Miguel Lopez Munive: 'Possible-words semantics for a partial necessity predicate'
Chair: Noah Waldschmidt
Abstract: Montague famously showed that the predicate version of the modal logic system T is inconsistent when combined with certain systems of arithmetic. Halbach et al. (2003, 2005) have shown, however, that it is actually possible to provide an operator-like possible-world semantics for necessity understood as a predicate if one restricts the definition of a possible-worlds model so as to avoid the frames (pairs of worlds and accessibility relation) that give rise to inconsistencies. In this talk I aim to clarify and further investigate the properties of this semantic framework when the underlying valuation scheme is non-classical. In particular, I define possible-world models for the language with the necessity predicate using Strong-Kleene valuation scheme and a construction which mirrors Kripke’s fixed point semantics for the truth predicate. Subsequently, I prove that in this semantics, and in contrast to the classical case, there is no need to restrict the definition of a PW-model: all frames admit PW-models. I briefly compare this framework to Kripke’s truth theory and finally show that it is adequate with respect to standard propositional modal logic, i.e. if we consider the same language, we obtain the exact same logical truths.
