Part 1 Fundamental ideas: the meaning of probability - probability in relation to the theory of knowledge - the measurement of probabilities - the principle of indifference - other methods of determining probabilities - the weight of arguments - historical retrospect - the frequency theory of probability - the constructive theory of part 1 summarized. Part 2 Fundamental theorems: introductory - the theory of groups, with special reference to logical consistence, inference, and logical priority - the definitions and axioms of inference and probability - the fundamental theorems of probable inference - numerical measurement and approximation of probabilities - observations on the theorems of chapter 14 and their developments, including testimony - some problems in inverse probability, including averages. Part 3 Induction and analogy: introduction - the nature of argument by analogy - the value of multiplication of instances, or pure induction - the nature of inductive argument continued - the justification of these methods - some historical notes on induction - notes on part 3. Part 4 Some philosophical applications of probability: the meanings of objective chance, and of randomness - some problems arising out of the discussion of change - the application of probability to conduct. Part 5 The foundations of statistical inference: the nature of statistical inference - the law of great numbers - the use of a priori probabilities for the prediction of statistical frequency - the mathematical use of statistical frequencies for the determination of probability a posteriori - the inversion of Bernoulli's theorem - the inductive use of statistical frequencies for the determination of probability a posteriori - outline of a constructive theory.

http://www.fitelson.org/confirmation/keynes_treatise.pdf

A Treatise on Probability

A Treatise on Probability.

Is the mind, by design, predisposed against performing Bayesian inference? Previous research on base rate neglect suggests that the mind lacks the appropriate cognitive algorithms. However, any claim against the existence of an algorithm, Bayesian or otherwise, is impossible to evaluate unless one specifies the information format in which it is designed to operate. The authors show that Bayesian algorithms are computationally simpler in frequency formats than in the probability formats used in previous research. Frequency formats correspond to the sequential way information is acquired in natural sampling, from animal foraging to neural networks. By analyzing several thousand solutions to Bayesian problems, the authors found that when information was presented in frequency formats, statistically naive participants derived up to 50% of all inferences by Bayesian algorithms. Non-Bayesian algorithms included simple versions of Fisherian and Neyman-Pearsonian inference. Is the mind, by design, predisposed against performing Bayesian inference? The classical probabilists of the Enlightenment, including Condorcet, Poisson, and Laplace, equated probability theory with the common sense of educated people, who were known then as "hommes eclaires." Laplace (1814/ 1951) declared that "the theory of probability is at bottom nothing more than good sense reduced to a calculus which evaluates that which good minds know by a sort of instinct, without being able to explain how with precision" (p. 196). The available mathematical tools, in particular the theorems of Bayes and Bernoulli, were seen as descriptions of actual human judgment (Daston, 1981,1988). However, the years of political upheaval during the French Revolution prompted Laplace, unlike earlier writers such as Condorcet, to issue repeated disclaimers that probability theory, because of the interference of passion and desire, could not account for all relevant factors in human judgment. The Enlightenment view—that the laws of probability are the laws of the mind—moderated as it was through the French Revolution, had a profound influence on 19th- and 20th-century science. This view became the starting point for seminal contributions to mathematics, as when George Boole

/pdf/how-to-improve-bayesian-reasoning-without-instruction-3nxesyrrx9.pdf

How to Improve Bayesian Reasoning Without Instruction: Frequency Formats

Procedures of statistical inference are described which generalize Bayesian inference in specific ways Probability is used in such a way that in general only bounds may be placed on the probabilities of given events, and probability systems of this kind are suggested both for sample information and for prior information These systems are then combined using a specified rule Illustrations are given for inferences about trinomial probabilities, and for inferences about a monotone sequence of binomial pi Finally, some comments are made on the general class of models which produce upper and lower probabilities, and on the specific models which underlie the suggested inference procedures

A generalization of bayesian inference

Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus

Self-taught mathematician and father of Boolean algebra, George Boole (1815–1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as a system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind.

An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities

An Investigation of the Laws of Thought

1. Nature and design of this work 2. Signs and their laws 3. Derivation of the laws 4. Division of propositions 5. Principles of symbolic reasoning 6. Of interpretation 7. Of elimination 8. Of reduction 9. Methods of abbreviation 10. Conditions of a perfect method 11. Of secondary propositions 12. Methods in secondary propositions 13. Clarke and Spinoza 14. Examples of analysis 15. Of the Aristotelian logic 16. Of the theory of probabilities 17. General method in probabilities 18. Elementary illustrations 19. Of statistical conditions 20. Problems on causes 21. Probability of judgements 22. Constitution of the intellect.

/pdf/an-investigation-of-the-laws-of-thought-on-which-are-founded-1xsetw61bg.pdf

An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities / by George Boole.

Self-taught mathematician George Boole (1815–1864) published a pamphlet in 1847 – The Mathematical Analysis of Logic – that launched him into history as one of the nineteenth century's most original thinkers. In the introduction, Boole closely adheres to two themes: the fundamental unity of all science and the close relationship between logic and mathematics. In the first chapter, he examines first principles of formal logic, and then moves on to Aristotelian syllogism, hypotheticals, and the properties of elective functions. Boole uses this pamphlet to answer a well-known logician of the day, Sir William Hamilton, who believed that only philosophers could study 'the science of real existence', while all mathematicians could do was measure things. In essence, The Mathematical Analysis of Logic humbly chides Hamilton and asks him to rethink his bias. Boole is compelling reading for anyone interested in intellectual history and the science of the mind.

George Boole

Papers

An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities

An Investigation of the Laws of Thought

An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities / by George Boole.

The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning

The mathematical analysis of logic