SALT 25
Jon Ander Mendia (UMass Amherst)
Complex Cardinals of Approximation in Spanish
I NTRODUCTION . In Spanish, various non–numerical expressions can form complex cardinal
numerals in order to (i) denote a range of possible values, and (ii) express uncertainty or vagueness
(Complex Cardinal Numerals of Approximation; CCNA). For example, all the statements below
are acceptable answers to a question like how much did you spend?
(1) a. 60 y {pico / tantos} euros
b. ciento y {pico / tantos} euros
60 and
“some”
euros
hundred and
“some”
euros
X
?
?
X
?
?
e≈[61,69]; e≤60; e≥70
e≈[101,199]; e≤100; e≥200
a. 60 y muchos euros
b. 60 y pocos euros
60 and many euros
60 and few euros
X
?
?
X
e≈[66,69]; e.66; e≥70
e≈[61,64]; ? e≤60; ? e&64
(1a) can denote integers from 61 to 69, not more, not less (cf. sixty-some euros). Similarly, (1b) has
the same interpretation with larger numbers, denoting integers between 101 and 199. The examples
in (2) are more restrictive: the possible numbers that (2a) denotes are limited to the set formed by
the upper half of (1a), that is, any integer from 66 to 69, maybe less, but not more. Similarly, (2b)
denotes an integer in the lower half of (1a), to the exclusion of any other number; i.e., some integer
from 61 to 64, maybe more, but not less. Thus, none of these expressions are simply approximate
(cf. Krifka 2009), their meaning is more limited. In addition, upon hearing any of (1a–b) to (2a–b),
a hearer may draw the inference that the speaker does not know what the exact number is, or that
she thinks it is irrelevant to provide an exact value.
A NALYSIS . In this paper I present the first analysis of these constructions. The analysis has three
main ingredients. First, I argue for a specific compositional account of the syntax/semantics of
cardinal numerals in decimal systems. Then I present an extension to CCNAs in Spanish, where
they are analyzed as determiners introducing subset selection functions over a set of numbers.
Finally, I introduce a pragmatic calculus that derives the uncertainty/vagueness component as a
quantity implicature that results in an inference of ignorance.
The syntax/semantics of cardinal numerals. I assume that cardinal numbers (simple and complex) are properties of degrees (3a) (Landman 2004). In addition, I assume that they have a
complex internal structure (cf. Hurford 1975): following the positional notation for numbers,
non–additive cardinals are the product of some integer i ∈ {1, 2, ..., 9} and the corresponding numerical BASE Bi , where Bi = 10i for any i ∈ N (e.g., 6 × B1 = 6 × 10 = 60). In the decimal
system, the BASE is a power of 10, and so its denotation is a property of degrees too: λd.[d = 10i ].
The multiplication is carried out by the head MUL (3c). Different BASE values are freely eligible by MUL; for example, the number 3000 has the structure [3[MUL B3 ]], which is interpreted
as 3 × B3 = 3 × 1000 = 3000. Complex additive cardinals are assumed to have an underlying
coordinate structure (Ionin & Matushanky 2006), headed by an additive head ADD, sometimes
spelled–out as “y”, with the semantics of addition (Anderson 2014; (3b)). ADD obeys two restrictions: (i) its arguments cannot be two cardinals formed by the same BASE (*veinte y treinta,
“twenty-thirty”), and (ii) the first argument must be smaller (*veinte y cien, “twenty-hundred”), as
indicated by the superscript on the property variables in (3b)). Thus, [80[ADD 9]] = 80 + 9 = 89.
(3) a. J60K = λd.[d = 60] b. JADDK = λDi λDi+n λd . ∃d0 d00 [d = d0 + d00 ∧ Di (d0 ) ∧ Di+n (d00 )]
c. JMULK = λD0 λD00 λd . ∃d0 d00 [d = d0 d00 ∧ D0 (d0 ) ∧ D00 (d00 )]
d. [[3[MUL B3 ]][ADD[6[MUL B2 ]][ADD[8[MUL B1 ]][ADD[9[MUL B0 ]]]]]]]]
⇔ [3000 [ADD [600 [ADD [80 [ADD [9 ]]]]]]] ⇔ 3689
(2)
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SALT 25
Jon Ander Mendia (UMass Amherst)
The case of “60 y pico/tantos”. The denotation of pico/tantos (JP / TK) is modeled after the epistemic indefinite algún in Spanish, as defined in (4a). JP / TK denotes a property of degrees, a subset
selection function over degrees f that selects some integer from a set of numbers (cf. AlonsoOvalle & Menéndez-Benito 2013, Anderson 2014). The size of the set restricting f depends
entirely on the BASE that JP / TK combines with: the minimum is determined by the BASE itself
(Bi ), whereas the maximum is the next power of the base: Bi+1 For concreteness, this constraint
is encoded as a presupposition on the value of f (cf. φ–features on pronouns; Heim & Kratzer
1998); if JP / TK does not combine with a BASE, the expression will be undefined. Combining JP / TK
with the BASE via MUL results in a property of degrees that can then be added to some other property of degrees, which, in turn, was also formed by the same process out of a bigger BASE (e.g.,
[6[MUL B1 ]] = λd . ∃d0 d00 [d = d0 d00 ∧ d0 = 10 ∧ d00 = 6], which is equal to 6 × 10 = 60, as in (4c)).
Note that the smallest BASE B0 is a special case, since the minimum required is 0, not B0 itself.
(4) a. JPICOS / TANTOSK = λd : there is a BASE Bi . d ∈ f ({n ∈ N : Bi < n < Bi+1 }) =JP / TK
b. JMULK(JB0 K)(JP / TK) = λD0 λD00 λd . ∃d0 d00 [d = d0 d00 ∧ D0 (d0 ) ∧ D00 (d00 )]
⇔ λd . ∃d0 d00 [d = d0 d00 ∧d0 = 1∧d00 ∈ f ({n ∈ N : 0 < n < 10})]
c. JADDK(J(4b)K)(J60K) ⇔ λDi λDi+n λd . ∃d0 d00 [d = d0 + d00 ∧
Di (d0 ) ∧ Di+n (d00 )](J(4b)K)(J60K)
⇔ λd.∃d0 d00 [d = d0 + d00 ∧
d0 ∈ f ({n ∈ N : 0 < n < 10}) ∧ d00 = 60]
The case of “ 60 y muchos/pocos”. These two CCNAs are special in that they further restrict
the range of possible values the CCNA can denote. The interpretation of both muchos and pocos is
proportional in these cases, since they are interpreted relative to the BASE they combine with: in 60
y muchos, muchos ranges from 6 to 9, but in ciento y muchos (“hundred and many”) it ranges from
51 to 99, and so on. Following common proposals on the semantics of proportional quantifiers
(cf. Partee 1988), I suggest that we define the lexical entry JMUCHOSK so that it further restricts
the lowest possible value of f ; this is done by introducing a fraction of the BASE+1 . Similarly,
JPOCOSK is defined so that the highest possible value of f is now a fraction of BASE+1 .
i+1
(5) a. JMUCHOSK = λd : there is a base Bi . d ∈ f ({n ∈ N : B 2 < n < Bi+1 })
i+1
b. JPOCOSK = λd : there is a base Bi . d ∈ f ({n ∈ N : Bi < n < B 2 })
The rest of the derivation proceeds as above: both pocos/muchos and BASE serve as arguments to
MUL , and the resulting property of degrees serves in turn as the first argument of ADD .
Calculating implicatures. Following the principle of Epistemic Implication (Hintikka 1962), I
assume that utterance of a sentence φ by a speaker S commits S to the knowledge of φ (KS φ). The
fact that f ’s domain is always restricted to a non–singleton set triggers the question as to why the
speaker did not restrict the domain to a singleton. As a consequence, Stronger Alternatives (SAs)
to the assertion (6b) are negated and this, together with the assertion, entails that for every SA [n],
¬KS [n] ∧ ¬KS ¬[n], i.e., the ignorance effect of CCNAs, (6c).
(6) a. I spent “60 y pico” euros ↔ KS [61, 69] b. SAs: KS [61], KS [62], ..., KS [69]
c. KS [61, 69] ∧ ¬KS [61] ∧ ¬KS ¬[61] ∧ ¬KS [62] ∧ ¬KS ¬[62], ..., ¬KS [69] ∧ ¬KS ¬[69]
C ONCLUSIONS . In this paper I present a semantic analysis of complex cardinals that is able to
account for CCNAs in Spanish. The approach is flexible enough to account for systems other than
the decimal by simply adjusting the BASE. In addition, it makes clear and empirically testable
predictions. If the analysis is on the right track, one would expect to find (i) other CCNAs in
non–decimal systems, and (ii) other logically plausible combinations of determiners and BASES.
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