Normalmente los órganos del Estado aplican ei derecho interno —aunque se trate de una cuestión que presente a la vez un aspecto

The results for particular sentences forjoint categories of tasks in which we could expect this kind of response (so excluding “all”- and “no”-models in which such responses did not occur at all) are characterized by quite high variance: v = 16.5 for “some”-sentences and v = 11.9 for “somenot”-sentences. We observed that the number of subjects generating the particular sentence “Some dots are black” decreased with the number of black dots on the picture. (The opposite effect was observed for “somenot”-conclusions). We already proved a significant difference in the number of generated “some/somenot”-sentences between tasks of Cat. 1. and Cat. 2. We further suspected a general correlation between

12_{To compare, parametric tests had similar results. For some: t(12) = 6.139; F(1,12) =}

some sn bdots Pearson Correlation -.493 .800 Sig. (1-tailed) .026 .000 N 16 16 proc Pearson Correlation -.564 .841 Sig. (1-tailed) .012 .000 N 16 16

Table 2.4: Correlation between the number of “some/somenot”-sentences and number/precentage of black dots in the model

the number of black dots and the number of “some”- (negative correlation) or “somenot”-sentences (positive correlation). The other possible correlation could be between the number of these conclusions and the percentage of black dots in the universe (which we will call a “model-dependent cardinality”). As we have already mentioned, both border-cases were excluded:

1. no black dots in the model, 2. all dots black.

In the first case a sentence “Some dots are black” is false and a sentence “Some dots are not black” does not appear because of the implicature effect (“somenot” as “not no”). On the other hand, in the second case a sentence “Some dots are black” does not appear because of the implicature (“some” as “not all”) and “Some dots are not black” is just false.

We checked whether the following correlations are significant:

• (negative) correlation between the variable some and the number of black dots (bdot) and/or the percentage of black dots (proc) in the universe.

• (positive) correlation between the variable (sn) and the number of black dots (bdot) and/or the percentage of black dots (proc) in the universe. According to our predictions, we proved a strong correlation (on the level of

α = .05) in all four cases. (See Tables 2.4 and 2.5.) The correlation coefficients were higher and the significance stronger (p < .01) for non-parametric tests. (We need this statistic at least for the variable some which shows a deviation from a normal frequency distribution.)

Further on we were interested in the nature of these dependencies. We carried out four different liner regression analyses to determine whether there is a linear relationship in all the above-mentioned cases. 13

13_{We are aware that our data are not appropriate for linear regression, especiallysomewhich}

2.4. Results: picture test 45 some sn bdots Spearman’s rho -.598 .820 Sig. (1-tailed) .007 .000 N 16 16 proc Spearman’s rho -.736 .849 Sig. (1-tailed) .001 .000 N 16 16

Table 2.5: Correlation between the number of “some/somenot”-sentences and number/precentage of black dots in the model – non-parametric

We regressed bothsomeandsn onbdot, or alternatively onproc, which served as independent variables. For the linear regression ofsome onbdot, ANOVA was not significant (p=.053) and the assumptions of the analysis were violated. The three remaining regressions: some onproc,sn onbdot andsn onproc were signif- icant (p < .05), however they were difficult to interpret because the assumptions were again violated. For more detailed reports of these analyses see AppendixA. Because of violation of the important assumptions of linear regression we can- not conclude that there is a linear dependence in the above cases. This situation can be caused by too small sample sizes, but can be also a signal that the consid- ered relationship is indeed non-linear. Curve estimation analysis was conducted to find out whether some other non-linear relationships are more adequate to model those dependencies. It turned out that for both some dependence on proc

and on bdot, cubic regression made the best fit, whereas the dependence of sn

on both predictors was better described by linear or logarithmic relation. (See AppendixA.)

One may also make conclusions based on analyzing graphs. Graph of the dependence of some on bdot (Figure 2.1) shows clearly that the score of the dependent variable is very low (but different from 0) for the value of bdot = 1

(we know also that for bdot= 0, which was an excluded case,some = 0), then it increases rapidly having the highest values forbdot = 2,3 and 4, and then begins to decrease. Similarly some depends on proc. The scores of some are first low (forproc = 10), then increase having highest values in the range between 20and

40, and then they decrease (Figure 2.2).

For dependencies of sn on proc and on bdot the curve estimation proved non-significant coefficients for cubic regression, whereas the linear regression was highly significant. Graphs (Figures 2.3 and 2.4) clearly show that sn scores in- crease regularly with the scores ofbdot and similarly with the scores of proc. Still we have to remember that the border case of all dots black in the model was should be treated carefully. We are however interested is the preliminary estimation of plausible dependencies.

Figure 2.1:

excluded. Otherwise we would observe the sudden decline (to 0!) of the scores of sn at some point and probably we could obtain a significant coefficient for polynomial regression.

Since all the statistics for linear relationship were significant (on the level

p < .0001) and as well the graphs proved its adequacy we could have agreed with such a model of dependencies of sn on proc and sn on bdot and explain the lack of satisfied assumptions by too small sample sizes. However, a similarly high significance was proved in those cases for logarithmic relationship. All the tests were significant on the level of p < .0001, and the explained variance was even higher for logarithmic models than for linear ones. It seems plausible that in reality the considered dependencies are not linear but the values of the dependent variable sn grow with the values of proc/bdot variables slower than linearly.

Summarizing, a subject’s inclination to generate “some”-sentences as true of given objects (measured by a number of subjects who generated such a sentence in the whole group) equals0 for models with 0elements of a referred quality, is low for models with only1such an object, then increases rapidly having highest values for2−3such objects and then decreases (even to0for models in which all elements have a referred quality). It works similarly if we consider not a cardinality of referred objects but a percentage of them in a universe. On the other hand, a subject’s inclination to generate “somenot”-sentences grows monotonically with the number of referred objects (provided that the model does not contain only this kind of elements) or alternatively with the percentage of objects of a referred quality in a universe (but of course would decrease to 0 for 100%). This growth seems to be, however, slower than linear and thus a logarithmic regression makes

2.4. Results: picture test 47

Figure 2.2:

Figure 2.4:

probably a better fit.

The above analysis gives us a strong base for presumptions concerning a kind of scalar implicature of the quantifier “some”. According to the above results we can conclude that “some” means “not too many” (or “not too many in a given domain”) and thus it is less willingly used when referring to bigger samples of objects (or bigger subsets of fixed domains). This “cardinality dependence” of the quantifier “some” is systematic.

Since our experiment was not oriented towards investigation of this kind of relationships, we suspect that one could get rid of some of the deficiencies of the analyses with bigger samples and a more suitable selection of models. We are of the opinion that an experiment that could confirm these results and would allow us to compare the differences between the cardinality dependence (bdot) and the “model” cardinality dependance (proc) is worth taking into consideration.

A few final words concerning the critical values of the independent variable

bdot that showed importance in our analyses. First of all number 2. It seems that “some” means “at least two”, at least in Polish. Less than half of subjects gave a response with “some” for models with only one black dot and this number increased considerably for 2 dots. This result may be connected with the plural form of a Polish quantifier “niektóre” and the sentence (in plural) given in the tutorial. “Niektóre” does not occur in Polish in singular at all.14 The other critical value is number 4. The curves for some had extrema close to 4 and displayed a decline beginning from 5. A noteworthy fact may be that this very number is

2.5. Results – direct inference 49