The representation of Takeuti's\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} -operator

Gaisi Takeuti has recently proposed a new operation on orthomodular latticesL, \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) :P(L)»L. The properties of \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) suggest that the value of \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) (A) (A) \( \subseteq \) L) corresponds to the degree in which the elements ofA behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular latticesL and the existence of two-valued homomorphisms onL.     

Local dependence latent structure models

A local independence latent structure model, which assumesm latent classes, requires a minimum of 2m-1 items for the solution of the 2m ² latent parameters. If one adds 3 items to the test and if one assumes local dependence between pairs of items, thereby adding additional latent parameters, _ij , representing the association between itemsi andj, then it is possible to obtain estimates for all of the latent parameters: latent class frequencies latent probabilities, and measures of association between pairs of items. The solution consists of (1) forming (m + 1) × (m + 1) matrices of manifest data, which are singular, (2) solving for the _ij in equations that result from the singularity of the data matrices, (3) correcting the manifest data by removing the contamination due to local dependence, and (4) estimating the remaining latent parameters from the corrected data, using methods outlined in earlier literature.     

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.     

Completeness theorems for some intermediate predicate calculi

We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].     

Dependence and Independence

We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using = ${(\vec{x}, \vec{y})}$ have.     

On Birkhoff’s Common Abstraction Problem

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a ??common abstraction?? that includes Boolean algebras and latticeordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of ${\mathcal{B} \mathcal{A}}$ and ${\mathcal{L} \mathcal{G}}$ their join ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ in the lattice of subvarieties of ${\mathcal{F} \mathcal{L}}$ (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ relative to ${\mathcal{F} \mathcal{L}}$ . Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ .     

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     

A symmetric and an asymmetric model for the equal-sensation function in olfaction

Data from various intramodel matching experiments in olfaction were analyzed with regard to a symmetric and an asymmetric model for the equal-sensation function. The asymmetric model was discussed in relation to the symmetric model. In all, 11 equal-sensation functions were investigated, and of these 9 were with different pairs of odorants. The following odorants were investigated: hydrogen sulfide, pyridine, dimethyl disulfide, and five odorants obtained by different combustion procedures of animal manure. It was found that the equal-sensation function can be written in the following asymmetric form: $$\varphi _i = b_{ik} \lambda \cdot \varphi _k ^{b_{ik} } ,\psi _i = \psi _k $$ or symmetric form: $$\varphi _i = C^{I - b_{ik} } \cdot \varphi _k ^{b_{ik} } ,\psi _i = \psi _k ,$$ where ?_i and ?_k are stimuli expressed in multiples of respective absolute thresholds, λ and C are general constants invariant of experimental matching method and matched attribute (perceived unpleasantness or intensity). The constants λ and C were calculated both for group data and individual data. The asymmetric form of the equal-sensation function was interpreted in terms of relativity and the symmetric form in terms of measurement.     

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).     

Expanding Quasi-MV Algebras by a Quantum Operator

We investigate an expansion of quasi-MV algebras ([10]) by a genuine quantum unary operator. The variety of such quasi-MV algebras has a subquasivariety whose members—called cartesian—can be obtained in an appropriate way out of MV algebras. After showing that cartesian . quasi-MV algebras generate ,we prove a standard completeness theorem for w.r.t. an algebra over the complex numbers. Presented by Heinrich Wansing     

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     

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.     

用于处理不努力作答的标准化残差系列方法和混合多层模型法的比较

文章采用模拟研究, 分别在混合多层模型假设满足和违背的情境下, 比较了混合多层模型方法与标准化残差系列方法在识别不努力作答和参数估计方面的表现。结果显示：(1)不存在不努力作答或其严重性低时, 各方法表现接近; (2)不努力作答严重性高时, 固定参数迭代标准化残差法普遍更优, 混合多层模型法仅在假设满足且两种作答反应时差异大的条件下表现较好。建议实际应用中优先选择固定参数迭代标准化残差法。     

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.     

What is Nominalistic Mereology?

Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language $\mathcal {H}_{\textsf {m}}$ is maximally acceptable for nominalistic mereology. In an extension $\mathcal {H}_{\textsf {gem}}$ of $\mathcal {H}_{\textsf {m}}$ , a modal analog for the classical systems of Leonard and Goodman (J Symb Log 5:45–55, 1940) and Le?niewski (1916) is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for $\mathcal {H}_{\textsf {gem}}$ -bisimulations.     

Generalization of Scott's formula for retractions from generalized Alexandroff's cube

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α?δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice. In the case when α=ω and δ=∞, this theorem becomes Scott's theorem: Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice. On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem: Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice. But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods. In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula. Namely it is proved that if α= 0 or δ= ∞ or α ? δ and \(F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n} \) and if L is an 〈α, δ〉-semidistributive lattice, then the function $$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$ such that for x ε ? ( \(\mathfrak{n}\) ): (*) $$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$ defines retraction, where C is a proper closure operator for \(B_{\alpha ,\delta }^\mathfrak{n} \) . It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice. Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].