“Gillies’ Philosophical Theories of Probability, Chapter 1,” August 29, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 2,” August 30, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 3,” August 31, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 4,” September 2, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 5,” September 3, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 6,” September 4, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 7,” September 10, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 8,” September 11, 2014.
“Gillies’ Philosophical Theories of Probability, Chapter 9,” September 12, 2014.
Showing posts with label Philosophical Theories of Probability. Show all posts
Showing posts with label Philosophical Theories of Probability. Show all posts
Wednesday, September 17, 2014
Gillies’ Philosophical Theories of Probability, Chapter by Chapter Summaries
Below are links to my chapter summaries of Donald Gillies’ book Philosophical Theories of Probability (2000):
Friday, September 12, 2014
Gillies’ Philosophical Theories of Probability, Chapter 9
Chapter 9 of Donald Gillies’ Philosophical Theories of Probability (2000) is his conclusion and sets out his views on a “pluralist” theory of probability.
Gillies argues that an epistemological theory of probability is appropriate for the social sciences, but an objective theory for the natural sciences (Gillies 2000: 187).
Gillies in a fascinating section (Gillies 2000: 188–194) considers probability in economics. He concludes that, since the actions of economic agents cannot all be considered as independent of one another, their actions are often very much characterised by reaction to other people’s action or expected actions (Gillies 2000: 190). But, since the independence assumption is fundamental for the objective theory of probability, it follows that an objective theory of probability cannot apply to many economic situations or phenomena (Gillies 2000: 190). For example, in the instance of financial markets, general expectations might appear to predict an outcome since they have influenced the outcome (Gillies 2000: 196).
Gillies concludes with some methodological points. The subjective and intersubjective theories of probability use operationalism, but the objective long-run propensity theory rejects operationalism (Gillies 2000: 200).
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Gillies argues that an epistemological theory of probability is appropriate for the social sciences, but an objective theory for the natural sciences (Gillies 2000: 187).
Gillies in a fascinating section (Gillies 2000: 188–194) considers probability in economics. He concludes that, since the actions of economic agents cannot all be considered as independent of one another, their actions are often very much characterised by reaction to other people’s action or expected actions (Gillies 2000: 190). But, since the independence assumption is fundamental for the objective theory of probability, it follows that an objective theory of probability cannot apply to many economic situations or phenomena (Gillies 2000: 190). For example, in the instance of financial markets, general expectations might appear to predict an outcome since they have influenced the outcome (Gillies 2000: 196).
Gillies concludes with some methodological points. The subjective and intersubjective theories of probability use operationalism, but the objective long-run propensity theory rejects operationalism (Gillies 2000: 200).
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Thursday, September 11, 2014
Gillies’ Philosophical Theories of Probability, Chapter 8
Chapter 8 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with intersubjective and pluralist views of probability.
Gillies (2000: 169) argues that there is an intermediate class of probability called intersubjective probability that lies between fully subjective and objective probability.
Furthermore, Gillies makes the case for a “pluralist” view of probability, a development of the two-concept view of probability by Ramsey and Carnap (Gillies 2000: 180–181). In Gillies’ “pluralist” theory of probability, probabilities can be conceptualised on a spectrum or continuum from the fully objective to fully subjective, and intermediate types of probability, as in the diagram below (which needs to be opened in a new window to be seen properly).
A fully objective probability is one that is found in natural phenomena or the material world and is independent of human beings, such as, for example, the probability of radioactive decay of atoms (Gillies 2000: 180).
An artefactual probability is objective in the sense that it is part of the material world, but is at the same time the result of human interaction with nature (Gillies 2000: 179). These include probabilities of games of chance, probabilities of repeatable scientific experiments, and certain statistical probabilities. If such probabilities can be established though stable long-run observed relative frequencies in trials or experiments, then they can be said to be objective (Gillies 2000: 180).
Between artefactual and intersubjective probabilities, Gillies sees some borderline cases such as in medicine or population studies (Gillies 2000: 180, 194).
An intersubjective probability is a measure of the degree of belief of a social group where a consensus has formed (Gillies 2000: 179). An intersubjective probability may be utterly subjective, or subjective but based on an underlying objective probability. If a group of people, though shared beliefs, can agree on a common betting quotient about some probability, then this can be seen as an intersubjective or consensus probability (Gillies 2000: 171).
A fully subjective probability is one of an individual and is a mere measure of a degree of belief, which has a non-objective status (Gillies 2000: 179).
Fully objective and artefactual probabilities are objective and properly described by the propensity theory of probability, but intersubjective and fully subjective probabilities are epistemological, and often such subjective probabilities have no real nor quantifiable objective “propensity” probabilities at all (Gillies 2000: 180–185).
Links
“Gillies on Probability and Uncertainty in Keynes’ Economic Thinking,” July 6, 2014.
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Gillies (2000: 169) argues that there is an intermediate class of probability called intersubjective probability that lies between fully subjective and objective probability.
Furthermore, Gillies makes the case for a “pluralist” view of probability, a development of the two-concept view of probability by Ramsey and Carnap (Gillies 2000: 180–181). In Gillies’ “pluralist” theory of probability, probabilities can be conceptualised on a spectrum or continuum from the fully objective to fully subjective, and intermediate types of probability, as in the diagram below (which needs to be opened in a new window to be seen properly).
A fully objective probability is one that is found in natural phenomena or the material world and is independent of human beings, such as, for example, the probability of radioactive decay of atoms (Gillies 2000: 180).
An artefactual probability is objective in the sense that it is part of the material world, but is at the same time the result of human interaction with nature (Gillies 2000: 179). These include probabilities of games of chance, probabilities of repeatable scientific experiments, and certain statistical probabilities. If such probabilities can be established though stable long-run observed relative frequencies in trials or experiments, then they can be said to be objective (Gillies 2000: 180).
Between artefactual and intersubjective probabilities, Gillies sees some borderline cases such as in medicine or population studies (Gillies 2000: 180, 194).
An intersubjective probability is a measure of the degree of belief of a social group where a consensus has formed (Gillies 2000: 179). An intersubjective probability may be utterly subjective, or subjective but based on an underlying objective probability. If a group of people, though shared beliefs, can agree on a common betting quotient about some probability, then this can be seen as an intersubjective or consensus probability (Gillies 2000: 171).
A fully subjective probability is one of an individual and is a mere measure of a degree of belief, which has a non-objective status (Gillies 2000: 179).
Fully objective and artefactual probabilities are objective and properly described by the propensity theory of probability, but intersubjective and fully subjective probabilities are epistemological, and often such subjective probabilities have no real nor quantifiable objective “propensity” probabilities at all (Gillies 2000: 180–185).
Links
“Gillies on Probability and Uncertainty in Keynes’ Economic Thinking,” July 6, 2014.
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Wednesday, September 10, 2014
Gillies’ Philosophical Theories of Probability, Chapter 7
Chapter 7 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with Gillies’ own long-run propensity theory of probability.
Gillies (2000: 138) rejects operationalist theories of the philosophy of science, and so regards von Mises’ frequency theory of probability as founded on an inadequate epistemology.
Nevertheless, Gillies also thinks that, with a “falsifying rule,” the link between observed frequency and probability in propensity theory can be established, and that the axioms of probability can be used derive the two empirical laws of probability, the law of statistical stability and the law of randomness (Gillies 2000: 150–153).
Furthermore, in his version of propensity theory, a process must be random in the sense that each outcome of that process is independent (Gillies 2000: 156)
Gillies (2000: 160–161) also argues that another axiom – the axiom of independent repetitions – must be added to the standard Kolmogorov axioms for a coherent version of the propensity theory.
Gillies concludes that propensity theory as a “mathematical science of randomness” has superseded von Mises’ frequency theory as the best theory of objective probabilities (Gillies 2000: 184).
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Gillies (2000: 138) rejects operationalist theories of the philosophy of science, and so regards von Mises’ frequency theory of probability as founded on an inadequate epistemology.
Nevertheless, Gillies also thinks that, with a “falsifying rule,” the link between observed frequency and probability in propensity theory can be established, and that the axioms of probability can be used derive the two empirical laws of probability, the law of statistical stability and the law of randomness (Gillies 2000: 150–153).
Furthermore, in his version of propensity theory, a process must be random in the sense that each outcome of that process is independent (Gillies 2000: 156)
Gillies (2000: 160–161) also argues that another axiom – the axiom of independent repetitions – must be added to the standard Kolmogorov axioms for a coherent version of the propensity theory.
Gillies concludes that propensity theory as a “mathematical science of randomness” has superseded von Mises’ frequency theory as the best theory of objective probabilities (Gillies 2000: 184).
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Thursday, September 4, 2014
Gillies’ Philosophical Theories of Probability, Chapter 6
Chapter 6 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with the propensity theory of probability.
The propensity theory of probability was developed by Karl Popper (1957, 1959 and 1990), although a number of versions exist today.
Popper criticised von Mises’ frequency theory as being unable to deal with the probability of single events (Gillies 2000: 115).
Popper held that a new theory was necessary and he conceptualised probability as a tendency or propensity of certain processes or phenomena to produce objective probabilities (Gillies 2000: 115–116), though Popper’s (1990) later version of the propensity theory differs from his earlier one (Gillies 2000: 126) (and this theory was developed by Miller 1994 and 1995).
Though Popper seemed to think that some single events can have objectively determined probabilities, Gillies (2000: 120–124) argues that many single event probabilities cannot be “fully objective,” owing to the reference class problem, except perhaps in games of chance. Therefore Gillies rejects Popper’s early propensity theory.
But Popper’s early propensity theory of both long-run and single-case probabilities is, however, not the only such theory, and Gillies (2000: 126) divides modern propensity theories into two classes:
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Miller, David. 1994. Critical Rationalism: A Restatement and Defence. Open Court, Chicago.
Miller, David. 1995. “Propensities and Indeterminism,” in A. O’Hear (ed.), Karl Popper: Philosophy and Problems. Cambridge University Press, Cambridge. 121–147.
Popper, Karl R. 1957. “The Propensity Interpretation of the Calculus of Probability, and the Quantum Theory,” in S. Körner (ed.), Observation and Interpretation: A Symposium of Philosophers and Physicists: Proceedings of the Ninth Symposium of the Colston Research Society, held in the University of Bristol, April 1st–April 4th, 1957. Butterworths, London. 65–70.
Popper, Karl R. 1959. “The Propensity Interpretation of Probability,” British Journal for the Philosophy of Science 10: 25–42.
Popper, Karl R. 1990. A World of Propensities. Thoemmes, Bristol.
The propensity theory of probability was developed by Karl Popper (1957, 1959 and 1990), although a number of versions exist today.
Popper criticised von Mises’ frequency theory as being unable to deal with the probability of single events (Gillies 2000: 115).
Popper held that a new theory was necessary and he conceptualised probability as a tendency or propensity of certain processes or phenomena to produce objective probabilities (Gillies 2000: 115–116), though Popper’s (1990) later version of the propensity theory differs from his earlier one (Gillies 2000: 126) (and this theory was developed by Miller 1994 and 1995).
Though Popper seemed to think that some single events can have objectively determined probabilities, Gillies (2000: 120–124) argues that many single event probabilities cannot be “fully objective,” owing to the reference class problem, except perhaps in games of chance. Therefore Gillies rejects Popper’s early propensity theory.
But Popper’s early propensity theory of both long-run and single-case probabilities is, however, not the only such theory, and Gillies (2000: 126) divides modern propensity theories into two classes:
(1) long-run propensity theories, andGillies (2000: 136) has developed and defended his own version of a long-run propensity theory of probability, which he sees as the best interpretation of objective probabilities.
(2) single-case propensity theories (Gillies 2000: 126).
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Miller, David. 1994. Critical Rationalism: A Restatement and Defence. Open Court, Chicago.
Miller, David. 1995. “Propensities and Indeterminism,” in A. O’Hear (ed.), Karl Popper: Philosophy and Problems. Cambridge University Press, Cambridge. 121–147.
Popper, Karl R. 1957. “The Propensity Interpretation of the Calculus of Probability, and the Quantum Theory,” in S. Körner (ed.), Observation and Interpretation: A Symposium of Philosophers and Physicists: Proceedings of the Ninth Symposium of the Colston Research Society, held in the University of Bristol, April 1st–April 4th, 1957. Butterworths, London. 65–70.
Popper, Karl R. 1959. “The Propensity Interpretation of Probability,” British Journal for the Philosophy of Science 10: 25–42.
Popper, Karl R. 1990. A World of Propensities. Thoemmes, Bristol.
Wednesday, September 3, 2014
Gillies’ Philosophical Theories of Probability, Chapter 5
Chapter 5 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with the relative frequency theory of probability.
The relative frequency theory was originally developed in the 19th century by John Venn (1834–1923) and Robert Leslie Ellis (1817–1859), and then by the early 20th century empiricists Hans Reichenbach and Richard von Mises (1919 and 1961), who were influenced by the Vienna circle (Gillies 2000: 88).
Richard von Mises saw probability theory as an empirical science dealing with repeated events of the same type, such as in games of chance, biological statistics, and natural phenomena (Gillies 2000: 89). Such repeatable events or mass phenomena were called “collectives” by von Mises (which could be “empirical collectives” or “mathematical collectives”), and a “sample space” is a set of possible outcomes in such a collective.
To establish objective probabilities by means of the relative frequency approach the phenomenon in question must yield stable relative frequencies for the outcomes it exhibits in the long run, and von Mises called this the “law of stability of statistical frequencies” (Gillies 2000: 92).
Richard von Mises thus believed that real numeric probabilities are confined to processes where stable relative frequencies can be obtained, and indeed “probability” in the scientific sense is limited to this (Gillies 2000: 97–98).
Critics of von Mises countered that his frequency theory approach does not explain the qualitative probabilities obtained in inductive arguments (Gillies 2000: 99), and thus seems to be a highly restricted definition of probability.
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Mises, Richard von. 1919. “Grundlagen der Wahrscheinlichkeitsrechnung,” Mathematische Zeitschrift 5: 52–99.
Mises, Richard von. 1961. Probability, Statistics and Truth (2nd edn.). Allen and Unwin, London.
The relative frequency theory was originally developed in the 19th century by John Venn (1834–1923) and Robert Leslie Ellis (1817–1859), and then by the early 20th century empiricists Hans Reichenbach and Richard von Mises (1919 and 1961), who were influenced by the Vienna circle (Gillies 2000: 88).
Richard von Mises saw probability theory as an empirical science dealing with repeated events of the same type, such as in games of chance, biological statistics, and natural phenomena (Gillies 2000: 89). Such repeatable events or mass phenomena were called “collectives” by von Mises (which could be “empirical collectives” or “mathematical collectives”), and a “sample space” is a set of possible outcomes in such a collective.
To establish objective probabilities by means of the relative frequency approach the phenomenon in question must yield stable relative frequencies for the outcomes it exhibits in the long run, and von Mises called this the “law of stability of statistical frequencies” (Gillies 2000: 92).
Richard von Mises thus believed that real numeric probabilities are confined to processes where stable relative frequencies can be obtained, and indeed “probability” in the scientific sense is limited to this (Gillies 2000: 97–98).
Critics of von Mises countered that his frequency theory approach does not explain the qualitative probabilities obtained in inductive arguments (Gillies 2000: 99), and thus seems to be a highly restricted definition of probability.
External Links
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Mises, Richard von. 1919. “Grundlagen der Wahrscheinlichkeitsrechnung,” Mathematische Zeitschrift 5: 52–99.
Mises, Richard von. 1961. Probability, Statistics and Truth (2nd edn.). Allen and Unwin, London.
Tuesday, September 2, 2014
Gillies’ Philosophical Theories of Probability, Chapter 4
Chapter 4 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with the subjective theory of probability, which was developed independently by Frank Ramsey (1903–1930) (1931) and Bruno de Finetti (1906–1985) (Gillies 2000: 50).
The subjective theory takes probability as a mathematical measure in the interval [0, 1] of a person’s degree of belief, but this belief is not necessarily rational not is the probability objective. Different people can hold different subjective beliefs about the probability of the same event.
Both Ramsey and de Finetti devised a method which uses betting to measure subjective belief (Gillies 2000: 54), and the related Ramsey-De Finetti theorem is the idea that a setting of betting quotients is only coherent if and only if they are able to satisfy the axioms of probability (Gillies 2000: 59). A person making a “coherent” bet ensures that a Dutch book cannot be made against him/her.
It is interesting that Ramsey and de Finetti disagreed about the existence of objective probabilities. As Gillies (2000: 69) points out, the fact that we can find real objective probabilities in the world was not denied by Ramsey, who admitted that probabilities are divided into objective and subjective classes. Other more extreme subjectivists like de Finetti insisted that all probabilities are ultimately subjective: indeed de Finetti held the view that objective probabilities are a type of illusion (Gillies 2000: 77).
Gillies (2000: 70–85) reviews the extreme subjectivist views of Finetti and finds them unacceptable: Gillies concludes that there seem to be real objective probabilities and a role for empirical statistics, using relative frequencies (Gillies 2000: 84).
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Ramsey, Frank Plumpton. 1931. The Foundations of Mathematics: and Other Logical Essays (ed. by R.B. Braithwaite). Kegan Paul, London.
The subjective theory takes probability as a mathematical measure in the interval [0, 1] of a person’s degree of belief, but this belief is not necessarily rational not is the probability objective. Different people can hold different subjective beliefs about the probability of the same event.
Both Ramsey and de Finetti devised a method which uses betting to measure subjective belief (Gillies 2000: 54), and the related Ramsey-De Finetti theorem is the idea that a setting of betting quotients is only coherent if and only if they are able to satisfy the axioms of probability (Gillies 2000: 59). A person making a “coherent” bet ensures that a Dutch book cannot be made against him/her.
It is interesting that Ramsey and de Finetti disagreed about the existence of objective probabilities. As Gillies (2000: 69) points out, the fact that we can find real objective probabilities in the world was not denied by Ramsey, who admitted that probabilities are divided into objective and subjective classes. Other more extreme subjectivists like de Finetti insisted that all probabilities are ultimately subjective: indeed de Finetti held the view that objective probabilities are a type of illusion (Gillies 2000: 77).
Gillies (2000: 70–85) reviews the extreme subjectivist views of Finetti and finds them unacceptable: Gillies concludes that there seem to be real objective probabilities and a role for empirical statistics, using relative frequencies (Gillies 2000: 84).
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Ramsey, Frank Plumpton. 1931. The Foundations of Mathematics: and Other Logical Essays (ed. by R.B. Braithwaite). Kegan Paul, London.
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)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
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.
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)
http://plato.stanford.edu/entries/probability-interpret/
BIBLIOGRAPHY
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.
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
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.
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Friday, August 29, 2014
Gillies’ Philosophical Theories of Probability, Chapter 1
Donald Gillies’ Philosophical Theories of Probability (2000) is an excellent overview of probability theory.
The book is of great interest, because Gillies (2000: xiv) has knowledge of Post Keynesian work on probability and uncertainty, and also sees his “intersubjective” theory of probability as a compromise between the theories of Keynes and Ramsey.
Probability has both a mathematical and philosophical/epistemic aspect.
The earliest “Classical” interpretation of probability of Pierre-Simon Laplace (1749–1827), which was based on earlier work from the 1650 to 1800 period, is now of historical interest only, and has no supporters today (Gillies 2000: 3).
Gillies (2000: 1) identifies five major modern interpretations of probability, which are in turn divided into two broad categories, as follows:
The epistemological/epistemic group of probability theories take probability to be a degree of belief, whether rational or subjective (Gillies 2000: 2).
The objective probability theories take probabilities to be an objective aspect of certain things or processes in the external world (Gillies 2000: 2).
Gillies (2000: 2–3) argues that all the major theories of probability may be compatible, as long as they are limited to their appropriate domains: for example, objective probabilities are usually appropriate for the natural sciences and epistemological/epistemic probabilities for the social sciences.
Serious study of probability began with mathematical theories of probability, often inspired by interest in gambling games (Gillies 2000: 4, 10), and these mathematical theories emerged in the 17th and 18th centuries, and famously in the correspondence between Blaise Pascal (1623–1662) and Pierre de Fermat (1601/1607–1665) in 1654 (Gillies 2000: 3), Jacob Bernoulli’s (1655–1705) treatise Ars Conjectandi (1713), the work of Abraham de Moivre (1667–1754), and of Thomas Bayes (c. 1701–1761) (Gillies 2000: 4–8).
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
The book is of great interest, because Gillies (2000: xiv) has knowledge of Post Keynesian work on probability and uncertainty, and also sees his “intersubjective” theory of probability as a compromise between the theories of Keynes and Ramsey.
Probability has both a mathematical and philosophical/epistemic aspect.
The earliest “Classical” interpretation of probability of Pierre-Simon Laplace (1749–1827), which was based on earlier work from the 1650 to 1800 period, is now of historical interest only, and has no supporters today (Gillies 2000: 3).
Gillies (2000: 1) identifies five major modern interpretations of probability, which are in turn divided into two broad categories, as follows:
(i) Epistemological/Epistemic probability theoriesThe “intersubjective” interpretation of probability is developed by Gillies (2000: 2) himself.(1) the logical interpretation;(ii) Objective probability theories
(2) the subjective interpretation (personalism, subjective Bayesianism);
(3) the intersubjective view.(4) the frequency interpretation;
(5) the propensity interpretation.
The epistemological/epistemic group of probability theories take probability to be a degree of belief, whether rational or subjective (Gillies 2000: 2).
The objective probability theories take probabilities to be an objective aspect of certain things or processes in the external world (Gillies 2000: 2).
Gillies (2000: 2–3) argues that all the major theories of probability may be compatible, as long as they are limited to their appropriate domains: for example, objective probabilities are usually appropriate for the natural sciences and epistemological/epistemic probabilities for the social sciences.
Serious study of probability began with mathematical theories of probability, often inspired by interest in gambling games (Gillies 2000: 4, 10), and these mathematical theories emerged in the 17th and 18th centuries, and famously in the correspondence between Blaise Pascal (1623–1662) and Pierre de Fermat (1601/1607–1665) in 1654 (Gillies 2000: 3), Jacob Bernoulli’s (1655–1705) treatise Ars Conjectandi (1713), the work of Abraham de Moivre (1667–1754), and of Thomas Bayes (c. 1701–1761) (Gillies 2000: 4–8).
BIBLIOGRAPHY
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
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