Possible Worlds Semantics

Origin. Carnap's state descriptions and intensions (Meaning and Necessity, 1947); Kripke's model theory for modal logic (1959, 1963); Hintikka's model sets (1961); Montague's intensional logic and "Universal grammar" (1970) and "PTQ" (1973); Lewis, "General semantics" (1970). Intensions as functions. Possible worlds as indices. Foundation of intensional semantics.

Models. Meaning as intension. Intension: function from worlds to extensions. Propositions as sets of worlds. Necessity as truth in all worlds. Foundation for modal and natural language semantics.

Formalism.

Extension vs intension: Extension: value at actual world. Intension: function from worlds to extensions. [[α]]^w: extension at w. [[α]]: intension (function).

Proposition: [[S]]: W → {0,1}. Set of worlds where S true. Proposition = set of worlds. Possible worlds proposition.

Individual concept: [[α]]: W → D. Function from worlds to individuals. "The president": different in different worlds. Rigid vs non-rigid.

Property: [[P]]: W → ℘(D). Function from worlds to sets. Intensional predicate. Extension varies.

Necessity: □φ true at w iff φ true at all accessible w'. Quantification over worlds. Different accessibility = different modality. Kripke frames.

Rigid designators: Same referent in all worlds. Proper names (Kripke). [[Hesperus]] = [[Phosphorus]] in all worlds. Identity across worlds.

Centered worlds: ⟨w, t, i⟩: world, time, individual. For indexicals. "I", "now", "here". Kaplan's contexts.

Symbols.

Symbol	Unicode	Name	Meaning
□	U+25A1	White square	Necessity: truth at every accessible world
℘	U+2118	Weierstrass p	Power set, as in the type W → ℘(D) of a property
→	U+2192	Function arrow	The type of an intension
α	U+03B1	Alpha	Metavariable over expressions
φ	U+03C6	Phi	Sentence variable
⟨	U+27E8	Angle bracket, left	Opens a centered world ⟨w, t, i⟩
⟩	U+27E9	Angle bracket, right	Closes a centered world
W	—	Set of worlds	The index set over which intensions are defined
D	—	Domain	The individuals
w, t, i	—	World, time, agent	Coordinates of an index or context
S	—	Sentence	Its intension is a proposition
P	—	Predicate	Its intension is a property
[[α]]	—	Intension	The function from worlds to extensions
[[α]]^w	—	Extension at w	The value of the intension at a world

Metatheory. Propositions taken as sets of worlds form a complete Boolean algebra under intersection, union, and complement, so entailment is inclusion and the logic of propositions is classical by construction. On the modal side, Kripke's relational semantics makes frame conditions correspond to axioms — reflexivity to T, transitivity to 4, symmetry to B — and Sahlqvist (1975) delimited a syntactic class of axioms for which the correspondence is uniform and completeness is automatic. The correspondence is not exhaustive: Thomason (1974) and Fine (1974) exhibited consistent normal modal logics complete for no class of Kripke frames at all, which is why general frames and algebraic duality are needed. For natural language, Montague's intensional logic reduces to a two-sorted type theory over worlds and individuals, so intensional constructions are ordinary functional application at a higher type. Kaplan (1989) separated character from content, which is what lets "I am here now" be true at every context while expressing a contingent proposition at each.

Applies to. Modal logic. Natural language semantics. Philosophy of language. Metaphysics.

Limitations. Identifying a proposition with a set of worlds makes all necessary truths one proposition and all contradictions another, so the framework cannot distinguish believing that 7 is prime from believing every arithmetical theorem — logical omniscience is a theorem of the model, not a defect of the modelling. Every counterpossible comes out vacuously true, so the conditionals that matter in mathematics and metaphysics are all equivalent. Substitution of necessary equivalents is validated everywhere, including in attitude and explanation contexts where it plainly fails, which is what forces the move to structured propositions or impossible worlds. The quantified extension is not neutral: constant domains validate the Barcan formula and its converse, so a choice of frame settles a metaphysical question about what could have existed. And the analysis is not reductive, since necessity is explained by quantification over worlds while possible world remains a modal notion.