The study tests the hypothesis that conditional probability judgments can be influenced by causal links between the target event and the evidence even when the statistical relations among variables are held constant. Three experiments varied the causal structure relating three variables and found that (a) the target event was perceived as more probable when it was linked to evidence by a causal chain than when both variables shared a common cause; (b) predictive chains in which evidence is a cause of (...) the hypothesis gave rise to higher judgments than diagnostic chains in which evidence is an effect of the hypothesis; and (c) direct chains gave rise to higher judgments than indirect chains. A Bayesian learning model was applied to our data but failed to explain them. An explanation-based hypothesis stating that statistical information will affect judgments only to the extent that it changes beliefs about causal structure is consistent with the results. (shrink)

In the first part of the paper we study arithmetical properties of Pascal triangles modulo a prime power; the main result is the generalization of Lucas' theorem. Then we investigate the structure N; Bpx, where p is a prime, α is an integer greater than one, and Bpx = Rem, px); it is shown that addition is first-order definable in this structure, and that its elementary theory is decidable.

Let $ be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of $\langle\mathbb{N};\bot, , is undecidable. Now denote by $ the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure $\langle\mathbb{N};\bot, . Furthermore, the structures $\langle\mathbb{N};\mid, and $\langle\mathbb{N};=,+,x\rangle$ are interdefinable.

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

Let θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

Letkandlbe two multiplicatively independent integers, and letL⊆ ℕnbe al-recognizable set which is not definable in 〈ℕ; +〉. We prove that the elementary theory of 〈ℕ; +,Vk, L〉, whereVk(x)denotes the greatest power ofkdividingx, is undecidable. This result leads to a new proof of the Cobham-Semënov theorem.