1. Squaring any Johan van der Auwera & Lauren Van Alsenoy
Centre for Grammar, Cognition and Typology
Antwerp University

2. The square of opposition
A three-leveled square of opposition
A three-leveled, three-dimensional square => a cube

3. 1. The square of opposition

4. 1. The square of oppositionM O A E N E G O I O

5. A E I O necessary impossible possible not necessary
1. The square of opposition
necessary
impossible A E I O
possible
not necessary

6. Languages do not lexicalize the O-value
1. The square of opposition
Larry Horn (1989)
Languages do not lexicalize the O-value

7. 1. The square of opposition
ALL NO
NOT ALL
SOME
*NALL

8. pas tous pas chaque tous, chaque ne …. aucun quelques
1. The square of opposition
tous, chaque
ne …. aucun
pas tous
pas chaque
quelques

9. pas tous pas chaque tous, chaque ne …. aucun quelques
1. The square of opposition
tous, chaque
ne …. aucun
pas tous
pas chaque
quelques
*netous, *nechaque

10. pas tous pas chaque tous, chaque ne …. aucun quelques
1. The square of opposition
tous, chaque
ne …. aucun
pas tous
pas chaque
quelques

11. pas tous pas chaque *naucun tous, chaque ne …. aucun quelques
1. The square of opposition
*naucun
tous, chaque
ne …. aucun
pas tous
pas chaque
quelques
*netous, *nechaque

12. pas tous pas chaque *naucun tous, chaque ne …. aucun quelques
1. The square of opposition
*naucun
nullus ↓
(ne …) nul
ne … nul Ø
tous, chaque
ne …. aucun
pas tous
pas chaque
quelques
*netous, *nechaque

13. Is there also a history of E?
1. The square of opposition
First problem:
Is there also a history of E?

14. There are however O-expressions
1. The square of opposition
Second problem:
There are however O-expressions

15. Some O-expressions are composed ones
1. The square of opposition
Some O-expressions are composed ones

16. Some expressions are composed ones
1. The square of opposition
Some expressions are composed ones

17. Some expressions are interesting!
1. The square of opposition
Some expressions are interesting!

18. Some expressions ARE interesting
1. The square of opposition
Some expressions ARE interesting
must, have to
can’t, mustn’t
must, have to
may, can
needn’t

19. 1. The square of opposition
Third problem:
Given that some O- expressions are interesting, one should retain the O-value

20. No projection to a triangle
1. The square of opposition
1. le carré des oppositions
Horn 1990
Jettison = reject, nadir = the other side of the earth, sun/moon below.
Vertex = highest point, apex = top of triagnle
No projection to a triangle

21. No three-cornered square
1. The square of opposition
Horn 1990
No three-cornered square

22. A three-leveled square
2. Three-leveled square of opposition
A three-leveled square

23. Three-leveled square necessary not necessary van der Auwera 1996
2. Three-leveled square of opposition
necessary
not necessary
Three-leveled square
van der Auwera 1996

24. Three-leveled square with five values
2. Three-leveled square of opposition
necessary 1 2
not necessary 3
Three-leveled square with five values

25. Three-leveled square with five values
2. Three-leveled square of opposition A I
necessary
possible
not necessary O E
Three-leveled square with five values

26. Three-leveled square with five values
2. Three-leveled square of opposition A I
necessary
at least possible, perhaps necessary
only
possible
not necessary O E
Three-leveled square with five values

27. Three-leveled square A I necessary not necessary O E contradiction
2. Three-leveled square of opposition A I
necessary
not necessary O E
contradiction
Three-leveled square

28. Three-leveled square contradiction contraries A I necessary
2. Three-leveled square of opposition A I
necessary
not necessary O E
contradiction
contraries
Three-leveled square

29. Three-leveled square contradictories contraries subcontraries A I
2. Three-leveled square of opposition A I
necessary
not necessary O E
contradictories
contraries
subcontraries
Three-leveled square

30. Three-leveled square contraries implication subcontraries
2. Three-leveled square of opposition A I
necessary
not necessary O E
contraries
implication
subcontraries
contradictories
Three-leveled square

31. Three-leveled square with five values
2. Three-leveled square of opposition A I
necessary
at least possible, perhaps necessary
only
possible
not necessary O E
Contraries
implication
Contradictories
subcontraries
Three-leveled square with five values

32. Three-leveled square for quantifiers in English
2. Three-leveled square of opposition
Three-leveled square for quantifiers in English
Aussi pour les quantificateurs

33. For quantifiers in French
2. Three-leveled square of opposition
For quantifiers in French

34. Fourth problem: etymologically very different forms in same region:
2. Three-leveled square of opposition
Fourth problem:
etymologically very different forms in same region:
English: someone/anyone in non-assertive contexts
Latin: nonnullus and nonnumquam

35. Same form in same context in different regions:
2. Three-leveled square of opposition
Same form in same context in different regions:
“strong” and “weak” indefinites:
in ENGLISH:
There are sm men in the garden.
SOME flags are green.
(Jaspers 2005: 178)

36. Different forms for “strong” and “weak” meaning: In Dutch
2. Three-leveled square of opposition
Different forms for “strong” and “weak” meaning:
In Dutch
Sommige regeringen waren niet veel beter als de bankiers.
Er waren enkele mensen ziek.

37. 2. Three-leveled square of opposition
NOT NO
(nonnullus)
Center is not some-but-not-all but both inclusive some-possibly-all and exclusive some
Former at-least-some area is now reserved for complex nonnullus-like indefinites or composed ones

38. What are these map points: contexts? Meanings? Meanings or contexts?
2. problems with the map itself
2. Three-leveled square of opposition
(Haspelmath 1997)
Problems with map:
What are these map points:
contexts?
Meanings?
Meanings or contexts?

39. Our solution: meanings-in-context
2. problems with the map itself
2. Three-leveled square of opposition
(Haspelmath 1997)
Our solution: meanings-in-context

40. For every value, there are several constructions
2. Three-leveled square of opposition
not no
Fifth problem:
For every value, there are several constructions
Aussi pour les quantificateurs
de l’anglais

41. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
not no
Fifth problem:
For every value, there are several constructions

42. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
any …
Did you see somebody/anybody?
not no
Fifth problem:
For every value, there are several constructions

43. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
any …
Did you see somebody/anybody?
I saw nobody.
I didn’t see anybody.
not no
not any …
Fifth problem:
For every value, there are several constructions

44. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
any …
Did you see somebody/anybody?
I saw nobody.
I didn’t see anybody.
not no
not any …
Fifth problem:
For every value, there are several constructions

45. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
any …
Did you see somebody/anybody?
any
I saw nobody.
I didn’t see anybody.
not no
Anybody/Everybody can do that.
not any …
Fifth problem:
For every value, there are several constructions

46. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
any …
any
not no
not any …
not any …
Fifth problem:
For every value, there are several constructions

47. For every value, there are several constructions
2. Three-leveled square of opposition
each
every …
any …
any
not no
not any …
not any …
Fifth problem:
For every value, there are several constructions

48. Sixth problem: The mystery of ANY
2. Three-leveled square of opposition
each
every …
any …
any
not no
not any …
not any …
Sixth problem:
The mystery of ANY

49. 2. Three-leveled square of opposition
some
any ∃

50. ∃ some specific reference any non-specific reference
2. Three-leveled square of opposition
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me?

51. ∃ some specific reference any non-specific reference
2. Three-leveled square of opposition
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me?

52. ∀ ∃ any some specific reference any non-specific reference
Somebody called me.
Did anybody call me?

53. ∀ ∃ any non-specific reference some specific reference
Anybody can call me.
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me?

54. ∀ ∃ any non-specific reference some specific reference
Anybody can call me.
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me?

55. ∀ ∃ ∀∼ ∼∃ any non-specific reference some specific reference
Anybody can call me.
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me? no ∀∼ ∼∃

56. ∀ ∃ ∀∼ ∼∃ any non-specific reference some specific reference
Anybody can call me.
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me?
no non-specific reference ∀∼ ∼∃
Don’t ask any old bloke for directions.

57. ∀ ∃ ∼∀ ∼∃ any non-specific reference some specific reference
Anybody can call me.
some specific reference
any non-specific reference ∃
Somebody called me.
Did anybody call me?
no non-specific reference ∼∀ ∼∃
Don’t ask any old bloke for directions.
I never ask anybody for directions.

58. Non-specific existential
3. The cube of opposition
univ
Non-specific existential
non zero
Specific existential
non univ
zero

59. Non-spec existential non zero non univ zero zéro
3. The cube of opposition
univ
Non-spec existential
non zero
Specific existential
non univ
zero
zéro

60. 3. The cube of opposition
univ
Non-spec exi
non univ
zero
English any

61. English any 3. The cube of opposition univ univ Non-spec exi
non univ
zero
zero
English any

62. English any 3. The cube of opposition univ univ Non-spec exi
non univ
zero
zero
zero
English any

63. Er lagen enige boeken op tafel. (2) Heb je enig idee?
3. The cube of opposition
univ
univ
Non-spec exi
Non-spec exi
Non-spec exi
non univ
zero
zero
zero
English any
Er lagen enige boeken op tafel.
(2) Heb je enig idee?
(3) Nooit heb ik enig succes gekend.
Non-spec exi
spec exi
zero
Dutch enig

64. English any Dutch enig allemand einig German einig
3. The cube of opposition
univ
univ
Non-spec exi
Non-spec exi
Non-spec exi
non univ
zero
zero
zero
English any
Non-spec exi
spec exi
spec exi
zero
Dutch enig
allemand einig
German einig

65. Conclusion

66. References
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HASPELMATH, Martin Indefinite pronouns. Oxford: Oxford University Press.
HOEKSEMA, Jack Dutch ENIG: From nonveridicality to downward entailment, In: ZEIJLSTRA, Hedde / SOEHN, Jan-Philipp (eds.), Proceedings of the Workshop on Negation and Polarity. , 8-15.
– (in print) Negative and positive polarity items: an investigation of the interplay of lexical meaning and global conditions of expression. In: HORN, Laurence R. (ed.), The expression of negation. Berlin: Mouton.
– / KLEIN, Henny Negative predicates and their arguments, Linguistic Analysis 25: ,
HORN, Laurence R Hamburgers and Truth: Why Gricean explanation is Gricean, In: HALL, Kira / KOENIG, Jean-Pierre / Meacham, Michael / REINMAN, Sondra / SUTTON, Laurel A. (eds.), Proceedings of the Sixteenth Annual Meeting of the Berkeley Linguistics Society. Berkeley: Berkeley Linguistics Society,
JÄGER, Agnes History of German Negation. Amsterdam: Benjamins.
JASPERS, Dany Operators in the Lexicon. On the Negative Logic of Natural Language.
VAN DER AUWERA, Johan Modality: the three-layered scalar square, Journal of Semantics 13:
– In defense of classical semantic maps,Theoretical Linguistics 34:
– in print. ‘On the diachrony of negation’, In: HORN, Laurence R. (ed.), The expression of negation. Berlin: Mouton.
– / TAEYMANS, Martine The need modals and their polarity”, In: BOWEN Rhonwen / MOBÄRG, Mats / OHLANDER, Sven (eds.), Corpora and Discourse – and Stuff. Papers in Honour of Karin Aijmer. Gothenburg: University of Gothenburg,
– / VAN ALSENOY, Lauren. (in print). Indefiniteness maps: problems, prospects and ‘retrospects’. Atti del Convegno della Società Italiana di Glottologia (Palermo 2008).
– / VAN ALSENOY, Lauren. (in print) Mapping indefiniteness: towards a Neo-Aristotelian approach, Proceedings 19th International Symposium on Theoretical and Applied Linguistics, Thessaloniki April 2009.
– / VAN ALSENOY, Lauren (submitted) Indefinite pronouns: synchrony and diachrony – comments on Willis.