Dual-Source Model of Conditional Reasoning

1. Conditional Reasoning: The Dual Source Model of Conditional Reasoning versus Minimizing the Kullback-LeiblerDivergence Henrik Singmann Sieghard Beller Karl Christoph Klauer

2. ConditionalReasoning • A conditionalis a statementwiththe form:Ifp, thenq. • Conditionalinferencesusuallyconsistoftheconditional(majorpremise), a minorpremise (e.g., p) andtheconclusion (e.g., q), e.g.: If a balloonisprickedwith a needle, thenit will pop. A balloonisprickedwith a needle. Therefore, theballoon will pop. Dual-Source Model

3. 4 ConditionalInferences Modus Ponens (MP): Ifp, thenq. p Therefore, q Affirmation oftheconsequent (AC): Ifp, thenq. q Therefore, p Modus Tollens (MT): Ifp, thenq. Not q Therefore, not p Denialoftheantecedent (DA): Ifp, thenq. Not p Therefore, not q Dual-Source Model

4. 4 ConditionalInferences Modus Ponens (MP): Ifp, thenq. p Therefore, q Affirmation oftheconsequent (AC): Ifp, thenq. q Therefore, p Denialoftheantecedent (DA): Ifp, thenq. Not p Therefore, not q Modus Tollens (MT): Ifp, thenq. Not q Therefore, not p NOT valid in standardlogic (i.e., truthofpremisesdoes NOT entailtruthofconclusion) valid in standardlogic (i.e., truthofpremisesentailstruthofconclusion) Dual-Source Model

5. Research Question: What is the effect of the presence of the conditional in everyday probabilistic conditional reasoning? Dual-Source Model

6. Example Item: Knowledge Phase Observation: A balloon is pricked with a needle. How likely is it that it will pop? Dual-Source Model

7. Example Item: Knowledge Phase Observation: A balloon is pricked with a needle. How likely is it that it will pop? X Dual-Source Model

8. Example Item: Rule Phase Rule: If a balloon is pricked with a needle, then it will pop. Observation: A balloon is pricked with a needle. How likely is it that it will pop? Dual-Source Model

9. Example Item: Rule Phase Rule: If a balloon is pricked with a needle, then it will pop. Observation: A balloon is pricked with a needle. How likely is it that it will pop? X Dual-Source Model

10. Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15) different linesrepresent different contentsoftheconditional ↑ conditional absent ↓ ↑ conditionalpresent ↓ newdata (n = 29) Dual-Source Model

11. Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15) different linesrepresent different contentsoftheconditional The presence of the rule increases participants' estimates of the probability of the conclusion. Especially so for the formally valid inferences MP and MT. ↑ conditional absent ↓ ↑ conditionalpresent ↓ newdata (n = 29) Dual-Source Model

12. Example Item: Knowledge Phase Observation: A balloon is pricked with a needle. How likely is it that it will pop? X Dual-Source Model

13. The Knowledge Phase • Participantsareaskedtoestimate a conclusiongiven a minorpremiseonly. E.g., Givenp, howlikelyisq? • The responseshouldreflecttheconditionalprobabilityoftheconclusiongivenminorpremise,e.g., P(q|p) Dual-Source Model

14. FormalizingtheKnowledge Phase • WehavethejointprobabilitydistributionoverP(p), P(q), andtheirnegations in theknowledgephase per content: • Fromthiswecanobtaintheconditionalprobabilities, e.g.: P(MP) = P(q|p) = P(p q) / P(q) • Weneedat least threeindependentparameters (e.g., P(p), P(q), and P(¬q|p), Oaksford, Chater, & Larkin, 2000) todescribethejointprobabilitydistribution. Dual-Source Model

15. How do we explain the effect of the conditional? • Whentheconditionalis absent, participantsusetheirbackgroundknowledgetoestimatetheconditionalprobabilityoftheconclusiongivenminorpremise.E.g., Givenp, howlikelyisq? P(q|p) • The presenceoftheconditionaladds a different type ofinformation: form-based evidence (i.e., the subjective probability to which an inference is seen as logically warranted by the form of the inference).E.g., How likely is the conclusion given that the inference is MP? • The dual-source model (Klauer, Beller, & Hütter, 2010) posits that people integrate these two types of information in the conditional inference task. Dual-Source Model

16. Formalizingthe Dual-Source Model • Observable response on oneinferenceWithconditional = λ{τ(x) × 1 + (1 – τ(x)) × ξ(C,x)} + (1 – λ)ξ(C,x)Withoutconditional = ξ(C,x) • τ(x) = form-basedevidence, subjectiveprobabilityforacceptinginference x (i.e., MP, MT, AC, DA) based on thelogical form. • ξ(C,x) = knowledge-basedevidence, subjectiveprobabilityforacceptinginferencexforcontentC based on thebackgroundknowledge. Dual-Source Model

17. Howelsecouldweexplainit? • A prominent alternative to our assumptions is that the presence of the conditional simply changes the knowledge base from which people reason. • So far we have only had one formalization of this approach, Oaksford, Chater, and Larkin's (2000) model of conditional reasoning. • In their model the joint probability distribution is defined by P(p), P(q), and P(¬q|p) (theexceptionsparameter), fromwhich all fourconditionalprobabilitiescanbecalculated.(Thisis also theparametrizationoftheknowledge in the dual-source model) • The presence of the conditional decreases possible exceptions, P'(¬q|p) < P(¬q|p), so onlyaffectingoneoftheunderlyingparameters. Dual-Source Model

18. An alternative formalization • AccordingtoHartmann andRafiee Rad (2012, SPP workshoporganizedby Niki Pfeifer) whenreasoningfromconditionalsandlearningthepremises, participants update theirjointprobabilitydistributionbyminimizingtheKullback-Leiblerdivergencewithrespecttothenewinformation. • Wethereforedescribethejointprobabilitydistributionbya = P(p), α = P(q|p), andβ = P(¬q|¬p). • Whenlearningtheconditional, participantsupdate theconditionalprobabilityα' > α, andtheotherparameters (a' andβ')update byminimizingtheKullback-Leiblerdistancetotheoriginal distribution. • Withonlyone additional freeparameter (α') wetherefore also update theotherparameters. Dual-Source Model

19. EmpiricalQuestion: Whichofthethreemodelsprovidesthebestaccounttotheempricialdata? Dual-Source Model

20. Klauer, Beller, & Hütter (2010) Experiment 1: emphasis on conditional • 15 participants; 4 different contents; participants responded to all four inferences per content. • Three measurements (at least one week in between): • without conditional • with conditional (conditional stated "as in everyday context") • with conditional (treat conditional as true without exceptions) Experiment 3: if-then versus only-if • 18 participants; 5 different contents • Three measurements (at least one week in between): • without conditional • with conditional: if-then • with conditional: only-if (actually contents and type of rule, if-then versus only-if, were counterbalanced across measurements 2 and 3) Dual-Source Model

21. Klauer, Beller, & Hütter (2010) Experiment 1, 48 data points: 4 × 3 knowledge parameters + • Dual-source (DS) model: 2 × 4 independent form-based parameters = 20 parameters • Kullback-Leibler (KL) model: 2 × 4 updating parameters (updating of previous measurement) = 20 parameters • Oaksford et al.'s (OAK) model: 2 × 4 updating parameters (increasing across measurements) = 20 parameters Experiment 3, 60 data points: 5 × 3 knowledge parameters + • Dual-source model: 2 × 4 independent form-based parameters = 23 parameters • Kullback-Leiblermodel: 2 × 5 updating parameters (independent) = 25 parameters • Oaksford et al.'s model: 2 × 5 updating parameters (independent) = 25 parameters Dual-Source Model

22. Dual-Source Model

23. all directcomparisonsaresignificant. onlyKullback-LeiblerandOaksford et al.'smodel do NOT differsignificantly. Dual-Source Model

24. Klauer, Beller, & Hütter (2010) Experiment 4: Negations • 13 participants; 4 contents • 5 measurements: • withoutconditional • withconditional (ifp, thenq) • withconditional (ifp, then not q) • withconditional (if not p, thenq) • withconditional (if not p, then not q) (as before, contents and negation of rule were actually counterbalanced so that at each measurement each content and each type of negation was presented once) Dual-Source Model

25. Klauer, Beller, & Hütter (2010) – Exp 4 80 datapoints: 4 × 3 knowledgeparameters + • Dual-sourcemodel: 4 form-basedparameters= 16 parameters • Kullback-Leiblermodel: 4 × 4 independentupdatingparameters= 28 parameters • Oaksford et al.'smodel: 4 × 4 independentupdatingparameters= 28 parameters * * Dual-Source Model

26. SupressionEffects Supressioneffects (datapresented last year): • 77 (dataset 1) and 91 (dataset 2) participants; 4 contents • 3 between-subjectsconditions: • Baseline (same asbefore) • Disablers (additional premisesunderminingthesufficiencyofpforq) • Alternatives (additional premisesunderminingthenecessityofpforq) • 2 measurements: • withoutconditional (but with additional premises) • withconditional (andwith additional premises) • All modelshave 4 × 3 knowledgeparameters plus 4 form-basedorupdatingparameters = 16 parametersfor 32 datapoints Dual-Source Model

27. SupressionEffects all directcomparisonsaresignificant. Dual-Source Model

28. Effectsof Speaker Expertise • 32 participants (still collecting); 7 contents • 3 & 3 contentsarerandomlyselected per participant • 2 measurements: • withoutconditionals • withconditionals (3 conditionalsutteredby expert, 3 conditionalsutteredby non-expert) Dual-Source Model

29. Effectsof Speaker Expertise • 48 datapoints: 6 × 3 knowledgeparameters + • DS: 4 form parameters+ 1weightingparameter= 21 parameters • KL: 6 updatingparameters= 22 parameters • OAK: 6 updatingps.= 22 parameters * * Dual-Source Model

30. Summary Results • The dual-source model providestheoverallbestaccountfor 6 datasets. Forthetwocaseswhereitsharesthefirstplace (withOaksford et al.'s model), ithaslessparameters. • The dual-source model providesthebestaccountfor 144 of 246 datasets (59%). • The Kullback-Leibler model (4 timessecondplace, 53 bestaccounts) andOaksford et al.'s model (2 timesfirstplace, one time second, 49 bestaccount) sharethesecondplace. • Thisis strong evidenceforourinterpretationoftheeffectoftheconditional. Itseemstoprovide formal informationthatcan not easilybeaccountedforbychanges in participants' knowledgebase. Dual-Source Model

31. Beyond Model Fit • The parameters of the dual-source model offer an insight into different underlying cognitive processes, e.g., a dissociation of different suppression effects (disablers decrease the weighting parameters, whereas alternatives decrease the knowledge base for AC and DA). • The dual-source model can be easily extended to other types of connectives such as e.g., or, as it is not strictly tied to conditional reasoning. Dual-Source Model