Some Theorems about a “Tree” System of Deontic Tense Logic

The purpose of this paper is to develop and to investigate some properties of a system of so called deontic tense logic, i.e. a logic which combines tense operators, or temporal modalities, with operators expressing obligation and permission. See sections 3–4 below. We call the proposed system DARB, where “D” suggests “deontic” and “ARB” the Latin arbor (meaning “tree”). DARB is interpreted semantically or model-theoretically by means of certain set-theoretical structures successively built up in a chain: “integral temporal frames”, “trees based on integral temporal frames”, “enriched trees” and “DARB-models”; naturally, this leads up to a well defined conception of validity in the system DARB. See sections 5–9 below. The DARB-models are closely related to the models used by Brian F. Chellas (in his work on imperatives) in Chellas (1969) ch. IV sect. 3 and in Chellas (1971) sect. 6 as well as to the models used in Aqvist’s work (1977) on action sentences.

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