Symbolic Logic as a Continuation of Traditional Formal Logic

  • Evert W. Beth


The aim of this essay is to call attention to the very close historical and systematical connections between traditional or ‘Aristotelian’ and modern or symbolic logic.


Formal Logic Propositional Logic Variable Element Ordinary Language Pure Reason 
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  1. 1.
    At the meeting where the paper which is the original version of this essay was read, it was concluded that the alternative use of the indicative ‘est’ and the subjunctive ‘sit’ has no special significance in scholastic Latin.Google Scholar
  2. 2.
    Cf. Theophrastus’ formulation of the first hypothetical syllogism: εἰ τό A, τό B, εἰ τό B; τό Г; εἰ ᾰρᾰ τό A, τό Г. If A, then B; if B, then C; therefore, if A, then C.Google Scholar
  3. 3.
    Strictly speaking we could consider this definition as per genus proximum et differentiam specificam, by taking the term ‘pair of numbers’ as the genus proximum. But then we would have to define the term ‘pair of numbers’, so that the difficulty returns at once.Google Scholar
  4. 4.
    An example of such a conclusion is: This is not an argument in the mood BARBARA, for then the minor would have to be: ‘The square of 6 is the square of an even number’. But the transition from the premiss ‘6 is an even number’ to this minor cannot be justified in traditional formal logic, since it has not at its disposal a theory of descriptions. A first commencement of such a theory can be found in John Stuart Mill, A System of Logic (Book i, Chapter ii, section 5), in his characterization of connotative individual names.Google Scholar
  5. 5.
    For example, the quality of the Logik that was edited by Jäsche and published under Kant’s name and with his authorization, is strikingly low. Thus modus ponens and modus tollens are introduced already in section 26, without the slightest systematic connection (though it appears from the notes left by Kant that he did not commit this blunder). J.G.C.C. Kiesewetter’s Grundriss einer reinen allgemeinen Logik nach Kantischen Grundsätzen (Outline of a Pure General Logic on Kantian Principles) is on a much higher level. — It is sufficiently known that Kant’s views on formal logic were very conservative (cf. Critique of Pure Reason, 2nd ed., p. viii). Yet now and then one comes across original and profound ideas on logic in his works. Thus in Book 2, Chapter 3, Section 4 of the Transzendentale Dialektik (Transcendental Dialectic), i.e., pp. 620ff. of the 2nd ed. of the Critique, he observes that existence cannot be a predicate. This was also the view held by the Scholastics, as appears from the fact that, following Avicenna, they distinguished between existence and essence. — The little progress that was at first made with the realization of Leibniz’ programme can be partly accounted for by the authority of Kant, who thought this programme quite unfeasible. The lack of a suitable system of symbols, referred to at the meeting, does not, I think, suffice as an explanation. Round about 1800 mathematical notation was already strongly developed, while the achievements of Boethius and Abelard in propositional logic show how much one can do even without many symbols. — Even in our days Kant’s authority is detrimental to symbolic logic.Google Scholar
  6. 6.
    On this controversy, see Lange (1877), pp. 99-100; Trendelenburg (1862), vol. 2, pp. 248 ff.; Ueberweg (1888), pp. 162-163.Google Scholar
  7. 7.
    Even in the more primitive stages of thought (magical, mythical, mystical, emotional thought) it seems that the identity relation does not meet with quite as much repugnance as other relations. This might account for the fact that in these stages of thought all relations were reduced to an identity relation. Even in philosophy various symptoms of this tendency are still perceptible: for example, in the identity philosophies in various periods of the history of philosophy (Parmenides, Spinoza, Schelling) and in the view that only judgments of identity are admissible (Stilpo, Antisthenes, Lotze).Google Scholar
  8. 8.
    In this concise formulation of the principles of propositional logic I have not aimed at ultimate precision. Thus I should, strictly speaking, have introduced special names for the signs by means of which propositional logic is formulated. Cf. pp. 31-2.Google Scholar
  9. 9.
    As far as I know this problem was first formulated by A. Tarski (Tarski, 1956; cf. also Carnap, 1934).Google Scholar
  10. 10.
    In Peano’s notation, appertaining (i.e., the pure inherence of the predicate in the individual substance) is indicated by ‘ε’, while inclusion (i.e., the subsumption of a general concept under a broader concept) is indicated by ⊆. The forms of the two arguments may now be represented as follows: The first argument-form is valid, while the second is not. The following argument-form is also valid: In traditional logic the judgments containing an ‘appartenance’ and an inclusion were distinguished as singular and general, respectively, but this distinction was thought to be of no consequence for the evaluation of arguments. Cf. Kant, Critique of Pure Reason, 1st ed., p. 71; Ueberweg (1888), pp. 170-1. Burkamp (1932, pp. 81-2) does not accept the distinction of appertaining and inclusion, but his counter-arguments tell strongly in favour of the distinction!Google Scholar
  11. 11.
    Compare, for instance, the ways in which Parmenides and Plato set forth their views. — About the origin of formal logic, cf. Kapp (1942).Google Scholar
  12. 12.
    During the discussion something like the following, not uncommon, situation was brought up. A defends, as against B, the aesthetic value of a painting by pointing out a number of peculiarities of this painting. B then shows a second painting manifesting these same peculiarities, though obviously possessing no aesthetic value at all. Has B now refuted A’s arguments, or could A reply: the beauty of my painting resides in the very peculiarities that render yours worthless? By accepting this reply we should indeed paralyse any discussion from the outset. — In this connection I might mention Gaunilo’s striking argument against the ontological proof of the existence of God. If somebody were to reply that you cannot argue about God in the same way that you can argue about an island, then there is no use in going on with the discussion; you may just as well simply establish the existence of God as a dogma. Cf. Beth (1955, 1958b).Google Scholar
  13. 13.
    A similar view is already to be found in Valla (it is discussed in Lange, 1877, pp. 30ff.). For example, Valla puts the ‘honesturn est’ in ‘honestum est pro patria pugnare’ (it is honourable to fight for your country) on a par with modal operators. From the view-point of formal logic, ‘oportet’ (one ought to) is of course completely equivalent to ‘honestum est’ Cp. Grelling (1939).Google Scholar

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© D. Reidel Publishing Company, Dordrecht, Holland 1968

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  • Evert W. Beth

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