It is argued that a number of important, and seemingly disparate, types of representation are species of a single relation, here called structural representation, that can be described in detail and studied in a way that is of considerable philosophical interest. A structural representation depends on the existence of a common structure between a representation and that which it represents, and it is important because it allows us to reason directly about the representation in order to draw conclusions about the (...) phenomenon that it depicts. The present goal is to give a general and precise account of structural representation, then to use that account to illuminate several problems of current philosophical interest — including some that do not initially seem to involve representation at all. In particular, it is argued that ontological reductions (like that of the natural numbers to sets), compositional accounts of semantics, several important sorts of mental representation, and (perhaps) possible worlds semantics for intensional logics are all species of structural representation and are fruitfully studied in the framework developed here. (shrink)

On the Lockean thesis one ought to believe a proposition if and only if one assigns it a credence at or above a threshold (Foley in Am Philos Q 29(2):111–124, 1992). The Lockean thesis, thus, provides a way of characterizing sets of all-or-nothing beliefs. Here we give two independent characterizations of the sets of beliefs satisfying the Lockean thesis. One is in terms of betting dispositions associated with full beliefs and one is in terms of an accuracy scoring system for (...) full beliefs. These characterizations are parallel to, but not merely derivative from, the more familiar Dutch Book (de Finetti in Theory of probability, vol 1, Wiley, London, 1974) and accuracy (Joyce in Philos Sci 65(4):575–603, 1998) arguments for probabilism. (shrink)

In both axiomatic theories and the practice of extensive measurement, it is assumed that a series of replicas of any given object can be found. The replicas give rise to a standard series, the "multiples" of the given object. The numerical value assigned to any object is determined, approximately, by comparisons with members of a suitable standard series. This prescription introduces unspecified errors, if the comparison process is somewhat insensitive, so that "replicas" are not really equivalent. In this paper, it (...) is assumed that the comparison process leads only to a semiorder, which allows for such insensitivity. It is shown that, nevertheless, extensive measurement can be carried out, provided that a certain set of (plausible) axioms is valid. Approximate measures, and their limits of error, can be derived from finite sets of semiorder observations. These approximate measures converge to ratio-scale exact measurement. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.

Axiomatizations of ordered preference differences typically assume that preferences are ordered. However, the mere assumption that preference differences can be ordered says nothing about whether preferences themselves are ordered. Utility representations for ordered differences without ordered preferences are investigated.

Part II: Suppes and the mature theory. Representation and uniqueness.