The Axiom of Choice implies the Partition Principle and the existence, uniqueness, and monotonicity of (possibly infinite) sums of cardinal numbers. We establish several deductive relations among those principles and their variants: the monotonicity follows from the existence plus uniqueness; the uniqueness implies the Partition Principle; the Weak Partition Principle is strictly stronger than the Well-Ordered Choice.

For a tree T of its height equal to or less than ω1, we construct a space XT by attaching a circle to each node and connecting each node to its successors by intervals. is the Hawaiian earring and H1T denotes a canonical factor of the first integral singular homology group. The following equivalences hold for an ω1-tree T: π1 is embeddable into π1, if and only if is embeddable into H1T, if and only if T is not an Aronzajn (...) tree. π1 is embeddable into if and only if H1T is embeddable into if and only if T is a special Aronzajn tree. π1 has a retract isomorphic to an uncountable free group, if and only if H1T has a summand isomorphic to an uncountable free abelian group, if and only if T has an uncountable anti-chain. (shrink)

In this article, 41 models of covariation detection from 2 × 2 contingency tables were evaluated against past data in the literature and against data from new experiments. A new model was also included based on a limiting case of the normative phi‐coefficient under an extreme rarity assumption, which has been shown to be an important factor in covariation detection (McKenzie & Mikkelsen, 2007) and data selection (Hattori, 2002; Oaksford & Chater, 1994, 2003). The results were supportive of the new (...) model. To investigate its explanatory adequacy, a rational analysis using two computer simulations was conducted. These simulations revealed the environmental conditions and the memory restrictions under which the new model best approximates the normative model of covariation detection in these tasks. They thus demonstrated the adaptive rationality of the new model. (shrink)

Causal induction in the real world often has to be quick and efficient as well as accurate. We propose that people use two different frames to achieve these goals. The A-frame consists of heuristic processes that presuppose rarity and can detect causally relevant factors quickly. The B-frame consists of analytic processes that can be highly accurate in detecting actual causes. Our dual frame theory implies that several factors affect whether people use the A-frame or the B-frame in causal induction: among (...) these are symmetrical negation, intervention and commitment. This theory is tested and sustained in two experiments. The results also provide broad support for dual process accounts of human thinking in general. (shrink)