assessing Gödei's Second Incompleteness Theorem and to science as a whole.

In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. -/- It is well known from universal algebra that, for every signature Sigma, there exist algebras over Sigma which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Sigma-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Sigma-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U, from the category of Sigma-multialgebras to Set, does not have a left adjoint. -/- In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint. (shrink)

The logic BKc1 is the basic constructive logic in the ternary relational semantics adequate to consistency understood as the absence of the negation of any theorem. Negation is introduced in BKc1 with a negation connective. The aim of this paper is to define the logic BKc1F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc1 and BKc1F are definitionally equivalent.

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection equation image containing at most κ+ many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in equation image. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot (...) fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord is (...) not ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. (shrink)

We show that the consistency of the theories “ holds below ” + “there is an injection ” + “both and fail” and + “ holds below ” + “there is an injection ” + “ satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long‐standing open questions concerning, weak square, approachability, and the tree property. There is nothing special about, and the injection into can be from any (...) ordinal λ (which means that pcf theory can fail badly in the models constructed if ). (shrink)

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \, there exist algebras over \ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms. Equivalently, the forgetful functor, from the category of \-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \, from the category of \-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices, there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \ does not have a left adjoint. (shrink)

Carruthers argues that an integrated faculty of metarepresentation evolved for mindreading and was later exapted for metacognition. A more consistent application of his approach would regard metarepresentation in mindreading with the same skeptical rigor, concluding that the “faculty” may have been entirely exapted. Given this result, the usefulness of Carruthers’ line-drawing exercise is called into question.

The meaning of sustainability is the subject of intense debate among environmental and resource economists. Perhaps no other issue separates more clearly the traditional economic view from the views of most natural scientists. The debate currently focuses on the substitutability between the economy and the environment or between “natural capital” and “manufactured capital”—a debate captured in terms of weak versus strong sustainability. In this article, we examine the various interpretations of these concepts. We conclude that natural science and economic (...) perspectives on sustainability are inconsistent. The market-based Hartwick-Solow “weak sustainability” approach is far removed from both the ecosystem-based “Holling sustainability” and the “strong sustainability” approach of Daly and others. Each of these sustainability criteria implies a specific valuation approach, and thus an ethical position, to support monetary indicators of sustainability such as a green or sustainable Gross Domestic Product. The conflict between “weak sustainability” and “strong sustainability” is more evident in the context of centralized than decentralized decision making. In particular, firms selling “services” instead of material goods and regarding the latter as “capital” leads to decisions more or less consistent with either type of sustainability. Finally, we discuss the implications of global sustainability for such open systems as regions and countries. Open systems have not been dealt with systematically for any of the sustainability criteria. (shrink)

I present an intractable puzzle for the currently popular view that belief is weak—the view that expressions like ‘S believes p’ ascribe to S a doxastic attitude towards p that is rationally compatible with low credence that p. The puzzle concerns issues that arise on considering beliefs in conditionals. I show that proponents of weak belief either cannot consistently apply their preferred methodology when accommodating beliefs in conditionals, or they must deny that beliefs in conditionals can be used (...) in reasoning. (shrink)

Philosophical orthodoxy identifies weakness of will with akrasia: the weak willed person is someone who intentionally acts against their better judgement. It is argued that this is a mistake. Weakness of will consists in a quite different failing, namely an over-ready revision of one's intentions. Building on the work of Bratman, an account of such over-ready revision is given. A number of examples are then adduced showing how weakness of will, so understood, differs from akrasia.

According to the Weak Supplementation Principle (WSP)—a widely received principle of mereology—an object with a proper part, p , has another distinct proper part that doesn't overlap p . In a recent article in this journal, Nikk Effingham and Jon Robson employ WSP in an objection to endurantism. I defend endurantism in a way that bears on mereology in general. First, I argue that denying WSP can be motivated apart from the truth of endurantism. I then go on to (...) offer an explanation of WSP's initial appeal, argue that denying WSP fails to have untoward consequences for the rest of mereology, and show that the falsity of WSP is consistent with a primary guiding thought behind it. (shrink)

Our last big theorem – Theorem 6 – tells us that if a theory meets certain conditions, then it must be negation incomplete. And we made some initial arm-waving remarks to the effect that it seems plausible that we should want theories which meet those conditions. Later, we announced that there actually is a consistent weak arithmetic with a first-order logic which meets the conditions (in which case, stronger arithmetics will also meet the conditions); but we didn’t say anything (...) about what such a weak theory really looks like. In fact, we haven’t looked at any detailed theory of arithmetic yet! It is high time, then, that we stop operating at the extreme level of abstraction of Episodes 1 and 2, and start getting our hands dirty. This Episode introduces a couple of weak arithmetics, before we tackle the canonical first-order arithmetic PA in the following instalment. By all means skip fairly lightly over some of the more boring proof details! But it is certainly worth getting just a flavour of how these two simple formal theories work. (shrink)

We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}, i.e. Chang’s Conjecture is consistent with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}.

In any probabilistic theory, we say that a bipartite state ω on a composite system AB steers its marginal state ω B if, for any decomposition of ω B as a mixture ω B =∑ i p i β i of states β i on B, there exists an observable {a i } on A such that the conditional states $\omega_{B|a_{i}}$ are exactly the states β i . This is always so for pure bipartite states in quantum mechanics, a fact (...) first observed by Schrödinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical. (shrink)

We clarify the significance of quasiprobability in quantum mechanics that is relevant in describing physical quantities associated with a transition process. Our basic quantity is Aharonov’s weak value, from which the QP can be defined up to a certain ambiguity parameterized by a complex number. Unlike the conventional probability, the QP allows us to treat two noncommuting observables consistently, and this is utilized to embed the QP in Bohmian mechanics such that its equivalence to quantum mechanics becomes more transparent. (...) We also show that, with the help of the QP, Bohmian mechanics can be recognized as an ontological model with a certain type of contextuality. (shrink)

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on (...) a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA.Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1.We refute various reasonable generalizations of these results to hypergraphs. (shrink)

This paper shows that our popular account of weakness of will is inconsistent with dilemmas. In dilemmas, agents judge that they ought to do one thing, that they ought to do something else, and that they cannot do both. They must act against either of their two judgments. But such action is commonly understood as weakness of will. An agent is weak-willed in doing something if she judges that she ought to and could do something else instead. Thus, it (...) seems that, in a dilemma, the agent is weak-willed by definition. But this is puzzling: clearly, the two are different phenomena. The puzzle may support scepticism about weakness of will or dilemmas. Here, I argue that the two are consistent on a revised understanding of weakness of will. To do so, I further distinguish the mental states of an agent in a dilemma from those of a weak-willed person. (shrink)

The Nash axiom is a basic property of consistency in choice. This paper proposes weaker versions of the axiom and examines their logical implications. In particular, we demonstrate that weak Nash axioms are useful to understand the relationship between the Nash axiom and the path independence axiom. We provide an application of weak Nash axioms to the no-envy approach. We present a possibility result and an impossibility result.

In this article we present a number of axiomatic theories of truth which are conservative extensions of arithmetic. We isolate a set of ten natural principles of truth and prove that every consistent permutation of them forms a theory conservative over Peano arithmetic.

Weak and strong consistency of the Abstract Principal Principle are defined in terms of classical probability measure spaces. It is proved that the Abstract Principal Principle is both weakly and strongly consistent. The Abstract Principal Principle is strengthened by adding a stability requirement to it. Weak and strong consistency of the resulting Stable Abstract Principal Principle are defined. It is shown that the Stable Abstract Principal Principle is weakly consistent. Strong consistency of the Stable Abstract (...) Principal principle remains an open question. (shrink)

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate (...) this by examining Easton's strengthening of Cohen's result:Theorem 1. There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals. (shrink)

Fine has shown that assigning any value to the Pasadena game is consistent with a certain standard set of axioms for decision theory. However, I suggest that it might be reasonable to believe that the value of an individual game is constrained by the long-run payout of repeated plays of the game. Although there is no value that repeated plays of the Pasadena game converges to in the standard strong sense, I show that there is a weaker sort of convergence (...) it exhibits, and use this to define a notion of ‘weak expectation’ that can give values to the Pasadena game and many others, though not to all games that fail to have a strong expectation in the standard sense. CiteULike Connotea Del.icio.us What's this? (shrink)

S There is a long tradition of representing causal relationships by directed acyclic graphs (Wright, 1934 ). Spirtes ( 1994), Spirtes et al. ( 1993) and Pearl & Verma ( 1991) describe procedures for inferring the presence or absence of causal arrows in the graph even if there might be unobserved confounding variables, and/or an unknown time order, and that under weak conditions, for certain combinations of directed acyclic graphs and probability distributions, are asymptotically, in sample size, consistent. These (...) results are surprising since they seem to contradict the standard statistical wisdom that consistent estimators of causal effects do not exist for nonrandomised studies if there are potentially unobserved confounding variables. We resolve the apparent incompatibility of these views by closely examining the asymptotic properties of these causal inference procedures. We show that the asymptotically consistent procedures are ‘pointwise consistent’, but ‘uniformly consistent’ tests do not exist. Thus, no finite sample size can ever be guaranteed to approximate the asymptotic results. We also show the nonexistence of valid, consistent confidence intervals for causal effects and the nonexistence of uniformly consistent point estimators. Our results make no assumption about the form of the tests or estimators. In particular, the tests could be classical independence tests, they could be Bayes tests or they could be tests based on scoring methods such as  or . The implications of our results for observational studies are controversial and are discussed briefly in the last section of the paper. The results hinge on the following fact: it is possible to find, for each sample size n, distributions P and Q such that P and Q are empirically indistinguishable and yet P and Q correspond to different causal effects. (shrink)

We prove that a form of the $Erd\H{o}s$ property (consistent with $V = L\lbrack H_{\omega_2}\rbrack$ and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle $\psi_{AC}$ holds, and therefore 2ℵ0 = ℵ2. We also prove that $\psi_{AC}$ implies that every function $f: \omega_1 \rightarrow \omega_1$ is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's (...) Maximum fails. (shrink)

We consider equational theories for functions defined via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S 1 2 . In particular this solves an open problem formulated by TAKEUTI (c.f. [5, p.5 problem 9.]).

Suzumura consistency is a necessary and sufficient condition for the existence of a weak-order extension. This paper provides some remarks on collective choice rules that generate Suzumura consistent social preferences. We examine the properties of such collective choice rules by introducing a procedural condition on collective choice rules. As applications of the procedural condition, we first investigate the decisive structure of a Paretian collective choice rule, and then consider the assignment of individual rights. In our analysis, the concept (...) of semi-decisiveness works effectively. (shrink)

We prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\omega_3, \omega_2) \twoheadrightarrow (\omega_2, \omega_1)}$$\end{document} implies there is an inner model with a weak repeat measure.

In his “Two Kinds of Moral Reasoning; Ethical Egoism as a Moral Theory”, Jesse Kalin defines ethical egoism as “the position that a person ought, all things considered, to do an action if and only if that action is in his overall self-interest”, by which he means that each person is ‘rationally justified in’ or ‘has conclusive reasons for’ acting thus, and not that ‘it is good', or that ‘it is desirable', or that ‘it conduces to any intrinsically desirable state (...) of affairs', for each to act only self-interestedly.In this way, Kalin seeks to formulate an egoism which he takes to be rational, insofar as, in the interests of consistency, the egoist holds a principle applicable to all, and yet is not committed, ex hypothesi, to valuing collective goods or states of affairs which would tend to compromise his own interests. (shrink)

This paper extends the theory of situated inference from Mares to treat two weak relevant logics, B and DJ. These logics are interesting because they can be used as bases for consistent naïve theories, such as naïve set theory. The concepts of a situation and of information that are employed by the theory of situated inference are used to justify various aspects of these logics and to give an interpretation of the notion of set that is represented in the (...) naïve set theories that are based on them. (shrink)

I defend a contextualist account of bare epistemic modal claims against recent objections. I argue that in uttering a sentence of the form ‘It might be that p,’ a speaker is performing two speech acts. First, she is (directly) asserting that in view of the knowledge possessed by some relevant group, it might be that p. The content of this first speech act is accounted for by the contextualist view. But the speaker's utterance also generates an indirect speech act that (...) consists in a weak suggestive that p. Since this second speech act is typically the main point of a bare epistemic modal utterance, our (negative or positive) responses to the utterance actually target this second speech act. I show how this two-speech-act account can explain the data recently adduced against contextualism. (shrink)

According to Hitchcock and Sober’s argument from overfitting for weak predictivism, the fact that a theory accurately predicts a portion of its data is evidence that it has been formulated by balancing simplicity and goodness-of-fit rather than overfitting data. The core argument consists of two likelihood inequalities. In this paper I show that there is a surprising accommodation-friendly implication in their argument, and contend that it is beset by a substantial difficulty, namely, there is no good reason to think (...) that their second likelihood inequality is true. (shrink)

The purpose of this article is to prove the relative consistency of certain statements about open colorings with 2ℵ0 < 2ℵ1. In particular both OCA and the statement that every 1–1 function of size ℵ1 is σ-monotonic are consistent with 2ℵ0 < 2ℵ1. As a corollary we have that 2ℵ0 < 2ℵ1 does not admit a ℙ max variation.

We give a consistency proof within a weak fragment of arithmetic of elementary algebra and geometry. For this purpose, we use EFA (exponential function arithmetic), and various first order theories of algebraically closed fields and real closed fields.

Recently, a number of experimental philosophers have converged on the position that the ordinary concept of weakness of will does not solely consist in “judgment” or “intention” violation but is more like a cluster concept in which each factor plays contributory roles in the application of the concept. This, however, raises the question as to which factor is more central or plays a more significant role in folk’s understanding of the concept. I contend that the ordinary concept of weakness of (...) will is primarily constituted by the “executive commitment” rather than the “evaluative commitment” practices. Drawing on extensive evidence from developmental psychology, I will argue that the executive commitment, which, as I will show, involve intention recognition and metarepresentation, is developmentally prior and more fundamental in our exercise and intuitive understanding of the concept. (shrink)

David Alexander has argued that “weak inferential internalism” (WII), a position which amounts to a qualified endorsement of Richard Fumerton’s controversial “principle of inferential justification,” is subject to a fatal dilemma: Either it collapses into externalism or it must make an arbitrary epistemic distinction between persons who believe the same proposition for the same reasons. In this paper, I argue that the dilemma is a false one, for weak inferential internalism does not entail internalism simpliciter. Indeed, WII is (...) compatible with modest externalism, and so is consistent with what Armstrong calls “Type II justification,” the rejection of which leads to the arbitrary epistemic distinctions to which he rightly objects. (shrink)