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Formal Models in the Study of Language pp 143–179Cite as

The Singular Square: Contrariety and Double Negation from Aristotle to Homer

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Abstract

While the traditional Square of Opposition was based on Aristotle’s logic, its first appearance postdates the Stagyrite by five centuries. In the Prior Analytics I, Aristotle himself lays out a somewhat different square, which I dub the Singular Square, to formalize his treatment of the interrelation of singular statements (it’s good, it isn’t good, it’s not-good, it isn’t not-good). Like the more familiar square, the Singular Square is based on the distinction between contradictory and contrary opposition. This paper focuses on the role of the Singular Square as a device for unmasking the conspiracy of MaxContrary, the natural language tendency for a formal contradictory (apparent wide-scope) negation ¬p to strengthen to a contrary of p in a variety of syntactic and lexical contexts. This conspiracy extends from the non-compositional narrow-scope readings of negation interacting with bare plurals, definite plurals, conjunctions, and neg-raising predicates to the prevalence of prohibitives and litotes, the contrary interpretations of affixal negation, and the seemingly illogical behavior of “logical” double negation.

I would by contraries/Execute all things.

—Tempest, II.i

The contrary bringeth bliss.

—Henry VI Part II, V.v

An earlier version of this paper was presented at the Logical Words workshop organized by Jacques Moeschler at CIL 19 in Geneva (July 2013). Some of the material discussed here was presented in other forms at other forums, including LNAT (Logic Now and Then) in Brussels (November 2008), ESSLLI in Ljubljana (August 2011), SCLP in Santa Cruz (November 2011), CRISSP in Brussels (December 2011), the LOT Summer School in Driebergen (July 2012), AMPRA 1 in Charlotte (October 2012), the Systematic Semantic Change Workshop in Austin (April 2013), and Go Figure in London (June 2013). I am grateful to commenters at those occasions and to Barbara Abbott, Chris Collins, Ashwini Deo, Elena Herburger, Dany Jaspers, Pierre Larrivée, Jacques Moeschler, and Paul Postal for helpful discussions and complaints. I am grateful to Philosophical Studies, its editor Wayne Davis, and Springer for granting permission to reprint material that appears in a slightly different form in “Lie-toe-tease: Double negatives and unexcluded middles” (Horn, to appear).

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Notes

  1. 1.

    What of existential import? Which of the four statement forms entail or presuppose that the set over which the quantifier ranges is non-null and how does this affect the subaltern and other relations? In particular, if all is import-free while some is not, doesn’t this vitiate the Square? The fact that other operators (binary connectives, adverbs, modals, deontics) for which existential import is irrelevant can be mapped onto the Square makes such a step as unappealing as it is unnecessary. This leaves a number of options for dealing with questions of import and quantification and their relation to the Square; see Horn (1997) for discussion.

  2. 2.

    Note that despite the standard dictionary entry for cannot in which it is equated to ‘can not’, the former—as a lexicalization—can only get the E reading, never the O, regardless of the modal flavor involved. The same is true for couldn’t and couldnae, both standardly but misleadingly glossed as contractions of ‘could not.’

  3. 3.

    The relative opacity of E (as opposed to O) negative values is further attested by negative indefinites and “n-words” like Eng. no and nary (a), Ger. nie, and Fr. personne, rien, jamais, and by the O > E drift illustrated by intensifying NPI adverbials like not very Adj, not too Adj, not (all) that Adj, and not at all.

  4. 4.

    Bypassed here is the extension of MaxContrary to the phenomena motivating the principle of Conditional Excluded Middle (cf. Stalnaker 1981; von Fintel 1997; Williams 2010 for discussion and references), based on the plausibility of the assumed disjunction ([if A then C] v [if A then ¬C]).

  5. 5.

    The negative prefix non- typically functions as a contradictory and often forms minimal pairs with other contrary-forming negative prefixes (non-scientific vs. unscientific, non-artistic vs. inartistic; cf. Horn 1989: Sect. 5.1). Thus we have cases in which the contradictory non-adjective is explicitly rejected as too weak:

    In Talmudic days myrtles were used at funerals. Nowadays, the custom of flowers at a funeral…is not merely non-jewish but positively unjewish.

    (Contribution to mail.jewish net list, April 1991)

    A scale of the form <x is non-Jewish, x is un-Jewish> is clearly in play here.

  6. 6.

    While most speakers can be counted on to grasp the nuance of weakness conveyed by the negated negative, those with impairments in Theory of Mind, like the high-functioning autistic savant Daniel Tammet have difficulty in determining the motivation for, and therefore the interpretation of, litotic structures:

    Certain sentence structures can be particularly hard for me to analyze, such as: “He is not inexperienced in such things,” where the two negatives (not and in-) cancel each other out. It is much better if people just say: “He is experienced in such things.” (Tammet 2006: 162).

  7. 7.

    See http://www.myspace.com/notnotlickingtoadstunes for the eponymous Austin-based rock band.

  8. 8.

    This is effectively a scale of sanity, although negatively expressed. Similarly nuanced instantiations of positively described sanity scales are not inextant:

    We stood watching him. It was strangely riveting. Half-Dead Fred hadn’t even acknowledged the presence of others in the room, so entranced was he with discovering the tools that would allow him to rescue the princess locked in a dungeon by a nefarious wizard in cyberworld…

    Is he mad? I asked Mike.

    As in crazy? No, I don’t think so. But sane would be a little too strong.

    (J. Martin Troost (2004), The Sex Lives of Cannibals, p. 260)

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  1. Professor Emeritus of Linguistics and Philosophy, 370 Temple St, New Haven, CT, 06511-6802, USA

    Laurence R. Horn

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  1. Dept of Linguistics, University of Geneva, Genève, Geneve, Switzerland

    Joanna Blochowiak

  2. Batiment Landolt, 7th Floor, Univ of Geneva, Dept of Linguistics, Geneva, Switzerland

    Cristina Grisot

  3. Dept. of Psycholinguistics, University of Geneva Dept. of Psycholinguistics, Geneva, Switzerland

    Stephanie Durrleman

  4. Department of Linguistics, University of Geneva, Genève, Geneve, Switzerland

    Christopher Laenzlinger

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Horn, L.R. (2017). The Singular Square: Contrariety and Double Negation from Aristotle to Homer. In: Blochowiak, J., Grisot, C., Durrleman, S., Laenzlinger, C. (eds) Formal Models in the Study of Language. Springer, Cham. https://doi.org/10.1007/978-3-319-48832-5_9

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