In one of these two, every known truth can be expressed. For the assertion made is either known to hold good of the subject in its whole extension, or not.

If it is known to hold good, we use the Universal proposition. If it does not hold good as regards the whole extension of the subject, or if, though it holds good, we do not know this to be the case, we use the Particular.

This distinction, combined with that based on quality, gives us the fourfold scheme, viz. Universal Affirmative, Particular Affirmative, Universal Negative, Particular Negative.

These are respectively given by the letters, A.I.E.O. These letters are the vowels of the two Latin words, Affirmo (I affirm) and Nego (I deny). The first vowel in each stands for the Universal, the second vowel for the Particular.

Another notation, which is found convenient, is SaP, SiP, SeP, SoP: this notation has symbols for the subject and predicate, as well as for quantity and quality. Hence, our four propositions may be thus expressed.

Analytic and Synthetic Propositions. This distinction is based on the fact that each of our Judgments is based on one or other of two very different motives.

The point will best be elucidated by a few examples. If we consider the following propositions, 'The angles of every triangle are equal to two right angles,' 'The whole is greater than its part,' 'Every square has four sides,' and compare them with such propositions as 'Water freezes at 320 Fahrenheit,' 'Some cows are black,' we shall at once recognize that there is a difference between the two classes.

We are, indeed, certain of the truth of all these propositions. But our certainty has a different motive in the first class, and in the second. In the case of the first class of Judgments, as soon as we consider the concepts of the subject and predicate, we see that they are necessarily bound together.

A triangle must have its angles equal to two right angles; otherwise it would not be what we mean by a triangle. Were we told of any figure that its interior angles were greater or less than two right angles, we should be justified in affirming that it was not, and could not under any circumstances be a rectilinear triangle. In the same way the intension of the concepts 'whole' and 'part' excludes the supposition of a whole that is not greater than its part; for the meaning of the term 'whole,' is 'that which consists of parts.' In regard to the second class, the motive of our assent is very different. It is experience that has led to my conviction that water freezes at 320 F., and that certain cows are black. There is nothing in my notion of ' cows which prescribes 'blackness,' nor in my notion of' water,' which compels me to think of it as possessing this particular freezing-point at the sea-level. In neither of these propositions are the two concepts linked together in virtue of their intension.

The former class of propositions is termed Analytic, the latter Synthetic.

The definition of Analytic and Synthetic propositions is differently given by Scholastic philosophers on the one hand, and by the greater number of Logicians since the days of Kant, on the other.

The difference is of primary importance in philosophy. We place the Scholastic definitions first.

An Analytic proposition is one, in which either the predicate is contained in the intension of the subject, or the subject in the intension of the predicate.

It will be seen that Analytic propositions are of two kinds. The first kind consists of those in which the predicate is a term signifying either the whole intension, or part of the intension of the subject. Such is the proposition, 'Every square has four sides.' The second kind consists of those in which the predicate is an attribute which results necessarily from the nature of the subject. For where this is the case the subject is found in the intension of the predicate. An example is fur nished by the proposition, 'A triangle is a figure having its interior angles equal to two right angles.' The predicate here is not found in the intension or definition of 'triangle’. But it is an attribute which necessarily results from and is involved in the characteristics of a triangle. And if we desire to define the attribute 'having its interior angles equal to two right angles,' we can only do so by stating that it is a quantitative measure proper to the angles of a triangle.²

It is not however necessary that the connection of the attribute with the subject should be evident on the first consideration. Many steps may be necessary.

Every geometrical theorem gives us an Analytical pro position as its conclusion.

The connection between subject and predicate is involved in the intension of the terms : but we must often take a long series of steps before the necessity of that connection becomes manifest to us.^³

The modern definitions are as follows

So prevalent have these definitions become, that in any public examination at the present day, a question, involving these terms, would certainly employ them in this latter sense.

• (a) We have spoken of the differences between the two definitions as of vital moment. The Kantian division of Analytic and Synthetic propositions relegates to one class propositions, our knowledge of which depends wholly on experience of the individual case, such as 'This book is bound in cloth,' and propositions such as, 'The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the remaining sides’. Neither of these, Kant tells us, can be discovered by analysis. For he considers only the case, in which the predicate is found by the analysis of the subject, and entirely ignores the case, in which the subject-term is revealed by an analysis of the predicate. What account then is to be given of our conviction as to the truth of such propositions as that relating to the square on the hypotenuse? If they are not analytic, two hypotheses only are possible. Either (1) we accept them on a dictate of our understanding, of which no account can be given. They are synthetic a priori. This is Kant's solution. Or (2) their truth is a conclusion, at which we arrive from an examination of individual instances, but we possess no ground for saying, that they must be true, that e.g. every right-angled triangle has the property described. This is Mill's solution.

(b) There have been three theories as to the object of Analytic propositions.

• Mill (following Hobbes) holds that they are concerned with the meaning of names only, and terms them Verbal propositions.
• Leibniz held that they are concerned with our concepts.
• The Scholastics taught that they are truths relating to things, though known through our concepts, and expressed in words.

Professor Case wisely says, "The division of propositions into verbal and real is defective. A verbal is not necessarily opposed to a real proposition, a predicate does not cease to be characteristic of a thing by becoming the meaning of its name, and there are some propositions which are verbal and real, such as, all bodies are extended, the whole is greater than its part. . . . Sometimes the same analytical judgment is at once 'real, notional and verbal, e.g. the whole is, is conceived, and means that which is greater than its part" (Physical Realism, p. 340).

(c) It is sometimes said, that every synthetic judgment becomes analytic with the growth of our knowledge, that e.g. 'George III. died in 1820,' is an analytic judgment to one who knows the history of that period. The argument is quite fallacious. The facts, which occur to an individual member of a class, are not necessary notes of his nature, forming the connotation of the concept which expresses it. In the first place, of individuals as such we have no concepts: all concepts are universal.

And secondly, even were it possible to have a concept expressing the essential nature of the individual, the purely contingent facts relating to him would not be part of it. Of course, if I form a complex concept applicable to George III, such as e.g. 'The King of England at the beginning of the nineteenth century,' and to this add the note 'who died in 1820,' then I may form the analytic proposition, 'The King of England at the beginning of the nineteenth century, who died in 1820, died in 1820.' But the value of an analytic proposition of this kind is not great.

(d) Analytic propositions are also termed Essential, Explicative, a priori, Verbal; and correspondingly, synthetic propositions are known as Accidental, Ampliative, a posteriori, Real. On this whole subject, see Professor Case's Physical Realism, pp. 334 — 353.