Given a groupoid hG, ⋆i, and k ≥ 3, we say that G is antiassociative if an only if for all x1, x2, x3 ∈ G, (x1 ⋆ x2) ⋆ x3 and x1 ⋆ (x2 ⋆ x3) are never equal. Generalizing this, hG, ⋆i is k-antiassociative if and only if for all x1, x2, . . . , xk ∈ G, any two distinct expressions made by putting parentheses in x1 ⋆ x2 ⋆ x3 ⋆ . . . ⋆ xk are never equal. We prove that for every k ≥ 3, there exist finite groupoids that are k-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
Current debates on the nature of explanatory understanding have converged on the idea that at least one of the core components of understanding is inferential. Philosophers have characterized the inferential dimension of understanding as consisting of several related cognitive abilities to grasp a given explanation and the nexus of complementing explanations to which it belongs. Whilst analyses of both the subjective epistemic abilities related to grasping and objective features of the inferential links within explanations have received much attention, both within theories of explanation and in the literature on understanding, the criteria for evaluating the specific structure and organization of explanatory clusters or nexuses have received much less attention. Nevertheless, two notable exceptions stand out—Khalifa’s characterization of an explanatory nexus and theories of explanatory unification. I take Khalifa’s ideas, together with the basic criteria of successful explanatory unification, as my starting point. To both, I make some corrections and additions, in order to arrive at a more robust notion of an explanatory nexus and ultimately show that its structural properties and the inter-explanatory relations it contains are relevant to the resulting understanding. I propose to represent such nexuses as directed graph trees and show that some of their properties can be related to the degree of understanding that such nested explanatory structures can offer. I will further illustrate these ideas by a case study on an eco-logical theory of predation.
In this paper significant challenges are raised with respect to the view that explanation essentially involves unification. These objections are raised specifically with respect to the well-known versions of unificationism developed and defended by Michael Friedman and Philip Kitcher. The objections involve the explanatory regress argument and the concepts of reduction and scientific understanding. Essentially, the contention made here is that these versions of unificationism wrongly assume that reduction secures understanding.