Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

One way to determine the quality and pace of change in a science as it undergoes a major transition is to follow some feature of it which remains relatively stable throughout the process. Following the chosen item as it goes through reinterpretation permits conclusions to be drawn about the nature and scope of the broader change in question. In what follows, this device is applied to the change which took place in logic in the mid-nineteenth century. The feature chosen as (...) the focal point is the categorical syllogism. (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)

Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different (...) formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences. (shrink)

This paper is concerned with De Morgan’s explanation of the validity of arguments that involve relational notions. It discusses De Morgan’s expansion of traditional logic aimed at accommodating those inferences, and makes the point that his endeavour is not successful in that the rules that made up his new logic are not sound. Nevertheless, the most important scholarly work on De Morgan’s logic, and contrary to that De Morgan’s mistake is not beyond repair. The rules that determine his new logic (...) are in fact monotonie replacement rules. And provided they are restricted in the correct way, these rules are demonstrably sound. (shrink)

The introduction has a brief statement, sufficient for the purpose of this paper, which describes in general terms the notion of probability logic on which the paper is based. Contributions made in the eighteenth century by Leibniz, Jacob Bernoulli and Lambert, and in the nineteenth century by Bolzano, De Morgan, Boole, Peirce and MacColl are critically examined from a contemporary point of view. Historicity is maintained by liberal quotations from the original sources accompanied by interpretive explanation. Concluding the paper is (...) a summary. (shrink)

In this note we reproduce the book reviews that De Morgan wrote on Boole's and Jevons's first logical works. The most notable property of these documents is the mere fact of their existence and the absence of any reference to them in the specialized literature.

Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa and Sikorski (1963) for relation algebras generated by a contact relation.