Proto-semantics for positive free logic

Journal of Philosophical Logic 29 (3):277-294 (2000)
Abstract
This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of "non-existing" objects. The models here introduced have only one (possibly empty) domain, and a partial reference function for the singular terms (that might be undefined at some arguments). Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non unlike the notion of "truth at world" found in modal logic. A model theory is developed, establishing compactness, interpolation (implying a strong form of Beth definability), and completeness (with respect to a particular axiomatization)
Keywords free logic  existence  denotation  semantics  Craig interpolation  Beth definability
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Jason Turner (2005). Strong and Weak Possibility. Philosophical Studies 125 (2):191 - 217.
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