Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for (...) number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)

Concentrating on Propertius 3.21 in particular, this article identifies a previously unnoticed network of allusions by three Roman poets (Catullus, Propertius and Ovid) to one another and to Book 1 of Apollonius Rhodius’ Argonautica. It shows that these intertextual links are pivoted on the three poets’ common use of the verse-ending lintea malo in scenes of departure by sea, and on their common interest in framing other aspects of the nautical context (especially the naval equipment involved and the presence of (...) a favourable wind) in specific ways. Highlighting the presence in all three cases of departing male lovers with traditionally compromised or otherwise dubious claims to heroism, the article argues that each of the three instances shows the poet in question interacting competitively and self-consciously with the usages of his predecessor(s) (and with those usages’ immediate contexts) and exploiting the choices made by them to serve his new context and to advertise his personal skill in the creative deployment of revered poetic models. (shrink)

In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such theories (...) will be reported in [4]. (shrink)

This chapter presents arguments positing that there is an important sense in which it takes more than all of the actual kinds to make a world, contrary to the popular saying that goes “it takes all kinds to make a world.” In a variety of ways, the various species of elementary particles are ideal cases of natural kinds since each belongs to exactly one of these natural kinds and it essentially belongs to that kind. There exists perfect uniformities within each (...) species and sharp distinctions between the species with respect to certain properties. Some of these properties are essential and suffice to give necessary and adequate conditions for species-membership, while others derive from the fundamental properties via exceptionless uniformities with no ceteris paribus escape clauses. (shrink)

In " Enn." IV, 3. 23 Plotinus presents a vindication of the well - known tripartition - cum - trilocation of the soul advanced by Plato in the " Timaeus." His version of the Platonic doctrine is marked by a strong spatial separation between the three parts of the soul -- reason in the brain, will in the heart and desire in the liver. This article addresses two related questions : Can this position be squared with the Plotinian key doctrine (...) that the soul is incorporeal and indivisible? What are the nature and provenance of this particular version of the tripartition, which, in this form, is not warranted by the Platonic text? Plotinus enlists the support of the anatomical researches presented by Galen in his " On the doctrines of Hippocrates and Plato." At the same time he adapts the latter ' s demonstration in such a way as to safeguard the unity and incorporeality of the soul. The parts of the soul are not " in " the three main bodily organs in an ordinary sense -- only their activity takes place there. Plotinus arrives at this position through a clarification of the concept of archê as employed by Galen. His understanding of the soul ' s localisation is heavily indebted to Alexander ' s " On the soul ". Meanwhile he dissociates himself from the hylomorphism typical of both Alexander and Galen. Other features of his argument are explicable as motivated by the wish to emasculate the arguments used by Alexander in favour of the cardiocentric theory. The upshot is a sophisticated and improved defence of the Platonic tripartition which not only is scientifically up - to - date but coheres with some of Plotinus ' most deeply held metaphysical convictions. (shrink)