Reasoning And Logic

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Reasoning And Logic

Criteria of reasoning

Difficulties in reasoning

Cognitive theories

Mathematical reasoning

Sequences of reasoning operations

Learning to reason

Inference of reasoning in problem solving

Reasoning abilities

BIBLIOGRAPHY

Reasoning is thinking enlightened by logic. It can be defined more broadly so that it becomes equivalent to problem solving or critical thinking, but it becomes a special kind of thinking worth separate discussion only when the logical component is stressed. Thus intuitive, imaginative, and trial-and-error thinking are eliminated, no matter how excellent the outcome of such activities may be.

Anyone who contemplates what man’s reasoning has accomplished may be moved to curiosity as well as rapture, and there have been many attempts, each in the fashion of its time, to locate the faculty of reason in the scheme of things. Thus to Plato reason was the messenger of the gods. To Augustine reason was the eye of the soul, given to man by God that he may comprehend God. Then, under the impact of the doctrine of evolution, reason, as the highest human capacity, became a critical issue in arguments about the continuity of the species. Although satisfactory methods for tracing the evolution of psychological capacities are lacking (no fossilized thoughts have been uncovered in the chalk beds) and speculations about preliterate man are untrustworthy, reasoning is usually included today with other activities of the organism, such as swallowing and perceiving, that have evolved over the centuries in the struggle for existence.

Logic is a social invention, but it too has a long history of progressive refinement and transmission from one generation to the next, along with such cultural acquisitions as language and clothing. Hence, the reasoning that occurs today can be considered an adaptive natural function operating with cultural products that have been developed to fit the minds that use them.

Logic apparently developed, in India as in Greece (Bocheński 1956), out of techniques for refuting arguments. Certainly it was devised by people for people. Thus it is not surprising that many philosophical psychologists of the nineteenth century treated logic as the laws of thought, as idealized statements of how people reason. Today most logicians assert that they are concerned with formal relations between propositions but not with the thinking processes that produce them. Psychology is concerned with thinking, including thinking enlightened by logic, as a fallible human activity. The empirical study of such logical thinking began in England early in the twentieth century (see Wheeler 1958) when psychologists and educators used syllogisms for tests of ability and as materials for the investigation and training of reasoning. Since then psychological research has been directed toward traditional more than modern logic and deductive more than inductive reasoning, although some attention is given to inductive reasoning in discussions of concept formation.

Criteria of reasoning

Since many logical expressions were taken originally from the resources of the general language, the verbal patterns of logical statements often resemble the verbal patterns of ordinary discourse. The famous syllogism “All men are mortal; Socrates is a man. Therefore Socrates is mortal” looks something like a sequence of sentences that can be read by anyone with ordinary reading ability. But a logical system also includes technical concepts that are not in the general vocabulary, rules of combination that are not rules of grammar, and a metalanguage for making statements about statements. These logical skills depend on reading habits and abstract thinking, but they are special formal achievements. When someone uses variables to stand for propositions and writes, “If P, then Q,” or rearranges the verbal patterns of ordinary discourse into a standard logical form, he has gone beyond the general language. When one uses these concepts and procedures to test an argument for inconsistency or to generate nonobvious inferences from a problematic situation, his thinking can be described as reasoning.

Difficulties in reasoning

Reasoning, as defined above, may occur during the solution of many problems, but it occurs most obviously during the solution of problems of formal logic—for example, syllogisms. Hence logic supplies materials for the psychological investigation of reasoning, as chemistry supplies materials for the investigation of smell. Such reasoning problems are difficult for the ordinary person, who solves most of his daily problems adequately. The laws of thought may be related, as ideals, to actual thinking, but the relation is tenuous; man does not often reach his ideals. Deductive reasoning especially is a cultivated performance, a kind of game played by rules that are not quite like those of life played in earnest. Correct solution of a logic problem requires a logical approach, a single-minded restriction of attention to the implications of sentences, although the habits acquired in years of reading and listening direct attention to connotations and other verbal and emotional associations. Children and intelligent adults without training in logic make many errors in working reasoning problems, and research on the nature of the errors illuminates the reasoning process itself.

Terminological difficulties

The terms of a logical statement may cause difficulty; syllogisms written in familiar terms are somewhat easier than the same syllogisms written in letters or unfamiliar terms (Wilkins 1928). The quantifier “some” is a special difficulty because many people include with it the notion of “but not all.” This is a reasonable extension of ordinary reading habits and is allowed in Indian logic (Bocheński [1956] 1961, p. 437) but leads to technical errors in the usual systems of Western logic.

Abstractness of materials

Obviously some of the difficulty in reasoning is due to the abstract nature of the materials. When children have to learn concepts and principles of different levels of abstraction and to use them according to instructions, more errors are made on the more abstract problems (Long & Welch 1942; Welch & Long 1940). As objects of thought, to be retained and processed in immediate memory despite interference from other objects of thought, classes are more difficult than perceived objects, and implications of propositions are more elusive than perceived relations.

Order of presentation

The order in which logical statements appear can also cause trouble because when analysis of the argument requires rearrangement of the statements, certain common habits of reading and thinking may interfere. Consider the following:

John is taller than Mary;
Mary is taller than David.
Is John taller than David?

This is an easy problem because the order in which the thinker gets the information is a convenient order for him to process it. Now consider

Mary is shorter than John;
Mary is taller than David.
Is John taller than David?

Problems of this second type of logical structure are more difficult, whether difficulty is measured in time required for correct conclusion (Hunter 1957) or in number of errors (De Soto et al. 1965). Unlike the first problem, which is so straightforward that the thinker can almost read the conclusion directly from the premises, the second has a more complex structure, and the relations have to be somehow rearranged before the logical implications become apparent. Many people do this by drawing little charts on paper, or by making gestures with their hands, or by some other kind of imaginary spatial representation. Thus problems stated in a relation that is easily visualized, such as “taller than,” are easier for English school children at age 11 than problems stated in terms of the relationship “warmer than” or “happier than.” This difference in difficulty is not found at age 16, when abstract skills presumably have replaced visualization (Hunter 1957).

Direction of relations

Even when the order of the argument is straightforward, one direction may be easier for solution than the other. American college students can read “A is to the left of B; B is to the left of C” and follow it more readily to a conclusion than the logically equivalent “C is to the right of B; B is to the right of A.” Similar problems based on the relation “better than” are easier than those based on “worse than,” and those stated in terms of the relation “above” are easier than those stated in terms of “below.” Habits of reading and writing make it relatively easy to represent “left” on the left and proceed rightward, or “better” or “above” or “taller” on top and proceed downward. The reverse procedure can be followed, but it is less habitual and more subject to error. Propositions stated in “lighter than” and “darker than” are equally difficult, however, because one can be represented spatially as easily as the other (De Soto et al. 1965).

Mixing direction of relations

The difficulty is increased when one premise is stated in one direction—for example, “shorter than“—while the other is stated in the other direction—”longer than.” Even though the proposition “Mary is shorter than John” is equivalent to “John is taller than Mary,” the first must be rearranged in the second form if it is to be aligned with “Mary is taller than David.” This rearrangement takes time and attention, and if the thinker has not memorized the rules he may make mistakes. Thus whenever such rearrangement is required by the structure of the premise combinations, time and errors increase (De Soto et al. 1965; Hunter 1957).

Location of the middle term

If C and A are related by their relation to B, many thinkers construct a three-term structure, either verbally or spatially, with the distributed term, B, in the middle. This is easier when the first term in a premise is an end of the structure to be attained rather than the middle. Hence, it is easy to relate A and C in these two cases: “A is better than B; C is worse than B.” “C is worse than B; A is better than B.” And in these two it is hard: “B is worse than A; B is better than C.” “B is better than C; B is worse than A” (De Soto et al. 1965).

Conversion and atmosphere errors

Research on arguments that involve categorical propositions of the form “All X is Y” has uncovered many errors due to invalid conversion. The converse of the proposition “All men are rational” would properly be a statement such as “The class of rational beings includes man.” However, when a sample of Frenchmen was asked to convert this proposition, only a few did it properly; many more wrote statements such as “Rationality is the property of man.” The conversions chosen for various propositions depended as much on the meaning of the proposition as on its logical form (Oléron 1964).

“All X is Y” is frequently converted to “All Y is X” because ordinary reading habits are carried over to a logical proposition. When the proposition is written Xy and read “All X is included in the class Y,” invalid conversion is less common. Consider the syllogism

All S isM;
All P is M.
Therefore, all S is P.

One hypothesis for the common acceptance of such an invalid argument is the “atmosphere effect.” The untrained reader who does not know the rules and cannot follow through the abstract implications seizes on nonlogical verbal cues. The word “all” in the premises suggests a universal affirmative atmosphere, so when he sees the same word in the conclusion he accepts it. The word “some” in the premises wafts a favorable atmosphere around a conclusion stated in “some.” This hypothesis about atmosphere effect predicts that certain invalid arguments will be accepted more often than others, and when Sells (1936) presented syllogisms written in terms of X, Y, and Z to college students, the results agreed with the predictions fairly well. Similar research using syllogisms written in familiar words as well as letters, with the conclusion to be chosen from five alternative conclusions, also confirmed most of the predictions from atmosphere effect (Morgan & Morton 1944).

Another interpretation of the results of such research emphasizes invalid conversion strengthened by the habit of probabilistic inference. Many statements may be converted simply, like arithmetic statements of the form “All 90° angles are right angles“; hence, acceptance of a conversion is often reinforced in ordinary problem solving, and one who has not been warned will transfer these intermittently reinforced habits to the syllogism game. For example, if the second premise in the above syllogism is converted from “All P is M” to “All M is P,” the conclusion is readily accepted as valid. This interpretation accounts for acceptance of invalid arguments approximately as well as predictions from atmosphere effect [see Chapman & Chapman 1959; see alsoProbability; and the biography ofBrunswik].

Although the atmosphere effect and the tendency toward invalid conversion were intended as alternative interpretations of the most common errors in syllogistic reasoning, recent evidence indicates that they are independent (Simpson & Johnson 1966). Those college students who make one type of error do not, as a rule, make the other type. Furthermore, differential training aimed specifically at the atmosphere effect reduced the atmosphere error but not the conversion error. Training aimed specifically at the conversion error tended to have a parallel effect on conversion errors.

Personal biases

Since a logical course through an argument to a conclusion is often hard to follow, the distracting influence of bias is easily demonstrated. Syllogisms written in controversial terms, such as “communists” and “war,” are generally more difficult to evaluate than syllogisms of the same form written in neutral terms, such as “whales” and “water” (Lefford 1946). The effects of bias on acceptance of specific conclusions has also been demonstrated by variation in the terms of the arguments. Graduate students with no training in logic judged syllogisms with respect to the validity of the arguments and later were asked if they personally agreed with the conclusions. Most of their errors were due to acceptance of conclusions they liked and rejection of conclusions they disliked (Janis & Frick 1943).

Cognitive theories

Current interest in cognitive theories among social scientists has encouraged more formal models of the interaction between logical reasoning and personal desires in the expression of attitudes and beliefs. One of these, the “logic model” of McGuire (1960), begins by asking people to estimate the probabilities of premises and conclusions, then studies whether these are related as the model predicts, and examines the discrepancies from the predictions. Probabilistic reasoning, according to this model, requires that the probability of a conclusion be quantitatively related to the combination of the probabilities of the premises. Research with high school and college students disclosed a moderate relation; even on controversial issues these people were somewhat logical. A moderate relation was also found between acceptance of conclusions and ratings of desirability of premises—which can be interpreted as wishful thinking (influence of desires on beliefs) or rationalization (influence of beliefs on desires), or both.

A more subtle logical influence was demonstrated by McGuire when new information increasing the probabilities of minor premises resulted in higher estimates of probabilities of logically related but not explicitly mentioned conclusions. However, this effect did not last long. Apparently beliefs represent a balance between desire and reason; new information may force logical re-examination of beliefs and thus temporarily increase logical consistency, but away from the influence of the new information desires weigh heavier in the balance, and logical consistency suffers. [SeeThinking, article oncognitive organization and processes.]

In these studies the psychologist has taken a system of logic as a model and has tried to describe how the thinking of ordinary people deviates from this model. But if they deviate so often in so many ways, why use this model to describe their behavior? The answer would be that no other model yet proposed describes the facts with fewer deviations. Ordinary people apparently do strive, with varying effort and skill, toward logical consistency as a goal, they construct visual aids to guide them, and they come closer to the goal with practice. The errors described above are due in general to (1) differences between the habitual intellectual activities of untrained people and the technical procedures of logical systems, (2) overloading of immediate memory by the necessity for rearranging and recoding abstract materials, and (3) bias for or against the conclusions.

Mathematical reasoning

Among logicians the prevalent view is that all inference, whether in chemistry, mathematics, or elsewhere, is logical. Among educators, however, mathematical reasoning appears to be a special practical skill. It is identified as such and taught to school children much more seriously than reasoning in other areas, such as social science. Errors of mathematical reasoning, analogous to the errors listed above, are exposed routinely. Precise statement is emphasized. Thus the intelligent adult who makes glaring errors in syllogistic reasoning, prompting the question whether man is naturally logical, has had years of experience in mathematical manipulations and does not make comparable mathematical blunders. The dependence of mathematical reasoning on mathematical training is obvious, so the question whether man is naturally capable of mathematical reasoning does not arise.

Sequences of reasoning operations

The time spent on the solution of problems has been used as a measure of reasoning difficulty, and some problems can be divided so that the time devoted to component operations can be recorded in sequence. Conventionally it is assumed that the solution of an analogy problem of the form “Feline is to canine as cat is to …?” begins with induction of the relation between the first two terms and proceeds to application of this relation to the third term for deduction of the solution. This assumption can be tested with the aid of suitable exposure apparatus that separates the time the thinker spends studying the first pair of words, called induction time, from the time he spends studying the third word and writing a solution word, called deduction time. Problems in which the first pair of words is difficult, as above, require relatively long induction time, and those in which the second pair is difficult—such as “Lose is to win as liability is to …?“—require relatively long deduction time (Johnson 1962). The two logical operations may overlap in time, but the obtained time differentials show that the overlap cannot be large.

Sequences of operations in solving problems of proof in modern logic, specifically in the sentential calculus of Alfred North Whitehead and Bertrand Russell, have been charted by requiring the thinker to state the rule for each operation as he performs it (Anderson 1957). Such research would hardly be worthwhile if reasoning always progressed deductively, in a straight line from premises to conclusions, but in these problems, as in others, many thinkers plot their course by the destination as well as by the starting point. In problems that required six steps for proof the first steps were relatively easy because the implications from the data were clear, and the last steps were relatively easy because the logical requirements for reaching the goal became apparent. But in the middle the sequence of steps was not so obvious, and irrelevant operations occurred frequently.

Computer programs also have been written for discovering proofs in the sentential calculus, and comparison of the computer’s behavior—in respect to intermediate steps, failures, and effects of sequences of problems—with people’s behavior may lead to a better understanding of human reasoning.

Learning to reason

A sound mind in a sound body is not drawn irresistibly to sound conclusions. Reasoning is a cultivated performance that depends on practice. A college course in logic improves skill in handling logic problems, and Navy enlisted men have acquired some facility in the sentential calculus in two hours of special instruction (Moore & Anderson 1954). Special instructions with diagrams and examples helped a group of adults to reach a high degree of accuracy and to demonstrate that the distorting effects of bias can be minimized (Henle & Michael 1956). When sequences of related reasoning problems were arranged so that a fallacious inference would contradict a preceding inference, college students learned to detect their own errors and made definite improvement (Wason 1964). Such improvement would not occur if formal reasoning were a routine performance that is practiced daily.

From the psychological point of view, training in logical reasoning departs from the general language and consists of learning special concepts, procedures, purposes, much as does training in other systematic disciplines. Practice in putting verbal patterns in a class—for instance, primitive constant—and naming them increases skill in dealing with the class as a unit. Rules for operations, such as conversion, are memorized and practiced. Ordinary discourse is analyzed and the arguments rearranged into formal patterns. The purposes and restrictions of formal logic are distinguished from those of ordinary communication.

Many of the difficulties listed above are reduced by techniques that make abstract relations more concrete, such as Euler’s circles, Venn diagrams, and truth tables. These techniques may be considered recoding techniques in that the information which appears as sentences, numerals, or shapes is transformed into some other kind of information that is more easily processed and matched to formal inference patterns. Psychologists have not yet worked out the principles that govern this high-level cognitive learning, except for the learning of concepts, and the terminology of the logician may not coincide with psychological processes, but the intuitions of interested teachers have been helpful to their students. Some people learn these skills by themselves, and some make better use of what they have been taught than others. Hence, when intelligent adults who have not studied logic are compared on reasoning tests with those who have, although the average scores are different, there is considerable overlap between the groups [see Morgan & Morgan 1953; see alsoConcept formation].

Transfer of training

Improvement on reasoning problems as a result of special practice is rather specific; the transfer to other problems is not large (see Johnson 1955). As logic continues to develop, its technical language will probably diverge even more than now from the general language, and the amount of transfer may be even less. Whereas the older logicians analyzed the inconsistencies in ordinary language, the modern logicians are creating formal languages for their own purposes which, though they include some familiar constants—“if,” “only,” “not,” “or”—are not adapted to the contexts and purposes of ordinary communication. Nevertheless, any school subject can be taught with emphasis on induction of generalizations, precise statements of relationships, organization of data into cognitive structures, logical inferences from these structures, separation of inferences from observations, and detection of common fallacies. Some teachers have tried to do this in the past, and post-Sputnik trends may reinforce the attempt.

Inference of reasoning in problem solving

When the thinker is faced by a logic problem of appropriate difficulty, he will try to use whatever logical patterns and procedures he has learned, and his performance will improve with training in logic; hence the psychologist may say that reasoning is a substantial part of his problem-solving effort. And it is a tenable hypothesis that reasoning also occurs during the solution of problems that are not logic problems per se, but the confirmation of this hypothesis is complicated. Throughout the history of this topic, problem solving that has been interpreted as reasoning has also been interpreted in terms of instinct or trial-and-error learning, or some alternative hypothesis.

Correct solution of a problem is not itself sufficient evidence that reasoning has occurred, nor, as Henle (1962) has argued, is failure good evidence for the absence of reasoning. But there are at least two more specific types of evidence. One is the demonstration that the solution is mediated by a specific logical structure, such as a syllogism. The hypothesis that a problem is solved through the mediation of a syllogism is supported when the thinker produces verbal comments or diagrams along with solutions, all of which can be better predicted from the syllogism hypothesis than from any alternative hypothesis. The other type of evidence is improvement in problem solving as a result of training in logic, which involves one of the standard transfer designs of experimental psychology. If training in logical reasoning improves performance in problem solving as compared to initial performance, the final problem solving must include something, such as logical forms and operations, carried over from the logical training. Scattered results to date indicate that the transfer is small and variable, depending on the nature of the training, the nature of the problems, and the abilities of the individuals.

Reasoning abilities

It is relatively easy to construct a reliable test of reasoning ability, simply by assembling a number of logic problems, such as syllogisms. This was done a half century ago in England (Burt 1919); since then reasoning has been considered an important component of general intelligence. People who score high on such tests of reasoning ability, with noncontroversial conclusions, are able to discount their own biases fairly well when judging the validity of arguments with controversial conclusions (Feather 1964). Tests of arithmetic reasoning ability can also easily be assembled, and this ability has long been distinguished from facility in simple arithmetic computations. When such reasoning tests and general intelligence tests are given to the same people, the scores are correlated because the abilities required for success on each have much in common. There are several plausible interpretations for the correlations obtained between pairs of tests, and many statistical analyses have been directed toward one or more of the following: (1) Reasoning is the central factor in intelligence. (2) Reasoning is one factor among several. (3) Reasoning itself can be split into several factors.

In the most sophisticated attack on this problem (Guilford 1959; Merrifield 1966) some abilities are identified according to the materials that are processed: figural, symbolic, and semantic. Others are identified according to the products that result: units, classes, relations, systems, transformations, and implications. In this formulation reasoning abilities could include classificatory, relational, and systemic induction as well as relational and implicational deduction.

Donald M. Johnson

[Directly related are the entriesConcept formation; Problem solving; Thinking; Simulation, article onindividual behavior. Other relevant material may be found inAttitudes; Decision Making; Decision Theory; Developmental Psychology, article onAtheory Of Development; Intelligence and intelligence testing; mathematics.]

BIBLIOGRAPHY

See Strawson 1952 for clarification of the relations between general language and the language of logic. See Kaplan 1964 for a discussion of “logic in use,” especially for the social scientist, as distinguished from a more formal or “reconstructed” logic. For a somewhat different analysis of reasoning abilities from the British point of view, see Wheeler 1958. For a French review, see Oleron 1957. For an analysis of reasoning abilities in a Philippine sample, see Guthrie 1963. For information on attainment of logical symbols by children of different ages, including deaf and retarded children, see Furth & Youniss 1965 and Youniss & Furth 1964.

Anderson, Scarvia B. 1957 Problem Solving in Multiple-goal Situations. Journal of Experimental Psychology 54:297-303.

Bochenski, I. M. (1956) 1961 A History of Formal Logic. Translated and edited by Ivo Thomas. Univ. of Notre Dame Press. → First published as Formale Logik.

Burt, Cyril 1919 The Development of Reasoning in School Children. Journal of Experimental Pedagogy 5:68-77, 121-127.

Chapman, Loren J.; and Chapman, Jean P. 1959 Atmosphere Effect Re-examined. Journal of Experimental Psychology 58:220-226.

De Soto, Clinton; London, Marvin; and Handel, Stephen 1965 Social Reasoning and Spatial Paralogic. Journal of Personality and Social Psychology 2:513-521.

Feather, N. T. 1964 Acceptance and Rejection of Arguments in Relation to Attitude Strength, Critical Ability, and Intolerance of Inconsistency. Journal of Abnormal and Social Psychology 69:127-136.

Furth, Hans G.; and Youniss, James 1965 The Influence of Language and Experience on Discovery and Use of Logical Symbols. British Journal of Psychology 56:381-390.

Guilford, J. P. 1959 Three Faces of Intellect. American Psychologist 14:469-479.

Guthrie, George M. 1963 Structure of Abilities in a Non-western Culture. Journal of Educational Psychology 54:94-103.

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Henle, Mary; and Michael, Miriam 1956 The Influence of Attitudes on Syllogistic Reasoning. Journal of Social Psychology 44:115-127.

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Johnson, Donald McEwen 1955 The Psychology of Thought and Judgment. New York: Harper.

Johnson, Donald McEwen 1962 Serial Analysis of Verbal Analogy Problems. Journal of Educational Psychology 53:86-88.

Kaplan, Abraham 1964 The Conduct of Inquiry: Methodology for Behavioral Science. San Francisco: Chandler.

Lefford, Arthur 1946 The Influence of Emotional Subject Matter on Logical Reasoning. Journal of General Psychology 34:127-151.

Long, Louis; and Welch, Livingston 1942 Influence of Levels of Abstractness on Reasoning Ability. Journal of Psychology 13:41-59.

McGuire, William J. 1960 A Syllogistic Analysis of Cognitive Relationships. Pages 65-111 in Attitude Organization and Change: An Analysis of Consistency Among Attitude Components, by Milton J. Rosenberg et al. Yale Studies in Attitude and Communication, Vol. 3. New Haven: Yale Univ. Press.

Merrifield, Philip R. 1966 An Analysis of Concepts From the Point of View of the Structure of Intellect. Pages 19-33 in Herbert J. Klausmeier and Chester W. Harris (editors), Analyses of Concept Learning. New York: Academic Press.

Moore, Omar K.; and Anderson, Scarvia B. 1954 Modern Logic and Tasks for Experiments on Problem Solving Behavior. Journal of Psychology 38:151-160.

Morgan, John J. B.; and Morton, James T. 1944 The Distortion of Syllogistic Reasoning Produced by Personal Convictions. Journal of Social Psychology 20: 39-59.

Morgan, William J.; and Morgan, Antonia B. 1953 Logical Reasoning: With and Without Training. Journal of Applied Psychology 37:399-401.

OlÉron, Pierre 1957 Les composantes de I’intelligence d’après les recherches factorielles. Paris: Presses Uni-versitaires de France.

OlÉron, Pierre 1964 Les activités intellectuelles. Paris: Presses Universitaires de France.

Sells, Saul B. 1936 The Atmosphere Effect: An Experimental Study of Reasoning. Archives of Psychology No. 200. → The entire issue is devoted to reasoning studies.

Simpson, Miles E.; and Johnson, Donald M. 1966 Atmosphere and Conversion Errors in Syllogistic Reasoning. Journal of Experimental Psychology 72:197-200.

Strawson, P. F. 1952 Introduction to Logical Theory. London: Methuen; New York: Wiley.

Wason, P. C. 1964 The Effect of Self-contradiction on Fallacious Reasoning. Quarterly Journal of Experimental Psychology 16:30-34.

Welch, Livingston; and Long, Louis 1940 The Higher Structural Phases of Concept Formation of Children. Journal of Psychology 9:59-95.

Wheeler, Dorothy 1958 Studies in the Development of Reasoning in School Children. British Journal of Statistical Psychology 11:137-159.

Wilkins, Minna C. 1928 The Effect of Changed Material on Ability to Do Formal Syllogistic Reasoning. Archives of Psychology No. 102. → The entire issue is devoted to reasoning studies.

Youniss, James; and Furth, Hans G. 1964 Attainment and Transfer of Logical Connectives in Children. Journal of Educational Psychology 55:357-361.