Superlative quantifiers and epistemic interpretation of disjunction Maria Spychalska Ruhr-University Bochum, Department of Philosophy Abstract. We discuss semantics of superlative quantifiers at most n and at least n. We argue that the meaning of a quantifier is a pair specifying a verification and a falsification condition for sentences with this quantifier. We further propose that the verification condition of superlative quanti- fiers 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 quantifiers in German. We discuss the results in the light of our proposal. 1 Introduction There is an ongoing debate (Look inter alia: (Geurts & Nouwen, 2007), (Koster- Moeller et al, 2008), (Geurts et al., 2010), (Cummins & Katsos, 2010), (Nouwen, 2010), (Cohen & Krifka, 2011)) concerning the right semantical interpretation of so-called superlative quantifiers, such as at most n and at least n, where n represents a bare numeral, e.g two. Generalized Quantifier Theory (referred here as a “standard account”) defines superlative quantifiers as equivalent to respective comparative quantifiers: fewer than n and more than n, that is: at most n (A, B) ⇐⇒ f ewer than n + 1(A, B)1 (1) at least n (A, B) ⇐⇒ more than n − 1(A, B) (2) 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 differences between comparative and superlative quantifiers involving their linguistic use, acquisition, processing and the inference patterns in which they occur. First of all, it seems that superla- tive and comparative quantifiers are not freely exchangeable in same linguistic contexts. Geurts & Nouwen (2007) provide, among many others, such examples: (a) I will invite at most two people, namely Jack and Jill. (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 quantifier 74 where (a) is considered a good sentence, while (b) is less felicitous. The con- trast between (a) and (b) suggest that while embedding an indefinite expression (two) in a superlative quantifier licenses a specific construal (namely Jack and Jill ), the same is not licensed in the case of a comparative modifier. Secondly, it has been demonstrated that superlative quantifiers are mastered later than the comparative ones during language development (Musolino, 2004), (Geurts et al., 2010). Furthermore, there is ample data concerning processing of those quantifiers. It has been for instance shown that verification of sentences with superlative quantifiers requires a longer time than verification of sentences with respective comparative quantifiers (Koster-Moeller et al, 2008),(Geurts et al., 2010). Moreover, the processing of quantifiers is influenced by their mono- tonicity. A quantifier Q(A, B) is upward monotone in its first argument A if it licences inferences from subsets to supersets, that is if Q(A, B) and A ⊆ A0 , then Q(A0 , B). A quantifier Q(A, B) is downward monotone in its first argument A if it licences inferences from supersets to subsets, that is if Q(A, B) and A0 ⊆ A, then Q(A0 , B). Understood as h1, 1i generalized quantifiers, at least n and more than n are upward monotone in both their arguments, while at most n and fewer than n are downward monotone. It has been shown that although the downward monotone at most n and fewer than n take a longer time to be verified than the upward monotone at least n and more than n, they are actually falsified faster (Koster-Moeller et al, 2008). Finally, important arguments against the semantical equivalence between the comparative and superlative quantifiers come from the analysis of people’s accep- tance of inferences with those quantifiers. 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 quantifiers (Geurts et al., 2010), (Cum- mins & Katsos, 2010). To illustrate it with an example: while people are unlikely to accept that if at most 5 kids are playing in this room, then at most 6 kids are playing in this room (2-14% in Cummins’ and Geurts’ experiments for this infer- ence scheme) they are more likely to accept that if fewer than 5 kids are playing in this room, then fewer than 6 kids are playing in this room (between 60-70% in Cummins’and Geurts’). Such data seem to directly contradict the standard account. 2 Modal semantics or clausal implicature? Several theories have been developed to explain why seemingly logically equiv- alent quantifiers show such big differences in the way people use them in the language. Geurts (2007), (2010) proposes modal semantics for superlative quan- tifiers and rejects the assumption that equivalences (1) and (2) hold in natural languages. According to this proposal (referred here as a “modal account”), while more than n and less than n have a conventional meaning defined in terms of inequality relation, at least n and at most n have a modal component, namely: 75 at least n A’s are B means that: a speaker is certain that there are n elements which are both A and B, and considers it possible that there are more than n. at most n A’s are B means that: a speaker is certain that there is no more than n elements that are both A and B, and considers it possible that there are n elements. According to this proposal, as semantically richer, superlative quantifiers are expected to be harder to process than the respective comparative quantifiers. Finally, defined as above, at most n A are B does not imply at most n+ A are B : The latter implies that it is possible that there are (exactly) n+ A that are B, which is contradicted by the semantics of at most n in the premise. There are strong arguments against the modal account. For instance this ac- count seems unsatisfactory with regard to superlative quantifiers embedded in conditional and various other contexts. Authors (Geurts & Nouwen, 2007),(Geurts et al., 2010) realize themselves this problem and illustrate it with the following example: If Berta had at most three drinks, she is fit to drive. Berta had at most two drinks. Conclusion: Berta is fit to drive. Such inferences, which are indeed licensed by the inference from at most n (2) to at most n+ (3), are commonly accepted by speakers (over 96% in Geurts’s experiment). Furthermore, while inferences from at most n to at most n+(1) are rejected by majority of people (ca 84% in Geurts’ experiment) there are subjects who do accept them (14% in Geurts’ and even more in our experiment — ca. 23%). 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. A slightly different account was proposed by Cummins & Katsos (2010), who 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. While sentence pairs, such as: Jean has at most n houses. Specifically she has exactly n+1 houses. get average coherence judgments very low, i.e. −4, in the scale from −5 (incoherent) to +5 (coherent), sentence pairs: Jean has at most n houses. Specifically she has n-1 houses 76 get already +1.9. This result speaks against the “modal account”, whose direct consequence is semantical incompatibility of at most n and exactly n-. Consequently, Cummins et al. agree with Geurts that at most n and at least 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 (Levinson, 1983). Clausal implicature is a quantity implicature inferred due to use of epistemically weak statement. Since the expressed statement with a superlative quantifier 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 quantifiers, 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 satisfied 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 quantifiers in different reasoning contexts. Our motivation to further experimentally investigate reasoning with superla- tive quantifiers 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 quantifiers: 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) finally, the lack of data concerning people’s acceptance of logically incorrect inferences with the quantifiers considered here. 3 Two semantic conditions for at most n Krifka (1999) points out that semantic interpretation of a sentence is usually a pair that specifies when the sentence is true and when it is false. Following Krifka, we propose to define meaning of a quantifier as a pair hCF , CV i, where CV is a verification condition (specifies how to verify sentences with this quantifier) and CF is a falsification condition (specifies how to falsify sentences with this quanti- fier). Furthermore, we propose that the interpretation of a quantifier depends on a semantic context in which this quantifier is used, namely whether the context requires the use of the verification condition or the falsification condition. Ver- ification and falsification 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 satisfied. From a perspective of classical logic, these conditions should be dual, namely if C is a CV condition for sentence φ, then C is a CF condition for sentence ¬φ, and vice versa. We further, however, observe that in the case of superlative quantifiers, there is a 77 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 quantifiers, in terms of algorithms, that have to be performed in order to verify (or falsify) sentences with those operators. (See also (Szymanik, 2009), (Szymanik & Za- jenowski, 2010), (Szymanik & Zajenowski, 2009)) Krifka (1999) observes, that a sentence at most n x: φ(x)2 says only that more than n x: φ(x) is false, and leaves a truth condition underspecified. In other words, the meaning of at most n provides an algorithm for falsifying sentences with this quantifier, but not (immediately) for verifying them. This corresponds with the experimental data showing that it is easier to falsify sentences with at most than to verify them (Koster-Moeller et al, 2008). Consequently, the primal semantical condition of at most n x: φ(x) could be understood as an algorithm: “falsify when the number of x that are φ exceeds n”, and would constitute what we understand by the falsification condition. Definition 1 ( falsification condition for at most) CF (at most x : φ(x)) := If ∃>n x(φ(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 defines in this case the truth-condition. However a negation of a falsification condition is in sense informationally empty: it does not describe any concrete situation in which the given sentence can be verified. As a result, in those contexts that require to directly verify a sentence, we refer to a verification condition, which is specified independently. As expressing a positive condition, at most n may be understood as a disjunction n or fewer than n (“disjunctive at most”). Definition 2 ( verification condition for at most) CV (at most n : xφ(x)) := If (∃=n xφ(x) ∨ ∃n xφ(x), as contradictory. In our analysis, however, we want to treat closure as a merely optional condition. This results from regarding the verification and falsification conditions as independent from each other. 3.2 Verification condition for at most If we assume that disjunctions in natural language are likely to be interpreted as conjunctions of epistemic possibilities, then we get the following verification condition for at most: Definition 4 ( epistemic interpretation of the verification 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 different domain of objects over which we quantify. We assume also standard semantics for modal operators, and we restrict to reflexive and transitive Kripke models. 79 CW (at most n x : φ(x)) := If (∃=n xφ(x) ∧ ∃n xφ(x), then f alsif y Where: n ^ (∃=n xφ(x) ∧ ∃n xφ(x)), i.e. ξ ξ, however the theorem holds only in reflexive Kripke frames. Furthermore, the optional character of the closure bases on our assumption that the falsification and verification conditions are in a sense independent and only as a pair constitute the full semantic interpretation. Since the falsification condition, as defined in 2, is sufficient to account for the right semantical criterion of when the sentence with at most n is false, the closure of the verification 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. When at most n is interpreted as in Definition 4, then the inference from at most n to at most n+1 is not valid. Namely, from ∃=n φ(x) ∧ ∃n−1 xφ(x). It is important to notice that without the closure the implication holds (if the epistemic reading of the verification condition is applied). In some aspects our proposal might seem similar to Geurts’ “modal” ap- proach, namely we define the verification condition of at most n and at least n (see below) in modal terms. The main difference is that, in our account, it is only the verification condition that is defined modally, while the falsification condition remains standard. This results in a specific split or ambiguity in the meaning of superlative quantifiers. 4 At least and bare numerals The quantifier at least n might seem perhaps less interesting than at most n, as 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 quantifier, at least n appears to provide a clear verification algorithm: “verify when n x (that are φ) are found”. Such a semantical interpretation would not, however, account for the linguistical differences between at least n and more than n-1. Let us start with defining a falsification condition for at least n as follows: Definition 5 ( falsification condition for at least) CF (at least n : xφ(x)) := If ∃n−1 xφ(x) ⇐⇒ ∃x1 ...∃xn [ φ(xi ) ∧ (xi 6= xj )] (5) i=1 1≤in xφ(x)), then verif y 81 The latter can be handled as a conjunctive list of possibilities. Definition 7 ( epistemic interpretation of the verification condition for at least n) CF (at least n : xφ(x)) := If (∃=n xφ(x) ∧ ∃>n xφ(x)), then verif y and (closure) n−1 _ If ( ∃=i xφ(x)), then f alsif y i=0 Having introduced the semantical conditions for at least n, we further ana- lyze how they affect inferences with this quantifier, 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 ψ = nx : φ(x) can simply mean that there are n x that are φ, without any further constraints on whether there are more. Then, ψ gets the reading as in formula (5). On the other hand, ψ could be understood with a kind of closure, that is that there are exactly n x’s are φ, 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 condi- tion exactly comes as a scalar implicture (Horn, 1972), (ii) the basic meaning of n is exactly n, while both at least and at most readings would be context-based (Breheny, 2008) (iii) n is ambiguous between at least n and exactly n (Geurts, 2006), (iv) n is underspecified and can receive at least, at most or exactly readings depending on the context (Carston, 1998). 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 n- 1. The premise which V is interpreted as in Definition 7 does not imply ∃=n−1 ∧ >n−1 n−2 ∃ (with closure i=0 ¬  ∃=i xφ(x)) While ∃>n−1 follows from both ∃>n and ∃=n , the problematic element is ∃=n−1 , which is directly contradicted by 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 counterintu- itive and allowed only in special contexts. 82 5 Experiment We conducted a pilot reasoning experiment (in German) to check how people reason with superlative quantifiers: at least n (mindestens n)6 and at most n (höchstens n), as well as with logically equivalent but linguistically different forms of those quantifiers: a comparative negative form and a disjunctive form. These were quantifiers: 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 infer- ences 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 quantifiers, in all forms (see Table 2). In the premises of our inferences we used four different quantifiers: 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 different. Ad- ditionally, to introduce more variation, sentences which had at most 4/at least 4 (or their equivalent forms) in the premise had a quantifier in the subject po- 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 quantifier in the object position (e.g. Arthur has at least three cars.) As fillers we used inferences with so-called bare numerals (e.g. four ): those 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 first 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 83 Inga has done three exercises. Inga has done more than two exercises. and one of an invalid inference, that requires a “no” response: Eva has done three exercises. Eva has done fewer than two exercises. Note that the examples were selected in such a way that their validity did 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. After reading the instruction subjects saw 40 randomly ordered reasoning 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 first 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? Tables: 1 and 2 summarize the inferences as well as the results. 5.2 Results Our first 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. The important result is that nearly 100% of responders did accept inference 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. Furthermore, while inferences from type B, namely the problematic (at most n → 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 difference was statistically significant: z = −2, 333, p = .02, r = −.4 8 Thus, it seems that paraphrasing at most n to the negative form not more than n facilitates the inference. 84 Table 1. Logically correct inferences Premise Conclusion Correct Responses Percentage A At least at least 4 at least 3 13 79% at least 3 at least 2 14 B At most at most 4 at most 5 3 23% at most 3 at most 4 5 C Not fewer than not fewer than 4 not fewer than 3 8 59% not fewer than 3 not fewer than 2 12 D Not more than not more than 4 not more than 5 7 44% not more than 3 not more than 4 8 E N or more than n 4 or more than 4 3 or more than 3 10 67% 3 or more than 3 2 or more than 2 13 F N or fewer than n 4 or fewer than 4 5 or fewer than 5 6 23% 3 or fewer than 3 4 or fewer than 4 2 G “At most” Equivalence not more than 3 at most 3 16 98% at most 3 not more than 3 17 not more than 4 at most 4 17 at most 4 not more than 4 17 H “At least” Equivalence not fewer than 3 at least 3 17 93% at least 3 not fewer than 3 17 not fewer than 4 at least 4 16 at least 4 not fewer than 4 13 K Numerical 5 4 9 65% 6 2 11 7 6 13 8 5 11 embedded at most 4 at most 6 17 100% L at most 2 at most 3 16 94% Table 2. Logically incorrect inferences Premise Conclusion Correct Responses Percentage A’ At least at least 4 at least 5 1 6% at least 3 at least 4 1 B’ At most at most 4 at most 3 0 6% at most 3 at most 2 2 C’ Not fewer than not fewer than 4 not fewer than 5 1 9% not fewer than 3 not fewer than 4 2 D’ Not more than not more than 4 not more than 3 1 9% not more than 3 not more than 2 2 E’ N or more than n 4 or more than 4 5 or more than 5 3 20% 3 or more than 3 4 or more than 4 4 F’ N or fewer than n 4 or fewer than 4 3 or fewer than 3 8 50% 3 or fewer than 3 2 or fewer than 2 9 K’ Numerical 4 5 0 1,5% 2 6 1 6 7 0 3 8 0 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 difficult (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. 85 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 interest- ing. First of all the response pattern for disjunctive counterpart of at least cor- responds 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 significantly lower, compared to the acceptance of the basic form (type A)), the invalid inferences (E’) were mostly rejected (only 20% accept). The opposite effect, however, we got for the disjunctive form of at most. The logically valid in- ferences (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 differences between acceptance rate of correct and incorrect forms were statistically significant, and were as follows: The differences 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 differences 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. The “correct” inferences with bare numerals were accepted in ca. 65% of cases. There was only one mistake in the incorrect inferences with bare numerals. Finally, both embedded at most inferences got nearly 100% correctness rate (one mistake only for question 2). 6 Discussion Although our results cannot be treated as a final evidence of our theory, our experiment certainly provides various important observations that support the plausibility of our proposal. First of all, all the implications between the negative comparative and su- perlative forms of considered quantifiers were almost without exceptions ac- cepted by our subjects. This result supports the assumption that those are se- mantically equivalent forms in natural language. 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 pre- dict, 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 effect 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 treat- ment 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” 86 reading of bare numerals, which resulted in their rejection of considered infer- ences. 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 significance, it was close to significant (Spearman’s rho= .426, p = .08) and we expect that with a bigger sample it could reach the significance level. A similar effect is presumably the reason why 20% of subjects rejected type A inferences: (at least n→ at least n-1 ). Namely, the application of the epis- temic verification condition of at least n together with an exact reading of bare numerals results in rejection of such inferences. This effect 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 verification condition is optional, since the falsification condition is sufficient to account for the right semantics. However, if context enforces applying one of the semantical conditions (verification or falsification), 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 defined via the first one), in the case of superlative quantifiers the epistemic reading of the verification condition creates the bifurcation in the meaning. This results in different inferential patterns in which those quantifiers occur, depending on what the context primarily enforced: the verification or falsification condition. In what follows, and as we have explained above, when the verification con- dition 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 verification 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 ac- count and in the epistemic account. Based on Definition 4, ∃=n xφ(x)∨∃n xφ(x)). But ∃n−1 xφ(x) which contradicts the as- sumption that ∃=n xφ(x) is ignored by subjects, which results in the high logical mistake ratio. 7 Conclusions We have argued that the meaning of a quantifier can be defined as a pair hCF , CV i, in which the verification (CV ) and the falsification (CF ) condition 87 for sentences with this quantifier are specified separately. Though from the log- ical point of view those conditions should be dual, in the case of superlative quantifiers they are not. Namely, pragmatic focus on the borderline n in both at least n and at most n enforces a disjunctive verification condition, which is further interpreted as a conjunctive list of epistemic possibilities. Finally, we would like to say few words about why we want to understand the verification and falsification conditions in terms of algorithms. Let us make an observation that semantical equivalence and procedural identity of algorithms are different things. Let us consider algorithms A1 and A2 : A1 : Count all x that are φ. If the number m of x that are φ is smaller than n − 1, then verif y. A2 : Count all x that are φ. If the number m of x that are φ equals n or is smaller than n, then verif y. A2 and A3 are semantically equivalent, namely they verify logically equiv- alent formulas, e.g. ψ1 , ψ2 and ψ3 , however in a sense of procedures that are executed they are not identical. ψ1 ⇐⇒ ¬∃>n xφ(x) ψ2 ⇐⇒ ∃