In Hailperin 1996 , in addition to its formal development of Probability Logic, there are many sections devoted to historical origins, illustrative examples, and discussion of related work by other authors. Here selected portions of its formal treatment are summarized and then used as a basis for a probability logic treatment of combining evidence.

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between a formal logic language (...) and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

In this paper the strategy for the eliminative reduction of the alethic modalities suggested by John Venn is outlined and it is shown to anticipate certain related contemporary empiricistic and nominalistic projects. Venn attempted to reduce the alethic modalities to probabilities, and thus suggested a promising solution to the nagging issue of the inclusion of modal statements in empiricistic philosophical systems. However, despite the promise that this suggestion held for laying the ‘ghost of modality’ to rest, this general approach, tempered (...) modal eliminativism, is shown to be inadequate for that task.<br><br>. (shrink)

There is a well-documented paradigm-shift in eighteenth century Jesuit philosophy and science, at the very least in Central Europe: traditional scholastic version(s) of Aristotelianism were replaced by early modern rationalism (Wolff's systematisation of Leibnizian philosophy) and early modern science and mathematics. In the field of probability, this meant that the traditional Jesuit engagement with probability, uncertainty, and truthlikeness (in particular, as applied to moral theology) could translate into mathematical language, and can be analysed against the background of the accounts of (...) probability, pre-mathematical Jesuit logic, Wolff's conceptual analysis, and Bernoullian mathematisation. The works of two Jesuit philosophers, Berthold Hauser and Sigismund Storchenau, can be related to this context. The core of their logic of (epistemic) probability is the account of negation (or ‘contradiction’) and implication (or ‘argument’), in particular, the algorithms for computing the reliability of one piece of evidence when compared to the respective counter-evidence and for computing the probability of a conclusion given the probability of its premises. (shrink)

Borel's concept of denumerable probability is described by means of three of his illustrative problems and their solution. These problems are then reformulated in contemporary terms and solved from the viewpoint of probability logic. A section compares Kolmogorov set-theoretic probability with probability logic. The concluding section describes a highly adverse criticism of Borel's conception for its not using something like Kolmogorov theory and, in support of Borel, this criticism is countered from the standpoint of quantifier probability logic.

Psychological research on people’s understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is (...) false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants’ truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people’s interpretations of natural language connectives. We find that de Finetti’s (1936/1995) three-valued system is the best approximation to participants’ truth tables. (shrink)

For Ivor Grattan-Guinness on the occasion of his retirement. The work of Augustus De Morgan on symbolic logic in the mid-nineteenth century is familiar to historians of logic and mathematics alike. What is less well known is his work on probability and, more specifically, the use of probabilistic ideas and methods in his logic. The majority of De Morgan's work on probability was undertaken around 1837???1838, with his earliest publications on logic appearing from 1839, a period which culminated with the (...) publication of his Formal Logic in 1847. This article examines the overlap between his work on probability theory and logic during the earliest period of his interest in both. (shrink)

Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features of subjectivist position. Indeed, (...) the Theorem's central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology. (shrink)

Students often study logic on the assumption that it provides a normative guide to reasoning in English. In particular, they are taught to associate connectives like “and” with counterparts in Sentential Logic. English conditionals go over to formulas with → as principal connective. The well-known difficulties that arise from such translation are not emphasized. The result is the conviction that ordinary reasoning is faulty when discordant with the usual representation in standard logic. Psychologists are particularly susceptible to this attitude.