A Formalization of Austrian Economics. Praxeological Foundations: The Base System and Its Derived Theorems

2026-06-15

AI Summary

This study presents the first axiomatization of Misesian praxeology using many-sorted first-order logic, constructing a formal system comprising five sorts and six primitive relations to capture purposeful action, revealed preference, and scarcity across three hierarchical layers. Implemented in the Lean theorem prover through type class encoding and model construction, the framework rigorously derives core propositions—including the asymmetry of revealed preference, the existence and subjectivity of opportunity cost, time-structured scarcity, diminishing marginal utility, and increasing marginal disutility of labor. The work not only explicates the implicit assumptions underlying classical results such as the law of diminishing marginal utility but also clarifies their logical dependencies through formal methods, thereby verifying the internal consistency of the theoretical system.

Abstract

This paper presents an axiomatization of Ludwig von Mises' praxeology in many-sorted first-order logic, isolating the foundational layer. We introduce a formal language with five sorts ({\sf Actors}, {\sf Actions}, {\sf Ends}, {\sf Things}, {\sf Times}) and six primitive relations ({\em Acts}, {\em Avail}, {\em EndOf}, {\em Use}, a preference order, and a time order), together with a base axiom system organised into three layers: the structure of action itself, the actor's preference order together with its revelation in choice, and material scarcity. The base system captures purposeful action in its bare praxeological form. Working entirely within the base system we derive the core classical Misesian propositions as Hilbert-style theorems: the asymmetry of revealed preference, the existence of opportunity cost, the structural scarcity of time, the subjectivity of opportunity cost, the law of diminishing marginal utility, and the increasing marginal disutility of labour. Where a theorem requires structure beyond the praxeological core -- as with diminishing marginal utility -- the additional premises are made explicit; identifying these hidden premises is one of the methodological payoffs of the approach. A self-contained {\em Lean} companion encodes the language as {\em Lean} type classes and constructs concrete models -- a three-period Robinson Crusoe economy and its infinite-time extension -- whose acceptance by the type-checker is a constructive consistency proof of the full base theory.

Problem

• praxeology
• axiomatization
• Austrian economics
• formal logic
• foundations of economics

Innovation

• praxeology
• axiomatization
• many-sorted first-order logic
• formal verification
• Lean