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	Meaning
W	—	set of possible worlds
[[α]]^w	—	extension at w
□	U+25A1	necessity
⟨w,t,i⟩	—	centered world

Metatheory. Intensions. World-indexed truth. Rigid designation. Foundation of intensional logic.

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

Limitations. Fine-grained meaning. Impossible worlds. Hyperintensionality. Actualism debates.