[1] Cf. Heidel, "The Logic of the Pre-Socratic Philosophy," in
Dewey's _Studies in Logical Theory_ (Chicago, 1903).
[2] Heraclitus, _Fragmm._ 107 (Diels, _Fragmente der Vorsokratiker_)
and 2, on which see Burnet, _Early Greek Philosophy_, p. 153 note
(ed. 2).
[3] e.g. Diog. Laërt. ix. 25, from the lost _Sophistes_ of Aristotle.
[4] _Plato and Platonism_, p. 24.
[5] Nothing is. If anything is, it cannot be known. If anything is
known it cannot be communicated.
[6] _Metaphys. [mu]. 1078b_ 28 sqq.
[7] Cf. Arist. _Top. [theta]_. i. 1 _ad fin_.
[8] For whom see Dümmler, _Antisthenica_ (1882, reprinted in his
_Kleine Schriften_, 1901).
[9] Aristotle, _Metaphys._ 1024b 32 sqq.
[10] Plato, _Theaetetus_, 201 E. sqq., where, however, Antisthenes is
not named, and the reference to him is sometimes doubted. But cf.
Aristotle, _Met._ H 3. 1043b 24-28.
[11] Diog. Laërt. ii. 107.
[12] Aristotle, _An. Pr._ i. 31, 46a 32 sqq.; cf. 91b 12 sqq.
[13] Athenaeus ii. 59c. See Usener, _Organisation der wissenschaftl.
Arbeit_ (1884; reprinted in his _Vorträge und Aufsätze_, 1907).
[14] Socrates' reference of a discussion to its presuppositions
(Xenophon, _Mem._ iv. 6, 13) is not relevant for the history of the
terminology of induction.
[15] _Theaetetus_, 186c.
[16] Timaeus, 37a, b (quoted in H. F. Carlill's translation of the
_Theaetetus_, p. 60).
[17] _Theaetetus_, 186d.
[18] Sophistes, 253d.
[19] _Ib. id._; cf. _Theaetetus_, 197d.
[20] Aristotle, _de An._ 430b 5, and generally iii. 2, iii. 5.
[21] For Plato's Logic, the controversies as to the genuineness of
the dialogues may be treated summarily. The _Theaetetus_ labours
under no suspicion. The _Sophistes_ is apparently matter for
animadversion by Aristotle in the _Metaphysics_ and elsewhere, but
derives stronger support from the testimonies to the _Politicus_
which presumes it. The _Politicus_ and _Philebus_ are guaranteed by
the use made of them in Aristotle's Ethics. The rejection of the
_Parmenides_ would involve the paradox of a nameless contemporary of
Plato and Aristotle who was inferior as a metaphysician to neither.
No other dialogue adds anything to the _logical_ content of these.
Granted their genuineness, the relative dating of three of them is
given, viz. _Theaetetus_, _Sophistes_ and _Politicus_ in the order
named. The _Philebus_ seems to presuppose _Politicus_, 283-284, but
if this be an error, it will affect the logical theory not at all.
There remains the _Parmenides_. It can scarcely be later than the
_Sophistes_. The antinomies with which it concludes are more
naturally taken as a prelude to the discussion of the _Sophistes_
than as an unnecessary retreatment of the doctrine of the one and the
many in a more negative form. It may well be earlier than the
_Theaetetus_ in its present form. The stylistic argument shows the
_Theaetetus_ relatively early. The maturity of its philosophic
outlook tends to give it a place relatively advanced in the Platonic
canon. To meet the problem here raised, the theory has been devised
of an earlier and a later version. The first may have linked on to
the series of Plato's dialogues of search, and to put the
_Parmenides_ before it is impossible. The second, though it might
still have preceded the _Parmenides_ might equally well have followed
the negative criticism of that dialogue, as the beginning of
reconstruction. For Plato's logic this question only has interest on
account of the introduction of an [Greek: Aristotelês] in a
non-speaking part in the _Parmenides_. If this be pressed as
suggesting that the philosopher Aristotle was already in full
activity at the date of writing, it is of importance to know what
Platonic dialogues were later than the début of his critical pupil.
On the stylistic argument as applied to Platonic controversies
Janell's _Quaestiones Platonicae_ (1901) is important. On the whole
question of genuineness and dates of the dialogues, H. Raeder,
_Platons philosophische Entwickelung_ (1905), gives an excellent
conspectus of the views held and the grounds alleged. See also PLATO.
[22] E.g. that of essence and accident. Republic, 454.
[23] E.g. the discussion of correlation, ib. 437 sqq.
[24] _Politicus_, 285d.
[25] _Sophistes_, 261c sqq.
[26] E.g. in _Nic. Eth._ i. 6.
[27] _Philebus_, 16d.
[28] Principal edition still that of Waitz, with Latin commentary, (2
vols., 1844-1846). Among the innumerable writers who have thrown
light upon Aristotle's logical doctrine, St Hilaire, Trendelenburg,
Ueberweg, Hamilton, Mansel, G. Grote may be named. There are,
however, others of equal distinction. Reference to Prantl, op. cit.,
is indispensable. Zeller, _Die philosophie der Griechen_, ii. 2,
"Aristoteles" (3rd ed., 1879), pp. 185-257 (there is an Eng. trans.),
and Maier, _Die Syllogistik des Aristoteles_ (2 vols., 1896, 1900)
(some 900 pp.), are also of first-rate importance.
[29] _Sophist. Elench._ 184, espec. b 1-3, but see Maier, _loc. cit._
i. 1.
[30] References such as 18b 12 are the result of subsequent editing
and prove nothing. See, however, ARISTOTLE.
[31] Adrastus is said to have called them [Greek: pro tôn topikôn].
[32] _Metaphys._ E. 1.
[33] _De Part. Animal._ A. 1, 639a 1 sqq.; cf. _Metaphys._ 1005b 2
sqq.
[34] _De Interpretatione_ 16a sqq.
[35] _De Interpretatione_ 16a 24-25.
[36] _Ib._ 18a 28 sqq.
[37] _Ib._ 19a 28-29.
[38] As shown e.g. by the way in which the relativity of sense and
the object of sense is conceived, 7b 35-37.
[39] _Topics_ 101a 27 and 36-b 4.
[40] _Topics_ 100.
[41] _Politics_ 1282a 1 sqq.
[42] 103b 21.
[43] _Topics_ 160a 37-b 5.
[44] This is the explanation of the formal definition of induction,
_Prior Analytics_, ii. 23, 68b 15 sqq.
[45] 25b 36.
[46] _Prior Analytics_, i. 1. 24a 18-20, [Greek: Syllogismos de esti
logos en ô tethentôn tinôn eteron ti tôn keimenôn ex anankês
snmbainei tô tauta einai]. The equivalent previously in _Topics_ 100a
25 sqq.
[47] _Prior Analytics_, ii. 21; _Posterior Analytics_, i. 1.
[48] 67a 33-37, [Greek: mê syntheôrôn to kath' ekateron].
[49] 67a 39-63.
[50] 79a 4-5.
[51] 24b 10-11.
[52] _Posterior Analytics_, i. 4 [Greek: kath auto] means (1)
contained in the definition of the subject; (2) having the subject
contained in its definition, as being an alternative determination of
the subject, crooked, e.g. is _per se_ of line; (3) self-subsistent;
(4) connected with the subject as consequent to ground. Its needs
stricter determination therefore.
[53] 73b 26 sqq., 74a 37 sqq.
[54] 90b 16.
[55] _Metaphys. Z._ 12, H. 6 ground this formula metaphysically.
[56] 94a 12, 75b 32.
[57] 90a 6. Cf. Ueberweg, _System der Logik_, § 101.
[58] 78a 30 sqq.
[59] _Topics_, 101b 18, 19.
[60] _Posterior Analytics_, ii. 13.
[61] _Posterior Analytics_, ii. 16.
[62] _Posterior Analytics_, i. 13 ad. fin., and i. 27. The form which
a mathematical science treats as relatively self-subsistent is
certainly not the constitutive idea.
[63] _Posterior Analytics_, i. 3.
[64] _Posterior Analytics_, ii. 19.
[65] _De Anima_, 428b 18, 19.
[66] _Prior Analytics_, i. 30, 46a 18.
[67] _Topics_, 100b 20, 21.
[68] _Topics_, 101a 25, 36-37, b1-4, &c.
[69] Zeller (_loc. cit._ p. 194), who puts this formula in order to
reject it.
[70] _Metaphys._ [Delta] 1, 1013a 14.
[71] _Posterior Analytics_, 72a 16 seq.
[72] _Posterior Analytics_, 77a 26, 76a 37 sqq.
[73] _Metaphys._ [Gamma].
[74] _Posterior Analytics_, ii. 19.
[75] _de Anima_, iii. 4-6.
[76] _Metaphys._ M. 1087a 10-12; Zeller loc. cit. 304 sqq.; McLeod
Innes, _The Universal and Particular in Aristotle's Theory of
Knowledge_ (1886).
[77] _Topics_, 105a 13.
[78] _Metaphys._ 995a 8.
[79] E.g., _Topics_, 108b 10, "to induce" the universal.
[80] _Posterior Analytics_, ii. 19, 100b 3, 4.
[81] _Topics_, i. 18, 108b 10.
[82] _Prior Analytics_, ii. 23.
[83] [Greek: Paradeigma], Prior Analytics, ii. 24.
[84] Sigwart, _Logik_, Eng. trans. vol. ii. p. 292 and elsewhere.
[85] Ueberweg, _System_, § 127, with a ref. to _de Partibus
Animalium_, 667a.
[86] See 67a 17 [Greek: ex hapantôn tôn atomôn].
[87] [Greek: Epiphora]. [Greek: Epi] = "in" as in [Greek: epagôgê],
inductio, and [Greek: -phora] = -ferentia, as in [Greek: diaphora],
dif_ferentia_.
[88] Diog. Laërt. x. 33 seq.; Sext. Emp. Adv. Math. vii. 211.
[89] Diog. Laërt. x. 87; cf. Lucretius, vi. 703 sq., v. 526 sqq. (ed.
Munro).
[90] Sextus Empiricus, _Pyrrhon. Hypotyp._ ii. 195, 196.
[91] Sextus, _op. cit._ ii. 204.
[92] _Op. cit._ iii. 17 sqq., and especially 28.
[93] The point is raised by Aristotle, 95A.
[94] See Jourdain, _Recherches critiques sur l'âge et l'origine des
traductions latines d'Aristote_ (1843).
[95] See E. Cassirer, _Das Erkenntnisproblem_, i. 134 seq., and the
justificatory excerpts, pp. 539 sqq.
[96] See Riehl in _Vierteljahrschr. f. wiss. Philos._ (1893).
[97] Bacon, _Novum Organum_, ii. 22, 23; cf. also Aristotle, _Topics_
i. 12. 13, ii. 10. 11 (Stewart, ad _Nic. Eth._ 1139b 27) and Sextus
Empiricus, _Pyrr. Hypot._ iii. 15.
[98] Bacon's _Works_, ed. Ellis and Spedding, iii. 164-165.
[99] A notable formula of Bacon's _Novum Organum_ ii. 4 § 3 turns
out, _Valerius Terminus_, cap. 11, to come from Aristotle, _Post.
An._ i. 4 _via_ Ramus. See Ellis in Bacon's _Works_, iii. 203 sqq.
[100] _De Civitate Dei_, xi. 26. "Certum est me esse, si fallor."
[101] Cf. Plato, _Republic_, 381E seq.
[102] _Elementa Philosophiæ_, i. 3. 20, i. 6. 17 seq.
[103] Hobbes, _Elementa Philosophiæ_, i. 1. 5.
[104] _Id. ib._ i. 6. 16.
[105] _Id. ib._ i. 4. 8; cf. Locke's _Essay of Human Understanding_,
iv. 17.
[106] _Id. Leviathan_, i. 3.
[107] _Id. Elem. Philos._ i. 6. 10.
[108] Condillac, _Langue des Calculs_, p. 7.
[109] Locke, _Essay_, iii. 3.
[110] _Id. ib._ iv. 17.
[111] _Loc. cit._ § 8.
[112] _Id. ib._ iv. 4, §§ 6 sqq.
[113] Berkeley, _Of the Principles of Human Knowledge_, § 142.
[114] Hume, _Treatise of Human Nature_, i. 1. 7 (from Berkeley, _op.
cit._, introd., §§ 15-16).
[115] _Essay_, iv. 17, § 3.
[116] Hume, _Treatise of Human Nature_, i. 3. 15.
[117] Mill, _Examination of Sir William Hamilton's Philosophy_, cap.
17.
[118] Cf. Mill, _Autobiography_, p. 159. "I grappled at once with the
problem of Induction, postponing that of Reasoning." _Ib._ p. 182
(when he is preoccupied with syllogism), "I could make nothing
satisfactory of Induction at this time."
[119] _Autobiography_, p. 181.
[120] The insight, for instance, of F. H. Bradley's criticism,
_Principles of Logic_, II. ii. 3, is somewhat dimmed by a lack of
sympathy due to extreme difference in the point of view adopted.
[121] Bacon, _Novum organum_, i. 100.
[122] Russell's _Philosophy of Leibnitz_, capp. 1-5.
[123] See especially remarks on the letter of M. Arnauld (Gerhardt's
edition of the philosophical works, ii. 37 sqq.).
[124] Gerhardt, vi. 612, quoted by Russell, _loc. cit._, p. 19.
[125] Ibid., ii. 62, Russell, p. 33.
[126] Spinoza, ed. van Vloten and Land, i. 46 (_Ethica_, i. 11).
[127] _Nouveaux essais_, iv. 2 § 9, 17 § 4 (Gerhardt v. 351, 460).
[128] _Critique of Judgment_, Introd. § 2, _ad. fin._ (_Werke_,
Berlin Academy edition, vol. v. p. 176, l. 10).
[129] _Kant's Introduction to Logic and his Essay on the Mistaken
Subtlety of the Four Figures_, trans. T. K. Abbott (1885).
[130] _Loc. cit._, p. 11.
[131] Or antitheses. Kant follows, for example, a different line of
cleavage between form and content from that developed between thought
and the "given." And these are not his only unresolved dualities,
even in the _Critique of Pure Reason_. For the logical inquiry,
however, it is permissible to ignore or reduce these differences.
The determination too of the sense in which Kant's theory of
knowledge involves an unresolved antithesis is for the logical
purpose necessary so far only as it throws light upon his logic and
his influence upon logical developments. Historically the question of
the extent to which writers adopted the dualistic interpretation or
one that had the like consequences is of greater importance.
It may be said summarily that Kant holds the antithesis between
thought and "the given" to be unresolved and within the limits of
theory of knowledge irreducible. The dove of thought falls lifeless
if the resistant atmosphere of "the given" be withdrawn (_Critique of
Pure Reason_, ed. 2 Introd. Kant's _Werke_, ed. of the Prussian
Academy, vol. iii. p. 32, ll. 10 sqq.). Nevertheless the
thing-in-itself is a problematic conception and of a limiting or
negative use merely. He "had woven," according to an often quoted
phrase of Goethe, "a certain sly element of irony into his method;
... he pointed as it were with a side gesture beyond the limits which
he himself had drawn." Thus (_loc. cit._ p. 46, ll. 8, 9) he declares
that "there are two lineages united in human knowledge, which perhaps
spring from a common stock, though to us unknown--namely sense and
understanding." Some indication of the way in which he would
hypothetically and speculatively mitigate the antithesis is perhaps
afforded by the reflection that the distinction of the mental and
what appears as material is an external distinction in which the one
appears outside to the other. "Yet what as thing-in-itself lies back
of the phenomenon may perhaps not be so wholly disparate after all"
(ib. p. 278, ll. 26 sqq.).
[132] _Critique of Judgment_, Introd. § 2 (_Werke_, v., 276 ll. 9
sqq.); cf. Bernard's "Prolegomena" to his translation of this, (pp.
xxxviii. sqq.).
[133] _Die Logik, insbesondere die Analytik_ (Schleswig, 1825).
August Detlev Christian Twesten (1789-1876), a Protestant theologian,
succeeded Schleiermacher as professor in Berlin in 1835.
[134] See _Sir William Hamilton: The Philosophy of Perception_, by J.
Hutchison Stirling.
[135] _Hauptpunkte der Logik_, 1808 (_Werke_, ed. Hartenstein, i. 465
sqq.), and specially _Lehrbuch der Einleitung in die Philosophie_
(1813), and subsequently §§ 34 sqq. (_Werke_, i. 77 sqq.).
[136] See Ueberweg, _System of Logic and History of Logical
Doctrines_, § 34.
[137] _Drei Bücher der Logik_, 1874 (E.T., 1884). The Book on Pure
Logic follows in essentials the line of thought of an earlier work
(1843).
[138] _Logic_, Eng. trans. 35 _ad. fin._
[139] _Logic_, Introd. § ix.
[140] For whom see Höffding, _History of Modern Philosophy_, Eng.
trans., vol. ii. pp. 122 sqq.; invaluable for the logical methods of
modern philosophers.
[141] _Wissenschaft der Logik_ (1812-1816), in course of revision at
Hegel's death in 1831 (_Werke_, vols. iii.-v.), and _Encyklopädie der
philosophischen Wissenschaften_, i.; _Die Logik_ (1817; 3rd ed.,
1830); _Werke_, vol. vi., Eng. trans., Wallace (2nd ed., 1892).
[142] _The Principles of Logic_ (1883).
[143] _Logic, or The Morphology of Thought_ (2 vols., 1888).
[144] _Logic_, Pref. pp. 6 seq.
[145] _Id._ vol. ii. p. 4.
[146] _Logik_ (1873, 1889), Eng. trans. ii. 17.
[147] _Op. cit._ ii. 289.
[148] _Introd. to Logic._, trans. Abbott, p. 10.
[149] _Ueber Annahmen_ (1902, &c.)
[150] _Logik_ (1880, and in later editions).
[151] Yet see _Studies in Logic_, by John Dewey and others (1903).
LOGOCYCLIC CURVE, STROPHOID or FOLIATE, a cubic curve generated by
increasing or diminishing the radius vector of a variable point Q on a
straight line AB by the distance QC of the point from the foot of the
perpendicular drawn from the origin to the fixed line. The polar
equation is r cos[theta] = a(1 ± sin[theta]), the upper sign referring
to the case when the vector is increased, the lower when it is
diminished. Both branches are included in the Cartesian equation (x² +
y²)(2a - x) = a²x, where a is the distance of the line from the origin.
If we take for axes the fixed line and the perpendicular through the
initial point, the equation takes the form y [root](a - x) = x [root](a
+ x). The curve resembles the folium of Descartes, and has a node
between x = 0, x = a, and two branches asymptotic to the line x = 2a.
[Illustration.]
LOGOGRAPHI ([Greek: logos], [Greek: graphô], writers of prose histories
or tales), the name given by modern scholars to the Greek
historiographers before Herodotus.[1] Thucydides, however, applies the
term to all his own predecessors, and it is therefore usual to make a
distinction between the older and the younger logographers. Their
representatives, with one exception, came from Ionia and its islands,
which from their position were most favourably situated for the
acquisition of knowledge concerning the distant countries of East and
West. They wrote in the Ionic dialect, in what was called the unperiodic
style, and preserved the poetic character of their epic model. Their
criticism amounts to nothing more than a crude attempt to rationalize
the current legends and traditions connected with the founding of
cities, the genealogies of ruling families, and the manners and customs
of individual peoples. Of scientific criticism there is no trace
whatever. The first of these historians was probably Cadmus of Miletus
(who lived, if at all, in the early part of the 6th century), the
earliest writer of prose, author of a work on the founding of his native
city and the colonization of Ionia (so Suïdas); Pherecydes of Leros, who
died about 400, is generally considered the last. Mention may also be
made of the following: Hecataeus of Miletus (550-476); Acusilaus of
Argos,[2] who paraphrased in prose (correcting the tradition where it
seemed necessary) the genealogical works of Hesiod in the Ionic dialect;
he confined his attention to the prehistoric period, and made no attempt
at a real history; Charon of Lampsacus (c. 450), author of histories of
Persia, Libya, and Ethiopia, of annals ([Greek: hôroi]) of his native
town with lists of the prytaneis and archons, and of the chronicles of
Lacedaemonian kings; Xanthus of Sardis in Lydia (c. 450), author of a
history of Lydia, one of the chief authorities used by Nicolaus of
Damascus (_fl._ during the time of Augustus); Hellanicus of Mytilene;
Stesimbrotus of Thasos, opponent of Pericles and reputed author of a
political pamphlet on Themistocles, Thucydides and Pericles; Hippys and
Glaucus, both of Rhegium, the first the author of histories of Italy and
Sicily, the second of a treatise on ancient poets and musicians, used by
Harpocration and Plutarch; Damastes of Sigeum, pupil of Hellanicus,
author of genealogies of the combatants before Troy (an ethnographic and
statistical list), of short treatises on poets, sophists, and
geographical subjects.
On the early Greek historians, see G. Busolt, _Griechische Geschichte_
(1893), i. 147-153; C. Wachsmuth, _Einleitung in das Studium der alten
Geschichte_ (1895); A. Schäfer, _Abriss der Quellenkunde der
griechischen und römischen Geschichte_ (ed. H. Nissen, 1889); J. B.
Bury, _Ancient Greek Historians_ (1909), lecture i.; histories of
Greek literature by Müller-Donaldson (ch. 18) and W. Mure (bk. iv. ch.
3), where the little that is known concerning the life and writings of
the logographers is exhaustively discussed. The fragments will be
found, with Latin notes, translation, prolegomena, and copious
indexes, in C. W. Müller's _Fragmenta historicorum Graecorum_
(1841-1870).
See also GREECE: _History, Ancient_ (section, "Authorities").