An Essay on Compositionality of Thoughts in Frege’s Philosophy

Abstract

In the paper, I propose a novel approach to Frege’s view on the principle of compositionality, its relation to the propositional holism and the formation of concepts. The main idea is to distinguish three stages of constructing a logically perfect language. At the first stage, only a sentence as a whole expresses a Thought. It is impossible to assign meaning to less complex units. This is the stage of an ordinary language. The second phase concerns the proper level of construction of a logically perfect language. We are forced to discriminate syntactic and semantic parts of sentences to account for the inference relations. We can distinguish senses and references of parts of sentences. Furthermore, it is possible here to choose between different ways of analysing the given Thought. Finally, at the third stage, every expression of the language has an unambiguous sense and this sense determines a unique reference. The logically perfect language is ready. We may view Thoughts as composed from primitive elements. Moreover, the senses of parts of a sentence correspond to the parts of a Thought, so that the structure of the sentence serves as the image of the structure of the Thought. The principle of compositionality is met and we can discern how understanding of the infinite numbers of Thoughts is possible and how languages are learnable. The main advantage of the presented view is that it allows accommodating some aspects of Frege’s philosophy that are often seen as mutually incompatible. Furthermore, I submit extensive textual data in favour of the discussed views and conceptions.

Acknowledgements

I would like to thank the editor and the anonymous reviewer for their suggestions and comments, which helped me to improve the manuscript.

Notes

1 This is a standard understanding of this matter. Additionally, I assume that if the principle of compositionality of a language is obeyed for sense, then it is true for reference (Bedeutung).

2 I will write (apart from quotations) ‘Thought’ with a capital T to mean a Thought in Frege’s sense as that part of the content of a sentence that is true or false and that is important for logic. Of course, a Thought is the sense of a declarative sentence in non-oblique contexts.

3 For these assumptions, see Pelletier (2001: 89).

4 This view is also known as the picture thesis or the building principle.

5 More on this topic later.

6 For an overview of the positions on the relation between the two principles, see Pelletier (2001: especially 87–103).

7 For a discussion on the role of the context principle in Frege’s philosophy of mathematics, see Dummett (1995). He also elaborates on the problems with interpretation of this principle in the light of introduction of the sense/reference distinction.

8 Frege did not distinguish between sense and reference in The Foundations of Arithmetic (Frege 1891a [1984]: 145; Frege 1893 [2013]: x). For more on this topic, see Dummett (1991: 15–17) and Sundholm (2001).

9 This shows that a very weak interpretation of the context principle that confines its role to demonstrating that quantifiers, for example, do not have a meaning in isolation is too narrow. On the detachment of syntactical phenomena from semantical, see Bell (1981). See also Kremer (1997: 115 (footnote 34)) for a similar view.

10 Propositional holism declares the priority of a Thought over its constituents. That means that a Thought is the prior and primitive phenomenon and we can obtain its elements only through the analysis of a propositional whole (see Bell 1981: 213; Sluga 1975: 481–482, 1980: 90–95).

11 The same is probably true of Heck (2012a). It seems to me that his view implies the repudiation of the recarving thesis.

12 The idea of distinguishing ‘three epistemic stages’ was outlined by Sluga (1977: 240) in one paragraph (he also cites the first quote from the next section). However, I do not know his views on the isomorphism thesis and other issues connected to the position I express. Also, I am not sure how to reconcile his views from the aforementioned essay with the views expressed in his later book (Sluga 1980: 130–137).

13 Frege’s discussion of senses of proper names in ‘On Sense and Meaning’ and in ‘Thoughts’—that I examine in the next section—shows that two speakers can connect a different sense with the same name; therefore, it is enough if the speaker and the hearer associate approximately the same Thought with a given sentence. As I claim below, Frege explicitly states this requirement in a letter to Peano (Frege 1896 [1980]).

14 Of course, these shortcomings are weaknesses of ordinary language only from the point of view of logic (see Frege 1879 [1967]: 6; Frege 1882b [1972]: 86).

15 In my view, it is legitimate to talk here of Thoughts since they have no intrinsic structure. Furthermore, as we will see, the essence of Frege’s recarving thesis is that the understanding of the content of a sentence (i.e. a Thought) allows us to express it in a different way. See below the discussion on §64 from The Foundations of Arithmetic (Frege 1884 [1960]: 21–23).

16 I used a simplification here. The proper requirement here is for the universal applicability of a function/argument analysis to complex expressions. In the case of atomic sentences, this analysis results in a division into a saturated and unsaturated part. However, some complex Thoughts may not have saturated components. Sentences containing expressions of generality, such as ‘All mammals are land-dwellers’, involve only expressions referring to concepts (first and second order) (see Frege (1892b [1984]: 187). I owe this observation to an anonymous reviewer.

17 For the truth or falsity of sequences of inferences, see Heis’ (2014: 276–279) discussion of Burge’s approach to the structure of Thoughts.

18 Pelletier (2001), Janssen (2001) and Dummett (1973, 1989) do not recognize the semantic differences between the concept script and an ordinary language. As a consequence, Pelletier and Janssen claim that Frege does not allow for the principle of compositionality. Dummett (1973, 1989) denies the idea that a Thought can be divided in many ways. As I am going to show in the next sections, however, it is plausible to ascribe to Frege both the principle of compositionality and the propositional holism.

19 Mark Textor (2017) claims that the isomorphism thesis is true for Frege’s logically perfect language but does not hold for a natural language. On the other hand, he does not link this view with the idea of building the concept script or the propositional holism.

20 According to Frege’s standards, when we want to treat people for, e.g., baldness, we have to develop a scientific theory that will determine the exact boundaries of the concept ‘x is bald’. Construction of such a theory requires precisification of our everyday concepts and methods. For instance, the Thought ‘Gary is bald’ would acquire clear and exact sense so the language of this theory—unlike our natural language—could be fully compositional. When I talk about a construction of a perfect language, I mean also developing a language accurate for some scientific theory. For a discussion of standards and construction of such a scientific language, see Weiner (1996: 262–268).

21 Frege’s semantic theories of ‘On Sense and Meaning’ and of Basic Laws of Arithmetic are good examples when such a talk of parts makes sense. As we will see, Frege formulates numerous provisos in the former text that indicate the limits of such a language.

22 See footnote 18 for some examples. Heck (2012a, 2012b) and Linnebo (2004, 2018, 2019) are instances of a weak interpretation of the context principle. They ignore the priority of judgments over concepts in their view and do not try to account for the formation of concepts in Frege’s doctrine. Unfortunately, there is not enough space to discuss their views here.

23 I describe this in more detail in Section 4.

24 At other times, their content is rather descriptive.

25 In this sense, when Frege had constructed his concept script of Basic Laws of Arithmetic, he could use it for the analysis of arithmetical sentences, properties and relations.

26 Throughout the essay, I will focus on Frege’s works published after 1891. This year marks, of course, the beginning of his mature doctrine. I cite earlier writings only to show continuity in his position or to explain his explicitly held view.

27 I slightly modified the translation here. I use uniformly the word ‘sentence’ instead of ‘proposition’ as a rendering of ‘Satz’. The choice is controversial, but it captures the current context better (see Frege 1896 [1976]: 183). Ebert and Rossberg have chosen ‘proposition’ as a translation of ‘Satz’ in Grundgesetze (see Ebert & Rossberg 2013: xxi).

28 Apart from this letter, Frege describes most explicitly the construction of a scientific language in the paper ‘Logic in Mathematics’, which I discuss in the next section. Note, however, that other scholars’ assurance of Frege’s unconditional acceptance of the principle of compositionality is empty without an explanation of Frege’s talk of logical defects of natural languages. Furthermore, it would be nonsensical to speak of requirements for the concept script unless there is such a thing as a language that does not meet these standards and a process that leads to the creation of an improved language.

29 The third formulation says: ‘[o]nly in a proposition have the words really a meaning’ (Frege 1884 [1960]: §60). I agree with Linnebo (2018) that the strongest interpretation, according to which words never have any meaning independent from the sentence, should be rejected, since it is clear that Frege has accepted in his mature doctrine the view that constituents of a sentence have sense and reference of their own.

30 Baker and Hacker (2005: 164–170) propose a view that (nearly) every formulation of the context principle performs a (slightly) different role in Frege’s philosophy. They do not aspire to any unified conception of this principle. According to them, the wording from introduction to The Foundations of Arithmetic serves to invoke anti-psychologism, §48 (Frege 1884 [1960]) states propositional holism, §60 proposes a literal and the strongest interpretation that words do not have any semantic values, and, finally, §64 identifies the meaning of a word with its contribution to the meaning of a sentence.

31 I elaborate on the last part of the passage in the section devoted to the proper stage of constructing the concept script.

32 Burge (1979 [2005]) argues in a resolute way that this view of proper names should be acknowledged as an official reading of Frege.

33 See Weiner (1996: 262) where she claims in Frege’s name that these ‘presuppositions are simply false’ and the failure to display sharp boundaries is almost universal among the concept-expressions of a natural language.

34 This quote comes from an important methodological essay published by Frege in the groundbreaking year 1891. The paper contains also an interesting passage resembling the context principle. Frege writes about his adversary—Ludwig Lange—that he wrongly ‘considers in separation from one another hypotheses that have a meaning (Bedeutung) only as a whole’ (Frege 1891b [1984]: 125). For further discussion, see Sluga (1980: 131–133).

35 In this article, Frege uses the notion of a proper name, which includes definite descriptions (Frege 1892a [1984]: 158).

36 This passage is located just after the first excerpt quoted by me from ‘On Sense and Meaning’.

37 See footnote 26.

38 In this sense of simplicity, the words ‘horse’ and ‘nag’ are primitive semantical items that are identical, since they have the same reference and sense. Similarly the logical constants ‘but’ and ‘and’ are not different semantical elements. On the other hand, the words ‘dog’ and ‘cat’, for instance, are independent simple items of language.

39 Frege’s reasoning is much longer here. For the sake of brevity, I only try to preserve the sense of it.

40 According to Pelletier (2001: 107), Frege’s claim in favour of the principle of compositionality is restricted in this excerpt to definitions only. I think Pelletier is wrong here, since Frege describes definitions as a special case of his line of reasoning but he postulates a compositional view of a (logically perfect) language in general. Moreover, it seems that Pelletier incorrectly disregards other passages concerning compositionality from ‘Logic in Mathematics’, since this paper was published only posthumously (Pelletier 2001: 107–108). See also footnote 74 below.

41 Frege had formulated his priority principle in regards to judgments and concepts at first. However, he later subscribed to the doctrine reformulated in terms of Thoughts, as is shown by fragments from papers cited below such as ‘A Brief Survey of My Logical Doctrines’, ‘Notes for Ludwig Darmstaedter’ and ‘On Concept and Object’, and the uncited ones: ‘Introduction to Logic’ (Frege 1906b [1979]: 187) and ‘Logic’ (Frege 1897 [1979]: 143). I focus on Frege’s mature period; therefore, I prefer to use the terminology of Thoughts. For a different discussion of primacy of truth and Thoughts in Frege’s mature doctrine, see Burge (1986 [2005]).

42 The range of texts that are important for the thesis in this point and in the next two extends from Frege (1879 [1967]: §9); 1882a [1980]: 101) through Frege (1897 [1979]: 143); 1906b [1979]: 187) up to Frege (1919 [1979]: 253) 1924/1925 [1979]: 269). They mostly correspond in content to support of these three views and I discuss them extensively in large part below; thus, the reader can confront them later with these theses.

43 Frege (1879 [1967]: §9).

44 I owe this reference to an anonymous reviewer.

45 Incidentally, Frege notes two pages later: ‘We can regard a sentence as a mapping of a thought; corresponding to the whole-part relation of a thought and its parts we have by and large, the same relation for the sentence and its parts’ (Frege 1919 [1979]: 253).

46 One of the consequences of that point of view is the rejection of the traditional subject-predicate account of judgments.

47 See footnote 17.

48 The article is the comparison of some aspects of Frege’s view expressed in Begriffsschrift (Frege 1879 [1967]) and George Boole’s position concerning the logic of propositions and of concepts contained in his book The Laws of Thought (Boole: 1854).

49 Hence the date at the end of the quotation.

50 Essentially, the same conception of concepts and functions appears in Basic Laws of Arithmetic, where Frege claims that the role of a function is to correlate arguments with values. They have an auxiliary and secondary function: ‘The essence of the function manifests itself rather in the connection it establishes between the numbers whose signs we put for “x” and the numbers that then appear as denotations of our expression […]’ (Frege 1893 [2013]: §1). See also Heck and May (2013). Incidentally, Heck and May’s discussion of unsaturatedness of functions (especially 888—892) seems to stand in conflict with Heck’s view of the context principle in Frege’s mature doctrine (see Heck 2012a). In Heck (2012a), this principle has a much more restricted scope than its discussion allows in Heck and May (2013).

51 For an interesting discussion, see Heis (2013: 118—122, 2014: 272–276).

52 See more on this in Beaney (1996: 125–131) and Bell (1996: 588–592).

53 Both examples come from Frege (1893 [1979]: 33).

54 For the sake of simplicity and availability to a wider public, I do not use Frege’s symbolic logic notation or its contemporary counterpart (apart from some exceptions; see footnotes 58, 59 and 70). However, it is a mistake to think that this choice diminishes our logical resources. We still use notation for generality and other concept scripts’ improvements even though they are written in their natural language counterparts (such as ‘for all … ’). See Carnap (1937 [2001]: 4–10).

55 For a similar example and discussion, see also Heck and May (2013). Heis entertains an example of the property of being hereditary in the ƒ-series (Heis 2013: 120–121).

56 I slightly modified the example and its description that can be found in Bell (1996: 590). Frege provides examples of the continuity of a function and of a limit and that of following a series and other (Frege 1893 [1979]: 21–27 and 34).

57 The sign ‘||’ symbolizes the relation of being parallel.

58 This principle can be formalized as follows: $\left(HP\right)\mathrm{\#}F=\mathrm{\#}G\leftrightarrow F\approx G$where ‘#F’ means the number of Fs, ‘#G’ means the number of Gs and F ≈ G is some second-order formalization of the claim that there is a relation that one-to-one correlates the Fs and the Gs (Linnebo 2018: 115).

59 In a contemporary notation: (ὲΦ(ϵ) = ὰΨ(α)) ↔ ∀x (Φ(x) ↔ Ψ(x)) (BLV).

60 See on this van Heijenort (1977: 103) and Sluga (1986: 59–61).

61 Frege claims there: ‘We introduce a new name by means of a definition by stipulating that it is to have the same sense and the same denotation as some name composed of signs that are familiar. Thereby the new sign becomes gleichbedeutend as that being used to define it’ (Frege 1893 [2013]: §27).

62 See Currie (1982) and Beaney (1996: 225–234) for a required notion of content here.

63 From now on I will call the latter view the recarving thesis, since Frege speaks of carving up a content in a different way in The Foundations of Arithmetic (see the passage quoted earlier from this work).

64 I stress this relation between developing the concept script, the concept formation, and the context principle, since it is usually overlooked. Actually, the only place where all three are in some way accounted for is in Beaney (1996: 127–136 and 143–148).

65 There are two possible cases here. Firstly, there are, at least, two sentences that have different logical forms, and non-isomorphic semantical structures, but they express the same Thought. Secondly, there are, at least, two sentences that express the same Thought and have non-isomorphic semantical structures, but they have the same logical form. Both cases are incompatible with the isomorphism thesis. According to the isomorphism thesis, a Thought built out of senses of an object and a first-order concept cannot be identical with a Thought containing a second-order concept and a first-order concept (having structures <O_s:C¹_s> and <C²_s:C¹_s>, respectively, where subscripts signal that we speak of senses of objects/concepts and superscripts indicate levels of concepts). Both cases are also incompatible with the view that Thoughts have an intrinsic structure. The pairs (IVa) & (IVb) and (Va) & (Vb) are instances of the first case. The sentences mentioned below i.e. (VIa) & (VIb) & (VIc) are examples of the second one (if we believe—as Frege seems to believe—that one form of the logical analysis is more basic than other in some instances. I have given reasons above for the thesis that both members of the pairs (IVa) & (IVb) and (Va) & (Vb) have the same sense. For a discussion, see Bell (1996); Beaney (1996: 237–242); Dummett (1989); Ebert (2015).

66 A Thought is composed of senses of objects and/or senses of functions/concepts. I call these components and their order the semantical structure of the Thought.

67 To be sure, the exact nature of this example (i.e. VIa, VIb, VIc) and the previous ones, i.e. (IIIa) & (IIIb), (IVa) & (IVb) and (Va) & (Vb) is different. All of them, however, presuppose the recarving thesis and the notion of sense that allows sentences being cognitively distinct to express the same Thought. It is a mistake to think that one of these theses is based on the grammatical subject-predicate analysis. On the contrary, Frege’s example of sentences VIa, VIb and VIc shows that his logic is superior to the traditional one since it enables a more flexible and effective analysis. This analysis reveals the function/argument structure of a sentence. For a view in a similar spirit, see Bell (1981: 221–224).

68 See on this Carl (1996: 110).

69 As we saw in the previous section, it is enough for an ordinary language if people engaged in communication connect approximately the same Thought with the same sentence.

70 In a contemporary notation, that form is: ∃ x (x² = 4).

71 I cited this fragment earlier, in Subsection 4.2.

72 I can now summarize the train of thought concerning the isomorphism thesis. I characterized this thesis in Section 1 as a view that ‘Thoughts contain senses as parts and that those parts of Thoughts correspond to parts of the sentence expressing them’. We should be familiar now with the view that a Thought consists of a definite set of parts and has a particular logical form only from the point of view of the given analysis. Therefore, the isomorphism thesis states that the structure of a sentence is isomorphic with a Thought expressed in the statement. A Thought has parts that are ordered in some way, but these components are ascribed to the Thought only in a particular language. On the other hand, Thoughts are always grasped by means of language; hence, they always posses some structure and components.

73 In fact, Dummett explicitly rejects the recarving thesis (Dummett 1989: 7).

74 After 1891, Frege states the view that a Thought can be expressed in different ways and hence it possesses the structure only relatively also in—for instance—‘Introduction to Logic’ (Frege 1906b [1979]: 187) and ‘Logic’ (Frege 1897 [1979]: 143).

75 In the next section, I will speak as if a Thought would have a structure tout court. However, it is necessary to remember that—strictly speaking—the Thought is always a statement under a particular logical analysis. In other words, only the Thought as expressed in the given statement contains senses as parts and these parts of the Thought correspond to the parts of the sentence expressing them. This issue is related to the circumstance that Thoughts cannot be apprehended except by means of some linguistic form, but this problem goes beyond the scope of my paper (see Frege 1924/1925 [1979]: 269).

76 It may seem that this quote and the next one concern languages in general rather than the logically perfect language. Nevertheless, the topic of Frege’s paper is formal languages such as the concept script, and I think it should be clear from this general topic of the paper that he does not mean a language in general (similarly to the second controversial essay, i.e. ‘Logic in Mathematics’). As we saw, the isomorphism thesis cannot be in the strict sense true for natural languages, and indeed, Frege warns us right after the cited fragment that we really speak of parts of Thoughts figuratively, because of ‘hitches’ or imperfections that occur from time to time (in natural languages). He uses a limited analogy only to emphasize some interesting properties of Thoughts. Frege reports an analogous proviso in the essay ‘Logic in Mathematics’ (Frege 1914 [1979]: 207). On the other hand, for some readers it can be misleading on Frege’s part to talk in this context of syllables or sounds, but this way of speaking has only a didactic aim.

77 In this paper, Frege considers the compositional view of a language four times. The first passage is cited in the previous section. The second follows directly the excerpt discussed below and it is analogous to the fragment from a letter to Russell. The last one is to be found on page 243.