In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Bayesian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations.

The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.