Reasoning (GDP)

When we think in propositions, the sequence of thoughts is organized. Sometimes the organization of our thoughts is determined by the structure of long-term memory. The thought of calling your father, for example, leads to a memory of a recent conversation with him at your house, which in turn leads to the thought of repairing the attic in your house. But memory associations are not the only means of organizing thought. Of interest is also the organization characteristic of those cases when we try to reason. Here the sequence of thoughts often takes the form of a justification, in which one statement represents the statement or conclusion that we want to draw. The remaining statements are the grounds for this assertion, or the premises of this conclusion.

deductive thinking

Logic rules

In logic, the most rigorous proofs have deductive certainty; this means that the conclusion of a proof cannot be false if all of its premises are true (Skyrms, 1986). Here is an example of such proof:

I.

1. If it’s raining, I’ll take an umbrella.
2. It’s raining.
3. Therefore, I will take an umbrella.

To what extent do the reasoning of ordinary people correspond to the reasoning of a logician? When we are asked whether a proof is deductively certain or not, we are quite accurate in assessing simple proofs. How do we make these kinds of judgments? Some theories of deductive reasoning assume that people act like intuitive logicians and apply logical rules in an attempt to justify that the conclusion of a proof follows from given premises. To illustrate, consider the following rule:

If there is a statement of the form «If p, then q» and another statement p, then the statement q can be deduced.

Apparently, adults know this rule (perhaps unconsciously) and use it to decide that the evidence given is reliable. In particular, they identify the first premise («If it’s raining, I’ll take an umbrella») with the «If p, then q» part of this rule. They identify the second premise («It’s raining») with the p part of this rule, and then they output the q part («I’ll take an umbrella»).

Following the rules becomes more conscious if you complicate the proof. We apply the above rule twice when we evaluate the following proof:

II.

1. If it’s raining, I’ll take an umbrella.
2. If I take an umbrella, I will lose it.
3. It’s raining.
4. Therefore, I will lose the umbrella.

Applying the familiar rule to statements 1 and 3, we can conclude that «I will take an umbrella»; and applying this rule once again to statement 2 and to the derived statement, one can conclude that «I will lose the umbrella.» One of the best indications that people use such rules is that the number of rules required for a proof determines the difficulty of the latter. The more rules required, the more likely a person will make a mistake and the longer it will take him to make the right decision (Rips, 1983, 1984).

Content Impact

Logical rules do not cover all aspects of deductive reasoning. These rules are determined only by the logical form of statements, but our ability to evaluate deductive evidence often also depends on the content of statements. This point is illustrated in the following experiment. Subjects are presented with 4 cards. In one version, each card has a letter on one side and a number on the other (top row in Figure 9.7). The subject must decide which cards to turn over to determine the correctness of the statement «If there is a vowel on one side of the card, then on the other side there is an even number.» While most of the subjects correctly chose the card with the letter «E», less than 10% of them also chose the card with the number «7», which is the second correct choice (to make sure that the «7» card is also the correct choice, note that if there is a vowel on the other side, the statement is refuted).

Significantly better, however, the subjects coped with another version of this task (bottom row of Fig. 9.7). In it, the subjects had to evaluate the statement «If a person drinks beer, he must be over 19.» Each card had a number on one side indicating the person’s age, and on the other side the name of the drink. From the point of view of logic, this version of the problem is identical to the first one (in particular, “Beer” corresponds to “E”, and “16” corresponds to “7”); but now most of the subjects found both correct answers (turned over the cards «Beer» and «16»). Thus, the content of statements affects our reasoning.

Such results mean that when we encounter deductive problems, we do not always use logical rules. Sometimes we apply pragmatic rules that are less abstract and more relevant to everyday tasks. An example is a permission rule that says, «If a certain action is to be taken, then a precondition must be met.» Most people know this rule, and they apply it when faced with the beer problem shown at the bottom of Fig. 9.7; i.e., they think about this problem from the point of view of the resolving condition. When activated, this rule will push them to look for cases where the corresponding precondition (being 19 years of age) has not been met, which in turn will lead them to choose the «16» card. In contrast, in the letter and number problem (top of Figure 9.7), the permission rule does not apply, so there is no reason to choose the «7» card. Thus, the content of the task determines whether or not the pragmatic rule will be activated, which in turn affects the correctness of the reasoning (Cheng, Holyoak, Nisbett & Oliver, 1986).

In solving the beer problem, subjects can not only use the rules, but also recreate in their minds a specific representation, or mental model, of the situation. One can, for example, imagine two people, each with a number on their back and a drink in their hand. You can then examine this mental model and see what happens, for example, if a drinker with the number 16 on his back has a beer in his hand. According to this view, a person reasons with the help of mental models offered by the task content (Johnson-Laird, 1989).

The two procedures just described—the application of pragmatic rules and the construction of mental models—have something in common: they are determined by the content of the problem. This distinguishes them from the application of logical rules, which should not be affected by the content of the task. Consequently, sensitivity to the content of the task often keeps us from acting according to intuitive logic.

inductive thinking

Logic rules. Logicians point out that a proof can be good even if it lacks deductive validity. This kind of evidence has the power of induction, which means that the conclusion is unlikely to be false if all the premises are true (Skyrms, 1986). Here is an example of a strictly inductive proof:

III.

1. In college, Mitch majored in accounting.
2. Mitch now works for an accounting firm.
3. Therefore, Mitch is an accountant.

This proof is not deductively valid (maybe Mitch got fed up with accounting courses and went to work as a night watchman in the only place where he had connections). Inductive rigor is thus a matter of probability, not certainty, and (as logicians hold) inductive logic must be based on the theory of probability.

We are constantly making and evaluating inductive proofs. Do we rely on the laws of probability theory, as logicians and mathematicians do? One related law of probability theory is the basis size rule, which states that the more likely something is to belong to a particular class (e.g., Mitch is in the accountant class) the greater the number of members of that class (i.e., the higher the size of the basis). this class). Thus, the above proof that Mitch is an accountant can be strengthened by adding to it the premise that Mitch became a member of a club in which 90% of the members are accountants.

Another probabilistic law relevant to our case is the rule of conjunction: the probability of a statement cannot be less than the probability of the same statement being combined with another statement. For example, the probability that «Mitch is an accountant» cannot be less than the probability that «Mitch is an accountant and earns more than $40 a year.» The basis volume rule and the conjunction rule are rational principles of inductive reasoning; they are backed by logic, and most people rely on them when these rules are explicit. However, in the turmoil of everyday thinking, people often break these rules, as we will soon see.

Heuristics is a simplified procedure that is fairly easy to apply and often produces a correct result, but the result obtained in this way is not necessarily correct. People often use heuristics in their daily lives because they find them useful. However, as follows from the further presentation, heuristics can not always be relied upon. ; In a series of simple experiments, scientists have shown (Tversky & Kahneman, 1983, 1973) that by making inductive judgments, people violate some basic rules of probability theory. Violations of the basis volume rule are especially frequent. In one experiment, a group of subjects were told that a psychological council interviewed 30 engineers and 70 lawyers (100 in total) and compiled descriptions of their personalities. The subjects were given several descriptions and asked to indicate for each of them the probability that this person is an engineer. Some of the descriptions were prototypes of the engineer (for example: «Jack is not interested in politics, he spends his free time in the workshop»); others were neutral (for example: «Dick is a very capable man and is promised real success»). Not surprisingly, these subjects were more likely to attribute the prototypical description to the engineer than the neutral one. Another group of subjects were given similar instructions and descriptions, and in addition they were told; that out of these 100 people, 70 were engineers and 30 were lawyers (the inverse proportion of the first group). Consequently, the scope of the basis of «engineers in these groups was significantly different. But this difference actually had no effect: the subjects of the second group gave basically the same ratings as the subjects of the first group. For example, in both groups, subjects were 50/50 likely to attribute neutral descriptions to engineers (whereas it would be rational for them to be more likely to attribute neutral descriptions to a profession with a higher basis volume). The subjects completely ignored information about basic volumes (Tversky & Kahneman, 1973);

People pay no more attention to the rule of conjunctions. In one study, subjects were presented with the following description:

Linda, 31, single, outspoken and very smart. She majored in philosophy in college…and took a serious interest in issues of discrimination.

Subjects then rated the likelihood of the following statements:

IV. Linda is a bank teller.

V. Linda is a bank teller and feminist activist.

Sentence V is a conjunction of sentence IV with «Linda is an activist in the feminist movement.» Clearly violating the conjunction rule, most of the subjects rated the probability V higher than the probability IV. Note that this is downright misleading, as every feminist bank teller is a bank teller, but some bank tellers are not feminists, and Linda may have been among the latter (Tversky & Kahneman, 1983).

The subjects in this study based their judgments on the fact that Linda is more like a bank teller and a feminist than just a bank teller: Although the subjects were asked to rate the likelihood, they instead rated Linda’s similarity to the prototypical concepts of «bank teller» and «feminist bank teller». «. Thus, the similarity score acts as a heuristic for probability estimation, where a heuristic is a shorthand procedure that is relatively easy to use and that can produce the correct answer often, though not always. That is, people use the similarity heuristic because similarity is often associated with probability and is also easier to calculate. The application of the similarity heuristic also explains why people ignore the volume of the basis. In the «engineer» and «lawyer» experiment, subjects apparently considered only the similarity of the presented description with their «engineer» and «lawyer» prototypes. Therefore, when the description suited both «engineer» and «lawyer» equally well, the subjects considered both to be equally likely. Using the similarity heuristic can lead to errors even for experts.

The principle of similarity manifests itself in another common case of reasoning, when, knowing that some members of a category have a certain property, one must decide whether members of another category have this property. In one study, subjects had to decide which of the following two pieces of evidence was stronger:

VI.

1. All robins have sesame bones.
2. Therefore, all sparrows have sesame bones.

VII.

1. All robins have sesame bones.
2. Therefore, all ostriches have sesame bones.

Not surprisingly, subjects found the first evidence stronger, apparently because robins are more like sparrows than ostriches. This reliance on similarity seems rational, especially since it is consistent with the notion that if things have many known properties in common, then it is likely that they also have unknown properties in common. However, the appearance of rationality fades when we move on to the subjects’ assessments of another pair of evidence:

VII.

1. All robins have sesame bones.
2. Therefore, all ostriches have sesame bones.

VIII.

1. All robins have sesame bones.
2. Therefore, all birds have sesame bones.

The subjects considered the second evidence to be stronger, apparently because robins are more similar to the bird prototype than to the ostrich prototype. But such a judgment is erroneous: based on the same premise (that robins have sesame bones), the presence of some property in all birds cannot be more likely than the presence of all ostriches, since ostriches are actually birds. Again we see that intuition based on similarity can sometimes be misleading (Oshersone et al., 1990).

Similarity is not the only kind of strong heuristic; in addition to it, there is also a causal heuristic. People judge the likelihood of a situation by the strength of the causal relationship between the events in that situation. For example, sentence 10 seems more likely to them than sentence 9:

IX. In the year 2000, there will be a severe flood in California, during which more than 1000 people will drown.

X. There will be an earthquake in California in the year 2000, which will cause severe flooding, during which more than 1000 people will drown.

Thinking that X is more likely than IX is another violation of the conjunction rule (and therefore another fallacy). This time the violation occurs because in sentence X the flood has a strong causal connection with another event, an earthquake; while sentence IX only mentions the flood and accordingly has no causal links.

So, using heuristics often leads us to ignore some of the basic rules of reasoning, including the basic volume rule and the conjunction rule. But don’t be too pessimistic about our level of rationality. First, the similarity and causality heuristics lead to correct decisions in most cases. Second, under appropriate circumstances, we are able to assess the appropriateness of certain logical rules for solving certain problems and apply them accordingly (Nisbett et al., 1983). So, by reading this material and thinking about it, you may have been able to convince yourself that the basic volume rule and the conjunction rule play an important role in solving problems.

Creative thinking

In addition to thinking in the form of statements, a person can also think in the form of images, especially visual images.

Many of us feel that part of our thinking is done visually. It often seems that we reproduce past perceptions or fragments of them and then operate on them as if they were real percepts. To appreciate this moment, try to answer the following three questions:

1. What shape are the ears of a German Shepherd?
2. What letter will you get if you rotate the capital N 90 degrees?
3. How many windows do your parents have in their living room?

In answer to the first question, most people say they form a visual image of a German Shepherd’s head and «look» at the ears to determine their shape. See →