Suppression Effects in the Dual-Source Model of Conditional Reasoning Henrik Singmann Sieghard Beller Karl Christoph Klauer

Research Goals How do humans reason with probabilistic relations? Which cognitive processes contribute to reasoning from everyday conditionals? How can we disentangle them and measure their contributions? How do different linguistic forms (of potentially the same logical relation) influence the reasoning process? 20.09.2018 Dual-Source Model

Conditional Reasoning A conditional is a rule with the form: If p, then q. Conditional inferences usually consist of the conditional rule (major premise), a minor premise (e.g., p) and the conclusion (e.g., q). e.g.: If a person drinks a lot of coke, then the person will gain weight. A person drinks a lot of coke. Conclusion: The person will gain weight. 20.09.2018 Dual-Source Model

4 Conditional Inferences Modus Ponens (MP): If p, then q. p Conclusion: q Affirmation of the consequent (AC): If p, then q. q Conclusion: p Modus Tollens (MT): If p, then q. Not q Conclusion: Not p Denial of the antecedent (DA): If p, then q. Not p Conclusion: Not q 20.09.2018 Dual-Source Model

4 Conditional Inferences Modus Ponens (MP): If p, then q. p Conclusion: q Affirmation of the consequent (AC): If p, then q. q Conclusion: p Modus Tollens (MT): If p, then q. Not q Conclusion: Not p Denial of the antecedent (DA): If p, then q. Not p Conclusion: Not q valid in standard logic (i.e., truth of premises entails truth of conclusion) NOT valid in standard logic (i.e., truth of premises does NOT entail truth of conclusion) 20.09.2018 Dual-Source Model

Content Effects Content can influence reasoning from conditionals via two types of counterexamples: disabling conditions and alternative antecedents. If a balloon is pricked with a needle, then it will quickly lose air. Few disablers for the consequent and many alternatives for the antecedent lead to strong endorsement of valid inferences (MP & MT) and weak endorsement of the invalid inferences (AC & DA) If a girl has sexual intercourse, then she will be pregnant. Many disablers for the consequent and few alternatives for the antecedent lead to weak endorsement of valid inferences (MP & MT) and strong endorsement of the invalid inferences (AC & DA) To tap everyday reasoning studies often use probabilistic instructions: "How likely is it that the conclusion holds?" 20.09.2018 Dual-Source Model

Explaining Content Effects Probabilistic theory (Oaksford et al., 2000) Reasoning is estimation of subjective probabilities based on background knowledge of the problem. Problem Type MP MT AC DA p → q p : q ¬q : ¬p q : p ¬p : ¬q Response reflects P(q|p) P(¬p|¬q) P(p|q) P(¬q|¬p) 20.09.2018 Dual-Source Model

Explaining Content Effects Probabilistic theory (Oaksford et al., 2000) Reasoning is estimation of subjective probabilities based on background knowledge of the problem. What is the effect of the rule? (e.g. Liu, 2003) Typically endorsement of MP and MT is heightened Problem Type MP MT AC DA p : q ¬q : ¬p q : p ¬p : ¬q Response reflects P(q|p) P(¬p|¬q) P(p|q) P(¬q|¬p) 20.09.2018 Dual-Source Model

Explaining Content Effects Probabilistic theory (Oaksford et al., 2000) Reasoning is estimation of subjective probabilities based on background knowledge of the problem. What is the effect of the rule? (e.g. Liu, 2003) Typically endorsement of MP and MT is heightened Problem Type MP MT AC DA p : q ¬q : ¬p q : p ¬p : ¬q Response reflects P(q|p) P(¬p|¬q) P(p|q) P(¬q|¬p) 20.09.2018 Dual-Source Model

What is the Effect of the Rule? Klauer, Beller, & Hütter (2010, Exp. 1): Participants work on conditional reasoning tasks with different contents on two sessions: Session (knowledge phase): Problems without conditional rule. Session (rule phase): Problems with conditional rule. Four contents cross disablers and alternatives: If a predator is hungry, then it will search for prey. (few disablers, few alternatives) If a balloon is pricked with a needle, then it will quickly lose air. (few disablers, many alternatives) If a girl has sexual intercourse, then she will be pregnant. (many disablers, few alternatives) If a person drinks a lot of coke, then the person will gain weight. (many disablers, many alternatives) 20.09.2018 Dual-Source Model

What is the Effect of the Rule? In both sessions participants work on all 4 contents and all four conditional inferences (MP, MT, AC, DA) To obtain better estimates of the perceived probability with which a conclusion holds, participants work on two versions of each inference, the original conclusion and the converse conclusion: MP (Original): p → q; p : q MP' (Converse): p → q; p : ¬q Dependent Variable: How likely does the conclusion hold given minor premise (Session 1) or given conditional rule and minor premise (Session 2)? Response on scale from 0% to 100% 20.09.2018 Dual-Source Model

Klauer et al. (2010): Experiment 1 predator: few disablers, few alternatives balloon: few disablers, many alternatives girl: many disablers, few alternatives coke: many disabler, many alternatives 20.09.2018 Dual-Source Model

The Dual-Source Model The finding that the presence of the rule affects reasoning from conditional problems is difficult for theories assuming that reasoning is a process of subjective probability estimation based on background knowledge. According to the dual-source model of conditional reasoning (Klauer, Beller, & Hütter, 2010) the effect of the rule is to provide evidence in addition to the background knowledge: form-based evidence (i.e., the subjective probability to which an inference is seen as logically warranted) 20.09.2018 Dual-Source Model

The Dual-Source Model The dual-source model is a mathematical measurement model that describes probabilistic conditional reasoning with three parameters: ξ (xsi): quantifies the knowledge-based component (formalized as in Oaksford et al.'s, 2000, model) τ (tau): form-based evidence λ (lambda): weighting factor The model is estimated by fitting individual data from both sessions to one set of parameters. 20.09.2018 Dual-Source Model

The Dual-Source Model Observable response on one inference: With rule = λ{τ(x) × 1 + (1 – τ(x)) × ξ(C,x)} + (1 – λ)ξ(C,x) Without rule = ξ(C,x) τ(x) = form-based evidence, subjective probability for accepting inference x (i.e., MP, MT, AC, DA) based on the logical form. ξ(C,x) = knowledge-based evidence, subjective probability for accepting inference x for content C based on the background knowledge. 20.09.2018 Dual-Source Model

Estimating the Dual-Source Model Participants work on conditional reasoning tasks with four different contents on two sessions: Session (knowledge phase): Problems without conditional rule. Session (rule phase): Problems with conditional rule. Four contents cross disablers and alternatives. In both sessions participants work on all 4 contents and all four conditional inferences (MP, MT, AC, DA) The model is fitted by minimizing the sum of the squared deviance from data and predictions. 20.09.2018 Dual-Source Model

Klauer et al. (2010): Experiment 1 predator: few disablers, few alternatives balloon: few disablers, many alternatives girl: many disablers, few alternatives coke: many disabler, many alternatives 20.09.2018 Dual-Source Model

Klauer et al. (2010): Experiment 1 ξ (xsi) & τ (tau) Model parameters τ (tau) predator: few disablers, few alternatives balloon: few disablers, many alternatives girl: many disablers, few alternatives coke: many disabler, many alternatives 20.09.2018 Dual-Source Model

Parameter Validation The knowledge parameter (ξ) generally maps on the responses from the knowledge phase (i.e., session without rule). The form parameter τ follows a conditional pattern: MP > MT > AC ≥ DA The form parameter τ was further validated in Experiment 3 (Klauer et al., 2010): An additional sentence with little background knowledge: “If the level of oxytocin in the blood is increased, lactation is elevated.” Two rules: "If p, then q" and "p only if q" Both rules were randomly and counterbalanced presented for all contents in two rule phases (seesions two and three). 20.09.2018 Dual-Source Model

Klauer et al. (2010), Experiment 3 predator: few disablers, few alternatives balloon: few disablers, many alternatives girl: many disablers, few alternatives coke: many disabler, many alternatives oxytocin: unkown content 20.09.2018 Dual-Source Model

Klauer et al. (2010), Experiment 3 The model was fitted to 60 data points per individual with 25 parameters: 8 τ parameters (4 per rule) 2 λ parameters (1 per rule) 15 ξ paramaters (3 per content) The additional rule only affected the form-parameters, the weighting parameters were identical (.88 versus .88). Knowledge parameters followed the knowledge phase (session without rule) Model fit was acceptable: .89 only if if then 20.09.2018 Dual-Source Model

Summary Dual-Source Model The dual-source model tries to explain the empirical differences between responses to conditional problems without rule and with rule. It describes probabilistic conditional reasoning as a weighted integration of two types of information, knowledge-based and form-based information. The three model parameters map onto those processes: ξ (xsi): quantifies the knowledge-based component (formalized as in Oaksford et al.'s, 2000, model) τ (tau): form-based evidence λ (lambda): weighting factor 20.09.2018 Dual-Source Model

Suppression Effects (Byrne, 1989) Counterexamples (disablers and alternatives) implicitly influence endorsement to inferences. Byrne (1989) showed that additional premises with explicit counterexamples defeat otherwise valid inferences (i.e., human reasoning is non-monotonic) 20.09.2018 Dual-Source Model

Suppression Effects Disabler Alternative If a predator is hungry, then it will search for prey. A predator is hungry. How likely is it that it will search for prey? Please note: A predator will only search for prey, if it is physically fit. If a predator is hungry, then it will search for prey. A predator is hungry. How likely is it that it will search for prey? Please note: A predator will also search for prey, if it needs to feed its offspring. Disabler Alternative 20.09.2018 Dual-Source Model

Research question: Which type of information (background knowledge or perceived validity) is affected by suppression effects? 20.09.2018 Dual-Source Model

Suppression Effects – Experiment We asked 41 participants to generate either alternatives or disablers for the four contents and used the three most frequently generated counterexamples for the next study. Participants work on original and converse conditional inferences (same 4 contents) in two phases: knowledge phase: without rule rule phase: with rule Three experimental conditions: baseline (no explicit counterexamples), n = 29 disablers (three additional disablers), n = 31 alternatives (three additional alternatives), n = 31 Six control conditions: 3 groups (baseline, …) with 2 knowledge phases (n = 92) 3 groups with only a rule phase (no prior knowledge phase) (n = 90) 20.09.2018 Dual-Source Model

Example Item (Disablers Condition) Observation: A predator is hungry. How likely is it that it will search for prey? Please note: A predator will only search for prey, if it is physically fit, it is living in the wild, it has no prey left. 20.09.2018 Dual-Source Model

Example Item (Disablers Condition) Rule: If a predator is hungry, then it will search for prey. Observation: A predator is hungry. How likely is it that it will search for prey? Please note: A predator will only search for prey, if it is physically fit, it is living in the wild, it has no prey left. 20.09.2018 Dual-Source Model

predator without rule with rule coke without rule with rule 20.09.2018

girl without rule with rule balloon without rule with rule 20.09.2018

Dual-Source Model We find suppression effects for both disablers and alternatives. The effect of the disablers seems to be weaker in the knowledge phase without rule. We now fit the dual-source model with 16 parameters to the individual 32 datapoints: 4 τ parameters (1 per inference) 12 ξ paramaters (3 per content) 1 λ parameters (note the λ is calculated from the τ parameters) 20.09.2018 Dual-Source Model

Fits, Weighting (λ), & Form (τ) Model Fit was good (R² = .92), no differences between the conditions. weight λ (lambda) form τ (tau) ** 20.09.2018 Dual-Source Model

Knowledge Paramaters (Xsi) 3-way interaction of condition × content × inference is significant (p < .001), as well as the 2-way interaction of condition × inference (p < .001): knowledge ξ (xsi) 20.09.2018 Dual-Source Model

predator xsi girl xsi coke balloon 20.09.2018

Summary Suppression Effects The dual-source model shows a dissociation for different suppression effects (results replicate from a prestudy): Explicit disabling conditions undermine the credibility of the rule: lower weighting λ undermine the perceived form-based evidence for the valid conditional inferences: lower τ(MP) and τ(MT) slight effects on knowledge parameter for MP: lower ξ(predator, MP) and ξ(balloon, MP) Explicit alternative conditions undermine the perceived form-based evidence for the invalid conditional problems: lower τ(AC) and τ(DA) undermine the perceived knowledge-based evidence for all contents on the invalid inferences: lower ξ(C, AC) and ξ(C, DA) 20.09.2018 Dual-Source Model

Summary Modeling suppression effects with the dual-source model shows that both disabling conditions and alternative conditions suppress the subjective probability with which the relevant inferences seem warranted by the logical form of the inference (i.e., lower τ parameters). Disabling conditions suppress the credibility of the conditional rule (i.e., lower λ parameters). Alternative antecedents suppress the subjective certainty that an inference seems warranted by background knowledge (i.e., lower ξ parameters). 20.09.2018 Dual-Source Model

Outlook: Next Study It seems desirable to disentangle the contribution of the causal direction from the contribution of the conditional form of the presented inferences. In the so far used materials the conditionals were all of the form "If cause, then effect". How will the role of disablers and alternatives to the conditional rule be, for conditional of the form "If effect, then cause" (diagnostic conditionals)? To run such a study we need to develop new materials in which disablers and alternatives can be kept constant across the direction of the conditional rule. 20.09.2018 Dual-Source Model

Research Goals How do humans reason with probabilistic relations? Usage and integration of two different types of information: background knowledge and logical form. Different linguistic forms alter the subjective probability to which an inference is seen as logically warranted. The dual-source model allows us to estimate the contribution of those processes. 20.09.2018 Dual-Source Model

Thank you. 20.09.2018 Dual-Source Model

Consistency weighting The model is originally fitted with a linear weighting function truncated at 40 and 160 (Klauer et al. 2010). Original R² = .92 We fitted the model with a number of different weighting functions (without truncation) : linear (R² = .89) logarithmic (R² = .92) quadratic (R² = .89) The results hold between the different weighting functions 20.09.2018 Dual-Source Model

Control conditions Exp. 2 # condition Phase 1 Phase 2 1 baseline without rule with rule 2 disabler 3 alternative 4 5 6 7 nothing 8 9 20.09.2018 Dual-Source Model

Suppression Effects: Prestudy Participants work on original and converse conditional inferences (4 contents) in two phases: knowledge phase: without rule rule phase: with rule Three conditions: baseline (no explicit counterexamples), n = 26 disablers (three additional disablers), n = 27 alternatives (three additional alternatives), n = 24 20.09.2018 Dual-Source Model

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Fits, Weighting (λ), & Form (τ) Model fit is good (R² = .91), no difference between conditions *** 20.09.2018 Dual-Source Model

Knowledge Paramaters (Xsi) 3-way interaction of condition × content × inference is not significant (p = .07), only the 2-way interaction of condition × inference (p < .001): 20.09.2018 Dual-Source Model

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Suppression Effects - prestudy The dual-source model shows a dissociation for different suppression effects: Explicit disabling conditions undermine the credibility of the rule: lower λ undermine the perceived form-based evidence for the valid conditional inferences: lower τ(MP) and τ(MT) Explicit alternative conditions undermine the perceived form-based evidence for the invalid conditional inferences: lower τ(AC) and τ(DA) undermine the perceived knowledge-based evidence for all contents on the invalid inferences: lower ξ(C, AC) and ξ(C, DA) 20.09.2018 Dual-Source Model