Deductive Logic Part 7

-- 190. We have laid down above that there are two kinds of copula, affirmative and negative: but some logicians have maintained that the copula is always affirmative.

-- 191. What then, it may be asked, on this view, is the meaning of negative propositions! To which the answer is, that a negative proposition a.s.serts an agreement between the subject and a negative term. When, for instance, we say 'The whale is not a fish,' this would be interpreted to mean 'The whale is a not-fish.'

-- 192. Undoubtedly any negative proposition may be exhibited in an affirmative form, since, by the law of excluded middle, given a pair of contradictory terms, wherever the one can be a.s.serted, the other can be denied, and vice versa. We shall find later on that this principle gives rise to one of the forms of immediate inference. The only question then can be, which is the more natural and legitimate form of expression. It seems simpler to suppose that we a.s.sert the agreement of 'whale' with 'not-fish' by implication only, and that what we directly do is to predicate a disagreement between 'whale' and the positive attributes connoted by 'fish.' For since 'not-fish' must apply to every conceivable object of thought except those which fall under the positive term 'fish,' to say that a whale is a 'not-fish,'

is to say that we have still to search for 'whale' throughout the whole universe of being, minus a limited portion; which is only a more clumsy way of saying that it is not to be found in that portion.

-- 193. Again, the term 'not-fish' must be understood either in its intension or in its extension. If it be understood in its intension, what it connotes is simply the absence of the positive qualities which const.i.tute a fish, a meaning which is equally conveyed by the negative form of proposition. We gain nothing in simplicity by thus confounding a.s.sertion with denial. If, on the other hand, it is to be taken in extension, this involves the awkwardness of supposing that the predicative power of a term resides in its extensive capacity.

-- 194. We therefore recognise predication as being of two kinds--affirmation and negation--corresponding to which there are two forms of copula.

-- 195. On the other hand, other logicians have maintained that there are many kinds of copula, since the copula must vary according to the various degrees of probability with which we can a.s.sert or deny a predicate of a subject. This view is technically known as the doctrine of

_The Modality of the Copula._

-- 196. It may plausibly be maintained that the division of propositions into affirmative and negative is not an exhaustive one, since the result of an act of judgement is not always to lead the mind to a clear a.s.sertion or a clear denial, but to leave it in more or less doubt as to whether the predicate applies to the subject or not. Instead of saying simply A is B, or A is not B, we may be led to one of the following forms of proposition--

A is possibly B.

A is probably B.

A is certainly B.

The adverbial expression which thus appears to qualify the copula is known as 'the mode.'

-- 197. When we say 'The accused may be guilty' we have a proposition of very different force from 'The accused is guilty,' and yet the terms appear to be the same. Wherein then does the difference lie? 'In the copula' would seem to be the obvious reply. We seem therefore driven to admit that there are as many different kinds of copula as there are different degrees of a.s.surance with which a statement may be made.

-- 198. But there is another way in which modal propositions may be regarded. Instead of the mode being attached to the copula, it may be considered as itself const.i.tuting the predicate, so that the above propositions would be a.n.a.lysed thus--

That A is B, is possible.

That A is B, is probable.

That A is B, is certain.

-- 199. The subject here is itself a proposition of which we predicate various degrees of probability. In this way the division of propositions into affirmative and negative is rendered exhaustive. For wherever before we had a doubtful a.s.sertion, we have now an a.s.sertion of doubtfulness.

-- 200. If degrees of probability can thus be eliminated from the copula, much more so can expressions of time, which may always be regarded as forming part of the predicate. 'The sun will rise to-morrow' may be a.n.a.lysed into 'The sun is going to rise to-morrow.'

In either case the tense belongs equally to the predicate. It is often an awkward task so to a.n.a.lyse propositions relative to past or future time as to bring out the copula under the form 'is' or 'is not': but fortunately there is no necessity for so doing, since, as has been said before (-- 188), the material form of the copula is a matter of indifference to logic. Indeed in affirmative propositions the mere juxtaposition of the subject and predicate is often sufficient to indicate their agreement, e.g. 'Most haste, worst speed,' chalepha tha kala. It is because all propositions are not affirmative that we require a copula at all. Moreover the awkwardness of expression just alluded to is a mere accident of language. In Latin we may say with equal propriety 'Sol orietur cras' or 'Sol est oriturus cras'; while past time may also be expressed in the a.n.a.lytic form in the case of deponent verbs, as 'Caesar est in Galliam profectus'--'Caesar is gone into Gaul.'

-- 201. The copula then may always be regarded as pure, that is, as indicating mere agreement or disagreement between the two terms of the proposition.

CHAPTER III.

_Of the Divisions of Propositions_.

-- 202. The most obvious and the most important division of propositions is into true and false, but with this we are not concerned. Formal logic can recognise no difference between true and false propositions. The one is represented by the same symbols as the other.

-- 203. We may notice, however, in pa.s.sing, that truth and falsehood are attributes of propositions and of propositions only. For something must be predicated, i.e. a.s.serted or denied, before we can have either truth or falsehood. Neither concepts or terms, on the one hand, nor reasonings, on the other, can properly be said to be true or false. In the mere notion of a Centaur or of a black swan there is neither truth nor falsehood; it is not until we make some statement about these things, such as that 'black swans are found in Australia,'

or 'I met a Centaur in the High Street yesterday,' that the question of truth or falsehood comes in. In such expressions as a 'true friend'

or 'a false patriot' there is a tacit reference to propositions. We mean persons of whom the terms 'friend' and 'patriot' are truly or falsely predicated. Neither can we with any propriety talk of true or false reasoning. Reasoning is either valid or invalid: it is only the premisses of our reasonings, which are propositions, that can be true or false. We may have a perfectly valid process of reasoning which starts from a false a.s.sumption and lands us in a false conclusion.

-- 204. All truth and falsehood then are contained in propositions; and propositions are divided according to the Quality of the Matter into true and false. But the consideration of the matter is outside the sphere of formal or deductive Logic. It is the problem of inductive logic to establish, if possible, a criterion of evidence whereby the truth or falsehood of propositions may be judged (-- 2).

-- 205. Another usual division of propositions is into Pure and Modal, the latter being those in which the copula is modified by some degree of probability. This division is excluded by the view which has just been taken of the copula, as being always simply affirmative or simply negative.

-- 206. We are left then with the following divisions of propositions--

Proposition according to Form Simple

Complex Conjunctive Disjunctive

Universal Singular General

according to Matter Verbal Real

according to Quant.i.ty Universal Singular General

Particular Indefinite (strictly) Particular

according to Quality Affirmative Negative

_Simple and Complex Propositions_.

-- 207. A Simple Proposition is one in which a predicate is directly affirmed or denied of a subject, e.g. 'Rain is falling.'

-- 208. A simple proposition is otherwise known as Categorical.

-- 209. A Complex Proposition is one in which a statement is made subject to some condition, e.g. 'If the wind drops, rain will fall.'

-- 210. Hence the complex proposition is also known as Conditional.

-- 211. Every complex proposition consists of two parts--

(1) Antecedent;

(2) Consequent.

-- 212. The Antecedent is the condition on which another statement is made to depend. It precedes the other in the order of thought, but may either precede or follow it in the order of language. Thus we may say indifferently--'If the wind drops, we shall have rain' or 'We shall have rain, if the wind drops.'

-- 213. The Consequent is the statement which is made subject to some condition.

-- 214. The complex proposition a.s.sumes two forms,