Lecture 11: psychology of proof

Question 1

What is proof?

Answer 1

Proof is an argument that establishes a fact or truth by drawing inferences.

Question 2

What are the three types of inference?

Answer 2

• Deduction: Specific conclusions from general premises.
• Induction: Generalizations from specific observations.
• Abduction: Inferring the most likely explanation.

Question 3

Give an example of each inference type.

Answer 3

• Deduction: If the UK leaves the EU, agreements will be easy. The UK left, so agreements should be easy.
• Induction: Parliament rejected five deals, so future deals may be rejected.
• Abduction: Regions receiving the most EU money voted to leave because they misunderstood its value.

Question 4

What is the Structural View of inference?

Answer 4

Inference depends on syntactic structure and formal logic.

Question 5

What are classical types of logical reasoning?

Answer 5

1. Syllogisms:
   All mammals are animals. X is a mammal. Therefore, X is an animal.
2. Conditionals:
   If X works hard, X gets a raise. X didn’t get a raise. Therefore…?
3. Transitives:
   Brighton beat Spurs, Spurs beat Man Utd. Therefore, Brighton > Man Utd.

Question 6

What are common errors in logical reasoning?

Answer 6

• Denying the antecedent: X is not true, so Y is not true (invalid).
• Affirming the consequent: Y is true, so X is true (invalid).

Question 7

What did Wason & Evans (1975) test with the selection task?

Answer 7

• Participants evaluated rules like If a card has a D, it must have a 5 on the other side.
• They often failed to select logically relevant cards (e.g., D and not-5).

Question 8

How did Cheng & Holyoak (1985) extend the selection task?

Answer 8

• Introduced pragmatic reasoning schemas using real-world contexts like checking cholera immunization on visas.
• Participants performed better when reasoning aligned with practical rules.

Question 9

What is the Semantic View (Mental Models Theory)?

Answer 9

Johnson-Laird (1983) proposed that inferences are drawn by constructing mental models based on possibilities.

Question 10

What constrains mental model reasoning?

Answer 10

1. Principle of Truth: Models only represent true possibilities.
2. Working Memory: Limits capacity to process multiple models.
3. Procedural Semantics: Models rely on the meaning of premises.

Question 11

How did Ormerod & Richardson (2003) test mental models?

Answer 11

Compared paraphrasing logical statements:
* Conditionals (If crash, then die) were easier than Disjunctions (Either not crash or die).
Results: Generating single models is easier than evaluating multiple models.

Question 12

What is Bayesian reasoning in inference?

Answer 12

Reasoning involves updating probabilities based on new evidence, using Bayes’ Rule:

Question 13

How was Bayes’ Rule applied to medical testing?

Answer 13

Example: Predicting Down syndrome births.
P(H): Probability of Down syndrome.
P(D|H): Reliability of the test.
Result: Positive tests predicted Down syndrome in ~8% of cases.

Question 14

What is information gain?

Answer 14

Oaksford & Chater (1994): Reasoning reduces uncertainty by maximizing information gain.
Rare events provide more valuable information.

Question 15

What evidence supports information gain?

Answer 15

Oaksford & Chater (2007): Varying the frequency of cards in selection tasks influenced participants’ choices based on rarity.

Question 16

What are the two systems in dual-system reasoning?

Answer 16

• System 1: Fast, heuristic, and pragmatic. Effortless but prone to illogical errors.
• System 2: Slow, analytical, and logical. Effortful but prone to incompletion errors.

Question 17

What did Kahneman & Klein (2009) contribute to dual-system theory?

Answer 17

Combined naturalistic decision-making with dual-system reasoning to explain varied responses in different contexts.

Question 18

How does dual-system theory explain inference?

Answer 18

• System 1: Handles heuristic reasoning.
• System 2: Engages for analytical and logical tasks

Question 19

How did the Chernobyl disaster illustrate inference failure?

Answer 19

• Rule: If the experiment continues, the reactor must be cool.
• Fact: The reactor wasn’t cool.
• Logical inference: The experiment must stop.
• Failure: The experiment wasn’t stopped, leading to disaster.

Question 20

Why don’t firefighters drop tools during evacuations?

Answer 20

System 1 dominance: Heuristic-driven decisions override logical instructions.

Question 21

How does each theory approach inference?

Answer 21

• Structural View: Focuses on logical form and syntax.
• Semantic View: Relies on mental models and meaning.
• Statistical View: Prioritizes probabilities and information gain.

Question 22

What’s the key limitation of each theory?

Answer 22

• Structural: Ignores semantics and context.
• Semantic: Constrained by working memory.
• Statistical: Requires accurate prior probabilities.

Question 23

What did Wason & Evans (1975) contribute, and what methodology did they use?

Answer 23

Developed the selection task to study logical reasoning.

• Methodology: Laboratory experiment with a problem-solving task involving cards. Participants evaluated the validity of conditional statements like “If a card has a D, it must have a 5 on the other side.”

Question 24

What was Cheng & Holyoak’s (1985) key finding, and what methodology did they use?

Answer 24

Introduced pragmatic reasoning schemas, showing real-world contexts improve reasoning.
* Methodology: Laboratory-based problem-solving tasks. Participants assessed practical rules (e.g., immigration officers ensuring cholera immunization compliance) in realistic scenarios.

Question 25

What did Johnson-Laird (1983) propose, and what methodology supported it?

Answer 25

Proposed Mental Models Theory, emphasizing that reasoning involves constructing possibilities constrained by working memory.
* Methodology: Lab experiments involving logical reasoning tasks where participants drew conclusions based on provided premises (e.g., If the plane crashes, the pilot dies).

Question 26

How did Oaksford & Chater (1994) challenge classical logic, and what methodology was used?

Answer 26

Argued that reasoning prioritizes maximizing information gain over strict logical validity.
* Methodology: Laboratory experiments incorporating Bayesian reasoning tasks where participants evaluated probabilities to reduce uncertainty in reasoning.

Question 27

What did Ormerod & Richardson (2003) demonstrate, and how did they test it?

Answer 27

Showed that paraphrasing logical statements depends on the complexity of mental models.
* Methodology: Lab-based reasoning experiments requiring participants to transform conditionals (e.g., If crash, then die) into disjunctions (e.g., Either not crash or die), testing ease of comprehension and accuracy.

Question 28

How did Kahneman & Klein (2009) advance dual-system theory, and what was their methodological approach?

Answer 28

Explained how heuristic (System 1) and analytic (System 2) systems operate in real-world decision-making.
* Methodology: Naturalistic decision-making studies, analyzing real-world cases (e.g., firefighter decisions in emergencies) and controlled experimental tasks to compare heuristic and analytic reasoning.

Question 29

What evidence supports Bayesian reasoning by Oaksford & Chater (2007)?

Answer 29

• Varying card frequencies in a reduced array selection task influenced participants’ choices based on rarity.
• Methodology: Lab experiments using Bayesian reasoning tasks with controlled variables.

Question 30

What practical example was used to highlight reasoning failures at Chernobyl?

Answer 30

Logical rule: If the experiment continues, the reactor must be cool. Real-life inference failure led to the disaster.
* Methodology: Case study analysis of events and decisions leading up to the disaster.

Question 31

How do the three inference theories interact?

Answer 31

• Logic (Structural): System 2.
• Mental Models (Semantic): System 1 → System 2.
• Statistics (Information Gain): System 1.

Question 32

What is the best approach to reasoning?

Answer 32

Combining all three perspectives (structure, semantics, and statistics) ensures robust reasoning.