There are two kinds of reduction, ostensive and ad absurdum. The initial letters in each of the modes (B, C, D, and F) indicate the modes of the first figure to which the modes of the other [figures] are to be reduced, i.e., those of which the initial letter is the same.11 S and P following a vowel show that that proposition is to be converted, S simpliciter, P per accidens. M shows that the propositions are to be transposed, K that the reduction is made per impossibile, of which more later. When this is done, the conclusion reached will be either the same as in reducing Cesare, Festino, etc., or [a conclusion] which implies the same conclusion, or the contradictory to the conceded premiss. The validity of an ostensive reduction is known from the rules of conversion and subalternation.
Reduction to the impossible is as follows. If it is denied that a given conclusion follows from true premisses, let the contradictory of the conclusion be substituted for the premiss whose symbol includes a K, like the major in Bokardo and the minor in Baroko; these premisses will then show in Barbara the truth of the contradictory of the premiss which was claimed to be true. If therefore the given premisses had been true, the conclusion would also have been true; for if it was not, its contradictory would have been true, and if that had been true, it will show (in Barbara) that the other premiss is false, contrary to the hypothesis.
For these rules of syllogisms to hold, we have to look carefully for the true subjects and predicates of the propositions, which are sometimes not at all obvious to beginners; and then we have to determine whether they are really affirmative or negative as they are used in the argument. For in complex [propositions], sometimes one part is negative, the other is affirmative, and occasionally it is the negative part (the less obvious part) which is chiefly in point. For example,
And the dictum de omni et nullo is so useful in proving a true argument and detecting a false one, that by its help any intelligent person may be able to see both true syllogistic force and its fallacious semblance, according to whether one of the premisses contains the conclusion or not, even before applying the special rules of syllogisms.
CHAPTER 7
With regard to the remaining forms of argument, it is evident that they are imperfect syllogisms or may be reduced to imperfect syllogisms.
1. The enthymeme12 or rhetorical syllogism is “when one of the premisses is unspoken because it is quite obvious”; it is for this reason that an enthymematic judgment has full syllogistic force.
2. Induction is “an inference from various examples,” of which the chief use is in physics, in politics, and in household matters. It does not generate the highest credit or exclude all fears of the contrary, unless it is clear that there are absolutely no contrary examples.
3. An epicheirema13 is “a complex syllogism in which a confirmation is attached to one or both of the premisses.”
4. Sorites is “discourse which contains several syllogisms which are connected with each other,” or where there are several middle terms which are connected with each other or with the extremes in several propositions of which if even one is negative, the conclusion will be negative, and if two are negative or any middle term is not distributed at least once, there will be no inference.
5. A dilemma is “a kind of epicheirema, where in making a division, that which is shown about the individual parts in the premisses is concluded of the whole.”
6. A hypothetical syllogism is “one in which one of the premisses is hypothetical”; when the minor is hypothetical, so also is the conclusion; these also serve to prove the inference in an enthymeme. More frequent are those in which the major is hypothetical, for example:
| Major: If this [is], that will be | Or, If this [is], that will be, | |
| Minor: But this [is] (con.), therefore also that. | But not that, therefore not this either. |
But since a more general predicate follows from any of the corresponding kinds (for example, If it is a man, if it is a horse, etc., it will also be an animal), but from a general predicate, no one particular species will follow (for from the fact that it is an animal, it does not follow that it will be a horse or an ass), it is evident that hypothetical syllogisms rightly proceed (1) from the positing of an antecedent to the positing of a consequent, or (2) from the removal of a consequent to the removal of an antecedent.
| If Titius is a man, he is also an animal, | 2) If it were a bird, it would fly, | |
| But he is a man, therefore he is an animal. | But it does not fly, therefore it is not a bird. |
It is a fallacious inference from the removal of an antecedent, or the positing of a consequent:
| If Titius is a horse, he is an animal, | Or, But he is an animal, | |
| He is not a horse, therefore he is not an animal. | Therefore he is a horse. |
The positing of a negative will be a negation, and the removal of it an affirmation.
Hypothetical [syllogisms] are reduced to categorical [syllogisms] by this general method: “every case which posits that Titius is a man, posits that he is an animal; but every case, or some case, posits that he is a man; therefore, etc.” But often it may be more easily and briefly done when there is either the same subject or the same predicate to the antecedent and the consequent; for example:
If man is an animal, he has sensation,
But every man is an animal, therefore he has sensation.
Every animal has sensation.
Every man is an animal; therefore,
Every man has sensation.
If every animal has sensation, every man has sensation;
But every animal has sensation; therefore,
Every man has sensation.
7. Disjunctive syllogisms are “those in which the major is disjunctive, [whether] affirmative or negative.” Either it is day, or it is night; but it is not day, therefore it is night. Or, it is not both night and day, but it is day; therefore it is not night. The force of the inference is obvious enough, when by positing an affirmative disjunctive major, an affirmative conclusion is drawn from a negative minor; or from a copulative negative major and an affirmative minor, the conclusion is negative. For in the former case the syllogism will be reduced to Barbara.
All time different from daytime is night;
But this time is different from daytime.
Therefore …
In the other case.
No daytime is night,
But this time is day.
Therefore …
There is no inference from an affirmative minor, in the former, or from a negative [minor] in the latter.
CHAPTER 8
As far as content is concerned, syllogisms are either certain or probable depending on their premisses.
A demonstration is “an argument duly reaching a conclusion from certain premisses,” and it is either ostensive, or leading to absurdity; the latter is the case when the contradictory of a proposition is shown to be false, from which it will be clear that it is itself true. The former is either a priori, or of a cause,14 “when an effect is shown from a known cause.” But there are causes of being and causes of knowing. The former are prior by nature and per se; the latter [are prior] in being known and in relation to us. Demonstrations drawn from both kinds of causes are called a priori, but especially those which are drawn from things prior by nature.15
“The