Sentences of the Begriffsschrift are constructed from basic (atomic) sentences. In a major break with the Aristotelian tradition, these are not necessarily of subject/predicate form. They are constituted by a verb phrase and the appropriate number of noun phrases, thus, for example: Sm, Ljm (which might express the claims, respectively, that Mary sings and that John loves Mary). (In the symbolism, and conventionally, the verb phrase is written at the start of the sentence.) The objective content of a noun phrase is the object it denotes. The objective content of a verb phrase, Frege calls a concept. This is a function in the mathematical sense. There are two special objects called truth values: the true, t, and the false, f. The content of a verb phrase is a function that maps the appropriate number of objects to one of these. Thus, the content of ‘S’ might be a function that maps an object to t iff that object is singing. And the content of ‘L’ might be a function that maps a pair of objects to t iff the first loves the second. The content of the whole sentence is the truth value you get when you apply the function which is the content of the verb phrase to the objects which are the contents of the noun phrases.
The rest of the sentences in the Begriffsschrift are generated from the atomic sentences by applying various grammatical constructions, which can be iterated recursively. The first kind of construction comprises connectives: ¬ (it is not the case that), ⊃ (if…then…34), ∧ (and), ∨ (or). (Some of these can be defined in terms of others; Frege takes ¬ and ⊃ as basic.) Traditional logic (though not medieval logic) recognizes only two binary connectives (∨ and ⊃) and does not iterate them. But Frege, following the algebraists, and mindful of what mathematicians need to express, was well aware that it makes perfectly good sense to say things of the form A ⊃ (B ⊃ C).
The objective content of a connective is a function, and the content of a sentence formed by a connective applied to some sentences is obtained by applying the function which is the content of the connective to the truth values which are the contents of the sentences. Thus, the content of ¬ is a function which maps t to f and vice versa. The content of ⊃ is a function that maps the pair to f iff a is t and b is f; other pairs of truth functions get mapped to the value f.
The other kind of grammatical construction involved in generating complex sentences comprises quantifiers. This constitutes another, and perhaps the most significant, break from traditional logic. For Aristotle, quantifier phrases such as ‘some man’ and ‘no woman’ are of the same grammatical kind as noun phrases such as ‘John’ and ‘Mary’. But once relations enter the picture this leads to problems. Thus, ‘every man loves some woman’ is ambiguous, depending on whether it means ‘every man loves some woman or other’ (maybe his mother), or it means that there is some woman whom every man loves (same woman in each case, maybe the Virgin Mary). How to account for this ambiguity?
Given any sentence, A(n), containing a noun phrase, n, we can replace this with a variable, x, to obtain A(x). We can then prefix this with a quantifier phrase ∀x, ∃x (all x are such that, some x is such that). ∀xA(x) is true (i.e. has the content t) just if whatever object we were to take x to refer to, A(x) would be true. Similarly, ∃xA(x) is true just if there is some object we can take x to refer to which would make A(x) true. The ambiguity noted is then explained by the order in which the quantifier phrases are applied. Thus, the difference is that between ∀x∃yRxy and ∃y∀xRxy.
It is worth noting that this sort of ambiguity had played havoc in mathematics in the period leading up to Frege. A (real-valued) function, f, is continuous (smooth) if for every ɛ, however small, some δ is such that if you make the difference between x and y less than δ, the difference between f(x) and f(y) will be less than ɛ. Note (as the italics show) that this is one of those ambiguous sentences containing a universal and a particular quantifier. The ambiguity corresponds to the difference between (plain) continuity and uniform continuity.35 These two notions have somewhat different mathematical properties, and mathematicians had been befuddled by the difference, though they had got it straight by Frege’s time. It may well be that Frege realized the need for his analysis of quantifiers by reflecting on this kind of situation.
The sorts of quantifiers I have been talking about so far are first-order, where we quantify over objects. The Begriffsschrift also has second-order quantifiers, where we quantify over concepts. Given any sentence, A(N), containing a verb phrase, N (let us suppose that this is monadic, to keep things simple), we can replace this with a different kind of variable, X, to obtain A(X). We can then prefix this with a quantifier phrase ∀X or ∃X. ∀XA(X) is true just if whatever concept we take X to refer to, A(X) is true; and ∃XA(X) is true just if there is some concept we can take X to refer to which makes A(X) true.
A word on notation. I have, in discussing Frege, as for the other thinkers I have discussed, used contemporary notation. Frege’s actual notation in the Begriffsschrift, though, is highly unusual. (An example is given in Figure 20.1.) In particular, it is two-dimensional. Thus he writes A ⊃ B as a horizontal line with B at the right-hand end of it; descending from the horizontal, there is a capital ‘L’ shape, with A at the bottom right-hand end of it. This is hard for most people with a standard training in mathematics to read, and it did not catch on. (The notation currently in use derives essentially from that developed by the Italian mathematician Giuseppe Peano.)
FIG. 20.1 An example of Begriffsschrift.
So much for the language of the Begriffsschrift. In virtue of its contents there will be some sentences that are true whatever the noun phrases and verb phrases in them refer to. These are the logical truths. Frege provided an axiom system for these. He specified a number of axioms, and rules of inference for inferring one sentence from another.36 For example, one axiom was A ⊃ (B ⊃ A), and modus ponens was a rule of inference: from A and A ⊃ B infer B. Frege was quite clear that axioms and rules of inference are different kinds of things. Logicians, even of the stature of Russell, however, standardly confused them until Hilbert and his school systematized the theory of formal systems in the 1920s.
Remarkably, it later turned out that Frege’s axiom system was complete. That is, if we ignore formulas with second order quantifiers, everything that is logically valid can be proved in Frege’s axiom system. The result was proved by Gödel in the 1920s. Frege had no way of addressing this question, though, or even of framing it properly, since it depends on a notion of validity developed only in the early twentieth century, essentially by Tarski (but pretty much that of Bolzano). A corollary of another of the significant results proved in the 1930s by Gödel established that once second order quantifiers are on board, no axiom system can do this completely. All this, however, belongs to the logical history of the twentieth century.
As the nineteenth century itself was coming to an end, we find Schröder and Frege debating which of them could best claim to have inherited Leibniz’ logical mantle.37 The answer, in the end, is, it seems to me, mainly of interest to Leibniz scholars: the facts about what each of them had achieved are clear enough.38
20.9 CONCLUSION
History rarely runs smoothly. As Hegel observed, the cunning of reason has strange ways of its own. Frege’s major project, to show that the truths about whole numbers, and hence about all sorts of numbers, were part of logic, crashed and burned spectacularly, due to the discovery of what has come to be known as Russell’s paradox. But nothing of this bore on the success of the Begriffsschrift in its own right. What was supposed to be but a means to an end turned out to be perhaps the most significant development in two millennia of logic. And even here: Frege’s work was transmitted into the twentieth century by Russell and Wittgenstein, and their overlay served to obscure it. It was not until the middle of the twentieth century that Frege’s achievements came to be generally appreciated.
Of course, nothing comes from nothing. And the developments Frege produced would have been impossible without all that had gone before, including the work of Leibniz, the turmoil in logic po
st-Hegel, the work of the algebraists, and developments in nineteenth-century mathematics. By the end of the century, however, the third great phase in the development of Western logic was well set in place. The nineteenth century had witnessed a fundamental rupture in logical history; and German thought had played a major role in this.
BIBLIOGRAPHY
Bell, J. L. (2012), ‘Infinitary Logic’, Stanford Encyclopedia of Philosophy, available at http://plato.stanford.edu/entries/logic-infinitary/.
Burbidge, J. (2004), ‘Hegel’s Logic’, in Gabbay, D., and Woods, J. (eds.), The Rise of Modern Logic from Leibniz to Frege, Oxford: Oxford University Press, pp. 131–75.
Bynum, T. (1972), Gottlob Frege. Conceptual Notation and Related Articles, Oxford: Oxford University Press.
Gabbay, D., and Woods, J. (eds.) (2004), The Rise of Modern Logic from Leibniz to Frege, Vol. III of Gabbay and Woods (eds.), Handbook of the History of Logic (2004–12), Amsterdam: Elsevier.
Gabbay, D., and Woods, J. (eds.) (2004–12), Handbook of the History of Logic, Vols. I–IX, Amsterdam: Elsevier.
Geach, P. and Black, M. (trans.) (1952), Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell.
George, R. (trans.) (1972), Bernard Bolzano. Theory of Science, Berkeley, CA: University of California Press.
Grattan-Guiness, I. (2000), The Search for Mathematical Roots, 1870–1940. Logics, Set Theories, and the Foundations of Mathematics from Cantor through Russell to Gödel, Princeton, NJ: Princeton University Press.
Guyer, P., and Wood, A. (trans.) (1998), Critique of Pure Reason, Cambridge: Cambridge University Press.
Haaparanta, L. (ed.) (2009), The Development of Modern Logic, Oxford: Oxford University Press.
Hailperin, T. (2004), ‘Algebraic Logic’, in Gabbay and Woods (2004), pp. 323–88.
Hartman, R. S. and Schwarz, W. (trans.) (1974), Kant: Logic, New York, NY: Bobbs-Merrill Company.
Hilpinen, R. (2004), ‘Peirce’s Logic’, in Gabbay and Woods (2004), pp. 611–58.
Kemp Smith, N. (trans.) (1929), Immanuel Kant’s Critique of Pure Reason, New York, NY: Macmillan. Reprinted with a new introduction by H. Caygill, New York, NY: Palgrave Macmillan, 2003.
Kneale, W. and Kneale, M. (1962), The Development of Logic, Oxford: Oxford University Press; reprinted in 1984.
Lenzen W. (2004), ‘Leibniz’ Logic’, in Gabbay and Woods (2004), pp. 1–83.
Miller, A. V. (trans.) (1969), Hegel’s Science of Logic, London: George Allen & Unwin.
Peckhaus, V. (2004), ‘Schröder’s Logic’, in Gabbay and Woods (2004), pp. 557–609.
Peckhaus, V. (2009), ‘Language and Logic in German Post-Hegelian Philosophy’, The Baltic International Yearbook of Cognition, Logic and Communication (200 Years of Analytic Philosophy) 4: 1–17.
Priest, G. (1982), ‘To Be and Not to Be: Dialectical Tense Logic’, Studia Logica 41: 249–68.
Priest, G. (1990), ‘Dialectic and Dialetheic’, Science and Society 53: 388–415.
Priest, G. (1998), ‘Number’, Vol. 7 of E. Craig (ed.), Encyclopedia of Philosophy, London: Routledge, pp. 47–54.
Priest, G. (1999), ‘Negation as Cancellation, and Connexivism’, Topoi 18: 141–8.
Priest, G. and Tanaka, K. (2009), ‘Paraconsistent Logic’, Stanford Encyclopedia of Philosophy, available at http://plato.stanford.edu/entries/logic-paraconsistent/.
Rusnock, P. and George, R. (2004), ‘Hegel as Logician’, in Gabbay and Woods (2004), pp. 177–205.
Sebestic, J. (2011), ‘Bolzano’s Logic’, Stanford Encyclopedia of Philosophy, available at http://plato.stanford.edu/entries/bolzano/.
Sluga, H. D. (1980), Gottlob Frege, London: Routledge & Kegan Paul (Arguments of the Philosophers Series).
Sullivan, P. (2004), ‘Frege’s Logic’, in Gabbay and Woods (2004), pp. 659–750.
Tyles, M. (2004), ‘From General to Transcendental Logic’, in Gabbay and Woods (2004), pp. 85–130.
Wallace, W. (trans.) (1873), Hegel’s Logic. Being Part One of the Encyclopedia of the Philosophical Sciences (1830), 3rd ed. 1975, Oxford: Oxford University Press.
Young, J. (ed.) (1992), Lectures on Logic (The Cambridge Edition of the Works of Immanuel Kant), Cambridge: Cambridge University Press.
Zalta, E. (2012) ‘Gottlob Frege’, Stanford Encyclopedia of Philosophy, available at http://plato.stanford.edu/entries/frege/.
* * *
1 There are some other preliminary remarks that need to be made. In a survey of this kind it is impossible to do justice to the richness and intricacies of the thought of any one of the writers we will meet, let alone all of them. For the same reason, there are people who would have to be mentioned in a longer treatise, but for whom there is no space in this. I have had to select what seem to me to be the most significant people, and the most significant features of their work. This introduces an ineliminable subjectivity into the chapter. A second source of subjectivity is the fact that history is not simply a catalogue of names and dates. It is a narrative which makes the names and dates meaningful. I would not wish to pretend that what I am doing here is anything other than telling a story about the history of logic as one contemporary logician sees it—though for the most part, I do not think there is anything particularly idiosyncratic about it. At the end of each main section of this chapter I will give references to places where the material covered in that section is discussed by others in greater detail.
2 It is hard to find a good book that covers the whole history of logic. Between them, Kneale and Kneale (1962) and Haaparanta (2009) give quite good coverage. The encyclopedic Gabbay and Woods (2004–12) contains detailed essays on most aspects of the history of logic. Lenzen (2004) can be consulted for an account of Leibniz’ views on logic.
3 When I reference books or articles that appeared in German, I shall give their original publication details, and then an accessible English translation if and where one exists. When dealing with symbolism, I have decided to write in the notation of modern logic. This is not because the notations actually used are without historical interest. And there is also a certain danger in this. One should not take it for granted that the writers we will meet meant by their symbols exactly what the modern logician means by theirs. However, the use of modern symbolism makes it easier to tell a uniform story, and one that is more intelligible for non-specialists. (Not to mention one that makes typesetting easier!) It should go without saying that, for someone who wants a detailed understanding of thinkers, their ideas, and their symbolism, there is no substitute for reading the primary texts.
4 Kritik der reinen Vernunft, Riga: Johann Friedrich Hartnoch. 1st ed. 1781; 2nd ed. 1786. There are several accessible translations. Kemp Smith (1923) is an old standard; Guyer and Wood (1998) is a good more recent translation.
5 Immanuel Kants Logik, ein Handbuch zu Vorlesungen, Königsberg: F. Nicolovius. English translation, Hartman and Schwarz (1974).
6 For further discussion, see Tyles (2004) and Young (1992).
7 Wissenshaft der Logik, Nürnberg: Schrag. Vol. 1, Pt. 1, 1812; Vol. 1, Pt. 2, 1813; Vol. 2, 1816. Translation, Miller (1969).
8 Enzyklopädie der philosophischen Wissenschaften, Heidelberg: Oßwald. 1st ed., 1817; 2nd ed., 1827; 3rd ed., 1830. Translated as Wallace (1873).
9 It is clear to readers of Hegel that he often struggles to fit material into his procrustean structure. It would appear that, in this case, he just gave up!
10 Herrn Eugen Dührings Umwälzung der Wissenschaft, Leipzig, 1878. Translated as Anti-Dühring: Herr Eugen Dühring’s Revolution in Science, Moscow: Progress Publishers, 1947. Notes for the Dialektik were compiled between about 1873 and 1883, but never completed. They were published posthumously (with a Russian translation), Moscow, 1935. This was translated into English as Dialectics of Nature, Moscow: Foreign Language Publishing House, 1954.
11 See Priest and Tanaka (2009).
12 On Hegel’s logic, see Burbidge (2004). For some steps towards dialectical logic, see Priest (
1982 and 1990).
13 ‘Zur Geschichte von Hegels Logik and dialektischer Methode. Die logische Frage in Hegels Systeme’, Neue Jenaische Allgemeine Literatur-Zeitung 1, 97: 405–8, 98: 409–12, 99: 413–14.
14 Logische Untersuchungen, Berlin: Bethge. 1st ed., 1840; 2nd ed., 1862; 3rd ed., 1870.
15 ‘Über Leibnizens Entwurf einer allgemeinen Charakteristik’, Philosophische Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin. Aus dem Jahr 1856: 36–69.
16 Wendepunkt der Philosophie im neunzehten Jahrhundert, Berlin: Reimer, 1834.
17 Neue Darstellung des Sensualismus, Leipzig, 1855.
18 Logik, Leipzig, 1843 and 1874.
19 He also anticipates two more Fregean themes, if somewhat inconsistently. One is the priority of the judgment over the concept; the other is the similarity between conceptual application and functional application in mathematics.
20 Further discussion of the matters in this section can be found in Peckhaus (2009) and Sluga (1980), chs. 1 and 2.
21 Wissenschaftslehre, Sulzbach: Seidel, 1837. Translated as George (1972).
22 See Priest (1999).
23 Further discussion of Bolzano and his logic can be found in Sebestic (2011) and Rusnock and George (2004).
24 I will often, as is standard in algebra, write things of the form a × b as ab.
25 Der Operationskreis des Logikkalküls, Leipzig: Teubner, 1877.
26 Vorlesungen über die Algebra der Logik, Leipzig: Teubner. Vol. 1, 1890; Vol. 2, 1891; Vol. 3, Pt. 1, 1895; Vol. 3, Pt. 2, 1905. The work has not been translated into English as far as I know. But a modern German version was published with New York, NY: Chelsea, 1966.
The Oxford Handbook of German Philosophy in the Nineteenth Century Page 77
