Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)
Jesus Christ may be regarded as the chief spirit of agitation and innovation. He himself declared, “I come not to bring peace, but a sword.” One cannot delve seriously into the centuries of activism and scholarship against racism, Jim Crowism, and the terrorism of lynching without encountering the legacies of Timothy Thomas Fortune and Ida B. Wells-Barnett. Black scholars from the 19th century to the present have been inspired by the sociological and economic works of Fortune and Wells. Scholars of (...) American philosophy, however, continue to ignore their writings, their theoretical contributions and their ethical aspirations, preferring instead the insipid declarations of white turn of the century .. (shrink)
These remarks are an obituary for William T. Scott who worked for many years on a biography of Michael Polanyi. In addition to providing an overview of Scott’s own life and work, his connection with Polanyi is reviewed.
For cases in which to remember that p is to have (strict) nonbasic, unmixed memory knowledge that p; in which there is at most one prior time, t, from which one remembers; in which one knew at t that p; and in which there can arise a sensible question whether one remembers that p from t — a person, B, remembers that p from t if and only if: (1) There is a set of grounds a subset of which consists (...) of (i) only those grounds B has at both t and the present for B to be sure that p, and (ii) enough such grounds to make it reasonable at both t and the present for B to be sure that p (I call any such subset a set of “adequate original grounds dating from t”), and (2) there is no time prior to t such that B has a set of adequate original grounds dating from that time. The way in which the crucial terms in this explication are being used is explained. And the explication is defended by showing how it can deal with cases that are counterexamples to explications recently offered by Malcolm and by Munsat. (shrink)
We formulate schemes and of the “typical” ∀∑ 1 b -sentences that are provable in T 2 1, respectively T 2 2. As an application, we reprove a recent result of Buss and Krajíček which describes witnesses for the ∀∑ 1 b -sentences provable in T 2 1 in terms of solutions to PLS-problems.
We are grateful for Roy T. Cook's attention to our work in his recent review of our book If A, Then B: How the World Discovered Logic. But Professor Cook leaves two misimpressions that we should like to correct. First, we have never maintained (as he phrases it) that "one's premises must be more certain than the conclusions that follow from them, ignoring the obvious logical fact that, if B logically follows from A, then B is provably at least as (...) probable as A." Instead, we assert that one must be *initially* more certain of one's premises than the conclusions that follow from them; otherwise, we contend, no argument that relies on those premises to prove such a conclusion can be rationally persuasive. On this view, one might still be equally certain of both the premises and the conclusion after being persuaded by the argument, especially in cases where the premises entail the conclusion. By analogy, Aristotle asserts in the Posterior Analytics that the premises of demonstration must be "better known" than the conclusion—meaning, in part, that the premises must be initially more convincing. But this hardly shows that Aristotle thinks that, if A is invoked to demonstrate B, then A and B can never be regarded as equally probable…. (shrink)
This essay examines the intellectual origins of Tocqueville's thoughts on political economy. It argues that Tocqueville believed political economy was crucial to what he called the ‘new science of politics’, and it explores his first forays into the discipline by examining his studies of J.-B. Say and T.R. Malthus. The essay shows how Tocqueville was initially attracted to Say's approach as it provided him with a rigorous analytical framework with which to examine American democracy. Though he incorporated important aspects of (...) Say's work in Democracy in America , he was troubled by elements of it. He was unable to articulate clearly these doubts until he began studying Malthus. What he learned from Malthus caused him to move away from the more formalised approach to political economy advocated by Say and his disciples and move towards an approach advocated by Christian political economists, such as Alban Villeneuve-Bargemont. This shift would have important consequences for the composition of Democracy in America. (shrink)
WILLIAM B. EWALD, From Kant to Hubert. A source book in the foundations of mathematics. Oxford:Clarendon Press, 1997. Two volumes, xviii + 1340pp. £175.00. ISBN 0 19 853271 7 DONALD GILLIES, Artificial Intelligence and Scientific Method. Oxford:Oxford University Press, 1996. xiii+176pp. £35.00 /£ 11.99. ISBN 0 19 875158 3/875159 1 N. VASSALLO, La depsicologizzazione délia logica. Un confronto tra Boole e Frege. Milano:Franco Angeli, 1995. 310 pp. 34,000 L G. SCHURZ, The Is-Ought Problem :An Investigation in Philosophical Logic. Dordrecht, (...) Boston and London : Kluwer Academic Publishers, 1997. x + 332 pp. $120/£72.00. ISBN 0792344103. (shrink)