An optimal property of least squares weights in prediction models

In predicting scores fromp > 1 observed scores in a sample of sizeñ, the optimal strategy (minimum expected loss), under certain assumptions, is shown to be based upon the least squares regression weights computed from a previous sample. Letting represent the correlation between and the predicted values , and letting represent the correlation between and a different set of predicted values , where w is any weighting system which is not a function of , it is shown that the probability of being less than cannot exceed .50. The relationship of this result to previous research and practical implications are discussed.     

A theorem in 3-valued model theory with connections to number theory,type theory,and relevant logic

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development).     

Free construction of time from events

Conclusions and conjectures Some may be of the opinion that one event can begin before another only by virtue of the existence of some event (a witness) which wholly precedes the other and does not wholly precede the one (and similarly for ends before and does not abut). Those would prefer ₀ to as a model for observers' apprehensions of events. Since G is a functor from to ₀, the current construction (restricted to ₀) remains applicable.This work supports a claim that the psychologically fundamental temporal relationships are wholly precedes, begins before, ends before and abuts. But only in a very weak sense. Any other set of relationships which is interdefinable with this one, using only quantifier-free formulas in the definitions, could be used to define a category which is indistinguishable from (because the same functions preserve and reflect the new relationships). This work equally supports the claim that those relationships are the psychologically fundamental ones, or the claim that it is just wholly precedes which is fundamental, and that we perceive begins before just by virtue of witnesses. But it refutes the claim that only wholly precedes is fundamental, and that we understand begins before only because we understand time as a linear ordering.     

Dominions in quasivarieties of universal algebras

The dominion of a subalgebra H in an universal algebra A (in a class ) is the set of all elements such that for all homomorphisms if f, g coincide on H, then a^f = a^g. We investigate the connection between dominions and quasivarieties. We show that if a class is closed under ultraproducts, then the dominion in is equal to the dominion in a quasivariety generated by . Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

Finite basis problems and results for quasivarieties

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection ₁ of finite algebras of the same signature, , such that SP( ₁) is finitely axiomatizable.We show also that if , then SP( ₁) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

On a problem of P(α, δ, π) concerning generalized Alexandroff s cube

Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of , -closure spaces. Alexandroff's cube. is an , -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power .Condition P(, , ) says that is a closure space of all , -filters in the lattice ( ), .Assuming that P (, , ) holds, in the paper [2], there are given sufficient conditions saying when an , -closure space is an absolute retract for the category of , -closure spaces (see Theorems 2.1 and 3.4 in [2]).It seems that, under assumption that P (, , ) holds, it will be possible to givean uniform characterization of absolute retracts for the category of , -closure-spaces.Except Lemma 3.1 from [1], there is no information when the condition P (, , ) holds or when it does not hold.The main result of this paper says, that there are examples of cardinal numbers, , , such that P (, , ) is not satisfied.Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition P (, ₁, 2 ) is not satisfied.Moreover it is shown that fulfillment of the condition is essential assumption in, Theorems 2.1 and 3.4 from [1] i.e. it cannot be eliminated.     

Empirical bayes estimates of domain scores under binomial and hypergeometric distributions for test scores

We introduce two simple empirical approximate Bayes estimators (EABEs)— and —for estimating domain scores under binomial and hypergeometric distributions, respectively. Both EABEs (derived from corresponding marginal distributions of observed test scorex without relying on knowledge of prior domain score distributions) have been proven to hold -asymptotic optimality in Robbins' sense of convergence in mean. We found that, where and are the monotonized versions of and under Van Houwelingen's monotonization method, respectively, the convergence rate of the overall expected loss of Bayes risk in either or depends on test length, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, and outperform their maximum likelihood counterparts over the middle range of domain scales. In terms of mean-squared error, we also found that: (a) given a unimodal prior distribution of domain scores, performs better than both and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterogeneity; (b) performs as well as when prior distribution is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimodal prior distribution exists.The authors are indebted to both anonymous reviewers, especially Reviewer 2, and the Editor for their invaluable comments and suggestions. Thanks are also due to Yuan-Chin Chang and Chin-Fu Hsiao for their help with our simulation and programming work.     

The Parents Matter! Program Interventions: Content and the Facilitation Process

The Parents Matter! Program (PMP) has developed three interventions for parents of 4th and 5th grade African-American children (9–12 years old). The overarching goal of all three interventions is to provide parents with knowledge, skills, and support for enhancing their efforts to raise healthy children. The interventions are: (1) Enhanced Communication and Parenting (five 2 -hour sessions), (2) Brief Communication and Parenting (single 2 -hour session), and (3) General Health (single 2 -hour session). This article discusses the development of these interventions, presents an overview of the content of each intervention, and discusses issues related to the facilitation/presentation of these interventions.     

Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.     

Isomorphism and Equational Equivalence of Continuous λ-Models

We will present several results on two types of continuous models of -calculus, namely graph models and extensional models. By introducing a variant of Engeler's model construction, we are able to generalize the results of [7] and to give invariants that determine a large family of graph models up to applicative isomorphism. This covers all graph models considered in the litterature so far. We indicate briefly how these invariants may be modified in order to determine extensional models as well.Furthermore, we use our construction to exhibit graph models that are not equationally equivalent. We indicate once again how the construction passes on to extensional models.     

Finite structural axiomatization of every finite-valued propositional calculus

In [2] A. Wroski proved that there is a strongly finite consequence C which is not finitely based i.e. for every consequence C ⁺ determined by a finite set of standard rules C C ⁺. In this paper it will be proved that for every strongly finite consequence C there is a consequence C ⁺ determined by a finite set of structural rules such that C(Ø)=C ⁺(Ø) and = (where , are consequences obtained by adding to the rules of C, C ⁺ respectively the rule of substitution). Moreover it will be shown that under certain assumptions C=C ⁺.     

More than the Terrible Twos: The Nature and Severity of Behavior Problems in Clinic-Referred Preschool Children

The primary goals of this study are to describe the nature and severity of disruptive behavior problems in clinic-referred preschoolers from low-income environments and to explore the validity of DSM-IV disruptive disorders for young children. We examine the relation between DSM-IV symptoms, standardized behavior checklists, and observational ratings as a means of exploring measurement validity in this age group. Seventy-nine clinic-referred preschoolers (ages 2 through 5 years) from low-income environments were assessed. To examine whether clinic-referred preschool children have symptoms that are consistent with DSM-IV disruptive behavior disorders, parents were administered a semistructured diagnostic interview, modified for developmentally appropriate usage. In addition, parents completed the Child Behavior Checklist (CBCL) and children's behavior problems were assessed with observational ratings during parent–child interaction. Nearly half of the sample met criteria for conduct disorder, and three quarters met criteria for oppositional defiant disorder. Preliminary evidence for the validity of DSM-IV disruptive disorders in preschool children was demonstrated through association with CBCL scores, behavior ratings, and significant levels of impairment. Future efforts aimed at validating these diagnoses in preschoolers and implications for prevention are discussed.     

A gentzen system for conditional logic

Conditional logic is the deductive system , where is the set of propositional connectives {, ,} and is the structural finitary consequence relation on the absolutely free algebra that preserves degrees of truth over the structure of truth values C, . HereC is the non-commutative regular extension of the 2-element Boolean algebra to 3 truth values {t, u, f}, andf<u<t. In this paper we give a Gentzen type axiomatization for conditional logic.Presented byJan Zygmunt     

Euclidean Hierarchy in Modal Logic

For a Euclidean space , let L _n denote the modal logic of chequered subsets of . For every n 1, we characterize L _n using the more familiar Kripke semantics, thus implying that each L _n is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics L _n form a decreasing chain converging to the logic L of chequered subsets of . As a result, we obtain that L is also a logic over Grz, and that L has the finite model property. We conclude the paper by extending our results to the modal language enriched with the universal modality.     

Tarski hierarchies

The general notions of object- and metalanguage are discussed and as a special case of this relation an arbitrary first order language with an infinite model is expanded by a predicate symbol T₀ which is interpreted as truth predicate for . Then the expanded language is again augmented by a new truth predicate T₁ for the whole language plus T₀. This process is iterated into the transfinite to obtain the Tarskian hierarchy of languages. It is shown that there are natural points for stopping this process. The sets which become definable in suitable hierarchies are investigated, so that the relevance of the Tarskian hierarchy to some subjects of philosophy of mathematics are clarified.It should be noticed that these terms object language and meta language have only a relative sense. If, for instance, we become interested in the notion of truth applying to sentences, not of our original object-language, but of our meta-language, the latter becomes automatically the object-language of our discussion; and in order to define truth for this language, we have to go to a new meta-language — so to speak, to a meta-language of a higher level. In this way we arrive at a whole hierarchy of languages.(Tarski, 1986, p. 674f)     

On the concept of language in some recent theories of meaning

The work on this paper was done in the research project V ra begrepp om spr k supported byHumanistisk-samhälls-vetenskapliga forskningsr det. I am indebted to Per Martin-Löf, Hans Ruin, Pär Segerdahl, and Sven Öhman for valuable comments on a previous version of this paper. Thanks to the comments of the anonymous referee I was able to improve some formulations in the paper.     

An algebraic study of well-foundedness

A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. These algebras, however, generate the variety of all ( ,f) with f a hemimorphism on ).Admitting a second hemimorphism corresponding to the transitive closure of R allows foundational algebras to be equationally defined, in a way that gives a refined analysis of the notion of diagonalisable algebra.The research reported in this paper was carried out while the author was at the Department of Mathematics and Statistics, University of Auckland, under a teaching exchange with the Victoria University of Wellington.     

Every world can see a reflexive world

Let be the class of frames satisfying the condition

Moral Conflicts between Groups of Agents

Two groups of agents, and , face a moral conflict if has a moral obligation and has a moral obligation, such that these obligations cannot both be fulfilled. We study moral conflicts using a multi-agent deontic logic devised to represent reasoning about sentences like ‘In the interest of group of agents, group of agents ought to see to it that .’ We provide a formal language and a consequentialist semantics. An illustration of our semantics with an analysis of the Prisoner’s Dilemma follows. Next, necessary and sufficient conditions are given for (1) the possibility that a single group of agents faces a moral conflict, for (2) the possibility that two groups of agents face a moral conflict within a single moral code, and for (3) the possibility that two groups of agents face a moral conflict.     

Connectives Stranger than Tonk

Many logical systems are such that the addition of Prior's binary connective to them leads to triviality, see [1, 8]. Since is given by some introduction and elimination rules in natural deduction or sequent rules in Gentzen's sequent calculus, the unwanted effects of adding show that some kind of restriction has to be imposed on the acceptable operational inferences rules, in particular if these rules are regarded as definitions of the operations concerned. In this paper, a number of simple observations is made showing that the unwanted phenomenon exemplified by in some logics also occurs in contexts in which is acceptable. In fact, in any non-trivial context, the acceptance of arbitrary introduction rules for logical operations permits operations leading to triviality. Connectives that in all non-trivial contexts lead to triviality will be called non-trivially trivializing connectives.