Contributions to the Formal Theory of Probability

  title={Contributions to the Formal Theory of Probability},
  author={Karl Raimund Sir Popper and David W. Miller},
Popper (1959), Appendices *iv and *v) has given several axiom systems for probability that ensure, without further assumptions, that the domain of interpretation can be reduced to a Boolean algebra. This paper presents axiom systems for subtheories of probability theory that characterize in the same way lower semilattices (Section 1) and distributive lattices (Section 2). Section 1 gives a new (metamathematical) derivation of the laws of semilattices; and Section 2 one or two surprising… Expand
Probabilistic Substitutivity at a Reduced Price
One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functorExpand
More Triviality
It is shown that even very weak axiom systems have only a very restricted set of models satisfying a natural generalisation of Adams' thesis, thereby casting severe doubt on the possibility of developing a non-Boolean semantics for conditionals consistent with it. Expand
Adams Conditionals and Non-Monotonic Probabilities
This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength, and shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, but that a standard complementation operation cannot be defined within them, on pain of collapsing probability into bivalence. Expand
Conditional Probability and Dutch Books
There is no set Δ of probability axioms that meets the following three desiderata: (1) Δ is vindicated by a Dutch book theorem; (2) Δ does not imply regularity (and thus allows, among other things,Expand
Can Bayes' Rule be Justified by Cognitive Rationality Principles?
The justification of Bayes' rule by cognitive rationality principles is undertaken by extending the propositional axiom systems usually proposed in two contexts of belief change: revising andExpand
Can Quantum Mechanics be shown to be Incomplete in Principle
The paper presents an argument for the incompleteness in principle of quantum mechanics. I introduce four principles (P0–P3) concerning the interpretation of probability, in general and in quantumExpand


Necessary and sufficient qualitative axioms for conditional probability
In a previous paper (Suppes and Zanotti, 1976) we gave simple necessary and sufficient qualitative axioms for the existence of a unique expectation function for the set of extended indicatorExpand
Probabilistic, truth-value, and standard semantics and the primacy of predicate logic
  • John A. Paulos
  • Mathematics, Computer Science
  • Notre Dame J. Formal Log.
  • 1981
A proof of the equivalence of these two semantic approaches is given which also demonstrates their equivalence to another nonreferential semantics, the truth-value (or substitution-theoretic) semantics of Leblanc, Dunn and Belnap, and others. Expand
Alternatives to Standard First-Order Semantics
Alternatives to standard semantics are legion, some even antedating standard semantics, among them: substitutional semantics, truth-value semantics, and probabilistic semantics, which are to be studied here. Expand
What Price Substitutivity? A Note on Probability Theory
Teddy Seidenfeld recently claimed that Kolmogorov's probability theory transgresses the Substitutivity Law. Underscoring the seriousness of Seidenfeld's charge, the author shows that (Popper'sExpand
Creative and Non-Creative Definitions in the Calculus of Probability
It was in 1935, in Paris, at a congress for ‘scientific philosophy’, that I met Woodger first - also Bertrand Russell, Susan Stebbing and Freddy Ayer. Woodger read a paper which I still rememberExpand
On Carnap and Popper Probability Functions
A comparison of Pr(A, A) and Pr(B, B) for at least one wff C of PC, B-requirements, where A is for A and B is for B. Expand
Birkhoff and von Neumann's Interpretation of Quantum Mechanics
Birkhoff and von Neumann's famous paper, “The Logic of Quantum Mechanics”, culminates in a proposal which clashes with each of a number of assumptions made by the authors. A thought experiment,Expand
The Logic of Scientific Discovery
Described by the philosopher A.J. Ayer as a work of 'great originality and power', this book revolutionized contemporary thinking on science and knowledge. Ideas such as the now legendary doctrine ofExpand
A “definitive” probabilistic semantics for first-order logic
  • K. Bendall
  • Mathematics, Computer Science
  • J. Philos. Log.
  • 1982