ABSTRACT
Co-authored with Jack Copeland, this essay is an edited and somewhat expanded version o f a typescript that Sylvan produced in November 1995 (some seven months before his death). Sylvan’s typescript was an early draft and Copeland has reorganised and abridged it, and has in several places attem pted to sharpen and expand the arguments (even where he does not agree with them). Nevertheless the words remain for the most part Sylvan’s own. While the typescript drew, in part, on Copeland’s contributions to their discussions and correspondence, he acknowledges that the work is largely Sylvan ’s alone. In particular, the material on (what Copeland has subsequently termed) paraconsistent computability theory is m ostly Sylvan’s. Had Sylvan lived he would no doubt have enlarged considerably upon his remarks on what he called ‘dialethic computing machines’. It is hoped that the sketch that he has provided (section 4) will inspire others working in the paraconsistent held to pursue his idea. (A related idea is described by Teixeira and Sarmento 1997). [Editors]
8.1 Introduction
Once upon a time, back in the golden age of recursive function theory, computability was an absolute. Further back still, things were different. While computability (by human hand) was, when it was thought about carefully, taken to depend on the resources available in a given framework or system, it was assumed that there was no upper bound to resources available. Indeed, absolutely sufficient resources were already available within classical logic, so some imagined. Within its comprehensive framework all problems were supposedly solvable mechanically, all mathematical questions were in principle decidable. As we are all supposed now to appreciate, these classical enthusiasts overreached themselves, ascended too high and sailed too near the classical sun, were badly burnt and took a stunning fall.
