Attentional Strategies and the Transition From Subitizing to Estimation in Numerosity Perception.

The common view of the transition between subitizing and numerosity estimation regimes is that there is a hard bound on the subitizing range, and beyond this range, people estimate. However, this view does not adequately address the behavioral signatures of enumeration under conditions of attentional load or in the immediate post-subitizing range. The possibility that there might exist a numerosity range where both processes of subitizing and estimation operate in conjunction has so far been ignored. Here, we investigate this new proposal, that people strategically combine the processes of subitizing and estimation to maximize accuracy and precision, given time or attentional constraints. We present a process-level account of how subitizing and estimation can be combined through strategic deployment of attention to maximize the precision of perceived numerosity given time constraints. We then describe a computational model of this account and apply it in two experimental simulations to demonstrate how it can explain key findings in prior enumeration research. While recent modeling work has argued that the behavioral signatures of enumeration can best be explained through a single numerosity system with a single form of representation, we argue that our model demonstrates how the traditional two-systems view of numerical representation accounts for behavioral data through coordination with a unified attentional mechanism, rather than a unified representation.

The role of attention in the enumeration of canonical patterns.

We report novel findings from experiments on the enumeration of canonical patterns under attentional load. While previous studies have shown that the process of enumerating randomized arrangements can be disrupted by attentional load, the effect of attentional load on canonical patterns has been unexplored. To investigate this case, we adapted a spatial dual-task paradigm previously used to study attentional disruption during the enumeration of randomized arrangements. We begin by replicating previous findings for randomized arrangements, with enumeration error increasing with cluster numerosity and attentional load. For dice patterns, enumeration error also increased under attentional load. However, contrary to findings from studies on single-task enumeration of dice patterns, we observed conflation of patterns with similar outlines. In subsequent experiments, we manipulated the spatial location of the enumeration task, placing the dot cluster in the center. With centrally located, canonical patterns that remained in the same location across trials, enumeration accuracy was more consistent with results from single-task studies. We hypothesize that participants may be using shape cues to inform guessing during enumeration tasks when unable to both localize and fully attend to target patterns.

Much Ado About Nothing: The Mental Representation of Omissive Relations.

When the absence of an event causes some outcome, it is an instance of omissive causation. For instance, not eating lunch may cause you to be hungry. Recent psychological proposals concur that the mind represents causal relations, including omissive causal relations, through mental simulation, but they disagree on the form of that simulation. One theory states that people represent omissive causes as force vectors; another states that omissions are representations of contrasting counterfactual simulations; a third argues that people think about omissions by representing sets of iconic possibilities - mental models - in a piecemeal fashion. In this paper, we tease apart the empirical predictions of the three theories and describe experiments that run counter to two of them. Experiments 1 and 2 show that reasoners can infer temporal relations from omissive causes - a pattern that contravenes the force theory. Experiment 3 asked participants to list the possibilities consistent with an omissive cause - it found that they tended to list particular privileged possibilities first, most often, and faster than alternative possibilities. The pattern is consistent with the model theory, but inconsistent with the contrast hypothesis. We marshal the evidence and explain why it helps to solve a long-standing debate about how the mind represents omissions.

Mental models and omissive causation.

Some causal relations refer to causation by commission (e.g., "A gunshot causes death"), and others refer to causation by omission (e.g., "Not breathing causes death"). We describe a theory of the representation of omissive causation based on the assumption that people mentally simulate sets of possibilities-mental models-that represent causes, enabling conditions, and preventions (Goldvarg & Johnson-Laird, 2001). The theory holds that omissive causes, enabling conditions, and preventions each refer to distinct sets of possibilities. For any such causal relation, reasoners typically simulate one initial possibility, but they are able to consider alternative possibilities through deliberation. These alternative possibilities allow them to deliberate over finer-grained distinctions when reasoning about causes and effects. Hence, reasoners should be able to distinguish between omissive causes and omissive enabling conditions. Four experiments corroborated the predictions of the theory. We describe them and contrast the results with the predictions of alternative accounts of causal representation and inference.