Does the structural approach to alternatives give us just enough
alternatives to solve the symmetry problem?
Richard Breheny
University College London
Nathan Klinedinst
University College London
Jacopo Romoli
Ulster University
Yasutada Sudo
University College London
Abstract The structural approach to alternatives (Katzir 2007, Fox & Katzir 2011)
aims at solving the symmetry problem of scalar implicatures by referring to a notion
of structural complexity of alternatives, but problematic data with ‘indirect’ and
particularised scalar implicatures have been raised (Romoli 2013, Trinh & Haida
2015). In an attempt to defend the structural approach from these problems, Trinh
& Haida (2015) recently propose to augment the theory with a constraint they call
‘atomicity.’ In this paper, it is shown that Trinh & Haida’s atomicity constraint
falls short of explaining simple variants of the original problems, and moreover
that it runs into trouble with the inference of sentences involving scalar adjectives
like full and empty. In addition, we point out two related problems, the problem of
‘too many lexical alternatives’ (Swanson 2010) and the problem of ‘too few lexical
alternatives.’ The three problems epitomise a general difficulty of constructing just
enough alternatives under the structural approach to solve the symmetry problem in
fully generality.
1
1.1
Introduction
Alternatives and the symmetry problem
Theories of scalar implicatures, while quite diverse, tend to have the following
shape: the scalar implicature of an assertion of S arises by deriving the negations
of alternatives to S. In the Neo-Gricean tradition this is derived in two stages. A
primary inference is that the speaker is not certain about any more informative,
relevant alternatives. This ‘ignorance’ implicature about the speaker is made on the
grounds that they will say the most informative relevant thing they believe to be
true. A secondary assumption of opinonatedness yields the stronger implicature that
the speaker thinks the alternative is not true and intends to communicate this (see
Soames 1982, Horn 1972, 1989, Gazdar 1979, Sauerland 2004 among others). Other
theories (Chierchia, Fox & Spector 2012, Fox 2007 among others) derive the strong
scalar implicature directly via a grammatical mechanism that is independent of the
1
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
mechanisms of ignorance implicature. Importantly, for either account the derivation
of strong scalar implicatures, such as (1b) for the sentence (1a), is assumed to involve
a set C of alternatives, as in (2).1
(1)
a.
b.
John did some of the homework.
John didn’t do all of the homework
(2)
C = {John did some of the homework, John did all of the homework}
Crucial to this explanation is the following question: given a sentence and a
context, how is the set of alternatives determined? Importantly, this must be answered
in a way that addresses the S YMMETRY P ROBLEM (Kroch 1972, Fox 2007, Katzir
2007, Fox & Katzir 2011 among others). Schematically, the symmetry problem
has the following form. Suppose that sentence S implicates ¬A, by hypothesis the
negation of some alternative A. Then a sentence equivalent to S ∧ ¬A must not be in
the set alternatives. For example, (3) must not count as an alternative to (1a).2 If it
did, the implicature (1b) could not be derived, on pain of contradicting the assertion
(1a) itself (or the opinionatedness assumption, at least, on the Neo-Gricean view).
(3)
John did some but not all of the homework.
Therefore, given a sentence S and its attested inference ¬A, one needs a principled
way to allow the alternative A and, at the same time, disallow the potential symmetric
alternative S ∧ ¬A.3 (Furthermore, that (1a) seemingly never implicates that John
did all the homework, implies that (3) cannot be an alternative on its own to the
exclusion of the (1a).)
1 There are approaches to scalar implicatures that do not make use of alternative sentences to derive
scalar inferences, e.g. Van Rooij & Schulz 2004, 2006 and Fine to appear. Our focus in this paper is
the former kind of approach and specifically the structural approach to alternatives.
2 This characterisation of the symmetry problem is based on the idea that the alternatives from which
scalar implicatures arise are only the ones that are strictly stronger than the assertion. Below, we
move to a definition which also excludes alternatives that do not stand in entailment relation with the
assertion, then we have more symmetric alternatives that we have to deal with. In particular, given
a sentence S that implicates the negation of some alternative A then ¬A should not be in the set of
alternatives. In the case above, (i) should not count as an alternative to (1a).
(i)
John didn’t do all of the homework.
3 The definition of symmetric alternatives from Fox & Katzir 2011:
(i)
Let S, S1 , S2 be sentences such that S1 and S2 are alternatives to S. We say that S1 and S2
are symmetric alternatives of S if both of the following are true.
a.
JS1 K ∪ JS2 K = JSK
b.
JS1 K ∩ JS2 K = 0/
2
A standard solution to the symmetry problem postulates that alternatives are
restricted lexically (Horn 1972 and much subsequent work). That is, lexical items
like some and all are marked in the lexicon to form a scale, and at the sentential level,
alternative sentences are constructed only by replacing these scalar terms with their
scale-mates. In this setting, the symmetry problem is avoided by the following lexical
assumption: Some and all are associated in a scale but some and some but not all
are not. Without independent motivation, it is unclear that postulating scales in this
way constitutes a genuine solution to the symmetry problem. To (partially) address
this concern, Horn 1989 and Matsumoto 1995, propose a general ‘monotonicity’
constraint on lexical scales, that scale-mates license entailments in the same direction
(i.e. all of them are upward entailing, all of them are downward entailing). However,
see Katzir (2007) for arguments that such a constraint is too strong, failing to account
for certain examples.
As an alternative to the lexical approach, Katzir (2007) and Fox & Katzir (2011)
advocate a STRUCTURAL APPROACH to alternatives, which appeals to an algorithm
for constructing alternatives based on a notion of structural complexity. Before
moving to illustrate their proposal in the next subsection we discuss arguments
for including not only alternatives that are stronger than the assertion, but also
alternatives that are logically independent.
1.2
The need for non-weaker alternatives
As outlined above scalar implicatures arise from reasoning over stronger alternatives
that the speaker might have said. We also said that the set of such alternatives should
be restricted in order to avoid the symmetry problem. At the same time, however,
there are arguments for expanding the set of alternatives for scalar implicatures
computation, to include alternative that are merely not weaker.
One main argument comes from non-monotonic contexts. A non-motonic operator embedding a scalar term like some, X(some) can implicate the negation of
X(all), i.e. logically independent alternative obtained by replacing all for some.
Chemla & Spector (2011) present experimental evidence for such implicatures from
sentences containing the non-monotonic operator, exactly one, embedding some.
Spector (2007) present cases involving exactly one and bare plurals. The example
that we find to be clearest is (4).
(4)
Every letter but A is connected to some of its circles.
The sentence in (4) involves a non-monotonic operator, Every letter but A, embedding
a scalar term, some. The question is whether the sentence in (4) can have a reading
with an inference corresponding to the negation of the non-weaker alternative in (5).
Such a reading can be distinguished from the literal reading of (4), by looking at the
3
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
connections that the letters other than A have.
(5)
Every letter but A is connected to all of its circles.
The two readings are paraphrased in (6a) and (6b). We illustrate two situations in
Figure 1 (modeled after Chemla & Spector 2011), one being only compatible with
the literal meaning and one with both. Intuitively, there is a clear difference between
the two situations, in that (4) appears harder to accept as a description of the picture
on the left than one of the right, which, in turn, suggests that the reading with the
inference exists.
(6)
a.
b.
Literal meaning: A is connected with none of its circles, all the other
letters are connected with some or all of their circles.
Meaning with inference: A is connected with none of its circles, all
the other letters are connected with some of their circles but not all of
the others are connected with all of their circles.
But if the reading with the inference exists, then we have a strong argument for the
need of non-weaker alternatives in the computation of scalar implicatures.
An additional argument for non-weaker alternatives come from the example of
contextual alternatives like the one we discuss below in (14). Therefore, in sum, we
think there are good arguments for expanding the set of alternatives from only the
stronger ones to the non-weaker ones. In the next section, we discuss the proposal
by Katzir (2007), Fox & Katzir (2011) and then we turn to the problems that this
proposal faces.
Figure 1
Situations in which either only the literal meaning of (4) is true (left) or
both the literal meaning and the meaning with the inference are (right)
4
2
The structural approach to alternatives and its problems
2.1
Katzir & Fox’s approach
As an alternative to lexical scales, Katzir (2007) and Fox & Katzir (2011) advocate
a structural approach to alternatives that appeals to structural complexity (relative
to what is salient in context). Formal alternatives are defined as in (7), with the
notion of substitution source in turn defined as in (8) (from Katzir (2007); Fox &
Katzir (2011) further employ focus relativity, something we suppress except where
relevant; see below).
(7)
The formal alternatives F(S) of sentence S in context c are the set of
sentences derivable by successive replacement of constituents of S with
items in the substitution source of S in c.
(8)
An item α is in the substitution source of S in c if
a. α is a constituent that is salient in c (e.g. by virtue of having been
mentioned); or
b. α is a sub-constituent of S; or
c. α is in the lexicon.
In line with the arguments discussed in subsection 1.2, Fox & Katzir (2011) assume
that alternatives are not restricted to strictly stronger propositions. A characterisation of scalar implicature compatible with these proposals can be given as follows
(Sauerland 2004, Fox 2007 among others).
(9)
Scalar Implicatures: The scalar implicatures of S are the negations of all
alternatives A of S such that A is an element of F(S) and A is RELEVANT
and NOT WEAKER than S ∧ ¬A does not contradict the negation of any
relevant A element of F(S) given S.
Given that there is no lexical restriction and all lexical items can be substituted for
each other, (modulo grammatically), if all formal alternatives were always employed
to derive scalar implicatures, the theory would lead to massive over-generation.
Therefore Fox & Katzir (2011) also assume that the context restricts F(S) into
a subset C that only consists of the formal alternatives that are relevant; where
contextual restriction is constrained by the closure condition in (10):
(10)
Closure condition on C:
a. C ∈ F(S)
b. S ∈ C and
c. There is no S0 ∈ (F(S) −C) s.t. S0 is in the Boolean closure of C.
5
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
The first clause requires C to be a subset of the formal alternative and the second
clause demands what is asserted to be in the set C. Finally, (10c) requires C to include
sentences which express all Boolean combinations of sentences in C, provided that
they are in the set of formal alternative F(S) in the first place.4 The idea behind
(10c) is the following intuition. If it is of interest in the context whether p is true
or false, then it is of interest whether ¬p is. Thus if the former is included then so
should be the latter, unless theres no formal alternative expressing it. Similarly, if
it is relevant in the context whether p is true and whether q is true, then it will be
relevant whether their conjunction is. Therefore, no formal alternative that expresses
the Boolean closure of C should be left out.
Going back to (1a), we can now see how formal alternatives as defined in (7)
derive the inference and circumvent the symmetry problem at the same time. The
most important part of Fox and Katzir’s analysis is that it excludes the problematic
alternative (3) by referring to its relative structural complexity. That is, since (3)
is structurally more complex than (1a), it simply cannot be obtained from (1a) by
successive replacements of constituents with items in the substitution source. (11a)
or (11b) on the other hand, can be generated by replacing some with all or many,
which are of equal structural complexity.
(11)
a.
b.
John did all of the homework.
John did many of the homework.
From the formal alternatives, we can then restrict a set of relevant alternatives C,
following the constraints in (10). For instance, (12) is a permissible set C of relevant
alternatives for (1a) and will yield the attested scalar implicature that John did not
do all of the homework.
(12)
C = {John did some of the homework, John did all of the homework}
Note that not all propositions that are in the Boolean closure of C must be
expressed by the sentences in C; (10) only demands this for those that are expressed
by formal alternatives. For example, the propositions expressed by the sentences
in (3) and that in (13) are in the Boolean closure of C, but these sentences are not
formal alternatives of (1a), as there is no way to construct them given (7) – nor,
apparently, equivalent ones, so they will not be in C.5
4 Here the Boolean closure of C is the smallest set of propositions such that it i) contains all the
propositions in C and ii) contains ¬p and p ∧ q if it contains p and q.
5 The definition of formal alternatives in (7) allows constituents that are salient in the context to be in
the substitution source. One question is whether one can make problematic symmetric alternatives
like (13) salient in the context. Katzir (2007), Fox & Katzir (2011) construct the following example
in (i), which is supposed to show that this is indeed possible. As they admit, however, (i) is unnatural,
arguably for independent pragmatic reasons, therefore it is difficult to conclude much from it.
6
(13)
John did not do all of the homework.
We end this section with a mention of “particularised” scalar implicatures. Given (9)
scalar implicatures are now allowed to arise also from alternatives that are logically
independent from the assertion. This way of framing the characterisation of scalar
implicatures implies a different perspective on particularised implicatures, like in
(14c):
(14)
a.
b.
c.
Mary got drunk. What did John do?
He smoked pot.
John didn’t get drunk
Given (7), (15) is an alternative of (14b) and we can obtain (14c) by negating it.
Notice that, as mentioned, (15) is not stronger than (14b), so it constitutes another
argument for non-weaker alternatives.
(15)
He got drunk.
This treatment of particularised scalar implicature superficially differs from the
widely discussed neo-Gricean treatment given in Hirschberg 1991, which develops
the idea that sets of alternative are created on an ad hoc basis according to what is
relevant in context and that these are partially ordered by entailment. In this example,
an ad hoc scale would give rise to an alternative that John smoked pot and got drunk.
Since for all particularised examples discussed in Hirschberg, the more informative
alternatives are formed by conjoining the asserted proposition with contextually
relevant additional propositions the effect is similar as that described here since Fox
& Katzir (2011) assume that contextual relevance constrains alternatives.6
(i)
??Yesterday John did some but not all of the homework and today he did some.
Nonetheless, to the extent that one can have judgements about sentences like (i), it appears clear that
(i) cannot have the implicature that John did all of the homework. This implicature would, however,
be possible if the contextual restriction of the formal alternatives was allowed to keep (13) or (3) and
not (11a). This is however not possible given the closure conditions on contextual restriction in (10).
6 A question is whether all cases that seem to require ad hoc scales can be handled using Fox & Katzir’s
(2011) complexity account. Take for example the following from Hirschberg 1991:
(i)
A: Do you have information on [Kathy M. for maternity] ...?
B: I don’t think she has delivered yet.
A: Then she HAS been admitted.
B: Yes.
Given A’s response to B’s first utterance, the latter arguably implicates that (B thinks that) the woman
has been admitted. Such an implicature would seem to require an alternative sentence [I don’t think
she has been admitted]. Whether this can be derived via (7) is moot since the form of the alternative
7
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
2.2
The problems of indirect and particularised scalar implicatures
The complexity-based approach, however, runs into problem with so-called indirect
implicatures, that is implicatures arising from a sentence containing a strong scalar
item in a downward entailing context like negation, as pointed out by Romoli (2013).
The crucial property of these cases is that unlike the examples we have seen so far,
the asserted sentence is structurally more complex than the problematic symmetric
alternative, and hence Katzir’s (2007) substitution procedure automatically includes
them as formal alternatives.
A case in point is given in (16a), which has a scalar implicature in (16b). To
obtain (16b), the following alternative in (17) needs to be negated.7
(16)
a.
b.
John didn’t do all of the homework.
John did some of the homework
(17)
John didn’t do some of the homework.
Fox & Katzir, however, wrongly predict that this alternative cannot be negated,
due to the presence of its symmetric alternative. Consider first the set of formal
alternatives of (16a) that can be constructed according to the definition in (7). To
avoid clutter in the examples, we will write alternatives in schematic form from now
on, adopting Trinh & Haida’s (2015) notation (whereby, for example, all = ‘John did
all of the homework’).
(18)
F(16a) = { ¬all, ¬some, all, some }
Here we would like to negate ¬some to obtain the inference (16b), but we cannot
because we also have the symmetric alternative some, generated from (16a) by
substituting the NegP in the assertion with its subconstituent VP, and substituting
some for all within the latter, as shown in (19).8
has a greater level of complexity than the assertion, containing a passive with the particle, [been] as
extra linguistic material. Since this is another manifestation of the problem of too few alternatives,
discussed below, we leave it open whether all cases involving ad hoc scales can be handled by Fox &
Katzir (2011).
7 (17) typically only has a reading in which some takes scope out of negation, probably due to the
positive polarity properties of some. As such, it is not clear that it should count as an alternative. If
so, the problem can still be framed in terms of the alternative (i). Thanks to an anonymous reviewer
for discussion on this point.
(i)
John didn’t do any of the homework.
8 For expository purposes, we represent the subject in the VP-internal position and also the verb in the
inflected form but nothing crucial hinges on this.
8
(19)
[NegP not [VP John did all of the homework]]
⇒ [VP John did all of the homework]
⇒ [VP John did [some] of the homework]
(16a)
subconstituent
all/some
Given that some is in C, the negation of ¬some contradicts the negation of some
and hence cannot be negated. (Note that it is not possible to use a subset C0 of the
formal alternatives containing ¬some but not some, i.e. C0 = { ¬all, ¬some, all } to
derive the desired scalar implicature. C0 violates the closure condition (10): some is
both in F(16a) and in the Boolean closure of C0 given that JsomeK = J¬¬someK.)
A similar problem arises with the slightly more complex example in (20) adapted
from Trinh & Haida (2015), a particularised implicature like (14), but arising from
an alternative containing negation.
(20)
Bill went for a run and didn’t smoke. What did John do?
John went for a run.
John smoked
As evidence for this inference, notice that (20) sounds unnatural in a context in which
it is known that John didn’t smoke. This parallels the unnaturalness of sentences like
(16a) and (14) in contexts in which their implicatures are false – ones which John
didn’t do any homework and ones in which he did get drunk, respectively.9
The implicature of (20) could be derived in principle by negating ¬smoke, or
run ∧ ¬smoke (which together with the assertion, run, entails smoke). But here
again, Fox & Katzir’s (2011) system under-generates, for essentially the same reason
as above. Assuming that the VPs in (20) are salient, ¬smoke, run ∧ ¬smoke or both
can be generated as formal alternatives, and we will have that:
(21)
F(20) ⊇ { run, ¬run, smoke, ¬smoke, run ∧ smoke, run ∧ ¬smoke }
However, the closure condition (10) on C requires that if C contains run ∧ ¬smoke
it also contains the symmetric alternative run ∧ smoke, since C must contain
run itself and any formal alternative in its Boolean closure (Jrun ∧ smokeK =
Jrun ∧ ¬(run ∧ ¬smoke)K.) For parallel reasons, if C contains ¬smoke, it must
9 An anonymous reviewer pointed out to us that for them (20) tends to have an ignorance inference that
the speaker is unsure as to whether John smoked, rather than the scalar implicature that John smoked.
We agree that this reading is possible and maybe even the most salient. However, variants of (20),
like (i), for which it is very plausible that the speaker is well informed about the relevant activity that
she herself has engaged with, the scalar implicature tends to arise more systematically.
(i)
Bill went for a run and didn’t smoke.
I went for a run.
I smoked
9
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
also contain the symmetric alternative smoke.
It should be noted here that Fox & Katzir (2011) assume an additional constraint
on the set of formal alternatives that requires replacements to be performed only on
focussed parts of the sentence. If focus is narrow enough, i.e. below negation, as in
(22a) or (22b), then the offending alternative some cannot be generated, as negation
cannot be replaced here.
(22)
a.
b.
John didn’t [do all of the homework]F
John didn’t do [all of the homework]F
Romoli (2013), however, points out that the scalar implicature is still observed, even
when the focus is broad enough to include negation, as in the following example.
(23)
What happened at school today?
[John (my favourite student) didn’t do all of the homework]F .
Thus the under-generation problem noted above persists for such cases (and of
course examples like (20) which contain no negation).10
In the following section we turn Trinh & Haida (2015)’s attempt to modify the
structural theory in a way the deals with the under-generation problem.
3
3.1
The addition of the atomicity constraint and its problems
Trinh and Haida’s addition
Adopting Katzir’s (2007) and Fox & Katzir’s (2011) structural approach to alternatives, Trinh & Haida (2015) propose to add one extra constraint on the construction
of formal alternatives which prohibits further substitutions on the expressions in the
substitution source. They state this constraint as follows:
10 At this point, one might propose that implicatures only arise from stronger members of C, rather than
all non-weaker ones. This would solve the problem because some, unlike its symmetric alternative
¬some, is not stronger than ¬all. As discussed in section 1.2, however, there are good arguments
against this idea.
A more sophisticated version of this move could allow negating non-weaker alternatives in
general, but still exploit the fact that the two symmetric alternatives some and ¬some differ in
their relation to the assertion (i.e., the latter but not the former entails the assertion). For example,
modify the conditions on contextual restriction in (10) in such a way that when there are symmetric
alternatives, where one entails the assertion and the other does not, then context can disregard the
latter from C. In other words, allow contextual restriction to distinguish between alternatives that are
strictly stronger versus merely not weaker. This move, however, would not deal with Trinh & Haida
(2015)’s (20), where the problematic symmetric alternatives (i.e., run ∧ smoke and run ∧¬ smoke)
both entail the assertion and thus cannot be distinguished.
10
(24)
Atomicity: Expressions in the substitution source are syntactically atomic.
What (24) effectively yields is that all constituents in the substitution source be
treated as if they were simple lexical items, whether they are or not. So for instance,
a complex constituent like [John did all of the homework], if in the substitution
source, behaves like e.g. [John] in that (once substituted in) it can only be substituted
for as whole.
3.2
Solving the problems with indirect and particularised scalar implicatures
Let us see how this solves the problem of indirect scalar implicatures. Remember
that the problem with (25a) is that Fox & Katzir’s (2011) procedure allows us to
construct both the needed alternative in (26a) and its symmetric alternative in (26b).
(25)
a.
b.
John didn’t do all of the homework.
John did some of the homework
(26)
a.
b.
John didn’t do some of the homework.
John did some of the homework.
The atomicity constraint correctly blocks the generation of the offending alternative
(26b). To see this, consider how it could be derived. We know the substitution source
includes the atomic item ‘some.’ Using this, let us try the following derivation.
(27)
a.
b.
c.
[NegP not [VP John did all of the HW]]
VP John did all of the HW
VP John did [some] of the HW
(25a)
subconstituent
*all/some
This derivation violates the atomicity constraint because after the second step, the
subconstituent VP is now treated as an atom (as indicated by the box here), and its
sub-part, ‘some,’ cannot be replaced in the third step.
Note that first substituting ‘some’ for ‘all’ and substituting the newly formed
VP = [VP did some of the homework] for the NegP is not possible, simply because
this VP is not in the substitution source. It is not a subconstituent of the assertion
and it is not salient in the context. Consequently, the symmetric alternative (26b)
(some) is not generated, and the correct indirect scalar implicature can be generated
by negating (26a) (¬some).11
11 What if this VP is contextually salient instead? Consider the following example.
(i)
Two weeks ago, John did all of his homework. Last week, he did some. And this week, he
didn’t do all of his homework.
John did some of his homework
11
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
Let us now also see how Trinh & Haida’s (2015) particularised scalar implicature
example (20) is accounted for. The solution is essentially identical. Consider the
first part of (20) in (28). Consider in particular the constituent α = ‘[T’ went for a
run and didn’t smoke].’ Without the atomicity constraint, the offending alternative
run ∧ smoke could be derived as follows:
(28)
Bill [α went for a run and didn’t smoke]
(29)
[John went for a run]]
⇒ [ John T’ went for a run and didn’t smoke ]
⇒ [ John T’ went for a run and [VP smoked] ]
the prejacent
T’/α
NegP/smoked
The atomicity constraint rules out this derivation, as the T’ in the second step is
atomic, and the replacement in the last step is forbidden. As a result, the formal
alternatives are not going to include the offending alternative run ∧ smoke and C
can be the following set, which gives rise to the correct scalar implicature.
(30)
C = { run, run ∧ ¬smoke }
We can also construct other formal alternatives from (28), repeated in (31). In
particular, we can use both β and γ to construct the alternatives in (32a) and (32b),
given that both β and γ are salient constituents.
(31)
Bill [went for a run and [β didn’t [γ smoke]]]
(32)
a.
John [β didn’t smoke]
b.
John [γ smoked]
Given the closure condition, we cannot have one of (32a) and (32b) in C without
having the other. However, neither is in the Boolean closure of (30) and so both can
be excluded.
In sum, the atomicity constraint accounts for the examples in (16a) and (20).
However, we now show with more examples that the atomicity constraint does not
provide a general solution to the problem of indirect scalar implicatures. We discuss
The discourse is perhaps not completely natural. However, if one can establish that the last sentence
has the inference that John did some of the homework, this would not be predicted, since ‘[VP John
did some of the homework]’ is in the substitution source by virtue of having been uttered in the
previous sentence, so the scalar implicature should be absent in this case.
Both Fox & Katzir (2011) and Trinh & Haida (2015) are aware of this potential problem, and they
propose that contextually salient alternatives are only optionally in the substitution source (see Fox &
Katzir 2011: fn. 23 and Trinh & Haida 2015: fn. 22). With this assumption, the scalar implicature is
optionally derived for (i), which appears to be a correct prediction, as the scalar implicature could be
seen as an optional inference.
12
two sets of data. One is based on simple variants of Trinh & Haida’s (2015) example
(20). The other has to do with the indirect scalar implicature that scalar adjectives
like full or empty give rise to when embedded under negation.
3.3
3.3.1
New problems
Particularised scalar implicatures without conjunction
We have seen above that the Atomicity constraint allows us to obtain the right
inference for (20). Before illustrating the problem with simple variants of (20),
let us emphasise what is crucial about (20) for Trinh & Haida’s (2015) solution to
work. The first part of (20) makes salient the constituent α in the box below in (33).
This constituent allows us to construct the alternative in (34), from the negation of
which we can derive the attested implicature that John did smoke. Crucially, given
atomicity, we cannot create the alternative John went for a run and smoked by further
substituting into it.
(33)
Bill [α went for a run and didn’t smoke]
(34)
John [T 0 went for a run]
⇒ John [α went for a run and didn’t smoke]
⇒ John [α went for a run and smoked ]
prejacent
T’/α
*NegP/VP
Therefore, Trinh & Haida’s (2015) solution crucially relies on the presence of this
conjunctive constituent which allows us to construct the alternative from which we
can derive the desired implicature. The question is, however, what happens with
simple variants of (20) where we do not have conjunction or where conjunction is
structurally in a different place. Consider for instance (35), which is very similar to
(20) and appears to have the same inference that John smoked.
(35)
Bill went for a run. He (also) didn’t smoke. What did John do?
John went for a run.
John smoked
The problem with (35), however, is that it does not contain a conjunctive constituent
out of which the inference could be derived as before. Consider some potential
candidate constituents (35) makes salient:
(36)
Bill [β went for a run]. He [γ didn’t [φ smoke]]
It is clear that none of β , γ or φ will be of help: by substituting them for T 0 in (37)
we can create the alternatives in (38). From (38), we cannot create a set C which
13
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
will allow us to conclude that John smoked.
(37)
John [T 0 went for a run ]
(38)
{ John [β went for a run] , John [φ smoked] , John [γ didn’t smoke] }
The desired scalar implicature would obtain if the alternative ¬smoke is negated, but,
as mentioned above, this cannot be done due to its symmetric counterpart smoke.
And smoke cannot be excluded from C due to the closure condition. In addition,
crucially, this time we do not have the conjunctive alternative run ∧ ¬smoke, which
allowed us to derive the implicature before.12
One might wonder if having multiple sentences as one alternative could help.
That is, if the solution lies in making use of the two sentences together ‘Bill went
for a run. He didn’t smoke’ as a single atomic constituent. While having multiple
sentences as an alternative is a plausible move, and probably an independently
justified one,13 the problem is that to create the necessary alternative ‘John went for
a run. He didn’t smoke,’ we would have to substitute into such alternative and this is
exactly what the atomicity constraint prohibits.
In fact, as an anonymous reviewer points out to us, the problem can even be
re-created by having conjunction as in the original (20) example, but placing it in a
different place in the structure. Consider (39), which is again is a minimal variant of
(20), with the same inference, but where conjunction appears higher in the structure.
(39)
Bill went for a run and he (also) didn’t smoke. What did John do?
John went for a run.
John smoked
The problem is that here the constituent α containing conjunction corresponds to the
whole sentence, (40). Therefore there is no way to construct the needed alternative,
without violating the atomicity constraint.
(40)
[α [Bill went for a run] and [he didn’t smoke]]
(41)
[John went for a run]]
⇒ [T P Bill went for a run and didn’t smoke]
the prejacent
TP/α
12 Notice that, as we mentioned above, Fox & Katzir (2011) and Trinh & Haida (2015) allow for the
possibility of ignoring contextually salient constituents. Would this help in any way here? It is easy
to see that it would not. In fact, it would actually make things worse, because if the salient constituent
‘didn’t smoke’ is ignored, we will end up having the following set of alternatives: C = { run, smoke }
From this set and the assertion we should conclude that John didn’t smoke, which is the opposite of
what we want to obtain.
13 As an anonymous reviewer points to us, there is independent evidence that two sentences might
underlyingly belong to the same syntactic structure, see Keshet 2007.
14
⇒
[T P John went for a run and didn’t smoke]
*Bill/John
To summarize, while the Atomicity constraint solves the problem for the original
example (20) by Trinh & Haida (2015), this solution crucially relies on the presence
of conjunction in the appropriate place in the structure. If we create simple variants
of (20), by either taking conjunction out, (35), or moving it around, (39), the problem
re-emerges and it is unclear to us how to extend Trinh & Haida’s (2015) solution to
these simple variants of their example.
3.3.2
Over and under-generation problems with adjectives
In this section, we show that while Trinh & Haida’s (2015) atomicity constraint
was able to solve the problem of indirect scalar implicatures like (25a), it runs into
a conundrum with similar cases generated by adjectives like full or empty under
negation. Consider the following example, for instance.
(42)
The glass isn’t full.
a.
The glass is not empty/is a bit filled
b. 6 The glass is empty
This sentence has a robust inference that the glass is not empty.14 Moreover, it does
not appear to implicate that the glass is empty. Similarly and symmetrically, for (43):
(43)
The glass isn’t empty.
a.
The glass is not full/not completely filled
b. 6 The glass is full
14 An anonymous reviewer asks us whether this could be a presupposition rather than an implicature.
While this is a possibility that we cannot exclude altogether, we think that the tests for presuppositionality, as imperfect as they are, suggest that this is not the case. For instance, a Hey wait a minute!
response to (ia) as in (ib) appears infelicitous (von Fintel 2004).
(i)
a.
The glass is full.
b. ??Hey wait a minute, I didn’t know it wasn’t empty!
Similarly, a sentence like (iia) doesn’t appear to us to project the inference that the glass is not empty,
or at least not more than a sentence like (iiia) projects the inference that some of the students came.
(ii)
a.
b.
Is the glass full?
? the glass is not empty
(iii)
a.
b.
Did all of the students come?
? some of the students came
15
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
How can we be sure that an inference like (42b) is absent, given that it is actually
compatible with the literal meaning of (42)? One argument comes from Hurford’s
constraint (see Chierchia et al. 2012 among others). The idea is constructing a
sentence where the second disjunct entails the first unless the first disjunct can have
the inference that we are investigating. Given these assumptions, the following
contrast indicates that the inference is not there.
(44)
a. ??Either the glass isn’t full or it is partially filled.
b. Either the glass is empty or it is partially filled.
Trinh & Haida (2015), however, predict exactly the opposite for these cases. That is,
they predict that a sentence like (42) should be able to have a scalar implicatures in
(42b), and not the one in (42a).
Starting from the latter problem, there is the question of how to predict the
actually attested inference in (42a). That is, the question is what is the alternative
that negated gives rise to the inference in (42a). This problem might be solvable
once we consider the syntax and semantics of these sentences more in detail. In
particular, one possibility is to assume that lexical replacement targets the silent
positive morpheme POS widely postulated for positive forms of adjectives (Bartsch
& Vennemann 1975, von Stechow 1984, Kennedy 1999):
(45)
The glass is [not [ POS full ]]
The main function of POS is to introduce the standard of fullness in the context,
and on the hypothesis that it can be replaced with other gradable modifiers, the
sentence can get stronger meanings that can be negated by the scalar implicature
algorithm. Suppose that POS is replaced by ‘half’, ‘1/3’, 1% or other lexical items,
like in (46). Then, the negation of any of these alternatives could gives us something
that resembles the inference above (i.e. ¬¬(1%/a third/half full)).
(46)
The glass is [not [ 1%/a third/half full ]]
This would solve the under generation problem for Trinh & Haida (2015) with cases
like (42) and (43) above.
However, there is a remaining question which is why the corresponding scalar
implicature seems considerably less available with other gradable adjectives. For
example, compared to (42) above, similar scalar inferences cannot be drawn from
the following sentences.15
15 Notice that the meaning of these gradable adjectives, and in particular their associated scale structure,
might help explaining the contrast here. However, notice that in (47) we picked three adjectives
cutting across the absolute/relative distinction and differing with respect to their scale structure: safe
has a upper closed scale, transparent has a fully closed one and tall a fully open scale (Kennedy 2007
16
(47)
a.
This neighbourhood is not safe.
This neighbourhood is not dangerous.
John is not tall.
6 John is not small.
The glass is not transparent.
6 The glass is not opaque.
6
b.
c.
If the replacement of POS with other gradable modifiers derives (42a) for (42), it
should, everything being equal, generate these scalar implicatures.
More problematic is the over-generation issue. Trinh & Haida’s (2015)’s system
seems to over-generate for (42) (similarly (43)), predicting the inference in (42b)
can arise by negating the alternative ¬empty which is obtained by simple lexical
substitution of empty for full.
Even if we consider a more sophisticated syntax and semantics for sentences
like (42), it is unclear to us how to block this inference.
(48)
The glass is [not [ POS full ]]
⇒ The glass is [not [ POS empty ]]
full/empty
Of course, this inference would be correctly blocked if the alternative empty was
available, but given the atomicity constraint, empty cannot be in the formal alternatives. i.e., there is no way to create empty out of ¬empty without violating the
atomicity constraint.
(49)
the glass is [not [ POS full ]]
⇒ [ the glass is [ POS full ]
NegP/AP
⇒ [ the glass is [ POS empty ]
*full/empty
Notice that this is precisely what gave Trinh & Haida (2015) the solution for the
earlier examples (16a) and (20) and now creates a problem here with the case in
(42). Fox & Katzir (2011), on the other hand, do not have this problem, but, as
we have seen, they fail to account for (16a) and (20), as explained above. These
observations are, therefore, connected in that Trinh & Haida (2015) solution for one
creates a problem for the other. In other words, the atomicity constraint appears to
be only a partial solution of the problem with indirect scalar implicatures.
among others). Thanks to an anonymous reviewer for discussion on this point.
17
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
3.4
Section summary
To summarise, Trinh & Haida’s (2015) atomicity constraint faces two issues. One
problem comes from a variant of the type of case they proposed as problematic
for Fox & Katzir (2011), without conjunction or with conjunction in a different
structural position. In either case, due to the lack of conjunction in the ‘right’
place, the alternative run ∧ ¬smoke that is necessary to derive the scalar implicature
cannot be in the substitution source. The other problem has to do with the indirect
scalar implicatures of gradable adjectives like full or empty. Here, the atomicity
constraint backfires as it prevents the symmetric alternative empty to be in the
substitution source. Consequently, the wrong scalar implicature is generated by
the alternative ¬empty that the glass is empty. Therefore, Trinh & Haida’s (2015)
proposal solves the original problem of indirect scalar implicatures of cases like
(16a) and (20) but it appears unable to deal with indirect scalar implicatures in the
general case. In sum, the problem of indirect scalar implicatures for the structural
approach to alternatives is not solved. In a general sense, this problem could be
characterised as arising from the central assumption of the theory that structurally
simpler alternatives always count. That is, the structural approach capitalises on
the difference in structural complexity between ‘some but not all’ and ‘all’ to solve
the original symmetry problem, but the same solution does not work for the case of
indirect scalar implicatures involving some and ¬some and other similar pairs of
symmetric alternatives where the offending alternative is structurally simpler. In the
next section, we will briefly discuss two further problems that illustrate the general
difficulty of deriving the correct set of relevant alternatives under the structural
approach, independently of the atomicity constraint.
4
Related problems
4.1
Too few lexical alternatives?
In this subsection, we discuss a potential problem of under-generation where it is not
clear that the needed formal alternative can be generated under the assumptions of
the structural approach to alternatives. For instance, in Japanese, deontic possibility
and necessity are expressed by constructions that are structurally clearly different.
Consider the following examples.
(50)
John-wa ki-te
yoi.
John-TOP come-GERUND good
‘John is allowed to come.’
(51)
a.
John-wa ko-naku-te-wa
nar-anai/ike-nai.
John-TOP come-NEG-GERUND-TOP become-NEG/go-NEG
18
b.
‘John must come.’
John-wa kuru hitsuyoo-ga
aru.
John-TOP come necessity-NOM exist
‘John needs to come.’
(50) expresses deontic possibility with the predicate yoi, which is morphologically
an adjective. On the other hand, the sentences expressing deontic necessity in (51) do
not involve adjectives. Specifically, (51a) involves a verbal stem (either nar- or ike)
with the negative suffix -(a)nai. This makes the main predicate morphologically more
complex than in (50). Generally, the structural approach to alternatives assumes that
adding negation complicates the structure, so it is unlikely that (51a) can be derived
from (50) by substitution and deletion alone. In addition, the topic marking on the
gerundive subject here is obligatory, while adding it would make (50) unacceptable.
This could be taken as suggesting that the gerundive clause is in syntactically
distinct positions in (50) and (51a). Similarly, (51b) is unlikely to be derivable
from (50) with substitution and deletion. It involves an existential construction with
the existential verb ar-, where the subject is not a gerundive clause but a nominal
hitsuyoo ‘necessity’ with a complement clause.
Despite this structural difference, however, (50) has the same scalar implicature
as the English translation, i.e. that John is not required to come. As evidence for this
inference, like its English counterpart, (50) is very unnatural in a context in which
John is required to come. However, it is not at all clear how (51) could be generated
in the structural approach from (50) (which presumably have a structure like (52)).
(52)
[John-wa] [ki-te [yoi]].
A possible response to this might be to assume that there actually is a grammatical
alternative to (50) that expresses deontic necessity (e.g. *[John-wa] [ki-te [doi]])
but is made unacceptable and practically unusable for some independent reasons,
to which computation of scalar implicatures is somehow oblivious to.16 Then, the
desired scalar implicature could be generated based on this unacceptable sentence.
However, a solution like this commits one to non-trivial assumptions about the theory
of lexicon and acquisition of lexical items.17,18
16 Thanks to Andreas Haida (p.c.) for suggesting a possibility along these lines, and related discussion.
17 For relevant discussion see Schlenker 2008, who considers the case of potential alternatives that are
not even grammatical, but that nonetheless appear to be used for the computation of implicatures.
18 A similar case of too few alternatives comes from French and was noted by Chemla (2007). The
observation is that ‘tous’, like its English counterpart all, cannot be used to quantify over a domain of
two individuals. The explanation for English relies on the existence of both which would compete
with all-sentences in such cases. Given that French has no lexical items corresponding to both, it is
unclear how to extend such explanation to tous.
19
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
4.2
Too many lexical alternatives
In the original symmetry problem created by ‘some but not all,’ the solution under the
structural approach to alternatives crucially relies on what is and is not lexicalised.
In particular, it is important that there is no constituent of the same or less structural
complexity as ‘some’ and ‘all’, which means ‘some but not all’ in the lexicon.
Swanson (2010), however, points out that once we move to a different domain,
there are examples of what appear to be symmetric alternatives of the same structural
complexity. He raises examples like the following examples where the scalar items
are permitted and sometimes.
(53)
a.
b.
c.
Going to confession is permitted.
Going to confession is optional.
6 Going to confession is required.
(54)
a.
b.
c.
d.
The heater sometimes squeaks.
The heater intermittently squeaks.
The heater occasionally squeaks.
6 The heater constantly squeaks.
These examples give rise to a symmetry problem, as indicated in the (c)-examples
above. And these versions of the symmetry problem cannot be dealt with by Fox
& Katzir’s (2011) or Trinh & Haida’s (2015) structural approach without further
ado.19 As Swanson (2010) himself suggests, one could try to supplement the theory
with a constraint that excludes the problematic lexical items in some principled way,
e.g., by resorting to their relative low frequency. This, however, appears a non-trivial
addition to the theory.
5
Concluding Remarks
The structural approach to alternatives advocated by Katzir (2007) and Fox & Katzir
(2011) is one of the few systematic accounts of the symmetry problem that are
available in the current literature. However, we argued that while it successfully
accounts for the certain instantiations of the symmetry problem, it does not handle
19 As an anonymous reviewer points, some of the examples above are more convincing than others. For
instance, it is unclear that occasionally really has the problematic meaning sometimes but now always:
it is unclear whether (i) is contradictory, as it should be predicted by such meaning. Intermittently, on
the other hand, appears more convincingly upper bounded: (ii) appears to be contradictory.
(i)
(ii)
All logicians occasionally make errors, and some do so constantly.
??It rains intermittently everywhere in the Netherlands and in Amsterdam it rains constantly.
20
others. In particular, we have discussed several problems for the structural approach:
(i) the problem of indirect and particularised scalar implicatures (Romoli 2013, Trinh
& Haida 2015) and (ii) the problem of too many lexical alternatives (Swanson 2010).
We claimed that Trinh & Haida’s (2015) atomicity constraint does not constitute a
general solution to the first problem, although it accounts for specific cases that have
been previously raised. Finally we discussed and (iii) the problem of too few lexical
alternatives. Both (ii) and (iii) are independent of the atomicity constraint.
All three problems may cast doubt on the central tenet of the structural approach.
Its core idea is to capitalise on differences in structural complexity. While this
works well for the original example of the symmetry problem involving symmetric
alternatives like ‘all’ and ‘some but not all’, the crucial imbalance in the structural
complexity may be an accidental property of these examples (problems (i) and (ii)).
Thus it may be worth pursuing alternative approaches to the question of if and what
alternatives are implicated in the generation of various implicatures, a point also
raised by problem (iii).
References
Bartsch, Renate & Theo Vennemann. 1975. The grammar of adjectives and comparison. Linguistische Berichte 20. 19–32.
Chemla, Emmanuel. 2007. French both: a gap in the theory of antipresupposition.
Snippets 15. 4–5.
Chemla, Emmanuel & Benjamin Spector. 2011. Experimental evidence for embedded scalar implicatures. Journal of Semantics 28(3). 359–400.
Chierchia, Gennaro, Danny Fox & Benjamin Spector. 2012. Scalar implicature as
a grammatical phenomenon. In Claudia Maienborn, Klaus von Heusinger &
Paul Portner (eds.), Semantics: An International Handbook of Natural Language
Meaning, 2297–2331. Berlin: de Gruyter.
Fine, Kit. to appear. Truth-maker semantics. In Blackwell philosophy of language
handbook, .
Fox, Danny. 2007. Free choice and the theory of scalar implicatures. In Uli Sauerland & Penka Stateva (eds.), Presupposition and Implicature in Compositional
Semantics, 71–120. Palgrave.
Fox, Danny & Roni Katzir. 2011. On the characterization of alternatives.
Natural Language Semantics 19(1). 87–107.
http://dx.doi.org/10.1007/
s11050-010-9065-3.
Gazdar, Gerald. 1979. Pragmatics: Implicature, presupposition and logical form.
New York: Academic Press.
Hirschberg, Julia Bell. 1991. A Theory of Scalar Implicature. New York: Garland.
21
Richard Breheny, Nathan Klinedinst, Jacopo Romoli, and Yasutada Sudo
Horn, Lawrence. 1972. On the semantic properties of logical operators in English:
UCLA dissertation.
Horn, Lawrence. 1989. A natural history of negation. Chicago: University of
Chicago Press.
Katzir, Roni. 2007. Structurally-defined alternatives. Linguistics and Philosophy
30(6). 669–690. http://dx.doi.org/10.1007/s10988-008-9029-y.
Kennedy, Chris. 2007. Vagueness and grammar: the semantics of relative and
absolute gradable adjectives. Linguistics and Philosophy 30. 1–45.
Kennedy, Christopher. 1999. Projecting the Adjective: The Syntax and Semantics of
Gradability and Comparison. New York: Garland Press.
Keshet, Ezra. 2007. Telescoping and scope economy. In C. B. Chang & H. J. Hayni
(eds.), Proceedings of the 26th west coast conference on formal linguistics,
324–331.
Kroch, Anthony. 1972. Lexical and inferred meanings for some time adverbs. In
Quarterly progress report of the research laboratory of electronics 104, MIT.
Matsumoto, Yo. 1995. The conversational condition on Horn scales. Linguistics and
Philosophy 18(1). 21–60.
Romoli, Jacopo. 2013. A problem for the structural characterization of alternatives.
Snippets 27. 14–15.
van Rooij, Robert & Katrin Schulz. 2004. Exhaustive interpretation of complex
sentences. Journal of Logic, Language and Information 13(4). 491–519. http:
//dx.doi.org/10.1007/s10849-004-2118-6.
van Rooij, Robert & Katrin Schulz. 2006. Pragmatic meaning and non-monotonic
reasoning: The case of exhaustive interpretation. Linguistics and Philosophy
29(2). 205–250. http://dx.doi.org/10.1007/s10988-005-3760-4.
Sauerland, Uli. 2004. Scalar implicatures in complex sentences. Linguistics and
Philosophy 27(3). 367–391. http://dx.doi.org/10.1023/B:LING.0000023378.
71748.db.
Schlenker, Philippe. 2008. Presupposition projection: Explanatory strategies. Theoretical Linguistics 34(3). 287–316. doi:10.1515/THLI.2008.021.
Soames, Scott. 1982. How presuppositions are inherited: a solution to the projection
problem. Linguistic Inquiry 13(3). 483–545.
Spector, Benjamin. 2007. Aspects of the pragmatics of plural morphology: On
higher-order implicatures. In Uli Sauerland & Penka Stateva (eds.), Presupposition and implicature in compositional semantics, Palgrave.
von Stechow, Arnim. 1984. Comparing semantic theories of comparison. Journal of
Semantics 3(1–2). 1–77. http://dx.doi.org/10.1093/jos/3.1-2.1.
Swanson, Eric. 2010. Structurally Defined Alternatives and Lexicalizations of
XOR. Linguistics and Philosophy 33(1). 31–36. http://dx.doi.org/10.1007/
s10988-010-9074-1.
22
Trinh, Tue & Andreas Haida. 2015. Constraining the derivation of alternatives. To
appear in Natural Language Semantics.
von Fintel, Kai. 2004. ‘would you believe it? the king of france is back! presuppositions and truth-value intuitions”. In Descriptions and Beyond, edited by Marga
Reimer and Anne Bezuidenhout. Oxford University Press. .
23