Modal Systems and Possible Worlds

A Semantics With Consequences

Possible-worlds semantics gave modal logic what it had lacked: a precise account of what modal claims mean, rather than a list of axioms. The payoff was immediate, questions about which axioms to adopt became questions about the structure of the accessibility relation, and those could be settled mathematically.

A frame is a set of worlds plus an accessibility relation; a model adds a valuation saying which atoms hold at each world. The correspondence results are exact: an axiom is valid on a frame precisely when the accessibility relation has the matching property. Reflexivity corresponds to $\Box P \rightarrow P$, transitivity to $\Box P \rightarrow \Box\Box P$, symmetry to $P \rightarrow \Box\Diamond P$.

S5 is the strongest of the standard systems, with accessibility an equivalence relation. Its distinctive feature is that iterated modalities collapse: $\Box\Diamond\Box P$ reduces to $\Box P$, and any string of operators reduces to its last one. This makes S5 attractive for logical necessity, where whether something is necessary should not itself be a contingent matter.

The framework extends well beyond alethic modality. Temporal logic reads the operators over times, epistemic logic over epistemic alternatives, deontic logic over ideal worlds, and counterfactuals — "if the match had been struck, it would have lit" — are analysed as claims about the closest worlds where the antecedent holds, which is why they are not material conditionals.

The metaphysical question is what possible worlds are. David Lewis defended modal realism: they are concrete, as real as the actual world, and "actual" is merely indexical, meaning "this one", exactly as "here" means wherever the speaker is. The theory is elegant and almost universally rejected as extravagant. The usual alternative is ersatzism: worlds are abstract objects — maximal consistent sets of sentences, or properties the universe might have had — which keeps the semantics while avoiding an infinity of concrete universes.

The disagreement matters because the semantics only explains modality if worlds are something we understand better than modality itself. If a possible world just is "a way things could have been", the analysis is circular, and that objection has never been fully answered.

Common pitfall: treating possible worlds as distant planets. They are not far away; they are not anywhere. Even on Lewis's realism, worlds are spatiotemporally isolated from one another, you cannot travel to one, and nothing in one can causally affect another.