Constitutivism without Normative Thresholds

Constitutivist accounts in metaethics explain the normative standards in a domain by appealing to the constitutive features of its members. The success of these accounts turns on whether they can explain the connection between normative standards and the nature of individuals they authoritatively govern. Many such explanations presuppose that any member of a norm-governed kind must minimally satisfy the norms governing its kind. I call this the Threshold Commitment, and argue that constitutivists should reject it. First, it requires constitutivists to restrict the scope of their explanatory ambitions, because it is not plausibly true of social kinds. Second, despite the frequent reliance on physical artifacts in constitutivists’ illustrations of the Threshold Commitment, it counter-intuitively entails that physical artifacts can cease to exist without being physically destroyed. Third, it misconstrues the normative force of authoritative norms on very defective kind-members because it locates this force not in the norm, but in the threat of non-existence. Fortunately, constitutivism can be decoupled from the Threshold Commitment, and I close by sketching a promising alternative account

Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

Decomposing 1-Sperner hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper

Maximum tolerance and maximum greatest tolerance

An important consideration when applying neural networks is the sensitivity to weights and threshold in strict separating systems representing a linearly separable function. Two parameters have been introduced to measure the relative errors in weights and threshold of strict separating systems: the tolerance and the greatest tolerance. Given an arbitrary separating system we study which is the equivalent separating system that provides maximum tolerance or/and maximum greatest tolerance.Postprint (author’s final draft