Contributions to the Formal Theory of Probability
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 theorems, previously derived only with the help of an axiom for complementation. The problem of the creativity of the axioms is explored in Section 3, enlarging on (1963). In conclusion, Section 4 explains how these systems, and the full system of (Popper 1959), provide generalizations of the relation of deducibility, contrasting our approach with the enterprise known as probabilistic semantics.
KeywordsBoolean Algebra Distributive Lattice Formal Theory Axiom System Probabilistic Semantic
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