According to Satosi Watanabe's "theorem of the ugly duckling", the number of predicates satisfied by any two different particulars is a constant, which does not depend on the choice of the two particulars. If the number of predicates satisfied by two particulars is their number of properties in common, and the degree of resemblance between two particulars is a function of their number of properties in common, then it follows that the degree of resemblance between any two different particulars is (...) also constant, which is absurd. Avoiding this absurd conclusion requires questioning assumptions about infinity in the proof or interpretation of the theorem, adopting a sparse conception of properties, or denying degree of resemblance is a function of number of properties in common. After arguing against both the first two options, this paper argues for a version of the third which analyses degree of resemblance as a function of properties in common, but weighted by their degree of naturalness or importance. (shrink)

This paper considers whether an analogy between distance and dissimilarlity supports the thesis that degree of dissimilarity is distance in a metric space. A straightforward way to justify the thesis would be to define degree of dissimilarity as a function of number of properties in common and not in common. But, infamously, this approach has problems with infinity. An alternative approach would be to prove representation and uniqueness theorems, according to which if comparative dissimilarity meets certain qualitative conditions, then it (...) is representable by distance in a metric space. I will argue that this approach faces equally severe problems with infinity. (shrink)

Probability measures can be constructed using the measure-theoretic techniques of Caratheodory and Hausdorff. Under these constructions one obtains first an outer measure over "events" or "propositions." Then, if one restricts this outer measure to the measurable propositions, one finally obtains a classical probability theory. What I argue is that outer measures can also be used to yield the structures of probability theories in quantum mechanics, provided we permit them to range over at least some unmeasurable propositions. I thereby show that (...) nonclassical probability theories can be seen to arise naturally within the framework of possible worlds semantics. (shrink)

This approach does not define a probability measure by syntactical structures. It reveals a link between modal logic and mathematical probability theory. This is shown (1) by adding an operator (and two further connectives and constants) to a system of lower predicate calculus and (2) regarding the models of that extended system. These models are models of the modal system S₅ (without the Barcan formula), where a usual probability measure is defined on their set of possible worlds. Mathematical probability models (...) can be seen as models of S₅. (shrink)

The idea that sentences can be closer or further apart in meaning is highly intuitive. Not only that, it is also a pillar of logic, semantic theory and the philosophy of science, and follows from other commitments about similarity. The present paper proposes a novel way of comparing the ‘distance’ between two pairs of propositions. We define ‘\ is closer in meaning to \ than \ is to \’ and thereby give a precise account of comparative propositional similarity facts. Notably, (...) our definition eschews metric assumptions, which are unrealistic in most applications of interest. (shrink)

An analogue of Arrow’s theorem has been thought to limit the possibilities for multi-criterial theory choice. Here, an example drawn from Toy Science, a model of theories and choice criteria, suggests that it does not. Arrow’s assumption that domains are unrestricted is inappropriate in connection with theory choice in Toy Science. There are, however, variants of Arrow’s theorem that do not require an unrestricted domain. They require instead that domains are, in a technical sense, ‘rich’. Since there are rich domains (...) in Toy Science, such theorems do constrain theory choice to some extent—certainly in the model and perhaps also in real science. (shrink)

Some have argued that chance and determinism are compatible in order to account for the objectivity of probabilities in theories that are compatible with determinism, like Classical Statistical Mechanics (CSM) and Evolutionary Theory (ET). Contrarily, some have argued that chance and determinism are incompatible, and so such probabilities are subjective. In this paper, I argue that both of these positions are unsatisfactory. I argue that the probabilities of theories like CSM and ET are not chances, but also that they are (...) not subjective probabilities either. Rather, they are a third type of probability, which I call counterfactual probability. The main distinguishing feature of counterfactual-probability is the role it plays in conveying important counterfactual information in explanations. This distinguishes counterfactual probability from chance as a second concept of objective probability. (shrink)

The contents of representations in non-human animals, human core cognition, and perception cannot precisely be characterized by sentences of a natural language. However, this fact does not stop us from giving imprecise characterizations of these contents through natural language. In this paper, I develop an account of the precision of content characterizations by appealing to possible-world semantics combined with set and measurement theory.

This article analyses the relationship between the concept of single aspect similarity and proposed measures of similarity. More precisely, it compares eleven measures of similarity in terms of how well they satisfy a list of desiderata, chosen to capture common intuitions concerning the properties of similarity and the relations between similarity and dissimilarity. Three types of measures are discussed: similarity as commonality, similarity as a function of dissimilarity, and similarity as a joint function of commonality and difference. Relative to the (...) desiderata, it is found that a measure of the second type fares the best. However, rather than recommend this measure alone as a measure of similarity, it is suggested that there are at least three separate concepts of single aspect similarity, corresponding to the three types of measures. In light of this proposal, three of the eleven measures (and variants of these) are deemed acceptable. (shrink)