Two-Dimensional Semantics

Origin. Kaplan (1977), Stalnaker, Chalmers. Context and circumstance. A priori vs necessary. Primary and secondary intensions. Foundation for epistemology of modality.

Models. Expressions evaluated at two indices. Character maps a context to a content; the content maps a circumstance to a truth value. Matrix semantics.

Formalism.

Two indices: c: context (world of utterance) w: circumstance (world of evaluation) ⟦φ⟧(c, w): value at context c, world w.

Diagonal (primary intension): λc. ⟦φ⟧(c, c) Kaplan's character is λc. λw. ⟦φ⟧(c, w); the diagonal collapses the two indices. A priori necessity.

Content (secondary intension): λw. ⟦φ⟧(c*, w) Fix actual context c*. Metaphysical necessity.

Matrix representation: w₁ w₂ w₃ c₁ = w₁ T F F c₂ = w₂ F T F c₃ = w₃ F F T

A priori: Diagonal all true. True in every context at that context. e.g., "I am here now."

Necessary: Row (for actual context) all true. True at every world of evaluation. e.g., "Water is H₂O."

Contingent a priori: Diagonal true, some row false. "The meter stick is one meter."

Necessary a posteriori: Diagonal not all true, row true. "Water is H₂O" (if water existed). Discovered, not knowable a priori.

Indexicals: "I": speaker at context. "Now": time at context. Content varies with context.

Symbols.

Symbol	Unicode	Name	Meaning
c	—	Context	World of utterance, the first index
w	—	Circumstance	World of evaluation, the second index
c*	—	Actual context	The context actually occupied
λ	U+03BB	Greek small lambda	Abstraction over an index
λc	—	Diagonal	Primary intension: collapse the two indices
λw	—	Content	Secondary intension: fix the context
⟦ ⟧	U+27E6, U+27E7	Mathematical white square brackets	Semantic value of an expression
@	—	Actually	Evaluates at the context's own world
□	U+25A1	White square / Box	Necessity: along a row
◇	U+25C7	White diamond	Possibility
F@	—	Fixedly actually	Davies–Humberstone operator: @ holding across all contexts
↔	U+2194	Left right arrow	Material biconditional, as in φ ↔ @φ
₁, ₂, ₃	U+2081, U+2082, U+2083	Subscript digits	Indices of contexts and circumstances in the matrix
T, F	—	Truth values	Entries of the two-dimensional matrix

Metatheory. The formal content of the framework is that validity and necessity come apart on the diagonal. "I am here now" and the schema φ ↔ @φ are true at every context taken at its own world, hence valid, yet false at some context-circumstance pair, hence not necessary; so diagonal consequence is not closed under the rule of necessitation, and the two-dimensional setting is where that failure is exhibited rather than assumed. Crossley and Humberstone (1977) drew the corresponding distinction between real-world and general validity for the logic of the actuality operator, and Davies and Humberstone (1980) added the "fixedly actually" operator F@, which is not definable from □ and @ alone and which they used to make the contingent a priori a matter of logical form. Segerberg (1973) gave a complete axiomatization of two-dimensional modal logic. The actuality operator is likewise not definable in the basic modal language, since no formula built from □ alone is sensitive to which world is the point of utterance.

Applies to. Philosophy of language. Epistemology. Consciousness (Chalmers). Indexicality.

Limitations. Primary intensions are not compositional: the diagonal of a compound is fixed by the full matrix of its parts, not by their diagonals, so two expressions with the same primary intension can be substituted into a context and change its primary intension. The failure of necessitation means the logic of diagonal validity is not normal, so the standard modal metatheory — canonicity, Sahlqvist correspondence, the usual completeness proofs — does not transfer to it without modification. Kaplan's prohibition on monsters, operators that shift the context index, is a stipulation about English rather than a constraint the semantics imposes, and languages with context-shifting operators falsify it. And the epistemic reading requires that every epistemic possibility correspond to a scenario describable in semantically neutral vocabulary; that assumption carries the philosophical weight of the framework and is not delivered by the formalism.