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. Partition of logical space.

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.