more complicated than the logical form of propositions; it is in order to prove them, or to use them in the proof of other propositions, that they are in Logic reduced as nearly as possible to such simple but explicit expressions as the above (tertii adjacentis). A Compound Proposition, reducible to two or more simple ones, is said to be exponible.
The modes of compounding sentences are explained in every grammar-book. One of the commonest forms is the copulative, such as Salt is both savoury and wholesome, equivalent to two simple propositions: Salt is savoury; Salt is wholesome. Pure water is neither sapid nor odorous, equivalent to Water is not sapid; Water is not odorous. Or, again, Tobacco is injurious, but not when used in moderation, equivalent to Much tobacco is injurious; a little is not.
Another form of Exponible is the Exceptive, as Kladderadatsch is published daily, except on week-days, equivalent to Kladderadatsch is published on Sunday; it is not published any other day. Still another Exponible is the Exclusive, as Only men use fire, equivalent to Men are users of fire; No other animals are. Exceptive and exclusive sentences are, however, equivalent forms; for we may say, Kladderadatsch is published only on Sunday; and No animals use fire, except men.
There are other compound sentences that are not exponible, since, though they contain two or more verbal clauses, the construction shows that these are inseparable. Thus, If cats are scarce, mice are plentiful, contains two verbal clauses; but if cats are scarce is conditional, not indicative; and mice are plentiful is subject to the condition that cats are scarce. Hence the whole sentence is called a Conditional Proposition. For the various forms of Conditional Propositions see chap. v. § 4.
But, in fact, to find the logical force of recognised grammatical forms is the least of a logician's difficulties in bringing the discourses of men to a plain issue. Metaphors, epigrams, innuendoes and other figures of speech present far greater obstacles to a lucid reduction whether for approval or refutation. No rules can be given for finding everybody's meaning. The poets have their own way of expressing themselves; sophists, too, have their own way. And the point often lies in what is unexpressed. Thus, "barbarous nations make, the civilised write history," means that civilised nations do not make history, which none is so brazen as openly to assert. Or, again, "Alcibiades is dead, but X is still with us"; the whole meaning of this 'exponible' is that X would be the lesser loss to society. Even an epithet or a suffix may imply a proposition: This personage may mean X is a pretentious nobody.
How shall we interpret such illusive predications except by cultivating our literary perceptions, by reading the most significant authors until we are at home with them? But, no doubt, to disentangle the compound propositions, and to expand the abbreviations of literature and conversation, is a useful logical exercise. And if it seem a laborious task thus to reduce to its logical elements a long argument in a speech or treatise, it should be observed that, as a rule, in a long discourse only a few sentences are of principal importance to the reasoning, the rest being explanatory or illustrative digression, and that a close scrutiny of these cardinal sentences will frequently dispense us from giving much attention to the rest.
§ 4. But now, returning to the definition of a Proposition given in § 2, that it is 'a sentence in which one term is predicated of another,' we must consider what is the import of such predication. For the definition, as it stands, seems to be purely Nominalist. Is a proposition nothing more than a certain synthesis of words; or, is it meant to correspond with something further, a synthesis of ideas, or a relation of facts?
Conceptualist logicians, who speak of judgments instead of propositions, of course define the judgment in their own language. According to Hamilton, it is "a recognition of the relation of congruence or confliction in which two concepts stand to each other." To lighten the sentence, I have omitted one or two qualifications (Hamilton's Lectures on Logic, xiii.). "Thus," he goes on "if we compare the thoughts water, iron, and rusting, we find them congruent, and connect them into a single thought, thus: water rusts iron—in that case we form a judgment." When a judgment is expressed in words, he says, it is called a proposition.
But has a proposition no meaning beyond the judgment it expresses? Mill, who defines it as "a portion of discourse in which a predicate is affirmed or denied of a subject" (Logic, Book 1., chap. iv. § 1.), proceeds to inquire into the import of propositions (Book 1., chap. v.), and finds three classes of them: (a) those in which one proper name is predicated of another; and of these Hobbes's Nominalist definition is adequate, namely, that a proposition asserts or denies that the predicate is a name for the same thing as the subject, as Tully is Cicero.
(b) Propositions in which the predicate means a part (or the whole) of what the subject means, as Horses are animals, Man is a rational animal. These are Verbal Propositions (see below: chap. v. § 6), and their import consists in affirming or denying a coincidence between the meanings of names, as The meaning of 'animal' is part of the meaning of 'horse.' They are partial or complete definitions.
But (c) there are also Real Propositions, whose predicates do not mean the same as their subjects, and whose import consists in affirming or denying one of five different kinds of matter of fact: (1) That the subject exists, or does not; as if we say The bison exists, The great auk is extinct. (2) Co-existence, as Man is mortal; that is, the being subject to death coinheres with the qualities on account of which we call certain objects men. (3) Succession, as Night follows day. (4) Causation (a particular kind of Succession), as Water rusts iron. (5) Resemblance, as The colour of this geranium is like that of a soldier's coat, or A = B.
On comparing this list of real predications with the list of logical relations given above (chap. i. § 5 (a)), it will be seen that the two differ only in this, that I have there omitted simple Existence. Nothing simply exists, unrelated either in Nature or in knowledge. Such a proposition as The bison exists may, no doubt, be used in Logic (subject to interpretation) for the sake of custom or for the sake of brevity; but it means that some specimens are still to be found in N. America, or in Zoological gardens.
Controversy as to the Import of Propositions really turns upon a difference of opinion as to the scope of Logic and the foundations of knowledge. Mill was dissatisfied with the "congruity" of concepts as the basis of a judgment. Clearly, mere congruity does not justify belief. In the proposition Water rusts iron, the concepts water, rust and iron may be congruous, but does any one assert their connection on that ground? In the proposition Murderers are haunted by the ghosts of their victims, the concepts victim, murderer, ghost have a high degree of congruity; yet, unfortunately, I cannot believe it: there seems to be no such cheap defence of innocence. Now, Mill held that Logic is concerned with the grounds of belief, and that the scope of Logic includes Induction as well as Deduction; whereas, according to Hamilton, Induction is only Modified Logic, a mere appendix to the theory of the "forms of thought as thought." Indeed, Mill endeavoured in his Logic to probe the grounds of belief deeper than usual, and introduced a good deal of Metaphysics—either too much or not enough—concerning the ground of axioms. But, at any rate, his great point was that belief, and therefore (for the most part) the Real Proposition, is concerned not merely with the relations of words, or even of ideas, but with matters of fact; that is, both propositions and judgments point to something further, to the relations of things which we can examine, not merely by thinking about them (comparing them in thought), but by observing them with the united powers of thought and perception. This is what convinces us that water rusts iron: and the difficulty of doing this is what prevents our feeling sure that murderers are haunted by the ghosts of their victims. Hence, although Mill's definition of a proposition, given above, is adequate for propositions in general; yet that kind of proposition (the Real) with regard to which Logic (in Mill's view) investigates the conditions of proof, may be more explicitly and pertinently defined
