Counterpart Semantics for Quantified Modal Logic

In this paper we deal with the semantics for quantified modal logic, QML in short, and their philosophical relevance. In the first part we introduce Kripke semantics for the first-order modal language L= with identity, then we consider some unsatisfactory features of this account from an actualist point of view. In addition, we show that the calculus QE .K + BF on free logic, with the Barcan formula is incomplete for this interpretation. In the second part of the paper we present counterpart semantics, as defined in (Brauner & Ghilardi, 2007; Corsi, 2001). We show that it faithfully formalizes Actualism, encompasses Kripke semantics, while analysing the modal properties of individuals in a more refined way. Quantified modal logic has always had a strong philosophical appeal, since it first appeared in papers by Barcan Marcus (Barcan, 1946a; 1946b; 1947), Hintikka (Hintikka, 1961; 1969), Prior (Prior, 1956; 1957; 1968) and Kripke (Kripke, 1959; 1963a; 1963b). Besides the topics of propositional modal logic – necessity and possibility, individual knowledge, obligations and permissions, programs and computations – quantified modal logic especially focuses on individuals: we can talk about actual and possible objects, the existence and the modal properties of individuals, as well as counterfactual situations. In the philosophy of QML we find dramatically relevant issues such as Actualism/Possibilism, realism about possible worlds, trans-world identity of individuals1. It is clear that the formal development of quantified modal logic will provide an useful tool to precisely define the concepts above.