Amirouche Moktefi and Stephen Read, Hugh MacColl after One Hundred Years. Special issue: Philosophia Scientiae, Studies in History of Sciences and Philosophy, Editions Kimé, Paris, vol. 15,...

Lewis Carroll offers an interesting perspective on the development of early symbolic logic. On the one hand, he makes a characteristic case of a logician who worked on symbolic methods...

It is well known that Charles L. Dodgson (alias Lewis Carroll, 1832–1898) worked on a logic treatise that would popularise the subject of symbolic logic. The first part appeared in 1896 but the next parts never appeared. It has been claimed that Carroll worked in isolation and did not read the main works of his time. The object of this paper is to inquire what Carroll’s private library teaches us on his readings. The content of this library is known thanks (...) to the sale catalogues that were issued when the library was auctioned at Carroll’s death. This paper provides an overview of the logic books owned by Carroll. Then, it investigates the extent to which Carroll was acquainted with the main logic works of his time. Finally, the paper considers some methodological issues related to the use of ‘library arguments’ in intellectual history. (shrink)

The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

In 1952, Quine showed that the problem of reducing a propositional formula to a simplest normal equivalent can be solved in two steps, viz., (i) express the given formula, Φ, equivalently as the disjunction of all its prime implicants, and (ii) find all non-redundant disjunctions of the latter that are equivalent to Φ (Quine 1952). However, it seems not generally known that an ingenious form of the same two-step process was published by Hugh McColl in 1878.

We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. (...) We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis. (shrink)

We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. (...) We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis. (shrink)