33. Two Kinds of Judgments in Necessary Matter. -- I. First kind: The connection is necessary because the subject, considered in its essential elements, is either the same term as the predicate (identical judgment), as: a square is an equilateral rectangle; 2 = 1 + 1 ; or includes the predicate, the latter being in this case part of the essence of the subject, as: a square is a rectangle; man is intelligent. In both cases the comparison of the two terms of the judgment reveals to the mind the necessity of their connection.

II. Second kind: The connection between the two terms of the judgment is necessary when the predicate necessarily presupposes the subject and, consequently, is not definable without bringing the essence of the subject into evidence. This case is where the predicate is a property (in the rigorous acceptation) of the subject.

The definition of the predicate (simple or disjunctive), put side by side with the essential notion of the subject, brings out the necessary connection of the two terms

(a) Example of a simple predicate:

A prime number is one out of which it is impossible to form several groups each containing the same number of objects.^{1}

This definition does not mention as a component part the number 5. But if we place the definition on one side, and on the other side the result of breaking up the number 5 into two groups of two units each and one of one unit, it will then appear that the definition of a prime number necessarily applies to the number 5. That is, a prime number is not the definition of the number 5, but to be prime is one of its properties

(b) Example of a disjunctive predicate: Every number is either even or odd.

The attribute even is not essential to number; it is not even a necessary property. Neither is the attribute odd of the essence of number or a property of it. The alternative even or odd forms no part of the definition of number, but it is a necessary consequence of that definition. Given that unity is not a number, but the principle of numbers, every number is or is not divisible by 2, is even or odd.

The Scholastics, following Aristotle, called the two kinds of necessary propositions which we have just been studying, duo modi dicendi per se, propositiones per se (two ways of saying by themselves, propositlons by themselves).^{2} kath' auto, and opposed to them modi dicendi per accidens, propositiones per accidens, kata symbebêkos.

It must be added that the necessity of the connection becomes apparent sometimes immediately, sometimes in a mediate way after more or less laborious analysis. This is an entirely subjective affair which nowise affects the nature of the connection.

{1} A prime number is usually defined as one which is divisible only by itself and unity.

{2} "Per se", says St. Thomas, "is used in a twofold sense. In one way a proposition is said to be per se when its predicate falls within the definition of the subject, as: Man is an animal; for animal is contained in the definition of man. And because that which is in the definition of a thing is in a way its cause, in these per se propositions the predicate is said to be the cause of the subject. A proposition is also said to per se in another way, when the subject is in the definition of the predicate, as: The nose is a hook; the number is even. For a hook [in this sense] is only a nose with a certain curve, and even is nothing but a number which has a half, and in these the subject is the cause of the predicate." De anima, bk. II, lect. 14.