Possible Worlds Semantics for Indicative and Counterfactual Conditionals? - A Formal Philosophical Inquiry into Chellas-Segerberg Semantics

Conditional logic is a sub-discipline of philosophical logic. It aims to provide an alternative account of conditionals in contrast to the traditional material implication analysis. The present thesis focuses on a specific possible worlds semantics for conditional logics, the Chellas-Segerberg (CS) semantics (Chellas, 1975; Segerberg, 1989), which has not been widely investigated, save by Nejdl (1992) and Delgrande (1987, 1988). The main thesis of this dissertation is that CS-semantics is an adequate framework for both (i) indicative and (ii) counterfactual conditionals. To argue for (i) and (ii) we, first, discuss the general need of a conditional logic approach, which goes beyond a material implication analysis. We address the difference between indicative and counterfactual conditionals. We focus, then, on two arguments brought forward by Bennett (2003) against accounts of indicative conditionals in terms of truth and falsehood, which arguably include possible worlds semantics such as CS-semantics: (a) D. Lewis’ (1976) triviality result and (b) Bennett’s (2003) Gibbardian stand-off argument, which goes back to Gibbard (1980). We, furthermore, investigate a lattice of conditional logics based on the basic proof theoretic system for CS-semantics plus 29 further axioms. This framework allows us to describe – as we shall show – a range of conditional logic systems, such as the indicative conditional logic system by Kraus, Lehmann, and Magidor (1990) and Lehmann and Magidor (1992) and the counterfactual system of D. Lewis (1973/2001). For our formal investigation we distinguish between Chellas frames (Chellas, 1975) on the one hand and Segerberg frames (Segerberg, 1989) on the other hand: While Chellas frames are generalizations of Kripke frames, Segerberg frames rather correspond to what is often called ‘general frames’. We give, then, correspondence proofs for the lattice of systems on the basis of Chellas frames and discuss the notion of trivial frame correspondence. We, then, provide a completeness result for the lattice of conditional logics for standard Segerberg frames. This type of Segerberg frames is solely based on structural conditions and is – unlike the notion of (simple) Segerberg frame completeness – not trivial in the sense that any conditional logic is complete w.r.t. some class of frames. We finally, provide an objective and a subjective interpretation of CS-semantics by drawing on the notion of alethic modality and the Ramsey-test, respectively. We, then, argue that our objective and subjective account of CS-semantics can serve as basis for indicative and counterfactual conditionals, respectively.