Reasoning

Reasoning: drawing inferences from pre-existing

knowledge

Deductive Reasoning: conclusions follow with

certainty from premises

Inductive Reasoning: conclusions follow from

premises with some probability

Example:

Fred is Mary’s brother.

Mary is Lisa’s mother.

Deductive: Fred is Lisa’s uncle.

Inductive: Fred is older than Lisa.

Deductive Reasoning

Conditional Reasoning

· Reasoning about conditional (if-then) relationships

If it’s raining, the game is cancelled.

It’s raining.

Therefore, the game is cancelled.

· More generally:

antecedent consequent

P ® Q

Premises

P cond. operator

Conclusion \ Q

· This is an example of a conditional syllogism.

Rules of deductive inference

· modus ponens: can infer the consequent if the antecedent is true

Given the premises

If it is February, there is snow on the ground.

It is February.

Then we can infer (by modus ponens) that

There is snow on the ground.

This is a valid deduction from the premises.

· modus tollens: can infer that an antecedent is false if the consequent is false

o general form:

P ® Q

not Q

\ not P

o e.g.,

If the light is green, the traffic will move.

The traffic is not moving.

Therefore, the light is not green.

Use of Logical Inference Rules

· how well do people apply these rules?

o Rips & Marcus (1977)

§ Is the conclusion always true, sometimes true, or never true?

§ modus ponens: no errors

P ® Q Always: 100%

P Sometimes: 0%

\ Q Never: 0%

P ® Q Always: 0%

P Sometimes: 0%

\ not Q Never: 100%

§ modus tollens: some errors

P ® Q Always: 57%

Not Q Sometimes: 39%

\ not P Never: 4%

P ® Q Always: 0%

Not Q Sometimes: 23%

\ P Never: 77%

· modus ponens is intuitive; easy to apply

· modus tollens is less intuitive

o (maybe) people treat reasoning as drawing inferences about causes and effects

o modus ponens: forward reasoning from cause to effect

o modus tollens: backward reasoning from effects to causes

· types of errors

o denying the antecedent

If the light is green, the traffic will move.

The light is not green.

Therefore, the traffic is not moving.

21% say this is always valid.

o affirming the consequent

If the light is green, the traffic will move.

The traffic is moving.

Therefore, the light is green.

23% say this is always valid.

· Why do people make these errors?

o one reason: most people interpret “if” differently than logicians do

o biconditional: if and only if

§ if either premise is true, the other will be true

· people aren’t necessarily illogical; they just interpret the premises in unexpected ways

· may also use pre-existing knowledge; content matters

If the horses had been to the waterhole, we

would see their tracks.

We see their tracks.

Therefore, the horses have been to the

waterhole.

If the horses had been to the waterhole, then

the food we left out would be gone.

The food we left out is gone.

Therefore, the horses have been to the

waterhole.

Wason Selection Task

E F 4 7

“If a card has a vowel on one side, it has an even number on the other side.”

· Should turn over “E” and “7”

o 89% of people pick E

o 62% pick 4

o 25% pick 7

o 16% pick F

· turning over E: modus ponens

· turning over 7: modus tollens

· turning over 4: affirming the consequent

· why?

o Matching hypothesis: match terms in conditional with states of affairs in the world (data)

· In general: performance depends on more than form of problem

· e.g., content effects:

o Griggs & Cox (1982)

“If a person is drinking beer, then the person must be over 21.”

Person 1: Drinking beer

Person 2: Drinking coke

Person 3: 16 years old

Person 4: 22 years old

74% identify correct two people

· Not just familiarity: Cheng & Holyoak (1985)

“If the form says “entering” on one side, then the other side lists “cholera” among the diseases.”

Condition 1: rule, no explanation

Condition 2: rule, with explanation

Form 1: Entering

Form 2: Transit

Form 3: cholera, typhoid

Form 4: typhoid

· performance much better in condition 2 (91%) than in condition 1 (60%)

Pragmatic Reasoning Schemas

· Cheng & Holyoak (1985)

· people don’t reason using content-free rules

· pragmatic reasoning schemas: sets of rules that are specific to particular goals or situations

· e.g., permission, obligation, authorization, causation

· permission schema: invoked when taking a particular action requires satisfaction of some precondition

Categorical Reasoning

· categorical syllogisms

o involve quantifiers (e.g., some, none, all)

All As are Bs.

All Bs are Cs.

\ All As are Cs.

Most people correctly recognize this as valid.

But most people think this is valid, too:

Some As are Bs.

Some Bs are Cs.

\ Some As are Cs.

This isn’t a valid conclusion. Consider:

Some men are lawyers.

Some lawyers are women.

\ Some men are women.

· People too readily accept invalid conclusions

· But not all invalid conclusions:

Some As are Bs.

Some Bs are Cs.

\ No As are Cs.

· Atmosphere hypothesis: people are more likely to accept conclusion if its quantifier matches the quantifiers in the premises

How do people actually reason? Competing views:

Rules-based reasoning (Rips)

· People use abstract inference rules

· Simple rules that can be chained together for more complex reasoning

· Three steps:

1. uncover the logical form of the premises

If it is raining then the game is cancelled.

It is raining.

Logical form:

If P then Q

2. derive the conclusion using stored rules

Therefore, Q.

3. translate the conclusion into the content of the premises

Therefore, the game is cancelled.

· Errors in reasoning:

o incorrect logical form is recovered from the premises

o difficult inferences require construction from simpler components

§ may be limited by working memory capacity, for example

· has trouble explaining content effects in a satisfying way

Model-based reasoning (Johnson-Laird)

· people construct a mental representation (model) of the world described by the premises

· inferences are drawn by inspecting the mental model

· models constructed of images or mental symbols

Example

Premise 1: All artists are beekeepers

Premise 2: All beekeepers are chemists

Conclusion: All artists are chemists

Premise 1:

artist = beekeeper

(beekeeper)

Premises 1 and 2:

artist = beekeeper = chemist

(beekeeper) = (chemist)

(chemist)

· more complicated inferences require more mental models to be constructed

· some support: reasoning is better when the premises contain concepts that are easy to imagine

· Errors in reasoning:

o Failure to construct correct mental model

o Memory limitations

· Can explain content effects

o Effects on model construction

o Effects on model memory