expressed by the copula 'is not.' These remarks may appear inconsistent with the form of the second example, but no rectilineal figure is, &c.' is really an abbreviated and unambiguous mode of stating the longer and ambiguous proposition 'All rectilineal figures are not, &c.'

The word 'predicated,' as equivalent to 'asserted or denied,' is here used in a wider than its ordinary signification. In common language, we say such and such an attribute cannot be predicated of such and such a term, using predicated' as equivalent to asserted' and as opposed to 'denied.' All ambiguity may be avoided by speaking of the predicate as predicated affirmatively or predicated negatively of the subject.

CHAPTER II

On the Copula

THE Logical Copula, its office being simply to serve as a sign of predication, is limited to the present tense of the verb 'to be,' with or without the addition of the negative particle, according as the proposition is negative or affirmative. This limitation follows from the fact that it is simply the office of the proposition to express my present judgment as to the agreement or disagreement of two terms. Hence all reference to time, past or future, and even to time present, as respects the terms themselves, and not my judgment as to their agreement, must be expressed in the predicate and not in the copula. I may, for brevity's sake, say 'fire burns,' 'Alexander was the son of Philip,' 'The guns will be fired to-morrow,' and, in conversation or discussion, it would undoubtedly be pedantic to express the propositions otherwise; but formally, for the purpose of being estimated logically, I must resolve them into their logical elements, and say 'Fire is burning,' 'Alexander is a person who was son of Philip,' 'The firing of the guns is an event which will take place to-morrow.'

Not only does the logical copula convey no notion of time with reference to the terms themselves (or, to speak more accurately, the things signified by them), but it is

also divested of the notion of existence. In other words, it is employed simply as a connecting particle, not as a substantive verb. Where the substantive verb is used in a logical proposition, it must be expressed in the predicate. Thus I am,' The king is not,' become 'I am existent,' 'The king is non-existent.' That the copula implies no notion of existence is evident from the fact that we can use such propositions as these: The labours of Hercules are a myth,' ' He is a nonentity.'

Can we modify the copula so as to express certainty, probability, possibility, or other modes of connexion between the subject and the predicate? This is the celebrated question of Modality, a question which has been the source of much difference of opinion amongst logicians. Even though it were granted that the proposition simply expresses our present judgment on the agreement or disagreement of two terms, it might be maintained that it should express the nature of our judgment and the degree of our assent or dissent, whether it be certain, approximating to certainty, or faltering. Thus it might be maintained that the following should be accepted as instances of the ultimate analysis of a logical proposition: 'This is certainly the man I saw yesterday,' 'This is probably the man I saw yesterday,' 'This is possibly the man I saw yesterday.' That we use these forms in conversation and discussion is unquestionable, but it is one main object of Logic to analyse our abbreviated inferences and statements into their full logical equivalents. Instead therefore of admitting various descriptions of copulæ

(other than the affirmative and negative), in order to adapt Logic to ordinary language, it seems simpler, as well as more scientific, to insist on the uniform character of the copula, and to represent propositions like the foregoing as predicating our degree of assent to or dissent from the sentence in question. Thus, after asking myself the question 'Is this the man I saw yesterday?' I may either answer simply 'This is the man I saw yesterday,' or I may describe the degree of my assent by stating 'That this is the man I saw yesterday is certain, probable, possible,' &c. In fact, such propositions seem to be the result of an act of reflexion on the degree of our own conviction.

I shall therefore regard the form A is or is not B as the ultimate and uniform logical analysis of all propositions, though I shall occasionally, for the sake of brevity, avail myself of the forms sanctioned by popular language.

Note. As regards the expression of time in the copula, the student will find the opposite opinion to that taken in the text adopted by Mr. Mill, Logic, vol. i. ch. iv. § 2. In support of my view, he may refer to Dr. Mansel's Prolegomena Logica, pp. 63, 64. On the question of expressing in the copula the degrees of assurance with which a proposition is entertained (certainly,' 'probably,' &c.), see Sir W. Hamilton's Discussions, pp. 145-7, and, for a more qualified view than that taken either by Sir W. Hamilton or myself, Dr. Mansel's Prolegomena Logica, note G.

CHAPTER III

Division of Propositions according to their Quantity and Quality

WE have already seen that propositions are either affirmative or negative, according as the copula used is of the form 'is' or 'is not.' This is called a division of propositions according to their Quality.

They are further divided, according to their Quantity, into Universal and Particular. For, in affirming or denying a predicate of a subject, it is obvious that I may either affirm or deny the predicate of all the individuals denoted by the subject, or of part only. Thus in affirming mortality of man, I may say 'All men are mortal,' or 'Some men are mortal'; in denying wisdom of man, I may deny it of all men or only of some men, I may say 'No men are wise,' or 'Some men are not wise.' When the predicate is affirmed or denied of all the individuals denoted by the subject, the proposition is called an Universal Proposition; when of part only, a Particular Proposition. A Singular Proposition, i. e. a proposition of which the subject is a singular term, ranks as an Universal, because the predicate is affirmed or denied of everything (i. e. in this

i. e.