# The psychology of proof Flashcards

Chernobyl- a (failed) deduction

- If the experiment continues, then the reactor must be cool.
- The reactor is not cool.

Therefore?

- The experiment must be stopped…
- But it was not stopped

There was a failure of inference- they failed to know what to do

Definition of proof

Evidence or argument establishing a fact or the truth of a statement = drawing an inference (you infer from a set of info what much follow from it)

List other tasks related to proof

- Explanation
- Diagnosis- medical diagnosis is going from symptoms to proof
- Prediction
- Imagination- inferring the possibilities available to you

What are the types of inference?

Deduction (Specific inference)

- Characterised by being given general rules and you draw a specific conclusion from predigood

- “When the UK leaves the EU, reaching an agreement will be the easiest deal in human history” – Liam Fox, 2017 . The UK is leaving the EU in January 2020 ….. Therefore?

Induction (General inference)

- You draw a general inference

- If you see one swan thats white and another swan thats white, you assume all swans are white

‘- The UK parliament has rejected deals negotiated between the UK prime

minister and the EU on five occasions (out of five)’. Therefore?….

- Will inductively predict it will fail again

Abduction (‘Best explanation available’)

- The info you have to the best available answer

- ‘The areas of the UK that voted most highly in favour of leaving the EU have received the most money from the EU over the past 25 years’. The reason why these regions voted to leave is….?

- They don’t follow from the structure or info you’ve been given.

- They use world knowledge to help you explain something.

How does the mind undertake deduction?

(Three ways you can draw the same argument)

E.g., ʻIf the plane crashes the pilot will die

and ʻThe plane crashed’

- Structure (form)

If crash then die. Given crash, die follows - Semantics (function)

Dying as a result of a crash is possible - Statistics (frequency)

Dying as a result of a crash is probable

Inference as logical reasoning:

Assumption and types of inference

Assumption:

Individuals draw conclusions from premises by applying stored rules of logic to derive a single valid inference.

Types of inference:

- Classical syllogisms: All artists are beekeepers, some beekeepers are chemists, therefore …?

- Conditional inferences: If I work hard, then I will get a pay rise. I didn’t get a pay rise. Therefore…?

- Transitive inferences: John is faster than Mike, John is slower than Bill, therefore…. ?

Conditional syllogisms

If X is a mammal, then X is an animal

Minor premises Potential conclusions

X is a mammal X is an animal -> Modus ponens (100%)

X is not a mammal. X is not an animal -> Deny antecedent

X is an animal X is a mammal -> Affirm consequent

X is not an animal X is not a mammal -> Modus tollens (50%)

In each of these cases, you can either draw valid or invalid logical influences

Depending on the rule, we will make mistakes

If given a modus ponens, you pretty much draw the right conclusion every time

If given modus tollens, roughly people will only draw it some of the time

The structural view

- What did Piaget say?

- formal logic?

Piaget said as small children we cannot reason logically but we move from a stage of concrete operations to formal operations (more generic type of thinking)

Formal logic → the use of syntactic structure (form) to determine the validity of an argument

Piaget

* Stage of ʻformal operational thinkingʼ

BRAINE & OʼBRIEN (1994); RIPS (1983) - Natural deduction

- what did they argue

- different interferences

They argued we do have formal rules, but its an incomplete system- we have some and don’t have some

- Direct inferences

When “p or q” and “not p” are held in memory, then conclusion “q” follows - Indirect inferences

When “if p then q” and “not q” are held in memory, “not p” is inferred by applying inference rules. This explains why some inferences are harder to draw.

Explain Watson’s selection task

I have a set of cards, each with a letter on one side and a number on the other.

There are four cards face down on the table, showing either a letter or a number. Select those cards, and only those cards, that will test the truth of the following rule:

- “If a card has a D on one side, then it has a 5 on the other side”

D H 5 8

The correct answer is D and H

Testing modus ponens inference

If you turn the 8 over and it does have a D, then you’re in trouble.

- “If a card has a B on one side, then it does not have a 7 on the other side”

W B 7 4

The correct answer is to select the B and 7

Implication of Wason’s selection task

People get it logically right just by adding the negative in. We’re showing a matching bias- matching the cards we select by the cards in the rule. (they select the B and 7)

When there is no negative, people are unlikely to get it right (they select the D and 5)

Cheng & Holyoak (1985) - Pragmatic reasoning schemas (rules of life)

You are an immigration officer, checking that your staff are applying a rule that passengers coming from areas where Cholera is epidemic have had an immunization injection. If a passenger comes from a Cholera epidemic area, it is marked on one side of their visa, and if they have had the immunization, it is marked on the other side. In front of you are four visas on the table. Which should you turn over to ensure that the staff are applying the rule correctly?

Cholera area

No Cholera

Immunized

Not immunized

People almost always get the right answer- they select the one that says if they come from a cholera area, we will check that

And they also need to check people who have not been immunized

The context added therefore makes it become easier

Inference as searching for mental models

Johnson-Laird (1983)

- Inferences are drawn by searching mental representations for

possibilities that have no counter-examples - Construction of and search for models is constrained by

i) Principle of truth- if you know in the rule that its true, thats the first possibility you come to- first thing you think is plane crashes and pilot dies

ii) working memory capacity- the more possibilities there are, the less likely you are to construct possibilities

ʻIf the plane crashes the pilot diesʼ allows….

Crash Dies

No crash Lives

No crash Dies

Evidence: Ormerod & Richardson (2003)

Paraphrasing between different kinds of logically equivalent conditional statement

Conditional : ʻIf the plane crashes the pilot diesʼ

Crash Die

(if, then)

Disjunction: ʻEither the plane does not crash or the pilot diesʼ

Not Crash ….

…. Die

(either, or)

ellipsis indicates more possibilities are available

The way you represent a disjunction is in 2 separate models.

Their test of WM capacity suggested that if you want to translate from conditional to disjunction and vice versa, the number of models that you have to do it will matter

Ormerod & Richardson (2003)

Another study

Gave people 2 different tasks.

In one task, we gave them a conditional task and they had to generate the disjunction or we gave them the disjunction and they had to generate the conditional.

In another task given C and D together and their task was to say whether they were the same or different logically.

Normally evaluation is easier than generation.

Their understanding was that generation would be harder than evaluation.

This is because if you give evaluation they have to compare one set of models with another (compare 2 sets of models)

But for generation you have to fill in gaps (compare one set of models)

When people were given the task of generation they were better than evaluation, but this is very rare in psychological research.

With generation, going from C to D is easier than D to C. Disjunction you have to have 2 models where as conditional you have one therefore conditional is easier than disjunction.