Provability, Computability and Reflection.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...) is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (shrink)

This paper focuses on two notions of effectiveness which are not treated in detail elsewhere. Unlike the standard computability notion, which is a property of functions themselves, both notions of effectiveness are properties of interpreted linguistic presentations of functions. It is shown that effectiveness is epistemically at least as basic as computability in the sense that decisions about computability normally involve judgments concerning effectiveness. There are many occurrences of the present notions in the writings of logicians; moreover, consideration of these (...) notions can contribute to the clarification and, perhaps, solution of various philosophical problems, confusions and disputes. (shrink)

New epistemic principles are formulated in the language of Shapiro's system of Epistemic Arithmetic. It is argued that some plausibility can be attributed to these principles. The relations between these principles and variants of controversial constructivistic principles are investigated. Special attention is given to variants of the intuitionistic version of Church's thesis and to variants of Markov's principle.

As to the misconceptions: In the first place, the existence of "undecidable propositions" or "unsolvable problems" has only remote connections with the failure of excluded middle. More precisely, from the fact that a certain problem is unsolvable, one cannot infer that the affirmative and negative answers to that problem are both incorrect. Both Gödel's and Church's theorems were originally proved for systems with the excluded middle, i.e. for systems in which 'p or not p' is provable for every proposition 'p'; (...) though in the light of Gödel's theorem there are propositions 'p' such that neither 'p' nor 'not p' is provable in those systems. Careful distinction should be made between the following locutions. (shrink)