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Frege’s Horizontal

Published online by Cambridge University Press:  01 January 2020

William C. Heck  and
William G. Lycan
William C. Heck
Affiliation:
Ohio State University
William G. Lycan
Affiliation:
Ohio State University
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Extract

Frege begins his exposition of the symbol system employed in his Begriffsschrift by introducing the sign ⟝, whereby, he says, “[a] judgment is always to be expressed” (p. 1 ).

[The judgment sign] stands to the left of the sign or complex of signs in which the content of the judgment is given. If we omit the little stroke at the left of the horizontal stroke, then the judgment is to be transformed into a mere complex of ideas; the author is not expressing his recognition or nonrecognition of the truth of this. (Pp. 1–2)

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 9 , Issue 3 , September 1979 , pp. 479 - 492
Copyright
Copyright © The Authors 1979

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References

1 Page references are to the extracts reprinted in Geach, and Black, (eds.), Translations from the Philosophical Writings of Gottlob Frege (Oxford, 1960).Google Scholar

2 Furth, (ed. and tr.), The Basic Laws of Arithmetic: Exposition of the System (Berkeley and Los Angeles, 1964).Google Scholar

3 “Function and Concept,” reprinted in Geach and Black, op. cit.

4 Since the result of applying the horizontal to a denoting expression is always a name of a truth-value, the horizontal itself fits Frege's definition of a “concept-expression” and so designates a concept. Frege goes on to describe some ways in which the horizontal may be “amalgamated” or elided with other truth-functional signs and quantifiers with which it has been concatenated, salva veritate.

5 By “sentence,” at this stage, we mean something that we would intuitively have counted as a sentence prior to the introduction of any of Frege's special apparatus, such as “Czuber is a mathematician” or “2 + 3 = 5.” In particular, we do not mean what Frege means by “Begriffsschriftsatz,” viz., a well-formed formula preceded by the Judgment sign, though “Begriffsschriftsatz” is typically translated as “sentence of Begriffsschrift” or “proposition of Begriffsschrift .”It seems to us that the latter translations are misleading, since a Begriffsschriftsatz is really the product of an illocutionary act of asserting; “statement” or “judgment” or even “assertion” would be better.

6 A more serious kind of grammatical question is raised by the fact that horizontals appear inside the scopes of other operators, as soon as any degree of semantical complexity obtains. We may grant Frege the harmlessness of his “amalgamation rules,” but we must assure ourselves that complex sentences in which horizontals occur within the scopes of other operators remain well-defined. And in the case of quantifiers, at least, this is not easy. (Edwin Martin has taken up this sort of problem for truth-functions generally in “A Note on Frege's Semantics,” Philosophical Studies 25 (1974).)

We might go on, after settling all these syntactic points, to investigate the horizontal's pragmatic properties as well. For example, when a logician or mathematician uses the horizontal in the course of working within Frege's logical object-language (as opposed to mentioning or displaying it while talking philosophically about the object-language), what speech act is he or she therein characteristically performing— that is, what is the horizontal's illocutionary role? Someone who tokens “—— 17 is prime” is asserting that 17 is prime; someone who tokens Just “17 is prime” is merely voicing or expressing the “thought” that 17 is prime without endorsing it, and simultaneously referring to the True. But what is someone doing who tokens the superficially intermediate “——17 is prime” (besides likewise referring to the True)? “—— 17 is prime” is not merely a name, in that it is true or false as a sentence is (and in Just the way that a sentence is); on the other hand, it cannot by itself be used to make an assertion, as an expression prefixed by the Judgment-stroke can be. Does it accordingly have an illocutionary function over and above naming, as befits its semantical status over and above being a name? The literature (so far as we have read) is mute on this. One might suggest that horizontalled expressions may characteristically be used by speakers in performing the acts of hypothesizing, assuming, supposing, and the like, which acts figure prominently in the larger process of producing arithmetic proofs. Of course, ordinary unhorizontalled sentences can be used to perform such acts also; but perhaps there is nothing more interesting to say about the horizontal's illocutionary properties.

7 Russell, The Principles of Mathematics (London, 1937).

8 Moreover, Russell adds:

Direct inspection seems to show that the relation of a proposition to the true or the false is quite different from that of (say), “the present King of England’ to Edward VII. Moreover, if Frege's view were correct on this point, we should have to hold that in an asserted proposition it is the meaning, not the indication, that is asserted, for otherwise, all asserted propositions would assert the very same thing, namely the true, (for false propositions are not asserted). Thus asserted propositions would not differ from one another in any way, but would be all strictly and simply identical. Asserted propositions have no indication (FuB. P. 21 ), and can only differ, if at all, in some way analogous to meaning. Thus the meaning of the unasserted proposition together with its truth-value must be what is asserted, if the meaning simply is rejected. But there seems no purpose in introducing the truth-value here: it seems quite sufficient to say that an asserted proposition is one whose meaning is true, and that to say the meaning is true is the same as to say the meaning is asserted. (pp. 504-5)

9 Dummett, Frege: Philosophy of Language (London, 1973).Google Scholar

10 Dummett concedes one kind of exception to this logical point: Frege intends the identity sign to double as a biconditional, but a formula such as does not hold in full generality, due to the fact that its right-hand side must denote a truth-value, while its left-hand side may denote anything.

11 Anscombe and Geach, Three Philosophers (Oxford, 1961).

12 Anscombe, An Introduction to Wittgenstein's Tractatus (New York, 1959).

13 Christian Thiel cites three advantages gained by introducing the horizontal as a functional sign (Sense and Reference in Frege's Logic (Dordrecht, 1968), p. 53): (1) “the greater precision which the sign connections and gain”; (2) “the making possible of a direction transition from objects Δ to truth-values and from functions to concepts or relations ; and (3) the subsumption under pre-existing logical laws of certain identities which “appeared in the Begriffsschrift as mere peculiarities of the system of signs,” which subsumption renders those previously unexplained identities “clear and provable” (one such identity would be …”).

These points are not as clear as they might be. The “greater precision” Thiel is alluding to, we take it, is that possessed by Frege's recursive truth-theoretic definition of the horizontal treated as a functional sign, as contrasted with his brief, murky, and somewhat psychologistic explanation of the content-stroke in Begr. This difference is indeed impressive, but does not help us with Question 1, since it does not show us why Fregean logic at any stage of development should require such an item as the content-stroke or the horizontal in the first place. Advantage (2) is also puzzling: Of course the introduction of the horizontal as a functor makes possible a “direct transition” from objects to truth-values, monadic functions to concepts, and so on; that is precisely what the horizontal is defined as doing. The question is, why does Frege find it necessary or desirable that such direct transitions be made at all? Advantage (3) shares (1 )'s drawback— the treatment of the horizontal as a functor is obviously superior to its Begr. handling, but that does not satisfy our curiosity as to the sign's original raison d'etre. So far, then, Anscombe's positive suggestion remains the only viable contender.

14 Kneale, The Development of Logic (Oxford, 1962).Google Scholar

15 Cf. the example given by Furth in the introduction to his edition of Grg., op. cit., pp. x-xi.

16 One small perplexity remains. We have explained why the standard negation sign , for example, is regarded as a composite in which the negation-stroke ‘ operates on a horizontal. But why does Frege understand as containing two horizontals? Why should he not Just have written his standard negation sign as ? We have no idea.

17 Walker, A Study of Frege (Oxford, 1965), p. 16.Google Scholar

18 It is interesting to note that we can know the truth-value of a Fregean expression, such as or without knowing what that expression means. In fact, it seems that we may know the truth-value of such an expression through all possible worlds, even though we do not know what that expression means because, intuitively, it does not mean anything (it expresses no proposition; it has no locutionary content; there is nothing that it says). This is a slight embarrassment to philosophers of language, including Davidson, David Lewis, Max Cresswell and others, who contend that a person knows the meaning of an expression if that person knows the expression's truth-value through all possible worlds. (This point is developed by W. Lycan in “Semantic Competence and Funny Functors,” Monist, in press.)

This paper was presented at the 1977 meeting of the American Philosophical Association (Western Division). We are grateful to Matthias Schirn for his helpful corrections on that occasion.