Deductive vs. Inductive Instructions: Evaluating the Predictive Powers of Cognitive Models for Conditional Reasoning

Abstract

Throughout the years, the question how humans reason with conditionals has been extensively researched by various disciplines due to its importance not only in science, but also our everyday life. A vast amount of cognitive models have been developed in order to get a better insight into how individuals interpret and reason with 'If'. Todorovikj and Ragni [21] proposed a predictive modeling testing paradigm for probabilistic conditional reasoning models. Instead of evaluating models based on their ability to fit aggregate data, the focus was shifted to the individual by challenging cognitive models to predict a participant's answer on a scale from 0 to 100. In this work, we continue the challenge of predicting probabilistic endorsements by taking data from Singmann and Klauer [19] who examine the influence of deductive vs. inductive instructions. We evaluate an established probabilistic cognitive model by Oaksford and Chater [11] and a proposed probabilistic approach based on mental models [21, 20]. Moreover, we take Oberauer's formalizations [12] of the Mental Model Theory [5] and the Suppositional Theory [4] and adapt them to represent probabilistic data in this new predictive testing paradigm. The models' parameter values are evaluated to examine their ability to represent the influence of instruction and content e↵ects.

Keywords

• Conditional Reasoning
• Predictive Modeling
• Individuals
• Instruction E↵ect
• Content E↵ect

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On a daily basis, we gather knowledge and information about ourselves and our surroundings that we use to infer conclusions by processes of reasoning. For many years, scientists work towards understanding how we reach these conclusions and what cognitive processes lead to them. In the 60s, a deductive path has been followed, assuming that logic is the basis for reasoning [ 4 ]. However, it has been repeatedly shown that humans are not always logical beings, the content of a premise often makes us question its certainty in a way that standard deductive logic cannot grasp [ 19, 8 ]. Acknowledging these uncertainties, interpreting them as probabilities and switching the focus on the inductive strength Copyright c 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). of arguments creates a major shift leading to a new Bayesian paradigm [ 13 ]. Proponents of both paradigms have developed a number of cognitive models and experiments that represent and examine reasoning mechanisms corresponding to the paradigm’s theories and assumptions. Singmann and Klauer [ 19 ] focus on the conditional reasoning domain. Through two experiments they provide evidence for a dissociation of the two reasoning modes: deductive and inductive and show the need for theories to explicitly account for the e↵ect of instruction.

Conditionals are statements of the form “If X then Y” (also written X ! Y, where X is the antecedent and Y the consequent ) that often describe a causal relationship between two propositions X and Y. Given a conditional (major premise) and a state of a proposition (minor premise), a conclusion can be inferred about the state of the other proposition. We consider the four major inference forms modus ponens (MP), modus tollens (MT), arming the consequent (AC) and denying the antecedent (DA), as shown in Table 1.

In causal conditionals the necessity and suciency of X for Y can be questioned and manipulated through two types of counterexamples: alternatives and disablers. Alternatives enable Y to occur even if X hasn’t and decrease the perceived necessity. On the other hand, disablers prevent Y from being true even if X is, decreasing the perceived suciency. Conditionals whose content has many alternatives but few disablers associated with it are called prological, ones with few alternatives but many disablers are counterlogical, and if a conditional has both many alternatives and disablers then it is called neutral [ 19 ].

In previous work [ 21 ], we evaluated conditional models in a new predictive evaluation setup, motivated by Riesterer et al.’s approach for syllogistic reasoning [ 16 ]. We showed that it is possible for models to predict a probabilistic endorsement of a conclusion in the range 0-100, which is more challenging than a dichotomous ‘yes’ or ‘no’ answer. The focus was to predict an individual’s answer, increasing the diculty due to the varying subjective influence of disablers and alternatives. Additionally we proposed a probabilistic approach based on mental models, ✏-MMT, which showed to be a valuable competitor among other Bayesian models.

Our main goal is to once again challenge the predictive capabilities of cognitive models, this time on Singman and Klauer’s data sets [ 19 ]. We take formalizations [ 12 ] of the Mental Model Theory [ 5, 18 ] and Suppositional Theory [ 4 ], adapt them to represent probabilistic data and compare them to Oaksford and Chater’s Bayesian model [ 11, 9 ] and ✏-MMT [ 21, 20 ]. Moreover, we will analyze whether the models’ parameter values can account for the instruction and content e↵ects. 2

Data and Cognitive Models We consider experimental data presented and analyzed by Singmann and Klauer [ 19 ]. There are two experiments, Exp. 1 and Exp. 2, aiming to establish a double dissociation between deductive and inductive instructions when validity and plausibility of the conditional problems are pitted against each other.

Under deductive instructions participants were asked to assume the truth of the premises and disregard background knowledge when judging the conclusion’s validity. Under inductive instructions, they were encouraged to use background knowledge when judging the conclusion’s probability. In both experiments participants were provided with prological and counterlogical conditionals. Additionally, in Exp. 2 they were also given neutral conditionals. 40 participants took part in Exp. 1, 20 received deductive instructions and 20 inductive. In Exp. 2, 27 participants received deductive instructions and 28 inductive, making it a total of 55 participants. In both experiments, they provided endorsements in the range 0-100 for the four inference forms for conditionals with everyday contents. Table 2 shows content examples under deductive and inductive instructions. The Mental Model Theory (MMT) assumes that when individuals are presented with some information, they build a mental representation of it using mental models. They aim to reach a conclusion based on the maintained information, and often, individuals would engage in a search for counterexamples to the conclusion. If their search is successful, they would no longer accept the conclusion [ 7 ]. A mental model consists of the truth states of the propositions in the premise. Given a conditional premise “If X then Y”, the initial mental model that an individual would construct is the one where both propositions are true, X Y.

The MMT assumes that once the initial model is created it triggers the recollection of relevant facts and knowledge [ 5 ]. Those facts can either serve as evidence that the initial model is correct or will stimulate a search for alternatives and lead to extending the mental model representation in a second process. The extended mental model representation is also called a fleshed-out representation, as shown in Table 3. It contains models that describe cases where X is false (written as ¬X). The fleshed-out representation consists of all the possible combinations of truth-values for the propositions X and Y for which the conditional “If X then Y” is true. Johnson-Laird and Byrne call representing only what is true and not false the principle of truth [ 6 ]. This coincides with the material implication definition which is the leading interpretation of conditionals in the deductive paradigm [ 2 ].

Later on we present and use a formalization of the MMT that is based on equations provided by Schroyens et al. [ 18 ] who revise the theory as presented by Johnson-Laird and Byrne [ 5 ] and present an alternative model within the mental models approach, focusing on the stage where individuals validate the conclusion based on their mental model representation. They assume that when looking for counterexamples individuals do not construct all possible alternative models but rather aim for the model that would falsify the inference. Based on the original model’s theory, when looking for counterexamples individuals only test whether alternative models are consistent with the premises. Schroyens et al. [ 18 ] on the other hand, take background knowledge about the content into consideration and suppose that counterexamples are retrieved from longterm memory, which is where this theory stops being a theory of deductive reasoning [ 12 ]. With this modification the material implication interpretation of conditionals within the theory is abandoned because now the X ¬Y model can actually be retrieved from long-term memory and become a part of the mental model representation. That leads to the possibility of refuting the logically valid MP and MT inferences, which is a consequence of the presence of disablers as described by the Suppression e↵ect [ 1 ].

Evans [ 3 ] has criticized the MMT for its lack of directionality. Specifically, comparing the di↵erences between reasoning problems involving “If X then Y” and the ones involving “X only if Y”. Namely, people are more likely to draw backward inferences (inferences from Y or ¬Y as a minor premise) in the “only if” case and they are more likely to draw forward inferences (inferences from X or ¬X as a minor premise) in the case of “if...then”. Assuming a directionality from the antecedent variable to the consequent variable, the only reachable conclusions are the ones following that direction. For example, having only the model X Y in the mental representation supports concluding Y from X (MP), but not X from Y (AC). 2.2

Suppositional Theory

The Suppositional Theory assumes that two systems of reasoning are used in order to make a conclusion for a given reasoning problem, called System 1 and System 2. System 1 describes the heuristic or pragmatic inferences, whereas System 2 is dedicated to abstract rule-based reasoning [ 4 ]. The influence of each system depends from one individual to another in terms of belief-based vs. logicbased responses.

Evans and Over [ 4 ] discuss early advancements towards the invalidity of the material implication interpretation of indicative conditionals, as primarily done by Ramsey [ 15 ]. He suggests that when people reason about a conditional “If X then Y”, they are judging the probability of Y, supposing X is true, i.e. P(Y|X), widely known as The Ramsey Test, which leads to the individual’s degree of belief in the conditional itself. Building up on this, System 1 is capable of only generating conclusions that follow directly from an individual’s belief in the conditional, leading to accepting only MP inferences, unless additional conditional premises are added through pragmatic implicature and the MP inference schemata is applied to them.

System 2, on the other hand, can accept MT by a suppositional line of proof [ 12 ]. The suppositional proof starts by supposing that X is true, which combined with the given conditional rule, “If X then Y”, leads to the conclusion that Y is true (MP inference). However, this contradicts the minor premise of MT, ¬Y, therefore the supposition must be false, so ¬X must be true. This leads to the acceptance of the MT inference form. Starting the suppositional proof with Y, combined with the converse, “If Y then X” (assuming that the converse of the conditional has been added by pragmatic implicature), leads to accepting the DA inference form. 2.3

Formalizations

Oberauer [ 12 ] provides formalizations of the MMT and Suppositional Theory as multinomial processing trees (MPTs). The MPT can be interpreted as a binary decision diagram with parameters on the edges. In the case of MMT (Fig. 1), the decisions are whether a human reasoner will add a mental model to their representation. For the Suppositional Theory (Fig. 2), the parameters describe the degree of belief in a conditional and/or its converses and inverses.

The leafs represent combinations of inference forms that are accepted if that specific path has been taken. It can be noted that there are combinations that the theories cannot express. In our setup we are not interested in the acceptance, but rather we aim to model individuals who provide probabilistic endorsements in the range 0-100. Towards that goal, we derived expressions from the MPTs for each inference form separately, by summing up and simplifying the probabilities of all possible paths leading to accepting said inference form.

For the MMT, Oberauer [ 12 ] presents three di↵erent formalization variants, and two variants for the Suppositional Theory. For each theory, we focus only on the variant that is best performing, so we adapted all five variants to model probabilistic data, examined their predictive performance and picked the best ones. In the following, we introduce the detailed technical background and endorsement expression derivation solely for them.

Mental Model Theory Formalizations Oberauer [ 12 ] presents three formalization variants for the MMT: original, deductive and with directionality. The best performing variant is MMT with Directionality.

The original variant is built directly on equations provided by Schroyens et al. [ 18 ]. The deductive variant is a special case of the original one, where the model X¬Y is impossible to be added to the mental representation.

e add X¬Y add ¬XY

a 1 e don’t add

X¬Y add ¬X¬Y

f 1 a add X¬Y e don’t add ¬XY

XY 1 e don’t add

X¬Y r 1 f

a add ¬XY e 1 e add don’t add X¬Y X¬Y

X!Y don’t add ¬X¬Y add X¬Y 1 a e don’t add ¬XY 1 r 1 e don’t add

X¬Y

None d 1 d d 1 d d 1 d None reverse ¬reverse reverse ¬reverse reverse ¬reverse None MP

d 1 d d 1 d reverse ¬reverse reverse ¬reverse MP, MT MP

AC, DA DA

All 4 MP, DA

AC

None MP, AC MP The parameters describe the probabilities of adding models to the representation. r is the probability of adding the model X Y, f for the model ¬X¬Y, a for ¬X Y, e for X ¬Y. Additionally, the parameter d describes the probability of reversing the directionality.

In total, MMT with Directionality needs five parameters to model one task.

Suppositional Theory Formalizations Evans and Over [4] do not explicitly

specify how the two systems interact, therefore Oberauer [ 12 ] provides two variants – an Exclusive and a Sequential one. The best performing variant is the Sequential one.

The Exclusive variant assumes that either System 1 or System 2 is used exclusively when reasoning.

The Sequential variant (Fig. 2) assumes that the two systems work sequentially. After the individual determines their degree of belief in the conditional X!Y, System 1 takes over and has the possibility of adding the converse, Y!X, and inverse, ¬X! ¬Y by pragmatic implicature. By combining the two, it can arrive to the contraposition, ¬Y! ¬X. Once System 1 is done with adding conditionals, it derives all the conclusions following from them. Next, System 2 takes the conditionals System 1 has added and with a certain probability and applies the procedure for suppositional proof. Two di↵erent suppositional proofs can take place. One supposes the truth of X and the original conditional and the other supposes the truth of Y and the converse.

We derived the following inference form endorsement expressions for Suppositional Sequential based on the MPT, shown in Fig. 2:

MP: b

DA: b · (c · s⇤ · (1

i) + i) AC: b · c

MT: b · ((1 s) · (2 · c · i c2 · i2) + s) The parameter b describes the belief in the conditional X!Y, c is the probability of the converse to be added (Y!X) and i, the inverse (¬X! ¬Y). s is the probability of the suppositional proof using X and the original conditional to succeed and s⇤ for the proof using Y and the converse.

In total, Suppositional Sequential needs five parameters to model one task.

¬X! ¬Y i + (1 i) · c ¬Y! ¬X (1 c) · (1 i) ¬Y!no¬X s 1 s All 4 Supp. X no Supp. X i

Y!X

c 1 i

no 1 i ¬X! ¬Y

X!Y i no 1 s ¬Y! ¬X s no Supp. X 1 s⇤ s⇤ All 4 MP, AC, DA no Supp.Y Supp.Y

Supp. X 1 s⇤ s⇤ no Supp. Y Supp. Y MP, AC, MT All 4 MP, AC MP, AC, DA MP, AC, MT All 4 b

1 c 1 s¬X!no¬Ys no Supp. X Supp. X

Reasoning

no 1 i Y!X

1 b i 1 c ¬X! ¬Y c 1 s⇤¬Y! ¬Xs⇤ 1 s¬Y!no¬Xs ¬Y! ¬X no Supp. Y Supp. Y MP MP, MT no Supp. X Supp. X MP, DA, MT no X!Y

None MP, DA MP, DA, MT 2.4

Oaksford-Chater Probabilistic Model

Oaksford and Chater [ 11, 9, 10 ] propose a probabilistic model for conditional reasoning. By using a 2 ⇥ 2 contingency table (Table 4), they represent conditional rules, where a = P(X) and b = P(Y), probabilities of the antecedent and consequent, respectively and ✏ = P(¬Y|X) is the exception parameter.

This model belongs to the new Bayesian paradigm – it assumes that inference form endorsement is proportional to the conditional probability of the conclusion given the minor premise. So, from Table 4, they derived expressions for the inference form endorsements as follows:

MP: P(Y|X) = 1 AC: P(X|Y) = ✏ b a(1 ✏)

The ✏-MMT: Mental Models and Probabilities In previous work [ 20, 21 ] we proposed a probabilistic cognitive model for conditional reasoning that is based on a mental model approach and showed that it is a valuable competitor among Bayesian models. We represent conditionals using mental models, interpret them as possible worlds, allowing for an application of a probabilistic assignment as described by ✏-semantics [ 14 ].

✏-semantics is a ‘formal framework for belief revision’, as introduced by Pearl [ 14 ]. Belief statements are interpreted as high probability statements and newly available evidence leads to belief revision. The following definition [ 14 ] lays the basis for the probabilistic aspect of ✏-MMT:

Let L be the language of propositional formulas, and let a truth-valuation for L be a function t, that maps the sentences in L to the set {1, 0}, (1 for ‘true’, 0 for ‘false’). To define a probability assignment over the sentences in L, we regard each truth valuation t as a world w and define P(w) such that Pw P(w) = 1. This assigns a probability measure to each sentence l of L. ✏-MMT takes the conditional’s mental model representation of all propositions’ truth state combinations, meaning that the material implication interpretation is abandoned by allowing the X ¬Y model to exist. From now on, we will refer to them as possible worlds. Given a conditional premise “If X then Y”, all possible worlds describing the states of X and Y along with the corresponding probability values assigned to them are shown in Table 5.

We follow the approach of previous accounts in the Bayesian paradigm [ 11 ], and interpret inference form endorsements as a conditional probability of the conclusion given the minor premise.

By applying the definition of conditional probability, as shown in Eq. 1, to Table 5, we derive the following inference form endorsement expressions: MP: P(Y|X) = AC: P(X|Y) =

p4 p3 + p4

p4 p2 + p4

Benchmark and Predictive Modeling Results We implemented a benchmark within the framework CCOBRA. The main goal is to examine the cognitive models’ ability to predict a value in the range 0100, by taking into consideration how close the model’s prediction is to the individual’s true answer. We evaluate a model’s performance by calculating the mean individual absolute di↵erence between a prediction and the corresponding true answer. A lower value indicates better performance.

The cognitive models are first exposed to existing data – a training set, from which it can learn about the individual’s reasoning behavior. Then, the model is challenged with predicting answers in unseen data – a test set. Here, we use Singmann et al.’s [ 19 ] experimental data for both training and testing. In this (1) case, the CCOBRA framework uses a leave-one-out cross-validation method – models are fit on every participant, except the one whose answers should be predicted. The same is done for each participant. The models are fit to the training data by optimizing their parameter values in order to minimize the absolute di↵erence between the prediction and the true answer. That was done using Python’s scipy.optimize.minimize3 with the method Sequential Least Squares Programming (SLSQP), which allows for constrained minimization.

In order to set a performance lower bound, our benchmark has a Random model, which simply gives a random value between 0-100 as a prediction. Since the models do not adapt to an individual, the upper bound is set by an empirical Most Frequent Answer (MFA) model, which gives the median of the responses as a prediction.

The results are shown in Figure 4. In the deductive case, the Suppositional Sequential model has an outstanding performance for both experiments (Exp. 1: 26.72, Exp. 2: 27.98), whereas the other models’ perform similarly, reaching an average absolute di↵erence of 30.57 (Exp. 1) and 29.70 (Exp. 2). In the inductive case, however, the Bayesian Oaksford-Chater model takes the lead with an absolute di↵erence of 16.08 in Exp. 1 and 17.90 in Exp. 2. Once again, a similar performance among the other models can be seen, having an average absolute di↵erence of 16.82 (Exp. 1) and 18.57 (Exp. 2).

We have two mental model based approaches, a probabilistic endorsement adaptation of Oberauer’s [ 12 ] formalization MMT with Directionality, and our proposed model ✏-MMT that follows the Bayesian paradigm. ✏-MMT outperforms MMT with Directionality in Exp. 1, but performs slightly worse in Exp. 2. Though, the di↵erence between the two models is no more than 0.55 in any scenario. It is noteworthy that the total number of free parameters that each model needs for one task is di↵erent. Here, ✏-MMT takes the lead with only three parameters, in contrast to five for MMT with Directionality.

We continue our analysis as we look into the models’ parameter values to learn more about the di↵erence between deductive and inductive instructions. For that purpose the main focus is on ✏-MMT’s p2 and p3, Oaksford-Chater’s ✏ and MMT with Direcionality’s a and e parameters. Their values for each instruction and conditional type are shown in Table 6.

Material implication states that a conditional “If X then Y” is false when X is true and Y is false. These three models are able to recognize this state through their parameters for counterlogical conditionals through low values under deductive instructions. More specifically, ✏-MMT’s p3 parameter describes the belief in the world ! 3, where X = 1 and Y = 0. Oaksford-Chater’s ✏ parameter describes the conditional probability P(¬Y|X). MMT with Directionality’s e parameter describes the probability of the mental model X ¬Y to be added to the mental representation. In both experiments, these parameters have an increase in value for individuals that had inductive instructions showing the material implication interpretation under deductive instructions. 3 https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize. minimize.html Exp.1: Deductive Exp.1: Inductive Exp.2: Deductive Exp.2: Inductive

Another point of interest is the influence of alternatives and disablers, under inductive instructions, when background knowledge is taken into consideration. In the following, we use X for the antecedent and Y for the consequent (“If X then Y”). The presence of many alternatives means that Y occurs even if X has not. ✏-MMT’s p2 parameter describes the belief in that world (! 2) and MMT with Directionality’s a parameter describes the probability of the corresponding mental model (¬X Y) to be added to the representation. In both experiments, for prological conditionals the values of these parameters are higher in comparison to counterlogical, reflecting the influence of alternatives. On the other hand, the presence of many disablers means that even if X has happened, Y has not. ✏MMT’s p3 parameter describes the belief in that world (! 3), Oaksford-Chater’s ✏ parameter describes the conditional probability for that scenario, P(¬Y|X) and MMT with Directionality’s e parameter describes the probability of the corresponding mental model (X ¬Y) to be added to the representation. Similarly, in both experiments for counterlogical conditionals, these parameter values are higher in comparison to prological ones, pointing to the e↵ect of disablers. Research approaches have been divided between emphasizing logical validity and focusing on content and background knowledge. Those are di↵erent types of reasoning that one single mechanism cannot cover, leading to arguments that one of the two approaches is wrong [ 19, 9, 17 ]. Singmann and Klauer [ 19 ] show a double dissociation of the two types of instructions. To further analyze this, we adapted MPT formalizations of cognitive theories such that they can now represent experimental data with probabilistic endorsements. Then, we compared 0.34 0.08 0.24 0.02 0.15 0.37 0.44 0.12 their predictive performance to an established Bayesian model and a newly proposed probabilistic approach based on mental models. We showed that after exposing conditional models to a training data set, they are capable of generating predictions on unseen data in the range 0-100. In previous work [ 21 ] we considered experiments where participants received inductive instructions and the models reached a mean deviation of ca. 17, which is what we encounter now as well. However, we learn that their performance drops to ca. 29 when dealing with deductive instructions. An important finding is that the Suppositional Sequential formalization stands out under deductive instructions, while the performance of the Oaksford-Chater model exceeds that of the others under inductive instructions. With this we show that only one single theory can not explain all reasoning process under all circumstances. The di↵erence in performance lays basis for future research to examine the di↵erence in reasoning processes under various conditions through cognitive modeling. Additionally, we showed that ✏MMT, Oaksford-Chater and MMT with Directionality are capable to account for the e↵ect of instruction, by showing the material implication interpretation under deductive instructions. More detailed research is necessary for Suppositional Sequential, especially given that it excels for deductive instructions. Additionally, we confirmed our finding [ 21 ] that ✏-MMT can successfully account for the influence of disablers and alternatives when background knowledge is taken into consideration. Both ✏-MMT and MMT with Directionality are based on mental models, have a similar performance and account for instruction and content effect in the same way through their parameter values. The advantage of ✏-MMT is that it needs only three free parameters in contrast to five. That indicates that the same information can be represented by ✏-MMT in a less complex and computationally more ecient way, while still preserving the same psychological interpretation and providing valuable insight. As mentioned before, the MPTs for MMT and the Suppositional Theory show that certain combinations of inference forms are lacking from the trees and cannot be accepted according to the theories. Even though we are not looking into combinations of accepted inferences, but rather their separate probabilistic endorsements, the parameter values still depend on the combinations. In future work we would look into how influential this is on the model’s predictive performance. Namely, is the lack of combinations hurting the model, or is it accurately describing human reasoning? Furthermore, an even more exciting next step would be to challenge the models to adapt to individuals as well. With this we continue the thrilling path of challenging the predictive capabilities of models and gaining insight about the individual reasoning processes under various e↵ects.

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