Proposition Algebra with Projective Limits

TL;DR

This paper introduces proposition algebra with projective limits, capturing the dynamic evaluation of sequential propositional logic through free valuations and algebraic specifications, extending classical Boolean algebra to handle infinite objects.

Contribution

It develops a new algebraic framework for sequential propositional logic using free valuations and inverse limits, generalizing Boolean algebra and enabling analysis of infinite propositional objects.

Findings

Proposition algebra captures dynamic evaluation in sequential logic.

Inverse limit construction connects finite and infinite propositional objects.

Algebraic specification provides a foundation for reasoning about sequential propositions.

Abstract

Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a propositional statement,atomic propositions may yield different Boolean values at repeated occurrences. We introduce `free valuations' to capture this dynamics of a propositional statement's environment. The resulting logic is phrased as an equationally specified algebra rather than in the form of proof rules, and is named `proposition algebra'. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The four axioms for free valuation congruence are then combined with other axioms in order define a few more valuation congruences that gradually identify more propositional statements, up to static valuation congruence (which is the setting of conventional propositional logic). Proposition algebra is developed in a fashion similar to the process algebra ACP and the program algebra PGA, via an algebraic specification which has a meaningful initial algebra for which a range of coarser congruences are considered important as well. In addition infinite objects (that is propositional statements, processes and programs respectively) are dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts about infinite ones while reducing all facts about infinite objects to an infinity of facts about finite ones in return.