Coherent, well organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition; Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories in both discussion and exercises. Ideal for undergraduates; no background in math or philosophy required.
First of all, I agree with much of what F.A. Muller says in his article ‘Reflections on the revolution in Stanford’. And where I differ, the difference is on the decision of what direction of further development represents the best choice for the philosophy of science. I list my remarks as a sequence of topics.
Quantum mechanical entangled configurations of particles that do not satisfy Bell’s inequalities, or equivalently, do not have a joint probability distribution, are familiar in the foundational literature of quantum mechanics. Nonexistence of a joint probability measure for the correlations predicted by quantum mechanics is itself equivalent to the nonexistence of local hidden variables that account for the correlations (for a proof of this equivalence, see Suppes and Zanotti, 1981). From a philosophical standpoint it is natural to ask what sort of (...) concept can be used to provide a “joint” analysis of such quantum correlations. In other areas of application of probability, similar but different problems arise. A typical example is the introduction of upper and lower probabilities in the theory of belief. A person may feel uncomfortable assigning a precise probability to the occurrence of rain tomorrow, but feel comfortable saying the probability should be greater than ½ and less than ⅞. Rather extensive statistical developments have occurred for this framework. A thorough treatment can be found in Walley (1991) and an earlier measurement-oriented development in Suppes (1974). It is important to note that this focus on beliefs, or related Bayesian ideas, is not concerned, as we are here, with the nonexistence of joint probability distributions. Yet earlier work with no relation to quantum mechanics, but focused on conditions for existence has been published by many people. For some of our own work on this topic, see Suppes and Zanotti (1989). Still, this earlier work naturally suggested the question of whether or not upper and lower measures could be used in quantum mechanics, as a generalization of.. (shrink)
Bayesian prior probabilities have an important place in probabilistic and statistical methods. In spite of this fact, the analysis of where these priors come from and how they are formed has received little attention. It is reasonable to excuse the lack, in the foundational literature, of detailed psychological theory of what are the mechanisms by which prior probabilities are formed. But it is less excusable that there is an almost total absence of a detailed discussion of the highly differentiating nature (...) of past experience in forming a prior. The focus here is on what kind of account, even if necessarily schematic, can be given about the psychological mechanisms back of the formation of Bayesian priors. The last section examines a detailed experiment relevant to how priors are learned. (shrink)
This article focuses on the role of statistical concepts in both experiment and theory in various scientific disciplines, especially physics, including astronomy, and psychology. In Sect. 1 the concept of uncertainty in astronomy is analyzed from Ptolemy to Laplace and Gauss. In Sect. 2 theoretical uses of probability and statistics in science are surveyed. Attention is focused on the historically important example of radioactive decay. In Sect. 3 the use of statistics in biology and the social sciences is examined, with (...) detailed consideration of various Chi-square statistical tests. Such tests are essential for proper evaluation of many different kinds of scientific hypotheses. (shrink)
In treatises or advanced textbooks on theoretical physics, it is apparent that the way mathematics is used is very different from what is to be found in books of mathematics. There is, for example, no close connection between books on analysis, on the one hand, and any classical textbook in quantum mechanics, for example, Schiff, , or quite recent books, for example Ryder, , on quantum field theory. The differences run a good deal deeper than the fact that the books (...) on theoretical physics are not written in the definition-theorem-proof style characteristic of pure mathematics. Although a good many propositions are proved in the books on physics, there are almost with exception no existential proofs, and consequently there is no really serious systematic use of quantifiers. Another important characteristic is the free use of infinitesimals. In fact, most results would not lose anything, from a physicist's point of view, by leaving them in approximate form, i.e., instead of strict equalities or inequalities, using equalities or inequalities only up to an infinitesimal.The discrepancy between the way mathematics is ordinarily done in theoretical physics and the way it is built up from a foundational standpoint in any of the standard modern views raises the question of whether it might be possible to construct quite directly a rigorous foundation that reflects a significant part of this standard practice in theoretical physics. Other parts of standard practice in physics, for example, the use of physically intuitive but nonrigorous arguments, are not present in our system. (shrink)
PREVIOUS WORK Theoretical discussion of the interval measurement of utility based upon theories of decision making under conditions of risk has been voluminous and will not be reviewed here. Those interested will find extensive ...
We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correlations that do not satisfy the Bell inequalities.
The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of (...) two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined. (shrink)
The role of the concept of invariance in physics and geometry is analyzed, with attention to the closely connected concepts of symmetry and objective meaning. The question of why the fundamental equations of physical theories are not invariant, but only covariant, is examined in some detail. The last part of the paper focuses on the surprising example of entropy as a complete invariant in ergodic theory for any two ergodic processes that are isomorphic in the measure-theoretic sense.
This volume broadens our concept of reasoning and rationality to allow for a more pluralistic and situational view of human thinking as a practical activity. Drawing on contributors across disciplines including philosophy, economics, psychology, statistics, computer science, engineering, and physics, _Reasoning, Rationality, and Probability_ argues that the search for strong theories should leave room for the construction of context-sensitive conceptual tools. Both science and everyday life, the authors argue, are too complex and multifaceted to be forced into ready-made schemata.
The fundamental problem considered is that of the existence of a joint probability distribution for momentum and position at a given instant. The philosophical interest of this problem is that for the potential energy functions (or Hamiltonians) corresponding to many simple experimental situations, the joint "distribution" derived by the methods of Wigner and Moyal is not a genuine probability distribution at all. The implications of these results for the interpretation of the Heisenberg uncertainty principle are analyzed. The final section consists (...) of some observations concerning the axiomatic foundations of quantum mechanics. (shrink)