Analysis and decomposition in Peirce


Peirce seems to maintain two incompatible theses: that a sentence is multiply analyzable into subject and predicate, and that a sentence is uniquely analyzable as a combination of rhemata of first intention and rhemata of second intention. In this paper it is argued that the incompatibility disappears as soon as we distinguish, following Dummett’s work on Frege, two distinct notions of analysis: ‘analysis’ proper, whose purpose is to display the manner in which the sense of a sentence is determined by the senses of its constituent parts, and ‘decomposition’, which is the process of dividing a sentence into a predicate and a subject, and whose purpose is to both to explain how quantified sentences are constructed and to evidence a pattern within a sentence which it shares with other sentences.

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  1. 1.

    The following abbreviations will be used for Peirce’s works: CP, followed by volume and paragraph number, stands for Peirce (1932–1958); SS, followed by page number, for Peirce (1977); R, followed by manuscript number and, when available, page number, for Peirce’s manuscripts (Charles S. Peirce Papers, 1787–1951, MS Am 1632, Houghton Library, Harvard University) as catalogued in Robin (1967).

  2. 2.

    See below, Sect. 2.

  3. 3.

    It has been objected, most notably by Currie (1985) and Levine (2002), that Dummett’s distinction between analysis and decomposition fails as a faithful exegesis of Frege’s thought. As I am only interested in the usefulness of Dummett’s distinction in the interpretation of Peirce, I will ignore the question whether that distinction provides an adequate interpretation of Frege.

  4. 4.

    This has been done partially but excellently by Hawkins (1993) and Anellis (2012).

  5. 5.

    This involves an explanation of how to treat a predicate with any number of places formed by attaching a quantifier to a predicate with one more place. Dummett (1981b, pp. 284–286) explain this by a substitutional account of quantification, which he argues was more or less what Frege had in mind.

  6. 6.

    Cf. Sullivan (2010, pp. 107–108).

  7. 7.

    On Peirce’s speech-act theoretical analysis of assertion see Bellucci (2017, pp. 295–321).

  8. 8.

    CP 1.559, 1867; R 484, p. 7, 1898; CP 2.357, 1901; RL 75, p. 21, 1902; CP 5.139, 1903; R 491, p. 9, 1903; R 7, p. 16, c. 1903. Hilpinen (1992) was the first to see that this is the standard definition of the proposition in Peirce; see also Stjernfelt (2015) and Bellucci (2017, pp. 95–96, 197–198, 287–288, 294).

  9. 9.

    Cf. e.g. R 599 (c. 1902); CP 2.252 (1903); R 517 (1904); CP 5.424n (1905); R 280, pp. 25–26 (1905).

  10. 10.

    Cf. also R 641, p. 24 (1909); R 659, p. 19 (1910).

  11. 11.

    Dummett (1991, p. 291) and Textor (2018) are in favour of such emendation of Frege’s principle of the uniqueness of analysis.

  12. 12.

    That Peirce regarded the subject of a relative proposition as an ordered set is evident, for example, from CP 2.230, 2.316, 4.453, 8.177.

  13. 13.

    There are passages that indicate that by 1908 he still admitted the possibility of multiple decompositions of one and the same sentence: “A proposition can be separated into a predicate and subjects in more ways than one” (R 278, 1908); “more or fewer objects may be regarded as subjects while the remainder of the assertion is the predicate” (R 339, p. 332, 1908).

  14. 14.

    Avicenna, in a passage frequently quoted at the end of the thirteenth century, had affirmed that second intentions are the subject matter of logic: “Subiectum vero logicae, sicut scisti, sunt intentiones intellectae secundo, quae apponuntur intentionibus intellectis primo” (Liber de philosophia prima, I, 2). In his questions on Porphyry, Scotus attributes this thesis to Boethius: “Aliter ponitur, quod [subiectum logicae] est de secundis intentionibus applicatis primis, sicut dicit Boethius, quia illae sunt communes omnibus in logica determinatis” (qu. III). It is probably to this passage that Peirce refers in the “New List of Categories” when he says that “Logic is said to treat of second intentions as applied to first” (CP 1.559). Peirce explicitly refers to Avicenna’s claim in the “Logical Tracts No. 2” in the context of his explanation of the difference between rhemata of first and of second intention (R 492, pp. 79–80).

  15. 15.

    Peirce speaks of “rhemata of second intention” also in “On Logical Graphs” (R 482) and in the fourth section of the Syllabus of 1903 (R 508, 478 = CP 4.394–417). In “Schroeder’s Logic of Relations” (an earlier draft of “The Logic of Relatives” of 1897) the distinction made is between indices (i.e. algebraical symbols) that denote first intentions and those that denote second intentions (R 521, p. 2).

  16. 16.

    For example, in R S-36 Peirce explains that a “spot” (about which more below) is an “undecomposed”, i.e. “unanalyzed”, element of a logical graph (p. 12). In a draft of a letter to his former student Christine Ladd-Franklin, probably written in 1901, he subsumes the question “what does the logical decomposition of a reasoning consist in” under the heading of “logical analysis” (RL 237, p. 6). In a 1909 letter to James we read: “there are concepts which, however we may attempt to analyze them, will always be found to enter intact into one or the other or both of the components into which we may fancy that we have analyzed them” (RL 224, emphases added). In discussing the inferential rules (“rules of transformation”) of the Existential Graphs, he says that “an indecomposable transformation is either an omission or an insertion, since any other may be analyzed into an omission followed by an insertion” (R 515, p. 1, 1904, emphases added); cf. also CP 4.564.

  17. 17.

    Cf. also CP 4.441 (1903); CP 4.403 (1903); R 295 (1906). Roberts, who was fully aware of Peirce’s doctrine of multiple analyses, did not differentiate between spots and rhemata (1973, p. 115); nor does Pietarinen, who thinks that a spot is simply the “iconic version” of a rhema (2006, p. 115; cf. also pp. 123–125).

  18. 18.

    On Peirce’s notion of “continuous predicate” see Bellucci (2013) and Bellucci (2017, pp. 331–334).

  19. 19.

    The question whether Peirce could be said to embrace the Fregean distinction between Sinn and Bedeutung is not easy to answer. With respect to general terms, which for Peirce are symbols, it is safe to say that they have both Sinn and Bedeutung: a symbol is a sign that both connotes and denotes, and denotes whatever satisfies the characters it connotes (CP 1.599, 2.344, 4.544). With respect to proper names, which for Peirce are indices, the question is more complex: on the one hand, he clearly says that proper names denote without connoting, i.e. have no signification (R 280); he thus seems to embrace a sort of Kripkean doctrine of proper names, and so he is usually taken to do by the commentators (see e.g. Hilpinen 1995). On the other hand, however, he clearly explains that the object of a proper name can only be given by “collateral observation” (also “collateral experience”, “collateral acquaintance”). The question is thus whether such collateral knowledge can be identified with the Fregean Sinn of the proper name, i.e. with a definite description that is associated to the name. Since I have no space to argue for it here, I limit myself to say that such identification is legitimate only if we restrict it to the referential use of definite descriptions (as opposed to their attributive use; see Donnellan 1966). In saying that the object of a proper name is given by collateral knowledge, Peirce is saying that its use is regulated by a description of that object that referentially determines the denotation, whether the object satisfies the description or not. It is in this restricted sense that I say above that in order to grasp the sense of a sentence we need to grasp the sense of the proper names it contains. In any case, Peirce’s notion of collateral knowledge may perhaps more easily be considered a variant of Russell’s “principle of acquaintance”: for Russell, anything, be it an object or a relation, can be a constituent of a proposition, provided we have acquaintance with it; cf. Russell (1905, p. 492); like for Russell, for Peirce not only subjects, but also predicates must be previously or collaterally known if a proposition is to function as such.

  20. 20.

    It must be observed that at some places Peirce regards proper names as spots (of first intention); cf. CP 3.469, 1897); R 491 (1903); R S51, p. 6. Evidence of such a treatment of proper names can also be found in Peirce’s papers on the logical graphs; cf. CP 3.471–477 (1897); CP 4.445 (1903); R 669, p. 11 (1911); R 670, p. 8 (1911). In Peirce’s 1903 taxonomy of signs, proper names are classified as indexical rhemata, and a rhema that thus serves as the subject of a proposition is called an “onome” (R 478, p. 89). But in this context “rhema” cannot be taken in the strict sense of the definition. For since a rhema is defined as that which remains of a sentence after something replaceable by a proper name is removed from it, and since a proper name is of course replaceable by a proper name, then in a sentence a rhema should be replaceable by a proper name, which is untrue: if in “Cain kills Abel” I replace “ξ kills Abel” with “Abel”, I obtain “Cain Abel”, which is not a sentence. This means that though it is harmless to regard proper names as spots of first intention (“The spots are of two kinds, rhemata and onomata, although the former are superfluities of which I make little use”, R 491), it is impossible to regard them as rhemata in the strict sense of the definition. Thus, the fact that proper names can be regarded as spots of first intention not only confirms that the “analysis” (in Dummett’s sense) of a sentence is into spots of first intention (whether “predicative” or “subjectal”) and spots of second intention (plus the quantifiers), but also, and more importantly, it supports the distinction between spots and rhemata.


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Bellucci, F. Analysis and decomposition in Peirce. Synthese 198, 687–706 (2021).

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  • Peirce
  • Logical analysis
  • Decomposition
  • Dummett
  • Frege