Showing papers in "History and Philosophy of Logic in 1988"

On the textual authenticity of kant's logic

Abstract: Philological background information is presented on the origin and composition of the text generally known as Kant's Logic. The text, which was not in the strict sense of the word written by Kant himself, but rather assembled by another writer whom Kant had authorized to do so on his behalf, is a mixture of materials, not all of which originate directly from Kant, and cannot claim full authenticity.

Beltrami's model and the independence of the parallel postulate

Abstract: E. Beltrami in 1868 did not intend to prove the consistency of non-euclidean plane geometry nor the independence of the euclidean parallel postulate. His approach would have been unsuccessful if so intended. J. Houel in 1870 described the relevance of Beltrami's work to the issue of the independence of the euclidean parallel postulate. Houel's method is different from the independence proofs using reinterpretation of terms deployed by Peano about 1890, chiefly in using a fixed interpretation for non-logical terms. Comparing the work of Beltrami and Houel with the treatment of non-euclidean geometry after the development of the axiomatic method in the 1890s indicates an important shift in mathematicians’ attitudes towards mathematical theories.

Die kontroverse um die intuitionistische logik vor ihrer axiomatisierung durch heyting im jahre 1930

Abstract: Brouwer's criticism of mathematical proofs making essential use of the tertium non datur had a surprisingly late response in logical circles. Among the diverse reactions in the mid 1920s and early 1930s, it is possible to delimit a coherent body of opinions on these questions: (1) whether Brouwer's denial of the tertium non datur meant only the abandonment of this classical law or, beyond that, the affirmation of its negation; (2) whether one or both of these alternatives were logically inconsistent; and (3) whether Brouwer's line of argument was forced to take resort to the very law it was designed to refute. The controversy centred around a series of articles by Marcel Barzin and Alfred Errera who fought against the intuitionistic critique, missed their victory because of conceptual confusions and fallacious reasoning, but emerged unconvinced from the debate in the late 1930s. The controversy is of interest to the historiography of formal logic since it stimulated the clarification not only of the conce...

The development of probability logic from leibniz to maccoll

Abstract: 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.

On continuity: aristotle versus topology?

Abstract: This paper begins by pointing out that the Aristotelian conception of continuity (synecheia) and the contemporary topological account share the same intuitive, proto-topological basis: the conception of a ‘natural whole’ or unity without joints or seams. An argument of Aristotle to the effect that what is continuous cannot be constituted of ‘indivisibles’ (e.g., points) is examined from a topological perspective. From that perspective, the argument fails because Aristotle does not recognize a collective as well as a distributive concept of a multiplicity of points. It is the former concept that allows contemporary topology to identify some point sets with spatial regions (in the proto-topological sense of this term). This identification, in turn, allows contemporary topology to do what Aristotle was unwilling to do: to conceive the property of continuity, as well as the properties of having measure greater than zero and having n- dimension, as emergent properties. Thus, a point set can be continuous (conn...

Functional operations in frege's begriffsschrift

Abstract: Frege uses Greek letters in two different ways in his Begriffsschrift. One way is the familiar use of bound variables, in conjunction with variable-binding operators, to mark and close argument-places. The other, which is quite unfamiliar, employs letters to mark places for operators to reach into, without thereby closing these places. Frege thereby invents a powerful and compact notation for functional operations which can be recommended even today. His regrettable double use of Greek letters obscured his invention, and this, together with the fact that in the Grundgesetze he no longer has need of function-valued functions, explains why the device was overlooked and has not passed into general use.

Brouwer’s Equivalence Between Virtual and Inextensible Order

Abstract: Brouwer’s theorem of 1927 on the equivalence between virtual and inextensible order is discussed. Several commentators considered the theorem at issue as problematic in various ways. Brouwer himself, at a certain time, believed to have found a very simple counterexample to his theorem. In some later publications, however, he stated the theorem in the original form again. It is argued that the source of all criticisms is Brouwer’s overly elliptical formulation of the definition of inextensible order , as well as a certain ambiguity in his terminology. Once these drawbacks are removed, his proof goes through.

Karl eugen müller (1865–1932) und seine rolle in der entwicklung der algebra der logik

Abstract: Karl Eugen Muller's contribution to the development of the algebra of logic is perhaps the most important part of his scientific work. Muller, who became Gymnasialprofessor after his university studies, was a student of Ernst Schroder's friend, the mathematician Jakob Luroth. As a result of publishing two papers on problems related to Schroder's monumental Vorlesungen iiber die Algebra der Logik, Muller was commissioned by the Deutsche Mathematiker- Vereinigung with the editing of the unpublished parts of the Vorlesungen from Schroder's Nachlaβ. Muller worked on Schroder's papers until 1910, but did not bring this work to a conclusion. Muller's own Nachlaβ, including those parts of Schroder's papers still in his possession, was destroyed in Frankfurt a.M. in 1943, so there remains no hope of finding through Muller any part of the missing Nachlaβ of Schroder.

Interpreting bradley: the critique of fact-pluralism

Abstract: The typically dismissive treatment of Bradleian idealism, to the extent that it is based on philosophical criticism rather than historical bias, suffers from a failure to distinguish Bradley's negative views from his positive doctrines. But the intermingling of the two plays havoc in Bradley's own presentation, so that proper interpretation requires a particularly aggressive approach to the texts. Specifically, in denying a real multiplicity of facts, Bradley, though he may seem to be, is not attacking the commonsense belief that there are many and disparate facts. His claim, as is confirmed by an examination of the analysis of judgement in The principles of logic, is that the facts ordinarily recognized are not those of the bona fide fact-pluralist, e.g. Mill. By getting Bradley's position straight, it becomes possible to tell an illuminating story about the early formation of ‘analytic’ philosophy, with its often bewildering faith in the ontological significance of logic.

A new symbolic representation of the basic truth-functions of the propositional calculus

Abstract: As with mathematics, logic is easier to do if its symbols and their rules are better. In a graphic way, the logic symbols introduced in this paper show their truth-table values, their composite truth-functions, and how to say them as either ‘or’ or ‘if … then’ propositions. Simple rules make the converse, add or remove negations, and resolve propositions.

How many syllogisms are there

Abstract: The incompleteness and artificiality of the ‘traditional logic’ of the textbooks is reflected in the way that syllogisms are commonly enumerated. The number said to be valid varies, but all the numbers given are of a kind that logicians should find irritating. Even the apparent harmony of what is almost invariably said to be the total number of syllogisms, 256, turns out to be illusory. In the following, it is shown that the concept of a distribution-value, which is related to the traditional theory of distribution, and the familiar concept of quantity together suffice to produce a far better way of enumerating syllogisms and a more complete understanding of the systematic features of syllogistic logic.