Complex propositions are such as have a complex term for their subject or their predicate. By a complex term is understood a many-worded term, consisting of two or more distinct parts, so that it expresses, not merely the nature of the thing denoted, but also one or more qualifications belonging to it, e.g. 'the white knight,' 'the roller which is in my garden,' and the like. These qualifications are often (as in the second of the instances just given), expressed by subordinate clauses, introduced by a relative. Yet it is manifest that, even if a complex term involves two or three such clauses, the term is but one, and constitutes a single subject or predicate, as the case may be.

Two forms of complex propositions are ordinarily distinguished by logicians. The distinction is grammatical, not logical, and is given in order to put us on our guard against ambiguity.

(1) Propositions with an explicative qualification. In these the qualification belongs to every individual signified by the general name, to which it belongs. Thus in the, proposition, 'Whales, which are mammals, are aquatic animals,' the relative clause is applicable to every individual, that is signified by the general name 'whales.'

(2) Propositions with a restrictive (or determinative) qualification. In these, the qualification restricts the signification of the general name to a certain part of its denotation. Thus in the sentence, 'All nations, that have been civilized, have cultivated philosophy,' the qualification does not belong to all the members of the class indicated. Not all nations are civilized.

The time determination involved in the use of the

past and future tenses of the verb, is a special form of

complexity in the proposition. This however, as we have noted, Logic is enabled to disregard. Another constantly recurring form is that produced by the employment of transitive verbs, followed by an object, e.g. 'Brutus slew his benefactor,' which gives as the logical predicate, the complex term 'a slayer of his benefactor.'

It often happens that what grammatically is a single assertion, is resolvable into two or more propositions, each with its own subject and predicate. In such cases, we have the Compound Categorical Proposition. Pro positions of this kind are divided into two classes — those whose character is apparent from their grammatical structure (aperté compositi), and those in which the grammatical form does not manifest their composite nature (occulté compositi). These latter are termed Exponibles.

1. Propositions compound in form.

Of these there are three classes

i. Copulative propositions. These are affirmative propositions, in which there are two or more subjects or predicates or both. Hence they are resolvable into a number of independent affirmative propositions: e.g. 'Peter and Paul ended their days at Rome.' This is equivalent to 'Peter ended his days at Rome. Paul ended his days at Rome.'

2. Remotive propositions. These are negations similarly united. The conjunctions employed will be such as the negative form demands. For example, 'Neither riches nor honours can banish anxiety'; this sentence may be resolved, ‘Riches cannot banish anxiety. Honours cannot banish anxiety.' 'No Moslem will eat swine's flesh or drink wine.' This, in its logical expression, becomes, 'No Moslem will eat swine's flesh. No Moslem will drink wine.'

3. Discretive or Adversative propositions. Here we have either tw6 affirmative propositions, or an affirmative and a negative proposition, connected by an adversative conjunction, such as but, although, yet. Thus:

‘William I. was brave but not magnanimous.' This

gives us the two propositions 'William I. was brave.

William I. was not magnanimous.'

2. Exponible propositions.

In these, as we have said, there is nothing in the grammatical structure of the sentence to indicate that it is equivalent to more than one logical proposition. Here also, three classes are ordinarily enumerated.

(1) Exclusive Propositions. These contain a word, such as 'alone,' attached to the subject, and thus excluding the predicate from any other subject than this one. Hence two propositions are necessary to declare the full meaning, one to affirm the predicate of this subject, and another to deny it of all others. For instance, 'God alone is omnipotent.' This is equivalent to 'God is omnipotent. No other is omnipotent.'

(2) Exceptive Propositions. In these, the subject term is restricted in its application by a word such as except — save, which excludes a portion of its denotation —e.g. 'All the crew save one were drowned.'

Here again, two exponent propositions are needed, the one denying the predicate of the excepted part, the other affirming it of the remainder. The example just given will become, 'One of the crew was not drowned. The remaining members were drowned.'

[If the order of the terms is altered, then both Exclusives and Exceptives may be expressed by a single exponent. 'Only God is omnipotent,' will become 'All that is omnipotent is God'; and 'All the crew save one were drowned,' will be 'The portion of the crew that was not drowned was one man.' But if the original order is to be preserved, two propositions are necessary. The reasons which justify a change of order will be dealt with later.]

(3) Inceptive and Desitive Propositions. In these a statement is made as to the commencement or ending of something; e.g. 'Printing became customary after 'the fifteenth century,' 'Paganism ceased in England 'about the year 700 A D.' These are resolved by two propositions, one relating to the state of things before the time indicated, and one relating to what occurred subsequently. Thus the first example will become 'Printing was not customary before the close of the fifteenth century. Printing was customary after that date.'

The Modal proposition affords us another case in which the traditional terminology differs from that in vogue since the days of Kant. Here too we shall first explain modality as understood by the Scholastic philosophers, and then deal with the Kantian account.

The characteristic of the Modal, is that the copula undergoes modification, in order to express the manner in which the predicate belongs to the subject. There are propositions, in which the attribute affirmed belongs to the subject by strict necessity. Thus 'mortality' is an attribute that is necessarily connected with the subject 'man.' In other cases the element of necessity is absent. 'To be learned' is affirmable of some men only. It is not an attribute belonging necessarily to the nature 'man.'

The pure categorical draws no distinction between these cases. We employ the same copula 'is,' whether the connection is necessary or contingent. But in the Modal proposition, the nature of the connection between attribute and subject receives expression.

It has been frequently objected, that this whole question belongs not to Logic but to Metaphysics. Thus Sir W. Hamilton says, "Necessity, Possibility, etc. do not relate to the connection of the subject and predicate . . . as terms in thought, but as realities in existence: they are metaphysical, not logical condi'tions."

This objection rests on a misconception as to the province of Logic. Necessity and Possibility as objective facts, belong to the real order. But as mentally expressed by us, they belong to the logical order; and a treatise on Logic would be incomplete without some mention of the manner in which the mental judgment represents these metaphysical conditions.

The relation of the attribute to the subject is, objectively, determined by one of three modes. These are

1. the Necessary, in which the attribute belongs necessarily to the subject. This is expressed by a proposition of the form, 'Men are necessarily mortal,' 'Equilateral triangles are necessarily equiangular.'
2. The Impossible: in this case the predicate is repugnant to the subject, e.g. 'It is impossible for irrational creatures to exercise free will.' And
3. the Possible (or Contingent). In this case the predicate belongs to the subject in some instances, while in other instances it is not found with it. Thus e.g. 'It is possible for a man to be a grammarian.' The conjunction of the two attributes involves no impossibility, but on the other hand is not necessary. This relation may be asserted from two points of view. We may assert the possibility of the connection between subject and predicate, and express the proposition as it is expressed above. Or we may declare the possibility of their separation: and in this case the proposition will take the form, 'It is possible for a man not to be a grammarian.' Hence though there are but three modes, there are four fundamental forms of Modal propositions, as there are four fundamental forms of Categorical.

A difficulty is occasioned by the fact that ambiguity attaches to the word 'possible.'

'Possible' may have the sense in which we have just explained it. It may however, include in its signification the Necessary also; for if a predicate belongs necessarily to a subject, we can say with truth that that subject is capable of receiving it.' If all triangles must have three angles, it is true to say that it is possible for a triangle to have three angles.

And similarly the assertion that it is possible for a subject not to have such and such a predicate, may have a sense in which it includes the Impossible.

The Modal may be expressed in two forms. In the first of these, the mode itself constitutes the predicate, having for its subject the proposition whose copula it affects, e.g. 'That man should be mortal is necessary,' 'that a bird should have gay plumage is possible.'

Modals of this form are all singular, since the subject is not a term, but a proposition taken as a whole. Nevertheless the modes of necessity and impossibility are a sure sign that the proposition in question is universal; while on the other hand the mode of possibility, in the sense of merely possible (as distinguished from the case in which possibility is predicated of a necessary judgment) is indicative of a particular proposition. In the second forms of the Modal the mode qualifies the copula itself: e.g. All triangles are necessarily three-angled. Modals of this form are not singular but take their quantity from their subject.

* Kant's division of Modals is based, not on the objective relation of the predicate to the subject, but on the subjective certainty of the thinker. He divides judgments into the Problematic, i.e. 'S may be P,' the Assertoric, i.e. 'S is P.' and the Apodictic, i.e. 'S must be P.' Of the problematic judgment he says that it expresses "a free choice of admitting such a proposition, and a purely optional admission of it into the under 'standing.' The assertoric judgment " implies logical reality or 'truth." The apodictic gives us the same judgment as the assertoric, when it is recognized as determined by the formal laws of the understanding, and therefore as subjectively necessary

In regard to this division it may be said in the first place that such a proposition as 'S may be P' is of no value to the logician. It is a mere declaration of ignorance, and not a judgment at all.

Secondly, since the apodictic judgment enunciates the same truth as the assertoric, merely involving that the speaker recognizes more clearly the subjective necessity under which he lies of thus judging, there is no reason why he should not express the assertoric in the same form as the apodictic 'S must be P.'

The root error of this view is the failure to see that the copula is not a mere mental act of union, but expresses the objective connection between the subject and its attribute in the real order

The influence of the Kantian system is to be seen in many recent logicians. We are not infrequently told that when a truth is styled 'necessary,' nothing more is meant than a 'necessity of thought,' and that the term has no reference to the real order. Mr. Bradley tells us, "a necessary truth is really an inference, and an inference is a necessary truth " (Principles, p. 225). Similarly Mr. Bosanquet writes, "Every necessary truth must, in so far as it is necessary, present itself as the conclusion from an antecedent" (Logic, II. 222).

Such a view as this must needs be fatal to any hope of attaining certitude in philosophy or science.

The existence of any necessary first principles is denied. But where there are no necessary principles, there can be no necessity in the conclusion derived from them.²

Reduction of Propositions to Logical Form.

The sentences employed in literature and in ordinary conversation exhibit considerable variety of form and complexity of structure. It is possible however to analyse them and express them in the shape of A E 1 0 propositions. This process is styled their reduction to logical form. By submitting sentences to this analysis we reach the simple elements of thought, which are contained in them. It is plain that this is very different from grammatical analysis into parts of speech. That process is concerned not with thoughts but with words. The preceding paragraphs should have rendered the task of reduction comparatively easy. Its essential feature is to obtain propositions, consisting of (1) a subject with the sign of quantity attached

(2) a copula, which must be of the form is or are (or is not, are not), and

(3) a predicate.

We find the subject by putting to ourselves the question, Of what or of whom is this statement made?

We find the quantity of the subject by asking, Is the assertion made of the whole extension of the subject, or of but part of it?

We find the predicate by enquiring, What is it that is asserted of the subject?

These three points must always be considered, whenever the analysis of a sentence is attempted. Two other cautions may be added. First, that it is well, whenever it is possible, to express the predicate as an attribute, i.e. adjectivally, in order to bring out the true meaning of the proposition: e.g. the form 'All flattery is to be avoided' is better than 'All flattery is a thing to be avoided.' Secondly, that wherever it is necessary to introduce a time determination, this must be done in the predicate as in No. (7) below. The copula must always be in the present tense.

A few examples will illustrate the process

(1) 'Fools despise wisdom.'

This will become, 'All fools are despisers of wisdom' (A).

(2) 'All's well that ends well.'

This will be, 'All that ends well is well' (A).

(3) 'Firm at his dangerous post he stood.'

This in logical form is, 'He is standing firm at his dangerous post' (A).

(4) 'As a man sows, so shall he reap.'

Here we have a relative sentence. The two clauses of these propositions give us the terms of a relation. Where the words 'As . . . so' are employed to introduce the clauses, the relation is one of likeness. The analysis gives us, 'In every instance, the character of a man's harvest is like the character of his sowing' (A).

If the words 'Where . . . there' are used, we have a relation of place: if 'When . . . then,' a relation of time.

(5) 'Where thy treasure is, there will thy heart be also.' Logically, this is, 'In every instance, the place of your treasure is the place of your heart' (A).

(6) 'Love is akin to madness.'

Here the subject is used without any sign of quantity, but clearly stands for the whole denotation of the term. The pro position becomes, 'All cases of love are akin to madness.'

Where we have compound or exponible propositions, they need resolving into their component parts, e.g.

(7) 'Lions and tigers once lived wild in Europe, but not now.'

This gives us four propositions.

'Some lions are animals, that once lived wild in Europe'

'Some tigers are animals, that once lived wild in Europe'

'No lions are living wild in Europe now' (E).

'No tigers are living wild in Europe now' (E).

(8) 'Only the just enjoy peace of mind.'

This is resolved into:

'Some of the just are enjoying peace of mind' (I). 'None, who are not just, are enjoying peace of mind' (E).

(9) 'All save he had fled.'

Here we have a case, where the full force of the proposition cannot be brought out in the analysis, since we have no universal term by which to designate all the remainder. The reduction gives

'He is not fleeing' (E).

'Some (the rest) are fleeing' (1).

(10) 'The great is not good, but the good is great.' Notice should be taken of the 'reduplicative' use of the word 'great' in the first clause. It signifies 'the great as such,' or 'the great, just in so far as it is great.' This must be expressed in the analysis 'The great, merely in virtue of its greatness, is not good' (E).

'The good is great' (A).

Other Signs of Quantity.

It will be useful to mention a few other modes of expressing Quantity besides those we have already

noticed.

A. The universal affirmative is occasionally denoted by the expressions, Any, Whoever, He who, Always, In every case.

I may be denoted by A few, Certain, Often. Generally, Most.

E may be expressed by the word Never.

O has equivalents in A few . . . not, Not all . . . are, All are not, Few, Certain . . . not.

The word Most has been placed as one of the equivalents of I. The proposition 'Most S's are P' signifies that 'Some (more than half) S's are P,' but does not necessarily imply in addition that 'Some S's are not P.' It merely signifies that the majority of instances have been examined, and found to possess the attribute P.

Thus we might say, 'Most English flowering-plants are dicotyledonous,' without desiring to commit ourselves to any opinion as to the whole flora: or again, after looking at seven cards out of a hand at whist, we might say, 'Most of the cards in this hand are court-cards,' knowing that it was possible they might all prove to be so.

Similarly we might say, 'Few English flowering-p]ants are monocotyledonous,' even if we were ignorant whether there were any of that character. Hence Few is commonly reckoned as merely a sign of the proposition O¹ The words Hardly any, Scarcely any are also regarded as equivalent to O. The use of All with a negative to signify O should be carefully noticed. 'Not all the crew were lost,' will be expressed 'Some of the crew were not lost.'

Special note should be taken as to whether the terms, to which words such as All, A few, etc. etc. are attached, are used distributively or collectively

Wherever the use is collective the proposition is singular. 'All the men built a raft' is a case in point. The proposition may be expressed, 'The whole body of men is building a raft.'