This is an advanced study of systems of propositional logic which offers a comprehensive account of a wide variety of logical systems and which encourages students to take a critical stance towards the subject. A great variety of systems and subsystems are defined and compared as regards their deductive power and relation to their model theory. Interesting features include a more refined treatment of modal logic and the special attention given to the weakenings of classical logic. Useful appendices provide a (...) topical bibliography and review of basic set theory. (shrink)

In books on the calculus of probability, there have been many accounts as to what is the meaning of the term “probable.” We can readily divide them into three groups. The first sometimes defines probability in terms of the ratio between the number of cases favorable to an event and the number of equally possible cases. Sometimes probability is defined in some way other than this, but the above formulation, or one similar to it is used to describe the “measure (...) of probability.” This concept is what is called “the classical concept” of probability and was held by the great workers in the field of probability from the beginning of the eighteenth and opening of the nineteenth centuries, including James Bernoulli, Thomas Bayes, and the Marquis de Laplace. The second conceives of probability as a logical relation between hypothesis and evidence. Following this conception, one could identify “probability” with “degree of confirmation.” We will term this the “logical concept.” The third group identified probability with the relative frequency with which a property occurs in a specified class of elements. We will call this the “frequency concept.”. (shrink)

This essay shows that some recent work by George Weaver can be reformulated in an especially perspicuous way within the theory of closure systems. Closure theoretic generalizations of some theorems of Robert Goldblatt are presented. And, more generally, the relation between closure systems and the deducibility relations of Goldblatt is explored.