Tr\`es courte enqu\^ete sur l'extension non-triviale de la logique de propositions \`a la logique du premier et deuxi\`eme ordre

TL;DR

This paper explores the non-trivial extension of propositional logic to first and second-order logic, examining their expressive power, philosophical implications, and the distinctions between different logical orders.

Contribution

It provides a brief philosophical and formal reflection on the nature, differences, and implications of extending propositional logic to higher-order logics.

Findings

Second-order logic cannot be reduced to first-order logic.

Certain predicates are inherently of higher order.

Logical order relates to natural language and philosophical positions.

Abstract

The formal construction of the second-order logic or predicate calculus essentially adds quantifiers to propositional logic. Why second-order logic cannot be reduced to that of the first order? How to demonstrate that certain predicates are of higher-order? What type of order matches the natural language? Is there a philosophical position behind every logic, even for classical ones? What philosophical position for what logic in connection to its expressive power? These are the questions we ask and that we very briefly sketch as a first reflection. (paper in French)