The Barcan Formula and its Converse: Early Developments

Abstract

Necessitists accept both the Barcan formula (or schema) and its converse in quantified modal logic; contingentists typically reject both formulas. Their controversial implications are explained. Both formulas first appeared in the work of Barcan Marcus, although they were informally anticipated by Ibn-Sina (Avicenna). Carnap endorsed both because they are validated by his intensional semantics for modal logic, based on a logical conception of modality, possible worlds, and individual concepts. The successes and failures of his approach are analysed. Only later did Prior bring out the contentiousness of the two formulas on a more metaphysical reading, by analogy with the corresponding formulas in quantified tense logic. The failure of his attempts to implement contingentism and temporaryism through the postulation of truth-value gaps in modal logic and tense logic manifests more general problems with non-bivalent approaches.