Arrow-Sen theory simplified

Abstract

The traditional Arrow--Sen Social Choice Theory $\bf{TSCT}$ is a mathematical theory built apparently on higher--order formal language. In this paper, we propose a reformulation and reclassification of the $\bf{TSCT}$ axioms in order to obtain a simpler theory based on the first--order language axioms, keeping the spirit of original ideas. This new theory, called Simplified Social Choice Theory, denoted by $\bf{SSCT}$, presents a sub--theory of $\bf{TSCT}$. Roughly speaking, we extract all quatifications over $n$--tuples of binary relations from the axioms of $\bf{TSCT}$ and move them to the meta--level obtaining a sub--theory $\bf{SSCT}$ of $\bf{TSCT}$. More accurately, we assign to each traditional higher--order axiom $\bf{TA}$ its simplified first--order version $\bf{SA}$ such that $\bf{TA}\vdash\bf{SA}$, i.e. $\bf{SA}$ can be logically derived from $\bf{TA}$. In this way we define a simpler and more accessible set of principles that provide a context in which we can prove many propositions analogous to well--known theorems including the Arrow's impossibility of Paretian non--dictatorship and Sen's impossibility of Paretian liberal. These simplifications are the result of decades of lecturing by the author with the aim of bridging the barriers between this beautiful complex theory and his students.