Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyzes a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of Knightian ambiguity without explicitly invoking any state space. We characterize a representation that integrates a monotone transformation of first order expected utility with respect to a second order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion and uniquely characterize absolute ambiguity neutrality. Finally, we discuss applications of the theory: reinsurance, games, and a mean–variance–ambiguity portfolio frontier.

Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the Church-Turing hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinate-free mathematical language which is both constructive and non-local to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of Church-Turing and quantum theories. It has constructively to bridge the non-local chasm between the weak anticipation of mathematics and the strong anticipation of physics.

In this paper we consider physical systems and the concept of their states in the context of the theory of fuzzy sets and systems. In section 1 we give a brief sketch on the fundamental difference between the theories of classical physics and quantum mechanics. In section 2 and 3 we introduce very shortly systems and their states in classical and quantum mechanics, respectively. Section 4 presents the concept of fuzzy systems. We propose to fuzzify the classical systems in section 5 and quantum systems in section 6. In section 7 we start to consider a fuzzy interpretation of the uncertainty principle.

The nondistributivity of compound quantum mechanical propositions leads to a theorem that rules out the possibility of microscopic deterministic hidden variables, the Logical No-Go Theorem. We observe that there appear in fact two distinct nondistributivity relations in the derivation: one with a semantics governed by an empirical conjunctive syntax, the other composed of conjunctive primitives in the quantum mechanical probability calculus. We venture to speculate how the two come to be confused in the derivation of the theorem. 1 introduction The 20th century witnessed a revolution in experimental instrumentation from the likes of the Plank’s black box apparatus to the Stern-Gerlach spin analyzer. From these there came a wealth of new and unusual data, much of which suggested a microscopic substructure [1] whose workings were not governed by the then prevailing Newtonian mechanics. Eventually, particles came to be seen no longer as entities with categorical properties but as carriers of properties that could only be