On Frege's Begriffsschrift Notation for Propositional Logic: Design Principles and Trade-Offs

Abstract

Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other.

Acknowledgements

Thanks to Roy Cook, George Englebretsen, Ansten Klev, Bernard Linsky, Daniel Lovsted, Paolo Mancosu and three anonymous reviewers for comments on an earlier draft.

Notes

1 On the background of Frege's Begriffsschrift, see Kreiser  2001 , in particular Ch. 3, pp. 135–275, and Sluga  1980 , Ch. II.

2 See, for example, Dudman  1971 , p. 30 and Frege  1980 , p. 148.

3 But, see the reviews of Frege's books (Vilkko  1998 ), the published parts of Frege's correspondence (Frege  1980 ), and recent philosophical reflections (Macbeth  2014 ).

4 See Barnes  2002  for various uses of the term ‘Begriffsschrift’.

5 See Cook  2013 for an excellent introduction to Frege's notation in Grundgesetze; for discussions of the differences between the 1879 and later versions, see Simons  1996 , Thiel  2005 , and the Introduction to Frege  2013 .

6 For some background on the audience of Frege's lectures, see Schlotter  2012 .

7 Macbeth  2005 argues for a close connection between Frege's notation and his philosophy of logic, but we leave that aside here, too.

8 The judgment stroke, mentioned above, will not be used in our discussion.

9 The numbering conventions for formulas are as follows: the label ‘()’, with a prime, denotes a formula in modern notation that corresponds to the Begriffsschrift formula (1), while subscripts, like ‘()’ and ‘()’, indicate formulas that are logically equivalent to (1) and (), respectively.

10 We follow here the terminology of Reck and Awodey ( 2004 , p. 52). The English translation of Grundgesetze uses ‘supercomponent’ and ‘subcomponent’ (Frege  2013 , p. 22).

11 In particular, Macbeth  2005 ; see also Thiel  2005 , pp. 15–16 and Moktefi and Shin  2012 , pp. 657–61.

12 See also Frege  1972 , p. 147 and Frege  2013 , p. 52.

13 This restriction is not explicitly discussed in the presentations of the Begriffsschrift by Macbeth  2005 and Cook  2013 , p. A-8.

14 See, for example, Miller  1956 and Chase and Simon  1973 .

15 See Frege  1879b , §7, Frege  1880/81 , p. 12, and the discussion of ‘and’, ‘neither—nor’, and ‘or’ in Frege  1893 , §12.

16 The interpretations of as conjunction and disjunction, are not to be confused with the primary and secondary readings of the Begriffsschrift discussed above; here, local combinations of conditional and negation strokes are interpreted as a unit (a complex symbol) representing a particular connective. For the secondary reading, whether a vertical stroke stands for an implication or a conjunction depends on the position of the stroke within the formula. This issue is taken up again in Section 2.3.

17 In his letter to Anton Marty (August 29, 1882), Frege transforms a formula by adding two negation strokes (Frege  1980 , pp. 101–2); see also Frege ( 2013 , p. 23).

18 On the notion of chunking, see the references in Footnote 14.

19 In fact, this rule of inference is sometimes referred to as ‘rule of detachment’ (Tarski  1994 ).

20 For a discussion of this and other metaphors for mathematics, see Schlimm  2016 .

21 See Łukasiewicz  1967 ; also Thiel  1968 , p. 21 and Frege  1972 , p. 73.

22 For example, in the reviews by Michaëlis ( 1880 , p. 218), Schröder ( 1881 , p. 83), and Venn ( 1880 ).

23 Frege  1880/81 , p. 39; adapted from Frege  1979 , p. 39.

24 Both Frege and Schröder traced their notations back to Leibniz. However, we leave this part of the debate aside, because both sides claimed their own system to be a lingua characteristica and criticized the other system to be merely a calculus ratiocinator. For a historically informed discussion of this issue, see Peckhaus  2004 .

25 Frege  1882b , p. 47; adapted from Frege  1979 , p. 47.

26 Boole  1854 , p. 6.

27 The term ‘logistic’ was introduced in 1904 in French as ‘logistique’ by Gregorius Itelson, André Lalande, and Louis Couturat at the 2nd Congress of Philosophy at Geneva (Peckhaus  2009 , p. 186). It figured prominently in a series of papers by Russell, Poincaré, and Couturat in the Revue de Métaphysique et de Morale in 1905–06. Couturat's contribution appeared in an English translation as Couturat  1912 .

28 Venn  1880 , p. 237.

29 Adapted from Frege  1980 , p. 102.

30 Letter to Frege, 11 February 1904; Frege  1980 , p. 13.

31 Letter to Frege, January 11, 1903; Frege  1980 , p. 105.

32 Frege  1980 , p. 26.

33 Bocheński  1961 , p. 268.

34 It is well known, for example, that each of the following sets of connectives is sufficient to express all other connectives of classical propositional logic: , , .

35 Because of the need for parentheses, the formula would also require more symbols in the modern notation: .

36 In Schröder's notation ‘+’ stands for disjunction, juxtaposition for conjunction, and subscripts for negation.

37 Frege  1880/81 , p. 35.

38 This issue is also discussed in Simons  1996 , p. 290.

39 We know that Sheffer had been in contact with Frege before publishing about the possibility of using a single symbol for a functionally complete system of propositional logic (Linsky  2011 , pp. 66–70), but we don't know about Frege's reactions to that. I presume Frege would not have found it very congenial to his own goals.

40 Frege  1880/81 , p. 37.

41 Frege discusses the four possible judgments and how they are denied or affirmed by the various operations also in Frege  1879b , p. 5 and Frege  1882b , pp. 48–9.

42 Frege  1880/81 , p. 37.

43 Frege  1882a , p. 6.

44 Frege  1879b , p. 13.

45 Thus, the claim that ‘[a]nother disadvantage of Frege's notation is that it does not allow us to introduce abbreviations for the other connectives’ (Gillies  1982 , p. 80) is overstated.

46 Frege  1880/81 , p. 12.

47 Frege  1879b , p. VII; adapted from Beaney  1997 , p. 51.

48 For an example, compare the Begriffsschrift Formula (9) with the corresponding representation in modern notation in Footnote 35.

49 Frege  1882/83 , pp. 7–8; adapted from Frege  1972 , p. 97.

50 Bynum gives a compelling example in his editorial comment to the passage from Frege quoted above (Frege  1972 , p. 97). An anonymous reviewer has pointed out that this makes the Begriffsschrift iconic in Peirce's sense: the truth of A pictorially rests on or is founded on the truth of B.

51 See also Frege  1879a .

52 Frege  1882/83 , p. 7; adapted from Frege  1972 , p. 97.

53 Frege  1896 ; translation adapted from Gillies  1982 , p. 82.

54 For other references to late 19th century literature, see Huey  1898 .

55 A more detailed analysis of various cognitive and pragmatic trade-offs of notations is currently in progress by the author.

Additional information

Funding

This research was supported by the Social Sciences and Humanities Research Council of Canada.