Cognitive ProcessesPSY 334

Chapter 10 – Reasoning

Logic vs Human Reasoning

Logic – a subdiscipline of philosophy and mathematics that formally specifies what it means for an argument to be correct. Human deviations from logic were thought

to be malfunctions of the mind. AI systems guided by logic are also

deficient, lacking common sense. Prescriptive or normative models do not

predict human behavior very well.

Demos of Human Irrationality

Four main areas of research have studied how humans deviate from prescriptive models: Reasoning about conditionals Reasoning about quantifiers Reasoning about probabilities Decision making

Two Kinds of Reasoning

Reasoning – the process of inferring new knowledge from what we already know.

Deductive reasoning – conclusions follow with certainty from their premises. Reasoning from the general to the specific.

Inductive reasoning – conclusions are probable (likely) rather than certain. Reasoning from the specific to the general. Probabilistic – based on likelihoods.

Syllogisms

Syllogism – a series of premises followed by a logical conclusion.

All poodles are pets Congruent 84%All pets have names

All poodles have names – T or F?All pets are poodles Incongruent 74%All poodles are viciousAll pets are vicious -- T or F?

Content-Free (Abstract)

Subjects did better judging syllogisms that were consistent with reality (congruent).

Content-free syllogisms use symbols instead of meaningful sentences:

All P are B Abstract 77%All B are C

All P are C – T or F?

Different Brain Regions Active

Logic problems can be approached in two different ways: Ventral prefrontal & parietal-temporal

areas active when reasoning about meaningful content

Posterior parietal active when reasoning about content-free material.

Different Brain Regions Active

Conditionals

If-then statements. Antecedent – the “if” part. Consequent – the “then” part.

Rules of inferences using conditionals: Modus ponens -- If A then B, observe A,

conclude B Modus tollens – If A then B, observe not-B,

conclude not-A Notation: negation, implication, therefore.

Table 10.1Anderson: Cognitive Psychology and Its Implications, Seventh EditionCopyright © 2010 by Worth Publishers

Modus Ponens and Tollens

If Joan understood this book, then she would get a good grade. If P then Q Joan understood .: she got a good grade. This uses modus ponens. P .: Q

If Joan understood this book, then she would get a good grade. If P then Q She did not get a good grade .: she did not

understand this book. ~Q .: ~P This uses modus tollens.

Logical Fallacies

Denial of the antecedent: If P then Q, not-P, conclude not-Q If P then Q, not-P, conclude Q

Affirmation of the consequent: If P then Q, Q, conclude P If P then Q, Q, conclude not-P

Subjects seem to interpret the conditional as a biconditional – “if” means “if and only if”

Denial of the Antecedent

If Joan understood this book, then she would get a good grade. If P then Q Joan did not understand .: she got a bad

grade. – This is not necessarily true. This is a fallacy. ~P .: ~Q

If it rains, then I will carry an umbrella. It is not raining .: I will not carry an

umbrella. But I may carry an umbrella for shade!

Affirmation of the Consequent

If Joan understood this book, then she would get a good grade. If P then Q Joan got a good grade .: she understood

the book. This is not necessarily true. This is a fallacy. Q .: P

If someone is abused as a child, then they will show certain symptoms. They show symptoms .: They were abused

as a child. Symptoms may not be of abuse!

Frequency of Acceptance

How People Reason

People may be reasoning in terms of conditional probabilities. Conditional probabilities can be found that

correspond to acceptance rates for fallacies.

Wason selection task – if there is a vowel on one side, then there must be an even number on the other side. Can be explained in terms of probabilities. Also explained by a permission schema

Sample Wason Task

E K 4 7

E 87%

K 16%

4 62% Affirming the consequent

7 25% Failure to apply modus tollens

Only 10% make all of the right choices

A Contextualized Version

In order to drink beer, someone must be 21 years of age:

DRINKING A BEER

DRINKING A COKE

22 YEARS OF AGE

16 YEARS OF AGE

Which ones would you check?

Explanations

Three proposed theories: Logic – people routinely fail to apply

modus tollens. Probabilistic – this task produces failures

only with certain underlying probabilities. Permission schema – the logical

connective is interpreted in terms of social contract.

A cheating context improves the results.

Conditional “If”

A strict logical construction means “If and only if”

People tend to adopt the more probabilistic condition if that means “under certain circumstances” In this case, the conditional statement is

interpreted in light of the probabilities and circumstances.

If a car has a broken headlight…

Which would you check?

1. Car with a broken headlight – check the taillight.

2. Car without a broken headlight – don’t check taillight.

3. Car with a broken taillight – check to make sure it has no broken headlight even though unnecessary.

4. Car without a broken taillight – reluctant to check every car in the lot although logically correct to check every car.

Quantifiers

Categorical syllogism – analyzes propositions with quantifiers “all,” “no,” and “some.”

Fallacies:Some A’s are B’s

Some B’s are C’s

Conclude: Some A’s are C’s Some women are lawyers, some lawyers are

men, conclude some women are men.

Atmosphere Hypothesis

People commit fallacies because they tend to accept conclusions with the same quantifiers as the premises. Some A’s are B’s Some B’s are C’s Conclude Some A’s are C’s, not No A’s are

C’s Universal premises go with universal

conclusions, particular with particular.

Two Forms of Atmosphere

People tend to accept a positive conclusion to positive premises, negative conclusion to negative premises. Mixed premises lead to negative

conclusions. People tend to accept universal

conclusions from universal premises (all, no), particular conclusions from particular premises (some, some not).

Limitations

Atmosphere hypothesis describes what people do, but doesn’t explain why.

People sometimes violate predictions of the atmosphere hypothesis. They are more likely to accept a syllogism

if it contains a chain leading from A to C. People should accept a syllogism with two

negative premises, but correctly reject it.

Process Explanations

People construct a mental model to think concretely about the situation and ask if it is possible, not if it is inevitable. Correct conclusions depend upon

choosing the correct mental model. Errors occur because people overlook

possible explanations of the premises: All the squares are shaded Some shaded objects have bold borders. .: Some of the squares have bold borders.

Possible Interpretations

Is it this way?

Or this?

B

Possible Meanings

A B AB

A B A AB A B

BA A B A B

All A are B

Some A are B

Some A are B

No A are B A B

Hypothesis Testing

Hypotheses are formed inductively from premises: Given this series: 1, 2, 4, ? What number comes next? 8th is a guess It exemplifies the rule: each number is

double the previous number. There is not usually a single conclusion

consistent with the premises.

Hypothesis Formation

Bruner, et al. (1956) presented series of boxes containing objects varying on four dimensions: Number of objects, number of borders,

shape, color. + identified a positive instance, -

negative (not an instance). Subjects had to identify the concept

exemplified by the stimuli.

Sample Sequences

+---+

Confirmation Bias

Focusing on instances that are consistent with a hypothesis instead of those that disconfirm it can lead to mistaken conclusions. This may not be a mistaken strategy if

selecting instances consistent with a hypothesis led to disconfirming it.

Drinking orange juice before an exam, if tested further, would quickly show itself to be wrong.