Abstract. The semantic paradoxes are often associated with self-reference or referential circularity. Yablo [1993], however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook [2004] and Yablo [2006] with respect to more restricted sentences systems, which we call F-systems.Keywords: Paradox, Hypodox, Reference structure, Circularity, Ungroundedness, Yablo's paradox, Liar paradox, Graph theory, Dangerous, Precarious, F-system, Kernel.The semantic paradoxes are often associated with self-reference or referential circularity. Yablo [1993] 1 , however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. 2 The attempts to purge the semantic antimonies by banning self-reference or by constructing sophisticated hierarchies only eliminate the class of paradoxes that rely on the circular reference structures-if cyclical reference is not essential to the semantic paradoxes, then the acyclical paradoxes remain unscathed. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. Since "circularity" has traditionally been assumed to be essential, this issue has been underrepresented in the literature on truth and semantic paradoxes 3 -but it is clear that no such theory can lay claim to comprehensiveness until this question is answered. The resolution of this general question, then, has great import for ...

We consider graphs G with ∆ = 3 such that χ ′ (G) = 4 and χ ′ (G − e) = 3 for every edge e, so-called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, P * , is such a graph and has average degree only 2 + 2 3 . He showed that every critical graph has average degree at least 2 + 2 3 , and asked if P * is the only graph where equality holds. We answer his question affirmatively. Our main result is that every subcubic critical graph, other than P * , has average degree at least 2 + 26 37 = 2.702.

We present the simplest solution ever to 'the hardest logic puzzle ever'. We then modify the puzzle to make it even harder and give a simple solution to the modified puzzle. The final sections investigate exploding god-heads and a two-question solution to the original puzzle. The simplest solution to the 'hard' puzzleThe puzzle. Three gods A, B, and C are called, in some order, 'True', 'False', and 'Random'. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for 'yes' and 'no' are 'da' and 'ja', in some order. You do not know which word means which. 1

Abstract:We prove that every graph G for which (G)

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