Vladimir A. Lefebvre, “The Golden Section and an Algebraic Model of Ethical
Journal of Mathematical Psychology, 1985, Vol. 29. 289-310.
In experimental investigations related to Kelly’s theory of constructs, it was
found that subjects on the average choose a positive over a negative pole with
the probability 0.62. In this paper, it is shown that the choice of this constant
can be mathematically explained with the algebraic model of ethical cognition.
Vladimir A. Lefebvre, “The Principle of Complementarity as the Basis for the
Model of Ethical Cognition”.
Journal of Social and Biological Structures, 1984, Vol. 3, pp. 129-175.
The analysis of the Boolean model of ethical cognition has demonstrated that
a law similar to the principle of complementarity is realized in the model:
‘interference of feelings’ is incompatible with the correct registration of inner
intentions. The entire class of the possible algebraic model is investigated, and
it is shown that if we postulate the existence in the human psyche of the
principle of complementarity and of analogues of parallel and sequential
locations of physical devices, then as a consequence we obtain the Boolean
Vladimir A. Lefebvre, “Modeling of Quantum-Mechanical Phenomena with
the Help of the Algebraic Model of Ethical Cognition,”
Mathematical Modeling, 1983, Vol. 4. pp. 361-366.
A brief description of an algebraic model of ethical cognition based on an
exponential representation of Boolean function is given in this paper. This
model allows us to connect the behavior of an individual with the structure of
his inner world. Unexpectedly, we found that the classical idealized experiment
on the interference of the electrons from two sleets can be simulated with the
help of the same model. Thus, we succeeded in establishing connection
between the phenomenon of cognition and the quantum phenomenon by
looking at quantum mechanics through a psychologist’s eyes.
William H. Batchelder, Vladimir A. Lefebvre, “A Mathematical Analysis of a
Natural Class of Partitions of a Graph.”
Journal of Mathematical Psychology, 1982, Vol. 24, No. 4, pp. 124-148.
Any finite graph (non directed, no loops) can be coupled with its comple-
mentary graph producing what we term a completed graph. This paper
studies completed graphs in the context of a structural concept called
stratification that is motivated by theoretical work in social psychology and
sociology. A completed graph is stratified in case its node (vertex) set can be
divided into two nonempty subsets with exactly one of the two types of ties
holding between every pair of nodes from distinct sets. A completed graph is
totally stratified in case every one of its nontrivial completed subgraphs is
stratified. The first part of the paper relates the concept of stratification to
the familiar graph property of connectedness. In particular, a completed graph
is totally stratified if and only if it does not have a four point completed
subgraph that is connected in both the original graph relation and its comple-
mentary relation. Using the concept of stratification, completed graphs can be
decomposed uniquely into taxonomic structures, that is nested sets of
partitions. An algorithm for the decomposition based on the work with the
concept of stratification is developed. The relationship between the notion of
stratification and the ideas in balance theory is examined, and stratification is
viewed as a generalization of Davis’ notion of clustering. It turns out that the
concepts of clique, status, and structural equivalence used throughout the
social network literature can be defined in an interesting way for completed
graphs. Cliques are sets of nodes with similar internal links and statuses are
sets of nodes with similar external links. Structurally equivalent sets are both
cliques and statuses. The concepts of status and structural equivalence are
closely related to the decomposition algorithm. In particular, for any totally
stratified completed graph, the set of all statuses and the set of all maximal
structurally equivalent sets can be generated by mathematical operations
performed on the taxonomic decomposition of the completed graph. Finally
some of our results are informally related to the block-modeling approach to
analyzing social network data discussed in several previous Journal of
Mathematical Psychology articles.