Individual-actualism and three-valued modal logics. I: Model-theoretic semantics (Q1092031)

Roughly speaking, an interpretation of a (first-order) modal language is `individual-actualist' if it refuses to countenance non-actual possible individuals; i.e. only actual individuals can be values for variables or designated by singular terms. The simplest ways to construe this requirement within a model-theoretic (possible world) semantics for modal languages would, however, validate such apparent falsehoods as `Everything necessarily exists'. A way to avoid this consequence is to define a three-valued semantics, or better, a semantics with truth-value gaps, whereby denotation functions and assignments of values to variables are not everywhere defined, and accordingly satisfaction and truth are also not everywhere defined. This paper presents the full machinery to make these notions formally precise; many variations on actualist and possibilist model-theoretic semantics for modal language are defined and compared. This is only the first part of a longer work.