This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, (...) the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties). (shrink)

We re-express a previous general result in a way which seems easier to remember, using the terminology of infinite games. We show how this can be applied to construct recursive linear orderings, showing, for example, that if there is a ▵ 0 2β + 1 linear ordering of type τ, then there is a recursive ordering of type ω β · τ.

We extend results of Harizanov and Barker. For a relation R on a recursive structure /oA, we give conditions guaranteeing that the image of R in a recursive copy of /oA can be made to have arbitrary ∑α0 degree over Δα0. We give stronger conditions under which the image of R can be made ∑α0 degree as well. The degrees over Δα0 can be replaced by certain more general classes. We also generalize the Friedberg-Muchnik Theorem, giving conditions on a pair (...) of relations R and S under which the images of R and S can be made ∑α0 and independent over Δα0 in a recursive copy of /oA. (shrink)

A subspace V of an infinite dimensional fully effective vector space V ∞ is called decidable if V is r.e. and there exists an r.e. W such that $V \oplus W = V_\infty$ . These subspaces of V ∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V ∞ ) and the set of decidable subspaces forms a lower semilattice S(V ∞ ). We analyse S(V ∞ ) and its relationship with L(V (...) ∞ ). We show: Proposition. Let U, V, W ∈ L(V ∞ ) where U is infinite dimensional and $U \oplus V = W$ . Then there exists a decidable subspace D such that U |oplus D = W. Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces. These results allow us to show: Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V ∞ )), is undecidable. This contrasts sharply with the result for recursive sets. Finally we examine various generalizations of our results. In particular we analyse S * (V ∞ ), that is, S(V ∞ ) modulo finite dimensional subspaces. We show S * (V ∞ ) is not a lattice. (shrink)

Let be a recursive structure, and let R be a recursive relation on . Harizanov isolated a syntactical condition which is necessary and sufficient for to have recursive copies in which the image of R is r.e. of arbitrary r.e. degree. We had conjectured that a certain extension of Harizanov's syntactical condition would be necessary and sufficient for to have recursive copies in which the image of R is ∑α0 of arbitrary ∑α0 degree, but this is not the case. Here (...) we give examples illustrating some restrictions on the possible ∑α0 degrees. In these examples, the image of R cannot be ∑α0 of degree d unless d possesses an “α-table”. (shrink)

Ash, C.J., Generalizations of enumeration reducibility using recursive infinitary propositional sentences, Annals of Pure and Applied Logic 58 173–184. We consider the relation between sets A and B that for every set S if A is Σ0α in S then B is Σ0β in S. We show that this is equivalent to the condition that B is definable from A in a particular way involving recursive infinitary propositional sentences. When α = β = 1, this condition is that B is (...) enumeration reducible to A. We establish further generalizations involving infinitely many sets and ordinals. (shrink)

In his celebrated masterpiece, Symposium, Plato imagines a high-society dinner-party in Athens in 416 BC at which the guests - including the comic poet Aristophanes and, of course, Plato's mentor Socrates - each deliver a short speech in praise of love. The sequence of dazzling speeches culminates in Socrates' famous account of the views of Diotima, a prophetess who taught him that love is our means of trying to attain goodness. And then into the party bursts the drunken Alcibiades, the (...) most popular and notorious Athenian of the time, who insists on praising Socrates himself rather than love, and gives us a brilliant sketch of this enigmatic character. The power, humour, and pathos of Plato's creation engages the reader on every page. This new translation is complemented by full explanatory notes and an illuminating introduction. (shrink)

"This edition, which replaces the original Loeb edition..., offers text, translation, and annotation that are fully current with modern scholarship"--Front flap of dust jacket, volume 1.

This edition of Martin Ostwald's revised version of J. B. Skemp's 1952 translation of _Statesman_ includes a new selected bibliography, as well as Ostwald's interpretive introduction, which traces the evolution in Plato's political philosophy from _Republic_ to _Statesman to Laws_--from philosopher-king to royal statesman.

Artikelen en lezingen uit de periode 1970-1982.

Deze korte studie biedt niet de zoveelste inleiding in het denken van Spinoza, maar doet verslag van de persoonlijke, vijfentwintig jaar durende zoektocht om de vaak moeilijke thema's van Spinoza onder de knie te krijgen. Om via Spinoza's filosofie meer te begrijpen van de wereld en van het eigen leven. Om de vreugde te ervaren die dit verbeterde inzicht, in oorzaken en gevolgen, en ook in de eigen beperktheid, ons biedt. Elk hoofdstuk gaat over een onderwerp dat de auteur bijzonder (...) aanspreekt. Bestaat er een levenswil, Spinoza's conatus, die mensen voort doet leven ook onder de meest bittere omstandigheden? Is kennis en inzicht gradueel van aard? Wat is het verschil tussen adequate en inadequate kennis van de oorzaken van ons handelen? Hoe ontwikkelt een machtsstaat zich tot een rechtsstaat? (shrink)

New natural law advocates are somewhat notorious for their loose action theory, having a track record of counterintuitive claims. In response to criticisms, advocates have entrenched, further defending their questionable action theory. This paper first rehearses the basic criticism against the new natural law action theory. It then examines four recent attempts to revive this action theory and finds these attempts wanting. Within these attempts, certain patterns arise. Given a certain means A to a goal C, a search is made (...) to determine whether any middle means B is implied by A. The standards of this search vary wildly, however. By some standards, a middle means can be found; by others, every middle means can be easily swept aside. The same author will sometimes use both kinds of standards, depending upon the situation. One great weakness of the action theory, then, is a lack of consistency in applying a universal standard. (shrink)

TheOctaviais, on the face of it, one of the most bizarre documents which have reached us from antiquity. If the news of its discovery had broken yesterday, there would certainly have been a sensation at the bare idea, whatever the literary merits of the work. A few years ago the publication of a 15-line fragment of a Greek play about Gyges caused discussion enough; but here we have acompleteRoman historical play, unlike any other ancient play in structure, featuring Nero, Octavia, (...) Poppaea, Seneca. …. (shrink)

_Verificationism_ is the first comprehensive history of a concept that dominated philosophy and scientific methodology between the 1930s and the 1960s. The verificationist principle - the concept that a belief with no connection to experience is spurious - is the most sophisticated version of empiricism. More flexible ideas of verification are now being rehabilitated by a number of philosophers. C.J. Misak surveys the precursors, the main proponents and the rehabilitators. Unlike traditional studies, she follows verificationist theory beyond the demise of (...) positivism to examine its reappearance in the work of modern philosophers. Most interestingly, she argues that despite feminism's strenuous opposition to positivism, verificationist thought is at the heart of much of contemporary feminist philosophy. _Verificationism_ is an excellent assessment of a major and influential system of thought. (shrink)

The Statesman is Plato's neglected political work, but it is crucial for an understanding of the development of his political thinking. In some respects it continues themes from the Republic, particularly the importance of knowledge as entitlement to rule. But there are also changes: Plato has dropped the ambitious metaphysical synthesis of the Republic, changed his view of the moral psychology of the citizen, and revised his position on the role of law and institutions. In its presentation of the statesman's (...) expertise, the Statesman modifies, as well as defending in original ways, this central theme of the Republic. (shrink)

Traditional accounts of the emergence of professional biology have privileged not only metropolis over province, but research over teaching and laboratory over museum. This paper seeks to supplement earlier studies of the ‘transformation of biology’ in the late nineteenth century by exploring in detail the developments within three biology departments in Northern English civic colleges. By outlining changes in the teaching practices, research topics and the accommodation of the departments, the authors demonstrate both locally contingent factors in their development and (...) continuities with existing traditions in natural history. The appointment of Arthur Milnes Marshall in preference to Louis Miall to the new zoology chair in Manchester in 1879 casts light on contemporary views of the laboratory and museum as ‘equal though different’. The transformation in biology, in Northern England at least, was shaped more by such local institutional changes than by a phoenix-like rise of the laboratory from the ashes of the museum—more by the rhetorical construction of a professional academic community than any dramatic shift in sites. In this period the biology laboratory supplemented, rather than eclipsed, the museum, and the dichotomy between the ‘naturalist’ and the ‘experimentalist’ was far from clear-cut. (shrink)