Proposition algebra with projective limits

Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a proposition, atomic propositions may yield different Boolean values at repeated occurrences. We introduce `free reactive valuations' to capture this dynamics of a proposition'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.
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 courser congruences are considered important as well. In addition infinite objects (that is propositions, 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.