Negated Antonyms: Creating and Filling the Gap

Krifka accounts for antonym quadruplets such as happy, not happy, unhappy, not unhappy, and in particular for the double negative, which reports a mild state of happiness rather than the middle ground that Horn's contrary analysis of antonyms predicts. The border between an antonym pair is sharp but its location is not fixed, as on Williamson's epistemic view of vagueness; the pair exhausts its scale, so happy and unhappy are literally contradictories; and the M principle reserves the more complex of two equivalent expressions for the non-stereotypical cases. Speakers use the simple forms only where every admissible border agrees, which opens the gap, and the complex forms where the simple form is true at the speaker's border but not safely usable, which fills it. The M principle is derived within Blutner's bidirectional optimality theory on McCawley's kill example, and the same evaluation over the quadruplet yields Krifka's assignment.

Implementation notes

• Literal meanings are the substrate's single-threshold AntonymForm.contradictoryDenotation; the admissible borders form a set of thresholds on a linear order, and the diagrams are read relative to a speaker's border within that set.
• Bidirectional evaluation is the substrate's weak optimality, (14), which compares two pairs only when they share a form or an interpretation, by the preference ordering on the other: the product of the two orderings, with no ranking between them. The quadruplet game is built from the literal semantics on the four regions ordered along the scale, with the border between the two middle ones, form complexity (AntonymForm.complexity) and the markedness of the border regions.
• The other uses of double negatives the paper sets aside (denial, irony, amplification) and the local strengthening of neither happy nor unhappy are not formalized.

References

• Krifka 2007b, Negated Antonyms: Creating and Filling the Gap
• Horn 1989, A Natural History of Negation
• Horn 1984, Toward a New Taxonomy for Pragmatic Inference: Q-based and R-based Implicature
• Williamson 1994, Vagueness
• Levinson 2000, Presumptive meanings: The theory of generalized conversational implicature
• Blutner 2000, Some Aspects of Optimality in Natural Language Interpretation formalized
• McCawley 1978, Conversational Implicature and the Lexicon