Impossible Worlds Logic

Origin. Routley's non-classical worlds (1980) and Rescher–Brandom (1979); Priest's development (1997, 2005); Nolan's defence of impossible worlds (1997); Berto and Jago's Impossible Worlds (2019) is the standard treatment. Extension of possible worlds. Worlds where contradictions hold. Non-trivial impossible reasoning. Hyperintensional semantics.

Models. Some worlds violate logic. Contradictions true at impossible worlds. Counterpossible conditionals meaningful. Distinguish logically equivalent content. Beyond possible worlds.

Formalism.

World types: W = Wₚ ∪ Wᵢ Possible worlds: classically behaved. Impossible worlds: logic may fail. Both in semantics.

Impossible worlds features: At some w: A ∧ ¬A true. At some w: B, B → C true, but C false. Logic "local" not global. Anything goes (at impossible worlds).

Counterpossibles: "If 2+2 were 5, then 2+2+1 would be 6." Classical: vacuously true. Impossible worlds: non-trivially true/false. Meaningful counterfactual reasoning.

Hyperintensionality: A, B necessarily equivalent, yet: "Believes A" ≠ "Believes B" Impossible worlds distinguish them. Fine-grained content.

Ternary accessibility: Routley-Meyer semantics. Rabc: from a, via b, access c. Impossible worlds: b may be impossible. Models relevant implication.

Applications: Belief about impossibilities. Fiction with impossible content. Counterpossible reasoning. Fine-grained propositions.

Realist vs instrumentalist: Are impossible worlds real? Or just semantic tools? Meinongianism: they exist. Ersatzism: they're representations.

Symbols.

Metatheory. Rantala (1982) gave the first systematic construction: a Kripke model is augmented with non-normal worlds whose valuation assigns truth values to all formulas directly rather than compositionally, so the logic of the epistemic operator becomes the logic of an arbitrary predicate — every closure principle fails, which is exactly why the construction blocks logical omniscience and exactly why it says nothing about which inferences an agent does make. Routley and Meyer's ternary-relation semantics (1972–73) puts impossible worlds in the same indispensable role, yielding completeness for the relevant logic R with the Routley star supplying the truth condition for negation. In counterfactual semantics the addition is precisely calibrated: extending the similarity ordering over impossible worlds keeps A □→ A valid and leaves the logic of possible antecedents untouched while making counterpossibles non-vacuous, at the cost of substitution of classically equivalent antecedents. Bjerring (2013) argued that the trade-off is forced — an impossible-worlds model of a non-omniscient agent closes belief under all trivially valid inferences or under none, with no principled position between.

Applies to. Modal logic. Philosophy of mind. Fiction logic. Metaphysics.

Limitations. Unrestricted impossible worlds trivialize the semantics: if every set of sentences counts as a world, the truth set of a formula is just the collection of sentence-sets containing it, and the account is a relabelling of a purely syntactic one that delivers no logic. Restriction is therefore mandatory, and no principled restriction is known — Jago's bounded-worlds problem is that any line drawn between the inferences a world must respect and those it may violate is stipulated rather than derived. Because impossible worlds are closed under nothing, none of the standard machinery survives: bisimulation invariance, filtration and the standard translation have no analogue, and completeness theorems constrain only the possible part of a model. Hyperintensionality is obtained by fiat rather than explained, since the distinction between necessarily equivalent propositions has no compositional source, and Nolan's strangeness-of-impossibility condition is a stipulation of the same kind.

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