Some may be of the opinion that one event can begin before another only by virtue of the existence of some event (a “witness”) which wholly precedes the other and does not wholly precede the one (and similarly for “ends before” and “does not abut”). Those would prefer $\mathbb{F}$ 0 to $\mathbb{F}$ as a model for observers' apprehensions of events. Since G is a functor from $\mathbb{M}$ to $\mathbb{F}$ 0, the current construction (restricted to $\mathbb{F}$ 0) remains applicable.This work supports (...) a claim that the psychologically fundamental temporal relationships are “wholly precedes”, “begins before”, “ends before” and “abuts”. But only in a very weak sense. Any other set of relationships which is interdefinable with this one, using only quantifier-free formulas in the definitions, could be used to define a category $\mathbb{F}$ ′ which is indistinguishable from $\mathbb{F}$ (because the same functions preserve and reflect the new relationships). This work equally supports the claim that those relationships are the psychologically fundamental ones, or the claim that it is just “wholly precedes” which is fundamental, and that we perceive “begins before” just by virtue of witnesses. But it refutes the claim that only “wholly precedes” is fundamental, and that we understand “begins before” only because we understand time as a linear ordering. (shrink)

In many of his writings Russell developed a theory of time,highly interesting bothfrom a philosophical and from a logical point of view.Strangely enough, this has not acquired generalattention. The most important relational propertiesof the duration and points of time will be presented.In addition, Russell's considerations on the existenceand density of time points will becritically analysed and systematically reconstructed.A. G. Walker's explication of the concept of timepoint is unlike that of Russell. His theory oftime reflects the Dedekindian concept of cut.Walker's constructions (...) will be modified in a suitableway and supplemented so that it becomes possileto prove the continuity of the temporal series. (shrink)

Axiomatizations of measurement systems usually require an axiom--called an Archimedean axiom--that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot--except in the most trivial cases--be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of "Archimedean axioms" is given and it is shown that (...) in all interesting cases "Archimedean axioms" are independent of other measurement axioms. (shrink)

In many of his writings Russell developed a theory of time, highly interesting both from a philosophical and from a logical point of view. Strangely enough, this has not acquired general attention. The most important relational properties of the duration and points of time will be presented. In addition, Russell's considerations on the existence and density of time points will be critically analysed and systematically reconstructed. A. G. Walker's explication of the concept of time point is unlike that of Russell. (...) His theory of time reflects the Dedekindian concept of cut. Walker's constructions will be modified in a suitable way and supplemented so that it becomes possible to prove the continuity of the temporal series. (shrink)