On testing hypotheses regarding a class of covariance structures

Let x be ap-component random variable having a multivariate normal distribution with covariance matrix . In this paper, we consider the problem of testing hypotheses of the formH ₀: =b ₁₁ + +b _{m m} , whereb _i 's are unknown scalars, and _i 's are a set of known and simultaneously diagonalizable matrices. This problem has both psychometric and statistical interest, and its basic theory is developed here. Besides, the problem of obtaining likelihood-ratio statistic for testingH ₀ is studied, and the statistic obtained in a special case.     

Syntactical results on the arithmetical completeness of modal logic

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the form p _i,m _i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic.     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

An n-Player Semantic Game for an n + 1-Valued Logic

First we show that the classical two-player semantic game actually corresponds to a three-valued logic. Then we generalize this result and give an n-player semantic game for an n + 1-valued logic with n binary connectives, each associated with a player. We prove that player i has a winning strategy in game if and only if the truth value of is t _i in the model M, for 1 ≤ i ≤ n; and none of the players has a winning strategy in if and only if the truth value of is t ₀ in M.     

A general nonmetric technique for finding the smallest coordinate space for a configuration of points

LetA ₁,A ₂, ...,A _n be anyn objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (A _i,A _i ), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculatingm coordinates {x _ia } for eachA _i for a semi-metricd of preassigned formd _ij =d(|x _i1 -x _j1 |, |x _i2 -x _j2|, ..., |x _im -x _jm |). The dimensionalitym is sought to be as small as possible, yet satisfy the monotonicity condition thatd _ij <d _kl whenever the observed data indicate thatA _i is closer toA _j thanA _k is toA _l . Minkowski and Euclidean spaces are special metric examples ofd. A general coefficient of monotonicity is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between hard-squeeze and soft-squeeze tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.This research was supported in part by the National Science Foundation of the United States Government, through Grant No. GS 929 to the University of Michigan.I am deeply indebted for many helpful comments on earlier drafts of this paper received from Joseph Kruskal, James Lingoes, and the managing editor.     

On sensitive rank tests for comparing the effects of two treatments on a single group

The rank testR _n of Cronholm and Revusky [1965] in effect doubles the number of comparisons carried out in the Mann-Whitney test between Treated and Control subjects chosen from a single group ofn and triples the asymptotic Pitman efficiency against shift alternatives by using a series ofn — 1 subexperiments. However, the full series can be too costly or time-consuming. It is shown thatj appropriately chosen subexperiments will permit one to make at leastj/(j+1) of the number of comparisons possible underR _n, with large sample efficiency ofj/(j+2) relative toR _n against shift alternatives and small sample efficiency greater than that,j=2, 3, ...,n — 2. The resulting test criterion is a sum ofj independent Mann-Whitney test statistics. Its null distribution is tabulated forn10 and small sample efficiency comparisons are carried out forn=10.     

Monotone regression: Continuity and differentiability properties

Least-squares monotone regression has received considerable discussion and use. Consider the residual sum of squaresQ obtained from the least-squares monotone regression ofy _i onx _i . TreatingQ as a function of they _i , we prove that the gradient Q exists and is continuous everywhere, and is given by a simple formula. (We also discuss the gradient ofd=Q ^1/2.) These facts, which can be questioned (Louis Guttman, private communication), are important for the iterative numerical solution of models, such as some kinds of multidimensional scaling, in which monotone regression occurs as a subsidiary element, so that they _i and hence indirectlyQ are functions of other variables.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

The probability of war in then-crises problem: Modeling new alternatives to Wright's solution

In hisStudy of War, Q. Wright considered a model for the probability of warP during a period ofn crises, and proposed the equationP=1–(1–p) ⁿ , wherep is the probability of war escalating at each individual crisis. This probability measure was formally derived recently by Cioffi-Revilla (1987), using the general theory of political reliability and an interpretation of the n-crises problem as a branching process. Two new, alternate solutions are presented here, one using D. Bernoulli's St. Petersburg Paradox as an analogue, the other based on the logic of conditional probabilities. Analysis shows that, while Wright's solution is robust with regard to the general overall behavior ofp andn, some significant qualitative and quantitative differences emerge from the alternative solutions. In particular,P converges to 1 only in a special case (Wright's) and not generally.C.C.-R. acknowledges support provided by the Merriam Laboratory for Analytic Political Research, the Research Board of the University of Illinois, and grant SES-84-00877 from the U.S. National Science Foundation.     

The number of paths and cycles in a digraph

An algorithm is presented for constructing from the adjacency matrix of a digraph the matrix of its simplen-sequences. In this matrix, thei, j entry,i j, gives the number of paths of lengthn from a pointv _i to a pointv _j ; the diagonal entryi, i gives the number of cycles of lengthn containingv _i . The method is then generalized to networks—that is, digraphs in which some value is assigned to each line. With this generalized algorithm it is possible, for a variety of value systems, to calculate the values of the paths and cycles of lengthn in a network and to construct its value matrix of simplen-sequences. The procedures for obtaining the two algorithms make use of properties of a line digraph—that is, a derived digraph whose points and lines represent the lines and adjacency of lines of the given digraph.The research reported here was supported by Grant NSF-G-17771 from the National Science Foundation. We wish to thank Professor Frank Harary for suggesting certain ways of improving an earlier draft of this paper.     

A mixture model for distributions of correlation coefficients

An old problem in personnel psychology is to characterize distributions of test validity correlation coefficients. The proposed model views histograms of correlation coefficients as observations from a mixture distribution which, for a fixed sample sizen, is a conditional mixture distributionh(r|n) = _{j j} h(r; _j ,n), whereR is the correlation coefficient, _j are population correlation coefficients and _j are the mixing weights. The associated marginal distribution ofR is regarded as the parent distribution underlying histograms of empirical correlation coefficients. Maximum likelihood estimates of the parameters _j and _j can be obtained with an EM algorithm solution and tests for the number of componentst are achieved after the (one-component) density ofR is replaced with a tractable modeling densityh(r; _j ,n). Two illustrative examples are provided.     

Reasons and Entailment

What is the relation between entailment and reasons for belief? In this paper, I discuss several answers to this question, and I argue that these answers all face problems. I then propose the following answer: for all propositions p ₁,…,p _n and q, if the conjunction of p ₁,…, and p _n entails q, then there is a reason against a person’s both believing that p ₁,…, and that p _n and believing the negation of q. I argue that this answer avoids the problems that the other answers to this question face, and that it does not face any other problems either. I end by showing what the relation between deductive logic, reasons for belief and reasoning is if this answer is correct.     

Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction

We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix \({\mathbb {A}}\) with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a _i,j =?. The method allows us to decide whether to declare a _i,j = 0 or a _i,j = 1 or a _i,j =? undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences.     

Depth relevance of some paraconsistent logics

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If _DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested 's required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M ₀ , M₁, ..., M}, where theM _i s are all characterized by Meyer's 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR.     

The Poznań school methodology of idealization and concretization from the point of view of a revised structuralist theory conception

My thesis is that some methodological ideas of the Pozna school, i.e., the principles of idealization and concretization (factualization), and the correspondence principle can be represented rather successfully using the relations of theoretization and specialization of revised structuralism.Let <n(i), t(j)> (i=1,...m, j=1,...k) denote the conceptual apparatus of a theory T, and a class M={} (i=1,...m, j=1,...k) the models of T. The n-components refer to the values of dependent variables and t-components to the values of independent variables of the theory. The n- and t-components in turn represent appropriate concepts. Consider T ^* as a conceptual enrichment of T with concepts <n(i ^*), t(j ^*)> (i<i ^* or j<j ^*) and models M ^*={<D ^*, n(i ^*), t(j ^*)>}. If the classes M and M ^* are suitably related, then the situation illustrates both the case of the theoretization-relation of (revised) structuralism and of the factualization-principle of the Pozna school.Assume now that the concepts n(i), t(j) of T for some i, j are operationalized using some special assumptions generating appropriate empirical values n and t for these concepts. Let M denote the class {<D,...n,...t,...>} which is formed by substituting n and t for values of concepts n(i), t(j) in the elements of M. If the classes M and M are related in a suitable way then the situation is an example of both the specialization-relation of (revised) structuralism and the concretization-principle of the Pozna school. The correspondence principle in turn can be represented as a limiting case of the theoretization-relation of (revised) structuralism.Many thanks to my anonymous referees for critical and fruitful comments and special thanks to Dr. Carol Norris for correcting the language of this paper.     

Finite axiomatization for some intermediate logics

LetN. be the set of all natural numbers (except zero), and letD _n ^* = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD _n ^* = 〈D _n ^* , ? _n , wherex? _ny iffx¦y for anyx, y∈D _n ^* , can easily be seen to be a pseudo-boolean algebra. In [5], V.A. Jankov has proved that the class of algebras {D _n ^* ∶n∈B}, whereB =,{k ∈N∶ ? \(\mathop \exists \limits_{n \in N} \) (n > 1 ≧n ² k)is finitely axiomatizable. The present paper aims at showing that the class of all algebras {D _n ^* ∶n∈B} is also finitely axiomatizable. First, we prove that an intermediate logic defined as follows: $$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$ finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ?〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ? is identical with. the set of formulas true in the Kripke modelH _B = 〈P(?), ?〉 (whereP(?) stands for the family of all prime filters in the algebra ?). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D _n ^* = 〈P (D _n ^* ), ?〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp ₃∨ [p ₃→(p₁→p₂)∨(p₂→p₁)] there is a structure of divisorsD _* ⁿ such that it is possible to define a strong homomorphism froomiH D _n ^* ontoH D _U . Exploiting, among others, this property, it turns out to be relatively easy to show that \(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\) .     

Interpretations of the first-order theory of diagonalizable algebras in peano arithmetic

For every sequence |p _n } _n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula ⁰ A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, ⁰ A is a theorem ofPA.     

Paraconsistent algebras

The prepositional calculiC _n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum's algebra forC _n . C. Mortensen settled the problem, proving that no equivalence relation forC _n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC _n , as well as the concept of paraconsistent closure system, are introduced in this paper.We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent.     

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.