Abstract
Frege frequently complains that others are ignorant of the distinction between “falling under” and “subordination”. This criticism is not only directed against the philosophers who are under the influence of Aristotelian logic but also against the mathematicians of his time. I shall show that this distinction must be the vantage point for understanding Frege in both historical and philosophical contexts. Strangely, this distinction is not studied extensively nowadays. There are some good reasons for this. First, ironically, it is so well established as to become a triviality. Secondly, some people think that Frege’s criticism of the aggregate view of sets is outdated. Consequently, we cannot understand why this distinction was so important to Frege. In what problem situation did Frege formulate this distinction? Were there any rival theories of predication? Was this distinction an ad hoc device for Frege in order to establish other important theses? What would happen if we lack this distinction? This chapter aims at a partial answer to these questions.
An earlier much shorter version was published in Korean as Park (1992), and also reprinted as Chap. 3 of Park (1997). But this published version completely deleted Sect. 3.1 of the original version, which was submitted as a final term paper to John Corcoran’s graduate seminar entitled “History of Logic, Spring 1986, State University of New York at Buffalo”. Such a cut was rather drastic, for deleted part had dealt with the lack of the distinction between “falling under” and “subordination” in traditional logic up until the 19th century. In this paper, I shall recover it entirely. Also, I shall attempt a thorough update by incorporating more recent advances in Frege scholarship.
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Notes
- 1.
Ignacio Angelelli and Nino B. Cocchiarella seem to be rare exceptional cases. Angelelli devotes a chapter on this distinction, and scrutinizes the question why it was lacking in traditional philosophy. As a consequence, this chapter is full of interesting information. However, in general, he tends to presuppose the correctness of the distinction rather than contesting its merit critically. He never presents clearly what the distinction itself is. Nor does he examine Frege’s criticism of 19th century mathematicians in detail. For example, he never quotes from Dedekind’s own works (Angelelli 1967). Cocchiarella’s analysis is full of logical and philosophical insights. Since he deals with all important issues related to Frege, however, it may not be a kind introduction to the distinction between “falling under” and “subordination” for beginners or general public.
- 2.
The distinction is of early origin in the development of Frege’s thought. In several places we can find it out. For example, at the beginning of §53, he drew the distinction between properties of a concept and the mark of the concept. And, immediately he added the following explanation: “These latter (the characteristics of which make up the concept) are properties of things which fall under the concept, not of the concept” (Frege 1884, \({64^{\text{e}} }_{7}\)). Moreover, at the end of the section, he made it clear that we should not confuse the relation of one concept’s falling under (within) another higher concept with the subordination of species to genus (Frege 1884, \({65^{\text{e}}}_{7}\)).
- 3.
As John Corcoran indicates, the idea that singulars are logically equivalent to universals is not the same as the idea that “falling under” is the same as “subordination”. Following Quine, we can treat Ps as logically equivalent to ∀x(x = s ⊃ Px). Even among the schoolmen we can find a thinker who interpreted “Socrates is running” as “Everything that is Socrates is running” (Prior 1963, p. 160). In this respect, it is by no means clear why traditional logicians (in particular in the post-Renaissance period) treated singular propositions as universal propositions. If it is the result of their theory which considers “falling under” as “subordination” (i.e., as will be examined in the next section), then it is still more dubitable why it was treated that way more clearly in post-Renaissance period.
- 4.
As Cocchiarella points out, here I am leaving unexplained how traditional logicians took predication to function in E, I, O type propositions (Cocchiarella 2015b).
- 5.
The following quote from Hintikka (1981) seems extremely informative: “The sharpest specific difference between Frege’s logical notation and the ideas of his predecessors lies in his treatment of verbs for being. Such verbs are, according to Frege and his followers, ambiguous in that they have to be translated into the logical notation in at least four different ways: (i) by the identity sign ‘=’ (the ‘is’ of identity); (ii) by the existential quantifier (the ‘is’ of existence); (iii) by predicative juxtaposition (the ‘is’ of predication or the copula), and (iv) by a general implication (the ‘is’ of class inclusion). In 1914 Bertrand Russell (1914) called this fourfold distinction ‘the first serious advance in real logic since the time of the Greeks’, and it has been incorporated in all the usual systems of first-order logic (lower predicate calculus, quantification theory). Hence everybody who has been using the notation of first-order logic for the purpose of semantical representation is committed to the ambiguity of ‘is’. This applies to linguists and philosophers otherwise as unlike each other as George Lakoff, Noam Chomsky, W. V. Quine, Donald Davidson, and Ludwig Wittgenstein. Frege’s distinction has thus become one of the most widely accepted inventions of his and an integral part of most current treatments of semantics (Hintikka 1981, p. 72).
- 6.
- 7.
This distinction must originate from Aristotle. As a matter of fact, in Topics 102a 32, we can find Greek equivalent of “in eo quod quid est”. Interestingly enough, I am unable to find out Aristotle’s equivalent of “in eo quod quale est”.
- 8.
Corcoran kindly reminded me of the relevance of this distinction to my discussion. He was careful enough not to identify the distinction with the fallacy of composition. The late Charles Lambros also helped me in various ways. He was so critical of Frege that he even thought that Frege (1884) would not have argued if he had known the distinction between collective and distributive predication: “Frege’s error here is glossing over the now familiar distinction between distributive and collective predicates” (Lambros 1976, 381).
- 9.
Of course, by ‘is a part of’ Dedekind meant ‘is a subset of’, i.e., class inclusion. So, one might think that Frege was being too fast here. It seems that Frege thought that the distinction between “class inclusion” and “class membership” is not different from the distinction between “subordination” and “falling under”: “We must keep separate from one another: (a) the relation of an object (an individual) to the extension of a concept when it falls under the concept (the subter relation); (b) the relation between the extension of one concept and that of another when the first concept is subordinate to the second (the sub relation) (Geach and Black 1952, p. 106). I will deal with this point in more detail below.
- 10.
For a rather detailed examination of Frege’s criticism of Schröder, see Cocchiarella (1987, pp. 89–90).
- 11.
“Und auch ein Individuum mögenwir bezichnen als eine Klasse, welche eben nur dieses Individuum selbst enthält. Ein jedes gedankending kann zu solchen Individuum gestempelt warden….Auch jene Klasse aber, die selber eine Menge von Individuen umfasst, kann wieder al sein Gedankending und demgemäss auch al sein “Individuum” (im weiteren Sinne, z.B. “relativ” in Bezug auf höhere Klassen) hinfestellt warden (Schröder 1890, 148).
- 12.
Here I largely depend on Gillies’ presentation. Frege’s own reasoning is much more complicated.
- 13.
See Frege (1895, 97). If we mark the existence of the item as o and non-existence as x, then there are only different possibilities.
- 14.
Cocchiarella disagrees with my opinion on the ground that these distinctions are equivalent “but that is because the distinction between membership and falling-under is one of the consequences of Frege’s double-correlation thesis” (Cocchiarella 2015b).
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Park, W. (2018). Frege’s Distinction Between “Falling Under” and “Subordination”. In: Philosophy's Loss of Logic to Mathematics. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-95147-8_2
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