Last Order Logic

An experimental logical language.

Based on paper Last Order Logic.

Motivation

In First Order Logic, the truth values of quantified expressions depend on evaluation. This means that an automated theorem prover must annotate expressions with their truth values in order to operate efficiently under modifications to the source. The user of the language has no direct access to this information.

Last Order Logic bridges the gap between usability and automated theorem proving.

• Increased readability and improved communication
• Efficient reuse of truth values
• Extensible to higher dimensional truth values

For example:

∀ x { ... } - It is not easy to see whether this is true or false.

With other words, First Order Logic is not computationally progressive.

Last Order Logic fixes this problem by having quantified expressions evaluate to themselves, while the truth value is encoded in the type.

∀ x { ... } : un(1) - It is easy to see this is true.

Types are used to communicate intentions of programs. Last Order Logic uses this feature to increase readability.

The un(..) syntax stands for "uniform" which is un(1) for ∀ and un(0) for ∃. Correspondingly, nu(..) stands for "non-uniform" which is nu(1) for ∃ and nu(0) for ∀.

Another reason is to express truth over paths, e.g. un(0 ~= 1). These are higher dimensional truth values, not expressible in First Order Logic.

The distinction between uniform and non-uniform sense of truth comes from the theory of Avatar Extensions. Only non-uniform truth has a meaningful example that shows its truth value.