Scott took up a post as assistant professor of mathematics, back at the
University of California, Berkeley, and involved himself with classical issues in
mathematical logic, especially
set theory and Tarskian
model theory. He proved that the
axiom of constructibility is incompatible with the existence of a
measurable cardinal, a result considered
seminal in the evolution of set theory. During this period he started supervising Ph.D. students, such as James Halpern (
Contributions to the Study of the Independence of the Axiom of Choice) and Edgar Lopez-Escobar (
Infinitely Long Formulas with Countable Quantifier Degrees).
Modal and tense logic Scott also began working on
modal logic in this period, beginning a collaboration with
John Lemmon, who moved to
Claremont, California, in 1963. Scott was especially interested in
Arthur Prior's approach to
tense logic and the connection to the treatment of time in natural-language semantics, and began collaborating with
Richard Montague (Copeland 2004), whom he had known from his days as an undergraduate at Berkeley. Later, Scott and Montague independently discovered an important generalisation of
Kripke semantics for modal and tense logic, called
Scott-Montague semantics (Scott 1970). John Lemmon and Scott began work on a modal-logic textbook that was interrupted by Lemmon's death in 1966. Scott circulated the incomplete monograph amongst colleagues, introducing a number of important techniques in the semantics of model theory, most importantly presenting a refinement of the
canonical model that became standard, and introducing the technique of constructing models through
filtrations, both of which are core concepts in modern Kripke semantics (Blackburn, de Rijke, and Venema, 2001). Scott eventually published the work as
An Introduction to Modal Logic (Lemmon & Scott, 1977). ==Stanford, Amsterdam and Princeton, 1963–1972==