Abstract
Since science detached itself from philosophy at the beginning of the modern period and took on the form of a multiplicity of single sciences, the idea of unified science and of a universal scientific method has continued to attract leading thinkers. Leibniz, who realized the importance not only of a certain and fruitful method of inference but also of a suitable symbolization for the progress of science, presented a famous program for the construction of such a scientia universalis. Therein he sought “a characteristic of reason, in function of which the truths of reason — like those of arithmetic and algebra — could be attained through a calculation, to the extent that they are subject to inference”.1
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References
Gerh. Phil., vii, 32. Bochexíski, FL, 38.10.
Ms. Phil. vii 185, and C 160.
Gerh. Phil., vii, 204.
Ibid., 206.
Gerh. Phil., vii, 14. Cf. ibid., 21f., 49, 64, 157, 198ff.
Raymundus Lullus, ‘Ars magna et ultima’, in Raymundi Lullii opera ea quae ad adinventam ab ipso artem universalem… pertinent, Strassburg 1617. — Athanasius Kircher, Polygraphia nova et universalis, ex combinatoria arte detecta, Roma 1663 (Named already in Leibniz’ Dissertatio de arte combinatoria; cf. Gerh. Phil., iv, 72. Leibniz also refers to an exchange of letters with Kircher in a 1670 letter to Oldenbourg; cf. Gerh. Phil., vii, 5). — George Dalgarno, Ars signorum, vulgo character universalis et lingua philosophica etc., London 1661 (cf. Gerh. Phil., vii, 7). — John Wilkins, Mercury, or the secret and swift Messenger: shewing how a Man may with Privacy and Spead communicate his Thoughts to a Friend at a Distance, London 1641. — Isdem: An Essay towards a Real Character and a Philosophical Language, with an alphabetical Dictionary, London 1668.
We see no reason for continuing today to stress the distinction between `logic’ and `logistic’. However, it is historically useful.
Scholz, Mathesis universalis, 146.
Ibid., 144. At the International Congress for Scientific Philosophy in Paris in 1935 Scholz remarked that “the exact logic which Leibniz… was the first to conceive essentially implies everything that we require today… from a logistic logic: a set of signs basically suitable for the presentation of all scientific statements (characteristica universalis) a system of transformation rules based on this set of signs (calculus ratiocinator) and a theory of definition which exactly controls the introduction of new signs (ars combinatoria)” (quoted from J. Ritter in Scholz, Mathesis universalis, 8f.). This interpretation of the ars combinatoria is, to my mind, not to be found in Leibniz himself.
We have in mind the otherwise excellent, short presentation `On the Early History of Logistic’, in Günther Jacoby, Die Ansprüche der Logistiker auf die Logik und ihre Geschichtschreibung [The Pretensions of the Logisticians on Logic and the Writing of its History], Stuttgart 1962.
Hermann Grassmann, Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik [Geometrical Analysis in Combination with the Geometrical Characteristic Invented by Leibniz], Leipzig 1847.
Ms. BRL, 3. Trendelenburg, from whom Frege seems to have taken the form ‘lingua characterica’ (instead of ‘characteristica’), presents the expression as Leibnizian, although it probably goes back to a heading of a Leibniz fragment, which was added by Raspe.
Scholz, Mathesis universalis, 271.
Ibid.
Halle a.S. 1879. Reprinted by I. Angelelli with the comments of E. Husserl and H. Scholz; Hildesheim (Georg Olms) or Darmstadt (Wiss. Buchgesellschaft) 1964.
Marcus Bierich, Freges Lehre von dem Sinn und der Bedeutung der Urteile und Russells Kritik an dieser Lehre [Frege’s Doctrine on the Sense and Reference of Judgements and Russell’s Critique Thereof], Hamburg 1951 (dissertation).
Frege assumes that the reader of the Begriffsschrift has some notion of what a judgement is in logic. Fortunately, the difficulty that traditional logic variously understood ‘judgement’ can be skirted in the present instance. There are many reasons for limiting the present case to Kant’s notion of judgement and to Lotze’s interpretation of Kant. This is what Bierich assumes when he tries to show that in the Begriffsschrift Frege uses ‘judgement’ for ‘assertory judgement’. The assumption is all the stronger since Frege indiscriminately uses the two expressions throughout.
In the Basic Laws of Arithmetic Frege introduces other forms of inference, but they are all derived from the modus ponens, which thus appears as basic and sufficient. Any limitation depends on the object one has in mind. Where sentences in the Begriffsschrift are to be proved, the process will be essentially shortened by the introduction of other forms. In investigations on the Begriffsschrift it will be simpler to have a single form of inference.
Used by Frege in GI., 83, but possibly not as a term (cf. WBBs., 52) so that any relation to the conceptual construction of the same name in Russell and Wittgenstein remains an open question. On the concept of ‘logical form’, see the critical comments of Y. Bar-Hillel, ‘Comments on Logical Form’, Philosophical Studies 2 (1951), 26–29.
We are referring to W. and M. Kneale, The Development of Logic, Oxford 1962. There (p.489) the transition There (p. 489) the transition with the limitation in the latter that a does not occur in A. If one adds to the Fregean kernel, in addition to the modus ponens, these two rules and the ‘ substitution rule, which was not formulated as such by Frege in the Begriffsschrift, one gets - according to Kneale - a complete system of axioms of a predicate calculus of the first degree.
Lukasiewicz has shown that Axiom 3 can be had from the others, and even from 1 and 2 alone (cf. J. Lukasiewicz, ‘Z historii zdan’, Przeglqd Filozoficzny 37 (1934), 417-437 (German’ ‘Zur Geschichte der Aussagenlogik’, Erkenntnis 5 (1935/ 36), 111-131). Of the remaining five axioms, none is to be had from the others. The reulting independence of the system of axioms (1, 2, 4, 5, 6) can be shown with the help of the following quasi-valuations where 0 is the designated value:
There is currently a widespread opinion that Frege’s logical symbolism is difficult and hard to handle. Strangely enough, one considers the two-dimensional character of the Begriffsschrift to be a disadvantage while it is precisely this that gives it the advantage over other logical systems of signs. It allows a presentation of the relations of predicate and propositional logic, free of brackets and points, which is of a clarity reached by linear systems only by putting the single lines under each other, i.e., by resorting to the same second dimension. One who is familiar with the ease of using Frege’s Begriffsschrift will not be surprised that it is finding employment in an essentilly two-dimensional field like the algebra of circuitry (W. Hoering, ‘Frege and die Schaltalgebra’, AMLG 3 (1957), 125-126). It is certain that, in addition to habit, only the ease of typesetting the linear system has led to its success (cf. H. Schnelle, Zeichenysteme zur wissenschaftlichen Darstellung. Ein Beitrag zur Entfaltung der Ars characeristica im Sinne von G. W. Leibniz [Sign-Systems for Scientific Presentation. A Conribution to the Development of the Ars Characteristica in the Meaning of G. W. Leibniz], Stuttgart-Bad Cannstatt 1962).
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Thiel, C. (1968). The Notion of the Begriffsschrift . In: Sense and Reference in Frege’s Logic. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2981-9_2
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