ORGANIZATION OF THE PROBLEM
In the process of problem solving, organization of the problem assumes an important role. For example, consider the candle problem, which has been discussed in previous article, The subjects who tried to solve the candle problem, organizing the materials as a box plus contents probably impaired the solution of the problem, for that organization centered around the container function of the new box.
This inappropriate organization tended to occur when the matches, candles, and tacks had been presented inside the box and the box had not been labeled. Thus, the manner of presenting the materials of a problem is a critical determinant of the organization achieved by the subjects.
Activity 1:
Try to solve the following problem:A man climbs a mountain on Saturday, leaving at daybreak and arriving at the top near sun down. He spends the night at the top. The next day, Sunday, he leaves at daybreak and heads down the mountain, following the same path that he climbed the day before.
The question is this: Will there by anytime during the second day when he will be at exactly the same point on the mountain as he was at that time on the first day?
If you try to solve this problem by using algebraic or verbal organization of the information, you will have a good deal of trouble. However, if you represent the problem with the kind of simple diagram illustrated in Figure 29.a, the solution becomes apparent.
Successful problem solving, then, requires that a person form an appropriate representation and organization of the problem. However, there is no one optimal way of representing and organizing material. Since that depends on the nature of the problem, sometimes simply restructuring a problem, from a verbal form to a pictorial or mathematical form for instance, can help out a direct solution.
SYLLOGISTIC REASONING
Thought or thinking refers to the general process of considering an issue in the mind, while logic is the science of thinking. Although two people may think about the same thing, their conclusions -both reached through thought -may differ, one being logical, the other illogical.
Thinking and logic has been the subject of speculation for a longtime. More than two thousand years ago, Aristotle introduced a system of reasoning or of validating arguments that is called the syllogism. Syllogistic reasoning is a part of deductive reasoning. Syllogism is a simple form of argument that contains three steps-a major premise, a minor premise, and a conclusion. Here premise refers to an underlying assumption. Types of premises and their descriptions are explained in Table 29 .1.
One example of a syllogism is as follows :
a) Major premise :All men are animals.
b) Minor premise : Some animals are aggressive.
Conclusion : Some men are aggressive.
Table-29.1 Categorical Syllogisms: Types of premises
Now, consider the following examples :
Major premise : All men are animals
Minor premise : Some animals are female
Conclusion Some men are female
Major premise : All A are B.
Minor premise : Some B are C.
Conclusion Some A are C
In the example b the conclusion is not true. The semantics of the conclusion are all wrong. Yet, if you examine the first example of syllogism (a) you will see that their form is identical.
Syllogism b and c appears to be valid.
The task that philosophers set before us is to ignore the semantics or the meaning of the premises altogether. The premises are simply assumed to be true for the sake of the argument.
The above examples, then, are really the syllogism. Because their syntax or structure is identical, their conclusions must be evaluated identically. Either the conclusion is valid in all three or it is invalid in all three. A valid conclusion is one that necessarily true, given that the premises are true.
TYPES OF SYLLOGISMS
There are two key types of syllogisms linear syllogisms and categorical syllogisms.
Ia a linear syllogism, the relationship among the terms is linear, involving a quantitative or qualitative comparison, in which each term shows either mere or less of a particular attribute or quantity. For example : the following problem represents linear syllogism.
You are smarter, than your best friend.
Your best friend is smarter than your roommate.
Which of you is the smartest ?
Here, each of the two premises describes a linear relationship between two items; People use either mental representations or proposition representations or combination of these two, to solve linear syllogisms.
Categorical syllogisms comprise two premises and a conclusion; in the case of the categorical syllogism, the premises state something about the category memberships of the terms. In fact, each term represents all, none or some of the members of a particular class or category.
An example of a categorical syllogism would be
All cognitive psychologists are pianists.
All pianists are athletes.
Therefore, all cognitive psychologists are athletes.
Errors do occur in deductive as well as syllogistic reasoning. Analyzing these errors has helped to reveal the psychological processes involved in deduction and problem solving.
ERRORS IN SYLLOGISTIC REASONING
People often err in reasoning tasks by responding on the basis of their prior knowledge and attitudes rather than on the basis of logic.
Atmosphere hypothesis
Atmosphere hypothesis explains the reason for errors people make in syllogistic reasoning. This hypothesis states that the premises of a syllogism establish a context-an atmosphere-favorable for the acceptance of a particular conclusion.
For example consider the following syllogisms:
Major premise : All A are B.
Minor premise : All B are C.
Conclusion : Therefore all C are A.
In this syllogism the occurrence of the word 'All' in both premises creates a favorable context or atmosphere for accepting or generating a conclusion containing ''All"
But, this conclusion is proved to be false (in some cases) by using Venn diagrams. Venn diagrams represent the meaning of each statement by classes symbolized by circles.
The following figure explains the above syllogism through Venn diagrams.
In the above figure each of the first two premises has two possible representations, which may be combined in four possible ways, as shown at the bottom of the figure. Since A and C are identical in only one of those representations (the one on the far left), the conclusion that "All C are A" cannot be accepted.
In a similar manner, the occurrence of some in the two premises creates an atmosphere that favors the acceptance of a conclusion containing some. Indeed. many people accept the following erroneous conclusion :
Major premise Some A are B
Minor premise Some B are C
Conclusion Some A are C
Luckily the atmosphere hypothesis is only partially correct. People can be swayed by the atmosphere of an argument, but they can also evaluate the logical relations within arguments. For example, subjects sometimes state that no valid conclusion can be drawn from a particular set of premises. Accordingly, the response (correctly) "can't say". Because the premises cannot establish an atmosphere for "can't say" responses, these responses must be based upon the evaluation of logical relations.
Premise Conversions
Chapman and Chapman (1959) proposed that errors in syllogistic reasoning stem from invalid premise conversions. They observed that some people converted the premise "All A are B" to "All B are A". This conversion is erroneous. For example, the fact that "all cats are animals does not imply the converse, that all animals are cats.
Chapman and Chapman pointed out that although people sometimes err by converting the premises illicitly, they may reason correctly from their interpretations of the premises. '
Analytic Representations and selective encoding
Analytic representations include abstracted properties and classes rather than specific class members. Erikson stated that representational process of the premises is selective i.e. the subjects do not form all of the possible representations for a particular premise.
For example, the premise "Some A are B" has four valid representations as shown in Figure 29.c.
People tend to represent "Some A are B" as partially overlapping circles in the left corner of the figure.
Knowledge of every day language and of non-artificial conceptual classes probably plays a role in influencing the people to choose this particular representation. This kind of selective representation sets the stage for the occurrence of errors.
SUMMARY
In the process of problem solving organization of the problem assumes an important role. Proper organization of the problem and its information facilitates problem solving~ A syllogism is ~ simple form of argument that contains three steps-a major premise, a minor premise and a conclusion. Premises are of four types viz. universal affirmative, universal negative, particular affirmative and particular negative. Errors in deductive reasoning often stem from the failure to construct all of the logically possible representations of a statement. For example, people tend to represent the statement some A are B in a manner that indicates that As are Bs, whereas other As are not Bs. But in fact, some A are B can be represented in several logically acceptable ways, each of which must be considered in order to derive the correct conclusion to a syllogism. Errors in syllogistic reasoning may occur because of atmosphere hypothesis, premise conversions, analytic representations and selective encoding.