Abstraction Logic: A New Foundation for (Computer) Mathematics

TL;DR

Abstraction logic introduces a unified, simpler foundation for mathematics that combines features of predicate and higher-order logic, enhancing expressivity and simplicity.

Contribution

It presents abstraction logic as a novel, unified foundation that maximizes simplicity and expressivity, capable of formalizing various logical systems.

Findings

Simpler terms and proof notions than other logics

Capable of formalizing both intuitionistic and classical abstraction logic

Sound and complete for a wide range of logics

Abstract

Abstraction logic is a new logic, serving as a foundation of mathematics. It combines features of both predicate logic and higher-order logic: abstraction logic can be viewed both as higher-order logic minus static types as well as predicate logic plus operators and variable binding. We argue that abstraction logic is the best foundational logic possible because it maximises both simplicity and practical expressivity. This argument is supported by the observation that abstraction logic has simpler terms and a simpler notion of proof than all other general logics. At the same time, abstraction logic can formalise both intuitionistic and classical abstraction logic, and is sound and complete for these logics and all other logics extending deduction logic with equality.