The Laplace-Jaynes approach to induction

An approach to induction is presented, based on the idea of analysing the context of a given problem into `circumstances'. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti's representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve `unknown probabilities' or `propensities'. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed.Comment: 38 pages, 1 figure. V2: altered discussion on some points, corrected typos, added reference

On the role of deduction in reasoning from uncertain premises

The probabilistic approach to reasoning hypothesizes that most reasoning, both in everyday life and in science, takes place in contexts of uncertainty. The central deductive concepts of classical logic, consistency and validity, can be generalised to cover uncertain degrees of belief. Binary consistency can be generalised to coherence, where the probability judgments for two statements are coherent if and only if they respect the axioms of probability theory. Binary validity can be generalised to probabilistic validity (p-validity), where an inference is p-valid if and only if the uncertainty of its conclusion cannot be coherently greater than the sum of the uncertainties of its premises. But the fact that this generalisation is possible in formal logic does not imply that people will use deduction in a probabilistic way. The role of deduction in reasoning from uncertain premises was investigated across ten experiments and 23 inferences of differing complexity. The results provide evidence that coherence and p-validity are not just abstract formalisms, but that people follow the normative constraints set by them in their reasoning. It made no qualitative difference whether the premises were certain or uncertain, but certainty could be interpreted as the endpoint of a common scale for degrees of belief. The findings are evidence for the descriptive adequacy of coherence and p-validity as computational level principles for reasoning. They have implications for the interpretation of past findings on the roles of deduction and degrees of belief. And they offer a perspective for generating new research hypotheses in the interface between deductive and inductive reasoning. Keywords: Reasoning; deduction; probabilistic approach; coherence; p-validit