Bernard Linsky

What is neologicism

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Freges [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

Is Lewis a meinongian

The views of David Lewis and the Meinongians are both often met with an incredulous stare. This is not by accident. The stunned disbelief that usually accompanies the stare is a natural first reaction to a large ontology. Indeed, Lewis has been explicitly linked with Meinong, a charge that he has taken great pains to deny. However, the issue is not a simple one. ‘Meinongianism’ is a complex set of distinctions and doctrines about existence and predication, in addition to the famously large ontology. While there are clearly non-Meinongian features of Lewis’ views, it is our thesis that many of the characteristic elements of Meinongian metaphysics appear in Lewis’ theory. Moreover, though Lewis rejects incomplete and inconsistent Meinongian objects, his ontology may exceed that of a Meinongian who doesn’t accept his possibilia. Thus, Lewis ex-

Leon Chwistek on the no-classes theory in Principia Mathematica

Leon Chwisteks 1924 paper ‘The Theory of Constructive Types’ is cited in the list of recent ‘contributions to mathematical logic’ in the second edition of Principia Mathematica, yet its prefatory criticisms of the no-classes theory have been seldom noticed. This paper presents a transcription of the relevant section of Chwisteks paper, comments on the significance of his arguments, and traces the reception of the paper. It is suggested that while Russell was aware of Chwisteks points, they were not important in leading him to the adoption of extensionality that marks the second edition of PM. Rudolf Carnap seems to have independently rediscovered Chwisteks issue about the scope of class expressions in identity contexts in his Meaning and Necessity in 1947.

Leon Chwistek’s Theory of Constructive Types

From the readily available sources in English one can learn that Leon Chwistek was born in 1884 in Zakopane, studied logic at Göttingen briefly during 1908 and 1909, at Krakow under Ślezyński and Zaremba, and then taught in a secondary school in Krakow for several years.1 After 1929 Chwistek was a Professor of Logic at the University of Lwów in a position for which Alfred Tarski had also applied. His interests in the 1930s were in a general system of philosophy of science, published in 1948 in English as The Limits of Science. Chwistek was also a painter in the Polish “Formist” school of expressionism and figure in the artistic scene of Poland between the world wars. He died in Moscow in 1944 having gone to Russia when the Germans invaded Poland in 1939. This broad outline of his career and work are all that would be known among English speaking philosophers who looked in the few familiar sources on Polish logic.2 For logicians, however, there is more material available. Chwistek’s “scientific correspondence” with Bertrand Russell has been published, and one of his papers is included in Storrs McCall’s collection Polish Logic: 1920-1939.3 Chwistek’s paper “The Theory of Constructive Types”, published in two parts in 1924 and 1925, has scarcely been read at all, though it is widely cited. Most prominent of these citations is in the Introduction to the second edition of Principia Mathematica where Chwistek is mentioned for his proposal to reject the axiom of reducibility. Chwistek is now known among logicians primarily for his unsuccessful argument that the

Propositional Logic from The Principles of Mathematics to Principia Mathematica

Bertrand Russell presented three systems of propositional logic, one first in Principles of Mathematics, University Press, Cambridge, 1903 then in “The Theory of Implication”, Routledge, New York, London, pp. 14–61, 1906) and culminating with Principia Mathematica, Cambridge University Press, Cambridge, 1910. They are each based on different primitive connectives and axioms. This paper follows “Peirce’s Law” through those systems with the aim of understanding some of the notorious peculiarities of the 1910 system and so revealing some of the early history of classical propositional logic. “Peirce’s Law” is a valid formula of elementary propositional logic: [(p ⊃ q) ⊃ p] ⊃ p This sentence is not even a theorem in the 1910 system although it is one of the axioms in 1903 and is proved as a theorem in 1906. Although it is not proved in 1910, the two lemmas from the proof in 1906 occur as theorems, and Peirce’s Law could have been derived from them in a two step proof. The history of Peirce’s Law in Russell’s systems helps to reconstruct some of the history of axiomatic systems of classical propositional logic.

Critical Notice of Richard Gaskin's The Unity of the Proposition

ion — Analysis: Proceedings of the 31st International Ludwig Wittgenstein Symposium in Kirchberg, 2008, Alexander Hieke and Hannes Leitgeb, eds. (Frankfurt: Ontos Verlag 2009) 259-72.