EXPRESS: Sometimes nothing is simply nothing: Automatic processing of empty sets

Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In the present study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, while no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as “zero”, but rather as “nothing”, when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

Modeling the latent structure of individual differences in the numerical size-congruity effect

When people are asked to choose the physically larger of a pair of numerals, they are often slower when relative physical size is incongruent with numerical magnitude. This size-congruity effect is usually assumed as evidence for automatic activation of numerical magnitude. In this paper, we apply the methods of Haaf and Rouder (2017) to look at the size-congruity effect through the lens of individual differences. Here, we simply ask whether everyone exhibits the effect. We develop a class of hierarchical Bayesian mixed models with varying levels of constraint on the individual size- congruity effects. The models are then compared via Bayes factors, telling us which model best predicts the observed data. We then apply this modeling technique to three data sets. In all three data sets, the winning model was one in which the size-congruity effect was constrained to be positive. This indicates that, at least in a physical comparison task with numerals, everyone exhibits a positive size-congruity effect. We discuss these results in the context of measurement fidelity and theory-building in numerical cognition.

Modeling response times in the size-congruity effect: Early versus late interaction

The size-congruity effect occurs when numerical magnitude interferes with judgments of physical size. Various accounts propose that this interference is either encoding-related or decision-related, though at present a clear consensus is lacking. In our study, we administered a single-digit physical comparison task (i.e., which digit is physically larger?) and applied four different mathematical models (ex-Gaussian, ex-Wald, shifted Wald and EZ-diffusion) to the observed response times. The aim of this modeling was to index the underlying cognitive processes via estimates of drift rate, response threshold, and non-decision time. The collection of estimates for each individual was then subjected to Bayesian paired samples t-tests. We found that the drift rate for incongruent trials was smaller than for congruent trials, indicating that congruent trials had a faster rate of information uptake. The response threshold for incongruent trials was generally larger than for congruent trials, indicating that for incongruent trials more information needed to be accumulated before a response could be initiated. Critically, we found evidence of an invariance in non-decision times between incongruent and congruent trials. This combination of results provides support for a late interaction account of the size-congruity effect, shedding further light onto models of decision making in number processing.

Two Attributes of Number Meaning

Many studies demonstrated interactions between number processing and either spatial codes (effects of spatial-numerical associations) or visual size-related codes (size-congruity effect). However, the interrelatedness of these two number couplings is still unclear. The present study examines the simultaneous occurrence of space- and size-numerical congruency effects and their interactions both within and across trials. In a magnitude judgment task physically small or large digits were presented left or right from screen center. The reaction times analysis revealed that space- and size-congruency effects coexisted in parallel and combined additively. Moreover, a selective sequential modulation of the two congruency effects was found. The size-congruency effect was reduced after size incongruent trials. The space-congruency effect, however, was only affected by the previous space congruency. The observed independence of spatial-numerical and within-magnitude associations is interpreted as evidence that the two couplings reflect different attributes of numerical meaning possibly related to ordinality and cardinality.