States, degrees, and additivity
Cara Feldscher
[email protected]
Michigan State University
1
JmoreK = J-erK(JmuchK)
JmoreK = λdλα.A(µ)(α) > d
Syntax and Semantics Workshop
of the American Midwest and Prairies
Northwestern University
What Thomas (2010) calls incremental more has also been called additive.
(3)
Roadmap
Here is the general procession planned:
1. Where I’m starting from: additivity in the literature, and additive
again
Analyses of incremental more include Greenberg (2012); Thomas (2010).2
Here are some commonalities:
2. Additive another and other further data – what generalizations
should be made?
• Tracking part-whole relations between events and individuals, or
events and degrees.
3. Comparing additivity to comparatives
• Adding subevents to create larger events.
4. What could we isolate as additivity?
2
a. (I have five papers to grade.) Last night I graded three, and
today I graded two more.
(5 total)
b. (My running goal is three miles.) I ran two miles yesterday,
and today I ran one more.
(3 total)
Additive again
2.2
Additivity in the literature
2.2.1
Data
Incremental More
2.1
Additive again is in (5), in comparison with the usual again data in (4).
More is most often seen in comparatives, as in (1).
(1)
a. I graded (five) more papers than John.
b. Suzie ran (five miles) more than Andrew did.
Wellwood (2015) breaks down comparative more into much + -er, with
much able to apply to multiple types:
(2)
(4)
Neville closed the door again.
(5)
a.
b.
c.
Neville is half again as tall as Pansy.
(1.5x)
This pile of loot is half again the size of the last! (1.5x)
Neville is a third again as tall as Pansy.
(1.3x)
The British word order from the OED:
J-erK = λghα,di λdλα.g(α) > d
Jmuchµ KA = A(µ)
(6)
2
Handout will be available at https://msu.edu/~feldsch3/.
1
a.
Lent shall be as long againe as it was.
Shakespeare
Denotation for Thomas’s incremental more in Appendix.
Henry VI, by
b.
2.2.2
(9) Jagainadd K = λfhd,hs,tii λdλs.∃s0 [s ≈ s0 ∧ f (d)(s0 ) ∧ µ(s) = µ(s0 ) + d]
Jhalf as tallK = λd.λs[tall(s) ∧ µ(s) = 21 d]
JagainK(Jhalf as tallK) = λd.λs.∃s0 [s ≈ s0 ∧ [tall(s0 ) ∧ µ(s0 ) = 12 d] ∧
µ(s) = µ(s0 ) + d]
Jhalf as tall againK = λd.λs.∃s0 [tall(s) ∧ tall(s0 ) ∧ µ(s) = 21 d + d]
The similarity relation ensures both are states of tallness.
The measure of s0 ( 12 d) is added to d to get the measure of
s to be 1 12 d.
Jhalf as tall againK(Jas PansyK) = λs.∃s0 [tall(s) ∧ tall(s0 ) ∧ µ(s) =
1
2 dPansy + dPansy ]
Brazil nut trees are emergent species, half as tall again
as most canopy trees.
Secret Life of Trees, by Colin Tudge
Analysis
Here is the analysis from Feldscher (2016):
• Adjectives have state arguments Parsons (1990). JtallK = λs.tall(s)
• A measure function (µ) takes a state and returns a degree (Wellwood, 2015). This can be thought of like a thematic role for an
event.
2.2.3
JNeville is 6ft tallK = λs.tall(s) ∧ µ(s) = 6f t ∧ Bearer(s) = Neville
Interim conclusions
An analysis of additive again supports a unification of stative and degree
theories of semantics.
(7) Jagainadd K = λfhd,hs,tii λdλs.∃s0 [s ≈ s0 ∧ f (d)(s0 ) ∧ µ(s) = µ(s0 ) + d]
• Adjectives have state arguments.
• Additive again calls for the repetition of a state of the same type
(using a similarity relation).
• Degrees (pos, measure phrases, etc.) are thematically related to
the state.
• The degree d being added is associated with this existentially quantified state of the same type.
• Additivity relates to repetition as the measure of one eventuality is
added to the other.
• The sentential state has a measure d0 which equals the measure of
the previous state, plus the degree d. Here is the additivity.
3
(8)
half as tall again as
Section goals:
hs, ti
hd, hs, tii
hd, hs, tii
Further additivity
Pansy 3
hdi
• Knock down the assumption that sets and events are working correctly.
hhd, hs, tii, hd, hs, tiii as Pansy
again
• Set up a different way to think about these relating to how they
pattern.
half as tall
3
For discussion of half and other multiplicative factor phrases, see Gobeski (2011).
Syntactic structure here built to match that, for convenience.
2
3.1
Data
Standard again repeats eventualities.
Comparative more deals only with degrees:
(10)
(11)
Eventive
a. I graded three more papers than John.
b. Harry won ten more Quidditch games than Draco.
Stative
a. I am more intelligent than John.
b. Neville is taller than Pansy.
Eventive
a. Last night I graded four papers, and today I graded two
more.
(6 total)
b. Yesterday I ran two miles, and today I ran one more.
(3
total)
(13)
Stative
a. Pansy was three feet tall at age 8, and now she’s two feet
taller.
(5ft tall)
b. The kitten weighed two pounds last week, and this week it
weighed one pound more.
(3 pounds)
Eventive
a. I graded half again as many papers as last week.
b. Harry won a third again as many games as Draco did.
(15)
Stative
a. Neville is half again as tall as Pansy.
b. This broom is a third again the price of the other one.
(17)
Stative
a. The door was open for the first time ever, so I shut it again.
b. The wizarding world is at peace again.
(18)
Eventive
a. I thought I could only grade two papers in one sitting, but I
managed to grade another three.
(5 in one go)
b. This time last year I could only run two miles, but as of today
I can run another three (miles).
(5 in one go).
(19)
Stative
a. I have two pieces of string. One is 10 inches, and the other is
another 15 (inches).
(second string 25in)
b. This kitten weighs one pound, and that one is another two
(pounds).
(K1 1lb, K2 3lb)
Thomas (2011) discusses what they call “additive another”, which sums
the measure of two eventualities to answer a QUD about a super eventuality.
Additive again sets the measure of the one state equal to the measure of
the other plus some d.
(14)
Eventive
a. I ran the race again.
b. Harry defeated Voldemort again.
What I called additive another sets the measure of an eventuality equal
to the measure of another plus some d.
Incremental more sums subevents into superevents.
(12)
(16)
(20)
(A: How wide are those two pieces of furniture together?)
B: The cabinet is 4 ft wide. The shelves are another 3 ft wide.
(3ft shelves, 7ft total)
His analysis:
3
• Normal another is adjectival other + existential quantification.4
b. The cake was supposed to be 10 inches wide, but it’s an additional three.
(13 in. wide together)
• Additive another presupposes this measure is a partial answer to
something in the QUD.
3.2
“Additional”– eventuality summing
a. All in all, I ran one mile yesterday, and then ran an additional two today.
(3 total)
b. The box is full, since the first cake was 10 inches wide, and
this one is an additional three.
(13 in. wide together)
(25)
“On top of” – degree addition
a. I only planned to run one mile, but then I ran two (more)
on top of that today.
(3 today)
b. The cake was supposed to be 10 inches wide, but it’s three
inches on top of that.
(13 in. wide)
(26)
“On top of”– eventuality summing
a. All in all, I ran one mile yesterday, and then ran two (more)
on top of that today.
(3 total)
b. The box is full, since the first cake was 10 inches wide, and
this one is three (more) on top of that. (13 in. wide
together)
Generalizations
Eventualities can be summed, events or states.
Adds only degrees
Comparative more (10),(11)
Additive again (14),(15)
Additive another (19),(18)
3.3
(24)
Sums eventualities
Incremental more (12),(13)
Standard again? (16),(17)
Thomas’s additive another (20)
Further data
Additivity is present in several other constructions, all with the same
ambiguity:
(21)
(22)
(23)
“Extra”– degree addition
a. I only planned to run one mile, but then I ran an extra two
today.
(3 today)
b. The cake was supposed to be 10 inches wide, but it’s an extra
three.
(13 in. wide)
Perhaps there are correlates lexically in verbs?
(27)
“Extra”– eventuality summing
a. All in all, I ran one mile yesterday, and then ran an extra
two today.
(3 total)
b. The box is full, since the first cake was 10 inches wide, and
this one is an extra three.
(13 in. wide)
4
4.1
“Additional” – degree addition
a. I only planned to run one mile, but then I ran an additional
two today.
(3 today)
Degree achievement verbs (inherent differentiality)
a. The gap widened (two feet).
b. The sea advanced (another meter).
Viewing degree additivity as a comparative
What about a comparative involves addition of degrees?
Comparative more:
4
As in Normally we go to Panera for lunch, but I thought we’d try another restaurant today.
(28) JmoreK = λdλα.A(µ)(α) > d
4
How to make this about addition.5
b. Look, if you spend $50, you get an extra book (?the books
you already bought) free!
(29) JmoreK = λdλα∃d0 .A(µ)(α) ≥ d + d0
Option 1:
Consider a differential comparative with explicit reference to a differential
degree (Morzycki, 2015).
(30)
(36) Jadd1 K = λdλα∃d0 .A(µ)(α) ≥ d + d0
Jadd2 K = λdλd0 λα.A(µ)(α) ≥ d + d0
Neville is three inches taller than Pansy.
If grabbing only one degree, it should be a salient one.
(31) JmoreK = λdλd0 λα.A(µ)(α) ≥ d + d0
4.2
(37) Jadd1 K = λdλα∃d0 : d0 ∈ C.A(µ)(α) ≥ d + d0
How does additivity mirror this?
Option 2:
Additive again again:
(38)
(32) Jagainadd K = λfhd,hs,tii λdλs.∃s0 [s ≈ s0 ∧ f (d)(s0 ) ∧ µ(s) = µ(s0 ) + d]
(39) JaddK = λdλd0 λα.A(µ)(α) ≥ d + d0
Can we unify all this degree additivity into one morpheme?
(33)
(34)
(35)
The cake was supposed to be 10 inches wide, but it’s an additional
three inches wider than it was supposed to be.
4.3
Two degrees mandated
a. The cake was supposed to be 10 inches, but it’s three inches
on top *(of that).
b. Last week I found ten dollars. This week I looked more carefully, and so I found half again ? (as much)!
Open issues
How to determine whether a morpheme gets add1 or add2 ?
Explaining other specificities between additive morphemes (ex: half v
two inches).
Optionally one or two
a. Neville is (ten inches) taller than Pansy.
Extending this to the event summing.
One mandated, local two impossible
a. A: Didn’t you only order ten dumplings?
B: They sent another two dumplings (* as/than/from ten
dumplings)! I got twelve!
5
Conclusion
A semantics for degree additivity is proposed, and the relation between
that and comparative more is shown, explaining patterns in additivity.
5
Note that this d0 must be a positive, nonzero degree. We could put this in the
denotation. But should we?
i. # Neville is negative three inches taller than Pansy.
ii. # A unicorn is zero inches tall, because it doesn’t exist.
5
References
Feldscher, C. (2016). States and degrees: additive again. Poster presented
at annual meeting of the North East Linguistic Society (NELS 47),
October 14–16, University of Massachusetts Amherst.
Gobeski, A. (2011). Twice versus two times in phrases of comparison.
Master’s thesis, Michigan State University.
Thomas’s additive another the QUD one:
(42) JanotherK = λd0 .λd.λPhshd,tii .λw : ∃Q ∈ QUD(c)[R(Q) = < ∧
B(Q)(w)(d0 + ιd00 [P (w)(d00 )])].P (d)
Wellwood’s full computation for Al’s coffee is hotter than Bill’s is:
Greenberg, Y. (2012). Event-based additivity in English and Modern
Hebrew. Verbal Plurality and Distributivity, 546:127.
S
Parsons, T. (1990). Events in the Semantics of English, volume 5. Cambridge, MA: MIT Press.
vSs
Thomas, G. (2011). Another additive particle. Proceedings of SALT 21.
Wellwood, A. (2016). States and events for s-level gradable adjectives. In
Proceedings of SALT 26.
Appendix
AP
hot
Thomas, G. (2010). Incremental more. Proceedings of SALT 20.
Wellwood, A. (2015). On the semantics of comparison across categories.
Linguistics and Philosophy, 38(1):67–101.
vs P
Al’s coffee
Morzycki, M. (2015). Modification. Cambridge University Press.
DegP
Deg0
(43)
thanP
muchµ -er
(44) JDeg0 KA = λdλα.A(µ)(α) > d
JDegPKA = λα.A(µ)(α) > δ
JAPKA = λs.hot(s)&A(µ)(s) > δ
Jvs PKA = λxλs.Holder(s)(x)&hot(s)&A(µ)(s) > δ
JSKS = λsHolder(s)(ac)&hot(s)&A(µ)(s) > δ
= T iff∃s[Holder(s)(ac)&hot(s)&A(µ)(s) > δ]
Gobeski’s denotation for half (in a Kennedian DegP):
(40) JhalfK = λFhd,he,tii λdλx[max{dX : F (dX )(x)} ≥ 12 d]
Thomas’s denotation for his incremental more:
(41) Jmoreinc K = λd.λe0 .λDhd.hv,tii .λe.∃d0 ∃D0 ∈ alt(D)[D0 (d0 )(e0 )]∧D(d)(e)∧
∃D00 ∈ alt(D)[D00 (d + δ)(e ⊕ e0 )]
where δ = ιd0 [∃D0 ∈ alt(D)[D0 (d0 )(e0 )]]