Age, schooling and conditional reasoning.

Cognitive Development 21 (2006) 131–145

Age, schooling and conditional reasoning

Lavee Artman a, Sorel Cahan b,∗, Dinah Avni-Babad b

a The David Yellin College, Jerusalem, Israelb School of Education, The Hebrew University, Jerusalem, Israel

Abstract

This study estimated the independent effects of age and schooling in grades 7–9 on scores obtained oninvalid conditional and class syllogisms. The results, which point to a negative, albeit small, effect of out-of-school experience and to a sizeable positive effect of schooling, replicate previous findings in a differentage range and support the counterintuitive hypothesis that accumulated daily experience with conditionalshas a negative effect on the development of conditional reasoning, and that improved performance on invalidproblems with age is entirely attributable to schooling. Contrary to most cognitive tasks, therefore, in whichschooling operates in the same direction as out-of-school experience, in this case, schooling breaks daily-lifeinterpretational habits, and therefore, is critical for development.© 2006 Elsevier Inc. All rights reserved.

Keywords: Conditional reasoning; Schooling; Cognitive development

Invalid conditional syllogisms include two argument patterns: denying the antecedent (DA; ifp then q, not p, not q) and affirming the consequent (AC; if p then q, q, p). The classical fallaciesregarding these invalid argument forms are to accept “not q” or “p”, respectively, as true, whichwould be in line with biconditional interpretation (“if and only if”) of the conditional (Braine &O’Brien, 1991; Braine & Rumain, 1983; O’Brien, 1987). These fallacies are common in earlychildhood and their frequency decreases with the increase of age (e.g., Evans, Newstead, & Byrne,1993).

Two major approaches have been used to explain the origin of these fallacies and the devel-opmental mechanism involved in their decrease with age: logical competency models (e.g.,Markovits, 1993; Markovits & Barrouillet, 2002) and comprehension theory (e.g., Braine &O’Brien, 1998). According to the first, the fallacies are due to inadequate logical ability orconstraints in premise representation, and their decrease with age reflects the development ofconditional reasoning abilities. Traditional competency models (e.g., Overton, 1990; Overton,

∗ Corresponding author. Tel.: +972 2 5882083; fax: +972 2 5882098.E-mail address: [email protected] (S. Cahan).

0885-2014/$ – see front matter © 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.cogdev.2006.01.004

132 L. Artman et al. / Cognitive Development 21 (2006) 131–145

Ward, Black, Noveck, & O’brien, 1987; Piaget, 1972) consider children to be limited interpretersof the conditional to cases in which both elements (p and q) are either present or absent. Conse-quently, increased success rates with age reflect continuous maturation of logical competence. Amore recent view (e.g., Markovits & Barrouillet, 2002) attributes the fallacies to children’s inabil-ity to represent the case of not p and q due to limited capacity of working memory or restrictedworld knowledge, and the decrease in their frequency with age to the development of workingmemory capacity and accumulation of world knowledge.

In contrast, comprehension theory (e.g., Rumain, Connel, & Braine, 1983) attributes logi-cal errors to the tendency to apply everyday discourse rules to reasoning tasks (Begg & Harris,1982; Braine & O’Brien, 1991; Braine & Rumain, 1983; Rumain et al., 1983). According tothis view, on problems in which neither the direct-reasoning routine nor any obvious availablestrategy leads to a solution, people will often base inferences on other resources, especially, con-versational implicatures (O’Brien, Dias, & Roazzi, 1998). These generalized implicatures followfrom extra logical meanings that are common to how a natural-language particle is construed(Grice, 1989) and are commonly made unless there are circumstances to inhibit them (Grice,1975).

Comprehension theory is especially relevant to the explanation of the classical fallacies inconditional reasoning. In natural language, the word ‘if’ is ambiguously represented as a condi-tional or a biconditional according to context (Evans et al., 1993, chap. 2). However, the commoninterpretation of a typical (i.e., circumstantial) ‘if’ is biconditional, that is, if not p then not q (e.g.,Geis & Zwicky, 1971). This is because these conditionals express either a sufficient condition (inwhich the not p case is irrelevant and no further interpretation is required)1 or a necessary condition(e.g., threats, intentions and promises) for which the biconditional interpretation is appropriate.Consequently, inferring the obverse of a conditional (i.e., “if not p then not q”) is in fact a correctconclusion and should be considered rational in daily context (e.g., Evans, 1993; Fillenbaum,1986). Indeed, Scholnick and Wing (1991, 1992) found that 70–90% of the conclusions derivedby adults from invalid conditional arguments in daily conversations were biconditional.

This also holds true with respect to everyday class statements of the form “all are”. Becausethe speaker in everyday life is expected to be as informative as possible, the correct interpretationof such statements is biconditional, that is “All A’s and only A’s” (see Cahan and Artman (1997)for a comprehensive discussion).

In contrast to the logic of conversation, in propositional logic the meaning of a conditionalstatement is decontextualized from its intended meaning in everyday discourse. In logic thereis only one correct interpretation of a conditional, namely, material implication: the meaningcommon to all conditional statements. Material implication between two propositions (P → Q)is a relation that is true in all cases but p and not q. That is, p is sufficient (but not necessary) toq, and q is necessary (but not sufficient) to p. Obviously, for this interpretation, the DA and ACargument forms are invalid by definition.

The fallacies in conditional and class syllogisms can thus be ascribed to overgeneralization ofthe logic of discourse. Subjects apply a biconditional interpretation to the major premise becausethis is the only one they know to be appropriate for those everyday situations in which the addresseeis required to make inferences. Hence, the generalization of the biconditional interpretation tolaboratory conditional inference tasks should not be a surprise:

1 Indeed, Newstead, Ellis, Evans, and Dennis (1997) found that on truth table tasks, FT cases are judged by 92–96% ofthe subjects as false or irrelevant for a variety of everyday conditionals (temporal, causal, promise, threat, tip, warning,universal, intentional).

L. Artman et al. / Cognitive Development 21 (2006) 131–145 133

. . . how is a subject to understand the meaning of an utterance in an artificial context such asthis [experimental reasoning tasks], except by reference to its normal everyday usage? . . .

[T]he DA inference is reasonably . . . made in most real-world contexts in which it appears.(Evans, 1993, p. 18)

The decrease in erroneous responses to invalid conditional syllogisms with age is explainedby comprehension theory as a growing dissociation from the everyday biconditional interpreta-tion which, in turn, is attributed to an improved ability to set aside the “logic of conversation”when solving laboratory reasoning problems (Rumain et al., 1983). If this is the case, the fol-lowing question arises: How, without assuming growth in logical competence, can we explainthe gradual abandonment of the biconditional interpretation, and hence the decline in classi-cal fallacies with age? Unfortunately and unsurprisingly, comprehension theory is not explicitregarding the causal model underlying this developmental process, due to the difficulties asso-ciated with its identification. That is, there is no specific explanation for how adults become“more able than children to set aside the logic of conversation when confronted by laboratoryreasoning problems” (Rumain et al., 1983, p. 421). In order to succeed on laboratory tasks, anaı̈ve reasoner must know in advance that in the laboratory reasoning context she/he is expectedto adopt a new and different interpretation to the premises, which is neither explicit in his/herinstructions (Evans, 2002) nor compatible with the correct everyday usage of the ‘if’ particles.That is, successful performance on laboratory tasks requires children to (a) become acquaintedwith the unnatural demands of the ‘logic game’ (decontextualization and implicative-like inter-pretation) and (b) learn to adapt interpretations to contexts (it will be inadequate if not bizarre,to subject everyday communication to the partial meaning common to all conditionals). Indeed,Politzer (1986) noted that the development of logical thought always competes with previouslyacquired, contradictory laws; it tends toward a dual system and it depends on differentiationbetween and coordination of the two subsystems: language communication laws and propositionallogic.

Clearly, everyday life cannot be expected to provide the experiences necessary for this learn-ing process. Daily experience cannot be expected to acquaint children with the implicative,asymmetric, interpretation of the conditional, nor can it be expected to provide them withpractice in its use. In fact, given that the only relevant everyday life interpretation is bicon-ditional, the accumulation of daily conversational experience should reinforce its usage ratherthan weaken it. That is, everyday experience should be dysfunctional to the disassociationfrom the biconditional interpretation and, therefore, also to success on invalid conditional syl-logisms. The differentiation between and the coordination of language communication lawsand propositional logic necessarily requires a different kind of experience, in which chil-dren have the opportunity to encounter and practice the ‘logic game’. According to Cahanand Artman (1997) this experience is provided by schooling. Mainly through mathematicsand science, schooling indirectly offers children what is perhaps their first encounter withthe “weak”, implicative, interpretation of conditional and class statements and provides themwith practice in recognizing the context in which this asymmetric interpretation is appropri-ate. The directionality of class and conditional statements is introduced explicitly in geometry,through the distinction between direct theorems and inverse theorems. In addition, school sci-ence involves systematic analysis of the necessary and sufficient conditions that determinethe occurrence of the phenomenon under study (Taylor, 1967). Many such scientific activi-ties (e.g., testing the properties of materials using diagnostic indicators) require students tomake conditional inferences, the correct formulation of which are constrained by the neces-


sity/sufficiency status of particular cause-effect relations (e.g., “If the bulb lights up, the mate-rial is a conductor, but it is not true that if it is a conductor, the bulb will necessarily lightup”).

In addition, schooling fosters the development of secondary skills (Braine, 1990): Childrenlearn that different situations require different interpretations. Literal analysis of texts, learning insettings isolated from the context of the objects referred to, and the taxonomic-like organizationof knowledge teach the child to confine interpretation to the information explicitly stated inthe text (Hildyard & Olson, 1978), to distinguish between what the sentence itself means andwhat one means by the same sentence (Braine, 1990), to construct verbal-hypothetical worlds(Applebee, 1984; Braine, 1990; Donaldson, 1987; Olson, 1977; Scribner & Cole, 1973), andorganize classes of objects through hierarchical classification (Ward, Byrnes, & Overton, 1990). Areasoner equipped with this knowledge should notice that the first premise of an invalid conditionalargument does not include information related to the second premise, to suspend judgmentsregarding conversational implicatures (i.e., to adopt ‘minimal commitment’ to the conditionalin the major premise) and to envision and represent the possibility of “not p and q” (Cahan &Artman, 1997).

In sum, according to this hypothesis, the decreased rates of the classical fallacies with ageare entirely attributable to a culture-specific experience, extrinsic to the developing individual,namely, schooling. The effect of aging and out-of-school experiences is expected to be nega-tive. These predictions regarding the effects of schooling and out-of-school experiences clearlycontradict those derived by the logical competency models, according to which the decrease inthe fallacies rates with age is considered to be a basic and universal process, resulting from thechild’s transactions with any normal environment, schooled or not. According to this approach,the effect of schooling is negligible and development is mainly due to aging and the associatedaccumulation of experience.

To date, Cahan and Artman’s (1997) study is the only available empirical evidence relevantto this issue. In their study, the amount of everyday experience was measured by chronologicalage and schooling was measured by grade level. The independent effects of age and schoolingon scores obtained on invalid conditional and class syllogisms were estimated in grades 4–6 (agerange 9.5–12.5). The results pointed to a negative, albeit very small, effect of chronological ageand to a sizeable positive effect of schooling.

The theoretical significance of these results is clearly dependent on their replicability acrossage range and tests. This study is a first step in this direction. The study estimated the uniqueeffects of schooling (the student’s grade level at the time of test administration) and out-of-schooldiscourse experience (i.e., chronological age) on success in solving invalid conditional and classproblems among adolescents (grades 7–9), using a broad range of uncertain forms of conditionalsyllogisms, including all possible combinations of premise type (if the; if a; all are) and argumentpattern.

1. Method

1.1. Participants

The sample included 1885 seventh, eighth and ninth graders in three Israeli junior highschools (about 600 in each grade) (see Table 1). Age means were 12.3,13.3, and 14.3,respectively.


Table 1Sample by grade and age status (N)

Age status Grade

7 8 9

Underage 6 13 9

“Normal” ageJanuary–October 465 495 510November–Decembera 62 63 70

Overage 58 67 67

Total 591 638 656

a These students were excluded from the data analysis.

1.2. Measures

The conditional syllogism test consisted of (a) test items – 12 randomly-ordered DA and ACclass and conditional problems and (b) camouflage items – 6 MP or MT items, 4 in the “if-then”form (2 MP and 2 MT) and 2 in the “all are” form (1 MP and 1 MT). These items (item nos. 1,6, 8, 9, 13, 17 in Appendix A) were included in the test in order to minimize the possibility of“response set” and were excluded from the data analysis.

Each of the 12 test items presented an argument composed of two premises and a conclusionwhich the subject had to evaluate as necessarily true, necessarily false or indeterminate (no wayto tell). For example:

The major premises included three types of sentence (four items each): “if the” (e.g., if the A is . . .

then it has to be a . . .), “if a” (if an A is . . . then it has to be a . . .) and “all . . . are” (all the A’s areF’s). For each type of premise, two items denied the antecedent and two affirmed the consequent.The test language was Hebrew (see Appendix A for an English translation of the original Hebrewversion).

1.3. Procedure

The test was administered on a classroom basis at the beginning of the school year. Eachparticipant responded to all the 18 items. Two testers (university students who had receivedspecial instruction) were present in each classroom. Before the start of the test, the children werepresented with two examples: one with an “all are” premise (MP), and the other with an invaliddisjunctive syllogism (A or B or both, A, is B true?). Before revealing the correct answers, thetester explained how the premises should be interpreted in order to arrive at the proper conclusion.


The rationale for this short training was to acquaint subjects with the test format and appropriateuse of the “no way to tell” response (i.e., as a judgment of indeterminacy rather than of subjectiveuncertainty). The test was administered without any time limitation.

1.4. Scoring

Each item was scored dichotomously (1 = correct, 0 = incorrect or no response). The percent-age of “no response” answers ranged between 1 and 4% across items with negligible differencesbetween grades. The proportion was higher, on the average, the higher the serial position of theitem, perhaps due to fatigue. Ninety percent of the subjects in each grade level responded toall the items and 95% in each grade level reached the last item. Among subjects who skippedat least one item, 50% did not respond to two items at the most. Three total scores (per-cent correct answers), one for each of the three types of premises, were calculated for eachexaminee.

1.5. Design and methodological considerations

1.5.1. The between-grades paradigmA between-grade regression discontinuity approach (Cahan & Cohen, 1989; Cahan & Davis,

1987) was employed to disentangle the independent effects of schooling and age on success inconditional reasoning tasks. This approach involves administering the same test to at least twoadjacent grade levels. It relies on the following assumptions: (1) the “allocation” of children tobirth dates is random; (2) grade level is solely a function of chronological age; that is, admissionto school is based on chronological age only according to some arbitrary cut-off point, andprogression through grades is automatic (i.e., there are no drop-outs and children are neither keptback nor advanced a grade).

On the basis of these assumptions, the net effect of chronological age and schooling areestimated by means of a regression discontinuity design (Cook & Campbell, 1979), in whichtest scores are regressed on chronological age within grades. In this design, the effect of ageis reflected in the slope of the within-grade regression of test scores on chronological age, andthe effect of schooling is reflected in the discontinuity between these regressions (see Fig. 1).

Fig. 1. The effects of age and schooling in the regression discontinuity design. Note: The dotted section in each regressionline represents extrapolation of the regression line for the months of birth which were excluded due to their high proportionof selective grade misplacement.


Specifically, the estimated effect of a one-year difference in chronological age in a given gradeequals the difference between the oldest and youngest students in that grade in mean predictedscores (see dotted arrows in Fig. 1), and the estimated effect of one year of schooling equals thedifferences in mean predicted scores between the youngest children in any given grade (X) andthe oldest children in the lower adjacent grade (X−1).

1.5.2. The truth of the assumptionsAs already stated, the between-grades paradigm makes two assumptions: (1) children are

randomly allocated to birth dates; (2) grade level is solely a function of chronological age. Certainreservations regarding the truth of these assumptions, that have been raised before (Cahan &Cohen, 1989; Cahan & Davis, 1987), should be reiterated.

The truth of the first assumption cannot be empirically tested. However, since we tested studentsin only three adjacent grades within a relatively homogeneous population, this assumption seemsreasonable with respect to between-grades variability. As far as within-grade randomization isconcerned, exceptions to this assumption can affect the estimation of age and schooling effectsonly if they are monotonically related to birth date, which is very unlikely.

The second assumption of the model is only partially true. While grade retention and gradeskipping are seldom practiced in the Israeli school system (Cahan & Cohen, 1989), admission toschool is sometimes delayed and sometimes accelerated. Consequently, in any given grade thereare “overage” and “underage” children, that is, children whose age should place them in a higheror lower grade, and there are also “missing” children (i.e., children who are learning in a higheror lower grade) (see Table 1).

More important, the delay or acceleration of school admission is not random. The childrenwhose admission was delayed are likely to be less developed intellectually than the other childrenin their age group, and those whose admission was accelerated are likely to be more developed.Hence, the children who remain in the “appropriate” grade are also selective (in the oppositedirection). In addition, the relative frequency of grade misplacement is related to the month ofbirth, being particularly high near the cut-off point: Delays are especially frequent among theyoungest children in each cohort and accelerations among the oldest (see Table 1). In particular,the relative frequency of delays is much higher (about 25%) among children born in Novemberand December relative to the other months (2–8%) (see Table 1). Cook and Campbell (1979) referto this possibility as a “fuzzy cutting point”.

There are two ways in which selective misplacement may affect the within-grade regressionslopes: (1) Due to the existence of underage and overage children in each grade. The direction ofthis effect cannot be established a priori since age and selection counteract each other in this case:the underage children are also brighter, while the overage ones are generally duller. (2) Due to themissing children in each grade. At the lower extreme of the age range, the missing children arethose that have been delayed; hence, the mean test score of the remaining children in the youngestgroups is higher than the true one. At the higher extreme of the age range, selection operates inthe opposite direction: the missing children are the brightest ones, whose admittance to schoolhas been accelerated. Consequently, the mean test score of the remaining children in the oldestgroup are lower than the true ones. Thus, the missing children at both extremities of the age rangeaffect the within-grade regression slopes in the same direction: The empirically obtained slope isattenuated, that is, smaller than the true one, thus leading to an underestimated age effect and anoverestimated schooling effect.

In order to cope with this problem, two groups of subjects were excluded from the computationof the within-grade regressions: (a) students who were under- or overaged, and (b) students born


Fig. 2. Within-grade regression lines of total scores on age (standardized scores).

in November or December, that is, the birth dates with the highest proportion of missing students(see Table 1). Thus, each within-grade regression was based only on children born betweenJanuary and October of the appropriate year for that grade. While in each of these months of birththere still was a small proportion of missing students (see Table 1), this proportion did not varyconsiderably between the months of birth and, therefore, was not likely to affect the within-gradeslope.

1.5.3. Linearity of the within-grade regressionsThe use of a linear model to examine development, which is generally asymptotic, may seem

surprising. However, the deviation from linearity occurs over a span of several years, whereasthe regressions in this research cover three one-year segments. Undoubtedly, even if devel-opment over several years is curvilinear, it can be satisfactorily approximated by short linearsegments.


Table 2Estimated effects of one year of age (grades 7–9) and one year of schooling (grades 8–9) on DA and AC problem scores(S.D. Units)a

Problem type Schooling Age

“If the” .19 −.07“If a” .15 −.04“All are” .23 −.09

a Standard errors range from .05 to .6.

1.6. Estimation of effects

This study examined the effects of schooling in two grade levels (eighth and ninth), and theeffects of age in three grade levels (seventh, eighth and ninth). To cope with random differencesbetween cohorts and the resulting possibility of age-grade interactions, the true effects of one yearof age and one year of schooling were estimated (following Cahan & Cohen, 1989) by averagingthe three grade-specific age effects and the two grade-specific schooling effects, respectively.These averages are equivalent to the coefficients of age and grade level in the across-grade multipleregression of test scores on age and grade level. This is a conservative approach, which attributesbetween-grade variability of the estimates to random error and is based on the generally incorrectassumption that the true effect does not vary between grades (i.e., that the within-grade regressionsare parallel and equidistant). We believe, nonetheless, that its adoption is the best course of action,because the information that these averages may fail to reveal is relatively minor, considering thatwe are dealing with only three adjacent grades.

2. Results

Fig. 2 presents the within-grade regression lines of total scores (per invalid problem type) onchronological age. The within-grade slopes are indicative of a slight negative effect of chronolog-ical age on the scores on all three problem types, whereas the discontinuity between regressionlines in adjacent grades is indicative of a marked schooling effect.

The quantitative estimates of the two effects (Table 2) confirm the visual impression: School-ing has a positive net effect on development, whereas age has a negative, albeit very small,effect. The consistency of this pattern of results across the three problem types (“If the”, “Ifa” and “all are”) helps increase our confidence in their validity and generalizability acrosstasks.

3. Discussion

The findings of this study – the minor negative effect of age, and the major positive effectof schooling – replicate the results of Cahan and Artman (1997) in a different age range. Theresults are thus consistent with the hypothesis of comprehension theory—namely, that dailyconversational experience is dysfunctional in the development of conditional and class reasoning.At the same time, they are inconsistent with “intrinsic” models of the development of reasoning(e.g., Markovits & Barrouillet, 2002), according to which conditional reasoning abilities develop


mainly as a consequence of psycho-physiological maturational factors, and constitute a universalaccomplishment, characterizing the natural course of the developmental process per se, regardlessof schooling.

On the contrary, our results support the hypothesis that the negative effect of everyday discourseexperience is compensated for by the much stronger positive effect of schooling. In other words,the results support the claim that the observed increase in success rates on DA and AC problemsas a function of age, documented in many studies, is entirely attributable to school learning. Thus,they support a non-universal, constructivist view, according to which cognitive developmentcannot be characterized independently of a particular culture. In this view, conditional reasoningis actively constructed in certain kinds of (non-universal) cultural contexts, of the sort associatedwith formal schooling (Gellalty, 1987; Morris & Sloutsky, 1998; for a discussion of related issues,see Vygotsky (1987), and Larivee, Normandeau, and Parent (2000)). Furthermore, schooling hasa unique status in the context of conditional reasoning. Contrary to other cognitive tasks, in whichschooling operates in the same direction as out-of-school experience (for a review, see Ceci(1991)), in this case, schooling apparently breaks daily-life interpretational habits, and therefore,is critical for development.

Some reservations regarding our results as well as Cahan and Artman’s (1997) should be keptin mind. First, because schooling in Western societies starts at a specific age (usually, 6–7), theschooling effect estimated by these studies is, in fact, the effect of the specific age × schoolingcombination rather than the main effect of schooling per se, regardless of age. Theoretically,this effect may, and is in fact likely to vary between different combinations of age and amountof schooling. Hence, it will be erroneous to conclude that much younger children (e.g., 3years old) or unschooled adults would benefit from schooling like those who started schoolat the “normal” age. There are of course psycho-physiological prerequisites to receptivenessof schooling inputs. Similarly, schooling may be less effective during adulthood. Second, thecritical contribution of schooling to the development of conditional reasoning found in ourstudy may be specific to “Western-style” schooling systems. In view of the possibility of con-siderable variability between school systems, the issue of the universality of the central roleof schooling in this realm is an empirical one, to be answered by additional studies. Third,our results, as well as those of Cahan and Artman (1997), are based on a specific popula-tion in a specific educational system (Hebrew speaking Israelis). The theoretical significanceof these results is clearly dependent on their replicability in other populations and educationalsystems.


Appendix A. The Syllogism Test


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Age, schooling and conditional reasoning.

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