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[8arg] The meaning of what the logicians call “an axiom,” and what it is called by our countrymen; and some other things which belong to the elements of the dialectic art.

WHEN I wished to be introduced to the science of logic and instructed in it, it was necessary to take up and learn what the dialecticians call εἰσαγωγαί or “introductory exercises.” 1 Then because at first [p. 159] I had to learn about axioms, which Marcus Varro calls, 2 now proposita, or “propositions,” and now proloquia, or “preliminary statements,” I sought diligently for the Commentary on Proloquia of Lucius Aelius, a learned man, who was the teacher of Varro; and finding it in the library of Peace, 3 I read it. But I found in it nothing that was written to instruct or to make the matter clear, but Aelius 4 seems to have made that book rather as suggestions for his own use than for the purpose of teaching others.

I therefore of necessity returned to my Greek books. From these I obtained this definition of an axiom: λεκτὸν αὐτοτελὲς ἀπόφαντον ὅσον ἀφ᾽ αὑτῷ. 5 This I forbore to turn into Latin, since it would have been necessary to use new and as yet uncoined words, such as, from their strangeness, the ear could hardly endure. But Marcus Varro in the twenty-fourth book of his Latin Language, dedicated to Cicero, thus defines the word very briefly: 6A proloquium is a statement in which nothing is lacking.”

But his definition will be clearer if I give an example. An axiom, then, or a preliminary proposition, if you prefer, is of this kind: “Hannibal was a Carthaginian”; “Scipio destroyed Numantia”; “Milo was found guilty of murder”; “pleasure is neither a good nor an evil”; and in general any saying which is a full and perfect thought, so expressed in words that it is necessarily either true or false, is called by the logicians an “axiom,” by Marcus Varro, as I have said, a “proposition,” but by Marcus Cicero 7 a pronuntiatum, or “pronouncement,” [p. 161] a word however which he declared that he used “only until I can find a better one.”

But what the Greeks call συνημμένον ἀξίωμα, or “a hypothetical syllogism,” 8 some of our countrymen 9 call adiunctum, others conexum. 10 The following are examples of this: “If Plato is walking, Plato is moving”; “if it is day, the sun is above the earth.” Also what they call συμπεπλεγμένον, or “a compound proposition,” we call coniunctum or copulatum; for example: “Publius Scipio, son of Paulus, was twice consul and celebrated a triumph, and held the censorship, and was the colleague of Lucius Mummius in his censorship.” But in the whole of a proposition of this kind, if one member is false, even if the rest are true, the whole is said to be false. For if to all those true statements which I have made about that Scipio I add “and he worsted Hannibal in Africa,” which is false, all those other statements which are made in conjunction will not be true, because of this one false statement which is made with them.

There is also another form, which the Greeks call διεζευγμένον ἀξίωμα, or “a disjunctive proposition,” and we call disiunctum. For example: “Pleasure is either good or evil, or it is neither good nor evil.” 11 Now all statements which are contrasted ought to be opposed to each other, and their opposites, which the Greeks call ἀντικείμενα, ought also to be opposed. Of all statements which are contrasted, one ought to be true and the rest false. But if none at all of them is true, or if all, or more than one, are true, or if the contrasted things are not at odds, or if those which are opposed to each other are not contrary, then that is a false contrast and is called [p. 163] παραδιεζευγμένον. For instance, this case, in which the things which are opposed are not contraries: “Either you run or you walk or you stand.” These acts are indeed contrasted, but when opposed they are not contrary; for “not to walk” and “not to stand” and “not to run” are not contrary to one another, since those things are called “contraries” which cannot be true at the same time. But you may at once and at the same time neither walk, stand, nor run.

But for the present it will be enough to have given this little taste of logic, and I need only add by way of advice, that the study and knowledge of this science in its rudiments does indeed, as a rule, seem forbidding and contemptible, as well as disagreeable and useless. But when you have made some progress, then finally its advantages will become clear to you, and a kind of insatiable desire for acquiring it will arise; so much so, that if you do not set bounds to it, there will be great danger lest, as many others have done, you should reach a second childhood amid those mazes and meanders of logic, as if among the rocks of the Sirens.

1 II. 194, Arn.

2 Fr. 29, G. and S.

3 Vespasian's Temple of Peace in the Forum Pacis.

4 p. 54. 19. Fun.

5 An absolute and self-evident proposition.

6 Fr. 29, G. and S.

7 Tusc. Disp. i. 14.

8 Literally, “a connected axiom.” See II. 213. Arn.

9 Aelius Stilo, Fr. 74, p. 75 Fun.

10 Two connected sentences of which the second follows as the result of the first. 4 II. 218. Arn.

11 aut s.d. sum, added by Hertz; aut s.d. est, Skutsch.

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