The (topo)logic of vagueness

Zeno's "dichotomy" paradox of the runner and the sorites paradox exhibit certain interesting similarities. Both of them involve a long series of steps, each of which seems legitimate, but which, taken together, apparently lead to an unacceptable conclusion. In this article, a particular interpretation of a common reply to Zeno's paradox is presented, which recognises that to defuse the paradox, it is necessary to assert that the number of stages that the runner has completed on Zeno's in nite sequence of times is not an appropriate measure of whether he nishes the race or not. Applying this style of reply to the sorites argument, one would reject the argument on the grounds of the inappropriateness of the number of hairs for reasoning about baldness. Such an attitude to the sorites argument implies a certain conception of the problem posed by vague terms, according to which the problem is to understand such relationships between terms as the appropriateness of one for reasoning about the other. Consequently, it poses a certain set of challenges to prospective theories of vagueness.Vagueness; theories of; problem of; sorites paradox; Zeno's paradox scale.

Confidence and ambiguity

This paper proposes a model of the decision-maker’s confidence in his probability judgements, in terms of an implausibility measure – a real-valued function on the set of probability functions. A decision rule is axiomatised according to which the decision-maker evaluates acts using sets of probability functions which vary depending on the agent’s implausibility measure and on what is at stake in the choice of the act. The framework proposed yields a natural notion of comparative aversion to lack of confidence, or ambiguity aversion, and allows the definition of an ambiguity premium. It is shown that these notions are equivalent and can be characterised in terms of the implausibility measure representing the agent’s confidence. A simple portfolio example is presented.Confidence; multiple priors; ambiguity aversion; ambiguity premium; implausibility measure

Awareness and equilibrium

There has been a recent surge of interest among economists in developing models of doxastic states that can account for some aspects of human cognitive limitations that are ignored by standard formal models, such as awareness. Epistemologists purport to have a principled reason for ignoring the question of awareness: under the equilibrium conception of doxastic states they favour, a doxastic state comprises the doxastic commitments an agent would recognise were he fully aware, so the question of awareness plays no role. The objective of this paper is to scrutinize this argument.Bounded rationality; awareness; doxastic states; cognitive equilibrium; belief change; formal epistemology.

Living without state-independence of utilities

This paper is concerned with the representation of preferences which do not satisfy the ordinary axioms for state-independent utilities.Elicitation; Subjective Probability; Subjective Expected Utility; Statedependent utility; Small worlds

Three analyses of sour grapes

The phenomenon of adaptive preferences – sometimes also known under the name of sour grapes – has long caused a stir in Social Theory. In this paper, the precise problem posed by adaptive preferences, as seen from the point of view of a theoretician who intends to model or understand the phenomenon, will be clarified, and three models of the phenomenon will be presented and compared. The general intention of the article is to sound out some of the wider consequences of the phenomenon for the project of modelling and understanding the relationship between decisions taken in different situations. Difficulties which arise when several decisions and several situations are involved shall be discussed, and an approach to these difficulties shall be suggested.Adaptive preferences; preference change; belief change; decision theory; belief and utility elicitation; representation theorems.

Confidence in preferences

Indeterminate preferences have long been a tricky subject for choice theory. One reason for which preferences may be less than fully determinate is the lack of confidence in one’s preferences. In this paper, a representation of confidence in preferences is proposed. It is used to develop an account of the role which confidence which rests on the following intuition: the more important the decision to be taken, the more confidence is required in the preferences needed to take it. An axiomatisation of this choice rule is proposed. This theory provides a natural account of when an agent should defer a decision; namely, when the importance of the decision exceeds his confidence in the relevant preferences. Possible applications of the notion of confidence in preferences to social choice are briefly explored.Incomplete preference; Revealed preference; Confidence in preferences; Deferral of decisions; Importance of decisions; Social choice

An additively separable representation in the Savage framework

This paper elicits an additively separable representation of preferences in the Savage framework (where the objects of choice are acts: measurable functions from an infinite set of states to a potentially finite set of consequences). A preference relation over acts is represented by the integral over the subset of the product of the state space and the consequence space which corresponds to the act, where this integral is calculated with respect to a “state-dependent utility” measure on this space. The result applies at the stage prior to the separation of probabilities and utilities, and requires neither Savage’s P3 (monotonicity) nor his P4 (likelihood ordering). It may thus prove useful for the development of state-dependent utility representation theorems in the Savage framework.Expected utility; additive representation; state-dependent utility; monotonicity

Dynamic Awareness

This paper proposes, firstly, a versatile model of awareness and awareness change, and secondly, logics for awareness and awareness change developed using this model.Awareness; knowledge; logic of awareness; awareness change; belief revision.

When is there state independence?

It has been noticed that whether a preference relation can be represented by state-independent utilities as opposed to state-dependent utilities may depend on which acts count as constant acts [Schervish et al., 1990]. Indeed, this remark underlies an extension of Savage’s expected utility theory to the state-dependent case that was proposed by Edi Karni [Karni, 1993]. This paper contains a characterisation of the preference relations that permit a choice of acts which can play the role of constant acts, and relative to which there is a representation involving a state-independent utility function. This result applies both in the Savage and in the Anscombe & Aumann frameworks. It has as an immediate corollary an extension of Karni’s representation theorem. Finally, it is of methodological interest, insofar that it harnesses techniques from mathematical logic to prove a theorem of interest to decision theorists and economists.Subjective expected utility; State-dependent utility; Monotonicity axiom

Avoiding Premature Oxidation During the Binding of Cu(II) to a Dithiolate Site in BsSCO. A Rapid Freeze-Quench EPR Study

The Bacillus subtilis version of SCO1 (BsSCO) is required for assembly of CuA in cytochrome c oxidase and may function in thiol-disulfide exchange and/or copper delivery. BsSCO binds Cu(II) with ligation by two cysteines, one histidine and one water. However, copper is a catalyst of cysteine oxidation and BsSCO must avoid this reaction to remain functional. Time resolved, rapid freeze-quench (RFQ) electron paramagnetic resonance of apo-BsSCO reacting with Cu(II) reveals an initial Cu(II) species with two equatorially coordinated nitrogen atoms, but no sulfur. We propose that BsSCO evolves from this initial sulfur free coordination of Cu(II) to the final dithiolate species via a change in conformation, and that the initial binding by nitrogen is a means for BsSCO to avoid premature thiol oxidation