Deductive Logic Part 11

-- 297. If this were so, it is plain that the number of possible propositions would be exactly doubled, and that, instead of four forms, we should now have to recognise eight, which may be expressed as follows--

1. All A is all B. ([upsilon]).

2. All A is some B. ([Lambda]).

3. No A is any B. ([Epsilon]).

4. No A is some B. ([eta]).

5. Some A is all B. ([Upsilon]).

6. Some A is some B. ([Iota]).

7. Some A is not any B. ([Omega]).

8. Some A is not some B. ([omega]).

-- 298. It is evident that it is the second of the above propositions which represents the original A, in accordance with the rule that 'No affirmative propositions distribute their predicate' (-- 293).

-- 299. The third represents the original E, in accordance with the rule that 'All negative propositions distribute their predicate.'

-- 300. The sixth represents the original I, in accordance with the rule that 'No affirmative propositions distribute their predicate.'

-- 301. The seventh represents the original O, in accordance with the rule that 'All negative propositions distribute their predicate.'

-- 302. Four new symbols are required, if the quant.i.ty of the predicate as well as that of the subject be taken into account in the cla.s.sification of propositions. These have been supplied, somewhat fancifully, as follows--

-- 303. The first, 'All A is all B,' which distributes both subject and predicate, has been called [upsilon], to mark its extreme universality.

-- 304. The fourth, 'No A is some B,' is contained in E, and has therefore been denoted by the symbol [eta], to show its connection with E.

-- 305. The fifth, 'Some A is all B,' is the exact converse of the second, 'All A is some B,' and has therefore been denoted by the symbol [Upsilon], which resembles an inverted A.

-- 306. The eighth is contained in O, as part in whole, and has therefore had a.s.signed to it the symbol [omega],

-- 307. The attempt to take the predicate in extension, instead of, as it should naturally be taken, in intension, leads to some curious results. Let us take, for instance, the u proposition. Either the sign of quant.i.ty 'all' must be understood as forming part of the predicate or not. If it is not, then the u proposition 'All A is all B' seems to contain within itself, not one proposition, but two, namely, 'All A is B' and 'All B is A.' But if on the other hand 'all' is understood to form part of the predicate, then u is not really a general but a singular proposition. When we say, 'All men are rational animals,' we have a true general proposition, because the predicate applies to the subject distributively, and not collectively. What we mean is that 'rational animal' may be affirmed of every individual in the cla.s.s, man. But when we say 'All men are all rational animals,' the predicate no longer applies to the subject distributively, but only collectively. For it is obvious that 'all rational animals' cannot be affirmed of every individual in the cla.s.s, man. What the proposition means is that the cla.s.s, man, is co-extensive with the cla.s.s, rational animal. The same meaning may be expressed intensively by saying that the one cla.s.s has the attribute of co-extension with the other.

-- 308. Under the head o u come all propositions in which both subject and predicate are singular terms, e.g. 'Homer was the author of the Iliad,' 'Virtue is the way to happiness.'

-- 309. The proposition [eta] conveys very little information to the mind. 'No A is some B' is compatible with the A proposition in the same matter. 'No men are some animals' may be true, while at the same time it is true that 'All men are animals.' No men, for instance, are the particular animals known as kangaroos.

-- 310. The [omega] proposition conveys still less information than the [eta]. For [omega] is compatible, not only with A, but with [upsilon]. Even though 'All men are all rational animals,' it is still true that 'Some men are not some rational animals': for no given human being is the same rational animal as any other.

-- 311. Nay, even when the [upsilon] is an identical proposition, [omega] will still hold in the same matter. 'All rational animals are all rational animals': but, for all that, 'Some rational animals are not some others.' This last form of proposition therefore is almost wholly devoid of meaning.

-- 312. The chief advantage claimed for the quantification of the predicate is that it reduces every affirmative proposition to an exact equation between its subject and predicate. As a consequence every proposition would admit of simple conversion, that is to say, of having the subject and predicate transposed without any further change in the proposition. The forms also of Reduction (a term which will be explained later on) would be simplified; and generally the introduction of the quantified predicate into logic might be attended with certain mechanical advantages. The object of the logician, however, is not to invent an ingenious system, but to arrive at a true a.n.a.lysis of thought. Now, if it be admitted that in the ordinary form of proposition the subject is used in extension and the predicate in intension, the ground for the doctrine is at once cut away. For, if the predicate be not used in its extensive capacity at all, we plainly cannot be called upon to determine whether it is used in its whole extent or not.

CHAPTER VI.

_Of the Heads of Predicables_.

-- 313. A predicate is something which is stated of a subject.

-- 314. A predicable is something which can be stated of a subject.

-- 315. The Heads of Predicables are a cla.s.sification of the various things which can be stated of a subject, viewed in their relation to it.

-- 316. The treatment of this topic, therefore, as it involves the relation of a predicate to a subject, manifestly falls under the second part of logic, which deals with the proposition. It is sometimes treated under the first part of logic, as though the heads of predicables were a cla.s.sification of universal notions, i.e. common terms, in relation to one another, without reference to their place in the proposition.

-- 317. The heads of predicables are commonly reckoned as five, namely,

(1) Genus.

(2) Species.

(3) Difference.

(4) Property.

(5) Accident.

-- 318. We will first define these terms in the sense in which they are now used, and afterwards examine the principle on which the cla.s.sification is founded and the sense in which they were originally intended.

(1) A Genus is a larger cla.s.s containing under it smaller cla.s.ses. Animal is a genus in relation to man and brute.

(2) A Species is a smaller cla.s.s contained under a larger one. Man is a species in relation to animal.

(3) Difference is the attribute, or attributes, which distinguish one species from others contained under the same genus. Rationality is the attribute which distinguishes the species, man, from the species, brute.

N.B. The genus and the difference together make up the Definition of a cla.s.s-name, or common term.

(4) A Property is an attribute which is not contained in the definition of a term, but which flows from it.

A Generic Property is one which flows from the genus.

A Specific Property is one which flows from the difference.

It is a generic property of man that he is mortal, which is a consequence of his animality. It is a specific property of man that he is progressive, which is a consequence of his rationality.

(5) An Accident is an attribute, which is neither contained in the definition, nor flows from it.

-- 319. Accidents are either Separable or Inseparable.

A Separable Accident is one which belongs only to some members of a cla.s.s.