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	Name	Meaning
°	U+00B0	Degree sign	Superscript marking the ordinary value
α	U+03B1	Alpha	Metavariable over expressions
λ	U+03BB	Lambda	Abstraction operator, which pointwise composition resists
∈	U+2208	Element of	Membership in an alternative set
∀	U+2200	Universal quantifier	Exhaustivity clause of only
¬	U+00AC	Negation	Exclusion of non-focused alternatives
∧	U+2227	Conjunction	Prejacent plus exclusion
≠	U+2260	Not equal to	Distinctness from the focus value
~	—	Squiggle	Rooth's focus-interpretation operator
[[α]]°	—	Ordinary semantic value	Standard denotation
[[α]]^A	—	Alternative semantic value	Set of alternatives, also written [[α]]^f
_F	—	F-marking	Syntactic focus feature
D_e	—	Domain of individuals	Source of the substituted alternatives

Metatheory. The two tiers are linked by two facts that hold throughout Rooth's system: the ordinary value is always a member of the alternative value, and the alternative value is the singleton {[[α]]°} exactly when α contains no F-marking, so the ordinary semantics is the F-free special case. Rooth, "A theory of focus interpretation" (Natural Language Semantics, 1992), makes focus interpretation presuppositional — ~ introduces a free variable over a contextually given set required to be a subset of the alternative value — which is what stops alternatives propagating without bound and localizes association. Kratzer and Shimoyama (2002) showed that pointwise Hamblin composition derives interrogative, indefinite, free-choice, and polarity readings of Japanese indeterminate pronouns from one lexical entry plus different operators. Von Stechow's comparison with structured meanings establishes an asymmetry: a background–focus pair determines the alternative set, but the set does not recover the pair. Shan (2004) showed that pointwise composition and variable binding are incompatible, which is why the alternatives must be recast monadically (Charlow) or the semantics made variable-free.

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

Limitations. Pointwise application does not compose with abstraction: a λ-binder cannot reach a variable inside an alternative set without collapsing the set, so binding into alternatives is unavailable in the plain system, and exceptional indefinite scope out of islands is not derivable from pointwise propagation alone. Because the alternative set forgets which constituent was F-marked, distinct focus structures can share a set, and multiple focus, second-occurrence focus, and contrastive topic need either the structured-meaning representation or additional indices. Where ~ attaches is not fixed by the syntax, so focus projection is stipulated rather than predicted. Hamblin sets encode neither exhaustivity nor the existence and uniqueness presuppositions of questions, which is what motivated Karttunen's true-answer denotations and Groenendijk and Stokhof's partitions. And the restriction to contextually relevant alternatives sits in a free variable, so truth conditions for only are only as constrained as the resolution of that variable.