Negated Antonyms and Approximative Number Words:

Two Applicationsof Bidirectional Optimality Theory

Manfred Krifka

Humboldt Universität Berlin

Zentrum für Allgemeine Sprachwissenschaft (ZAS)Berlin

krifka@rz.hu-berlin.dehttp://amor.rz.hu-berlin.de/~h2816i3x

Topics of this talk: Two Applications of Bi-OT

1. Interpretation of measure terms with round and non-round numbers, e.g.

The distance between Amsterdam and Vienna is one thousand kilometers.The distance is roughly one thousand kilometers.

The distance between Amsterdam and Vienna is nine hundred sixty five kilometers.The distance is exactly nine hundred sixty five kilometers.

cf. Krifka (2002), ‘Be brief and vague! And how bidirectional optimality theory allows for verbosity and precision’, in D. Restle & D. Zaefferer (eds.), Sounds and systems. Studies in structure and change. A Festschrift for Theo Vennemann, Berlin: Mouton de Gruyter)

2. Interpretation of antonyms and their negation, e.g.

John is happy. John is not unhappy. John is unhappy.

John is not happy.

cf. Blutner (2000), ‘Some aspects of optimality in natural language interpretation’, Journal of Semantics 3.

Part I: Interpretation of measure terms

How much precision is enough?From the land of bankers and watchmakers.

Street sign in Kloten, Switzerland.

Measure Terms: Round Numbers, Round Interpretations

The Round Numbers / Round Interpretation Principle (RN/RI):Round (simple) numbers suggest round (vague or imprecise) interpretations.

Krifka (2002) suggests an explanation by general pragmatic principles:

1. A preference for simple expressions(cf. Zipf’s law; Speaker economy; R-Principle of Horn 1984, I-Principle of Levinson 2000)one thousand > nine hundred sixty-five (where a > b: ‘a preferred over b’)

2. A preference for vague, approximate interpretations(cf. P. Duhem 1904, balance between precision an certainty; Ochs Keenan 1976, vagueness helps to save face; reduction of cognitive effort) aprrox. > precise

Cognitive motivation: St. Dehaene 1997, The number sense: How the mind creates mathematics

distinguishes between an older, approximate sense of numerosity (in animals, babies, many adult uses of number) and a precise sense of counting.

Optimal expression-interpretation pairs

one hundred, precise

one hundred, approx.

one hundred and three, approx.

one hundred and three, precise

Non-optimal Non-optimal

Optimal

Optimal, as the other

comparable pairsare non-optimal.

notcom-peting

not competing, cannot be used in same situation,

except in caseslike one hundredvs. one hundred point zero

3. Interaction of the two principles following Weak Bidirectional OT (Blutner, Jäger):

An expression-interpretation pair F, M is optimal iffthere are no other optimal pairs F’, M or F’, M’ such that F’, M > F, M or F’, M > F, M.

Bidirectional Optimization as a general explanationof M-Implicatures

Neo-Gricean pragmatics: Horn, Levinson, in particular Levinson (2000), Presumptive Meanings, MIT Press.

Three basic principles:Q-Principle (Quantity)

Speaker chooses the maximally informative expression of a set of alternative expressionsthat is still true (provided there is no reason not to do so)Q-Implicatures: John ate seven eggs. ≈≈> ¬ John ate eight eggs.

I-Principle (Informativity)Addressee enriches the literal information to a normal, stereotypical interpretation (provided there is no reason not to do so)I-Implicature: Mary turned the switch, and it became dark. ≈≈> temporal order, cause, purpose.

M-Principle (Modality / Manner / Markedness)Non-normal, non-stereotypical interpretations are interpreted in non-normal ways, i.e. unmarked expressions ⇔ unmarked interpretations marked expressions ⇔ marked interpretationsM-Implicature:Mary turned the switch, and it also became dark.

Example: kill vs. cause to die

Problem (McCawley, Generative Semantics:kill means cause to die, but: Black Bart killed the sheriff. direct killing

Black Bart caused the sheriff to die. indirect killing

Explanation by M-implicature: marked (complex) form ⇔ marked meaning

Derivation by bidirectional OT:

• Form preference: kill > cause to die

• Meaning preference: direct killing > indirect killing

• Application of evaluation algorithm

〈 kill, direct killing 〉

〈 kill, indirect killing 〉 〈 cause to die, direct killing 〉

〈 cause to die, indirect killing 〉

A conditional preference for approximate interpretations?Another take on the RN/RI phenomenon:• Assume that precise and approximate interpretation ranked equally,

i.e. hearer can expect precise and imprecise interpretation with p = 0.5• Under approximate interpretation, round numbers are preferred (brevity)

0 10 20 30 401 2 3 4 5 6 7 8 9

range of approximate interpret.

twenty

shortest expression within range

Rule: Choose least complex number expression within the range of interpretation! If interpretation is precise, there is only one possible number expression.

precise interpretation

thirty-seven

Assume: probability of reported values [0,...100]: 0.01, range of approximate interpretation 4, (cf. error margin)

For twenty: probability of use under approximate interpretation = 0.5 * 0.09 = 0.045 probability of use under precise interpretation = 0,5 * 0.01 = 0.005

For thirty-seven: only precise interpretation; under approximate interpretation, forty would be chosen.

Conditional preferences for short expressions and vague interpretations

one hundred, precise.

one hundred, approx.

one hundred and three, approx.

one hundred and three, precis.

Non-optimal Non-optimal

Optimal

Optimal, no other

competitor

Complex constraint rankings: short > long under vague interpretation, i.e. short, approx. > long, approx. approx. > precise for short expressions, i.e. short, approx. > short, precise

Theoretical Background: Game Theory

Game-theoretic approaches to Pragmatics:

• Dekker & van Rooy (2000) “Bi-directional optimality theory: An application of game theory”, Journal of Semantics:

Optimal form/interpretation pairs are Nash equilibria (local optima);any unilateral deviation from these optima is dispreferred.

• Parikh (2001), The Use of Language, CSLI Publ:

General game-theoretic approach to communication, “strategic communication”

Is preference for short expressions sufficient?

Preference for short expressions cannot explain all interpretation preferences:

I did the job in twenty-four hours. vague I did the job in twenty-three hours. preciseI did the job in twenty-five hours. precise

The house was built in twelve months. vagueThe house was built in eleven months. preciseThe house was built in thirteen months. precise

Two dozen bandits attacked him. vagueTwenty-four bandits attacked him. precise

... and sometimes even makes the wrong predictions:

Mary waited for forty-five minutes. vagueMary waited for forty minutes. precise

I turned one hundred and eighty degrees. vagueI turned two hundred degrees. precise

Her child is eighteen months. vagueHer child is twenty months. precise

John owns one hundred sheep. vagueJohn owns ninety sheep. precise

Alternative theory: A preference for simple, coarse-grained representations?

A preference for coarse-grained representations!Round numbers as cognitive reference points: E. Rosch (1975), Cognitive reference points, Cognitive Psychology

Scales of different granularity P. Curtin (1995), Prolegomena to a theory of granularity, U Texas Master Thesis

0 5 10 15 20 25 30 35 40 45 50 55 60

Finer-grained scale of minutes of an hour

0 15 30 45 60

Coarser-grained scale of minutes of an hour

Possible durations

Application of values

Application of values

In interpreting a measure report, assume the most coarse-grained scale compatible with the chosen number word!

twenty minutes: 20min, assume scale: 5min–10min–15min–20min–25min–...

fifteen minutes: 15min, use scale: 15min–30min–45min-60min

A preference for coarse-grained representations

A priori assumptions:

• Assume that durations come with equal frequency within one hour (p = 1/60)

• Assume that fine-grained and coarse-grained representationsare selected with same probability (p = 1/2)

1. On hearing fifteen minutes, interpreted as 15min:Task of hearer: Find out which scale the speaker used, fine-grained [5...10...15...20min...] or coarse-grained [15...30...45...60min].

• A-priori probability for fine-grained scale: p = 1/2Possible values for 15min: (13, 14, 15, 16, 17min): p = 5/60Total probability of real value + encoding: p = 5/120.

• A-priori probability for coarse-grained scale: p = 1/2Possible values for 20min: (8, 9, 10, ... 15, ..., 22min): p = 15/60Total probability of real value + encoing: p = 15/120

Hence assumption of coarse-grained scale (vague interpretation) is safer.

2. On hearing twenty minutes, interpreted as 20min:This term does not exist on the coarse-grained scale, hence fine-grained scale must be assumed (precise interpretation)

Bidirectional OT and preference for coarse-grained representations

15min, [5-10-15-20-25min-...]

15min, [15-30-45-60min]

20min, [15-30-45-60min]

20min, [10-15-20-25min-...]

Non-optimal Not generated!

Preferred as more probably interpretation,

Optimal!

Cannot be truein the same situation,

not comparedwith the other cases

-- Optimal!

Compare pairs of values (meanings) and levels of granularity of representation

An evolutionary perspective on brevity?

It cannot be an accident that for many, perhaps most scales,coarse-grained scales have expressions of reduced complexity(cf. Krifka 2002)

Example: Complexity by average number of syllablesa. one, two, three, four, ... one hundred: 273/100 = 2.73

b. one, five, ten, fifteen, ... one hundred: 46/20 = 2.3

c. one, ten, twenty, thirty, ... one hundred: 21/10 = 2.1

Suspicion: Scales develop in a way to enable complexity-based optimization, expressions of coarse-grained scales tend to be simpler.

An evolutionary perspective on brevity?The optimization of scales.

Scales and hierarchies of scales of different granularityhave to satisfy certain requirements to be useful for communication:

1. Requirement for scales: Equidistance of units (additive, sometimes logarithmic, cf. decibel; kilo/mega/giga)

2. Requirements for scale hierarchies of different granularity:Scales of increasing granularity Sn Sn+1 Sn+2

should increase granularity by the same factor, [10, 20, 30, 40, 50, 60, ...][100, 200, 300, 400, 500, ..]

[1000, 2000, 3000, 4000, ...]: powers of 10where the most natural step is decrease granuality by factor 1/2:

[1, 2, ...][1/2, 1, 11/2, 2, ...][1/4, 1/2, 3/4, 1, ...]

cf. hour scale:[1h, 2h, ...], [30min, 1h, 1h30min, ...],[15min, 30min, 45min, 60min, ...]

Evidence for preferred reference points: Frequency of number words

If fine-grained / coarse-grained scales are used to report measurements, and if with coarse-grained scales, only certain number words occur, then these number words should occur more likely in a natural linguistic corpus containing measurement reports.

Cf. Dehaene & Mehler (1992), Cross-linguistic regularities in the frequency of number words, Cognition 43, Jansen & Pollmann (2001), On round numbers: Pragmatic aspects of numerical expressions. Journal of Quantitative Linguistics 8, Corpora of English, French, Dutch, Japanese, Kannada:

• Between 10 and 100, the powers of ten occur most frequently• Frequency decreases with higher powers of 10, but local maximum for 50• Between 10 and 20, local maxima at 15, also at 12 (“dozen”)Example: Occurrences of number words in British National Corpus,

after H. Hammarström (2004), Properties of lower numerals and their explanation: A reply to Paweł Rutkowski (ms.)

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12 15 50

Evidence: Frequency of number words

Frequency of round numbers on -aine in FrenchFrench web sites of Google, April 11, 2005, search for strings “une quarantaine de”

dixaine 4.230 vingtaine 737.000

onzaine 19 trentaine 866.000

douzaine 262.000 quarantaine 272.000

treizaine 16 cinquantaine 490.000

quatorzaine 47 soixantaine 159.000

quinzaine 540.000 septantaine 614

seizaine 6 quatre-vingtaine 85

dix-septaine 1 quatre-vingtdixaine 0

dix-huitaine 11 centaine 548.000

dix-neuvaine 1

An evolutionary perspective on brevity?The optimization of scales

The expressions of values of scales align with the optimization of scales

Example: Expression of half points between powers of ten

Roman number writing (also motivated iconically, by shape of hand) I II III IV V VI VII VIII IX X X XX XXX XL L LX LXX LXXX XC C

Simplification of number word ‘five’:

English: fifteen (*fiveteen), fifty (*fivety): loss of diphthong, shortening

OE fi:f as word vs. fif- as prefix; vowel shift only affected i: (> ai)

Colloquial German fuffzehn (fünfzehn), fuffzig (fünfzig): unrounding ü > u, loss of n, shortening (3 morae to 2 morae)

Simplification of ‘half’:German anderthalb ‘one and a half’, lit. ‘the second half’ vs. regular eineinhalb

Still an effect of complexity of expression?

In vigesimal number systems, ‘50’ is more complex than ‘40’/’60’Question:

Is ‘50’ nevertheless used as approximate number word?Conflict cognitive preference / communicative preference

Cf. Hammarström (2004), Number bases, frequencies and lengths cross-linguistically.

Inspired by that: Investigation of occurrernces of number words on Norwegian vs. Danish web sites of Google (March 4, 2005):

Number Norwegian Occurren ces Danis h Occurren ces 20 tjue 61300 tyve 121000 30 trett i 43700 tred ive 25400 40 førti 39200 fyrre 26800 50 femt i 81200 halvtreds 15500 60 sekst i 19400 tres 36400 70 sytt i 10200 halvfjerds 581 80 åtti 13100 firs 3740 90 nitti 13500 halvfems 540

Complexity matters: Common belief: Grammars do best what speakers do most (DuBois 1987) But: Sometimes speakers do most what grammars do best!

Part II: Antonyms and their Negation

Larry Horn (1991), ‘Duplex negatio affirmat: The economy of double negation’;(1992), ‘Economy and redundancy in a dualistic model of natural language’

Basic phenomenon:Mary is not unhappy implicates: Mary is quite happy.

Implicature cancelling:She is happy, or at least she is not unhappy.*She is not happy, or at least she is not unhappy.**She is unhappy, or at least she is ot unhappy.

Gael Green (1982), Doctor Love:“These days all marriages seem to be doomed”, Barney said.”Who’s happy?”“I’m not unhappy”, Mike offered.

Otto Jespersen (1924), The Philosophy of Grammar:“The two negatives [...] do not exactly cancel one another so that the result [not uncommon, not infrequent] is identical with the simple common, frequent: the longer expression is always weaker: “this is not unknown to me” [...] means: ‘I am to some extent aware of it’.“

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Parteitag der Bündnisgrünen in Münster wählt mit großer Harmonie mit Renate Künast und Fritz Kuhn ein neues Führungsgremium umd beschließt die Unterstützung des Atomkonsens der Regierung.

Grüne Harmonie: Glücklich (ganz links): Fraktionschefin Kerstin Müller. Glücklich (darunter): Fraktionschef Rezzo Schlauch. Glücklich (rechts daneben): Gesund-heitsministerin AndreaFischer. Glücklich (darüber): Schleswig-Holsteins Um-weltminister Klaus Müller. Glücklich (verdeckt): Um-weltminister Jürgen Trittin. Nicht unglücklich (vor Trittin): AußenministerJoschka Fischer. Überglücklich: die neue Parteichefin RenateKünast. (TAZ 26.6.2000)

Reasons for being doubly negative: Various proposals in the literature.

Psychological exhaustion? Jespersen, ibid.: “The psychological reason for this is that the détour through the two mutually destructive negatives [not uncommon, not unknown] weakens the mental energy of the listener and implies a hesitation which is absent from the blunt, outspoken common or known.”

Pomposity? Orwell (1946), ‘Politics and the English language’“Banal statements are given an appearance of profundity by meansof the not un- formation.”

Being English? Fowler (1927), A dictionary of modern English usage“The very popularity of the idiom in English is proof enough that there is something it it congenial to the English temperament, & it is pleasant to believe that it owes its success with us to a stubborn national dislike of putting things too strongly”.

Completely unnecessary? Frege (1919), ‘Die Verneinung’“Wrapping up a thought in double negation does not alter its truth value.”

Reasons for being doubly negative:Proposals by Horn

Horn gives the following taxanomy of motives for saying not un-A

a. Quality: S is not sure A holds, or is sure it doesn’t.

b. Politeness: S strongly believes A holds, but is too polite, modest, or wary to mention it directly.

c. Weight or impressiveness of style: S violates brevity precisely to avoid brevity.

d. Absence of a corresponding positive (e.g., not unfounded)

e. Parallelism of structure

f. Minimization of precessing, in contexts of direct rebuttal or contradiction.

Here: Mainly Reason (a).

A first pragmatic OT treatment: Blutner 2001

happy and unhappy are contraries; their lexical meanings do not apply to to emotional states in middle range.

happy unhappy

The literal meaning of the negations not happy and not unhappy:

happy unhappy

not unhappy not happy

Negated forms compete with shorter forms and are pragmatically restricted:

happy unhappy

not unhappy not happy

Problem:-- Unclear how different interpretation of not happy and not unhappy comes about,-- prediction: not unhappy gets blocked because it is more complex than not happy!

A Weak Bi-OT Theory about Happiness

Assume that antonyms are literally interpreted in an exhaustive way, i.e. they are contradictories.

Initial situation: Antonym pairs and their negations.

not unhappy not happy

happy unhappy

I-Implicature: Restriction of simpler expressions to prototypical uses.

not unhappy not happy

happy unhappy

not unhappy not happy

happy unhappyM-Implicatures: Restriction of complex expressions to non-prototypical uses.

Weak Bidirectional-OT on Being not Unhappy

happy,

not unhappy, happy,

not unhappy,

unhappy,

not happy, unhappy,

not happy,

Preference for stereotypical interpretations:

> >

Preference for simple expressions: happy > unhappy > not happy > not unhappy

A reason to prefer extreme interpretations

With exhaustive interpretation of antonyms: It is unclear where to draw the border, cf. epistemic theory of vagueness of Timothy Williams (1994).

happy unhappy

Saying that someone is happy or unhappy may not very informativeif the person’s state is close to the borderline;this is a motivation for restricting the use of happy/unhappy to the clear cases, the ones on which speaker and hearer definitely should agree upon.

happy unhappy

unhappyhappy

happy unhappy

Further phenomena in the interpretation of evaluative adjectives

Litotes (understatement): This is not bad for ‘This is good’, I’m not unhappy about it for ‘I’m happy about it’Avoidance of positive values from the range of expressions. Reason: showing off critical attitude that nothing can be really positive.

not unhappy not happy

happy unhappy

Avoidance of negative valuesThis is not good for ‘This is bad’I’m not happy about it for ‘I am unhappy about it’Avoidance of negative values from the range of expressionsReason: Politeness, attempts to save face.

not unhappy not happy

happy unhappy

Conclusion

Round numbers / round interpretations:

-- Refined theory of interpretation preferences,extending usual forms of bidirectional optimization.

-- Simplification of forms a secondary factorafter preference for coarse-grained scales with round numbers?

-- Development of scales to allow for round numbers / round interpretations

Interpretation of antonyms:

-- Standard application of bidirectional optimization gets the facts rightif we assume basic exhaustive interpretation of antonyms

-- Explanation of tendency for stereotypical interpretationsin Williamsons’ epistemic theory of vagueness

-- Litotes as avoidance of extreme values.

krifka@rz.hu-berlin.de

http://amor.rz.hu-berlin.de/~h2816i3x