Comparing the Logistic Regression Model and the Dual Source Model

The purpose of this section is to consider some of the ways in which the logistic regression model could itself be used directly to model the effect of the presence of the rule found in Klauer et al. (2010). That is, we will now consider the possibility of using the logistic regression equations as not only providing a model of the content mode of reasoning in equation (2) but as a direct competitor of equation (2).

Before we begin, we must notice that since we already have the results, methodologically we are in a peculiar situation, where we are designing a model to retrodict known findings rather than predicting them in advance. It is always easier to modify a model to existing findings. Hence, such “predictions” should not be allowed to count much in favor of the model, unless they are based on hard to vary aspects of the underlying theory (as opposed to be based on auxiliary hypotheses).

On the face of it, the main finding in Klauer et al. (2010) was that the endorsement rates of MP and MT were boosted in the presence of the rule. The logistic regression model would be able to account for these findings, if we assume that the effect of introducing the rule is not merely to raise b₁ (i.e. to increase expectations about the epistemic relevance), but specifically to increase the perceived sufficiency (i.e. increasing b₁ and decreasing b₀^*). For in section III 3.5, the latter has been shown to have the effect of increasing MP and MT. However, whereas the former effect is straightforwardly motivated by the ranking-theoretic approach to conditionals, it might seem that the latter lacks a principled justification.

We will now show that this is not the case. In Spohn (forthcoming) it is said that ‘the circumstances are such that’ reading of conditionals is the most interesting, expressive

121

function of the 8 main expressive functions he analyzes. As it turns out, this reading also lays the foundation for his account of causal conditionals, which happens to be the kind of stimulus material used in Klauer et al. ( 010). The idea behind Spohn’s ( 01 a, forthcoming) analysis of ‘the circumstances are such that’ reading of conditionals is that the positive, epistemic relevance of the antecedent for the consequent is itself based on a condition, which is assumed to obtain, when the conditional is asserted. As a result, the speaker is portrayed as expressing an unconditional belief about the fulfillment of this condition (in addition to the features of his conditional beliefs expressed by the conditional itself) about which there can be a factual dispute. In the case of causal conditionals, this takes the form of having an unconditional belief about the actual history being such that the antecedent was somehow required to bring about the consequent. Epistemically, what this means is that the positive relevance of the antecedent for the consequent is itself grounded in the actual history up until the occurrence of the consequent at time t¹ (where we have excluded the occurrence of the antecedent at time t). So according to this analysis, causes can be thought of as “reasons given the history” (Spohn, forthcoming).

Paraphrased in terms of the logistic regression model, we can understand ‘history not being as it actually was up until the effect occurred’ as a disabler for the positive relevance of the antecedent for the consequent in causal conditionals. Accordingly, ‘history being as it actually was up until the effect occurred’ can be analyzed as the non-occurrence of a disabler, which we might demarcate terminologically as an enabling condition. Hence, the unconditional belief that causal conditionals express, according to Spohn’s analysis, is that the enabling condition for the epistemic relevance of the antecedent for the consequent is fulfilled. And as we have already seen in chapter III how the absence of disablers has the effect of increasing MP and MT, it turns out that we are able to account for the main finding of Klauer et al. (2010) after all, once we apply Spohn’s account of causal conditionals to their stimulus material (which consisted of … causal conditionals). That is, the conditional rule raises the perceived sufficiency of the antecedent for the consequent in virtue of conveying the information that an enabling condition for the positive, epistemic relevance of the antecedent for the consequent is fulfilled. Hence, it turned out to be possible to give a principled account of the main effect in that study after all.

122

However, at this point, we need to return to a possible objection (cf. endnote 65). In chapter III, it was said that the effect of the absence of disablers is to lower P(X=1, Y=0) and the effect of the absence of alternative antecedents is to lower P(X=0, Y=1). However, more changes will have to be made to the probability distribution to ensure that we end up with a probability measure in the end, where the probabilities sum up to one.⁸⁰ The most natural changes are that the downward adjustment of P(X=1, Y=0) is accompanied by an upward adjustment of P(X=1, Y=1) in the absence of disablers and that the downward adjustment of P(X=0, Y=1) is accompanied by an upward adjustment of P(X=0, Y=0) in the absence of alternative antecedents.

This extension does not affect the predictions derived in chapter III. But it has implications for what further predictions can be derived on the basis of the logistic regression model. In particular, it implies that the increases to the perceived sufficiency affected by the absence of the disablers will also raise AC in addition to MP and MT and that the increase to MP should be larger than the increase to MT as: (a) P(X=1, Y=1) figures in the equation for AC, and (b) MP will be affected by both the changes to P(X=1, Y=1) and P(X=1, Y=0).

To make this more perspicuous, the equations for the parameters from chapter III have been inserted in the logistic regression model in the equations below (and terms that cancel out have been removed):

(7)

(8)

(9)

(10)

Keeping this prediction in mind, we now inspect the results cited in Klauer et al.

(2010) for a second time. For the results reported in their figure 1 and 3, it holds that MP is

123

boosted more than MT and that AC is boosted in the presence of the rule. This pattern fits with our predictions. For the results reported as figure 4 and 5 in Klauer et al. (2010), we see essentially the same pattern with the only difference being that there are also now some minor increases to DA as well. As these changes to DA were not consistent across all experiments, it appears in comparison to be a more negligible effect, which should be accounted for in terms of auxiliary hypotheses.

The following candidates suggest themselves: (i) that the participants sometimes fail to take alternative antecedents into account, because they are engaging in what has been called closed-world reasoning (cf. Beller & Kuhnmünch, 2007), where they assume that they have taken all the relevant factors into account, when they have processed both the major and the minor premise in evaluating the conclusion of MP, MT, AC, and DA. Or (ii) that the participants sometimes fail to take alternative antecedents into account due to cognitive limitations. Generating alternative antecedents, and taking their contribution to the problem at hand into account, is itself a process that takes up cognitive resources. So perhaps the participants are already so preoccupied with processing the major and minor premises in the conditional inference task that they simply fail to invest further cognitive resources in initiating the former process (although they could easily be brought to realize that they should have known better). The second auxiliary hypothesis is plausible, because it is known from a number of experiments that ordinary subjects tend to be ‘cognitive misers’ that will substitute resourceful analytical thinking for rough heuristic approximation whenever they can (Stanovich, 2011: 21, 29, Kahneman, 2012: ch. 5). But clearly, the two auxiliary hypotheses can also be combined.

However, in accordance with our introductory remarks to this section, such explanations that take a recourse to auxiliary hypotheses should not be taken to count much in favor of the ranking-theoretic approach to conditionals (what they do, if successful, is rather to show that the data need not be considered as evidence against the latter account). Yet, what does count in favor of our model is its ability to retrodict the pattern that MP is boosted more than MT and that AC is boosted in the presence of the rule, insofar as it has received a principled justification.

124