OT syntax explores the ways languages generate optimal forms for a particular input meaning. OT semantics maps a given input form onto its optimal interpretation. Bidirectional OT combines the two, and evaluates pairs of forms and meanings. Bidirectionality turns out to be an interesting perspective to study cross-linguistic variation in meaning. We exemplify this with the issue of negation. In some languages, the combination of two negative indefinites leads to a double negation meaning, as in the English ‘Nobody said nothing’. In other languages, the combination of two expressions that look like negative indefinites leads to a single negation meaning, as in the Spanish ‘Nadie maraba a nadie’. This phenomenon is known as negative concord. Because of this contrast, nothing has been labelled as a negative quantifier, and nadie as a so-called n-word. But the difference between negative quantifiers and n-words can only be detected in context: in isolation both types of expressions mean ‘nothing’.

We assume here that the contextual difference in meaning requires a grammatical, rather than a lexical explanation. More specifically, we argue that double negation languages and negative concord languages involve the same constraints in production and interpretation, but rank them differently in a bidirectional grammar. N-words reflect the marking of arguments in the scope of negation (production oriented). Negative quantifiers reflect a fully compositional interpretation (interpretation oriented). A typology of negation is the result of this setup of the syntax-semantics interface.

More variation in forms arises out of the interaction of negative indefinites with sentential negation. But in general, this does not affect the semantics, as a result of the constraint ranking in concord languages. There is one interesting exception, namely marked instances of double negation readings in negative concord languages. We show that these interpretations can be accounted for with Blutner’s notion of weak bi-directional optimality (or ‘superoptimality’).

The approach developed here establishes a clear distinction between syntactic and semantic questions, but links them in the grammar. The expression and interpretation of negation in natural language is thus an example of the explanatory force of bi-directional OT.