Linear programming techniques for multidimensional analysis of preferences

This paper offers a new methodology for analyzing individual differences in preference judgments with regard to a set of stimuli prespecified in a multidimensional attribute space. The individual is modelled as possessing an ideal point denoting his most preferred stimulus location in this space and a set of weights which reveal the relative saliences of the attributes. He prefers those stimuli which are closer to his ideal point (in terms of a weighted Euclidean distance measure). A linear programming model is proposed for external analysisi.e., estimation of the coordinates of his ideal point and the weights (involved in the Euclidean distance measure) by analyzing his paired comparison preference judgments on a set of stimuli, prespecified by their coordinate locations in the multidimensional space. A measure of poorness of fit is developed and the linear programming model minimizes this measure overall possible solutions. The approach is fully nonmetric, extremely flexible, and uses paired comparison judgments directly. The weights can either be constrained nonnegative or left unconstrained. Generalizations of the model to consider ordinal or interval preference data and to allow an orthogonal transformation of the attribute space are discussed. The methodology is extended to perform internal analysis,i.e., to determine the stimuli locations in addition to weights and ideal points by analyzing the preference judgments of all subjects simultaneously. Computational results show that the methodology for external analysis is unbiased—i.e., on an average it recovers the true ideal point and weights. These studies also indicate that the technique performs satisfactorily even when about 20 percent of the paired comparison judgments are incorrectly specified.The authors wish to express their deep appreciation to J. Douglas Carroll and Joseph B. Kruskal for their most valuable comments on an earlier version of this paper.     

Reformatory blame

It is argued that blame is an ambiguous and misused term leading to ineffective, harmful consequences, such as hate, guilt, revenge (retaliation, or retribution), or punishment for its own sake. A clarification of blame is given by showing that it is an open-context, value term and, so, meaningless in itself. For intelligibility, it must be reduced to naturalistic and pragmatic terms. The definition almost universally given is based on the fallacious beliefs that (1) we can change the past, and (2) we can have done otherwise than we did. But can here is seen also to be used in a pseudo-scientific and subjunctive, rather than in a factual sense. It is shown that retributive blame makes no sense because, as Marcus Aurelius said, we must accept what is and we can only do what is within our power. We cannot change past behavior, but we can try to change present behavior. For effectiveness and intelligibility, we are forced to substitute rehabilitative blame for retributive blame. Thus, in the prevention and elimination of such negative emotions as guilt, inferiority, hatred, revengefulness, a thorough understanding of the term blame is required.     

The American plan completed: Alternative classical-style semantics,without stars,for relevant and paraconsistent logics

American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 (A, a) iff 0 (A, a) and 0 (A, a) iff 1 (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost of these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa.     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

On a certain class ofm-address machines

Conclusion It follows from the proved theorems that ifM =Q, (whereQ={0,q ₁,q ₂,...,q }) is a machine of the classM _F then there exist machinesM _i such thatM _i(1,c)=M (q _i,c) andQ _i={0, 1, 2, ..., +1} (i=1, 2, ..., ).And thus, if the way in which to an initial function of content of memorycC a machine assigns a final onecC is regarded as the only essential property of the machine then we can deal with the machines of the formM ={0, 1, 2, ..., }, and processes (t) (wheret=1,c,cC) only.Such approach can simplify the problem of defining particular machines of the classM _F , composing and simplifying them.Allatum est die 19 Januarii 1970     

A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.     

Utility theory with inexact preferences and degrees of preference

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.     

Children's judgment and recall of causal connectives: A developmental study of “because,” “so,” and “and”

The present study examined the acceptability judgments and recall performance of children aged 6,9, and 11 years with sentences expressing psychological cause-effect relations. Thirty-two sentences containing because medially, because initially, so, and and were generated from four clause pairs. Both probable and improbable event orders were used. The results indicated that children preferred because to so or and for statements of psychological causality. However, on both tasks, first and third graders frequently failed to attend to the temporal ordering specified by each sentence construction. The younger children also tended to judge all sentences acceptable, suggesting that they were concerned only with the probable association of event pairs, and not with the usual order of the events. Order of mention strategies did not occur in any group, suggesting that they arise only when children cannot make interpretations based on probable order of events. The data also indicated that the recall task is an unreliable index of rules for comprehension and production.This article is based on research conducted by the first author as partial fulfillment of the Ph.D. requirements for the Department of Educational Policy Studies, University of Wisconsin.     

Supererogation in Deontic Logic: Metatheory for DWE and Some Close Neighbours

In "Doing Well Enough: Toward a Logic for Common Sense Morality", Paul McNamara sets out a semantics for a deontic logic which contains the operator It is supererogatory that. As well as having a binary accessibility relation on worlds, that semantics contains a relative ordering relation, . For worlds u, v and w, we say that u w v when v is at least as good as u according to the standards of w. In this paper we axiomatize logics complete over three versions of the semantics. We call the strongest of these logics DWE for Doing Well Enough.     

A semantical investigation into Leśniewski's axiom of his ontology

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure).     

Six-year-old children's understanding of sentences adjoined with time adverbs

The adjoining of clauses with temporal links is the basis for many sentences that convey sequence of events. The present study attempts to delineate 6-year-old children's (N=30) understanding of the meaning sequences imparted by sentences adjoined with after, before, and until. Their performance of the meaning sequence for each of 24 carefully constructed sentences is compared to an adult model. Analysis of the results (using a Wilcoxon Matched Pairs Signed Rank Test) indicated that: (1)Ss understood sentences adjoined with after according to an adult model more frequently than before adjoined sentences (P<0.01); (b) Until adjoined sentences with a negative marker in the main clause were understood according to an adult model more often than until adjoined sentences with no such negative element, but the difference was not significant at a=0.01; (3)Ss understood before adjoined sentences according to an adult model more often than until adjoined sentences, but the difference was not significant at a=0.01. In general, the results indicated that 6-year-olds have not yet completed development of an adult grammar with respect to adjoining clauses with temporal links, after, before, and until.     

Suppositions in argumentation

The atheist who begins to argue his case by saying, Suppose there is an omniscient Being of the sort in which Christians believe ... is employing a very familiar move in argumentation. However, most books on argumentation theory ignore suppositions completely. Searle omits suppositions entirely from his taxonomy of speech acts and this appears to lead to a similar omission in Speech Acts in Argumentative Discussions by van Eemeren and Grootendorst.This paper argues that suppositional argument is elegant, powerful and extremely common, that the correct way to understand it is based on Gottlob Frege's distinction between asserted and unasserted propositions and hence that suppositions are neither assertions nor (and this is more important) assertives. The paper discusses the connections between suppositions and conditionals; it argues that argumentation theory which ignores suppositions is systematically misleading; and it concludes by indicating some possible developments in argumentation theory.Revised version of a paper delivered at the University of Amsterdam, April 3, 1987.     

John Nash's Postdelusional Period: A Case of Transformed Narcissism

This article concludes the psychoanalytic study of mathematical genius John Nash begun in previous articles (Capps 2003a, 2003b) by focusing on his recovery from paranoid schizophrenia after more than a decade of being under control of his delusions. I develop the idea that Nash was a highly narcissistic personality, the primary focus of which was his beautiful mind, in the years preceding his mental breakdown. I attribute his recovery primarily to the transformation of his narcissistic personality and support this attribution by means of Heinz Kohut's identification of five major expressions of transformed narcissism in his classic essay on the forms and transformations of narcissism.     

Quantitative Versuche über die Prägnanz von Punktmustern

Summary The intention of the paper is to describe Prägnanz-phenomena in terms of information theory. Two different concepts for the application of informational variables to perceptual processes are discriminated. For several reasons, Attneave's concept (1951) is preferred: In the sense of information theory single figures can be treated as signal structures; thus the degree of regularity in them should be specifiable in terms of informational variables.Two quasi-informational parameters are used to describe the degree of regularity in simple dot patterns.It was expected that perceived figural quality of dot patterns would increase with the redundancy of spatially distributed messages. At a constant word-length (= constant number of dots), the quality perceived should be the better the lower the entropy (or uncertainty) of the figure.In several experiments the information measures used turned out to be closely related to different operationally defined response-variables for figural quality, e.g. figural goodness-ratings and exactness of the reproduction of dot-positions. Besides, an inverse relationship is observed between interestingness-ratings and redundancy measures.The results are discussed in terms of information theory and psychological concepts. It is shown, that it has important implications on the psychological theory of perception to regard the human perceiver as an information-decoding-system. Apparently there must be some dynamically active system which enables the perceiver to utilize the redundancy of spatial brightness-distributions for sure and quick recognition.     

Action incompleteness

The author has previously introduced an operator into dynamic logic which takes formulae to terms; the suggested reading of A was the bringing about of A or the seeing to it that A. After criticism from S. K. Thomason and T. J. Surendonk the author now presents an improved version of his theory. The crucial feature is the introduction of an operatorOK taking terms to formulae; the suggested reading of OK is always terminates.     

Three prepositional calculi of probability

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic ₀ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.The second calculus MP is a usual modal propositional calculus. It has the modal rules x x, x y x y, x x, x y (y x), (y x), in addition to the rules of classical propositional logic. One may read x as x is probable. Imbeddings of NP and of ₀ into MP are given.The third calculus P is a modal extension of ₀. It may be obtained by adding the rule ((xy)y) xy to the modal logic of quantum mechanics Q [5]. One may read x in P as x is observed. An imbedding of NP into P is given.     

Intuitionistic uniformity principles for propositions and some applications

This note deals with the prepositional uniformity principlep-UP: p x N A (p, x) x N p A (p, x) ( species of all propositions) in intuitionistic mathematics.p-UP is implied by WC and KS. But there are interestingp-UP-cases which require weak KS resp. WC only. UP for number species follows fromp-UP by extended bar-induction (ranging over propositions) and suitable weak continuity. As corollaries we have the disjunction property and the existential definability w.r.t. concrete objects. Other consequences are: there is no non-trivial countable partition of;id is the only injective function from to; there are no many-place injective prepositional functions; card () is incomparable with the cardinality of all metric spaces containing at least three elements.     

The good I will,I do not

For I do not do the good I want, but the evil I do not want is what I do-Now if I do what I do not want, it is no longer I that do it, but sin which dwells within me. (Romans 7:19–20)A sermon delivered on November 16, 1958, in James Memorial Chapel. Reprinted by permission from Union Seminary Quarterly Review.     

On a new property of partially balanced association schemes useful in psychometric structural analysis

In an earlier paper [Psychometrika,31, 1966, p. 147], Srivastava obtained a test for the HypothesisH ₀ : = ₀₀ + ... + _ll, where i are known matrices,i are unknown constants and is the unknown (p ×p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p ×p) covariance matrix of a random variablex (withp components) the condition that ₀, ₁, ..., l form a commutative linear associative algebra and a certain vector, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where ₀, ₁, ..., l are the association matrices of a partially balanced association scheme.This research was partially supported by the U. S. Air Force under Grant No. AF33(615)-3231, monitored by the Aero Space Research Labs.Now at Colorado State University.