Linear Logical Economics: Proof-Theoretic Foundations of Resource Consumption

Title: Linear Logical Economics: Proof-Theoretic Foundations of Resource Consumption Abstract: This paper reconstructs the foundations of neoclassical microeconomics using Jean-Yves Girard's Linear Logic (1987), addressing the fundamental mismatch between classical logic—where knowledge is non-consumable via structural rules of contraction and weakening—and economic reality, where resources are consumed. By adopting a resource-sensitive logical framework, we provide a unified proof-theoretic formulation for consumer theory, producer theory, general equilibrium, and game theory. Key economic concepts are reinterpreted as logical connectives: the multiplicative tensor (⊗) as asset portfolios, linear implication () as market transactions, additive internal choice (&) as opportunity costs, and additive external choice (⊕) as uncertainty. We demonstrate that utility maximization and cost minimization correspond to proof search algorithms, and that market equilibrium is equivalent to the normalization of proof nets (Cut-Elimination). This approach establishes "resource consumption" not as an external assumption, but as a syntactic necessity of the economic logic itself.