Abstract
This chapter systematically expounds Vasil’ev’s logic of concepts, that is, a logic in which the law of excluded middle does not hold. Sigwart, especially with his concept of the forms of judgment and his critique of particular judgement, exercised a considerable influence on Vasiliev’s development of such a logic. Taking up Sigwart’s analysis, Vasil’ev gives a strong interpretation of the particular judgment as ‘Only some (not all) S are P,’ while the form ‘Some, and maybe all, S are P’ would correspond to the Aristotelian indefinite proposition. According to Vasil’ev, the strong particular affirmative judgment presupposes the particular negative judgment ‘Some (the remaining) S are not P,’ and vice versa. These constitute one sole judgment, the accidental one. By means of an analysis of the square of opposition, Vasil’ev shows that for the judgments about concepts, which he distinguishes from the judgments about facts, there are three kinds of universal judgments (affirmative, negative, and universal) among which only the relation of contrariety holds, and therefore the law of the excluded fourth holds, not that of excluded middle, as is shown in the triangle of oppositions. The chapter closes with a historical excursus on the principle of excluded middle followed by a debate on particular propositions between Louis Couturat and Salomon Ginzberg, who discuss ideas which show very strong affinities to Vasil’ev’s on strong particular judgment.
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Notes
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.
Sigwart ’s Logik went through four editions (Tübingen 1873–18781, 1889–18932, 19043, 19114). Since the Russian version Vasil’ev consulted was based on the third edition (cf. Sigwart 1908–1909), we take this as our reference out of concern for uniformity. For further details on Sigwart ’s Logik see Raspa (1999b: 99 ff., 271 ff.), Stelzner & Kreiser (2004: 71 ff., 99 ff., 147 ff. and passim).
- 8.
Sigwart (1904 3: i, § 1, p. 1 [1895: i, 1]).
- 9.
Cf. Sigwart (1904 3: i, § 1, pp. 9–10; § 5, pp. 27–28 [1895: i, 9–10, 25–26]).
- 10.
According to the English translation of Sigwart ’s Logik (1895), the German word ‘Vorstellung’ is here translated as ‘idea.’ However, when Vasil’ev uses the term ‘представление’ (predstavlenie), I have adopted the word ‘representation,’ which is closer to the sense of ‘Vorstellung.’
- 11.
- 12.
- 13.
- 14.
- 15.
- 16.
- 17.
Sigwart (1904 3: i, § 34, p. 276 [1895: i, 207]).
- 18.
Cf. Sigwart (1904 3: i, § 27, pp. 220–222; § 30, p. 234 [1895: i, 163–165, 175]).
- 19.
Sigwart (1904 3: i, 235 [1895: i, 176]).
- 20.
- 21.
Sigwart (1904 3: i, § 31, p. 237 [1895: i, 177]) writes: “The formula so often used ‘A may be B’ is ambiguous and misleading, for it expresses both the objective ‘can’ (δύνασθαι) and subjective hesitation.”
- 22.
Sigwart (1904 3: i, § 31, p. 239 [1895: i, 179]).
- 23.
- 24.
Cf. Hamilton (1861–18662: iv, 283–284): “The designation of indefinitude or particularity, some (ˏ or ˎ) may mean one or other of two very different things. 1°, It may mean some and some only, being neither all nor none, and, in this sense, it will be both affirmative and negative, (ˎ ˏ). 2°, It may mean, negatively, not all, perhaps none, — some at most; affirmatively, not none, perhaps all, — some at least, (ˏ ˎ). Aristotle and the logicians contemplate only the second meaning. The reason of this perhaps is, that this distinction only emerges in the consideration of Opposition and Immediate Inference, which were less elaborated in the former theories of Logic; and does not obtrude itself in the consideration of Mediate Inference, which is there principally developed.”
- 25.
Cf. Minto (1893: 63): “Some stands for any number short of all: it may be one, few, most, or all but one.”
- 26.
Cf. Venn (1894 2: 11–13, 180–185, 277, 485–487).
- 27.
Cf. Bain (1870: i, 81–82).
- 28.
- 29.
Cf. Aristotle, An. pr. i 1, 24a16–22: “A proposition, then, is a statement affirming or denying something of something; and this is either universal or particular or indefinite. By universal I mean a statement that something belongs to all or none of something; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark of being universal or particular, e.g. ‘contraries are subjects of the same science,’ or ‘pleasure is not good’.”
- 30.
- 31.
Cf. Aristotle, An. pr. i 4, 26a30; 6, 28b27–29; 7, 29a27–29; 20, 39b2–3.
- 32.
- 33.
- 34.
- 35.
Ibid.
- 36.
- 37.
- 38.
- 39.
The general form of an Aristotelian sentence is SxP, where S denotes the subject, P the predicate and x the copula; this can be replaced by a, e, i, o — taken from the Latin words affirmo and nego — expressing quality and quantity of judgments. Therefore, SaP stands for ‘All S are P,’ SeP for ‘No S is P,’ SiP for ‘Some S are P’ and SoP for ‘Some S are not P.’
- 40.
- 41.
As Lossky (1927: 172) had already noted.
- 42.
We shall see that the need to eliminate the vagueness of the weak form of the particular judgment in favour of the univocity of the strong form lies at the heart of the controversy between Salomon Ginzberg and Louis Couturat (see Sect. 3.4).
- 43.
- 44.
Cf. Trendelenburg (1870 3: ii, 291).
- 45.
Cf. Lotze (1880 2: i, 67).
- 46.
- 47.
- 48.
- 49.
As Bradley (1883: 46–47, 82) has already pointed out, every universal judgment of the type ‘All A are B’ is really a hypothetical judgment, which asserts: ‘If anything is A, then it is B.’ The same holds for the negative universal judgment and for the accidental one: ‘If anything is A, then either it is B or is not B.’
- 50.
Cf. Aristotle, Int. 7. 17b16–20, 26–27.
- 51.
Cf. Aristotle, Int. 7. 17b3–6, 20–23.
- 52.
Cf. Aristotle, An. pr. ii 11. 61b6, 62a17–19.
- 53.
Cf. Aristotle, Int. 7. 17b23–26.
- 54.
Cf. Aristotle, Top. ii 1. 109a3–6.
- 55.
- 56.
- 57.
Cf. Boethius , In librum Aristotelis de interpretatione. Libri duo, editio prima, seu minora commentaria, i, 321 b; In librum Aristotelis de interpretatione. Libri sex, editio secunda, seu majora commentaria, ii, 471 a–b.
- 58.
Cf. Kneale & Kneale (1962: 56 ff.). The chapter on Aristotle is by Martha Kneale.
- 59.
Cf. Sainati (1968: 226–240). In more recent times, the same thesis has been sustained by Wedin (1990) and Parsons (1997; 2008; 2014). Menne & Öffenberger (1980/1981) offer an interpretation of the square of opposition from a four-valued perspective. In their opinion, whilst a two-valued interpretation is not exhaustive, but leaves some inconclusive links in the theory of oppositions, the four-valued perspective involves and extends the two-valued one, without setting off conflict.
- 60.
Sainati (1968: 229).
- 61.
Cf. Prior (1962 2: 165), who refers also to the Schoolmen.
- 62.
Cf. Aristotle, Int. 5. 17a8–9; An post. I 25. 86b34–36.
- 63.
Cf. Aristotle, Int. 7. 17b26, 18a5.
- 64.
Wedin (1990: 134).
- 65.
- 66.
Parson s (2008: 8–9).
- 67.
Today, the square of opposition has become object of renewal studies, as a series of recent publications and conferences has demonstrated. For more information, see the website: http://www.square-of-opposition.org/
- 68.
- 69.
- 70.
- 71.
In Russian, the word ‘some’ can be translated both as несколько (neskol’ko) and некоторые (nekotorye), whereas neskol’ko is equivalent to an indefinite numeral adjective (some, several, a few) and is used to express the idea of a moderate generic quantity, the use of nekotorye, in turn an indefinite but not numeral adjective, implies a comparison, a relation between one part (some, several) and the whole. Therefore, Vasil’ev uses neskol’ko (some, several, a few) as a sign of indefinite-numerical judgment, nekotorye (some, several) as a sign of accidental judgment (see above the examples on pp. 37 and 43). This peculiarity of the Russian language can be translated into English only with figurative phrases, thus both neskol’ko and nekotorye have been translated by ‘some.’
- 72.
Cavaliere (1992–1993: 118).
- 73.
- 74.
Cf. Łukasiewicz (1910a/1987: 139–141) and (1910b: 37 [1971: 508–509]): “At a time of the political decline of Greece, Aristotle became the founder and investigator of systematic, scientific, cultural work. […] Denial of the principle of contradiction would have opened door and gate to every falsity and nipped the young, blossoming science in the bud. Hence, the Stagirite turns against the opponents of the principle with forceful language in which one can trace an internal fervor, against the eristic thinkers of Megara, the cynics of the school of Antisthenes , the disciples of Heraclitus , the partisans of Protagoras ; and he battles with all of them for a theoretical principle as if for personal goods. He might well have himself felt the weaknesses of his argument, and so he announced his principle a final axiom, an unassailable dogma.”
- 75.
- 76.
- 77.
Aristotle, Metaph. γ 7. 1011b23–24.
- 78.
The formulation of the principle of excluded middle in the Logique of Port-Royal is different to Vasil’ev’s: “Contradictories are never both true or both false, but if one is true the other is false, and if one is false the other is true” (Arnauld & Nicole 1662/1965: ii partie, ch. iv, p. 117 [1996: 85]).
- 79.
Cf. Wolff (1740/1962: ii, § 532, p. 401): “Propositionum contradictoriarum altera necessario vera; altera necessario falsa [Of two contradictory propositions one is necessarily true; the other necessarily false].”
- 80.
Vasil’ev refers to Kant (1800: Ak. ix, 53 [1992: 560]): “[…] the principle of the excluded middle (principium exclusi medii inter duo contradictoria), on which the (logical) necessity of a cognition is grounded — that we must necessarily judge thus and not otherwise, i.e., that the opposite is false — for apodeictic judgments.”
- 81.
Cf. Schopenhauer (1859 3/1988: ii, Chap. 9, p. 122 [1966: ii, 103]): “It seems to me that the doctrine of the laws of thought could be simplified by our setting up only two of them, namely the law of the excluded middle, and that of sufficient reason or ground. The first law thus: ‘Any predicate can be either attributed to or denied of every subject.’ Here already in the ‘either, or’ is the fact that both cannot occur simultaneously, and consequently the very thing expressed by the law of identity and of contradiction. Therefore these laws would be added as corollaries of that principle, which really states that any two concept-spheres are to be thought as either united or separated, but never as both simultaneously; consequently, that where words are joined together which express the latter, such words state a process of thought that is not feasible. The awareness of this want of feasibility is the feeling of contradiction.”
- 82.
According to Lotze , the law of excluded middle is a particular case of the disjunctive law of thought (disjunktives Denkgesetz). Moreover he states: “Der Gedanke, den die Form des disjunctiven Urtheils ausdrückt, wird gewöhnlich in zwei gesonderten Denkgesetzen, dem Dictum de omni et nullo und dem Principium exclusi tertii inter duo contradictoria ausgesprochen; ihre Verschmelzung in ein einziges drittes Grundgesetz ist indessen nicht nur leicht, sondern nothwendig” (Lotze 1880 2: i, 94–95).
- 83.
Cf. Troitsky (1886: 101): “What is called the principle of excluded middle is the axiom which establishes direct evidence of the incompatibility of contradictory propositions (contradictoriae). The principle is expressed as follows: ‘The contradictory propositions A and O, E and I, by excluding each other, do not allow for even a middle term between them’.”
- 84.
Cf. Hamilton (1861–18662: iii, 83): “The principle of Excluded Third or Middle — viz. between two contradictories, (principium Exclusi Medii vel Tertii), enounces that condition of thought, which compels us, of two repugnant notions, which cannot both coexist, to think either the one or the other as existing. Hence arises the general axiom, — Of contradictory attributions, we can only affirm one of a thing; and if one be explicitly affirmed, the other is implicitly denied. A either is or is not. A either is or is not B.”
- 85.
Cf. Wundt (1893 2: i, 565–567): “Schon Aristoteles hat dem Satz des ausgeschlossenen Dritten eine selbständige Bedeutung zuerkannt. Später hat man ihn meist für entbehrlich angesehen, indem man meinte, er ergebe sich von selbst, wenn man das Identitätsgesetz mit dem Satz des Widerspruchs verbinde. Wäre aber dies richtig, so müsste in der Formel „A = B und A = non-B widersprechen sich “unmittelbar der Satz des ausgeschlossenen Dritten enthalten sein: „A ist entweder B oder non-B“. Dies ist aber nicht der Fall; die Erklärung, dass B und non-B sich widersprechen, schliesst nicht aus, dass es neben beiden noch ein Drittes gebe. Ebenso wenig folgt dies aus der Aufhebung der doppelten Verneinung. Denn diese zeigt nur an, dass man durch die Häufung der Verneinungen keine neue logische Function neben Bejahung und Verneinung erzeugen kann; es bleibt aber dahingestellt, ob nicht neben der Verneinung noch eine andere Form der Aufhebung eines positiven Begriffs existirt. Dass dies nicht der Fall ist, sagt eben erst der Satz des ausgeschlossenen Dritten. Dagegen setzt dieser die Gesetze der Identität und des Widerspruchs voraus, und wenn es daher durchaus darauf ankäme die drei logischen Axiome auf eines zurückzuführen, so wäre dazu, wie Schopenhauer richtig erkannt hat, kein anderes als der Satz des ausgeschlossenen Dritten geeignet. Gleichwohl würde sich diese Reduction kaum empfehlen. […] Der Satz des ausgeschlossenen Dritten kann als das Grundgesetz der disjunctive Urtheile betrachtet werden […] Gerade der Satz des ausgeschlossenen Dritten ist mehr als die beiden vorigen Axiome in seiner abstracten logischen Form als Regel der wirklichen Eintheilung, selbst der Erfahrungsobjecte, verwendet worden, indem man die Eintheilung nach dem contradictorischen Gegensatze wegen ihrer nie mangelnden logischen Richtigkeit bevorzugte.”
- 86.
Cf. Minto (1893: 29): “Every thing is A or not-A; or A is either b or not-b.”
- 87.
Cf. Ueberweg (1882 5: § 78, p. 265): “A ist entweder B oder ist nicht B; jedem Subjecte kommt jedes fragliche Prädicat entweder zu oder nicht.” Ueberweg gives also an Aristotelian formulation: “contradictorisch einander entgegengesetzte Urtheile (wie: A ist B, und: A ist nicht B) können nicht beide falsch sein und lassen nicht die Wahrheit eines dritten oder mittleren Urtheils zu, sondern das eine oder andere derselben muss wahr sein, und aus der Falschheit des einen folgt daher die Wahrheit des anderen. Oder: die Doppelantwort: weder ja noch nein, auf eine und dieselbe in dem nämlichen Sinne verstandene Frage ist unzulässig” (p. 254).
- 88.
Cf. Krug (1819 2: § 19, p. 51 ff.).
- 89.
Krug (1819 2: § 19, Anm. 3, p. 54): “Denn ein Triangel überhaupt ist doch wohl ein logischer oder denkbarer Gegenstand. In dem Begriffe des Triangels überhaupt aber ist weder das Merkmal rechtwinkelig noch das Merkmal nicht rechtwinkelig enthalten. Der Gegenstand bleibt nämlich in dieser Hinsicht unbestimmt. […] Also nur unter der Voraussetzung, dass ein Ding als durchgängig bestimmt gedacht werden soll, muss ihm von jedem Paar widersprechender Merkmale Eins zukommen.”
- 90.
- 91.
- 92.
Mill (1872 8/1973: ii, vii, § 5, p. 278).
- 93.
Cf. Sigwart (1904 3: i, § 25, p. 202 [1895: i, 150]): “It follows of itself from the principles of contradiction and of twofold negation that of two contradictorily opposed judgments one is necessarily true; hence that there is no third statement besides affirmation and negation which would imply the falsity of both. This is the principle of the excluded middle, which, like the two previous principles, aims only at interpreting more fully the nature and meaning of the negation.”
- 94.
- 95.
- 96.
Vasil’ev (1910: 41, 44 = 1989: 48, 50; 1912–1913a: 64 = 1989: 106 [1993: 337]) appears to adopt as his own the idealistic thesis according to which reality consists of perceptions and representations. Since he does not follow through with any arguments on this matter, we are unable to examine his point of view.
- 97.
- 98.
- 99.
- 100.
- 101.
Cf. Lossky (1927: 169–173). Stelzner (2001: 280–283; cf. also Stelzner & Kreiser 2004: 253–256) points out a parallelism between Lossky and Vasil’ev regarding the denial of the laws of contradiction and of excluded middle, although textual evidence that would indicate Vasil’ev was familiar with Die Grundlegung des Intuitivismus (1908) is lacking.
- 102.
Ginzberg (1913: 102).
- 103.
Ginzberg (1913: 103).
- 104.
Ginzberg (1913: 103–104).
- 105.
On this matter see Sect. 6.4. Suchoń (1999: 133) speculates that Vasil’ev must have been disappointed by the exiguous number of valid modes one might construct (only six, if four figures are considered). Stelzner & Kreiser (2004: 170) also draw attention to the fact that, in the logic of concepts, only five valid modes can be constructed (if the first three figures are considered), and maintain this to be an argument that a traditional logician might raise as an objection to the reform Vasil’ev proposed. If this were so, the argument would rebound back against whoever used it, because fourteen modes are not too many more than five. Ginzberg (1913: 106) on the other hand maintains that researchers ought to be left free to choose either interpretation of ‘some,’ corresponding to the states of knowledge, since the restricted interpretation is adapted to the exposition of the truths that have been acquired. Yet one more affinity with Vasil’ev.
- 106.
Cf. Couturat (1913: 257–258).
- 107.
Cf. Ginzberg (1914).
- 108.
Cf. Gergonne (1816/1817).
- 109.
Couturat (1914: 260).
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Raspa, V. (2017). The Logic of Concepts. In: Thinking about Contradictions. Synthese Library, vol 386. Springer, Cham. https://doi.org/10.1007/978-3-319-66086-8_3
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