We discuss semantics of superlative quanti ers at most n and at least n. We argue that the meaning of a quanti er is a pair specifying a veri cation and a falsi cation condition for sentences with this quanti er. We further propose that the veri cation condition of superlative quantiers should be interpreted in an epistemic way, that is as a conjunctive list of possibilities. We also present results of a reasoning experiment in which we analyze the acceptance rate of inferences with superlative and comparative quanti ers in German. We discuss the results in the light of our proposal.
Moeller et al, 2008),
It has been observed that in natural languages those equivalences (1) and (2)
might not hold, or at least they might not be accepted by language users based
on pragmatical grounds. There are numerous di erences between comparative
and superlative quanti ers involving their linguistic use, acquisition, processing
and the inference patterns in which they occur. First of all, it seems that
superlative and comparative quanti ers are not freely exchangeable in same linguistic
contexts.
(b) I will invite fewer than three people, namely Jack and Jill. 1 Q(A,B) means Q A's are B, where Q is a h1; 1i generalized quanti er
where (a) is considered a good sentence, while (b) is less felicitous. The contrast between (a) and (b) suggest that while embedding an inde nite expression (two) in a superlative quanti er licenses a speci c construal (namely Jack and
Secondly, it has been demonstrated that superlative quanti ers are mastered
later than the comparative ones during language development
Finally, important arguments against the semantical equivalence between the
comparative and superlative quanti ers come from the analysis of people's
acceptance of inferences with those quanti ers. Empirical data show that a majority
of responders usually reject inferences from at most n to at most n+, (where n+
denotes any natural number greater than n), although they accept presumably
equivalent inferences with comparative quanti ers
Several theories have been developed to explain why seemingly logically
equivalent quanti ers show such big di erences in the way people use them in the
language.
expected to be harder to process than the respective comparative quanti ers.
count seems unsatisfactory with regard to superlative quanti ers embedded in
conditional and various other contexts. Authors
If Berta had at most three drinks, she is t to drive. Berta had at most two drinks. Conclusion: Berta is t to drive.
If at most n logically implies possible that n and not possible that more than n, then the inference from at most n to at most n+ should be inaccessible (except for cases of random mistakes) for any language users, due to the apparent contradiction between the premise and the conclusion. Last but not least, to say that possible that n is a part of the semantics of at most n, implies that at most n cannot be paraphrased by not more than n. However such a paraphrase seems totally eligible.
observe that the considered linguistical phenomena could be better explained on pragmatical grounds. The authors show that people do not evaluate at most n and exactly n-1 as equally semantically incoherent as cases of obvious logical incoherence, e.g. at most n and exactly n+1 or more than n and exactly n-1.
get average coherence judgments very low, i.e. (incoherent) to +5 (coherent), sentence pairs:
get already +1:9. This result speaks against the \modal account", whose direct consequence is semantical incompatibility of at most n and exactly n-.
n both imply possible that n, but they claim that this is a pragmatical rather
than a logical inference, namely a co-called clausal implicature
weak statement. Since the expressed statement with a superlative quanti er e.g. at most n A are B, as equivalent to a disjunctive statement there are exactly n or fewer than n elements that are both A and B, does not imply the truth of its subordinate proposition p = there are exactly n elements that are A and B, the possibility that p might or might not be true is inferred.
Although we agree with the intuitions concerning a modal component in the reasoning with superlative quanti ers, we reject the assumption by Geurts et al. that this component is a part of their meaning. Furthermore, although we agree with Cummins et al. that the mechanism that results with the observed inference patterns is more of a pragmatical nature, we are not satis ed with the \causal implicature" account. What we lack is a deeper insight into the source of this kind of a pragmatical inference and how it interacts with the logical meaning of those quanti ers in di erent reasoning contexts.
Our motivation to further experimentally investigate reasoning with superlative quanti ers is based inter alia on: (i) the lack of data concerning whether people accept inferences: at least n ! at least n -, (ii) the lack of satisfactory data about how people accept inference with logically equivalent forms of superlative quanti ers: such as not more than n/not fewer than n or n or fewer than n/n or more than n, as well as how they accept mutual equivalences between these forms, (iii) nally, the lack of data concerning people's acceptance of logically incorrect inferences with the quanti ers considered here. 3
pair that speci es when the sentence is true and when it is false. Following Krifka, we propose to de ne meaning of a quanti er as a pair hCF ; CV i, where CV is a veri cation condition (speci es how to verify sentences with this quanti er) and CF is a falsi cation condition (speci es how to falsify sentences with this quantier). Furthermore, we propose that the interpretation of a quanti er depends on a semantic context in which this quanti er is used, namely whether the context requires the use of the veri cation condition or the falsi cation condition. Veri cation and falsi cation conditions are to be understood algorithmically, with the \else" part of the conditional instruction being empty - thus, they verify (or falsify) the formulas only if their conditional test is satis ed. From a perspective of classical logic, these conditions should be dual, namely if C is a CV condition for sentence , then C is a CF condition for sentence : , and vice versa. We further, however, observe that in the case of superlative quanti ers, there is a split between these two conditions. We suggest, that this split is a result of a pragmatic focus on the expressed borderline n.
One can think of the meaning of logical operators, thus also quanti ers, in
terms of algorithms, that have to be performed in order to verify (or falsify)
sentences with those operators. (See also
De nition 1 ( falsi cation condition for at most)
CF (at most x : (x)) := If 9>nx( (x)); then f alsif y But how can we know when a sentence with at most n is true? From the point of view of an algorithm it is a so-called \otherwise" condition that de nes in this case the truth-condition. However a negation of a falsi cation condition is in sense informationally empty : it does not describe any concrete situation in which the given sentence can be veri ed. As a result, in those contexts that require to directly verify a sentence, we refer to a veri cation condition, which is speci ed independently. As expressing a positive condition, at most n may be understood as a disjunction n or fewer than n (\disjunctive at most "). De nition 2 ( veri cation condition for at most)
CV (at most n : x (x)) := If (9=nx (x) _ 9<nx (x)); then verif y The disjunction in 2 could be further broken down to: Win=1 9=ix (x) _ :9x (x), in short: Win=0 9=ix (x), where the disjunct 9=ox (x) means that :9x (x).3
And 9=nx (x) means precisely n x are , that is:
n 9=nx (x) () 9x1:::9xn[ ^ i=1 (xi)^
^ 2 at most n x are 3 Let us observe that 9<nx (x) is a short notation that can be misleading, since it is not an existential sentence. As existential, fewer than n would imply that there has to be at least one (though less than n) such x that is . However we would like a sentence less than n x: (x) to be also true if no x's are . Therefore, in fact, such a downward entailing sentence is a disguised universal sentence: 8x1::xn(Vin=1 (xi) ! W1 i<j n(xi = xj ))
Epistemic interpretation of disjunction
in natural language are likely to get so-called epistemic reading, that is they are interpreted as conjunctive lists of epistemic possibilities. According to the proposed solution a disjunction P1 or:::or Pn is interpreted as an answer to a question: Q: What might be the case? and, thus, is paraphrased as a (closed) list L:
L: P1 (might be the case) [and]... Pn (might be the case) [and (closure) nothing else might be the case].
De nition 3 (Zimmermann, 2000)
P1 _ ::: _ Pn ()
P1 ^ ::: ^ Pn and (closure):
8P [ P ! [P \ P1 = ; _ ::: _ P \ Pn = ;]]
The character of the closure requires a bit of our attention.
De nition 4 ( epistemic interpretation of the veri cation condition for at most)4 4 A detailed description of the modal predicate logic needed for providing semantics of this kind of sentences is beyond the scope of this paper. For our present purposes it is just enough to assume that, for each possible world, we have a di erent domain of objects over which we quantify. We assume also standard semantics for modal operators, and we restrict to re exive and transitive Kripke models.
CW (at most n x : (x)) := If ( 9=nx (x) ^ 9<nx (x)); then verif y and (closure)
If
9>nx (x); then f alsif y ( 9=nx (x) ^ 9<nx (x)) () 9=ix (x) (4) n ^ i=o
dition of at most n. While the falsi cation condition speci es precisely when the sentence has to be rejected as false, the veri cation condition (in the epistemic reading) provides a conjunctive list of epistemic possibilities that should all be the case in order to verify it.
The epistemic reading of the veri cation condition is also what di erentiates superlative quanti ers from the comparative ones. We propose that the disjunctive form of the veri cation condition in the case of superlative quanti ers is a result of the focus on the borderline n. The n mentioned in the superlative quanti er constitutes the borderline of the truth-conditions, however the numeral n that occurs in a comparative quanti er is not a part of the truth-conditions. The borderline of truth-conditions (that is n-1 for fewer than n, and n+1 for more than n) remains silent. Consequently, the disjunctive form of comparative quanti ers (and hence also the epistemic reading), though logically possible, is not pragmatically justi ed. In principle, a comparative quanti er can be as well interpreted in the epistemic way (as a conjunction of epistemic possibilities). Such a reading is, however, not equally likely to occur as in the case of a superlative quanti er, since the borderline is not explicitly expressed.
Let us observe that closure of the veri cation condition is stronger than the falsi cation condition. Intuitively, if : (equivalently : ), then as well : (here: = 9>nx (x)), i.e. , however the theorem holds only in re exive Kripke frames. Furthermore, the optional character of the closure bases on our assumption that the falsi cation and veri cation conditions are in a sense independent and only as a pair constitute the full semantic interpretation. Since the falsi cation condition, as de ned in 2, is su cient to account for the right semantical criterion of when the sentence with at most n is false, the closure of the veri cation condition is redundant and might or might not be considered in the reasoning process. The optional character of closure turns out crucial in evaluating validity of inferences with at most n.
most n to at most n+1 is not valid. Namely, from 9=n (x) ^ 9<nx (x) one cannot infer 9=n+1x (x) ^ 9<n+1x (x). It is easy to observe that the conjunct 9=n+1x (x) cannot be proven based on the premise, though it can be excluded only if the closure of the premise is applied.
On the other hand, the inference: at most n ! at most n-1, which is invalid in the standard account, in the epistemic interpretation is blocked only due to closure of the conclusion. That is the 9=nx (x) implied by the premise is contradicted by the closure of the conclusion, i.e. : 9>n 1x (x). It is important to notice that without the closure the implication holds (if the epistemic reading of the veri cation condition is applied).
In some aspects our proposal might seem similar to Geurts' \modal" approach, namely we de ne the veri cation condition of at most n and at least n (see below) in modal terms. The main di erence is that, in our account, it is only the veri cation condition that is de ned modally, while the falsi cation condition remains standard. This results in a speci c split or ambiguity in the meaning of superlative quanti ers. 4
it seems, as is also evident from our results (see Section (5)), less problematic from the point of view of reasoning. As an upward monotone quanti er, at least n appears to provide a clear veri cation algorithm: \verify when n x (that are ) are found". Such a semantical interpretation would not, however, account for the linguistical di erences between at least n and more than n-1.
De nition 5 ( falsi cation condition for at least)
CF (at least n : x (x)) := If 9<nx (x); then f alsif y
range of epistemic possibilities in which the sentence with at least n can be true. Understood as in formula (2), thus as equivalent to more than n-1, a sentence with at least n can be expressed as an existential sentence that merely says that there are n (x that are ), but does not exclude the possibility that there are more: n 9 nx (x) () 9>n 1x (x) () 9x1:::9xn[ ^ i=1 (xi) ^
^
comparative more than n-1, does not outline the borderline n, which is emphasized in the superlative at least n. Therefore we further break down formula (5) into a disjunctive formula: (exactly) n or more than n x are , which constitutes our veri cation condition for at least n.
De nition 6 ( veri cation condition for at least n)
CV (at least n : x (x)) := If (9=nx (x) _ 9>nx (x)); then verif y
De nition 7 ( epistemic interpretation of the veri cation condition for at least n)
CF (at least n : x (x)) := If ( 9=nx (x) ^ 9>nx (x)); then verif y and (closure)
n 1 If ( _ i=0
9=ix (x)); then f alsif y
lyze how they a ect inferences with this quanti er, in particular we mean here the inference (at least n! at least n-1 ), as well as its presumably equivalent disjunctive form: (n or more than n ! n-1 or more than n-1 ). We propose that the way people handle these inferences depends on how they interpret bare numerals, such as n.
From a logical perspective, a bare numeral n (e.g. \two") can be interpreted as denoting any set of at least n elements, or a set of exactly n elements.5 Thus = nx : (x) can simply mean that there are n x that are , without any further constraints on whether there are more. Then, gets the reading as in formula (5). On the other hand, could be understood with a kind of closure, that is that there are exactly n x's are , thus as in formula (3).
It has been a matter of a wide debate in formal semantics and pragmatics
what is the right approach to interpreting bare numerals in natural languages. It
has been proposed that: (i) the literal meaning of n is at least n, while the
condition exactly comes as a scalar implicture
Let us now show how the interpretation of n interacts with the validity of inferences (n or more than n)! (n-1 or more than n-1 ), given the epistemic interpretation of disjunction. Suppose now that n is interpreted with a closure: exactly n. It is easy to observe that, in such a case, possible that n and possible that more than n does not imply possible that n-1 or possible that more than na19n.d>Tnhe91=p(nrwe,mitthhiseecplworsohubircleehmVisaint=iinc02tee:rlepmr9ee=tneitdxias(sxi)9n)=WDneh1i,nlewithio9inc>hn7isd1odfeiorslelcontwoltys ifcmroonpmtlyrabdo9itch=tned91>b^ny the closure of the premise. But suppose that n does not get the \exact" reading, but it is interpreted barely as there are n, so as (5). Then from possible that n we can infer possible that n-1, since the latter does not exclude the possibility that there is a bigger set of elements. 5 or a set of at most n elements, however this interpretation seems to be counterintuitive and allowed only in special contexts.
reason with superlative quanti ers: at least n (mindestens n)6 and at most n (hochstens n), as well as with logically equivalent but linguistically di erent forms of those quanti ers: a comparative negative form and a disjunctive form. These were quanti ers: not more than n (nicht mehr als n) and n or fewer than n (n oder weniger als n) (as logically equivalent to at most n), and not fewer than n (nicht weniger als n) and n or more than n (n oder mehr als n) (as equivalent to at least n).
We were particularly interested in comparing subjects' acceptance of inferences from at most n to at most n+1 (type B: see Table 1) with their acceptance of logically equivalent forms of this inference: type D and type F. We were also interested in people's willingness to infer at most n from not more than n, and vice versa (type G). Furthermore, we checked the respective inferences with at least n, that is: at least n ! at least n-1 (type A), and its equivalent forms: type C and type E. Finally, we checked the inferences between not fewer than n and at least n (type H). Last but not least we tested the respective incorrect inferences with the considered quanti ers, in all forms (see Table 2).
In the premises of our inferences we used four di erent quanti ers: at least three, at most three, at least four, at most four, and their equivalent forms. All the numbers were throughout spelled out in words according to the requirements of German grammar. There was only one example for each distinct inference relation (i.e. two per inference type). Every sentence content was di erent. Additionally, to introduce more variation, sentences which had at most 4/at least
sition (e.g. At least four computers are broken in the lab), sentences which had at least 3 /at most 3 in the premise had a quanti er in the object position (e.g. Arthur has at least three cars.)
whose correctness depends on the \at least" reading of bare numerals, i.e. n ! n (e.g. 4 ! 3); and those that are logically incorrect independently of the presumed reading, such as n ! n+ (e.g. 4 ! 5). 5.1
Procedure The experiment was conducted on German native speakers, mainly students of philosophy, psychology, neuroscience and computer sciences. There were 17 subjects (7 male). Subjects were asked to respond \yes" or \no" to the question whether the second sentence (below the line) has to be true, assumed that they know that the rst sentence (above the line) is true. For a better understanding of the task two examples were given: one of a valid inference, that should be given a \yes" response: 6 We give in brackets the German translation used in the experiment and one of an invalid inference, that requires a \no" response:
Inga has done three exercises.
Inga has done more than two exercises.
Eva has done three exercises.
Eva has done fewer than two exercises.
not depend on the understanding of any of the tested inference relations, that is the examples served as an instruction of what is an inference in general, but not how to evaluate the inferences, that were tested in the experiment.
tasks: one task at a time, displayed on a computer screen. There was no time limit in the test.
At the end of the experiment two additional control questions were asked, in which the inference from at most n to at most n+1 was embedded in a deontic context. Note that the logically correct response to rst question is \yes", while to the second is \no".
(1) Erika promised to drink at most six beers. She drank at most four. Did she keep her promise? (2) Thomas is allowed to eat at most three cookies. He ate at most two. Did he break the rule? Our rst observation is that people accepted the logically correct inferences much more frequently than the logically incorrect ones. The incorrect inferences (apart from disjunctive inferences: E' and F' which turned out specially problematic) were mostly rejected and their acceptance rate was low enough (1 9%) to be considered as a result of random mistakes (Table 2). On the other hand all correct inferences were accepted on the level of at least 20%7, with high variance depending on the form of an inference, in this: inferences of type B and F seemed the most problematic.
of type G and H, that is (at most n ! not more than n) as well as (not more than n ! at most n), and respective inferences between at least n and not fewer than n, which suggests that they do see those expressions as equivalent.
! at most n+1 ) were accepted only by 23% of responders, the inferences of type D (not more than n ! not more than n+1 ), were already accepted by 44%. The di erence was statistically signi cant: z = 2; 333; p = :02; r = :4 8
The inferences of type A, that is at least n ! at least n-1, turned out relatively unproblematic for subjects, who accepted them in ca. 80% of cases. Interestingly a paraphrase to the negative comparative form not fewer than n, made the task more di cult (59% accepted; means comparison: z = 2:07; p = :038; r = :355). However, it is worth to note that inferences of type A were still rejected 7 We give an overall result for a given type of an inference 8 In all the cases we used Wilcoxon Signed Ranks test to compare means. by ca. 20%, which suggests that there is some, at least pragmatic, mechanism suppressing this inference.
The results for the disjunctive inferences (E and F) are especially interesting. First of all the response pattern for disjunctive counterpart of at least corresponds with the predictions of classical logic: While logically valid inferences (E) were accepted on a relatively high level of 67% (which is lower, though not signi cantly lower, compared to the acceptance of the basic form (type A)), the invalid inferences (E') were mostly rejected (only 20% accept). The opposite e ect, however, we got for the disjunctive form of at most. The logically valid inferences (F) were mostly rejected (only 23% accept), whilst invalid inferences (F') were accepted in exactly 50% of cases. Interestingly the acceptance rate of (F) inferences was similar to the acceptance rate of the basic form of inferences with at most (B). In both cases the di erences between acceptance rate of correct and incorrect forms were statistically signi cant, and were as follows: The di erences between correct and incorrect inferences with \disjunctive at most " (F and F'): z = 2:491; p = :013; r = :43 and correct and incorrect \disjunctive at least " (E and E'): z = 2:165; p = :030; r = :37. The di erences between disjunctive at most and at least : incorrect (E' and F') z = 2:057; p = :040; r = :36: and correct (E and F) :z = 2:697; p = :007; r = :46.
cases. There was only one mistake in the incorrect inferences with bare numerals.
6
Secondly, the inferences (at least n ! at least n-), although accepted by a majority of responders, were not as obvious as the standard theory would predict, and 20% of subjects rejected them (A, Table 1). This suggests existence of some, at least pragmatic, mechanism interfering in subject's reasoning with at least. What is also worth reminding, valid inferences with \disjunctive at least " were rejected even more often (F, Table 1). We consider that this e ect can be explained in terms of the epistemic interpretation of at least n and its interaction with the reading of bare numerals. Let us notice that our results provide a weak evidence for the interplay between the reading of bare numerals and the treatment of \disjunctive at least " inferences. In our experiment inferences of type K: n ! n , which base on the \at least" reading of numerals were accepted in ca. 65% of cases, thus our responders in 35% cases integrated the \exact" reading of bare numerals, which resulted in their rejection of considered inferences. However, \disjunctive at least " inferences (type E) ware rejected also in ca. 33%. We suggest that rejection of type E inferences was as well a result of an exact reading of a bare numeral n, as we have explained above. Although a correlation between subject's acceptance of type E and type K inferences failed to reach signi cance, it was close to signi cant (Spearman's rho= :426, p = :08) and we expect that with a bigger sample it could reach the signi cance level.
A inferences: (at least n! at least n-1 ). Namely, the application of the epistemic veri cation condition of at least n together with an exact reading of bare numerals results in rejection of such inferences. This e ect might be however weaker and less likely to occur than in the case when the disjunctive form is given explicitly.
Thirdly, the surprising result that subjects accepted the invalid inferences with \disjunctive at most " more frequently than the valid ones can be explained by our proposal. As we have proposed above, closure in the veri cation condition is optional, since the falsi cation condition is su cient to account for the right semantics. However, if context enforces applying one of the semantical conditions (veri cation or falsi cation), then the other one tends to be ignored. While, from the perspective of classical logic it should be enough to use only one of the conditions (since the other can be de ned via the rst one), in the case of superlative quanti ers the epistemic reading of the veri cation condition creates the bifurcation in the meaning. This results in di erent inferential patterns in which those quanti ers occur, depending on what the context primarily enforced: the veri cation or falsi cation condition.
In what follows, and as we have explained above, when the veri cation condition is used, then n or fewer then n does not imply n+ or fewer then n+ due to the epistemic interpretation. Though, it also does not exclude it if no closure is applied. However, n or fewer then n does imply n- or fewer then n- if the veri cation condition is used but no closure is applied. Now we can explain why inferences (F', Table 2): (n or fewer than n) ! (n-1 to fewer than n-1 ) got a 50% rate of acceptance, although they are invalid both in the standard account and in the epistemic account. Based on De nition 4, 9=nx (x) _ 9<nx (x) is interpreted as 9=nx (x) ^ 9<nx (x) (with closure: : 9>nx (x)). But 9<nx (x) can be broken down to: = ( 9n 1x (x) ^ 9<n 1x (x)) Now implies 9=n 1x (x) ^ 9<n 1x (x) (here we use the assumption that the world accessibility relation is transitive). In such a case it might happen that the closure of the conclusion, that is: : 9>n 1x (x) which contradicts the assumption that 9=nx (x) is ignored by subjects, which results in the high logical mistake ratio. 7
We have argued that the meaning of a quanti er can be de ned as a pair hCF ; CV i, in which the veri cation (CV ) and the falsi cation (CF ) condition for sentences with this quanti er are speci ed separately. Though from the logical point of view those conditions should be dual, in the case of superlative quanti ers they are not. Namely, pragmatic focus on the borderline n in both at least n and at most n enforces a disjunctive veri cation condition, which is further interpreted as a conjunctive list of epistemic possibilities.
veri cation and falsi cation conditions in terms of algorithms. Let us make an observation that semantical equivalence and procedural identity of algorithms are di erent things. Let us consider algorithms A1 and A2:
A1 : Count all x that are . If the number m of x that are is smaller than n 1, then verify.
A2 : Count all x that are . If the number m of x that are equals n or is smaller than n, then verify.
A2 and A3 are semantically equivalent, namely they verify logically equivalent formulas, e.g. 1, 2 and 3, however in a sense of procedures that are executed they are not identical.
1 () :9>nx (x) 2 () 9<n+1 (x) 3 () 9=nx (x) _ 9<nx (x) 1 () 2 () 3 Furthermore, A3 A3 : When the number m of x that are is bigger then n, then falsify.
is dual to both A1 and A2, namely adding an otherwise verify condition to A3 and otherwise falsify condition to A1 and A2 would make A3 semantically equivalent to both A1 and A2. However, without the \otherwise" condition, A3 does not allow to verify any of the given sentences, while A2 or A1 do not allow to falsify them. Then, hA1; A3i, or hA2; A3i could be considered pairs of partial algorithms. Each pair could constitute a full semantical interpretation of each of the sentences 1, 2, 3.
We consider that logically equivalent, but linguistically di erent, natural language sentences may trigger di erent kinds of such partial algorithms, or pairs of partial algorithms. First of all two equivalent sentences that di er in the linguistical form can trigger as primary only one of the algorithmic conditions: veri cation or falsi cation, while the complement condition is ignored. Secondly, they might trigger non-identical veri cation/falsi cation procedures. Consequently, it might happen that the executed procedures di er in complexity. Additionally, if we take into account that some extra mechanisms, e.g. the above-discussed epistemic interpretation of disjunctive conditions, might play a role, we not only obtain di erent algorithms in the procedural sense but also semantically non-equivalent algorithms for logically equivalent or even same sentences. For instance, A2 would be replaced by:
A02: if [it is possible that there are exactly n x that are AND it is possible that there are fewer than n x that are ], then verify.
While A2 is dual to A3, A02 is not anymore. However a pair hA3; A02i could constitute a semantical interpretation of a natural language sentence at most n : x (x), which would explain the non-semantically coherent (from the point of view of classical semantics) inference patterns in which this quanti er occurs. Appendix: The complete list of pairs of sentences used in the experiment (premise, conclusion), translated from German. At least four Anna's dress are red. At least three Anna's dresses are red.
Arthur has at least three cars. Arthur has at least two cars.
At most four books were stolen from the library. At most ve books were stolen from the library. Markus ate at most three pieces of cake. Markus ate at most four pieces of cake. Not fewer than four cards are missing in the deck. Not fewer than three cards are missing in the deck.
A child has painted not fewer than three pictures. A chid has painted not fewer than two pictures. Not more than four students came today to the philosophy seminar. Not more than ve students came today to the philosophy seminar.
Sabine got not more than three presents. Sabine got not more than four presents. Four or more than four students were sick this week. Three or more than three students were sick this week.
Christopher speaks three or more than three languages. Christopher speaks two or more than two languages.
Four or fewer than four students have passed the course. Five or fewer than ve students have passed the course.
Beate has three or fewer than three children. Beate has four or fewer than four children. Five people came to the party. Four person came to the party.
Monika invited six guests to her birthday party. Monika invited two guests to her birthday party. Seven fruits in the basket have spoilt. Six fruits in the basket have spoilt.
Alicia bought eight bottles of beer. Alicia bought ve bottles of beer.
At least four of Carolina's scarfs are blue. At least ve of Carolina's scarfs are blue. Thomas has read at least three books. Thomas has read at least four books.
At most four computers in the lab are broken. At most three computers in the lab are broken. Andrea baked at most three pizzas. Andrea baked at most two pizzas.
Not fewer than four professors attended the meeting. Not fewer than ve professors attended the meeting.
Hans had not fewer than three glasses of wine. Hans had not fewer than four glasses of wine. Not more than four people have applied for this job. Not more than three people have applied for this job.
Natalie wrote not more than three exercises. Natalie wrote not more than two exercises. Four or more than four girls in the class are good in arts. Five or more than ve girls in the class are good in arts.
Christina's cat gave birth to three or more than three kittens. Christina's cat gave birth to four or more than four kittens.
Four or fewer than four students failed in the exam. Three or more than three students failed in the exam.
Tanja trains three or fewer than three times a week. Tanja trains two or fewer than two times a week.
Four new students joined the chess club this week. Five new students joined the chess club this week. Stephanie baked two cakes for her birthday. Stephanie baked six cakes for her birthday. Six members of the library club came to the meeting. Seven member of the library club came to the meeting.
Frank gave his mother three roses. Frank gave his mother eight roses.
Not more than three children have done their homework for today. At most three children have done their homework for today.
At most three girls took part in the maths competition. Not more than three girls took part in the maths competition.
Erika has not more than four necklaces. Erika has at most four necklaces.
Lena takes at most four courses at the university. Lena takes not more than four courses at the university.
Not fewer than three new animals were born in the city zoo. At least three new animals were born in the city zoo.
At least three exotic three have died in our botanic garden. Not fewer than three exotic threes have died in our botanic garden.
Daniel plays not fewer than four times a week football. Daniel plays at least four times a week football.
Albert has at least four exams this semester. Albert has not fewer than four exams this semester.