Non-constructive reasoning The non-constructive aspect · Non-constructive inference and...

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1 Non-constructive inference and conditionals David Over Psychology Department Durham University Thanks to: The organizers Nagoya University Japan Society for the Promotion of Science Non-constructive reasoning Modified example from Toplak & Stanovich (2002): Jack is looking at Ann, and Ann is looking at George. Jack is a cheater, but George is not. Is a cheater looking at a non-cheater? A) Yes B) No C) Cannot tell The non-constructive aspect Jack is looking at Ann but Ann is looking at George. Jack is a cheater but George is not. Is a cheater looking at non-cheater? Ann is either a cheater or not . If she is, then a cheater (Ann) is looking at a non- cheater (George). If she is not, then a cheater (Jack) is looking at a non-cheater (Ann). Therefore, the answer is “Yes”. A constructive approach Jack is looking at Ann but Ann is looking at George. Jack is a cheater but George is not. Is a cheater looking at non-cheater? We get hold of Ann and try to cooperate with her in reciprocal altruism. She does not cooperate. Our “cheater detection mechanism” fires, and we conclude she is a cheater. Therefore, the answer is “Yes”. The distinction and dual process theory Non-constructive inference is the purest example of a type 2 analytic process. It is an inference from “above”, using logic. Constructive inference is from “below”: it is grounded in type 1 heuristic processes, such as those of perception.

Transcript of Non-constructive reasoning The non-constructive aspect · Non-constructive inference and...

Page 1: Non-constructive reasoning The non-constructive aspect · Non-constructive inference and conditionals David Over Psychology Department Durham University Thanks to: • The organizers

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Non-constructive inference

and conditionals

David Over

Psychology Department

Durham University

Thanks to:

• The organizers

• Nagoya University

• Japan Society for the Promotion of Science

Non-constructive reasoning

Modified example from Toplak &

Stanovich (2002):

Jack is looking at Ann, and Ann is

looking at George. Jack is a cheater, but

George is not. Is a cheater looking at a

non-cheater?

A) Yes B) No C) Cannot tell

The non-constructive aspect

• Jack is looking at Ann but Ann is looking

at George. Jack is a cheater but George is

not. Is a cheater looking at non-cheater?

• Ann is either a cheater or not. If she is,

then a cheater (Ann) is looking at a non-

cheater (George). If she is not, then a

cheater (Jack) is looking at a non-cheater

(Ann). Therefore, the answer is “Yes”.

A constructive approach

• Jack is looking at Ann but Ann is looking

at George. Jack is a cheater but George is

not. Is a cheater looking at non-cheater?

• We get hold of Ann and try to cooperate

with her in reciprocal altruism. She does

not cooperate. Our “cheater detection

mechanism” fires, and we conclude she is

a cheater. Therefore, the answer is “Yes”.

The distinction and dual

process theory

• Non-constructive inference is the purest

example of a type 2 analytic process. It

is an inference from “above”, using logic.

• Constructive inference is from “below”:

it is grounded in type 1 heuristic

processes, such as those of perception.

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Jonathan Evans’s list of dual process theories

Perception & attention: Egeth & Yanis (1997), stimulus &

goal driven attention.

Skilled performance: Anderson (1983), procedural &

declarative knowledge.

Learning & memory: Reber (1993), implicit & explicit

learning.

Social cognition: Strack & Deustch (2004), impulsive &

reflective.

Reasoning & decision making: Evans (2006) and Evans &

Over (1996), heuristic & analytic; Barbey & Sloman (in

press), associative & rule based; Stanovich (1999) and

Kahneman & Frederick (2002), System 1 & System 2.

System 1 mental processes

(Stanovich, 2004)

• Associative

• Holistic

• Parallel

• Automatic

• Undemanding of cognitive capacity

• Fast

• Highly contextualized

• “Old” in evolutionary terms

System 2 mental processes

(Stanovich, 2004)

• Rule based

• Analytical

• Serial

• Controlled

• Demanding of cognitive capacity

• Slow

• Decontextualized

• “New” in evolutionary terms

Jonathan Evans’s characterization

Type 1 processes: Fast and automatic, with

high capacity and low effort.

Type 2 processes: Slow and controlled, with

limited capacity and high effort. These

processes make use of working memory.

Dual process theory is opposed to

the massive modularity hypothesis

• Many leading evolutionary psychologists

have argued for what has been called the

massive modularity hypothesis.

• Some leading evolutionary psychologists

do not accept this hypothesis, but the

following support it: Cosmides & Tooby,

Buss, Pinker, and Gigerenzer.

Massive modularity implies:

• There is no mental logic: no formal

system for performing valid inferences.

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Massive modularity implies:

• There are no content independent

mechanisms for inference or learning.

Massive modularity implies:

• There are only content specific or domain

specific mechanisms – the modules - for

solving adaptive problems.

Massive modularity metaphor:

• The mind is a Swiss army knife - it has

many special blades for solving adaptive

problems but no general purpose blade.

Dual process theory implies:

• Type 1 processes result from content

specific mechanisms for perception,

memory, and heuristic inference - the

modules.

Dual process theory implies:

• Type 2 processes result from general

purpose mechanisms, including a means

of logical inference.

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Dual process theory implies:

• That the mind has two systems, System 1

and System 2, or at least has two kinds of

processes, type 1 and type 2. The Swiss

army knife metaphor could also be used

for dual process theory.

My claim:

• The best example of a type 2 process is

non-constructive reasoning. A good

example of this reasoning is inferring a

disjunction, “p or q”, from “above”.

Validly inferring a disjunction

from “above”

• We may infer, “Ann is a cheater or not a

cheater”, from “above” using pure logic –

in this case we cannot say which disjunct

is true. We do not know which property

Ann has: being a cheater or not one.

Justifiably inferring a disjunction

from “above”

• We may infer, “Ann is a cheater or Jack

is a cheater”, from “above” using

probabilistic inference. Resources are

missing. Someone is taking more than

their share. Other general considerations

point to Ann or Jack, but we do not know

which is the cheater.

What use is non-constructive

reasoning?

• We may infer, “Ann is a cheater or Jack

is a cheater”, from “above”. This enables

us to infer, “If Ann is not the cheater then

Jack is.” That is a useful conditional to

infer. For when we later get evidence that

Ann is not the cheater, we may infer that

Jack is.

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Constructive reasoning is not

useful in this way

• Suppose we infer, “Ann is a cheater or

Jack is a cheater”, from “below”, from

Ann is the cheater. We now cannot infer,

“If Ann is not the cheater then Jack is.”

If it does turn out that our information

from “below” was wrong, we do not have

any reason to suspect Jack.

Is non-constructive inference “old”?

• It is often said that type 2 processes are “new”

in evolutionary terms. Is this true of non-

constructive inference? Do any other animals

show signs of this kind of reasoning? The Stoic

logician Chrysippus claimed that a dog could

know that an animal went down one of three

roads and infer that, if it did not go down the

first two, then it went down the third. Is there

any scientific evidence of such inference?

The logical form of the inference

The form is that of inferring “if not-p then q”

from “p or q” or, equivalently inferring “if p

then q” from “not-p or q”. My claim is that

such inferences are justified only when the

disjunction is inferred non-constructively. But

in elementary logic, “if p then q” just means

“not-p or q” and so such inferences are always

justified, that is also so in the main

psychological theory of conditional reasoning.

The material conditional

In elementary extensional logic, “if p then

q” just means “not-p or q”, and so “if

not-p then q” means “p or q”. This kind

of conditional is the material conditional.

The mental model theory of Johnson-

Laird & Byrne (2002) implies that the

ordinary conditional of natural language

is the material conditional.

What the mental model theory of

Johnson-Laird & Byrne implies

• Johnson-Laird & Byrne (2002, p. 650) hold

that the following inference is valid:

• In a hand of cards, there is an ace or a king or

both. So if there isn’t an ace in the hand, then

there is a king.

• Now if the above were valid, then the ordinary

conditional would be the material conditional,

which Johnson-Laird & Byrne deny in places,

but that is logically implied in what they write

down as their theory.

The form of the inference

referred to by Johnson-Laird

& Byrne (2002)

• Inferring “if not-p then q” from “p or q”.

• From “not-p or q”, we get “if not-not-p

then q” by the form, from which we infer

by double negation “if p then q”.

• So one could equally well study inferring

“if p then q” from “not-p or q”.

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More logical points

For all conditionals we must have that

“if p then q”

logically implies

“not-p or q”

But only for the material conditional,

can the converse hold, as the material

conditional just means “not-p or q”.

If Johnson-Laird & Byrne

(2002) are right

• Then “if p then q” is logically equivalent

to the truth function material conditional,

“not-p or q”.

• And those of us who deny the equivalence

are wrong.

• But we deny the equivalence and so must

show that Johnson-Laird & Byrne (2002)

are wrong.

Our theory of ordinary conditionals in

natural language

• In our account, “if p then q” does not

mean “not-p or q”. According to us,

people evaluate “if p then q” by

supposing that p holds and judging q

under that supposition. This process is

called the Ramsey test in philosophical

logic.

The Ramsey test

• Ramsey (1931) suggested that people

could judge “if p then q” by “...adding

p hypothetically to their stock of

knowledge …” They would thus fix

'...their degrees of belief in q given

p…”, which is their conditional

subjective probability of q given p,

P(q/p).

What the Ramsey test implies

(Over, Hadjichristidis, Evans,

Handley, & Sloman, 2007)

• The probability of an indicative

conditional, P(if p then q), is the

conditional subjective probability,

P(q/p).

P(q/p) high implies high

P(not-p or q)

Suppose we find that P(q/p) is high.

Then we will find that P(not-p or q),

the material conditional, is high.

P(not-p or q) =

P(not-p) + P(q) - P(not-p & q) =

P(not-p) + P(q/p) - P(not-p)P(q/p)

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P(not-p or q) high does not

imply high P(q/p)

Suppose we find that P(not-p) is high.

Thus P(not-p or q) is high, but recall:

P(not-p or q) =

P(not-p) + P(q/p) - P(not-p)P(q/p)

And that means that P(q/p) can be low

when P(not-p or q) is high.

Validity and strength

• Inferring “if not-p then q” from

“p or q”is not logically valid,

as P(p or q) can be higher than

P(q/not-p).

• However, the inference can be a

strong probabilistic inference in

non-constructive reasoning.

Constructive example

• We think that we see Ann going into

the library. We infer with high

confidence that Ann is in the library or

the computer lab. But we could not

infer from this that, if she is not in the

library, then she is in the computer

lab.

Constructive details

P(library & lab) = 0

P(library & not-lab) = .9

P(not-library & lab) = .01

P(not-library & not-lab) = .09

P(library or lab) = .91

P(lab/not-library) = .01/.1 = .1

Non-constructive example

• We infer from reading the module

guide that everyone in the class is

in the library or the lab. Ann is in

the class. So Ann is in the library

or the lab. And so, if Ann is not in

the library, then she is in the lab.

Non-constructive details 1

P(library & lab) = 0

P(library & not-lab) = .5

P(not-library & lab) = .5

P(not-library & not-lab) = 0

P(library or lab) = 1

P(lab/not-library) = .5/.5 = 1

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Non-constructive details 2

P(library & lab) = 0

P(library & not-lab) = .45

P(not-library & lab) = .45

P(not-library & not-lab) = .1

P(library or lab) = .9

P(lab/not-library) = .45/.55 = .81

Non-constructive details 3

P(library & lab) = 0

P(library & not-lab) = .7

P(not-library & lab) = .2

P(not-library & not-lab) = .1

P(library or lab) = .9

P(lab/not-library) = .2/.3 = .66

Non-constructive details 4

P(library & lab) = 0

P(library & not-lab) = .8

P(not-library & lab) = .1

P(not-library & not-lab) = .1

P(library or lab) = .9

P(lab/not-library) = .1/.2 = .5

Ramsey test example

• Hypothetically suppose I buy a lottery ticket.Under this supposition, I can use knowledge ofthe lottery and probability to infer I willprobably lose my money. Using the Ramseytest, I disbelieve, If I buy a lottery ticket, I willwin millions.

• Johnson-Laird & Byrne (2002) imply that Ishould believe, If I buy a lottery ticket, I willwin millions, as I will not buy a lottery ticket,and If I buy a lottery ticket, I will win millionssupposedly means I do not buy a lottery ticketor I will win millions.

The Ramsey test and heuristics

• The Ramsey test can be compared to the

simulation heuristic (Kahneman & Tversky,

1982). Both are high level processes that have

to be implemented by more specific ones.

• The availability heuristic (Tversky &

Kahneman, 1972) could be used to judge P(p &

q) is probable than P(p & not-q), i.e. P(q/p) is

relatively high.

Over, Hadjichristidis, Evans, Handley,

& Sloman (2007) show:

• For an indicative conditional, If global

warming continues, London will be flooded.

• The subjective probability of such indicative

conditionals, P(if p then q), is the conditional

subjective probability of q given p, P(q/p).

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Over, Hadjichristidis, Evans, Handley,

& Sloman (2007)

• People explicitly assess P(if p then q).

• They also explicitly judge:

• P(p & q).

• P(p & not-q).

• P(not-p & q).

• P(not-p & not-q).

What we can get from the

probabilistic truth table

• P(p) = P(p & q) + P(p & not-q)

• P(q/p) = P(p & q)/[P(p & q) + P(p & not-q)]

• P(q/not-p) = P(not-p & q)/[P(not-p & q) +

P(not-p & not-q)]

The analysis

• We performed multiple regression

analyses on P(if p then q) using P(p)

and P(q/p) as predictors

• If “If p then q” is the material

conditional and so means “not-p or q”,

then P(p) should have a significant

negative loading.

The results for participants

• Analyses across individual participants.

Cells = beta weights

P(q/p)

P(p)

.51*-.38*.42*

.16*.02.02

FalseTrue

EXP2

(indicatives)

EXP1

(indicatives)

The results for items

• Analyses of item means on item means.

Cells = beta weights

P(q/p)

P(p)

.93*-.93*.90*

.14*.00.05

FalseTrue

EXP2

(indicatives)

EXP1

(indicatives)

Summary of results

• P(q/p) was by far the strongest predictor.

• There was no negative effect of P(p) as there

would have to be if “if p then q” means “not-p

or q”.

• There was a smaller negative effect of

P(q/not-p). This could suggest a relation to the

delta-p rule, which takes P(q/p) - P(q/not-p) to

measure the degree of covariation between p

and q. Does this mean that a counterfactual

states a causal relation between p and q?

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The results on disjunction

(not published so far)

• For all 81 participants in these

experiments:

mean P(not-p or q) > mean P(if p then q)

• The same was true in the analyses by

items. For all 64 items:

mean P(not-p or q) > mean P(if p then q)

Explicit probability judgments about

disjunction

• Have more recently studied participant’s

explicit subjective probability judgments

about disjunctions and conditionals.

• Order bias and the suppositional disjunction,

end of ESRC grant report, Shira Elqayam

Jonathan Evans, David Over, & Eyvind Ohm.

Experiment: Probabilistic evaluation task 1

Variation on the probability of conditionals task (Evans et

al., 2003; Oberauer & Wilhelm, 2003).

A pack contains cards which are either blue or yellow

and have either a triangle or a circle printed on them. In

total there are:

10 blue triangles

40 blue circles

40 yellow triangles

10 yellow circles

How likely is the following claim to be true of a card

drawn at random from the pack?

The card is either blue or has a triangle printed on it

1 - - - - - 2 - - - - - 3 - - - - - 4 - - - - - 5 - - - - - 6 - - - - - 7

Very unlikely Very likely

Experiment: Probabilistic evaluation task 2

Variation on the probability of conditionals task (Evans et

al., 2003; Oberauer & Wilhelm, 2003).

A pack contains cards which are either blue or yellow

and have either a triangle or a circle printed on them. In

total there are:

10 blue triangles

40 blue circles

40 yellow triangles

10 yellow circles

How likely is the following claim to be true of a card

drawn at random from the pack?

If the card is not blue then it has a triangle printed on it

1 - - - - - 2 - - - - - 3 - - - - - 4 - - - - - 5 - - - - - 6 - - - - - 7

Very unlikely Very likely

Experiment: Probabilistic evaluation task

• Do participants rate disjunctions as more

probable than conditionals?

• Mean rate (1-7 scale) across 16 items of

each linguistic form

Experiment: Probabilistic evaluation task

Comparisons of mean probability estimates,

disjunctions vs. conditionals (standard deviations

in parentheses).

Ratings for disjunctions significantly higher than ratings for the

extensionally equivalent conditionals

Disjunctions Conditionals

p or q 5.2 (0.6) If not-p then q 3.7 (0.4)

not-p or q 4.3 (1.0) if p then q 3.5 (0.6)

p or else q 4.8 (0.8) If and only if not-p then q 3.5 (0.5)

not-p or else q 4.3 (0.9) if and only if p then q 3.5 (0.6)

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Evidence that the natural

language conditional is not the

material conditional

• People do not judge the probability of

“if p then q”, P(if p then q), to be the

probability of the material conditional,

P(not-p or q).

• People judge P(p or q) to be higher

than P(if not-p then q).

• People judge P(not-p or q) to be higher

than P(if p then q).

Experiment: Probabilistic evaluation task

• Is this a fallacy?

• Compare to conjunction fallacy

• Does not behave like one

– Ps were given transparent frequency

distributions

– Typically makes conjunction fallacy

disappear

– In our results the pattern persisted

Belief versus assertion

• In people’s beliefs, P(p or q) is often

greater than P(if not-p then q).

• People will often infer “if not-p then q”

when “p or q” is asserted.

• How is this possible? People acquire

extra, pragmatic information from

assertions.

A pragmatic inference from an

assertion

• Suppose you ask me where Linda is, and I

reply, “She is in her office or the Library.”

You will think you are justified in inferring,

“If Linda is not in her office, she is in the

Library.” You will assume I have a non-

constructive justification for what I assert.

Why pragmatic?

• Suppose I know that Linda is in her

office and nothing about why she is

there. From this, I can infer that she is

in her office or the Library. If I assert

only this disjunction, however, I will

violate Grice’s Maxim of Quantity,

and you will be misled.

Return to belief

• When can we infer “if p then q” from

“not-p or q” in our beliefs?

• When can we infer “if not-p then q”

from “p or q” in our beliefs?

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Justifications of “p or q”

• “p or q” could be justified from

“below”, constructively. Then we

cannot believe “if not-p then q”.

• “p or q” could be justified from

“above”, non-constructively. Then

we can believe “if not-p then q”.

Conclusion

• Inferring “if not-p then q” from

“p or q” is sometimes justified and

sometimes not. It is justified when

P(if not-p then q) is close to P(p or

q), and that is so when we have a

non-constructive justification of

“p or q”, inferred using a type 2

process.