A superlative analysis of superlative scalar modifiers
Doris Penka, Universität Konstanz
Background. The scalar modifiers at least and at most have recently received growing interest in
the semantic literature (Geurts & Nouwen 2007, Büring 2008, Nouwen 2010). What proves to be
particularly challenging is the fact that they generally involve an epistemic meaning component,
which vanishes in certain contexts: when at least and at most occur in episodic contexts without
an overt modal, they imply that the speaker is uncertain of the precise value, making (1a) an odd
thing to say. But when at least co-occurs with a necessity modal (cf. 1b), and at most with a
possibility modal (cf. 1c), a reading not conveying speaker uncertainty is salient. The same holds
for generic/habitual statements (cf. 1d).
(1)
a. I have at least / at most three children.
‘The speaker isn’t sure how many children (s)he has but considers three to be the lower/
upper bound.’
b. You must have at least three children (to be eligible for child benefit).
c. You may check out at most ten books at the same time.
d. Computers of this kind have at least/at most 2GB of memory.
Nouwen (2010) argues that the speaker uncertainty effect arising in episodic contexts is due to
a covert epistemic possibility operator, which is inserted to rescue an utterance otherwise ruled
out. So far, none of the existing accounts of at least and at most has taken into account the fact
that they (and their counterparts in a great number of languages) involve superlative
morphology. This paper shows that analysing at least and at most as superlatives contributes to
solving the puzzles surrounding them.
Superlative temporal modifiers. Our analysis starts from the observation that the empirical
generalisations about at least and at most carry over to the superlative temporal modifiers at the
latest and at the earliest, cf. (2). This strongly suggests that they are related to superlative
semantics.
(2)
a. Peter arrived at 6 pm at the earliest/ at the latest.
b. You may leave at 6 pm at the earliest.
c. You must leave at 6 pm at the latest.
d. On weekends, Ann gets up at 8 am at the latest/ at the earliest.
speaker uncertainty
no speaker uncertainty
no speaker uncertainty
no speaker uncertainty
Building on the analysis of the temporal adverbials early and late in von Stechow (2007), cf. (3),
and the analysis of superlatives in Heim (1995/1999), adapted to temporal superlatives in (4),
(2a) has the LF (5) and is associated with the truth conditions and presuppositions in (6).
(3)
a. [[ spät / late ]] = λdi. λti. t ≥ d
b.
[[ früh / early ]] = λdt. λti. t < d
(4)
[[ -estC ]] = λR<i,<i,t>>.λti: t ∈ C & ∀t’ [ t’ ∈ C → ∃d R(d)(t’) ] & ∃t’ [ t’ ∈ C & t’ ≠ t ].
∃di [ R(d)(t) & ∀t’ [ t’ ∈ C & t’ ≠ t → ¬R(d)(t’) ]]
(5)
6pm -estC λd λt2 [ PAST [<i,t> [<i,t> d-late] [<i,t> λt3 [ [t3 at t2] λt1 [ Peter arrived t1 ]]]]]
(6)
a. truth conditions:
∃d [ ∃t [ t < s* & t < d & t = 6 pm & Peter arrives at t ] &
∀t’ [t’ ∈ C & t’ ≠ 6 pm → ¬∃t’’[t’’< s* & t’’ < d & t’’=t’ & Peter arrives at t’’]]]
b. presuppositions: (i) 6 pm ∈ C
(ii) ∀t [ t ∈ C → ∃d ∃t’ [t’ < s* & t’ < d & t’ = t & Peter arrives at t’ ]
(iii) ∃t [ t ∈ C & t ≠ 6 pm ]
Crucially, superlatives presuppose that the comparison class C consists of at least two elements,
in this case times at which Peter arrived. The presuppositions (ii) and (iii) together entail that
Peter arrived at more than one time (in the contextually relevant interval), which leads to a
presupposition failure. Presupposition failure can be prevented by assuming a covert epistemic
possibility operator in the spirit of Nouwen (2010). If -estC takes wide scope with respect to
epistemic possibility, the comparison class C consists of times at which Peter arrives in some
epistemic alternative, which might be different. Also in habitual sentences like (2d), C is made
up of different times.
Analysis of at least and at most. Extending our analysis of the superlative temporal modifiers,
we analyse at least and at most as the superlative forms of adverbial much/little with the
semantics in (7). Illustrating for sentence (8a), we propose LF (8b), where -estC-much is merged
countercyclically adjoining to a predicate derived by QR of the degree phrase 2m (cf.
Nissenbaum 1998), followed by QR of -estC. The presuppositions associated with this LF are
contradictory (requiring that John has more than one height), if a functional meaning for
adjectives is assumed (e.g. [[tall]] = λdd. λxe. Height(x) = d). Again, inserting a covert epistemic
possibility operator prevents presupposition failure.
(7)
a. [[much ]] = λdd. λd’d. d’ ≥ d
b.
[[ little ]] = λdd. λd’d. d’ < d
(8)
a. John is at most 2m tall.
b. 2m [ -estc λd [d-much] λd’ [ John d’-tall ]]
(9)
a. truth conditions:
∃d [ 2m ≥ d & Height(j) = 2m &
∀d’ [ d’ ∈ C & d’ ≠ 2m → ¬ [ d’≥ d & Height(j) = d’ ]]]
b. presuppositions: (i)
2m ∈ C
(ii) ∀d’ [ d’ ∈ C → ∃d [ d’≥ d & Height(j) = d’ ]
(iii) ∃d [ d ∈ C & d ≠2m]
Summary. The observation that at least /at most share certain puzzling behaviour with at the
earliest/at the latest strongly suggests that it is related to superlative semantics. We propose a
fully compositional analysis of at the earliest/at the latest and at least/at most in terms of
superlative semantics. The fact that superlative modifiers yield a reading of speaker uncertainty
in non-modal episodic contexts follow from presuppositions standardly attributed to the
superlative, if covert epistemic modal operators are assumed to be available.
References
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