Abstraction Logic

A New Foundation for Reasoning, Computing, and Understanding
Download the book for free, or support Practal and pay whatever you think is fair!

This is the first book about Abstraction Logic. It presents Abstraction Logic in its most recent and comprehensive form, and supersedes all previous publications on Abstraction Logic.

With the purchase of this electronic version of the book, you also receive all future updates to this e-book. Currently, the book consists only of the first two chapters: “The Mathematical Universe”, and “Abstraction Algebra”. Chapter 3, “Abstraction Logic”, is scheduled for August 2024. Further chapters are planned to be released throughout 2024 and 2025. Note that existing chapters might change in future versions of the book.

The following material is outdated, but remains accessible here. It contains information not yet incorporated into the book and shows the evolution of Abstraction Logic.
April 1, 2023 https://arxiv.org/abs/2304.00358 Logic is Algebra
Logic really is just algebra, given one uses the right kind of algebra, and the right kind of logic. The right kind of algebra is abstraction algebra, and the right kind of logic is abstraction logic.
February 5, 2023 https://practal.com/press/aair/1 Axioms Are Inference Rules
As it turns out, Abstraction Logic's axioms should be more general than initially thought. Instead of just terms, they should be inference rules.
July 13, 2022 https://arxiv.org/abs/2207.05610 Abstraction Logic: A New Foundation for (Computer) Mathematics
This was a submission to the CICM 2022 conference.
April 10, 2022 https://doi.org/10.47757/al.crete.1 Video: Abstraction Logic (UNILOG 2022)
This is a rerecording of a talk about Abstraction Logic at the UNILOG 2022 conference in Crete.
March 23, 2022 https://doi.org/10.47757/aal.1 Automating Abstraction Logic
We describe how to automate Abstraction Logic by translating it to the TPTP THF format.
December 21, 2021 (last revised December 23, 2021) https://doi.org/10.47757/pal.2 Philosophy of Abstraction Logic
Abstraction Logic has been introduced in a previous, rather technical article. In this article we take a step back and look at Abstraction Logic from a conceptual point of view. This will make it easier to appreciate the simplicity, elegance, and pragmatism of Abstraction Logic. We will argue that Abstraction Logic is the best logic for serving as a foundation of mathematics.
October 22, 2021 (last revised November 14, 2021) https://doi.org/10.47757/abstraction.logic.2 Abstraction Logic
Abstraction Logic is introduced as a foundation for Practical Types and Practal. It combines the simplicity of first-order logic with direct support for variable binding constants called abstractions. It also allows free variables to depend on parameters, which means that first-order axiom schemata can be encoded as simple axioms. Conceptually abstraction logic is situated between first-order logic and second-order logic. It is sound with respect to an intuitive and simple algebraic semantics. Completeness holds for both intuitionistic and classical abstraction logic, and all abstraction logics in between and beyond.