The dictum de omni is the highest principle of affirmative syllogisms. It says: Whatever is universally affirmed of a concept is also affirmed of everything contained under it. This is grounded on the rule of affirmative ratiocination. A concept that contains other concepts has been abstracted from them and is a predicate. Whatever belongs to this concept is a predicate of other predicates and therefore a predicate of the subject.
The dictum de nullo says: Whatever is denied of a concept is also denied of everything that is contained under it. The concept is a predicate that has been abstracted from the concepts that are contained under it. Whatever is inconsistent with this concept is inconsistent with the subject and therefore also with the predicates of the subject. This is based on the rule of negative ratiocination.
Section III - Of Pure and Mixed Ratiocination
If one judgment can be immediately discerned from another judgment without the use of a middle term, then the inference is not a ratiocination. A direct, non-ratiocinative inference would, for example, be: "from the proposition that all airplanes have wings, it immediately follows that whatever has no wings is not an airplane."
Pure ratiocination occurs by means of three propositions. Mixed ratiocination occurs by more than three propositions. A mixed ratiocination is still a single ratiocination. It is not compound, that is, consisting of several ratiocinations.
An example of a mixed ratiocination is:
Nothing immortal is a man,
Therefore, no man is immortal; (this is a negative conversion of the preceding premise)
Socrates is a man,
Therefore, Socrates is not immortal.
A mixed ratiocination interposes an immediate inference, resulting in more than three propositions. However, a mixed ratiocination may show only three propositions if the fourth proposition is unspoken, unexpressed, and merely thought. For example, the ratiocination
Nothing immortal is a man,
Socrates is a man,
Therefore, Socrates is not immortal
is only valid if the fourth proposition Therefore, no man is immortal is covertly thought. This unspoken proposition should be inserted after the first proposition and is merely its negative converse.
Section IV
In the so-called First Figure only Pure Ratiocinations are possible, in the remaining Figures only mixed Ratiocinations are possible.
Pattern of First Figure:
Subject...............Predicate
Middle Term........Major Term........Major Premise
Minor Term.........Middle Term........Minor Premise
Minor Term........Major Term...........Conclusion
A ratiocination is always in the first figure when it accords with the first rule of ratiocination: A predicate B of a predicate C of a subject A is a predicate of the subject A. This is a pure ratiocination. It has three propositions: