La búsqueda de contraejemplos como estrategia metalógica. Un estudio sobre la influencia de su entrenamiento en el razonamiento silogístoco

Resumen

El objetivo del presente trabajo ha sido estudiar la influencia de los contraejemplos en el desarrollo del razonamiento lógico. Partíamos de la hipótesis de que un entrenamiento en “búsqueda de contraejemplos” produciría en los sujetos experimentales un aumento de su rendimiento en la resolución de silogismos, así como una mejora en la seguridad lógica de sus deducciones. Participaron 189 estudiantes (98 chicos y 91 chicas) de edades comprendidas entre 15–18 años. Usamos un diseño de dos factores: entrenamiento y nivel educativo. Los niveles fueron 2°BUP, 3°BUP y COU, dividiéndose aleatoriamente los sujetos de cada nivel en un grupo control, que no recibió entrenamiento y un grupo experimental, que sí lo recibió. Después de un pretest, realizamos dos fases de entrenamiento, tras las que pasamos dos pruebas postest. Los resultados muestran diferencias muy significativas entre los grupos controles y experimentales debidas al entrenamiento en las dos pruebas (F 1,23=33.29, p=0.0000 y F 1,29 = 176.34, p=0.0000). Las ganancias en rendimiento fueron de un 60% para los sujetos entrenados y un—1% para los no entrenados. Estos resultados parecen mostrar una estrecha relación entre la elaboración de contraejemplos y la mejora del rendimiento y seguridad lógica de los sujetos en la resolución de silogismos.

Abstract

The aim of this paper is to study the influence of counterexamples on the development of logical reasoning. We started with the hypothesis that training students to search for counterexamples would produce an improvement in their performance solving syllogisms, as well as an improvement in the logical security of their deductions. A total of 189 students aged 15 to 18 years were tested (98 males and 91 females). A two factor design: training and educational level (grades 10–12) was used. Subjects were randomly divided into two groups: a control group receiving no training and an experimental group who reveived the training. The experimental sequence involved a pretest, two training phases, and two posttests. The results show highly significant differences in both tests between the control and experimental groups due to the training (F 1,23=33.29, p<0.0001, and 1,29=176.34, p<0.0001). Improvement in performance increased by 60% for the trained subjects and by—1% for the non—trained subjects. It is therefore concluded that there is a close relationships between searching for counterexamples, improvement in performance, and the logical security of solving syllogistic tasks.

Extended Summary

Metalogic strategies are conscious reasoning strategies—or at least accessible to consciousness—essential for succesfully solving complex logical tasks. An important strategy in deductive reasoning is “searching for counterexamples” and is used to check the validity of an initial conclusion. In general, it is agreed that counterexamples play an important role in rational thinking. In fact, a generalization is only true if there is no counterexample to contradict it.

One of the basic pillars of the mental models theory about deductive reasoning is the search for counterexamples. To solve complex logical tasks, such as categorical syllogisms of medium and high difficulty, according to mental models theory, it is necessary to carry out an exhaustive search for counterexamples which assure that the final conclusion will not be refuted by another situation that might derive from the information in the premises.

Despite the apparent relevance of counterexamples, few studies have been carried out on the influence and importance of counterexamples for human reasoning, its development through instruction and its influence on logical reasoning. This has been one of the main targets of our work: to study the effects of training students to search for counterexamples on the development of metalogical reasoning. In short, we have tried to demostrate how instruction on how to elaborate counterexamples influences (a) the subject's capacity to metalogically understand syllogisms, and (b) the development of strategies to solve complex syllogistic problems. our hypotheses stated the following:

1.

An intervention study training subjects to search for counterexamples will increase their metalogical knowledge of them. Thus, trained experimental groups will show improved metalogic understanding of deductive task, and this will give way to greater security in the logical necessity of their responses. This security will manifest itself in an increase, with training, in the percentage of correct responses marked as “necessarily valid” with respect to those marked “probably valid”.

2.

Training subjects to search for counterexamples will significantly improve their capacity to elaborate metalogical strategies. This in turn will improve their performance solving logical reasoning tasks, such as categorical syllogisms. That is to say, the overall percentage of correct conclusions should be higher in the experimental groups compared with those of the control groups, in every form. This superiority will be more evident with respect to difficult syllogisms, where a greater capacity for elaborating these strategies are necessary for their correct solution.

A total of 189 students were tested (98 males and 91 females) between 15 and 18 years of age. A two factor design: training and educational level (grades 10–12) was used. Subjects in each form were randomly divided into two groups: a control group receiving no training and a trained experimental group. The material given took the form of a booklet of general instructions that included a description of a syllogism together with some examples. The booklet also included ten combinations of premises which the subjects had to work out (the subjects had to write the conclusion). After each of this problems, they were asked to answer whether the syllogism had a valid conclusion or not. If they answered “no conclusion”, they had to explain the reason for their response. If they considered that there was a valid conclusion, they had to construct it and write it down. Inmediately after, they were asked to state if they were absolutely certain of the logical validity of their conclusion; that is, if they thought that their conclusion “necessarily” derived from the premises. If this was the case, they were asked to tick that their conclusion was “necessarily valid”. If they were not sure of the logical necessity of their conclusion and they thought that only one conclusion was probable, they had to tick “probably valid”.

First, a pretest with syllogisms of varying levels of difficulty was administered to all the groups in their own class—rooms. The experimental groups then received the training which was divided into two phases, with a posttest after each. The first posttest included easy, medium and difficult syllogisms while the second only included difficult syllogisms.

Results show significant differences between experimental and control groups due to the training after the first posttest (p=0.0174 if all syllogisms in the task are considered, and p<0.0001 if only difficult syllogisms are included). The second posttest produced highly significant differences (p<0.0001). The performance of each experimental group was superior, not only with respect of their control group, but also to the other control groups (e.g., 10th grade experimental group students, 15 years of age, obtained 63% correct responses versus 13% for 12th grade's control group).

The security of the logical validity of responses follows a similar rule for performance, with a remarkable improvement in the experimental groups due to the two-phased training received. It is important to underline that training has a greater effect in the group of older subjects, due to their initial higher level of metalogical development. Control subjects had to solve the same problems as experimental groups during training, and simple practice did not improve their performance.

These results empirically support the study hypotheses. The study therefore demostrates the influence and importance which the strategy of searching for counterexamples has on syllogistic reasoning. Their training produces a clear improvement, both in logical and in metalogical components of subjects' behaviour.

Palabras clave:

• Metalógica
• contraejemplos
• razonamiento silogístico

Keywords:

• Metalogic
• counterexamples
• syllogistic reasoning