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A unified account of the distributive and free choice inferences of disjunction under modals

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A uniﬁed account of the distributive and free choice inferences of disjunction under modals
Disjunction embedded into epistemic modal expressions like
possible
,
likely
, and
certain
give riseto the inferences that the disjuncts are epistemically possible. While identical, these inferencesare classiﬁed and treated differently, with the ones in (1a/b) labeled ‘free choice’ inferences whilethose in (2a/b), ‘distributive’ ones. Distributive inferences are routine predictions of most standardaccounts of implicature (Sauerland 2004, Fox 2007, a.o.), while free choice inferences are notori-ously problematic (Kratzer and Shimoyama 2002, Fox 2007, Chierchia 2013 a.o.). We show that adegree-based semantics for modals can easily predict all these inferences via the same mechanism.Our proposal dovetails well with (though does not require endorsing) recent probability-based se-mantics for epistemic modals (Yalcin 2010, Lassiter 2011, 2014 a.o.), and hence can be seen as anindirect argument for analyses in this vein. On the other hand, we show that our analysis can alsobe implemented by accommodating Klecha’s 2014 worries against probabilistic theories of thesemodals. We conclude by showing how the proposal can be extended to deontic modals, givenreasonable assumptions about a degree semantics for these modals.(1) It’s possible that it will rain or snow (2) It’s likely/certain that it will rain or snow
(
a
)
it is possible that it will rain
(
a
)
it is possible that it will rain
(
b
)
it is possible that it will snow
(
b
)
it is possible that it will rain
Degree semantics for modal adjectives.
We adopt a standard analysis of degree adjectives (Heim2000 a.o.) and assume that the meaning of
possible/likely/certain
is that in (4), mapping a propo-sition to the set of degrees up to which that proposition is probable.
1
In addition, we make thestandard assumption that in the positive form these adjectives combine with a silent morpheme
POS
, which contributes an operation of existential closure binding the degree variable as in (5),where
s
is the standard function (Kennedy & McNally 2005, Kennedy 2007 a.o.)(4)
[[
possible/certain/likely
]]
=
λpλd
[
Pr
(
p
)
≥
d
]
(5)
[[
POS
]]
=
λG
d,t
∃
d
[
s
(
G
)(
d
)
∧
G
(
d
)]
The standard function
s
gives different results depending on the type of adjectives it combines with.In particular, the result of combining
POS
with absolute adjectives like
possible
and
certain
is thatthe probability of raining is not 0 and equals to 1, respectively. In the case of a relative adjectivelike
likely
, the standard varies contextually (cf. Yalcin 2010, Lassiter 2011, 2014).(6)
[[
it is
POS
[possible [that it is raining]]
]]
=
∃
d
[
d >
0
∧
Pr
(
rain
)
≥
d
]
(7)
[[
it is
POS
[likely [that it is raining]]
]]
=
∃
d
[
d >
s
(
likely
)
∧
Pr
(
rain
)
≥
d
]
(8)
[[
it is
POS
[certain [that it is raining]
]]
=
∃
d
[
d
= 1
∧
Pr
(
rain
)
≥
d
]
Scalar implicatures.
For concreteness, we adopt an exhaustiﬁcation-based account of scalar im-plicatures (Chierchia et al. 2012, Fox 2007 a.o.). Scalar implicatures are generated by an exhaus-tivity operator
EXH
.
EXH
takes as arguments a sentence
S
and a set of alternatives
A
lt
(
S
)
, andreturns the conjunction of
S
with the negation of the ‘excludable’ alternatives in
A
lt
(
S
)
. An alter-native is excludable just in case (a) negating it doesn’t contradict the assertion
S
, and (b) negatingit doesn’t force us to accept any other alternative in
A
lt
(
S
)
(Sauerland 2004, Fox 2007).
1
More precisely, given any propositional argument
φ
, we write (i): where
e
is a relevant epistemic state; epistemicstates are pairs
E,Pr
of a set of possible worlds
E
and a probability measure
Pr
deﬁned over
E
.(i)
[[
possible/certain/likely
φ
]]
e,w
=
λd
[
Pr
e,w
(
{
w
′
: [[
φ
]]
e,w
′
}
)
> d
]
(8)
[[
EXH
[
φ
][
A
lt
(
φ
)]]]
w
=
[[
φ
]]
w
∧∀
ψ
∈
excl
(
φ,
A
lt
(
φ
))[
¬
[[
ψ
]]
w
](i)
excl
(
φ,
A
lt
(
φ
)) =
{
ψ
∈ A
lt
(
φ
) :
φ
ψ
∧¬∃
χ
[
χ
∈ A
lt
(
φ
)
∧
(
φ
∧¬
ψ
)
⊆
χ
]
}
Deriving the inferences
The inferences of
certain
follow straightforwardly from exhaustiﬁcationand standard assumptions about alternatives. The exhaustiﬁcation of (2a) (with
certain
) involvesnegating the alternatives in (9) giving rise to the strengthened meaning in (10). Informally, exhaus-tiﬁcation has the effect of ‘distributing’ the probability over the two disjuncts: the strengthenedmeaning of (2) says that the probability of rain or snow is 1 and that the probabilities of each of rain and snow is smaller than 1. Given standard assumptions about probability measures, it followsthat the probabilities of both rain and snow are positive. But then, given our analysis of
possible
in(6), this, in turn, entails the inferences in (2a) and (2b).(9)
{
it is certain that it will rain, it is certain that it will snow
}
(10)
[[
it is
EXH
[
POS
certain that it will rain or snow]
]]
=
Pr
(
r
∨
s
) = 1
∧¬
(
Pr
(
r
) = 1)
∧¬
(
Pr
(
s
) = 1)
⇒
Pr
(
r
)
>
0
∧
Pr
(
s
)
>
0
The inferences with
likely
are easily accounted for in a completely analogous way. Assuming forconcreteness that
s
(
likely
) is .5 in the context, the alternatives negated are in (11) and the strength-ened meaning is (12), which entails the inferences in (2a) and (2b), in the same way as above.(11)
{
it is likely that it will rain, it is likely that it will snow
}
(12)
Pr
(
r
∨
s
)
> .
5
∧¬
(
Pr
(
r
)
> .
5)
∧¬
(
Pr
(
s
)
> .
5)
⇒
Pr
(
r
)
>
0
∧
Pr
(
s
)
>
0
Finally, the inferences of
possible
in (1a) and (1b) are also captured in the same way. The onlyextra assumption required is that the relevant exhaustivity operator scopes between
POS
and themodal (as in (13)). Analogous assumptions about alternatives yield that the meaning of the con-stituent headed by
EXH
(before combing with
POS
) is (14):(13)
[[
it is
POS
[
EXH
[possible that it will rain or snow]]
]]
(14)
[[
EXH
possible that it will rain or snow
]]
=
Pr
(
r
∨
s
)
> d
∧
Pr
(
r
)
≤
d
∧
Pr
(
s
)
≤
d
Applying
POS
we get (15), which entails (16). But then, again, on our semantics, (16) amountsprecisely to the assertion conjoined with its inferences in (1a) and (1b).(15)
∃
d
[
d >
0
∧
Pr
(
r
∨
s
)
> d
∧
Pr
(
r
)
≤
d
∧
Pr
(
s
)
≤
d
]
(16)
∃
d
[
d >
0
∧
Pr
(
r
∨
s
)
> d
]
∧ ∃
d
[
Pr
(
r
)
> d
]
∧ ∃
d
[
Pr
(
s
)
> d
]
In sum: a degree-based semantics for modal adjectives like
possible
,
certain
and
likely
provides auniﬁed account of the inferences in (1a) and (1b) and the similar inferences in (2a) and (2b) as asimple scalar implicature.
Extensions
: Freechoiceinferencesarealsotriggeredbyothermodals, inparticulardeonticmodals.Our account can be extended to these cases, provided that the relevant operators is given a seman-tics based on a degree measure that vindicates the property in (16) (Holliday and Icard 2013,Cariani 2013, Wedgwood 2015).(16) A
DDITIVITY
Degree
(
p
)
≥
Degree
(
q
) iff
Degree
(
p
∧¬
q
)
≥
Degree
(
q
∧¬
p
)
Selected References:
•
Fox, D. 2007
.
Free choice and the theory of scalar implicatures
.
•
Holl-iday, W. and Icard, T. 2013
.
Measure semantics and qualitative semantics for epistemic modals
.
•
Lassiter, D.: 2011
.
Measurement and Modality
.
•
Yalcin, S.: 2010
.
Probability operators
.
References
Chierchia, G.: 2013,
Logic in Grammar: Polarity, Free Choice, and Intervention
, Oxford Univer-sity Press.Chierchia, G., Fox, D. and Spector, B.: 2012, The grammatical view of scalar implicatures andthe relationship between semantics and pragmatics,
in
C. Maienborn, K. von Heusinger andP. Portner (eds),
Semantics: An International Handbook of Natural Language Meaning volume3
, Mouton de Gruyter, Berlin.Fox, D.: 2007, Free choice and the theory of scalar implicatures,
in
U. Sauerland and P. Stateva(eds),
Presupposition and Implicature in Compositional Semantics
, Palgrave, pp. 71–120.Holliday, W. and Icard, T.: 2013, Measure semantics and qualitative semantics for epistemicmodals,
Proceedings of SALT 23
, pp. 514–534.Klinedinst, N.: 2007,
Plurality and Possibility
, PhD thesis, UCLA.Kratzer, A. and Shimoyama, J.: 2002, Indeterminate pronouns: The view from Japanese,
in
Y. Otsu(ed.),
Proceedings of the Tokyo conference on psycholinguistics
, Vol. 3, Hituzi Syobo, Tokyo,pp. 1–25.Lassiter, D.: 2011,
Measurement and Modality: The scalar basis of modal semantics
, PhD thesis,NYU.Lassiter, D.: 2014, Epistemic comparison, models of uncertainty, and the disjunction pussle,
Jour-nal of Semantics
doi: 10.1093/jos/ffu008
.Moss, S.: in press, On the semantics and pragmatics of epistemic vocabulary,
Semantics and Prag-matics
.Sauerland, U.: 2004, Scalar implicatures in complex sentences,
Linguistics and Philosophy
27
(3), 367–391.Swanson, E.: 2015, The application of constraint semantics to the language of subjective uncer-tainty,
Journal of Philosophical Logic
pp. 1–26.
URL:
http://dx.doi.org/10.1007/s10992-015-9367-5
Yalcin, S.: 2010, Probability operators,
Philosophy Compass
5
(11), 916–937.

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