Scalar Implicature

Origin. Grice, "Logic and conversation" (1975), for the maxim of Quantity and the notion of a conversational implicature; Laurence Horn's dissertation On the Semantic Properties of Logical Operators in English (UCLA, 1972), which introduced the scales; Gazdar (1979) for the first formal implementation; Levinson's Presumptive Meanings (2000) for generalized implicature as a default. The grammatical turn is Chierchia (2004), Fox (2007), and Chierchia, Fox, and Spector (2012), where exhaustification becomes a covert operator in the syntax rather than a pragmatic inference.

Models. A speaker who uses a weaker expression when a stronger one was available invites the inference that the stronger one is false. The account therefore has two moving parts: a set of alternatives to the uttered sentence, and a rule that negates the stronger members. The dispute is over where the rule lives. On the Gricean view it is reasoning about the speaker, applies to whole utterances, and is defeasible; on the grammatical view it is a silent operator EXH, adjoinable at any propositional node, and its output is part of the sentence's meaning — which is what makes implicatures under embedding a diagnostic between the two.

Formalism.

Gricean reasoning: Maxim of Quantity: be as informative as required. Speaker said "some." If "all" were true, would have said it. So: not all.

Scalar alternatives: ⟨some, most, all⟩. ⟨or, and⟩. ⟨possible, necessary⟩. ⟨warm, hot⟩. Ordered by strength.

Basic implicature: "Some students passed" +> not all passed. "John or Mary came" +> not both. Weaker asserted → stronger denied.

Exhaustification operator: EXH(S) = S ∧ ∧{¬S' : S' ∈ Alt(S), S' stronger}. Assert S, deny all stronger alternatives. Grammatical approach. Fox, Chierchia.

Embedded implicatures: "Every student read some book." Embedded: each read not all books? Debate: local vs global. Grammatical vs pragmatic.

Obligatory vs optional: Some implicatures obligatory (seem lexicalized). Others cancellable. "Some, in fact all, passed." Scalar variability.

Innocent exclusion: Don't exclude alternatives that entail asserting others. "John or Mary or both" — don't exclude "both." Formal constraint. Fox's approach.

The epistemic step (Sauerland 2004): Quantity alone yields ¬K(all): the speaker does not know that all passed. The stronger K(¬all) needs a further assumption, Competence — that the speaker is opinionated about the alternative. Without it the implicature is only an ignorance inference.

Symmetry problem: "Some but not all" is as good an alternative as "all", and excluding both is contradictory. So alternatives cannot be defined by logical strength alone. Katzir (2007): alternatives are structural — obtained by substitution and deletion on the utterance itself. Fox and Katzir (2011) for the general statement.

Free choice: "You may take an apple or a pear" +> you may take either. Bar-Lev and Fox (2020) derive it by adding innocent inclusion to innocent exclusion: after excluding what can be excluded, include what can consistently be included.

Hurford's constraint (1974): A disjunction is odd if one disjunct entails the other — *"She is French or European." "She read some or all of the books" is fine, so some must have been strengthened locally. This is the standard argument for embedded exhaustification.

Symbols.

Symbol	Unicode	Name	Meaning
∧	U+2227	Conjunction	Joins the assertion to the negated alternatives
¬	U+00AC	Negation	Exclusion of a stronger alternative
∈	U+2208	Element of	Membership in the alternative set
⟨	U+27E8	Angle bracket, left	Opens a Horn scale ⟨some, most, all⟩
⟩	U+27E9	Angle bracket, right	Closes a Horn scale
+>	—	Implicates	S +> P: S conversationally implicates P
EXH	—	Exhaustification operator	Asserts S and denies its innocently excludable alternatives
Alt(S)	—	Alternative set	The formal alternatives to S
K	—	Knowledge operator	The speaker's epistemic state, in Sauerland's derivation
S, S′	—	Sentence variables	The utterance and an alternative to it

Metatheory. Sauerland (2004) showed that Quantity reasoning by itself delivers only the weak inference that the speaker does not know that the stronger alternative holds, and that the strong inference — that the speaker knows it fails — requires a separate Competence assumption; the two are empirically distinguishable and the difference is what the ignorance readings track. Fox (2007) gave EXH a precise semantics through innocent exclusion: negate every alternative belonging to every maximal consistently negatable subset, which is what prevents the contradiction that arises when incompatible alternatives are excluded together. Bar-Lev and Fox (2020) added innocent inclusion, deriving free choice from the same operator. The symmetry problem is what forces a definition of the alternative set independent of strength, and Katzir (2007) supplied one: alternatives are what can be obtained from the utterance by substituting or deleting subconstituents, so the alternative set is fixed by structure and a lexical scale.

Applies to. Pragmatics. Scalar terms. Disjunction. Modals.

Limitations. The theory's predictions rest on the alternative set, and no independent criterion fixes it: structural definitions overgenerate and lexical scales are stipulated, so a wrong prediction can always be repaired by adjusting the scale. Scales are not uniform in strength — van Tiel and colleagues (2016) found implicature rates ranging from near-zero to near-ceiling across scalar pairs, which no theory that treats all scales alike predicts. The embedded cases that motivate the grammatical account are contested experimentally: Geurts and Pouscoulous (2009) found them largely absent under direct testing. Free choice remains a repair, since innocent inclusion is added precisely to derive it and has no independent motivation. And the grammatical view makes exhaustification freely insertable, so any node may host EXH and the theory generates readings that must be filtered by economy conditions external to it.