The first illustration of the A proposition, “All men are mortal,” may be represented by two circles, a larger circle standing for the predicate, _mortal_, and a smaller circle entirely inside the larger representing the subject, _men_. Thus: Illustration: FIG. 1. It is evident that all of the smaller circle belongs to the larger. This diagram will then fit any proposition where it may be said that all of the subject belongs to a part of the predicate, or which may be symbolized as “All S is some P.” (All the subject is some of the predicate.) The student knows that circles are plane surfaces and when such a statement as “All men are mortal” is given, reference is made to only that part of the “mortal” circle which is _directly underneath_ the “men” circle. Nothing has been said relative to the remaining part of the “mortal” circle. “_A_” propositions which may be interpreted as meaning “All S is all P” are called co-extensive A’s because the subject and predicate are exactly equal in extension. Such propositions are best illustrated by definitions; e. g.: 1. “A man is a rational biped.” 2. “A trigon is a polygon of three sides.” 3. “Teaching is the art of occasioning those activities which result in knowledge, power and skill.” To represent the meaning of the co-extensive A by the Euler diagram, two circles of the same size may be drawn, one coinciding at every point with the other. If the first circle is drawn heavily in black and the second dotted in red, it will make clear to the eye that there are two circles. (2) _The Universal Negative or E Proposition._ _“No S is P” best symbolizes the E proposition_, though sometimes the universal negative is written “All S is not P.” This latter form, as has been explained, is ambiguous and therefore illogical. “No S is P” surely means that no part of the subject belongs to any part of the predicate and no part of the predicate belongs to any part of the subject. The subject and predicate are mutually exclusive. The following illustrate the E proposition: 1. “No man is immortal.” 2. “No true teacher works for money.” 3. “No thorough student can remain unwise.” The E proposition may be represented by two circles, the one entirely without the other as in Fig. 2: Illustration: FIG. 2. (3) _The Particular Affirmative or I Proposition._ _This may be symbolized as “Some S is P,”_ and considered as meaning that a _part_ of the subject belongs to a _part_ of the predicate. It has already been noted that “some” is ambiguous and that its logical signification is “some at least_.” (It may be all or it may not be all.) For example, the only logical interpretation which can be placed on “Some men are wise” is, that the investigation has resulted in finding only a _part_ of the man family wise. Whether or not all are wise is unknown as the entire field has not received attention. In no case can it be assumed that all the others are _not_ wise.