The horizontal in Frege’s Begriffsschrift


This paper addresses an issue with the sign ‘’ in Frege’s mature version of Begriffsschrift, i.e., the version in ‘Function and Concept’ and Grundgesetze. The sign is a performative for asserting in that writing down ‘’ is equivalent to asserting that p. Frege further says that writing ‘’ is also equivalent to identifying the reference of ‘p’ with the truth-value True. It looks as if he holds that asserting that p consists in identifying the True with the reference of ‘p’. Frege’s commitment to it, however, seems to encounter a number of tensions. This paper aims to show that these tensions can be avoided by endorsing a non-assertive conception of identification under which making an identification is not making an identity assertion. The suggested reading leads to an entirely different understanding of the compositionality of the sign ‘’ as well as Frege’s conception of assertion and judgment.

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  1. 1.

    To avoid too much use of italicization, I do not italicize ‘Begriffsschrift’.

  2. 2.

    The shape of the symbol in the early version of Frege (1879) is slightly different from that in the mature version in Grundgesetze (1893). I follow the typesetting of the latter.

  3. 3.

    The translation of the cited work is displayed in the bibliography. For the translation of Frege’s articles, I depend on Frege (1970, (1979, (1997). For Grundgesetze, I depend on Ebert and Rossberg’s translation (Frege 2013).

  4. 4.

    Smith (2000) deepens the conception of performative that underlies ‘’. But I do not introduce his view here because all I need is a performative in the sense Frege specifies in the above passage from ‘Function and Concept’.

  5. 5.

    In the mature Begriffsschrift, the horizontal should be longer when it comes alone (Frege 1893, p. 9).

  6. 6.

    Heck and Lycan (1979) is a pioneering work that examines this question. They seek an answer by focusing more on the logical/mathematical aspects of Frege’s work. Here, I am seeking an answer by focusing more on Frege’s philosophical insights. I believe that Heck and Lycan’s answer is compatible with my answer in this paper.

  7. 7.

    Frege explains why the content-stroke must follow the judgment-stroke in \(\S 2\) of Frege (1879). The content-stroke combines the ‘ideas’ of the symbols that follow the content-stroke into a judgeable whole. In Grundgesetze Frege writes: ‘Earlier I called [the horizontal] the content-stroke, when I combined under the expression “judgeable content” that which I now have learnt to distinguish as truth-value and thought.’ (1893: \(\S 5\) fn.2). Thus, the change from the content-stroke to the horizontal is related to the development of Frege’s understanding of content. Frege regards the change from the content-stroke to the horizontal as one of the ‘consequences of a deep-reaching development of [his] logical views’ (1893: X).

  8. 8.

    I do not intend this way of explanation to suggest that Frege’s conception of truth-values as objects is grounded in his semantic framework. Here, I am merely following the way Frege himself explains why truth-values are objects in ‘Function and Concept’ (Frege 1891).

  9. 9.

    Since sentences are names of truth-values for Frege, they can take name position in other sentences. ‘Grass is green is the True’ or ‘Snow is purple is identical to there is a largest prime number’ is difficult to parse, however, and arguably ungrammatical in ordinary English. I’m using the double bars to aid parsing. They mark where names of truth-values start and end within a sentence, but have no semantic significance. They are grouping devices, similar to parenthesis in arithmetic or formal logic. In particular, they do not denote a function.

  10. 10.

    Greimann (2007, (2008), for instance, can reasonably be taken to accept the AI explanation. According to him, writing down ‘’ is (i) identifying \(\Vert p\Vert \) with the True and (ii) asserting that \(\Vert p\Vert \) is the True.

  11. 11.

    The horizontal is a first-level functor because the H-concept maps an object to an object. A second-level functor maps a first-level functor to an object and a third-level functor maps a second-level functor to an object and so on.

  12. 12.

    See p. 184 of Frege (1979).

  13. 13.

    Exactly the same argument is given in ‘Thought’ (Frege 1918a).

  14. 14.

    In ‘Thought’ (Frege 1918a), Frege talks about truth definitions that make truth have characteristic marks. Only properties have characteristic marks.

  15. 15.

    Whether or not the above infinite regress is the absurdity to which the indefinability argument appeals, Frege’s conception of judgment produces this regress if truth is a property. Thus, in any event, this infinite regress is a reason why Frege must avoid the AI explanation of ‘’ if he takes assertion to be constituted by identification.

  16. 16.

    This argument is also found in Frege’s other mature works such as Frege (1892b, 1914). In the latter, Frege takes this argument to show that ‘truth is not a property of sentences or thoughts’ (1914: 234).

  17. 17.

    The sudden change of locution from ‘\(\langle p\rangle \) is true’ to ‘It is true that p’ can be ignored. The same argument in ‘On Sense and Reference’ (Frege 1892b, p. 64) consistently uses the locution ‘\(\langle p\rangle \) is true’.

  18. 18.

    In \(\S 11\), Frege stipulates that is the True after he establishes that it is possible to stipulate that a truth-value is an arbitrary value-range. See Heck (2012): Ch.3’s discussion which provides an excellent explanation about why Frege makes this stipulation. I here use this particular non-sentential name ‘’ to bypass the alleged factivity of judgment in Frege. The factivity of judgment does not matter for our discussion.

  19. 19.

    Dummett (1973, p. 315) also writes that the horizontal ‘in effect turns any singular term into a sentence.’

  20. 20.

    I acknowledge that in Frege the act of judging ought to be distinguished from the epistemic status we assign to a thought by judging. The distinction is important to understanding Frege’s anti-psychologism of logic. As ‘Thought’ clearly shows, what is manifested by an assertion is not a judgment as a special epistemic status, but a judgment as an act. In the following discussion, ‘judgment’ uniquely means ‘the act of judging’.

  21. 21.

    Frege (1970) uses ‘a’ for the translation instead of ‘its’. The original word is ‘seinem’ that means ‘its’.

  22. 22.

    Frege takes ‘’ to be a ‘Begriffsschrift representation of a judgment’ (1893: \(\S 5\)). One might wonder if ‘judgment’ here means a judgment qua an epistemic status. See Footnote 19. First, it is possible for Frege to mean both the act of judging and the epistemic status. Hence, choosing one reading is not necessarily rejecting the other reading. Secondly, Frege calls ‘’ a representation of a judgment right after he says a judgment is the acknowledgment of the truth of a thought that is taken by him to be ‘the act of judgment’ (1918a, p. 329).

  23. 23.

    Scholars like Heck (2012) and Textor (2010) even claim that, for Frege, to acknowledge the truth of \(\langle p\rangle \) is to ‘refer to the True via \(\langle p\rangle \)’. Although they do not specify what kind of reference they mean by ‘refer’ there, it must be speaker-reference.

  24. 24.

    Specifically, Millikan denies the existence of assertive/judgmental identification because she denies the existence of the relationship of identity. But we do not have to be committed to such a controversial claim in order to argue for the notion of non-assertive/non-judgmental identification.

  25. 25.

    In the rest of the paper, ‘predication’ (or ‘predicating’) without further qualification always refers to the act of committal predication.

  26. 26.

    This point only applies to Frege’s mature career. His early conception of judgment and assertion is different. In the early Begriffsschrift, Frege takes judgment and assertion to be predication of truth or facthood (Frege 1879: \(\S 2\)). The indefinability argument Frege develops in his mature career shows that his conception of judgment/assertion has changed.

  27. 27.

    Unlike Millikan’s view, this view admits that we can make an assertive identification, i.e., assert an identity.

  28. 28.

    There is a similarity between Frege and Millikan relevant to this point. Millikan says that when we accept ‘\(o_{1}\) is \(o_{2}\)’, what we do is not to make an identity judgment but to realize the ‘overlap’ (2000, p. 144) of two contents on a single object. Realizing the overlap of contents, which Millikan calls ‘co-identifying’, seems to be quite similar to Frege’s deciding whether a sense belongs to the reference of a known name.

  29. 29.

    My reading can’t explain other uses of the horizontal, e.g., why the negation-stroke in ‘’ (the vertical line) must accompany horizontals (the lines on the sides of the negation-stroke). I do not take it to be a problem with my reading. Different uses of the horizontal are underwritten by different rationales. Then, what is the rationale for the negation-stroke and horizontals? First, the negation-stroke alone is not a function. Say it is a function. It must be one that can have an argument of the form ‘’, i.e., a first-level function that maps an object to an object. For Frege a function must always be totally defined (cf. 1891: 32–33). The negation-stroke must also be defined for, say, 2 and ‘’ must be a legitimate expression of Begriffsschrift. But it is not. Thus the negation-stroke is a special sign that constitutes a function only with horizontals attached. What is it that the negation-stroke embodies? I think it embodies what (Frege 1918b, p. 357) calls ‘a negating word’. ‘A negating word may occur anywhere in a sentence’ (1918b, p. 353), and depending on which part of a sentence it is attached to, a resulting thought varies. So a negating word alone cannot denote a function. The negation-stroke qua a negative word is always attached to the horizontal ‘\(\cdots \) is the True’ on its right side and thereby constitutes ‘\(\cdots \) is not the True’. Thus ‘’ refers to ‘a concept under which all objects fall with the sole exception of the True’ (1893: \(\S 6\)). The fact that the negation-stroke can only have the horizontal on its right side means just that ‘\(\cdots \) is not the True’ is the only negation necessary for logic. Now, the horizontal on the left side of the negation-stroke is related to the point that logic does not need negative judgment (1918b, pp. 355–357). ‘’ is not a performative for negative judgment due to the left horizontal: ‘’ comes from ‘’ by the fusion of horizontals that combines consecutive horizontals into one. In this way, I believe, different rationales can be given for different usages of the horizontal.


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Correspondence to Junyeol Kim.

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Kim, J. The horizontal in Frege’s Begriffsschrift. Synthese (2020).

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  • Gottlob Frege
  • Begriffsschrift
  • Grundgesetze der Arithmetik
  • Horizontal
  • Judgment-stroke
  • Truth
  • Truth-values