Alternative Semantics

Origin. Hamblin (1973), Rooth (1985). Sets of alternatives. Questions as alternative sets. Focus interpretation. Foundation of question semantics.

Models. Expressions denote sets of alternatives. Questions: set of possible answers. Focus: introduces alternatives. Composition over sets.

Formalism.

Ordinary vs alternative denotation: [[α]]° : ordinary semantic value. [[α]]^A : alternative semantic value (set). Two-tiered interpretation.

Questions (Hamblin): [[who came]]^A = {that x came : x ∈ D_e}. Set of propositions. Possible answers. Not a partition: these propositions overlap, since "that Ann came" and "that Bob came" are jointly true when both came, and they need not cover the space. Carving logical space into mutually exclusive exhaustive answers is Groenendijk and Stokhof's partition semantics, a rival account rather than a gloss on this one.

Focus (Rooth): [[John]_F ate]]^A = {that x ate : x ∈ D_e}. Focused element generates alternatives. Set of alternatives. F-marking.

Composition: [[α β]]^A = {f(a) : f ∈ [[α]]^A, a ∈ [[β]]^A}. Pointwise function application. Alternatives propagate. Hamblin semantics.

Focus sensitive operators: only, even, also. Quantify over alternatives. [[only [John]_F came]] = John came ∧ ∀x[x≠j → ¬came(x)]. Alternatives matter for truth.

Association with focus: "John only [ate]_F the beans." vs "John only ate [the beans]_F." Different alternatives. Different truth conditions.

Squiggle (~): Rooth's ~ operator. Introduces alternative requirement. Focus antecedent. Anaphoric to alternatives.

Symbols.

Symbol	Unicode	Meaning
[[α]]°	—	ordinary value
[[α]]^A	—	alternative value
_F	—	focus marking
~	—	squiggle

Metatheory. Alternative sets. Focus. Questions. Two-dimensional.

Applies to. Question semantics. Focus. Only/even. Information structure.

Limitations. Two-tiered complexity. Alternative set proliferation. Focus projection. Theory variants.