‘Horrent with Mysterious Spiculæ’. Augustus De Morgan’s Logic Notation of 1850 as a ‘Calculus of Opposite Relations’

Abstract

The present paper expounds the logic notation proposed by Augustus De Morgan in 1850 from within the original context of De Morgan’s account of syllogistic logic and his approach to quantification. The notational system of 1850 is shown to be a flexible tool to state inferences, to prove their validity and to derive formulæ of the respective system by ‘blind’ application of transformation rules. These pertain to the swapping of operator signs, which are of inverse ‘character’ in a two-fold sense: The signs both represent inverse operations and exemplify oppositeness due to their very shapes. Thus, while De Morgan’s initial choice of symbol shapes is due to certain diagrammatic features, his notational system of 1850 leads to an account of logical operations as effected by manipulation of symbol tokens. It is concluded that as an instantiation of what De Morgan terms a ‘calculus of opposite relations’, his notation of 1850 may also be conceived of as a ‘graphicism’ affording for the working of a ‘symbolic machine’ on (intermediate) ‘de-semantification’.

Acknowledgements

I am grateful to Amirouche Moktefi, Wolfgang Kienzler, and Dirk Schlimm for novel perspectives, and indebted to anonymous referees for additional information, comments and suggestions as well as thorough criticism.

Notes

1 Not to be confounded with the Irish mathematician Sir William Rowan Hamilton, the inventor of quaternions.

2 ‘Spicula’ is the Latin word for ‘parenthesis’ or ‘bracket’.

3 De Morgan 1850, p. 83.

4 De Morgan’s notational system of 1850 has rarely been presented by a reconstruction from an internal historical perspective. There is, however, a brief outline in Heath 1966 (pp. xxvi–xxvii) and an illuminative overview in Richards and Hobart 2008 (pp. 297–306). Merrill 1990 (pp. 149–69) invokes it systematically to expound some fundamental ideas of De Morgan’s steps towards a logic of relations. In a similar fashion, it is employed in Panteki 1991 (pp. 440–50). However, it is omitted in other expositions of the subject such as Hawkins 1995 as well as Sánchez Valencia 1987 and Sánchez Valencia 1997. In some overviews of the history of algebraic logic, only an earlier notation of De Morgan’s is referred to. This applies, for example, to Hailperin 2004 (pp. 346–9) and Sánchez Valencia 2004 (p. 408), who gives a (quasi) set-theoretic paraphrase of what he holds to be De Morgan’s approach to quantification of the terms of syllogisms (pp. 409–10).

5 This is important because earlier versions of De Morgan’s logic notation differ substantially from that proposed in De Morgan 1850. In that the earlier notation has one spicula in an absolute position, whereas in the later systems, there are as many parentheses as there are terms, and they are applied as relative to the respective terms.

6 In his 1850, De Morgan refers to his systems as the ‘system of contraries’ (De Morgan 1850, pp. 101, 102) the ‘numerically definite system’ (De Morgan 1850, p. 79), and the ‘exemplar system’ (De Morgan 1850, p. 101) as opposed to a ‘cumular’ one (De Morgan 1850, p. 102). The first is propounded by a somewhat clumsier notation in De Morgan 1847a already; the second is suggested in the ‘Addition’ to 1847a and presented in chapter VIII of De Morgan 1847b. The third is developed in De Morgan 1850.) As to their conceptual foundation, the system of contraries, the numerically definite system and the exemplar system share what in his 1850, De Morgan calls a ‘language of numeration of instances’ of terms (De Morgan 1850, p. 96). However, De Morgan’s systems differ in their respective structural features, prominently in the number of valid inferences and their suspending different rules of traditional syllogistic.

7 The ‘system of contraries’ will be referred to in a technical sense without quotation marks in the remainder of the present paper. On De Morgan’s notion of ‘contraries’ see Section 2.2. For a tabular exposition of the system’s 48 forms of valid inferences by De Morgan’s spicular notation see Table 2 in Section 2.7.2.

8 De Morgan 1850, p. 85.

9 This wording invokes an analogy to the principle of ‘separation of symbols’, namely of symbols of quantity and of operations, which is an issue of methodical importance to De Morgan’s mathematical background, especially in Symbolic Algebra. For De Morgan’s mathematical background see Panteki 2003 or 2008 and of course her extensive 1991, especially chapters 3 and 6. On De Morgan’s views on algebra see also Pycior 1983. In chapter 6.9 of her 1991, Panteki argues that there is an analogy between De Morgan’s earlier treatment of the composition of functions and their inverses and his later approach to the composition of relations. These might be considered as further instantiations of De Morgan’s ‘calculus of opposite relations’. But to make this point, it would be necessary to engage into a comparative survey of De Morgan’s views on the ‘form’ vs. ‘matter’ distinction and the role of symbols in mathematics and in logic, taking into account his earlier papers on the foundation of algebra and their echoes in De Morgan 1858, too. For reasons of space, this task cannot be met in the present paper.

10 By 1847a, I am referring to De Morgan’s first paper on the syllogism, which was read before the Cambridge Philosophical Society in November 1846, but published in the Society’s Transactions in 1847. There are altogether five papers in De Morgan’s syllogism series, of which his 1850 is normally counted as the second, and his 1858 as the third.

11 ‘Contraries’ will be employed as a technical term without quotation marks in the remainder of the present paper.

12 In his own explanation (De Morgan 1850, p. 91), De Morgan invokes the notion of induction to explain this meaning, which makes a direct quote problematic.

13 De Morgan 1850, pp. 81–2.

14 De Morgan 1850, p. 82.

15 Ironically, he adopted it from Hamilton in the course of their controversy over what Hamilton dubbed ‘the quantification of the predicate’. Hamilton, however, has quite a different approach to logical quantity than does De Morgan. In De Morgan’s case, the action-like inflection is appropriate, but not in Hamilton’s. The present paper omits further reference to this debate. Its historical course is outlined in Heath 1966 (pp. xi–xxiv); an overview is also given in Laita 1979 (pp. 51–60), and a detailed historical reconstruction may be found in Heinemann 2015. Bonevac 2012, however, omits reference to De Morgan altogether, to the exception of a casual remark stating that De Morgan adopted Hamilton’s scheme of quantification (Bonevac 2012, p. 94) – which I think he did not.

16 De Morgan 1850, p. 92.

17 At least systematically speaking. Genetically, De Morgan’s involvement in the debate over the quantification of the predicate seems to have caused him to extend his notational system as compared with the earlier version in his 1847a and 1847b, which does not have a quantifier sign for each term.

18 I owe this point to Amirouche Moktefi. A similar suggestion can be found in Richards and Hobart 2008 (p. 298), where reference is made to later Venn diagrams.

19 De Morgan 1850, p. 86.

20 De Morgan 1850, p. 80.

21 De Morgan 1850, p. 80.

22 This wording may also be read in the light of De Morgan’s 1858, where De Morgan expounds his version of the distinction between ‘form’ and ‘matter’. Here, De Morgan makes plain that on his view, ‘form’ (with which, of course, logic is concerned) is in fact the ‘law’ of thought, which in turn can be ‘detected when we watch the machine in operation without attending to the matter operated on’. To do so, one must first detect the ‘modus operandi’ (De Morgan 1858, p. 174) which is common to several actions of ‘thought’. Thus, making out logical forms is in fact not a case of abstraction of qualities, but ‘an abstraction of the instrument from the material’, as often seen in mathematics (De Morgan 1858, p. 176).

23 De Morgan 1850, p. 92.

24 De Morgan 1850, p. 92.

25 Hence certain redundancies in the system of contraries.

26 In his 1860, De Morgan names it the ‘rule of contraversion’ (De Morgan 1860, p. 15), and he states it in a language pertaining to the interpretation of the logical operations rather than to the symbols: On ‘changing a name into its contrary without altering the import of the proposition’, the rule ‘is, Change also the quantity of the term, and the quality of the proposition. […] When both names are contraverted, change both quantities, and preserve the quality’ (De Morgan 1860, pp. 15–16). The rule of contraversion is to be distinguished from the ‘rule of conversion’; the latter is just: ‘Write or read the proposition backwards’ (De Morgan 1860, p. 16).

27 The traditional account is that negative propositions, whether universal or particular, exclude the subject from all of the predicate (while affirmatives apply to only some of the predicate term). To state that the subject is not among some of the predicate apparently requires a specification of the intended portion from which it is excluded. But then, the subject term is not among any of specified some of the predicate term.

28 De Morgan 1850, p. 92.

29 De Morgan 1850, p. 92.

30 De Morgan 1850, p. 92.

31 De Morgan here speaks of ‘[c]ontrary propositions’ (De Morgan 1850, p. 92). However, in order not to confound the use of the term ‘contrary’ as applied to terms a with a denomination of the relation holding between propositions mutually denying each other, I retain its common labelling as ‘contradictory’. Note that this not equivalent to the proposition’s ‘contranominal’, which is derived by replacing all term symbols by contraries.

32 De Morgan himself says that one ‘coexists’ with the other.

33 The original text has X(.)Y instead of X()Y here. This must be an erratum.

34 De Morgan 1850, p. 95.

35 De Morgan 1850, p. 93.

36 De Morgan 1850, p. 93. De Morgan here assumes that there are two sides to a proposition—quantity and quality. His idea is that successive transformations regarding quantity (on the one side) and quality (on the other side) may be abbreviated by numbers independently. However, on De Morgan’s scheme, any single transformation on the one side will require (or prompt) one on the other side. If there is one change in quantity—indicated by a rotation of a quantifier sign—, but no change in quality, one counts 1 on the first side, as in 1|0. But if successively, there is a change in quality—as indicated by the withdrawal of a dot or the substitution of an upper-case by a lower-case letter (or vice versa)—, the second side will also count 1; hence 1|1. And so forth. Due to the symmetries in De Morgan’s spicular notation, completion of all possible concatenated transformations restores the original expression.

37 De Morgan 1850, p. 87.

38 De Morgan 1850, p. 94.

39 De Morgan 1850, p. 94.

40 De Morgan’s ‘numerically definite’ system suspends with the rule of the distributed middle. But the numerically definite system does not need quantifier signs (it has numbers of instances instead). Hence the spicular notation is not applicable to the numerically definite system. All systems to which the spicular notation is applicable, however, require a distributed middle in the sense of the canon of validity to be explained above.

41 The table given above omits the headings which De Morgan adds because they contain technical terms from syllogistic, which would require an extra explanation. The indicated way of arrangement of spicular expressions is to be described in the subsequent paragraphs. The table also omits a second notational scheme which De Morgan inserts independently of the spicular forms. This notation uses sub- and superscripted accents to specify the traditional abbreviations A, E, I, and O according as they are instantiated on (positive) terms or (negative) contraries. In fact, the other system does not come up to De Morgan’s own design principles for a versatile notation as expounded in Section 2.1. De Morgan’s use of accents to differentiate propositional forms in the system of contraries should not be confounded with that propounded to specify the forms of the ‘exemplar system’ (De Morgan 1850, p. 102). In this context, ‘the accents refer to the example, subaccent to subject, superaccent to predicate’ (De Morgan 1850, p. 101): A subaccented universal has a universal subject, a superaccented universal has a universal predicate. A subaccented particular has a particular subject and a superaccented particular has a particular predicate (cf. De Morgan 1850, p. 101). Accents are also used in De Morgan’s discussion of the ‘theory of the copula’ (De Morgan 1850, p 104) as a relation in the abstract (cf. De Morgan 1850, p. 105) to differentiate according as a relation between instances is marked by ‘transitivity’ and/or ‘convertibility’ (De Morgan 1850, p. 104) under certain conditions (De Morgan 1850, p. 114).

42 Obviously, just like traditional syllogistic, the system of contraries requires that at least one premise be universal. However, it suspends the rule that at least one premise be affirmative. De Morgan’s numerically definite system, however, allows for inferences from two particular premises.

43 Only recently I learned that De Morgan’s ‘zodiac’ was inscribed on the back side of the ‘De Morgan Medal’, which has been awarded to mathematicians residing in the UK by the London Mathematical Society every third year since 1884.

44 Going beyond De Morgan’s own logic, there are at least two points relating to his spiculæ as tested against the history of logic and logic notations. The first concerns the general question whether De Morgan himself ever stated what today is known as ‘De Morgan’s Laws’. (I am grateful to Wolfgang Kienzler for directing my attention to this question.) The second relates to the history of notations for propositional logic, and the impact which Wilfrid Hodges in a talk suggested that De Morgan’s spiculæ had – inter alia – on Russell and the horseshoe sign of implication. I hope to make this addition to the present exposition of De Morgan’s spicular notation at another occasion.

45 De Morgan 1858, p. 184.

46 Thanks to Dirk Schlimm for re-referring me to Krämer’s works.

47 One example would be a Turing-machine (cf. Krämer 1988, p. 180, cf. pp. 170–71). Here as in other cases, any result which may be obtained symbolically within the workings of the machine cannot itself apply to (or speak of) the machine itself (Krämer 1988, p. 180).