Deductive Logic Part 8

(1) If A is B, C is D.

This is known as the Conjunctive or Hypothetical proposition.

(2) Either A is B or C is D.

This is known as the Disjunctive proposition.

-- 215. The disjunctive proposition may also appear in the form

A is either B or C,

which is equivalent to saying

Either A is B or A is C;

or again in the form

Either A or B is C,

which is equivalent to saying

Either A is C or B is C.

-- 216. As the double nomenclature may cause some confusion, a scheme is appended.

Proposition ____________|_____________ | | Simple Complex (Categorical) (Conditional) ___________|__________ | | Conjunctive Disjunctive.

(Hypothetical)

-- 217. The first set of names is preferable. 'Categorical' properly means 'predicable' and 'hypothetical' is a mere synonym for 'conditional.'

-- 218. Let us examine now what is the real nature of the statement which is made in the complex form of proposition. When, for instance, we say 'If the sky falls, we shall catch larks,' what is it that we really mean to a.s.sert? Not that the sky will fall, and not that we shall catch larks, but a certain connection between the two, namely, that the truth of the antecedent involves the truth of the consequent. This is why this form of proposition is called 'conjunctive,' because in it the truth of the consequent is conjoined to the truth of the antecedent.

-- 219. Again, when we say 'Jones is either a knave or a fool,' what is really meant to be a.s.serted is--'If you do not find Jones to be a knave, you may be sure that he is a fool.' Here it is the falsity of the antecedent which involves the truth of the consequent; and the proposition is known as 'disjunctive,' because the truth of the consequent is disjoined from the truth of the antecedent.

-- 220. Complex propositions then turn out to be propositions about propositions, that is, of which the subject and predicate are themselves propositions. But the nature of a proposition never varies in thought. Ultimately every proposition must a.s.sume the form 'A is, or is not, B.' 'If the sky falls, we shall catch larks' may be compressed into 'Sky-falling is lark-catching.'

-- 221. Hence this division turns upon the form of expression, and may be said to be founded on the simplicity or complexity of the terms employed in a proposition.

-- 222. In the complex proposition there appears to be more than one subject or predicate or both, but in reality there is only a single statement; and this statement refers, as we have Seen, to a certain connection between two propositions.

-- 223. If there were logically, and not merely grammatically, more than one subject or predicate, there would be more than one proposition. Thus when we say 'The Jews and Carthaginians were Semitic peoples and spoke a Semitic language,' we have four propositions compressed into a single sentence for the sake of brevity.

-- 224. On the other hand when we say 'Either the Carthaginians were of Semitic origin or argument from language is of no value in ethnology,'

we have two propositions only in appearance.

-- 225. The complex proposition then must be distinguished from those contrivances of language for abbreviating expression in which several distinct statements are combined into a single sentence.

_Verbal and Real Propositions_.

-- 226. A Verbal Proposition is one which states nothing more about the subject than is contained in its definition, e.g. 'Man is an animal'; 'Men are rational beings.'

-- 227. A Real Proposition states some fact not contained in the definition of the subject, e.g. 'Some animals have four feet.'

-- 228. It will be seen that the distinction between verbal and real propositions a.s.sumes a knowledge of the precise meaning of terms, that is to say, a knowledge of definitions.

-- 229. To a person who does not know the meaning of terms a verbal proposition will convey as much information as a real one. To say 'The sun is in mid-heaven at noon,' though a merely verbal proposition, will convey information to a person who is being taught to attach a meaning to the word 'noon.' We use so many terms without knowing their meaning, that a merely verbal proposition appears a revelation to many minds. Thus there are people who are surprised to hear that the lion is a cat, though in its definition 'lion' is referred to the cla.s.s 'cat.' The reason of this is that we know material objects far better in their extension than in their intension, that is to say, we know what things a name applies to without knowing the attributes which those things possess in common.

-- 230. There is nothing in the mere look of a proposition to inform us whether it is verbal or real; the difference is wholly relative to, and const.i.tuted by, the definition of the subject. When we have accepted as the definition of a triangle that it is 'a figure contained by three sides,' the statement of the further fact that it has three angles becomes a real proposition. Again the proposition 'Man is progressive' is a real proposition. For though his progressiveness is a consequence of his rationality, still there is no actual reference to progressiveness contained in the usually accepted definition, 'Man is a rational animal.'

-- 231. If we were to admit, under the term 'verbal proposition,' all statements which, though not actually contained in the definition of the subject, are implied by it, the whole body of necessary truth would have to be p.r.o.nounced merely verbal, and the most penetrating conclusions of mathematicians set down as only another way of stating the simplest axioms from which they started. For the propositions of which necessary truth is composed are so linked together that, given one, the rest can always follow. But necessary truth, which is arrived at 'a priori,' that is, by the mind's own working, is quite as real as contingent truth, which is arrived at 'a posteriori,' or by the teachings of experience, in other words, through our own senses or those of others.

-- 232. The process by which real truth, which is other than deductive, is arrived at 'a priori' is known as Intuition. E.g. The mind sees that what has three sides cannot but have three angles.

-- 233. Only such propositions then must be considered verbal as state facts expressly mentioned in the definition.

-- 234. Strictly speaking, the division of propositions into verbal and real is extraneous to our subject: since it is not the province of logic to acquaint us with the content of definitions.

-- 235, The same distinction as between verbal and real proposition, is conveyed by the expressions 'a.n.a.lytical' and 'Synthetical,' or 'Explicative' and 'Ampliative' judgements.

-- 236. A verbal proposition is called a.n.a.lytical, as breaking up the subject into its component notions.

-- 237. A real proposition is called synthetical, as attaching some new notion to the subject.

-- 238. Among the scholastic logicians verbal propositions were known as 'Essential,' because what was stated in the definition was considered to be of the essence of the subject, while real propositions were known as 'Accidental.'

_Universal AND PARTICULAR Propositions_.

-- 239. A Universal proposition is one in which it is evident from the form that the predicate applies to the subject in its whole extent.

-- 240. When the predicate does not apply to the subject in its whole extent, or when it is not clear that it does so, the proposition is called Particular.

-- 241. To say that a predicate applies to a subject in its whole extent, is to say that it is a.s.serted or denied of all the things of which the subject is a name.

-- 242. 'All men are mortal' is a universal proposition.

-- 243. 'Some men are black' is a particular proposition. So also is 'Men are fallible;' for here it is not clear from the form whether 'all' or only 'some' is meant.

-- 244. The latter kind of proposition is known as Indefinite, and must be distinguished from the particular proposition strictly so called, in which the predicate applies to part only of the subject.

-- 245. The division into universal and particular is founded on the Quant.i.ty of propositions.

-- 246. The quant.i.ty of a proposition is determined by the quant.i.ty in extension of its subject.