Logic, Metaphysics, and the Natural Sociability of Mankind. Francis Hutcheson
[propositions] are “two particular [propositions], one affirmative, the other negative”; they too are in conflict by quality alone, like I and O. Since they are often both true at the same time, namely when they have contingent content, they are not truly opposed.
The rules of opposition are: (1) “of contradictories, one is always true, the other false”; this is the major opposition. (2) “Contraries are never at the same time true, but are sometimes both false at the same time,” i.e., with contingent content. (3) “Subcontrary [propositions] are never false at the same time.” If it is false that some man is learned, it will not be false that some man is not learned, since the contradictory of the former is true.12
CHAPTER 1
When the relation or connection of two ideas or terms cannot be directly perceived, the relation between them will often be able to be seen by a comparison of both of them with some third or middle [idea or term] or with several middle [ideas or terms] which are clearly connected with each other. This mental process is dianoetic judgment or discourse.
When there is only one middle, we are said to have a syllogism; when there are several middles connected with each other, by which the comparison of the terms is made, it is a sorites, or complex form of reasoning.1 First, therefore, we must deal with the simple and categorical syllogism, for the other more complex forms may be reduced to syllogisms.
A syllogism is “discourse in which a third proposition is inferred from two propositions rightly arranged.”
Before a proof is given by means of a syllogism, there is a question or problem of showing the relationship between two terms. These terms are called the Extremes; they are the Major term and the Minor term. The Major term is “the predicate of the question” or of the conclusion, and the Minor term is “the subject of the question.” The Middle Term is that which is compared with both of the extreme terms in the premissed propositions.
Irrespective of the content of the syllogism, there are these three terms: the Major, the Minor, and the Middle Terms. Taking account of the content, there are three propositions: the Major Proposition, the Minor Proposition (these are also called the Premisses), and the Conclusion. They are distinguished not by their order but by their nature.
1. The major proposition “is that in which the major term is compared with the middle term” and is called the proposition par excellence.
2. The minor proposition is that “in which the minor term is compared with the middle term” and is called the assumption or subsumption.
3. The conclusion is that “in which the extremes are compared with each other,” and the middle term never appears here.
CHAPTER 2
The whole force of the syllogism may be explained from the following axioms.2
Axiom 1. “Those things which agree with a single third thing agree with each other.”
2. “Those of which one agrees and the other does not agree with one and the same third thing, do not agree with each other.”
3. “Those which agree in no third thing, do not agree with each other.”
4. “Those which do not disagree with any third thing, do not disagree with each other.” From these [axioms] the general rules of syllogisms are deduced. The first three are about the quality of propositions.
Rule 1. If one of the premisses is negative, the conclusion will be negative (by axiom 2).
Rule 2. If both the premisses are affirmative, the conclusion will be affirmative (axiom 1).
Rule 3. From two negative [premisses] nothing follows because those which agree with each other and those which disagree with each other may both be different from a third.
Two [rules] on the Quantity of Terms:
Rule 4. The middle must be distributed once, or taken universally; for a common term often contains two or more species which are mutually opposed to each other, and from which predication may be made according to different parts of its own extension; therefore terms do not truly agree with a third term, unless one at least agrees with the whole of the middle.
Rule 5. No term may be taken more universally in the conclusion than it was in the premisses, because an inference from particular to universal is not valid.
On the Quantity of Propositions:
Rule 6. “If one of the premisses is particular, the conclusion will be particular.” For (i) suppose the conclusion is affirmative: therefore (by rule 1) both premisses are affirmative; but no term is distributed in a particular [premiss]; therefore (by rule 4) the middle term has to be distributed in the other one; it is therefore the subject of a universal affirmative; therefore the other extreme is also taken particularly, since it is the predicate of an affirmative, ergo, the conclusion will be particular (by rule 5). (ii): Suppose the conclusion is negative: therefore, its predicate is distributed; hence (by rules 5 and 4) both the major term and the middle term have to be distributed in the premisses, but (rule 3) when one premiss is negative, the other is affirmative. If one [premiss] is particular, only these two terms can be distributed; when one premiss is affirmative, the other should be particular. Therefore the minor extreme, the subject of the conclusion, is not distributed in the premisses; therefore (by rule 5) it is not distributed in the conclusion.
Rule 7. “From two particulars nothing follows,” at least in our normal way of speaking, according to which the predicate of a negative is taken to be distributed. For (i) if the conclusion is affirmative and both premisses are affirmative, no term in the premisses is distributed (contrary to rule 4). (ii) Suppose the conclusion is negative; therefore some predicate is distributed, but the predicate is distributed only in particular premisses; it will therefore be invalid (contrary to rule 4 or 5).
Rules 1 and 7 are thus reduced to one rule. The conclusion follows the weaker side, i.e., the negative or particular. All the rules are contained in these verses:3
You must distribute the middle, and there should be no fourth term.
Both premisses should not be both negative and particular.
The conclusion should follow the weaker side;
And it may not be distributed or negative, except when a premiss is.4
In a curious and unusual manner of speaking, a certain negative conclusion may be reached, with the predicate undistributed, as in this example:
Certain Frenchmen are learned,
Certain Englishmen are not learned,
Therefore,
Certain Englishmen are not certain Frenchmen.
CHAPTER 3
A figure of a syllogism is “the proper arrangement of the middle in the premisses”; there are only four figures.
1. That in which the middle is the subject of the major and the predicate of the minor.
2. That in which the middle is the predicate of both.
3. That in which the middle is the subject of both.
4. That in which the middle is the predicate of the major and the subject of the minor.
In the first [the middle is] sub[ject and] pre[dicate]; in the second [it is] twice a pre[dicate]; in the third [it is] twice a sub[ject]; and in the fourth [it is] pre[dicate and] sub[ject].
The mood of the syllogism is “the correct determination