Now of categorems, some are accidents … as for instance, “The sailing through a rock.” … And of categorems, some are direct, some indirect, and some neither one nor the other. Now those are correct which are construed with one of the oblique cases, in such a manner as to produce a categorem, as for instance, “He hears, he sees, he converses.” And those are indirect which are construed with the passive voice, as for instance, “I am heard, I am seen.” And those which are neither one nor the other are those which are construed in a neutral kind of manner, as for instance, “To think, to walk.” And those are reciprocal which are among the indirect ones without being indirect themselves. Those are effects, ἐνεργήματα, which are such words as “He is shaved;” for then, the man who is shaved, implies himself.
The oblique cases are the genitive, the dative, and the accusative.
An axiom is that thing which is true, or false, or perfect in itself, being asserted or denied positively, as far as depends upon itself; as Chrysippus explains it in his Dialectic Definitions; as for instance, “It is day,” “Dion is walking.” And it has received the name of axiom, ἀξίωμα, because it is either maintained, ἀξιοῦται, or repudiated. For the man who says, “It is day,” appears to maintain the fact of its being day. If then it is day, the axiom put before one is true; but if it is not day, the axiom is false. And an axiom, a question, and an interrogation, differ from one another, and so does an imperative proposition from one which is adjurative, or imprecatory, or hypothetical, or appellative, or false. For that is an axiom which we utter, when we affirm anything positively, which is either true or false. And a question is a thing complete in itself, as also is an axiom, but which requires an answer, as for instance “Is it day?” Now this is neither true nor false; but, as “It is day” is an axiom; so is “Is it day?” a question. But an interrogation, πύσμα, is a thing to which it is not possible to make an answer symbolically, as in the case of a question, ἐρώτημα, saying merely “Yes,” but we must reply, “He does live in this place.”
The imperative proposition is a thing which we utter when we give an order, as for instance this:
Do you now go to the sweet stream of Inachus. ⋮
The appellative proposition is one which is used in the case in which when a man says anything, he must address somebody, as for instance:
Atrides, glorious king of men, Most mighty Agamemnon.
A false judgment is a proposition, which, while it has at the same time the appearance of a real judgment, loses this character by the addition, and under the influence of, some particle, as for instance:
The Parthenon at least is beautiful. How like the herdsman is to Priam’s sons.
There is also the dubitative proposition, which differs from the judgment, inasmuch as it is always uttered in the form of a doubt; as for instance:
Are not, then, grief and life two kindred states?
But questions and interrogations, and things like these, are neither true nor false, while judgments and propositions are necessarily one or the other.
Now of axioms, some are simple and others are not simple; as Chrysippus, and Archedemus, and Athenodorus, and Antipater, and Crinis, agree in dividing them. Those are simple which consist of an axiom or proposition which is not ambiguous (or of several axioms, or propositions of the same character), as for instance the sentence “It is day.” And those are not simple, which consist of an axiom or proposition which is ambiguous, or of several axioms or propositions of that character. Of an axiom or proposition which is ambiguous, as “If it is day;” of several axioms or propositions of that character, as “If it is day, it is light.”
And simple propositions are divided into the affirmative, the negative, the privative, the categorical, the definite, and the indefinite; those which are not simple, are divided into the combined, and the adjunctive, the connected and the disjunctive, and the causal and the augmentative, and the diminutive. That is an affirmative proposition: “It is not day.” And the species of this is doubly affirmative. That again is doubly affirmative, which is affirmative of an affirmative, as for instance: “It is not not day;” for this amounts to “It is day.” That is a negative proposition, which consists of a negative particle and a categorem, as for instance “No one is walking.” That is a privative proposition which consists of a privative particle and an axiom according to power, as “This man is inhuman.” That is a categorical proposition, which consists of a nominative case and a categorem, as for instance “Dion is walking.” That is a definite proposition, which consists of a demonstrative nominative case and a categorem, as for instance “This man is walking.” That is an indefinite one which consists of an indefinite particle or of indefinite particles, as for instance “Somebody is walking,” “He is moving.”
Of propositions which are not simple, the combined proposition is, as Chrysippus states, in his Dialectics, and Diogenes, too, in his Dialectic Art, that which is held together by the copulative conjunction “if.” And this conjunction professes that the second member of the sentence follows the first, as for instance “If it is day, it is light.” That which is adjunctive is, as Crinis states in his Dialectic Art, an axiom which is made to depend on the conjunction “since” (ἐπεὶ), beginning with an axiom and ending in an axiom, as for instance “Since it is day, it is light.” And this conjunction professes both that the second portion of the proposition follows the first, and the first is true. That is a connected proposition which is connected by some copulative conjunctions, as for instance “It both is day, and it is light.” That is a disjunctive proposition which is disconnected by the disjunctive conjunction, “or” (ἤτοι), as for instance, “It is either day or night.” And this proposition professes that one or other of these propositions is false. That is a causal proposition which is connected by the word “because;” as for instance “Because it is day, it is light.” For the first is, as it were, the cause of the second. That is an augmentative proposition which explains the greater, which is construed with an augmentative particle, and which is placed between the two members of the proposition, as for instance “It is rather day than night.” The diminutive proposition is, in every respect, the exact contrary of the preceding one; as for instance “It is less night than day.” Again, at times, axioms or propositions are opposed to one another in respect of their truth and falsehood, when one is an express denial of the other; as for instance “It is day” and “It is not day.”
Again, a conjunctive proposition is correct when it is such that the opposite of the conclusion is contradictory of the premise; as for instance the proposition “If it is day, it is light,” is true; for “It is not light,” which is the opposite to the conclusion expressed, is contradictory to the premise “It is day.” And a conjunctive proposition is incorrect when it is such that the opposite of the conclusion is not inconsistent with the premise, as for instance “If it is day, Dion is walking.” For the fact that Dion is not walking is not contradictory of the premise “It is day.”
An adjunctive proposition is correct which begins with a true premise and ends in a consequence which follows of necessity, as for instance “Since it is day, the sun is above the earth.” But it is incorrect when it either begins with a false premise or ends with a consequence which does not follow properly; as for instance “Since it is night, Dion is walking,” for this may be said in the daytime.
A causal proposition is correct when it begins with a true premise and ends in a consequence which necessarily follows from it, but yet does not have its premise reciprocally consequent upon its conclusion; as for instance “Because it is day, it is light.” For the fact of its being light is a necessary consequence of its being day; but the fact of its being day is not necessarily a consequence of its being light. A causal proposition is incorrect which either begins with a false premise or ends with a conclusion that does not follow from it, or which has a premise which does not correspond to the conclusion; as for instance “Because it is night, Dion is walking.”
A proposition is persuasive, which leads to the assent of the mind, as for instance “If she brought him forth, she is his mother.” But still this is a falsehood, for a hen is not the mother of an egg. Again, there are some propositions which are possible, and some which are impossible; and some which are necessary, and some which are not necessary. That is possible which is capable of being true, since external circumstances are no hindrance to its being true; as for instance “Diocles lives.” And that is impossible which is not capable of being true; as for instance “The earth flies.” That is necessary which, being true, is not capable of being false; or perhaps is intrinsically capable of being false, but still has external circumstances which hinder its being false, as for instance “Virtue profits a man.” That again, is not necessary, which is true, but which has a capacity of being false, though external circumstances offer no hindrance to either alternative; as for instance “Dion walks.”
That is a reasonable or probable proposition which has a great preponderance of opportunities in favor of its being true; as for instance, “I shall be alive tomorrow.” And there are other different kinds of propositions and conversions of them, from true to false, and re-conversions again; concerning which we must speak at some length.
An argument, as Crinis says, is that which is composed of a lemma or major premise, an assumption or minor premise, and a conclusion; as for instance this: “If it is day, it is light;” “But it is day, therefore it is light.” For the lemma, or major premise, is “If it is day, it is light.” The assumption, or minor premise, is “It is day.” The conclusion follows: “Therefore it is light.” The mode of a proposition is, as it were, a figure of an argument, as for instance such as this: “If it is the first, it is the second; but it is the first, therefore it is the second.”
A conditional syllogism is that which is composed of both the preceding arguments, as for instance “If Plato is alive, Plato breathes; but the first fact is so, therefore so is the second.” And this conditional syllogism has been introduced for the sake, in long and complex sentences, of not being forced to repeat the assumption, as it was a long one, and also the conclusion; but of being able, instead, to content oneself with summing it up briefly thus, “The first case put is true, therefore so is the second.”
Of arguments, some are conclusive, others are inconclusive. Those are inconclusive which are such that the opposite of the conclusion drawn in them is not necessarily incompatible with the connection of the premises. As for instance such arguments as these: “If it is day, it is light; but it is day, therefore Dion is walking.” But of conclusive arguments, some are called properly by the kindred name conclusions, and some are called syllogistic arguments. Those then are syllogistic which are either such as do not admit of demonstration, or such as are brought to an indemonstrable conclusion, according to some one or more propositions; such for instance as the following: “If Dion walks, then Dion is in motion.” Those are conclusive which infer their conclusion specially, and not syllogistically; such for instance, as this: “The proposition it is both day and night is false. Now it is day; therefore, it is not night.”
Those again are unsyllogistic arguments which have an air of probability about them, and a resemblance to syllogistic ones, but which still do not lead to the deduction of proper conclusions. As for instance, “If Dion is a horse, Dion is an animal; but Dion is not a horse, therefore, Dion is not an animal.”
Again, of arguments, some are true and some are false. Those are true which deduce a conclusion from true premises, as for instance “If virtue profits, then vice injures.” And those are false which have some falsehood in their premises, or which are inconclusive; as for instance “If it is day, it is light; but it is day, therefore Dion is alive.”