Wilfred Sieg. Relative Consistency and Accesible Domains.

We present a relative consistency proof for second order systemRRC* and for certain important extensions of this system. The proof proceeds as follows: we prove first the equiconsistency of the strongest of such extensions (viz., systemH RRC*+(/CP**)) with second order systemT * . Now, N. Cocchiarella has shown thatT * is relatively consistent to systemT*+Ext; clearly, it follows thatH RRC*+(/CP**) is relatively consistent toT*+E xt. As an immediate consequence, the relative consistency ofRRC* and the other extensions (...) also follows, being all of them subsystems ofH RRC*+(/CP**). (shrink)

LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. We could, (...) of course, obtain such a contradiction by proving the theorems ofCone by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem inCfrom its proof inC′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.The advantage of the proof may be stated more specifically by arithmetizing the syntax ofCandC′. (shrink)

We prove the consistency result from the title. By forcing we construct a model of g = ℵ l , b = cf(Sym(ω)) = ℵ 2.

We introduce an Absolute (Relative) Time-consistent Axiom of Revealed Preference which characterizes the consistency of a choice function with the property of absolute (relative) time-consistency and impatience. The axiom requires that the absolute (relative) time-consistent and impatient closure of the revealed preference relation does not conflict with the strict revealed preference relation.

In a recent article in this journal, Kingsley has tried to show that the postulates of special relativity contradict each other. Here we show that the arguments of Kingsley are invalid because of an erroneous appeal to symmetry in a non-symmetric situation. The consistency of the postulates of special relativity and the relativistic kinematics deduced from them is restated.

It is common in the literature on electrodynamics and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the hypothesis that the relativity principle (RP) applies for Maxwell’s electrodynamics. As it will turn out from our analysis, these derivations raise several problems, and certain steps are logically questionable. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following questions: (1) Is (RP) a true (...) law of nature for electrodynamical phenomena? (2) Are, at least, the transformation rules of the fundamental electrodynamical quantities, derived from (RP), true? (3) Is (RP) consistent with the laws of electrodynamics in one single inertial frame of reference? (4) Are, at least, the derived transformation rules consistent with the laws of electrodynamics in one single frame of reference? Obviously, (1) and (2) are empirical questions. In this paper, we will investigate problems (3) and (4). First we will give a general mathematical formulation of (RP). In the second part, we will deal with the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. In the third part of the paper, applying what J. S. Bell calls “Lorentzian pedagogy”—according to which the laws of physics in any one reference frame account for all physical phenomena— we will show that the transformation rules of the electrodynamical quantities are identical with the ones obtained by presuming the covariance of the coupled Maxwell–Lorentz equations, and that the covariance is indeed satisfied. As to problem (3), the situation is much more complex. As we will see, the relativity principle is actually not a matter of the covariance of the physical equations, but it is a matter of the details of the solutions of the equations, which describe the behavior of moving objects.. (shrink)

The logical theory of branching space-times, which is intended to provide a framework for studying objective indeterminism, remains at a certain distance from the discussion of space-time theories in the philosophy of physics. In a welcome attempt to clarify the connection, Earman has recently found fault with the branching approach and suggested ``pruning some branches from branching space-time''. The present note identifies the different---order theoretic vs. topological---points of view of both discussion as a reason for certain misunderstandings, and tries to (...) remove them. Most importantly, we give a novel, topological criterion of modal consistency that usefully generalizes the order-theoretic criterion of directedness, and we introduce a differential-geometrical version of BST based on the theory of non-Hausdorff manifolds. (shrink)

It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the hypothesis that the relativity principle applies to Maxwell's electrodynamics. As it will turn out from our analysis, these derivations raise several problems, and certain steps are logically questionable. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following questions: Is the RP a true (...) law of nature for electrodynamical phenomena? Are, at least, the transformation rules of the fundamental electrodynamical quantities, derived from the RP, true? Is the RP consistent with the laws of ED in a single inertial frame of reference? Are, at least, the derived transformation rules consistent with the laws of ED in a single frame of reference? Obviously, and are empirical questions. In this paper, we will investigate problems and. Abstract First we will give a general mathematical formulation of the RP. In the second part, we will deal with the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. In the third part of the paper, applying what J. S. Bell calls “Lorentzian pedagogy”---according to which the laws of physics in any one reference frame account for all physical phenomena---we will show that the transformation rules of the electrodynamical quantities are identical with the ones obtained by presuming the covariance of the equations of ED, and that the covariance is indeed satisfied. Abstract As to problem, the situation is much more complex. As we will see, the RP is actually not a matter of the covariance of the physical equations, but it is a matter of the details of the solutions of the equations, which describe the behavior of moving objects. This raises conceptual problems concerning the meaning of the notion “the same system in a collective motion”. In case of ED, there seems no satisfactory solution to this conceptual problem; thus, contrary to the widespread views, the question we asked in the title has no obvious answer. (shrink)

The special theory of relativity is based on two postulates: (I) All inertial frames are equivalent for the performance of physical experiments; and (II) the velocity of light is a universal constant. It is shown that if postulate I is true, then postulate II is not true, and if postulate II is true, then postulate I is not true. Further, it is shown that the only possible velocity consistent with postulate I follows the rule that the velocity of light with (...) respect to the observer is equal to the velocity of light with respect to the source plus the velocity of the source with respect to the observer. (shrink)

The error in Kingsley's attack on special relativity is shown to arise from his failure to actually use the Lorentz transformation. He has instead applied a fallacious symmetry argument to justify the use of time dilation in a situation for which this special case formula is not valid.

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square _{\lambda}^{\ast}$ together with 2 λ = λ⁺ implies $\lozenge _{S}$ for every S ⊆ λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, (...) then $\text{NS}_{\lambda ^{+}}$ ↾ S is non-saturated; 2. if I[S; λ] is fat and 2 λ = λ⁺, then $\lozenge _{S}$ holds; 3. $\square _{\lambda}^{\ast}$ implies that I[S; λ] is fat for every S ⊆ λ⁺ that reflects stationarily often; 4. it is relatively consistent with the existence of a supercompact cardinal that $\square _{\lambda}^{\ast}$ fails, while I[S; λ] is fat for every stationary S ⊆ λ⁺ that reflects stationarily often. The stronger principle $\lozenge _{\lambda ^{+}}^{\ast}$ is studied as well. (shrink)

Lawrence et al. have presented an argument purporting to show that “relative facts do not exist” and, consequently, “Relational Quantum Mechanics is incompatible with quantum mechanics”. The argument is based on a GHZ-like contradiction between constraints satisfied by measurement outcomes in an extended Wigner’s friend scenario. Here we present a strengthened version of the argument, and show why, contrary to the claim by Lawrence et al., these arguments do not contradict the consistency of a theory of relative (...) facts. Rather, considering this argument helps clarify how one should not think about a theory of relative facts, like RQM. (shrink)

This paper argues that contrary to what is commonly claimed, presentism is perfectly consistent with the special theory of relativity. More precisely, this paper provides a formulation of a novel relativistic version of presentism that preserves the core “metaphysical stance” of classical presentism, and is fully compatible with special relativity. Others have tried to relativize presentism, but the view put forward here is different from the views that have been proposed in the past.

In a previous work, we have shown that the null result of the Michelson–Morley experiment in vacuum is deeply connected with the notion of time. The same is true for the postulate of constancy of the two-way speed of light in vacuum in all frames independently of the state of motion of the emitting body. The argumentation formerly given is very general and has to be true not only within Special Relativity and its “equivalence” of all inertial frames, but as (...) well as in Lorentz-Poincaré scenario of a preferred reference frame. This paper is the second of a trilogy intending to revisit the foundations of Special Relativity, and addresses the question of the constancy of the one-way speed of light and of the differences and similarities between both scenarios. Although they manifestly differ in philosophy, it is debated why and how the assumption of a “special system of reference experimentally inaccessible” is indeed compatible with Einstein’s Special Relativity, as beautifully outlined and discussed by Bell [Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1988)]. This rather trivial statement is still astonishing nowadays to a big majority of scientists. The purpose of this work is to bring such assertion into perspective, widening the somewhat narrow view of Special Relativity often presented in textbooks and scientific papers. (shrink)

Retrocausal quantum mechanics (RQM) provides a local causal explanation of Bell correlations. It is widely thought that RQM is consistent with special relativity. In this paper, I point out that this view is not wholly right. It is argued that RQM violates the Lorentz invariance of the temporal relation between cause and effect for certain spacelike separated events in Bell-type experiments.

It is common in the literature on classical electrodynamics (ED) and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the hypothesis that the relativity principle (RP) applies to Maxwell’s electrodynamics. As it will turn out from our analysis, these derivations raise several problems, and certain steps are logically questionable. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following questions: (1) Is the (...) RP a true law of nature for electrodynamical phenomena? (2) Are, at least, the transformation rules of the fundamental electrodynamical quantities, derived from the RP, true? (3) Is the RP consistent with the laws of ED in a single inertial frame of reference? (4) Are, at least, the derived transformation rules consistent with the laws of ED in a single frame of reference? Obviously, (1) and (2) are empirical questions. In this paper, we will investigate problems (3) and (4). First we will give a general mathematical formulation of the RP. In the second part, we will deal with the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. In the third part of the paper, applying what J. S. Bell calls “Lorentzian pedagogy”—according to which the laws of physics in any one reference frame account for all physical phenomena— we will show that the transformation rules of the electrodynamical quantities are identical with the ones obtained by presuming the covariance of the equations of ED, and that the covariance is indeed satisfied. As to problem (3), the situation is much more complex. As we will see, the RP is actually not a matter of the covariance of the physical equations, but it is a matter of the details of the solutions of the equations, which describe the behavior of moving objects. This raises conceptual problems concerning the meaning of the notion “the same system in a collective motion”.. (shrink)

Results from two self-paced reading experiments in English are reported in which subject- and object-extracted relative clauses (SRCs and ORCs, respectively) were presented in contexts that support both types of relative clauses (RCs). Object-extracted versions were read more slowly than subject-extracted versions across both experiments. These results are not consistent with a decay-based working memory account of dependency formation where the amount of decay is a function of the number of new discourse referents that intervene between the dependents (...) (Gibson, 1998; Warren & Gibson, 2002). Rather, these results support interference-based accounts and decay-based accounts where the amount of decay depends on the number of words or on the type of noun phrases that intervene between the dependents. In Experiment 2, presentation in supportive contexts was directly contrasted with presentation in null contexts. Whereas in the null context the extraction effect was only observed during the RC region, in a supportive context the extraction effect was numerically larger and persisted into the following region, thus showing that extraction effects are enhanced in supportive contexts. A sentence completion study demonstrated that the rate of SRCs versus ORCs was similar across null and supportive contexts (with most completions being subject-extractions), ruling out the possibility that an enhanced extraction effect in supportive contexts is due to ORCs being less expected in such contexts. However, the content of the RCs differed between contexts in the completions, such that the RCs produced in supportive contexts were more constrained, reflecting the lexical and semantic content of the preceding context. This effect, which we discuss in terms of expectations/lexico-syntactic priming, suggests that the enhancement of the extraction effect in supportive contexts is due to the facilitation of the subject-extracted condition. (shrink)

Physical Relativity explores the nature of the distinction at the heart of Einstein's 1905 formulation of his special theory of relativity: that between kinematics and dynamics. Einstein himself became increasingly uncomfortable with this distinction, and with the limitations of what he called the 'principle theory' approach inspired by the logic of thermodynamics. A handful of physicists and philosophers have over the last century likewise expressed doubts about Einstein's treatment of the relativistic behaviour of rigid bodies and clocks in motion in (...) the kinematical part of his great paper, and suggested that the dynamical understanding of length contraction and time dilation intimated by the immediate precursors of Einstein is more fundamental. Harvey Brown both examines and extends these arguments, after giving a careful analysis of key features of the pre-history of relativity theory. He argues furthermore that the geometrization of the theory by Minkowski in 1908 brought illumination, but not a causal explanation of relativistic effects. Finally, Brown tries to show that the dynamical interpretation of special relativity defended in the book is consistent with the role this theory must play as a limiting case of Einstein's 1915 theory of gravity: the general theory of relativity.Appearing in the centennial year of Einstein's celebrated paper on special relativity, Physical Relativity is an unusual, critical examination of the way Einstein formulated his theory. It also examines in detail certain specific historical and conceptual issues that have long given rise to debate in both special and general relativity theory, such as the conventionality of simultaneity, the principle of general covariance, and the consistency or otherwise of the special theory with quantum mechanics. Harvey Brown' s new interpretation of relativity theory will interest anyone working on these central topics in modern physics. (shrink)

Define z to be the smallest cardinality of a function f: X → Y with X. Y ⊆ 2ω such that there is no Borel function g ⊇ f. In this paper we prove that it is relatively consistent with ZFC to have b < z where b is, as usual, smallest cardinality of an unbounded family in ωω. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists X ⊆ (...) 2ω such that the Borel order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin. (shrink)

It is a central aspect of our ordinary concept of time that history unfolds and events come into being. It is only natural to take this seriously. However, it is notoriously difficult to explain further what this `becoming' consists in, or even to show that the notion is consistent at all. In this article I first argue that the idea of a global temporal ordering, involving a succession of cosmic nows, is not indispensable for our concept of time. Our experience (...) does not support the existence of global simultaneity and arguments from modern physics further support the conclusion that time should not be seen as a succession of cosmic nows. Accordingly, I propose that if we want to make sense of becoming we should attempt to interpret it as something purely local. Second, I address the question of what this local becoming consists in. I maintain that processes of becoming are nothing but the successive happening of events, and that this happening of events consists entirely in the occurring of these events at their own spacetime locations. This leads to a consistent view of becoming, which is applicable even to rather pathological spacetimes. (shrink)

A speed-accuracy trade-off (SAT) in behavioural decisions is known to occur in a wide range of vertebrate and invertebrate taxa. Accurate decisions often take longer for a given condition, while fast decisions can be inaccurate in some tasks. Speed-accuracy tactics are known to vary consistently among individuals, and show a degree of flexibility during colour discrimination tasks in bees. Such individual flexibility in speed-accuracy tactics is likely to be advantageous for animals exposed to fluctuating environments, such as changes in predation (...) threat. We therefore test whether individual speed-accuracy tactics are fixed or flexible under different levels of predation threat in a model invertebrate, the bumblebee Bombus terrestris. The flexibility of speed-accuracy tactics in a foraging context was tested in the laboratory using a ‘meadow’ of artificial flowers harbouring ‘robotic’ crab spider predators. We found that while the ranking of bees along the speed and accuracy continuums was consistent across two levels of predation threat, there was some flexibility in the tactics used by individual bees – most bees became less accurate at colour discrimination when exposed to predation threat when flower types were rewarding. The relationship between decision speed and accuracy was influenced by predator detectability and the risk associated with making incorrect choices during the colour discrimination task. Predator crypsis resulted in a breakdown in the relationship between speed and accuracy, especially when making an incorrect floral choice incurred a distasteful quinine punishment. No single speed-accuracy tactic was found to be optimal in terms of foraging efficiency under either predation threat situation. However, bees that made faster decisions achieved higher nectar collection rates in predator free situations, while accurate bees achieved higher foraging rates under predation threat. Our findings show that while individual bees remain relatively consistent in terms of whether they place greater emphasis on speed or accuracy under predation threat, they can respond flexibly to the additional time costs of detecting predators. (shrink)

Two widely different conceptions of electromagnetic radiation were compared by Einstein in his 1916 papers, namely, the classically diffuse form and the fully directional form. The emission process of each kind of radiation is examined hereafter from a relativistic point of view. The necessary covariance of any physical description, further specifying the original directional property, points to a strictly quantized form of radiation as the only one consistent with relativity, thus showing a deep relationship between this latter and quantum theory.

It is widely accepted that when ‘might’ expresses certain kinds of relative modality, the sentence ‘p and it might not be the case that p’ is in some sense inconsistent. It has proven difficult to define a formal semantics that explicates this inconsistency while meeting certain other desiderata, in particular, that p does not imply ‘Must p’. This paper presents such a semantics. The key idea is that background contexts have to have multiple levels, including an inner set consisting (...) of worlds that represent what might be true and an outer set of worlds such that a sentence must be true only if it is true in all of them. This is an open access publication. (shrink)

This volume consists of the first of the John Dewey Lectures delivered under the auspices of Columbia University's Philosophy Department as well as other essays by the author. Intended to clarify the meaning of the philosophical doctrines propounded by Professor Quine in 'Word and Objects', the essays included herein both support and expand those doctrines.

Lanczos, C. Einstein's path from special to general relativity.--Balazs, N. L. The acceptability of physical theories: Poincaré versus Einstein.--Ellis, G. F. R. Global and non-global problems in cosmology, by G. F. R. Ellis and D. W. Sciama.--Ehlers, J. The geometry of free fall and light propagation, by J. Ehlers, F. A. E. Pirani and A. Schild.--Trautman, A. Invariance of Lagrangian systems.--Penrose, R. The geometry of impulsive gravitational waves.--Exact solutions of the Einstein-Maxwell equations for an accelerated charge.--Taub, A. H. Plane-symmetric similarity (...) solutions for self-gravitating fluids.--Robinson, I. Equations of motion in the linear approximation, by I. Robinson and J. R. Robinson.--Florides, P. W. Rotating bodies in general relativity.--Chandrasekhar, S. A limiting case of relativistic equilibrium.--Israel, W. The relativistic Boltzmann equation.--Thompson, W. B. The self-consistent test-particle approach to relativistic kinetic theory. (shrink)

We study the coupled dynamics of two populations of random replicators by means of statistical mechanics methods, and focus on the effects of relative population size, strategy correlations and heterogeneities in the respective cooperation pressures. To this end we generalise existing path-integral approaches to replicator systems with random asymmetric couplings. This technique allows one to formulate an effective dynamical theory, which is exact in the thermodynamic limit and which can be solved for persistent order parameters in a fixed-point regime (...) regardless of the symmetry of the interactions. The onset of instability can be determined self-consistently. We calculate quantities such as the diversity of the respective populations and their fitnesses in the stationary state, and compare results with data from a numerical integration of the replicator equations. -/- . (shrink)

This paper aims to explain how the insights Weyl gleaned from Husserl played an important role in his scientific work, and then how Einstein’s major work exhibit important parallels to Weyl’s work, thereby establishing phenomenology both as an indirect historical influence and a systematic underpinning for Einstein’s work in theoretical physics. In so doing, this paper seeks to show how some of the most basic problems that Einstein addresses have a kinship not just to problems addressed in a completely different (...) context by Edmund Husserl’s phenomenology and his circle, but also to perennial problems in ontology and epistemology that go back to Kant, Hume and Leibniz. The conclusion seems to suggest that it not only shows how phenomenology both historically and systematically provides a backdrop for Einstein’s work; my thesis actually situates issues in twentieth-century scientific thought against the backdrop of a philosophical development, and perhaps the most original idea of this study consists not just in showing how phenomenology influenced Einstein, but also how Einstein’s work on relativity had an important influence on the work of Edmund Husserl in Crisis of European Sciences and in his later works. (shrink)

Relativism is one of the most tenacious theories about truth, with a pedigree as old as philosophy itself. Nearly as ancient is the chief criticism of relativism, namely the charge that the theory is self-refuting. This paper develops a logic of relativism that (1) illuminates the classic self-refutation charge and shows how to escape it; (2) makes rigorous the ideas of truth as relative and truth as absolute, and shows the relations between them; (3) develops an intensional logic for (...) relativism; (4) provides a framework in which relativists can consistently promote ethical, mathematical, scientific, religious, and political truths (among others) as being relative; (5) argues that the notion of incommensurability is far less troubling than is commonly thought; and (6) argues that the concept of a perspective as needed by the theory is not prey to Davidson's well-known critique of conceptual schemes. The paper will not defend relativism as the correct theory of truth, nor will it provide a fully satisfying theory about the nature of a perspective. The logic of relativism is primarily meant to provide a formal framework in which relativists can consistently develop their theories. This alone is a considerable step forward, since the debate about relativism often founders upon the rock of self-refutation. It is argued that while 'everything is relative' is inconsistent, 'everything true is relatively true' is not. The latter is all a relativist really needs. (shrink)

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ (...) , or $\aleph _{\omega _{2}}$ , we show that injective failures at $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , or $\aleph _{\omega _{2}}$ can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell. (shrink)

The indiscernibility of identicals is incompatible with geach's theory of 'relative' identity, But consistent with the view that x is identical with y iff x is the same a as y, For some count-Noun 'a'. 'x is the same a as y' expresses identity only if x is an a, Otherwise it is merely an equivalence relation.

People's treatment of others (humans, nonhuman animals, or other entities) often depends on whether they think the entity is worthy of moral concern. Recent work has begun to investigate which entities are included in a person's moral circle, examining how certain target characteristics (e.g., species category, perceived intelligence) and judge characteristics (e.g., empathy, political orientation) shape moral inclusion. However, the relative importance of target and judge characteristics in predicting moral inclusion remains unclear. When predicting whether a person will deem (...) an entity worthy of moral consideration, how important is it to know who is making the judgment (i.e., characteristics of the judge), who is being judged (i.e., characteristics of the target), and potential interactions between the two factors? Here, we address this foundational question by conducting a variance component analysis of the moral circle. In two studies with participants from the Netherlands, the United States, the United Kingdom, and Australia (N = 836), we test how much variance in judgments of moral concern is explained by between‐target differences, between‐judge differences, and by the interaction between the two factors. We consistently find that all three components explain substantial amounts of variance in judgments of moral concern. Our findings provide two important insights. First, an increased focus on interactions between target and judge characteristics is needed, as these interactions explain as much variance as target and judge characteristics separately. Second, any theoretical account that aims to provide an accurate description of moral inclusion needs to consider target characteristics, judge characteristics, and their interaction. (shrink)

This study investigates the relative roles of frequency and transparency in native speakers’ comprehension of Spanish object relative sentences by comparing performance with two variants of the construction that differ in these properties. Experiment 1 suggests that seven- to eight-year-old children’s processing of object relative sentences is consistently facilitated by variant frequency and likely relies on separate representations for each variant as two different, although related constructions. Experiment 2 turns to adult comprehension of the two variants, showing (...) similar accuracy and reaction time patterns for both variants for this population. This second outcome could be interpreted as a decrease in frequency effects over time, in line with the power law of practice, as an experience-based increase in the ability to make use of redundant markers, or both. (shrink)

In 5, Tait identifies a set of reflection principles called equation image-reflection principles which Peter Koellner has shown to be consistent relative to the existence of κ, the first ω-Erdős cardinal 1. Tait also defines a set of reflection principles called equation image-reflection principles; however, Koellner has shown that these are inconsistent when m > 2, but identifies restricted versions of them which he proves consistent relative to κ 2. In this paper, we introduce a new large-cardinal property, (...) the α-reflective cardinals. Their definition is motivated by Tait's remarks on parameters of third or higher order in reflection principles. We prove that if κ is ℶκ + α + 1-supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that α-reflective cardinals relativize to L, and that if κ exists then the set of cardinals λ < κ, such that λ is α-reflective for all α such that 0 < α < λ, is a stationary subset of κ. We show that an ω-reflective cardinal satisfies some restricted versions of equation image-reflection, as well as all the reflection properties proved consistent in 2. (shrink)

The paper "Partitions of Products" [DiPH] investigated the polarized partition relation $\begin{pmatrix}\omega\\\omega\\\omega\\\vdots\end{pmatrix} \rightarrow \begin{pmatrix}\alpha_1\\\alpha_1\\\alpha_2\\\vdots \end{pmatrix}$ The relation is consistent relative to an inaccessible cardinal if every α i is finite, but inconsistent if two are infinite. We show here that it consistent (relative to an inaccessible) for one to be infinite. Along the way, we prove an interesting proposition from ZFC concerning partitions of the finite subsets of ω.

I argue for the consistency of frege's treatments of identity and number. Specifically, I argue that geach is wrong in suggesting that frege's insights about number should have led him to the doctrine of relative identity.