Deductive Reasoning and Decision Making

Deductive Reasoning “Inductive” reasoning allows one to draw general conclusions or make judgments given evidence that we encounter. If, for example, you observe you observe an individual wearing a police officer’s uniform, carrying a sidearm, and getting into a patrol car, you would likely conclude he or she is a police officer going on duty. In this way, we have generated some new “knowledge” or “belief.” What about those situations, however, in which we already have some knowledge or belief about something, like the situation above? Where can we go from there?

Deductive Reasoning (con’t) “Deductive” reasoning allows you to draw inferences or predictions from general statements. 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, since logic is a formal method of specifying what it means for an argument to be correct.

Deductive Reasoning (con’t) Deductive reasoning is that in which conclusions follow with certainty from their premises and is sometimes described as reasoning from the “general to the specific.” For example, 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 that you are correct.

Reasoning About Quantifiers 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) Considerable effort has focused on our reasoning about “quantifiers” (e.g., all, some, no, and some not). For example:

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.

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: AB 2. “all” A’s are B’s may be represented by: A B ABor

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

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: From the diagram, we see the syllogism is valid. A B C

Sources Of Errors The errors commonly observed typically fall into one of two categories: Belief bias – the tendency to accept conclusions that are consistent with one’s beliefs and to not accept conclusions that differ from one’s beliefs. Some men are jerks.All murderers are reprehensible. Some jerks hit women.Some children are murderers.  Some men hit women.  Some children are reprehensible.

Sources Of Errors (con’t) 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. ABC Simple “matching” strategies: The Atmosphere Hypothesis.

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: 1.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. AB C

The Atmosphere Hypothesis (con’t) 2.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 CA

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: ValidInvalid 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. (yet subjects are more likely to accept the valid case above, showing some ability to evaluate a syllogism correctly).

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. (yet subjects are more willing to erroneously accept the first syllogism above).

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.

Reasoning About Conditionals The conditional takes the form “If, then.” 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.” How do people reason about such statements?

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

Evaluating Conditional Syllogisms (con’t) There are four such combinations: If P  QIf P  QIf P  QIf 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.

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 and denying the antecedent.

Wason’s Selection Task Given four cards, each with a letter on one side and a number on the other, EK47 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.” Many subjects correctly select the “E,” but incorrectly select the “4” and fail to correctly select the “7.”

Wason’s Selection Task (con’t) 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. Beer21 yrs 18 yrs Coke

Mental Models Again, we see several reasoning strategies can coexist in our minds and are used depending on the “triggers” that are present. “Mental model theory” maintains that subjects create a mental model of the premises and inspect the model to see if the conclusion is satisfied. That is, they construct a “world” and then search through it to see what is true about it.

Mental Models For example, given the premises, “All the squares are striped. Some of the striped objects have bold borders,” a subject might construct a mental model in the following way: 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. For example,

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

Decision-Making Do you want what’s behind Door 1, Door 2, or Door 3? Should you go to graduate school right after getting your bachelor’s degree or go out and work for a couple years? Do you order the BBQ ribs or wings? Should I continue asking inane questions or just move on? We are required to make choices every day; some simple, some very complex. So, how do people go about making decisions?

Utility Theory We all have likes and dislikes (values) and things we hope to achieve (goals). We also recognize there are tradeoffs, pros and cons, upsides and downsides… that is costs and benefits to every decision. In considering our choices, we compare the costs and benefits and then decide on the action that minimizes the costs and maximizes the benefits. Simple, right?

Utility Theory (con’t) Consider the tradeoff between going right on to grad school after graduation or working for a few years. Can you easily compare “getting a head start” on your advanced degree with “paying down some school loans”? Many judgments require such “apples-to-oranges” evaluations. Yet we do so on a regular basis by means of determining a subjective utility – the personal/informal value a person places on something, whether positive or negative (i.e., “disutility”). Decisions are also often accompanied by uncertainty or risk. For example, how easy will it be to go back to school and give up that source of income from your job? We must, then, also consider the probabilities of the events in our calculations.

Utility Theory (con’t) The prescriptive answer is to select the alternative with the greatest expected value (EV). That is, multiply the probability of an event times the value of the outcome (i.e., the utility): EV = (Probability of outcome) x (Utility of outcome) For example, imagine you are presented an offer with a certainty of $400 or a 50% chance of $1,000. Utility theory suggests you choose the second option: Choice 1: 400 * 1.00 = 400 Choice 2: 1000 *.50 = 500 An EV can be calculated for each factor in a decision and those EVs can be summed for overall EV.

Subjective Utility In the above example, most people choose the $400 with certainty. Why? When psychologists have examined subjects’ subjective utility regarding money, they have found a non-linear function, rather than a linear function. LossesGains Value (Subjective Utility)

Subjective Utility (con’t) 1. It is curvilinear such that it takes more than doubling the amount of money to double its utility. 2. The function is steeper in the loss region and, therefore, people weigh losses more strongly than gains of equal amounts. LossesGains Value (Subjective Utility)

Subjective Utility (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: Furthermore, people’s decisions may vary depending on where they perceive themselves to be on the utility curve. Objective probability Subjective probability

Subjective Utility (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? Value (Subjective Utility) LossesGains

Framing In many cases we are influenced by psychological factors that are independent of any utilities. Framing – how the problem is phrased – can have a significant impact on our decisions. Consider the following:

Framing (con’t) Given the first version, in which the outcomes are phrased as “gains” (i.e., lives saved), 72% select Program A. Given the second version, in which the outcomes are phrased as “losses” (i.e., lives lost), 78% select Program B. In general, when the choices are framed as “losses,” we tend to be more “risk-seeking” and gamble, perhaps in the hope of avoiding the loss. However, when the choices are framed as “gains,” we tend to be more “risk-averse” and want to hold on to a “sure thing.”

Framing Questions In addition to changing how the outcomes are phrased, we also see the effects of framing when the phrasing of the question is changed. Consider the following: On a 1-10 scale, with 1 = very bad and 10 = very good… How would you rate our government’s economic recovery efforts knowing that 91.5% of the labor force is employed? How would you rate our government’s economic recovery efforts knowing that 8.5% of labor force is unemployed?