Sunday, August 31, 2014

Gillies’ Philosophical Theories of Probability, Chapter 3

Chapter 3 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with Keynes’ logical theory of probability, which was taken up by the Vienna circle and logical positivists like Carnap (Gillies 2000: 25).

Keynes’ logical theory was partly inspired by the lectures of W. E. Johnson at Cambridge university, which were also attended by Harold Jeffreys who later formulated his own logical theory of probability (Gillies 2000: 25).

Keynes had finished the proofs of his Treatise on Probability in 1913, but it was not published until 1921.

Keynes’ views on probability were influenced by the intellectual climate at Cambridge university, particularly the ethical work of G. E. Moore and Bertrand Russell’s logicist work on mathematics (Gillies 2000: 27). Gillies (2000: 27) sees Keynes’ attempts to provide a “logical” foundation for probability, and particularly inductive reasoning, as inspired by Russell and Whitehead’s attempts to found mathematics on logic.

For Keynes, if the evidence h justifies a conclusion a to some degree α, then there is a probability relation of degree α between a and h, so that inductive probability is a degree of partial entailment or degree of rational belief (Gillies 2000: 31)

Keynes thought that we have knowledge of this probability relation by logical intuition or direct acquaintance, but this is a problematic part of Keynes’ theory (Gillies 2000: 31–32).

Gillies (2000: 32–33) argues that underlying Keynes’ idea that probability is objective and known by logical intuition is a problematic concept derived from Platonic ontology: that the probability relation is objective and belongs to some Platonic realm.

Keynes also thought that mathematical probabilities are expressible as numbers on a range in the interval [0, 1]; he thought that not every probability had a numerical value, and also that some probabilities can only be arranged in an ordinal ranking, while others cannot even be compared at all (Gillies 2000: 33–34).

Numerical “point” probabilities are possible when the relevant outcomes involved are finite, exclusive and equiprobable (Gillies 2000: 35).

The “principle of indifference,” which was introduced by Bernoulli as the “principle of non-sufficient reason,” is an a priori principle invoked by Keynes to determine when cases are equiprobable (Gillies 2000: 35–36). Gillies (2000: 37–46) sees insuperable difficulties with the a priori principle of indifference, such as the book paradox and wine–water paradox, which Keynes did not really solve.

While the “principle of indifference” might in some cases be a useful heuristic, it cannot be a sound logical principle, and so, since Keynes’ logical theory of probability requires it to be a successful theory, Gillies (2000: 48–49) concludes that Keynes’ overall logical theory is flawed, though important aspects of it may be salvaged, provided that the theory’s Platonist view of how the probability relation is understood and the a priori principle of indifference are abandoned (Gillies 2000: 35, 49; Runde 1994).

Further Reading
“Bibliography on Keynes’s Theory of Probability (Updated),” July 6, 2014.

External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)

Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.

Runde, J. 1994. “Keynes After Ramsey: In Defence of A Treatise on Probability,” Studies in History and Philosophy of Science 25.1: 97–121.

Saturday, August 30, 2014

Gillies’ Philosophical Theories of Probability, Chapter 2

Chapter 2 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with the Classical interpretation of probability theory.

As Gillies notes, the “Classical” interpretation was the earliest theory of probability and its most important statement was by Pierre-Simon Laplace (1749–1827) in his Essai Philosophique sur les Probabilités [A Philosophical Essay on Probabilities] (1814). However, it is largely of historical interest now, and has no supporters today (Gillies 2000: 3).

Laplace’s Essai Philosophique sur les Probabilités (1814) made the assumption of universal determinism on the basis of Newtonian mechanics (Gillies 2000: 16). Laplace argued that an agent with perfect knowledge of Newtonian mechanics and all matter could predict the future state of the universe. It is only human ignorance that prevents perfect forecasting, and leads us to calculate probabilities (Gillies 2000: 17). Thus probability, according to Laplace, is a measure of human ignorance (Gillies 2000: 21).

Laplace’s formula for calculating probabilities is the familiar one where the probability P(E) of any event E in a finite sample space S, where all outcomes are equally likely, is the number of outcomes for E divided by the total number of outcomes in S.

But there is an obvious limitation with this, as pointed out by the later advocates of the frequency theory of probability like Richard von Mises: what if our outcomes are not equiprobable? (Gillies 2000: 18). Thus the Classical interpretation of probability has a serious shortcoming.

As probability theory came to be increasingly applied to phenomena in the natural and social sciences in the 19th century, its limiting assumption of equiprobable outcomes was exposed as a problem, and the relative frequency approach was developed as a new and alternative theory.

Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.