Corrigendum

This article refers to:

Scope ambiguities and conditionals

Over, D., Douven, I., and Verbrugge, S. (2013). Scope ambiguities and conditionals. Thinking & Reasoning, 19(3–4), pp. 284–307. http://dx.doi.org/10.1080/13546783.2013.810172

When the above article was published online and in print Over, Douven, and Verbrugge (2013) (hereafter ODV) reported experimental findings on the scope of modal operators in indicative conditionals. As they noted, these findings are relevant to a prominent claim in philosophical logic that the probability of an indicative conditional equals the conditional probability, $\text{Pr}\left(\mathit{\text{if}}\mathit{\text{p}}\mathit{\text{then}}\mathit{\text{q}}\right)=\text{Pr}\left(q\right|p)$. This relation has been called the Equation because of its deep implications for theories of the conditional, and it has been supported as descriptive of people’s probability judgements for a range of conditionals in natural language. In the authors’ first two experiments, inference studies were used to confirm that people give wide scope to the probability operator in conditionals, and they explained how their findings provide further support for the Equation. In their Experiment 3, the authors asked whether other modal operators of relevance to the psychology of reasoning—for credibility, plausibility, and acceptability—had corresponding equations that would be confirmed in people’s judgements. The purposes of this note are to correct a coding error that was made in the analysis of Experiment 3 and to re-analyse its data.

Consider again the Equation. It is of the general form $X\left(\mathit{\text{if}}\mathit{\text{p}}\mathit{\text{then}}\mathit{\text{q}}\right)=X\left(q\right|p)$, where $X$ could one of the modals: Probability, credibility, plausibility, or acceptability. Does this general form hold when credibility, plausibility, or acceptability is substituted in it for $X$? If it does not, then that the Equation holds is a significant fact about people’s specific understanding of probability and conditionals. In particular, the Equation could not be an instance of a general trivialising tendency for people to give any modal $X$ supposedly narrow scope, or to attach it to the consequent of the conditional (as argued by Girotto & Johnson-Laird, 2004, 2010; see our paper for a reply). In addition to answering this trivialising argument, ODV reported materials effects. Due to space limitations, these materials were just flagged and not explored further. But after correcting the coding error, it is possible to say more about these effects and their importance.

ODV did note that the materials effects were not surprising given that the conditional sentences used in the experiment embodied different types of inferential connections between antecedent and consequent. ODV referred to Douven and Verbrugge (2010), which introduced a distinction between three types of conditionals: Deductive inferential (DI) conditionals, in which the consequent is deductively inferred from the antecedent (given background information); inductive inferential (II) conditionals, in which (roughly) the consequent follows on statistical grounds from the antecedent; and abductive inferential (AI) conditionals, in which the consequent follows via an inference to the best explanation from the antecedent. Each of these types was equally represented in the set of materials used in Experiment 3 of ODV, but this typology was not analysed in that paper. The following makes good this omission.

According to the trivialising argument, all modal operators are given narrow scope regardless of their place in the sentence, and there should be no order effects. In contrast, we predicted differential treatment of modal operators, leading to an interaction between order and operator. To test this prediction, we ran two analyses, both presented in Table 1. Where sphericity was violated, we used the Greenhouse–Geisser correction. First, we re-ran the analysis reported in ODV, Experiment 3, with the amended data: A $15\times 4\times 2$ mixed analysis of variance (ANOVA) with sentence as a within-participants variable, and operator (probability/credibility/plausibility/acceptability) and order (conditional $X$/$X$ of conditional) as between-participants variables. Similar to the pattern reported in ODV, the main effects of order, operator, and sentence were all significant (respectively, $F(1,489)=33.96$, MSE = 5.06, $p<.0001$, ${\eta }_p^2=.065$; $F(3,489)=22.48$, MSE = 5.06, $p<.0001$, ${\eta }_p^2=.121$; and $F(11,5496)=129.6$, MSE = 1.14, $p<.001$, ${\eta }_p^2=.209$); sentence interacted with operator ($F(33,5496)=6.70$, MSE=1.14, $p<.001$, ${\eta }_p^2=.039$) and with order ($F(11,5496)=20.3$, MSE = 1.14, $p<.001$, ${\eta }_p^2=.025$), indicating materials effects. No other effects were significant. Specifically, the two-way interaction between order and operator, and the three-way interaction between sentence, order, and operator, reported significant in ODV, were not significant for the amended dataset. This failed to support our interaction hypothesis, although the main effect of order is still incompatible with the trivialising hypothesis.

Table 1 Mean endorsement as a function of operator, order, and sentence type

	DI conditionals	II conditionals		AI conditionals		All conditionals
	$M$ (SE)	$M$ (SE)	$d$	$M$ (SE)	$d$	$M$ (SE)
Conditional probability	6.29 (0.08)	5.87 (0.10)	0.49	5.60 (0.09)	0.55	5.92 (0.07)
Probability of conditional	6.29 (0.08)	5.55 (0.10)		5.29 (0.09)		5.71 (0.07)
Conditional credibility	6.13 (0.08)	5.57 (0.10)	1.10	5.30 (0.09)		5.67 (0.07)
Credibility of conditional	6.24 (0.09)	4.67 (0.11)		5.01 (0.10)		5.31(0.08)
Conditional plausibility	6.29 (0.09)	5.98 (0.10)	0.75	5.74 (0.09)	0.58	6.00 (0.08)
Plausibility of conditional	6.37 (0.08)	5.50 (0.10)		5.39 (0.09)		5.75 (0.07)
Conditional acceptability	6.33 (0.09)	5.04 (0.11)	0.72	5.34 (0.10)		5.57 (0.08)
Acceptability of conditional	6.22 (0.08)	4.30 (0.10)		4.99 (0.09)		5.17 (0.07)

To explore the interaction hypothesis more systematically, we analysed the data with conditional type (in the sense of Douven & Verbrugge, 2010)¹ instead of sentence. This yielded a $3\times 4\times 2$ mixed ANOVA with conditional type (DI, II, AI) as within-participants variable, and operator (probability/credibility/plausibility/acceptability) and order (conditional $X$/$X$ of conditional) as between-participants variables. The main effects of order, operator, and conditional type were all significant (respectively, $F(1,489)=33.96$, MSE = 1.01, $p<.0001$, ${\eta }_p^2=.065$; $F(3,489)=22.48$, MSE = 1.01, $p<.0001$, ${\eta }_p^2=.121$; $F(2,939)=475.9$, $p<.001$, ${\eta }_p^2=.493$). Conditional type interacted with operator ($F(6,939)=24.8$, $p<.001$, ${\eta }_p^2=.132$) and with order ($F(6,939)=24.8$, $p<.001$, ${\eta }_p^2=.132$), both explained by a three-way interaction between conditional type, order, and operator ($F(6,939)=2.7$, $p=.014$, ${\eta }_p^2=.016$). We found significant order effects in II conditionals for all four operators, and in AI conditionals for the probability and plausibility operators, all with moderate to large effect sizes ($t\left(124\right)⩾2.7$, $p⩽.007$, $d⩾.49$). The order effect for the probability operator must be studied further, if only because Douven and Verbrugge (2010) found no difference for the probability operator in any type of conditional. We also note as worth further study the large order effect for credibility in II conditionals. There were no other significant order effects.

ODV examined the claims that there is a general tendency to give modal operators narrow scope in conditionals, and that this tendency makes, for example, the probability of a conditional trivially equivalent to the probability of its consequent given its antecedent. Such a general trivialising tendency should be found for all types of conditional and other modal operators. We compared a difference in order for the modal operators of probability, credibility, plausibility, and acceptability, predicting—and finding—an interaction between order and operator. After amending for a coding error, this interaction was lost in the re-analysis. However, the follow-up analysis, which included the type of conditional, revealed a conceptually similar significant three-way interaction between order, operator, and type of conditional. This interaction pattern is incompatible with the trivialising claim.²

Notes

¹ In the order in which they appear in Appendix B of ODV, sentences 1–5 are DI conditionals, sentences 6–10 are II conditionals, and sentences 11–15 are AI conditionals.

² We owe a great debt to Shira Elqayam for all her help with the re-analysis.