In a process of reasoning available information is taken into account in form of
premises
. Through a process of inferencing a
conclusion
is reached on the base of these premises. The conclusion’s content of information goes beyond the one of the premises. To make this clear consider the following consideration Knut makes before planning his holiday:
The conclusion in this example follows directly from the premises but it entails information which is not explicitly stated in the premises. This is a rather typical feature of a process of reasoning. In the following it is decided between the two major kinds of reasoning, namely
inductive
and
deductive
which are often seen as the complement of one another.
Deductive reasoning
Deductive reasoning
is concerned with
syllogisms
in which the conclusion follows logically from the premises. The following example about Knut makes this process clear:
1.Premise: Knut knows: If it is warm, one needs shorts and T-Shirts.
2.Premise: He also knows that it is warm in Spain during summer.
Conclusion: Therefore, Knut reasons that he needs shorts and T-Shirts in Spain.
In the given example it is obvious that the premises are about rather general information and the resulting conclusion is about a more special case which can be inferred from the two premises.
Hereafter it is differentiated between the two major kinds of syllogisms, namely categorical and conditional ones.
Categorical syllogisms
In categorical syllogisms the statements of the premises begin typically with “all”, “none” or “some” and the conclusion starts with “therefore” or “hence”. These kinds of syllogisms fulfill the task of describing a relationship between two categories. In the example given above in the introduction of deductive reasoning these categories are
Spain
and
the need for shorts and T-Shirts
. Two different approaches serve the study of categorical syllogisms which are the
normative approach
and the
descriptive approach
.
The normative approach
The normative approach is based on logic and deals with the problem of categorizing conclusions as either valid or invalid. “Valid” means that the conclusion follows logically from the premises whereas “invalid” means the contrary. Two basic principles and a method called
Euler Circles
(Figure 1) have been developed to help judging about the
validity
. The first principle was created by Aristotle and says “If the two premises are true, the conclusion of a valid syllogism must be true” (cp. Goldstein, 2005). The second principle describes that “The validity of a syllogism is determined only by its form, not its content.” These two principles explain why the following syllogism is (surprisingly) valid:
All flowers are animals. All animals can jump. Therefore, all flowers can jump.
Even though it is quite obvious that the first premise is not true and further that the conclusion is not true, the whole syllogism is still valid. Applying formal logic to the syllogism in the example, the conclusion is valid.
Figure 1, Euler Circles
Due to this precondition it is possible to display a syllogism formally with symbols or letters and explain its relationship graphically with the help of diagrams. There are various ways to demonstrate a premise graphically. Starting with a circle to represent the first premise and adding one or more circles for the second one (Figure 1), the crucial move is to compare the constructed diagrams with the conclusion. It should be clearly laid out whether the diagrams are contradictory or not. Agreeing with one another, the syllogism is valid. The displayed syllogism (Figure 1) is obviously valid. The conclusion shows that everything that can jump contains animals which again contains flowers. This agrees with the two premises which point out that flowers are animals and that these are able to jump. The method of Euler Circles is a good device to make syllogisms better conceivable.
The descriptive approach
The descriptive approach is concerned with estimating people´s ability of judging validity and explaining judging errors. This psychological approach uses two methods in order to determine people`s performance:
Method of evaluation: People are given two premises, a conclusion and the task to judge whether the syllogism is valid or not.
(preferred one)
Method of production: Participants are supplied with two premises and asked to develop a logically valid conclusion.
(if possible)
While using the
method of evaluation
researchers found typical misjudgments about syllogisms. Premises starting with “All”, “Some” or “No” imply a special atmosphere and influence a person in the process of decision making. One mistake often occurring is judging a syllogism incorrectly as valid, in which the two premises as well as the conclusion starts with “All”. The influence of the provided atmosphere leads to the right decision at most times, but is definitely not reliable and guides the person to a rash decision. This phenomenon is called the
atmosphere effect
.
In addition to the form of a syllogism, the content is likely to influence a person’s decision as well and causes the person to neglect his logical thinking. The
belief bias
states that people tend to judge syllogisms with believable conclusions as valid, while they tend to judge syllogisms with unbelievable conclusions as invalid. Given a conclusion as like “Some bananas are pink”, hardly any participants would judge the syllogism as valid, even though it might be valid according to its premises (e.g. Some bananas are fruits. All fruits are pink.)
Mental models of deductive reasoning
It is still not possible to consider what mental processes might occur when people are trying to determine whether a syllogism is valid. After researchers observed that Euler Circles can be used to determine the validity of a syllogism, Phillip Johnson–Laird (1999) wondered whether people would use such circles naturally without any instruction how to use them. At the same time he found out that they do not work for some more complex syllogisms and that a problem can be solved by applying logical rules, but most people solve them by imagining the situation. This is the basic idea of people using mental models – a specific situation that is represented in a person’s mind that can be used to help determine the validity of syllogisms – to solve deductive reasoning problems. The basic principle behind the Mental Model Theory is: A conclusion is valid only if it cannot be refuted by any mode of the premises. This theory is rather popular because it makes predictions that can be tested and because it can be applied without any knowledge about rules of logic. But there are still problems facing researchers when trying to determine how people reason about syllogisms. These problems include the fact that a variety of different strategies are used by people in reasoning and that some people are better in solving syllogisms than others.
Effects of culture on deductive reasoning
People can be influenced by the content of syllogisms rather than by focusing on logic when judging their validity. Psychologists have wondered whether people are influenced by their cultures when judging. Therefore they have done cross–cultural experiments in which reasoning problems were presented to people of different cultures. They observed that people from different cultures judge differently to these problems. People use evidence from their own experience (empirical evidence) and ignore evidence presented in the syllogism (theoretical evidence).
Conditional syllogisms
Another type of syllogisms is called “conditional syllogism”. Just like the categorical one, it also has two premises and a conclusion. In difference the first premise has the form “If … then”. Syllogisms like this one are common in everyday life. Consider the following example from the story about Knut:
1. Premise: If it is raining, Knut`s wife gets wet.
2. Premise: It is raining.
Conclusion: Therefore, Knut`s wife gets wet.
Conditional syllogisms are typically given in the abstract form: “If p then q”, where “p” is called the
antecedent
and “q” the
consequent
.
Forms of conditional syllogisms
There are four major forms of conditional syllogisms, namely
Modus Ponens
,
Modus Tollens
,
Denying The Antecedent
and
Affirming The Consequent
. These are illustrated in the table below (Table 1) by means of the conditional syllogism above (i.e. If it is raining, Knut`s wife gets wet). The table indicates the premises, the resulting conclusions and it shows whether these are valid or not. The lowermost row displays the relative number of correct judgements people make about the validity of the conclusions.
Table 1, Different kinds of conditional syllogisms
|
|
Modus Ponens
|
Modus Tollens
|
Denying the Antecedent
|
Affirming the Consequent
|
|
Description
|
The antecedent of the first premise is affirmed in the second premise.
|
The consequent of the first premise is negated in the second premise.
|
The antecedent of the first premise is negated in the second premise.
|
The antecedent of the first premise is affirmed in the second premise.
|
|
Formal
|
If P then Q.
P
Therefore Q.
|
If P then Q.
Not-Q
Therefore Not-P.
|
If P then Q.
Not-P
Therefore Not-Q.
|
If P then Q.
Q
Therefore P.
|
|
Example
|
If it is raining, Knut's wife gets wet.
It is raining.
Therefore Knut's wife gets wet
|
If it is raining, Knut's wife gets wet.
Knut's wife does not get wet.
Therefore it is not raining.
|
If it is raining, Knut's wife gets wet.
It is not raining.
Therefore Knut's wife does not get wet.
|
If it is raining, Knut's wife gets wet.
Knut's wife gets wet
Therefore it is raining.
|
|
Validity
|
VALID
|
VALID
|
INVALID
|
INVALID
|
|
Correct Judgements
|
97% correctly identify as valid.
|
60% correctly identify as valid.
|
40% correctly identify as invalid.
|
40% correctly identify as invalid.
|
Obviously, the validity of the syllogisms with valid conclusions is easier to judge in a correct manner than the validity of the ones with invalid conclusions. The conclusion in the instance of the modus ponens is apparently valid. In the example it is very clear that Knut`s wife gets wet, if it is raining.
The validity of the modus tollens is more difficult to recognize. Referring to the example, in the case that Knut`s wife does not get wet it can`t be raining. Because the first premise says that if it is raining, she gets wet. So the reason for Knut`s wife not getting wet is that it is not raining. Consequently, the conclusion is valid.
The validity of the remaining two kinds of conditional syllogisms is judged correctly only by 40% of people. If the method of denying the antecedent is applied, the second premise says that it is not raining. But from this fact it follows not logically that Knut`s wife does not get wet – obviously rain is not the only reason for her to get wet. It could also be the case that the sun is shining and Knut tests his new water pistol and makes her wet. So, this kind of conditional syllogism does not lead to a valid conclusion.
Affirming the consequent in the case of the given example means that the second premise says that Knut`s wife gets wet. But again the reason for this can be circumstances apart from rain. So, it follows not logically that it is raining. In consequence, the conclusion of this syllogism is invalid.
The four kinds of syllogisms have shown that it is not always easy to make correct judgments concerning the validity of the conclusions. The following passages will deal with other errors people make during the process of conditional reasoning.
The Wason Selection Task
The
Wason Selection Task
is a famous experiment which shows that people make more errors in the process of reasoning, if it is concerned with abstract items than if it involves real-world items (Wason, 1966).
In the abstract version of the Wason Selection Task four cards are shown to the participants with each a letter on one side and a number on the other (Figure 3, yellow cards). The task is to indicate the minimum number of cards that have to be turned over to test whether the following rule is observed: “If there is a vowel on one side then there is an even number on the other side”. 53% of participants selected the ‘E’ card which is correct, because turning this card over is necessary for testing the truth of the rule. However still another card needs to be turned over. 64% indicated that the ‘4’ card has to be turned over which is not right. Only 4% of participants answered correctly that the ‘7’ card needs to be turned over in addition to the ‘E’. The correctness of turning over these two cards becomes more obvious if the same task is stated in terms of real-world items instead of vowels and numbers. One of the experiments for determining this was the beer/drinking-age problem used by Richard Griggs and James Cox (1982). This experiment is identical to the Wason Selection Task except that instead of numbers and letters on the cards everyday terms (beer, soda and ages) were used (Figure 2, green cards). Griggs and Cox gave the following rule to the participants: “If a person is drinking beer then he or she must be older than 19 years.” In this case 73% of participants answered in a correct way, namely that the cards with “Beer” and “14 years” on it have to be turned over to test whether the rule is kept.
Figure 2, The Wason Selection Task
Why is the performance better in the case of real–world items?
There are two different approaches which explain why participants’ performance is significantly better in the case of the beer/drinking-age problem than in the abstract version of the Wason Selection Task, namely one approach concerning permission schemas and an evolutionary approach.
The regulation: “If one is 19 years or older then he/she is allowed to drink alcohol”, is known by everyone as an experience from everyday life (also called
permission schema
). As this permission schema is already learned by the participants it can be applied to the Wason Selection Task for real–world items to improve participants` performance. On the contrary such a permission schema from everyday life does not exist for the abstract version of the Wason Selection Task.
The evolutionary approach concerns the important human ability of
cheater-detection
. This approach states that an important aspect of human behaviour especially in the past was/is the ability for two persons to cooperate in a way that is beneficial for both of them. As long as each person receives a benefit for whatever he/she does in favour of the other one, everything works well in their social exchange. But if someone cheats and receives benefit from others without giving it back, some problem arises (see also chapter
Cognitive Psychology and Cognitive Neuroscience/Evolutionary Perspective on Social Cognitions#3. Evolutionary Perspective on Social Cognitions
). It is assumed that the property to detect cheaters has become a part of human`s cognitive makeup during evolution. This cognitive ability improves the performance in the beer/drinking-age version of the Wason Selection Task as it allows people to detect a cheating person who does not behave according to the rule. Cheater-detection does not work in the case of the abstract version of the Wason Selection Task as vowels and numbers do not behave or even cheat at all as opposed to human beings.
Inductive reasoning
In the previous sections deductive reasoning was discussed, reaching conclusions based on logical rules applied to a set of premises. However, many problems cannot be represented in a way that would make it possible to use these rules to get a conclusion. This subchapter is about a way to be able to decide in terms of these problems as well:
inductive reasoning
.
Figure 3, Deductive and inductive reasoning
Inductive reasoning is the process of making simple observations of a certain kind and applying these observations via generalization to a different problem to make a decision. Hence one infers from a special case to the general principle which is just the opposite of the procedure of deductive reasoning (Figure 3). A good example for inductive reasoning is the following:
Premise: All crows Knut and his wife have ever seen are black.
Conclusion: Therefore, they reason that all crows on earth are black.
In this example it is obvious that Knut and his wife infer from the simple observation about the crows they have seen to the general principle about all crows. Considering Figure 4 this means that they infer from the subset (yellow circle) to the whole (blue circle). As in this example it is typical in a process of inductive reasoning that the premises are believed to support the conclusion, but do not ensure it.
Figure 4
Forms of inductive reasoning
The two different forms of inductive reasoning are "
strong
" and "
weak
" induction. The former describes that the truth of the conclusion is very likely, if the assumed premises are true. An example for this form of reasoning is the one given in the previous section. In this case it is obvious that the premise ("All crows Knut and his wife have ever seen are black") gives good evidence for the conclusion ("All crows on earth are black") to be true. But nevertheless it is still possible, although very unlikely, that not all crows are black.
On the contrary, conclusions reached by "weak induction" are supported by the premises in a rather weak manner. In this approach the truth of the premises makes the truth of the conclusion possible, but not likely. An example for this kind of reasoning is the following:
Premise: Knut always hears music with his IPod.
Conclusion: Therefore, he reasons that all music is only heard with IPods.
In this instance the conclusion is obviously false. The information the premise contains is not very representative and although it is true, it does not give decisive evidence for the truth of the conclusion.
To sum it up, strong inductive reasoning gets to conclusions which are very probable whereas the conclusions reached through weak inductive reasoning on the base of the premises are unlikely to be true.
Reliability of conclusions
If the strength of the conclusion of an inductive argument has to be determined, three factors concerning the premises play a decisive role. The following example which refers to Knut and his wife and the observations they made about the crows (see previous sections) displays these factors:
When Knut and his wife observe in addition to the black crows in Germany also the crows in Spain, the
number of observations
they make concerning the crows obviously increases. Furthermore, the
representativeness of these observations
is supported, if Knut and his wife observe the crows at all different day- and nighttimes and see that they are black every time. Theoretically it may be that the crows change their colour at night what would make the conclusion that all crows are black wrong. The
quality of the evidence
for all crows to be black increases, if Knut and his wife add scientific measurements which support the conclusion. For example they could find out that the crows' genes determine that the only colour they can have is black.
Conclusions reached through a process of inductive reasoning are never definitely true as no one has seen all crows on earth and as it is possible, although very unlikely, that there is a green or brown exemplar. The three mentioned factors contribute decisively to the strength of an inductive argument. So, the stronger these factors are, the more reliable are the conclusions reached through induction.
Processes and constraints
In a process of inductive reasoning people often make use of certain heuristics which lead in many cases quickly to adequate conclusions but sometimes may cause errors. In the following, two of these heuristics (
availability heuristic
and
representativeness heuristic
) are explained. Subsequently, the
confirmation bias
is introduced which sometimes influences people's reasons according to their own opinion without them realising it.
The availability heuristic
Things that are more easily remembered are judged to be more prevalent. An example for this is an experiment done by Lichtenstein et al. (1978). The participants were asked to choose from two different lists the causes of death which occur more often. Because of the availability heuristic people judged more “spectacular” causes like homicide or tornado to cause more deaths than others, like asthma. The reason for the subjects answering in such a way is that for example films and news in television are very often about spectacular and interesting causes of death. This is why these information are much more available to the subjects in the experiment.
Another effect of the usage of the availability heuristic is called
illusory correlations
. People tend to judge according to stereotypes. It seems to them that there are correlations between certain events which in reality do not exist. This is what is known by the term “prejudice”. It means that a much oversimplified generalization about a group of people is made. Usually a correlation seems to exist between negative features and a certain class of people (often fringe groups). If, for example, one's neighbour is jobless and very lazy one tends to correlate these two attributes and to create the prejudice that all jobless people are lazy. This illusory correlation occurs because one takes into account information which is available and judges this to be prevalent in many cases.
The representativeness heuristic
If people have to judge the probability of an event they try to find a comparable event and assume that the two events have a similar probability. Amos Tversky and Daniel Kahneman (1974) presented the following task to their participants in an experiment: “We randomly chose a man from the population of the U.S., Robert, who wears glasses, speaks quietly and reads a lot. Is it more likely that he is a librarian or a farmer?” More of the participants answered that Robert is a librarian which is an effect of the representativeness heuristic. The comparable event which the participants chose was the one of a typical librarian as Robert with his attributes of speaking quietly and wearing glasses resembles this event more than the event of a typical farmer. So, the event of a typical librarian is better comparable with Robert than the event of a typical farmer. Of course this effect may lead to errors as Robert is randomly chosen from the population and as it is perfectly possible that he is a farmer although he speaks quietly and wears glasses.
Figure 5, Feminist bank tellers
The representativeness heuristic also leads to errors in reasoning in cases where the
conjunction rule
is violated. This rule states that the conjunction of two events is never more likely to be the case than the single events alone. An example for this is the case of the feminist bank teller (Tversky & Kahneman, 1983). If we are introduced to a woman of whom we know that she is very interested in women’s rights and has participated in many political activities in college and we are to decide whether it is more likely that she is a bank teller or a feminist bank teller, we are drawn to conclude the latter as the facts we have learnt about her resemble the event of a feminist bank teller more than the event of only being a bank teller.
But it is in fact much more likely that somebody is just a bank teller than it is that someone is a feminist in addition to being a bank teller. This effect is illustrated in Figure 5 where the green square, which stands for just being a bank teller, is much larger and thus more probable than the smaller violet square, which displays the conjunction of bank tellers and feminists, which is a subset of bank tellers.
Confirmation bias
This phenomenon describes the fact that people tend to decide in terms of what they themselves believe to be true or good. If, for example, someone believes that one has bad luck on Friday the thirteenth, he will especially look for every negative happening at this particular date but will be inattentive to negative happenings on other days. This behaviour strengthens the belief that there exists a relationship between Friday the thirteenth and having bad luck. This example shows that the actual information is not taken into account to come to a conclusion but only the information which supports one's own belief. This effect leads to errors as people tend to reason in a subjective manner, if personal interests and beliefs are involved.
All the mentioned factors influence the subjective probability of an event so that it differs from the actual probability (
probability heuristic
). Of course all of these factors do not always appear alone, but they influence one another and can occur in combination during the process of reasoning.
Why inductive reasoning at all?
All the described constraints show how prone to errors inductive reasoning is and so the question arises, why we use it at all?
But inductive reasons are important nevertheless because they act as shortcuts for our reasoning. It is much easier and faster to apply the availability heuristic or the representativeness heuristic to a problem than to take into account all information concerning the current topic and draw a conclusion by using logical rules.
In the following excerpt of very usual actions there is a lot of inductive reasoning involved although one does not realize it on the first view. It points out the importance of this cognitive ability:
The sunrise every morning and the sunset in the evening, the change of seasons, the TV program, the fact that a chair does not collapse when we sit on it or the light bulb that flashes after we have pushed a button.
All of these cases are conclusions derived from processes of inductive reasoning. Accordingly, one assumes that the chair one is sitting on does not collapse as the chairs on which one sat before did not collapse. This does not ensure that the chair does not break into pieces but nevertheless it is a rather helpful conclusion to assume that the chair remains stable as this is very probable. To sum it up, inductive reasoning is rather advantageous in situations where deductive reasoning is just not applicable because only evidence but no proved facts are available. As these situations occur rather often in everyday life, living without the use of inductive reasoning is inconceivable.
Induction vs. deduction
The table below (Table 2) summarises the most prevalent properties and differences between deductive and inductive reasoning which are important to keep in mind.
Table 2, Induction vs. deduction
|
|
Deductive Reasoning
|
Inductive Reasoning
|
|
Premises
|
Stated as
facts
or general principles ("It is warm in the Summer in Spain.")
|
Based on
observations
of specific cases ("All crows Knut and his wife have seen are black.")
|
|
Conclusion
|
Conclusion is more
special
than the information the premises provide. It is reached directly by applying
logical rules
to the premises.
|
Conclusion is more
general
than the information the premises provide. It is reached by
generalizing
the premises' information.
|
|
Validity
|
If the premises are true, the conclusion
must be true
.
|
If the premises are true, the conclusion
is probably true
.
|
|
Usage
|
More difficult to use (mainly in logical problems). One needs
facts
which are definitely true.
|
Used often in everyday life (fast and easy).
Evidence
is used instead of proved facts.
|