Generalized approaches to the maxbet problem and the maxdiff problem,with applications to canonical correlations

Van de Geer has reviewed various criteria for transforming two or more matrices to maximal agreement, subject to orthogonality constraints. The criteria have applications in the context of matching factor or configuration matrices and in the context of canonical correlation analysis for two or more matrices. The present paper summarizes and gives a unified treatment of fully general computational solutions for two of these criteria, Maxbet and Maxdiff. These solutions will be shown to encompass various well-known methods as special cases. It will be argued that the Maxdiff solution should be preferred to the Maxbet solution whenever the two criteria coincide. Horst's Maxcor method will be shown to lack the property of monotone convergence. Finally, simultaneous and successive versions of the Maxbet and Maxdiff solutions will be treated as special cases of a fully flexible approach where the columns of the rotation matrices are obtained in successive blocks.The author is obliged to Henk Kiers for computational assistance and helpful comments.     

Involutions defined by monadic terms

We prove that there are two involutions defined by monadic terms that characterize Monadic Algebras. We further prove that the variety of Monadic Algebras is the smallest variety of Interior Algebras where these involutions give rise to an interpretation from the variety of Bounded Distributive Lattices into it.Research funded by Fondecyt, project N 764-1987.     

Arithmetic based on the Church numerals in illative combinatory logic

In the early thirties, Church developed predicate calculus within a system based on lambda calculus. Rosser and Kleene developed Arithmetic within this system, but using a Godelization technique showed the system to be inconsistent.Alternative systems to that of Church have been developed, but so far more complex definitions of the natural numbers have had to be used. The present paper based on a system of illative combinatory logic developed previously by the author, does allow the use of the Church numerals. Given a new definition of equality all the Peano-type axioms of Mendelson except one can be derived. A rather weak extra axiom allows the proof of the remaining Peano axiom. Note. The illative combinatory logic used in this paper is similar to the logic employed in computer languages such as ML.The author wishes to thank Dr. Roger Hindley for proofreading this paper and for making some useful suggestions. This paper was presented at the 1986 Annual Conference of the Australasian Association of Logic in Auckland, 9–12 July, 1986.     

A representation theorem for languages with generalized quantifiers through back-and-forth methods

We obtain in this paper a representation of the formulae of extensions ofL by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraïssés theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.     

Fuzziness of concepts and concepts of fuzziness

It has been a vexing question in recent years whether concepts are fuzzy. In this paper several views on the fuzziness of concepts are pointed out to have stemmed from dubious concepts of fuzziness. The underlying notions of the roles feasibly played byprototype, set, andprobability in modeling concepts strongly suggest that the controversy originates from a vague relation between intuitive and mathematical ideas in the cognitive sciences. It is argued that the application of fuzzy sets cannot resolve this vagueness since they are one sided,viz., defined on sets. An alternative definition based on classes (in the sense of axiomatic set theory) is proposed.     

Closed-class words as first syllables do interfere with lexical decisions for nonwords: implications for theories of agrammatism

It has been proposed that a principal cause of the agrammatism of some Broca's aphasics is that such patients, unlike normal subjects, are unable to make use of a special retrieval mechanism for closed-class ("function") words (D. C. Bradley, 1978, Computational distinctions of vocabulary type, Unpublished Ph.D. thesis; D. C. Bradley, M. F. Garrett, & E. B. Zurif, 1980. In D. Caplan (Ed.), Biological studies of mental processes). The main evidence for the existence of such a mechanism consisted of two observations: (1) the recognition of open-class words was observed to be frequency-sensitive, but that of closed-class words was not; and (2) lexical decisions for nonwords which began with open-class words were delayed, whereas there was no such interference for nonwords which began with closed-class words. However, the first of these observations has proved nonreplicable (e.g., B. Gordon & A. Caramazza, 1982, Brain and Language, 15, 143-160, 1983, Brain and Language, 19, 335-345; J. Segui, J. Mehler, W. Frauenfelder, & J. Morton, 1982, Neuropsychologia, 20, 615-627), and in the present paper, three lexical decision experiments are reported in which it is found that, when certain confounding variables are controlled, nonwords which begin with closed-class words are subject to interference. Moreover, contrary to a suggestion of Kolk and Blomert (1985, Brain and Language, 26, 94-105) the interference is independent of the presence of closed-class items in the lexical decision word list. It seems, then, that closed-class words are not qualitatively different from open-class words with respect either to frequency sensitivity or to nonword interference, and in consequence, the above proposed explanation of agrammatism is left without major empirical support.