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)

Kant asserts that we cannot represent to ourselves the non-existence of space. In his discussion of the Ontological Argument he maintains that there is nothing whose non-existence is inconceivable. He thus seems to contradict himself. If the non-existence of space is unthinkable, so is the non-existence of a part of space — a place. Indicating a particular place, we might say "There are no objects there", but it would be nonsense to say "There doesn't exist". We can say, as Aristotle (...) saw, "There is a place where there was water and where there is now air"; but to do so is to bind an adverbial variable with a quantifier, not to attach "exists" to the name of a place. To assert of a place, or of space, that it exists or that it does not exist would be nonsense, and the unthinkable in that sense is not something whose negation is, as Kant thought, a necessary truth. (shrink)

Kant asserts that we cannot represent to ourselves the non-existence of space. In his discussion of the Ontological Argument he maintains that there is nothing whose non-existence is inconceivable. He thus seems to contradict himself. If the non-existence of space is unthinkable, so is the non-existence of a part of space — a place. Indicating a particular place, we might say "There are no objects there", but it would be nonsense to say "There doesn't exist". We can say, as Aristotle (...) saw, "There is a place where there was water and where there is now air"; but to do so is to bind an adverbial variable with a quantifier, not to attach "exists" to the name of a place. To assert of a place, or of space, that it exists or that it does not exist would be nonsense, and the unthinkable in that sense is not something whose negation is, as Kant thought, a necessary truth. (shrink)

PAPERS IN LOGIC AND ETHICS by A. N. Prior. Duckworth, 1976. 233 pp. £8.95.THE DOCTRINE OF PROPOSITIONS AND TERMS by A. N. Prior. Duckworth, 1976. 143 pp. £5.95.

New technology has a prominent place in the theory and practice of innovation, but the association between high tech and innovation is not inevitable. In this paper, we discuss six metaphorical heroes of agricultural innovation, starting with the dominant hero of frontier science and technology. At first sight, our six heroes can be divided in those who are pro- and those who are anti-technology. Yet in the end technology, and more specifically GM technology, does not emerge as the main issue. (...) Empowering the poor, finding solutions for urgent climate problems, and enhancing the quality of our daily relations to food and the environment – these are the issues the heroes are fighting for. Relations between innovation and technology are better seen as a matter of pragmatic consideration, we will argue. (shrink)

The Battle of Drepana in 249 b.c. marks the most significant defeat of Roman naval forces at the hands of their Carthaginian opponents during the First Punic War. Attempting to take the Punic fleet in the harbour of Drepana by surprise, the consul P. Claudius Pulcher sailed with his ships from Lilybaeum about midnight, and reached Drepana at dawn. Yet, owing to swift and level-headed counter-measures taken by the Punic commander, Adherbal, the unfolding fight – partly in the harbour, mostly (...) off the shore – turned into a fiasco for the Romans. The consul got away; he returned to Rome, where the Senate instructed him to appoint a dictator. How soon he returned, and by what route, is the question. (shrink)

This paper is concerned with the structure of texts in which aproof is presented. Some parts of such a text are assumptions, otherparts are conclusions. We show how the structural organisation of thetext into assumptions and conclusions helps to check the validity of theproof. Then we go on to use the structural information for theformulation of proof rules, i.e., rules for the (re-)construction ofproof texts. The running example is intuitionistic propositional logicwith connectives , and. We give new proofs of some (...) familiar results aboutthe proof theory of this logic to indicate how the new techniques workout. (shrink)