Understanding Conditionals

Gernot D. Kleiter

Department of Psychology

University of Salzburg, Austria

Supported by the Austrian Science Foundation

within the LogICCC programme of the European Science Foundation

LcpR.

Motivation & Preview

Importance of conditionals, if—then

Empirically : 2/3 interpret conditionals as conditional eventsand 1/3 as conjunctions.

Present contribution

Extends and improves our approachInvestigates the role of working memoryInvestigates a developmental hypothesisImplications for reasoning: Wason Task

The Card Task

Here you see ten cards showing houses and cars. They are red, blue, or green.

The Card Task

Here you see ten cards showing houses and cars. They are red, blue, or green.

The Card Task

Here you see ten cards showing houses and cars. They are red, blue, or green.

How sure can you be that the following sentence holds?

If the card shows a car, then the card shows blue

Dice Task

Die with six sides, red and blue, circle and square

Probability is used as a vehicle to infer the interpretation

n = 65 students, 32 female, 33 make, m = 71 items, entity/feature

Convergence to conditional event interpretation during the course ofitems 1, 2, . . . 71

Still: Some participants stick to conjunction

There are practically no material implication and no biconditionals

Card task – Method

Improvements: unique classification, thematic objects, design

n = 80 participants, m = 52 items, 40 male, 40 female, 40 objectfirst, 40 feature first

computer controlled individual sessions, payed

n-back task

Working Memory: n-back Task

5 6

Working Memory: n-back Task

5 6 4

Working Memory: n-back Task

5 6 4 6

Working Memory: n-back Task

5 6 4 6 4

Working Memory: n-back Task

5 6 4 6 4 3

Working Memory: n-back Task

5 6 4 6 4 3 3

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct —

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO NO

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO NO NO

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO NO NO

3 back lure

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO NO NO

3 back lure YES

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO NO NO

3 back lure YES

1 back lure

Working Memory: n-back Task

5 6 4 6 4 3 3 4

Correct — NO YES YES NO NO NO

3 back lure YES

1 back lure YES

Working memory

Explaining the transition from conjunction to conditional eventinterpretations

The Mental Models approach has stressed the role of workingmemory

Not specific for mental models, plausible candidate to explainconjunction versus conditional event

WM = set of mechanisms for retrieving and maintaininginformation during processing (Baddeley, 1986), limited

Performance suffers when complexity and demands exceed supply

Lovett et. al. (2000, ACT-R theory): available activation W isdivided among n items (models)

Continuous updating

Activation strategy (familiarity, matching) versus update strategy(list maintenance)

Participants

80 students, 40 female, 40 male, 40 object-color, 40 color-object(2 × 2)52 items, binary 0/1 variable for “conditional event = yes / noResponse time for each itemn-back task after the card task

Modal response (card task)

Card task: Modal response

Response

Per

cent

0

20

40

60

80

100

CE CONJ OTHER REV

factor(RESPONSE)

CE

CONJ

OTHER

REV

Conditional events, histogram (card task)

Card task: Conditional event interpretation

CE−Score

coun

t

0

5

10

15

20

25

30

0 10 20 30 40 50 60

Conditional events, histogram, gender (cardtask)

Card task: Conditional event interpretation

CE−Score

coun

t

0

10

20

30

40

Female

0 10 20 30 40 50 60

Male

0 10 20 30 40 50 60

factor(Sex)

Female

Male

Change point

Card task – Results

35 Ss have a maximum greater than 0.20, that is 10 times higher thanuniform16 Ss have the max after trial 5 and before trial 50

Latencies

Gender effects

Entity-feature effects

n-back

Rating Scales

Card task (Bayesian generalized linear model)

Conditional Event Response Time

Constant -0.518 0.836 8.969 0.000Item position 0.019 0.000 -0.012 0.000Male Gender 1.537 0.000 0.102 0.000EFObject-Color -0.301 0.000 -0.003 0.9181 — ID 0.000 1.000 0.000 1.000Conditional event 0.208 0.000ITEM × EFObject-Color 0.002 0.015

AIC 4921.476 5069.955BIC 4953.143 5120.621N 4160 4160

Conditional Event:bayesglm(c ∼ ITEM + SEX + EF + (1 | ID), family = binomial)Response Time:

bayesglm(log(T1 + T2) ∼ ITEM + SEX + EF + EF:ITEM + (1|ID))

Card task – Conditional Event (bayesglm)

Regression Estimates

−4 −2 0 2 4

ITEM

SEXMale

EFObject−Color

1 | IDTRUE

Card task – Response Time (bayesglm)

Regression Estimates

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

c

ITEM

ITEM:EFObject−Color

c:SEXMale

1 | IDTRUE

Card task – Conditional Event and Gender

Conditional Event, Gender

Item

Pro

port

ion

of C

E r

espo

nses

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353637383940414243444546474849505152

factor(SEX)

female

male

Card task – Conditional Event, Gender andEntity/Feature

Conditional EventP

ropo

rtio

n

0.0

0.2

0.4

0.6

0.8

1.0

Color−Object Object−Color

factor(SEX)

Female

Male

Card task

Response time

log(

t)

8.0

8.5

9.0

9.5

10.0

10.5

11.0

Color−Object Object−Color

factor(SEX)

Female

Male

Card task and working memory (n-back)

No correlation between the interpretation of conditionals andn-back performance—with one exception:

Lure-3 correlates with conjunction responses, r = .30, but

not with lure-1 → simple familiarity versus up-date strategydoes not hold

Weak evidence for a “matching” and conjunctioninterpretation

Serial information, non-commutive conditional versuscommutative conjunction

Interpretation of natural language conditionals does notrequire high working memory efforts.

No gender effects in the n-back

Does n-back really measure working memory? r(memory span,n-back) = .22 (Kane et al., 2007)!

Rating scales

r(confidence of being correct, number of CE-responses) = .47,speaks for the competence model.

Female participants are slightly less confident, r = −.27

Developmental Approach

Developmental hypothesis: conjunction → biconditional →conditional event (Barrouillet, Gauffroy & Lecas, 2008; Gauffroy &Barrouillet, 2009)

Biconditional with material implication: A ⊃ B ∧ B ⊃ A

With conditional event B|A ∧ A|B

B = 1 B = 0A = 1 A|B = 1 A|B =?

B|A = 1 B|A = 0A = 0 A|B = 0 A|B =?

B|A =? B|A =?

Mental Models & Conditional Event

Age init flesh-o incompat Interpretationtrue indet rest Probability

1 11 10,01,00 ConjunctionP(11)/[P(11) + P(10) + P(01) + P(00)]

2 11 00 10,01 BiconditionalP(11)/[P(11) + P(10) + P(01)]

3 11 01,00 10 Conditional EventP(11)/[P(11) + P(10)]

P(biconditional) =P(true possibilities)

P(true) +∑

P(incompatible possibilities)

Do not include the indeterminate possibilities

Results (Bayesian generalized linear model)

Conditional Event Response Time

Constant -1.781 0.477 9.360 0.000Item position 0.019 0.000 -0.010 0.000Age groups 0.961 0.000 -0.138 0.000Male Gender 1.067 0.000 0.074 0.000EFObject-Color -0.322 0.051 -0.126 0.000ITEM × EFObject-Color -0.009 0.098 0.001 0.1451 — ID -0.000 1.000 -0.000 1.000Conditional event 0.109 0.000

AIC 3415.570 3612.957BIC 3458.005 3667.516N 3172 3172

Conditional Event:bayesglm(c ∼ ITEM + GR + SEX + EF + ITEM:EF (1|ID), family =binomial)Response Time

bayesglm(log(T1 + T2) ∼ ITEM + c + GR + SEX + EF + (1|ID))

Conditional Events

Regression Estimates−4 −2 0 2 4

ITEM

GR

SEXMale

EFObject−Color

ITEM:EFObject−Color

1 | IDTRUE

Response Time

Regression Estimates−0.2 −0.1 0.0 0.1 0.2

ITEM

c

GR

SEXMale

EFObject−Color

1 | IDTRUE

Conditonal Event and Gender

Weak trend

Males give more conditional event responses

Conditional Event

Item position

Pro

port

ion

0.0

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50

factor(SEX)

Female

Male

Conditonal Event and Age

Age groups

Conditional Event

Item position

Pro

port

ion

0.0

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50

factor(GR)

1

2

3

Conditonal Event: Age, gender & object-color

Convergence to CE with increasing age

Color-first facilitates CE in girls

Conditional Event

Age 12, 15, adult

Pro

port

ion

0.0

0.2

0.4

0.6

0.8

1.0

Color−Object

1 2 3

Object−Color

1 2 3

factor(SEX)

Female

Male

Age: conditonal event, conjunction, and rest

Increasing conditional events, decreasing conjunctions

Even 12 years old give many conditional event responses

Practically no biconditionals, 22 biconditionals out of 3172 responses

Practically no material implications, 8 out of 3172 responses

Interpretation of Conditionals

Age Group

Pro

port

ion

0

20

40

60

80

A12 A15 ADT

factor(RE)

CE

CON

OTH

Response time

Women are faster

Weak trend

Response time

Item position

log(

t)

8.0

8.5

9.0

9.5

10.0

10.5

10 20 30 40 50

factor(SEX)

Female

Male

Response time: Age, gender & object-color

Response time is decresing with age

Male take more time for object-color order

Response time

Age 12, 15, adult

log(

t)

8.0

8.5

9.0

9.5

10.0

10.5

Color−Object

1 2 3

Object−Color

1 2 3

factor(SEX)

Female

Male

Wason task

Age, Gender, Wason task, and modal CE response

Gender

Pro

port

ion

of m

odal

CE

par

ticip

ants

0

20

40

60

80

0

20

40

60

80

A12

ADT

Female Male

A15

Female Male

factor(WASON)

Right

Wrong

Wason task

Wason given CE−Interpretation

Conditional Event

Fre

quen

cy

0

5

10

15

20

25

CE−NO

WASON−NOWASON−YES

CE−YES

WASON−NOWASON−YES

factor(GR)

A12

A15

ADT

Discussion

Overwhelming empirical support for the conditional eventinterpretation

Gender differences

Working memory, serial processing

We could not replicate the biconditionals

Wason task

What is next?

Conditional in inference

Consequence relation

Extending to FOL ...

Fugard, A. J. B., Pfeifer, N., Mayerhofer, B. & Kleiter, G. D.(in press). How people interpret conditionals: Shifts towardsthe conditional event Journal of Experimental Psychology:

Learning, Memory, and Cognition.

Fugard, A. J. B., Pfeifer, N., Mayerhofer, B. & Kleiter, G. D.(in preparation). The conditional event interpretation ofconditionals

Gansbiller, J. & Kleiter, G. D. (in preparation). Understandingconditionals: A developmental perspective

Gauffroy, C. & Barrouillet, P. (2009). Heuristic and analyticprocesses in mental models for conditionals: An integrativedevelopmental theory. Developmental Review, 29, 249–282.