ACTL 2014 Introduction to Formal Semantics
1
Tutorial 3: Monotonicity and NPIs
Review: Generalised Quantifier Theory
• Assumption: NPs and VPs denote sets of individuals.
(1)
a.
b.
vsemanticistwM “ vsemanticistswM “ t x | x is a semanticist u “ semanticist
vsmokewM “ vsmokeswM “ t x | x smokes u “ smoke
For the sake of simplicity, we do not distinguish singular and plural NPs (but this is
problematic, unsurprisingly, once you consider plurality). We also ignore mass nouns.
• Proposal (strong version): All quantificational determiners express relations between
sets of individuals. (Weak version: some quantificational determiners do.)
vevery semanticist smokeswM ô semanticist Ď smoke
vsome semanticist smokeswM ô semanticist X smoke ‰ H
ô Dxpx P semanticist ^ x P smokeq
ô |semanticist X smoke| ě 1
M
vno semanticist smokesw
ô semanticist X smoke “ H
v(exactly) five semanticists smokewM ô |semanticist X smoke| “ 5
vfive or more semanticists smokewM ô |semanticist X smoke| ě 5
vno more than five semanticists smokewM ô |semanticist X smoke| ď 5
vat least three semanticists smokewM ô |semanticist X smoke| ě 3
v(exactly) 60% of the semanticists smokewM ô |semanticistXsmoke|
“ 0.6
|semanticist|
2
Monotonicity
• One of the linguistically relevant properties of quantificational determiners is their
monotonicity. We have four types of monotonic properties:
1.
2.
3.
4.
2.1
Left upward monotonic
Left downward monotonic
Right upward monotonic
Right downward monotonic
Left Monotonicity
A quantificational determiner D is left upward monotonic if
in every model M , for any VP and for any pair NPS and NPL such that
vNPS wM Ď vNPL wM , if vD NPS VPwM “ 1 then vD NPL VPwM “ 1.
D is left downward monotonic if
in every model M , for any VP for any pair NPS and NPL such that
vNPS wM Ď vNPL wM , if vD NPL VPwM “ 1 then vD NPS VPwM “ 1.
D is left non-monotonic if it is neither left upward monotonic nor left downward
monotonic.
1
• Example of NPS and NPL : semanticist Ď linguist
• ‘Some’ is left upward monotonic
semanticist
–
–
–
–
‘Some semanticist smokes’ ñ ‘Some linguist smokes’
linguist
‘Some semanticist smokes’ is true iff semanticist X smokes ‰ H
‘Some linguist smokes’ is true iff linguist X smokes ‰ H
Because semanticist Ď linguist, whenever semanticist X smokes ‰ H, it must be the
case that linguist X smokes ‰ H.
– More generally, if S Ď L and S X P ‰ H, then L X P ‰ H.
• ‘Some’ is not left downward monotonic.
– ‘Some linguist smokes’ œ ‘Some semanticist smokes’
2.2
Right Monotonicity
A quantificational determiner D is right upward monotonic if
in every model M , for any NP and for any pair VPS and VPL such that
vVPS wM Ď vVPL wM , if vD NP VPS wM “ 1 then vD NP VPL wM “ 1.
D is right downward monotonic if
in every model M , for any NP and for any pair VPS and VPL such that
vVPS wM Ď vVPL wM , if vD NP VPL wM “ 1 then vD NP VPS wM “ 1.
D is right non-monotonic if it is neither right upward monotonic nor right downward monotonic.
• Example of VPS and VPL : live.in.Shadwell Ď live.in.London
• ‘Some’ is right upward monotonic.
– ‘Some linguist lives in Shadwell’ ñ ‘Some linguist lives in London’
– The reasoning is the same as before (notice that ‘some’ is symmetric)
• ‘Some’ is not right downward monotonic.
– ‘Some linguist lives in London’ œ ‘Some linguist lives in Shadwell’
Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Complete the following table by filling in the blanks with ‘Up’, ‘Down’ or ‘Non-monotonic’.
Some
Every
No
At most five
At least 50% of the
Exactly five
Most
Left
Up
Right
Up
.......................................................................................
2
2.3
Monotonicity of Generalised Quantifiers
• The monotonicity of quantificational DPs (generalised quantifiers) can be defined in
parallel ways.
A generalised quantifier Q is upward monotonic if
in every model M , for any pair VPS and VPL such that vVPS wM Ď
vVPL wM , if vQ VPS wM “ 1 then vQ VPL wM “ 1.
Q is downward monotonic if
in every model M , for any pair VPS and VPL such that vVPS wM Ď
vVPL wM , if vQ VPL wM “ 1 then vQ VPS wM “ 1.
Q is non-monotonic if it is neither upward monotonic nor downward monotonic.
Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the monotonic properties of the following generalised quantifiers?
• ‘Everyone’
• ‘John’ (as a generalised quantifier)
• ‘Only John’
• ‘Everyone but John’
.......................................................................................
3
NPI Licensing
• The Fauconnier-Ladusaw Licensing Hypothesis
Weak NPIs like ever and any are licensed if it is in a downward monotonic context.
• An item α is in a downward monotonic context if α is in the following configuration, where vβwM is a left downward monotonic determiner or a downward monotonic
generalised quantifier.
γ
β
...α...
– If α occurs in the NP argument of a left downward monotonic determiner β, α is
in a downward monotonic context.
– If α occurs in the VP argument of a right downward monotonic determiner/a
downward monotonic quantifier β, it is in a downward monotonic context.
– (One can also define negation as downward monotonic by generalising the notion
of monotonicity)
Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Which of the following examples are problematic for the Fauconnier-Ladusaw hypothesis?
(2)
a. Everyone who ever read a book got an A.
b. At most five books have ever been read.
c. Exactly two people have ever been on the moon.
d. *Ten or more books that have ever been written are interesting.
3
e. Only John has ever said anything.
f. *Everybody but John has ever said anything.
.......................................................................................
4
Strong NPIs
• Certain NPIs like in weeks are only licensed in a subset of contexts where weak NPIs
like ever are licensed. They are called strong NPIs.
(3)
Ever
a. *Every doctor has ever seen John.
b. No doctor has ever seen John.
c. At most five doctors have ever seen John.
(4)
In Weeks
a. *Every doctor has seen John in weeks.
b. No doctor has seen John in weeks.
c. *At most five doctors have seen John in weeks.
• Hypothesis: Strong NPIs are licensed in anti-additive contexts.
– A quantificational determiner D is left anti-additive iff in every model M , for any
NP1 , NP2 and VP, vD NP1 or NP2 VPwM ô vD NP1 VPwM and vD NP2 VPwM .
– (similarly for right anti-additivity)
– A generalised quantifier Q is anti-additive iff in every model M , for any VP1 and
VP2 , vQ VP1 or VP2 wM ô vQ VP1 wM and vQ VP2 wM .
• E.g. vno doctorwM is an anti-additive generalised quantifier.
– No doctor sang or danced ô (no doctor sang) and (no doctor danced).
• E.g. at most five doctors is not anti-additive.
– At most five doctors sang or danced ñ at most five doctors sang and at most five
doctors danced
– But the converse does not hold!! (e.g. when three doctors sang and three different
doctors danced).
Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
• Prove that every anti-additive generalised quantifier is downward-monotonic. (This
result generalises to all anti-additive items)
• Which of the following sentence is problematic for the above hypothesis?
(5)
a. No boy who has seen John in weeks likes him.
b. *Every boy who has seen John in weeks likes him.
c. *Between five and ten boys who have seen John in weeks like him.
.......................................................................................
4