1. Reasoning and Decision Making

2. Reasoning
Reasoning is the process by which people infer new knowledge from what they already know. Decision making is the process of choosing among alternatives.
Some examples:
“Sale: 2 for 1.” We can reason it would also be “4 for 2.”
If you try a light bulb in a socket you know to be working and it does not light, you can infer the bulb is burned out.
Do you want to invest your money in a lower-interest, but safer account, or a higher-interest, yet more risky account?
What college major will you pursue?

3. Reasoning (con’t)
Much work in reasoning has been tied to “logic” in terms of “how to” think as well as a standard against which we compare “how correctly” we think.
As we will see, however, applying formal logic to the real world does not always work well. In fact, logic will often result in an answer that contradicts our understanding of the world.

4. Types of Reasoning There are two types of reasoning we will discuss:
Deductive reasoning -- that in which conclusions follow with certainty from their premises.
If you were told “all living mammals have blood and all blood contains red blood cells,” you could deductively conclude your blood has red blood cells in it and be certain you are correct.

5. Types of Reasoning (con’t)
Inductive reasoning -- that in which conclusions probabilistically follow from their premises.
If you were told “all of our graduates from the welding program have found welding jobs after graduation,” you could inductively conclude you will probably find a welding job after graduation, but it is not a certainty.

6. Reasoning About Conditionals
The conditional takes the form “If <something>, then <something else>.”
For example: If you were born in the U.S.A., then you are an American citizen.
The “if” part is called the “antecedent” and the “then” part is the “consequent.”

7. Evaluating Conditional Syllogisms
A considerable amount of research has focused attention on how people reason about conditional “syllogisms” -- a set of two or more statements and a conclusion.
A conditional syllogism takes the following (formal) form:
1st premise: If P  Q
2nd premise: P
Conclusion:  Q

8. Evaluating Conditional Syllogisms (con’t)
There are four such combinations:
If P  Q If P  Q If P  Q If P  Q
P ~P Q ~Q
 Q  ~Q  P  ~P
The first and last forms represent “valid” arguments while the middle two forms represent “fallacies” in reasoning.
1. modus ponens -- given P, one may conclude Q.
2. denying the antecedent -- given ~P, you cannot conclude ~Q.
3. affirming the consequent -- given Q, you cannot conclude P.
4. modus tollens -- given ~Q, one may conclude ~P.

9. Evaluating Conditional Syllogisms (con’t)
Let us examine those four possibilities using the following syllogism:
“If it is raining, then the streets are wet.”
Most people are able to correctly apply the modus ponens inference. However, people have considerable difficulty applying modus tollens correctly and often incorrectly use affirming the consequent.

10. Wason’s Selection Task
Given four cards, each with a letter on one side and a number on the other, E K 4 7
subjects are instructed to select only those cards necessary to turn over in order to determine if the following rule is true or false:
“If there is a vowel on one side, then there is an even number on the other.”

11. Wason’s Selection Task (con’t)
Many subjects correctly select the “E,” but incorrectly select the “4” and fail to correctly select the “7.”
A considerable number of studies have investigated the reasons subjects make those errors:
1. confirmation bias -- tendency to select cards that could confirm the rule is true.
2. matching bias -- tendency to select cards mentioned in the rule.
3. permission schema -- invoke real-life experience to solve the problem in terms of what would be allowed.

12. Reasoning About Quantifiers
In addition to the conditional connective ‘if,” considerable effort has focussed on our reasoning about “quantifiers” (e.g., all, some, no, and some not). For example:
No one who jumps from a plane can survive the fall. (universal statement)
All people love a parade. (universal statement)
Some men are lazy. (particular statement)
Some people are not friendly. (particular statement)

13. Reasoning About Quantifiers (con’t)
Research on quantifiers has utilized the “categorical syllogism.” Typically, a categorical syllogism has two premises and a conclusion. Subjects are asked to determine the conclusion follows from the premises.
For example,
All A’s are B’s.
All B’s are C’s.
 All A’s are C’s.

14. Reasoning About Quantifiers (con’t)
Such syllogisms can be solved using Venn Diagrams -- system of circles used to represent the different categories.
Each quantifier can be represented in different ways:
1. “no” A’s are B’s is represented by: A B
2. “all” A’s are B’s may be represented by: A B AB or

15. Reasoning About Quantifiers (con’t)
3. “some” A’s are B’s may be represented by: A B AB or
4. “some” A’s are “not” B’s may be represented by: A B or

16. Reasoning About Quantifiers (con’t)
For the syllogism
All A’s are B’s.
All B’s are C’s.
 All A’s are C’s.
we could have: A B C
From the diagram, we see the syllogism is valid.

17. The Atmosphere Hypothesis
The most common problems people have with categorical syllogisms, is that they are too willing to accept invalid conclusions.
Thus, they are likely to accept as valid:
Some A’s are B’s.
Some B’s are C’s.
 Some A’s are C’s. A B C

18. The Atmosphere Hypothesis (con’t)
The “Atmosphere Hypothesis” has been proposed to explain subjects’ responses. It is assumed the quantifiers create an “atmosphere” that predisposes subjects to accept conclusions with the same terms. The hypothesis has two parts:
Subjects tend to accept positive conclusions to positive premises and negative conclusions to negative premises.
When the premises are mixed, subjects tend to prefer negative conclusions.
No A’s are B’s.
All B’s are C’s.
 No A’s are C’s. A B C

19. The Atmosphere Hypothesis (con’t)
Subjects tend to accept universal conclusions with universal premises and particular conclusions with particular premises.
When the premises are mixed, subjects tend to prefer particular conclusions.
All A’s are B’s.
Some B’s are C’s.
 Some A’s are C’s. B C A

20. Limitations Of Atmosphere Hypothesis
While the atmosphere hypothesis describes what subjects do somewhat accurately, it is not entirely accurate. For example, subjects should be equally likely to accept valid and invalid conclusions:
Valid Invalid
Some A’s are B’s. All A’s are B’s.
All B’s are C’s. Some B’s are C’s.
 Some A’s are C’s.  Some A’s are C’s.
(yet subjects are more likely to accept the valid case above, showing some ability to evaluate a syllogism correctly).

21. Limitations Of Atmosphere Hypothesis (con’t)
Furthermore, subjects should be equally likely to erroneously accept invalid conclusions:
Some A’s are B’s. Some B’s are A’s.
Some B’s are C’s. Some C’s are A’s.
 Some A’s are C’s.  Some A’s are C’s.
(yet subjects are more willing to erroneously accept the first syllogism above).

22. Limitations Of Atmosphere Hypothesis (con’t)
Nor does the atmosphere hypothesis predict what subjects do with two negatives:
No A’s are B’s.
No B’s are C’s.
 No A’s are C’s.
The atmosphere hypothesis would predict subjects would accept that conclusion, but most do not.

23. Process Explanations
More recent attempts to understand subjects’ reasoning with categorical syllogisms involve examination of the processes involved.
One theory, the “mental model theory,” maintains that subjects create a mental model of the premises and inspect the model to see if the conclusion is satisfied.
While this may often lead to a successful response, it, likely any heuristic, can lead one astray. In most cases, the failure comes from subjects constructing a mental model based on one interpretation of the premises, while ignoring other possible interpretations.

24. Process Explanations (con’t)
For example,
All the squares are striped.
Some of the striped objects have bold borders.
 Some of the squares have bold borders.
One model interpretation A second model interpretation
While the first representation indicates the conclusion is true, the second possible model indicates it is not valid.

25. Process Explanations (con’t)
In general, we may suspect that people’s problems with categorical syllogisms may stem from the (mistaken) expectation that “logic” maps well on to the real world, or that it is “probabilistic” and not quite so precise as logicians interpret it.
For example,
All men are scum.
Some scum hit women.
 Some men hit women.
Scum
Women
Hitters
Men
While the above strikes us as “correct,” it is not a valid conclusion in logic.

26. Inductive Reasoning
Whereas deductive conclusions are meant to be certain, inductive conclusions are probable.
Inductive reasoning is often described as reasoning from the specific to the general: given certain facts or observations, we are to draw a general (probabilistic) conclusion.

27. Bayes’s Theorem
Bayes’s theorem is a prescriptive model for reasoning about inductive problems.
It provides a means of combining a “prior probability” for the truth of a hypothesis with a “conditional probability” to yield a “posterior probability” that the hypothesis under consideration is true (e.g., H: a student cheated on an exam).
prior probability – probability that the hypothesis is true before considering any evidence.
For example, the probability of students cheating, based on an anonymous survey, is 1 in Thus P(H) = .01
The alternative hypothesis,-H: student did not cheat, is equal to 1-P(H). Thus, P(-H) = .99

28. Bayes’s Theorem (con’t)
conditional probability – the probability a particular type of evidence (E) is true if a particular hypothesis is true (e.g., E: student was looking over at next student).
That would be likely if H is true: we might estimate P(E | H) = .8
However, is would be less likely if -H were true: we might estimate P(E | -H) = .3

29. Bayes’s Theorem (con’t)
posterior probability – the probability the hypothesis is true after considering the evidence: P(H | E)
P(H| E) = P(E | H) * P(H) P(E | H) * P(H) + P(E | -H) * P(-H)
Thus,
P(H| E) = P(.8) * P(.01) = P(.8) * P(.01) + P(.3) * P(.99)
While Bayes’s theorem allows us to evaluate such hypotheses in light of evidence, the question is how do people compare to what Bayes’s theorem prescribes?

30. Bayes’s Theorem (con’t)
As it turns out, not very well:
Imagine you are a personnel manager just starting at a new engineering corporation. You study the employment records in an attempt to learn as much as you can about each employee. You discover there are 30 engineers, 70 lawyers, mostly male, ranging in age from 27 to 62 years of age, as well as a number of other facts.
After some hours of studying the files, you decide to test yourself and select one file at random. The character profile summary reads as follows:
Jack is a 45-year-old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political or social issues and spends most of his free time on his many hobbies which include home carpentry, sailing, and mathematical puzzles.
Do you think Jack is more likely an engineer or a lawyer?

31. Bayes’s Theorem (con’t)
Base Rate Neglect – people often ignore base rate (i.e., prior probability) information and give too much weight to evidence.
Conservatism – subjects sometimes show they do not give evidence enough weight when revising posterior probabilities.
People do not seem very much like Bayesian calculators,but they do get better with practice, even though they are unable to give explicit descriptions of how they arrived at their answers.

32. Judgment of Probability
In general, subjects are fairly good at judging probabilities when they do not have to rely on their memory. When memory is necessary, we often see a number of problems:
availability heuristic – probability estimates based on those instances easily available in memory.
Do people have a greater chance of being killed by a shark or pig each year?

33. Judgment of Probability (con’t)
representativeness heuristic – tendency to assign higher probabilities to instances the more similar they are to some “ideal” or “central members” of a category.
Which is more likely the result of a sequence of 6 coin tosses?
H H H H H H or H T T H T H
gambler’s fallacy – mistaken belief about the “law of averages.”
Given eight consecutive “heads,” what is the probability the next flip will be a “tail?”

34. Decision Making
How do people go about choosing between alternatives? The prescriptive answer is to select the alternative with the greatest “expected utility.” That is, multiply the probability times the value.
For example, would you choose a certainty of $400 or a 50% chance of $1,000?
Choice 1: 400 * 1.00 = 400
Choice 2: * .50 = 500
Most people would choose $400 with certainty. Why?

35. Decision Making (con’t)
subjective utility – the value a person places on something.
One’s subjective utility does not necessarily correspond in a one-to-one fashion with the prescriptive model.
When psychologists have examined subjects’ subjective utility regarding money, they have found a non-linear function, rather than a linear function.
Losses
Gains
Value (Subjective Utility)

36. Decision Making (con’t)
There are two things to point out in this function:
It is curvilinear such that it takes more than doubling the amount of money to double its utility.
This does seem to make sense in the real world… if you earn $100,000 a year, a raise of $1,000 is not viewed as much as it would if you made $12,000 a year.
The function is steeper in the loss region so subjects weight losses more strongly than gains of equal amounts.
This also seems to make sense in the real world… losing your paycheck for the month would be worse than possibly winning that much with the same probability.

37. Decision Making (con’t)
In addition, subjects’ subjective probabilities are not identical to objective probabilities… we tend to over-weight very low probabilities relative to high probabilities:
Objective probability
Subjective
probability

38. Framing Effects
We must also consider other non-normative issues, even in light of using subjective utilities and probabilities… how the choices are put to us can make a difference.
framing effects – people’s decisions may vary depending on where they perceive themselves to be on the utility curve.
Losses
Gains

39. Framing Effects: Some Examples (con’t)
You want to buy an Indians shirt and K-Mart is selling it for $15. Before you buy it, a customer in line tells you the same shirt is on sale at Wal-Mart for $10. Do you put the shirt back and got to Wal-Mart?
Suppose you are faced with a similar problem, except the suit you are buying is $125 and you find out from a customer in line you can get it on sale for $120 at another store. Do you go to the other store?
In such cases, there is not a clear basis for judging. People tend to choose the alternative which is most easily justified (to themselves or to others). Often, whether the choices are framed to focus on the positives or the negatives will influence our choices.

40. Framing Effects: Some Examples (con’t)
Such bases for deciding can lead to strange behavior: if asked to choose an alternative, you might choose A, but if asked to eliminate an alternative, you might also choose A! That is especially the case if A has extreme positives and negatives.
For example,
Imagine choosing between two houses when one has many pluses, but is in a bad neighborhood, while the other is kind of “average.”

41. Conclusions
While the previous discussion suggests people are quite “non-logical” in their reasoning and decision making, we must always consider prescriptive models use abstract, non-real content. We, however, must deal with real-world content and considerations.
Although we reason more “concretely” than such models would have us, it is exactly that foundation in reality and experience which makes us superior reasoners and decision makers when compared with computers.