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  • 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, ifthen

    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

    CEScore

    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

    CEScore

    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

    EFObjectColor

    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:EFObjectColor

    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

    ColorObject ObjectColor

    factor(SEX)

    Female

    Male

  • Card task

    Response time

    log(

    t)

    8.0

    8.5

    9.0

    9.5

    10.0

    10.5

    11.0

    ColorObject ObjectColor

    factor(SEX)

    Female

    Male

  • Card task and working memory (n-back)

    No correlation between the interpretation of conditionals andn-back performancewith 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 Estimates4 2 0 2 4

    ITEM

    GR

    SEXMale

    EFObjectColor

    ITEM:EFObjectColor

    1 | IDTRUE

  • Response Time

    Regression Estimates0.2 0.1 0.0 0.1 0.2

    ITEM

    c

    GR

    SEXMale

    EFObjectColor

    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

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