Many people understand the expression “formal logic” as meaning modern mathematical logic by opposition to traditional logic before the revolution that happened in the second part of the 19th century with Boole, Frege and others. But in fact this expression was created by Kant (see Scholz 1931). Some people like to quote a excerpt of the preface of the second edition of the Critic of pure reason (1787), where Kant says that formal logic is a finished and closed science: (...) “logic … has not been able to advance a single step, and hence is to all appearances closed and complete”. Retrospectively, this remark by Kant seems pretty ridiculous. One may wonder how such a wise man could have been so wrong. On the other hand it is quite ironic that the expression created by this philosopher has turned to be used to name the new logic that he was not able to prophesy. Of course “formal logic” is not the only expression used to denote the new logic but it is quite popular and widely spread, maybe because it means several things at the same time. (shrink)
This lecture is cpncerned with the expected-utility or Bayesian model of rationality, with particular attention both to the strengths and limitations of the model. The alternative market and legal models of rationality are examined and rejected as less satisfactory than the expected-utility model. The role of intuitive judgement in the context of actual decision making is stressed. The fundamental place of intuitive judgement in science, especially in the performance of experiments and the analysis and presentation of results is analyzed. Errors (...) of measurement naturally arise in application of the expected-utility model, but there is a long history of theory and practice for dealing with such errors. The existence of such errors constitutes a limitation, not a prohibition, on the use of expected-utility theory as a fundamental framework for rational behaviour. (shrink)
First of all, I agree with much of what F.A. Muller (Synthese, this issue, 2009) 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.
The thesis of this article is that the nature of probability is centered on its formal properties, not on any of its standard interpretations. Section 2 is a survey of Bayesian applications. Section 3 focuses on two examples from physics that seem as completely objective as other physical concepts. Section 4 compares the conflict between subjective Bayesians and objectivists about probability to the earlier strident conflict in physics about the nature of force. Section 5 outlines a pragmatic approach to the (...) various interpretations of probability. Finally, Sect. 6 argues that the essential formal nature of probability is expressed in the standard axioms, but more explicit attention should be given to the concept of randomness. (shrink)
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)
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)
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)
Ordinary measurement using a standard scale, such as a ruler or a standard set of weights, has two fundamental properties. First, the results are approximate, for example, within 0.1 g. Second, the resulting indistinguishability is transitive, rather than nontransitive, as in the standard psychological comparative judgments without a scale. Qualitative axioms are given for structures having the two properties mentioned. A representation theorem is then proved in terms of upper and lower measures.
The author argues for the importance of non-Markovian causality in the social sciences because Markovian conditions often cannot be satisfied. Two theorems giving conditions for non-Markovian causes to be transitive are proved. Applications of non-Markovian causality in psychology and economics are outlined.
This article is mainly concerned with the methodology of public policy studies and how this methodology compares with that of standard scientific studies. The main systematic section of the article develops a concept of justified policy which is related to the concept of justified procedure that originates in ancient Greek mathematics. The last section sketches some ways in which philosophers can make a methodological contribution to policy analysis. Possible contributions are discussed under four headings: numerical models, statistical methodology, philosophy of (...) applications, and analysis of predictions. (shrink)
Three main lines of arguments are presented as a defense of randomization in experimental design. The first concerns the computational advantages of randomizing when a well-defined underlying theoretical model is not available, as is often the case in much experimentation in the medical and social sciences. The high desirability, even for the most dedicated Bayesians, of physical randomization in some special cases is stressed. The second line of argument concerns communication of methodology and results, especially in terms of concerns about (...) bias. The third line of argument concerns the use of randomization to guarantee causal inferences, whether the inference consists of the identification of a prima facie or a genuine cause. In addition, the relation of randomization to measures of complexity and the possibility of accepting only random procedures that produce complex results are discussed. (shrink)
The retreat from the paradise of deterministic causation and the general principles involved in this retreat, which has been forced upon us by quantum mechanics, is described in more or less successive stages.
The perfect fit of syntactic derivability and logical consequence in first-order logic is one of the most celebrated facts of modern logic. In the present flurry of attention given to the semantics of natural language, surprisingly little effort has been focused on the problem of logical inference in natural language and the possibility of its completeness. Even the traditional theory of the syllogism does not give a thorough analysis of the restricted syntax it uses.My objective is to show how a (...) theory of inference may be formulated for a fragment of English that includes a good deal more than the classical syllogism. The syntax and semantics are made as formal and as explicit as is customary for artificial formal languages. The fragment chosen is not maximal but is restricted severely in order to provide a clear overview of the method without the cluttering details that seem to be an inevitable part of any grammar covering a substantial fragment of a natural language. (Some readers may feel the details given here are too onerous.). (shrink)
ELEMENTARY LOGIC GR. C. MOISIL Institute of Mathematics, Rumanian Academy, Bucharest, Rumania 1. We shall consider a typified logic of propositions. ...
Griinbaum's own article sets forth his views on the ontology of the curvature of empty space, especially in the geometrodynamics of Clifford and Wheeler. ...
This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving (...) operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. The second part is concerned with physical space and space-time and deals mainly with topics concerning the axiomatic theory of bodies, the operational foundations of special relativity and the conceptual foundations of elementary physics. (shrink)
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)
This introduction to rigorous mathematical logic is simple enough in both presentation and context for students of a wide range of ages and abilities. Starting with symbolizing sentences and sentential connectives, it proceeds to the rules of logical inference and sentential derivation, examines the concepts of truth and validity, and presents a series of truth tables. Subsequent topics include terms, predicates, and universal quantifiers; universal specification and laws of identity; axioms for addition; and universal generalization. Throughout the book, the authors (...) emphasize the pervasive and important problem of translating English sentences into logical or mathematical symbolism. 1964 edition. Index. (shrink)
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)
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.
