Indistinguishability, Choices, and Logics of Agency

This paper deals with structures ${\langle{\bf T}, I\rangle}$ in which T is a tree and I is a function assigning each moment a partition of the set of histories passing through it. The function I is called indistinguishability and generalizes the notion of undividedness. Belnap’s choices are particular indistinguishability functions. Structures ${\langle{\bf T}, I\rangle}$ provide a semantics for a language ${\mathcal{L}}$ with tense and modal operators. The first part of the paper investigates the set-theoretical properties of the set of indistinguishability classes, which has a tree structure. The significant relations between this tree and T are established within a general theory of trees. The aim of second part is testing the expressive power of the language ${\mathcal{L}}$ . The natural environment for this kind of investigations is Belnap’s seeing to it that (stit). It will be proved that the hybrid extension of ${\mathcal{L}}$ (with a simultaneity operator) is suitable for expressing stit concepts in a purely temporal language.     

Agrammatism and the Psychological Reality of the Syntactic Tree

Syntactic trees, or phrase markers, have originally been suggested as a representation of syntax in the mind based on purely linguistic grounds. In this paper, the psychological reality of syntactic trees and hierarchical ordering is explored from another perspective—that of the neuropsychology of language breakdown. The study reported here examined several syntactic domains that rely on different nodes in the tree—tense and agreement verb inflection, subordinations, interrogatives, and verb movement, through a study of 14 Hebrew- and Palestinian Arabic-speaking agrammatic aphasics and perusal of the cross-linguistic literature. The results show that the impairment in agrammatic production is highly selective and lends itself to characterization in terms of a deficit in the syntactic tree. The complex pattern of dissociations follows from one underlying deficit—the inaccessibility of high nodes of the syntactic tree to agrammatic speakers. Structures that relate to high nodes of the tree are impaired, while lower structures are spared.     

The Category of Node-and-Choice Preforms for Extensive-Form Games

It would be useful to have a category of extensive-form games whose isomorphisms specify equivalences between games. Since working with entire games is too large a project for a single paper, I begin here with preforms, where a “preform” is a rooted tree together with choices and information sets. In particular, this paper first defines the category \(\mathbf {Tree}\), whose objects are “functioned trees”, which are specially designed to be incorporated into preforms. I show that \(\mathbf {Tree}\) is isomorphic to the full subcategory of \(\mathbf {Grph}\) whose objects are converging arborescences. Then the paper defines the category \(\mathbf {NCP}\), whose objects are “node-and-choice preforms”, each of which consists of a node set, a choice set, and an operator mapping node-choice pairs to nodes. I characterize the \(\mathbf {NCP}\) isomorphisms, define a forgetful functor from \(\mathbf {NCP}\) to \(\mathbf {Tree}\), and show that \(\mathbf {Tree}\) is equivalent to the full subcategory of \(\mathbf {NCP}\) whose objects are perfect-information preforms. The paper also shows that many game-theoretic entities can be derived from preforms, and that these entities are well-behaved with respect to \(\mathbf {NCP}\) morphisms and isomorphisms.     

“Minimax length links” of a dissimilarity matrix and minimum spanning trees

A theorem is proved stating that the set of all minimax links, defined as links minimizing, over paths, the maximum length of links in any path connecting a pair of objects comprising nodes in an undirected weighted graph, comprise the union of all minimum spanning trees of that graph. This theorem is related to methods of fitting network models to dissimilarity data, particularly a method called Pathfinder due to Schvaneveldt and his colleagues, as well as to single linkage clustering, and results concerning the relationship between minimum spanning trees and single linkage hierarchical trees.Acknowledgments: The author thanks Phipps Arabie, Lawrence J. Hubert, and K. Christoph Klauer for a number of helpful suggestions and comments on various aspects of this paper.     

On Intermediate Predicate Logics of some Finite Kripke Frames,I. Levelwise Uniform Trees

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.     

Tree inference with factors selectively influencing processes in a processing tree

We show how to systematically construct a processing tree using data from an experiment in which two experimental factors are intended to selectively influence two different processes. Only two arrangements are possible for the two selectively influenced processes: ordered and unordered. We show that there are only two standard trees that need be considered, one for each arrangement. Necessary and sufficient conditions are given for each tree to be applicable. If the conditions do not hold, no tree is possible. If the selectively influenced processes are ordered, their order is sometimes determined by the data. We consider two factors that selectively influence two processes in every tree in a set of trees, such that on a given trial, a tree is selected from the set at random. We show that if the influenced processes are ordered in every tree or unordered in every tree, then the mixture of trees is equivalent to one of the two standard trees. To illustrate the method, a tree is constructed from simulated data from a model of Batchelder and Riefer [Batchelder, W.H., Riefer, D.M. (1980). Separation of storage and retrieval factors in free recall of clusterable pairs. Psychological Review, 87, 375–397].     

The estimation of ultrametric and path length trees from rectangular proximity data

A least-squares algorithm for fitting ultrametric and path length or additive trees to two-way, two-mode proximity data is presented. The algorithm utilizes a penalty function to enforce the ultrametric inequality generalized for asymmetric, and generally rectangular (rather than square) proximity matrices in estimating an ultrametric tree. This stage is used in an alternating least-squares fashion with closed-form formulas for estimating path length constants for deriving path length trees. The algorithm is evaluated via two Monte Carlo studies. Examples of fitting ultrametric and path length trees are presented.G. De Soete is Aspirant of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek at the University of Ghent, Belgium. W. S. DeSarbo and J. D. Carroll are Members of Technical Staff at AT&T Bell Laboratories, Murray Hill, N.J. G. W. Furnas is Member of Technical Staff at Bell Communications Research, Murray Hill, N.J.     

Tree representations of ternary relations

In recent years there has been a growing interest in representing data by trees. Most of the literature has been concerned with inferring trees from pairwise data. In this paper, trees are constructed from ternary relations; the model represents each object of an empirical set by a node of a tree so that a betweenness relation among the nodes (the node b is on the path from the node a to the node c) in the graph reflects a ternary relation among the objects. Various systems of formal properties that lead to three different kinds of tree representation are investigated.     

Joint action and group action made precise

The paper argues that there are two main kinds of joint action, direct joint bringing about (or performing) something (expressed in terms of a DO-operator) and jointly seeing to it that something is the case (expressed in terms of a Stit-operator). The former kind of joint action contains conjunctive, disjunctive and sequential action and its central subkinds. While joint seeing to it that something is the case is argued to be necessarily intentional, direct joint performance can also be nonintentional. Actions performed by social groups are analyzed in terms of the notions of joint action (basically DO and Stit).A precise semantical analysis of the aforementioned kinds of joint action is given in terms of time-trees. With each participant a tree is connected, and the trees are joined defining joint possible worlds in terms of state-expressing nodes from the trees. Sentences containing DO and Stit are semantically evaluated with respect to such joint possible worlds. Intentional joint actions are characterized in terms of the notion of we-intention (joint intention), characterized formally by means of a special operator.     

Constructing a Tree for Community Leaders: Contexts and Processes in Collaborative Inquiry

This paper presents collaborative research between a university-based team and a grass roots community organization on the south side of Chicago. The purpose of the work is to document the nature of community leadership as expressed by members of the host organization. Using a semistructured interview, 77 community members nominated by the host organization were asked about various aspects of their community work. Qualitative analyses of interview text generated a set of 56 codes concerning different aspects of community leadership as reported by grassroots leaders. These codes were subsequently grouped together into five conceptually linked dimensions of community leadership. A leadership tree that simultaneously analyzes and visually displays each of these five dimensions of community leadership was created. Implications of the methodology used to create the leadership tree are discussed for the salience of qualitative methods in community research.     

Free trees and bidirectional trees as representations of psychological distance

Two types of graph-theoretic representations of psychological distances or dissimilarities are proposed: weighted free trees and weighted bidirectional trees. A weighted free tree is a generalization of the sort of graph representation used in hierarchical clustering. A weighted bidirectional tree is a further generalization which allows for asymmetric dissimilarities. The properties of these structures are discussed and numerical methods are presented that can be used to derive a representation for any given set of dissimilarities. The applicability of these structures is illustrated by using them to represent the data from experiments on the similarities among animal terms and on memory for sentences. An analysis based upon free trees is compared and contrasted with analyses using hierarchical clustering and multidimensional scaling.     

Trees with structure

We investigate M-trees, that is, trees with structure possible at each node or level. M is a mathematical structure such as a set or a Cartesian product. An extension of Pólya's theorem is proved which allows the number of M-trees for a given number of nodes to be counted. The special case of componential trees is investigated. Here M is a Cartesian product of 0's and 1's. A componential analysis is a componential tree of depth 1, that is, there is no hierarchy. We prove that for any componential tree there exists a componential analysis which makes the same predictions on a triad test of judged similarity. A brief empirical example is given, in which a componential tree is applied as a model of a sernantic domain.     

Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers p, p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most , the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel–Dummett logics with quantifiers over propositions.     

Nice Embedding in Classical Logic

It is shown that a set of semi-recursive logics, including many fragments of CL (Classical Logic), can be embedded within CL in an interesting way. A logic belongs to the set iff it has a certain type of semantics, called nice semantics. The set includes many logics presented in the literature. The embedding reveals structural properties of the embedded logic. The embedding turns finite premise sets into finite premise sets. The partial decision methods for CL that are goal directed with respect to CL are turned into partial decision methods that are goal directed with respect to the embedded logics.     

Semantic organization in deaf and hearing subjects

Hierarchical cluster analysis of data from the sorting of noun words was used to compare semantic structures in 63 profoundly deaf and 63 hearing adolescents. In the first study, performance differed only for a set of words referring to sounds, where deaf persons have no experience, and not for a set of common noun words and pictures. In the second study, differences between matched sets of high- and low-imagery words were comparable for 63 deaf and 63 hearing subjects. It is concluded that deaf subjects manifested abstract hierarchical relations and were not dependent on visual mediators or hindered by the absence of acoustic mediators.This investigation was supported in part by National Institutes of Health Research Grant NS-09590-03 from the National Institute of Neurological Diseases and Stroke and in part by a Faculty Research Grant from Bowling Green State University.Portions of Study 1 were previously reported at the Eighty-first Annual Convention of the American Psychological Association, Montreal, 1973.     

Multidimensional scaling of measures of distance between partitions

The techniques of multidimensional scaling were used to study the numerical behavior of twelve measures of distance between partitions, as applied to partition lattices of four different sizes. The results offer additional support for a system of classifying partition metrics, as proposed by Boorman (1970), and Boorman and Arabie (1972). While the scaling solutions illuminated differences between the measures, at the same time the particular data with which the measures were concerned offered a basis both for counterexamples to some common assumptions about multidimensional scaling and for some conjectures as to the nature of scaling solutions. The implications of the latter findings for selected examples from the literature are considered. In addition, the methods of partition data analysis discussed here are applied to an example using sociobiological data. Finally, an argument is made against undue emphasis upon interpreting dimensions in nonmetric scaling solutions.     

Fitting the factor analysis model

When the covariance matrix (p×P) does not satisfy the formal factor analysis model for m factors, there will be no factor matrix (p×m) such that =(-) is diagonal. The factor analysis model may then be replaced by a tautology where is regarded as the covariance matrix of a set of residual variates. These residual variates are linear combinations of discarded common factors and unique factors and are correlated. Maximum likelihood, alpha and iterated principal factor analysis are compared in terms of the manner in which is defined, a maximum determinant derivation for alpha factor analysis being given. Weighted least squares solutions using residual variances and common variances as weights are derived for comparison with the maximum likelihood and alpha solutions. It is shown that the covariance matrix defined by maximum likelihood factor analysis is Gramian, provided that all diagonal elements are nonnegative. Other methods can define a which is nonGramian even when all diagonal elements are nonnegative.A modified version of this paper forms part of a Ph.D. thesis submitted to the University of South Africa.Presently at the National Institute for Personnel Research, South Africa.     

Cut-free sequent and tableau systems for propositional Diodorean modal logics

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the superformulae involved are always bounded by a finite set of formulaeX* _L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive.Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems. The techniques are due to Hintikka and Rautenberg.Presented byDov M. Gabbay