Practal — Practical Logic

Recursive teXt (RX) is a new general-purpose text format.

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.

As it turns out, Abstraction Logic's axioms should be more general than initially thought. Instead of just terms, they should be inference rules.

The first release of Practal as a Visual Studio Code Extension is out! It is an early and experimental release without its most crucial functionality of proving and computing, but it has syntax highlighting and the ability to define your own syntax. It is already possible to state declarations, definitions and axioms, and definitions are properly checked.

This was a submission to the CICM 2022 conference.

This is a rerecording of a talk about Abstraction Logic at the UNILOG 2022 conference in Crete.

This is a short note about combining indentation-sensitivity and pyramid grammars. We describe the syntax of indentation-sensitive pyramid grammars by one such grammar.

We describe how to automate Abstraction Logic by translating it to the TPTP THF format.

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.

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.

These are preliminary notes about Practal’s logic and its system of practical types. Practal Light is a research prototype implementing (part of) what is presented in this document.

This is a first sketch of the design of Practal, an interactive theorem proving system for practical logic.

A revised semantics for Local Lexing is presented that entirely avoids priority conflicts between non-empty tokens. Pyramid Grammars are introduced, and their semantics is given in terms of Local Lexing. It is demonstrated that they are more expressive than Context-Free Grammars, and at least as expressive as Parsing Expression Grammars.