Grattan-Guinness The application of mathematical method was made possible by the discovery of structural similarities between classical logic and algebra. Insofar as algebraic logic applies a method borrowed from mathematics, it subsumes logic under mathematics. Due to this mathematical treatment, logical propositions figure in algebraic logic mainly in an uncommon form, namely as equations. Its main subject hence is the theory of equations and the development of functions. Relational logic was essential for the logicist programme of reducing mathematics to logic, i.

In so far, the rise of relational logic turned the relation of logic and mathematics upside down.

In the years from onward Couturat became an ardent advocate of the logicist programme. The study of algebraic logic was nevertheless of philosophical importance for Couturat, as I will show in the following sections. Introduction 9 V. His concern was not to invent, but to identify the results of recent research in order to keep up a vivid thinking against the conservatism of the institutions.

This image indeed seems to be fair in a number of its characterisations. Couturat avoided these latter problems simply by abandoning any absolute meaning of the notion of universe. According to him the universe is always to be understood relatively to a given problem ch. In ch. The necessity of the distinction between inclusion 10 Introduction must have been quite exceptional at that time1.

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He left out all the doubts and difficulties, all the places where consistency had led me into paradox, and everything which he imagined calculated to shock, and at the same time put the thing forward as a dogmatic doctrine finally solving a host of difficulties.

In consequence he made me appear absurd, and took to the international language, which now occupies him wholly. Huntington and Cesare Burali-Forti. A remarkable exception to this absence of modern logic is presented by Louis Liard — who published a book on contemporary English logicians in Couturat a, p. Dassen , p. I am grateful to Kenneth Blackwell and Carl Spadoni for transcribing the quotation and for granting the permission to publish.

For a French translation cf. Schlaudt b, p. Introduction 11 tions cf. Since Couturat himself constantly spells out these debts, itemising them here in detail does not promise to yield unforeseen insights. It is governed by several particular preferences or decisions, which are held together by a general aspect. The particular preferences are 1. This framework allowed Couturat, as we will see, to conceive the study of logic as a way to study the mind itself.

Measurement theory had become an important issue during the nineteenth century, primarily in consequence of two important developments in science: on the one hand the emergence of quantitative psychology due to Herbart and Fechner, and on the other hand the growing interest in the foundations of mathematics that led, among other things, to an axiomatization of quantity.

Erkenntnistheoretisch betrachtet from cf. Schlaudt a, pp. In this book Couturat studied the concept of number on the one hand and the concept of quantity or magnitude on the other hand. In his reply b Couturat showed that the alleged antinomies can in large part be resolved simply by attentively distinguishing between kinds of quantities, determinate states of quantity, numerical expressions of quantities, and concrete instantiations of abstract quantities.

The basic idea of this book is, that the concepts of number and quantity are independent of one another and that both are independent of experience. This comparative study offered to Couturat the idea of alternative ways of formalisation which, though they do not proceed in a quantitative manner, permit rigorous reasoning.

These formulae characterise two different algebras proceeding according to different rules: The first characterises the quantitative calculus, in which taking a two times gives 2a, thus leading to numerical expressions; the second characterises the logical calculus, in which considering a a second time does not add anything to a ch. Classical logic can therefore rightly be considered as a qualitative algebra. In turn traditional algebra could be considered as the logic of quantity. In Leibniz this endeavour was coupled 1 d, p. Peckhaus Introduction 13 with a critique of Spinoza and Descartes, who, according to Leibniz, had failed to extend the geometrical method to metaphysics a, p.

The subject of his critique was nota bene not the attempt to subject other topics to a rigorous and secure method, but the failure to examine the preconditions of these efforts.

This lack of methodology resulted in a procedure more geometrico, which merely imitated the mathematical method without establishing a real calculus in metaphysics. One of the noticeable traits of this work is its distinction between a logical and a rational point of view: According to Couturat, concept formation is essentially underdetermined by logic; there are always various ways to form concepts and to extend theories which are equally possible from the point of view of logic.

They thus demand a supplementary justification from a rational point of view cf. Bowne , p. This justification is provided by the criterion of applicability to quantity. Whole numbers, for example, logically introduced as sets of pairs of integers, can be regarded as numbers in so far as there are quantities adequately representable by them. The same reasoning holds, following Couturat, with regard to the logical calculus. In his review of Whitehead d, p. This is the task Spinoza and Descartes missed. I will now examine the particular consequences that Couturat drew from this general idea for the conception of his treatise on symbolic logic.

Calculus vs. This mechanical reasoning not only rules out error, but also permits us to save human intellectual power or, as one may put it, permits us to effectuate complicated logical inferences which surpass our limited natural powers cf. Ordinary languages on the contrary are afflicted with equivocality, inexactness, and ambiguity. This is the opposition as Couturat sees it. Johnson: As a material machine is an instrument for economising the exertion of force, so a symbolic calculus is an instrument for economising the exertion of intelligence.

And, employing the same analogy, the more perfect the calculus, the smaller would be the amount of intelligence applied as compared with the results produced. According to Grossmann, concepts gained from technique can in turn be applied to nature, as the editors point out p. As regards our case, we find a similar situation: The logical calculus as well as logic machines thus provide a model for the human intellect as regards its faculty to reason logically. Couturat , p. Introduction 15 ical and formal reasoning.

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This idea becomes even more evident in regard to early clocks and in regard to machines from antiquity whose inventors did not succeed in mechanising the whole procedure to be effectuated. In this case of partial mechanisation, a repeated intervention of a keeper is necessary, i.

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Here a lack of mechanical performability is interpreted as indicating an imperfect formalisation. Mechanical and formal reasoning are thus equated. As regards the advantages of the calculus compared to ordinary language — mechanical rigour vs. His distinction between ordinary language and calculus however rests on a remarkable philosophical conception. His arguments against psychologism are at first sight common too: Logic is the normative science of correct reasoning whereas psychology can at best be the natural history of the soul.

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Semidecidability of predicate logic, formulated by Thoralf Skolem and Jacques Herbrand in the s, nevertheless provided a basis for further research on the mechanization of deductive reasoning. The appearance of computer technology in the s additionally stimulated this research and efforts towards automated theorem provers cf. Marciszewski and Murawski , p. Kusch As Kusch points out, Husserl observed that the normative-descriptive-distinction alone does not provide a conclusive argument against psychologism, since thought as it ought is a special case of thought as it in fact occurs p.

In Couturat too this distinction is a mere preliminary to the key argument. Couturat drew the conclusion that reasoning cannot be studied empirically unless it has been externalised, for example in the form of language, i. It is however a difficult question whether Couturat performed a linguistic turn, i. We can thus state a first result concerning the philosophical significance of algorithmic logic for Couturat: Algorithmic logic was indeed a promising candidate for studying the objective structure of the mind.

However, this approach does not fit the idea of a linguistic turn. Remember that at that time Couturat conceived logic as opposed to language.

Couturat considered natural language rather as an obstacle to precise reasoning, and it is the aim of his analysis to show the advantages of the calculus as compared to ordinary language. Introduction 17 — for all that — the most adequate. He expresses the conviction that the mind esprit indeed impresses its pattern on language by continuously reshaping it.