One Form of Successive Approximation Method and Choice Problem

TL;DR

The paper develops a two-preference decision framework where internal and external preferences interact through an evaluation mapping $\psi:S\to L$ and an internal representation $\varphi:S\to M$, formalizing a successive-approximation approach that decomposes $\psi$ into monotone components using connectives to minimize reliance on external information. It proves existence theorems for approximating forms that express $\psi$ as compositions of monotone mappings, and introduces $ heta$-functions and rank concepts to refine representations. The work then connects this framework to logic by showing the boolean case yields classical propositional logic and implicative normal forms, and presents Lefebvre's choice model as an instance of approximating forms, replacing certain axioms with an ensemble-based justification via $L$-ensembles and a three-element internal order. The results offer a unified view in which logic, probabilistic ensembles, and decision processes emerge from a common principle of successive approximation over posets, with potential generalizations to broader mappings and logics beyond Lefebvre’s setup.

Abstract

A mathematical model of Subject behaviour choice is proposed. The background of the model is the concept of two preference relations determining Subject behaviour. These are an "internal" or subjective preference relation and an "external" or objective preference relation. The first (internal) preference relation is defined by some partial order on a set of states of the Subject. The second (external) preference relation on the state set is defined by a mapping from the state set to another partially ordered set. The mapping will be called evaluation mapping (function). We research the process of external preference maximization in a fashion that uses the external preference as little as possible. On the contrary, Subject may use the internal preference without any restriction. The complexity of a maximization procedure depends on the disagreement between these preferences. To solve the problem we apply some kind of the successive approximations methods. In terms of evaluation mappings this method operates on a decomposition of the mapping into a superposition of several standard operations and "easy" mappings (see the details below). Obtained in such way superpositions are called approximating forms. We construct several such forms and present two applications. One of them is concerned with a hypothetic origin of logic. The other application provides a new interpretation of the well known model of human choice by Lefebvre. The interpretation seems to suggest a justification different from the one proposed by Lefebvre himself.