Reconstructing Propositional Calculus in Database Semantics

Propositional calculus is reconstructed as a simplified version of natural language used for communication. This requires a number of changes with respect to the traditional approach, First, the method must be changed from the meta-language definitions of old logic to a declarative specification for computer software. Second, the ‘realist ontology ’ of old logic, treating meaning as a direct relation between sentences and the ‘world ’ (defined as a set-theoretic model) must be replaced by a functional system comprising an agent’s cognitive operations.1 Third, the syntax of propositional calculus must be adapted to the time-linear nature of language. Fourth, the truth conditional semantics of propositional calculus must be integrated into the time-linear process of communication. The latter task requires solutions to the following problems: First, whether an expres-sion like p & p̃... is contradictory cannot be decided in a time-linear interpretation until the continuation, e.g., p & p ̃ _ q..., is known; second, using truth conditions for check-ing consistency must avoid the SAT problem, which is known to be computationally intractable. The reconstruction is presented as the declarative specification of a computer pro-gram, comprising (i) a data structure for storing formulas of propositional calculus as concatenated propositions, and (ii) LA-grammars for conceptualization, production, and interpretation. Written at a high level of abstraction, the reconstruction provides a com-parison between the metalanguage-based handling of truth in traditional logic and the procedural model of communication in Database Semantics.