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Deductive Logic by St. George Stock

S >> St. George Stock >> Deductive Logic




(1) Universal,

(2) Particular,

(3) Indefinite.

But logicians anxious for simplification asked, whether a predicate in
any given case must not either apply to the whole of the subject or
not? And whether, therefore, the third head of indefinite propositions
were not as superfluous as the so-called 'common gender' of nouns in
grammar?

§ 256. It is quite true that, as a matter of fact, any given predicate
must either apply to the whole of the subject or not, so that in the
nature of things there is no middle course between universal and
particular. But the important point is that we may not know whether
the predicate applies to the whole of the subject or not. The primary
division then should be into propositions whose quantity is known and
propositions whose quantity is unknown. Those propositions whose
quantity is known may be sub-divided into 'definitely universal' and
'definitely particular,' while all those whose quantity is unknown are
classed together under the term 'indefinite.' Hence the proper
division is as follows--

Proposition
__________|____________
| |
Definite Indefinite
_____|_______
| |
Universal Particular.

§ 257. Another very obvious defeat of terminology is that the word
'universal' is naturally opposed to 'singular,' whereas it is here so
used as to include it; while, on the other hand, there is no obvious
difference between universal and general, though in the division the
latter is distinguished from the former as species from genus.



_Affirmative and Negative Propositions._


§ 258. This division rests upon the Quality of propositions.

§ 259. It is the quality of the form to be affirmative or negative:
the quality of the matter, as we saw before (§ 204), is to be true or
false. But since formal logic takes no account of the matter of
thought, when we speak of 'quality' we are understood to mean the
quality of the form.

§ 260. By combining the division of propositions
according to quantity with the division according to quality,
we obtain four kinds of proposition, namely--

(1) Universal Affirmative (A).

(2) Universal Negative (E).

(3) Particular Affirmative (I).

(4) Particular Negative (O).

§ 261. This is an exhaustive classification of propositions, and any
proposition, no matter what its form may be, must fall under one or
other of these four heads. For every proposition must be either
universal or particular, in the sense that the subject must either be
known to be used in its whole extent or not; and any proposition,
whether universal or particular, must be either affirmative or
negative, for by denying modality to the copula we have excluded
everything intermediate between downright assertion and denial. This
classification therefore may be regarded as a Procrustes' bed, into
which every proposition is bound to fit at its proper peril.

§ 262. These four kinds of propositions are represented respectively
by the symbols A, E, I, O.

§ 263. The vowels A and I, which denote the two affirmatives, occur in
the Latin words 'affirmo' and 'aio;' E and O, which denote the two
negatives, occur in the Latin word 'nego.'



_Extensive and Intensive Propositions._


§ 264. It is important to notice the difference between Extensive and
Intensive propositions; but this is not a division of propositions,
but a distinction as to our way of regarding them. Propositions may be
read either in extension or intension. Thus when we say 'All cows are
ruminants,' we may mean that the class, cow, is contained in the
larger class, ruminant. This is reading the proposition in
extension. Or we may mean that the attribute of chewing the cud is
contained in, or accompanies, the attributes which make up our idea of
'cow.' This is reading the proposition in intension. What, as a matter
of fact, we do mean, is a mixture of the two, namely, that the class,
cow, has the attribute of chewing the cud. For in the ordinary and
natural form of proposition the subject is used in extension, and the
predicate in intension, that is to say, when we use a subject, we are
thinking of certain objects, whereas when we use a predicate, we
indicate the possession of certain attributes. The predicate, however,
need not always be used in intension, e.g. in the proposition 'His
name is John' the predicate is not intended to convey the idea of any
attributes at all. What is meant to be asserted is that the name of
the person in question is that particular name, John, and not
Zacharias or Abinadab or any other name that might be given him.

§ 265. Let it be noticed that when a proposition is read in extension,
the predicate contains the subject, whereas, when it is read in
intension, the subject contains the predicate.



_Exclusive Propositions._


§ 266. An Exclusive Proposition is so called because in it all but a
given subject is excluded from participation in a given predicate,
e.g. 'The good alone are happy,' 'None but the brave deserve the
fair,' 'No one except yourself would have done this.'

§ 267. By the above forms of expression the predicate is declared to
apply to a given subject and to that subject only. Hence an exclusive
proposition is really equivalent to two propositions, one affirmative
and one negative. The first of the above propositions, for instance,
means that some of the good are happy, and that no one else is so. It
does not necessarily mean that all the good are happy, but asserts
that among the good will be found all the happy. It is therefore
equivalent to saying that all the happy are good, only that it puts
prominently forward in addition what is otherwise a latent consequence
of that assertion, namely, that some at least of the good are happy.

§ 268. Logically expressed the exclusive proposition when universal
assumes the form of an E proposition, with a negative term for its
subject

No not-A is B.

§ 269. Under the head of exclusive comes the strictly particular
proposition, 'Some A is B,' which implies at the same time that 'Some
A is not B.' Here 'some' is understood to mean 'some only,' which is
the meaning that it usually bears in common language. When, for
instance, we say 'Some of the gates into the park are closed at
nightfall,' we are understood to mean 'Some are left open.'



_Exceptive Propositions._


§ 270. An Exceptive Proposition is so called as affirming the
predicate of the whole of the subject, with the exception of a certain
part, e.g. 'All the jury, except two, condemned the prisoner.'

§ 271. This form of proposition again involves two distinct
statements, one negative and one affirmative, being equivalent to 'Two
of the jury did not condemn the prisoner; and all the rest did.'

§ 272. The exceptive proposition is merely an affirmative way of
stating the exclusive--

No not-A is B = All not-A is not-B.

No one but the sage is sane = All except the sage are mad.



_Tautologous or Identical Propositions_


§ 273. A Tautologous or Identical proposition affirms the subject of
itself, e.g. 'A man's a man,' 'What I have written, I have written,'
'Whatever is, is.' The second of these instances amounts formally to
saying 'The thing that I have written is the thing that I have
written,' though of course the implication is that the writing will
not be altered.




CHAPTER IV.

_Of the Distribution of Terms._


§ 274. The treatment of this subject falls under the second part of
logic, since distribution is not an attribute of terms in themselves,
but one which they acquire in predication.

§ 275. A term is said to be distributed when it is known to be used in
its whole extent, that is, with reference to all the things of which
it is a name. When it is not so used, or is not known to be so used,
it is called undistributed.

§ 276. When we say 'All men are mortal,' the subject is distributed,
since it is apparent from the form of the expression that it is used
in its whole extent. But when we say 'Men are miserable' or 'Some men
are black,' the subject is undistributed.

§ 277. There is the same ambiguity attaching to the term
'undistributed' which we found to underlie the use of the term
'particular.' 'Undistributed' is applied both to a term whose quantity
is undefined, and to one whose quantity is definitely limited to a
part of its possible extent.

§ 278. This awkwardness arises from not inquiring first whether the
quantity of a term is determined or undetermined, and afterwards
proceeding to inquire, whether it is determined as a whole or part of
its possible extent. As it is, to say that a term is distributed,
involves two distinct statements--

(1) That its quantity is known;

(2) That its quantity is the greatest possible.

The term 'undistributed' serves sometimes to contradict one of these
statements and sometimes to contradict the other.

§ 279. With regard to the quantity of the subject of a proposition no
difficulty can arise. The use of the words 'all' or 'some,' or of a
variety of equivalent expressions, mark the subject as being
distributed or undistributed respectively, while, if there be nothing
to mark the quantity, the subject is for that reason reckoned
undistributed.

§ 280. With regard to the predicate more difficulty may arise.

§ 281. It has been laid down already that, in the ordinary form of
proposition, the subject is used in extension and the predicate in
intension. Let us illustrate the meaning of this by an example. If
someone were to say 'Cows are ruminants,' you would have a right to
ask him whether he meant 'all cows' or only 'some.' You would not by
so doing be asking for fresh information, but merely for a more
distinct explanation of the statement already made. The subject being
used in extension naturally assumes the form of the whole or part of a
class. But, if you were to ask the same person 'Do you mean that cows
are all the ruminants that there are, or only some of them?' he would
have a right to complain of the question, and might fairly reply, 'I
did not mean either one or the other; I was not thinking of ruminants
as a class. I wished merely to assert an attribute of cows; in fact, I
meant no more than that cows chew the cud.'

§ 282. Since therefore a predicate is not used in extension at all, it
cannot possibly be known whether it is used in its whole extent or
not.

§ 283. It would appear then that every predicate is necessarily
undistributed; and this consequence does follow in the case of
affirmative propositions.

§ 284. In a negative proposition, however, the predicate, though still
used in intension, must be regarded as distributed. This arises from
the nature of a negative proposition. For we must remember that in any
proposition, although the predicate be not meant in extension, it
always admits of being so read. Now we cannot exclude one class from
another without at the same time wholly excluding that other from the
former. To take an example, when we say 'No horses are ruminants,' the
meaning we really wish to convey is that no member of the class,
horse, has a particular attribute, namely, that of chewing the
cud. But the proposition admits of being read in another form, namely,
'That no member of the class, horse, is a member of the class,
ruminant.' For by excluding a class from the possession of a given
attribute, we inevitably exclude at the same time any class of things
which possess that attribute from the former class.

§ 285. The difference between the use of a predicate in an affirmative
and in a negative proposition may be illustrated to the eye as
follows. To say 'All A is B' may mean either that A is included in B
or that A and B are exactly co-extensive.

[Illustration]

§ 286. As we cannot be sure which of these two relations of A to B is
meant, the predicate B has to be reckoned undistributed, since a term
is held to be distributed only when we know that it is used in its
whole extent.

§ 287. To say 'No A is B,' however, is to say that A falls wholly
outside of B, which involves the consequence that B falls wholly
outside of A.

[Illustration]

§ 288. Let us now apply the same mode of illustration to the
particular forms of proposition.

§ 289. If I be taken in the strictly particular sense, there are, from
the point of view of extension, two things which may be meant when we
say 'Some A is B'--

(1) That A and B are two classes which overlap one another, that is
to say, have some members in common, e.g. 'Some cats are black.'

[Illustration]

(2) That B is wholly contained in A, which is an inverted way of
saying that all B is A, e.g. 'Some animals are men.'

[Illustration]

§ 290. Since we cannot be sure which of these two is meant, the
predicate is again reckoned undistributed.

§ 291. If on the other hand 1 be taken in an indefinite sense, so as
to admit the possibility of the universal being true, then the two
diagrams which have already been used for A must be extended to 1, in
addition to its own, together with the remarks which we made in
connection with them (§§ 285-6).

§ 292. Again, when we say 'Some A is not B,' we mean that some, if not
the whole of A, is excluded from the possession of the attribute B. In
either case the things which possess the attribute B are wholly
excluded either from a particular part or from the whole of A. The
predicate therefore is distributed.

[Illustration]

From the above considerations we elicit the following--

§ 293. Four Rules for the Distribution of Terms.

(1) All universal propositions distribute their subject.

(2) No particular propositions distribute their subject,

(3) All negative propositions distribute their predicate.

(4) No affirmative propositions distribute their predicate.


§ 294. The question of the distribution or non-distribution of the
subject turns upon the quantity of the proposition, whether universal
or particular; the question of the distribution or non-distribution of
the predicate turns upon the quality of the proposition, whether
affirmative or negative.




CHAPTER V.

_Of the Quantification of the Predicate._


§ 295. The rules that have been given for the distribution of terms,
together with the fourfold division of propositions into A, E, 1, 0,
are based on the assumption that it is the distribution or
non-distribution of the subject only that needs to be taken into
account in estimating the quantity of a proposition.

§ 296. But some logicians have maintained that the predicate, though
seldom quantified in expression, must always be quantified in
thought--in other words, that when we say, for instance, 'All A is B,'
we must mean either that 'All A is all B' or only that 'All A is some
B.'

§ 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 quantity of the predicate
as well as that of the subject be taken into account in the
classification 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 assigned 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 quantity '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 class,
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 class, man. What the proposition
means is that the class, man, is co-extensive with the class, rational
animal. The same meaning may be expressed intensively by saying that
the one class 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
analysis 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 classification 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 classification 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
classification is founded and the sense in which they were originally
intended.

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

(2) A Species is a smaller class 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 class-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
class.

An Inseparable Accident is one which belongs to all the members of a
class.

Blackness is a separable accident of man, an inseparable accident of
coals.

§ 320. The attributes which belong to anything may be distinguished
broadly under the two heads of essential and non-essential, or
accidental. By the essential attributes of anything are meant those
which are contained in, or which flow from, the definition. Now it may
be questioned whether there can, in the nature of things, be such a
thing as an inseparable accident. For if an attribute were found to
belong invariably to all the members of a class, we should suspect
that there was some causal connection between it and the attributes
which constitute the definition, that is, we should suspect the
attribute in question to be essential and not accidental. Nevertheless
the term 'inseparable accident' may be retained as a cloak for our
ignorance, whenever it is found that an attribute does, as a matter of
fact, belong to all the members of a class, without there being any
apparent reason why it should do so. It has been observed that animals
which have horns chew the cud. As no one can adduce any reason why
animals that have horns should chew the cud any more than animals
which have not, we may call the fact of chewing the cud an inseparable
accident of horned animals.

§ 321. The distinction between separable and inseparable accidents is
sometimes extended from classes to individuals.

An inseparable accident of an individual is one which belongs to him
at all times. A separable accident of an individual is one which
belongs to him at one time and not at another.

§ 322. It is an inseparable accident of an individual that he was born
at a certain place and on a certain date. It is a separable accident
of an individual that he resides at a certain place and is of a
certain age.

§ 323. There are some remarks which it may be well to make about the
above five terms before we pass on to investigate the principle upon
which the division is based.

§ 324. In the first place, it must of course be borne in mind that
genus and species are relative terms. No class in itself can be either
a genus or a species; it only becomes so in reference to some other
class, as standing to it in the relation of containing or contained.

§ 325. Again, the distinction between genus and difference on the one
hand and property on the other is wholly relative to an assumed
definition. When we say 'Man is an animal,' 'Man is rational,' 'Man is
progressive,' there is nothing in the nature of these statements
themselves to tell us that the predicate is genus, difference, or
property respectively. It is only by a tacit reference to the accepted
definition of man that this becomes evident to us, Similarly, we
cannot know beforehand that the fact of a triangle having three sides
is its difference, and the fact of its having three angles a
property. It is only when we assume the definition of a triangle as a
three-sided figure that the fact of its having three angles sinks into
a property. Had we chosen to define it, in accordance with its
etymological meaning, as a figure with three angles, its
three-sidedness would then have been a mere property, instead of being
the difference; for these two attributes are so connected together
that, whichever is postulated, the other will necessarily follow.

Why girls' books still build their dreams around home
CS Lewis built the Chronicles of Narnia around medieval cosmology, it is claimed

Letter: Gender roles in the Cinderella story

Doctors assure us that wherever you find an elderly, pompous old writer long past his prime you will find a bottle of scotch nearby. If only that were the case. Hilly hid mine after I fell up the stairs when I came home from the Garrick yesterday, and I've had to make do with a bottle of Blue Nun I found in the maid's parlour. Not that I am an alcoholic. Dipsomaniacs are a breed of the lower orders you meet on street corners: people like myself are bon viveurs who happen to like a drink. Or 12.

My primary observation is that drinking makes the daily grind of dealing with people so much easier. You drink a pint of whisky and become the life and soul of the party. You then start insulting people, before sweating heavily and wetting yourself involuntarily. You will usually find that everyone quickly avoids you, thereby relieving you of the need to make conversation. This is why I prefer to do much of my drinking at home. It saves so much time.

There are a great many drinks on the market - spirits, wines and beers - and I've probably drunk them all. Usually in some kind of combination with one another. Mixing cocktails is one of my favourite hobbies. Here's one I invented last week for my great sycophant, Christopher Hitchens.

The Hitch

One bottle of Babycham

One bottle of absinthe

Five shots of Angostura very bitters

Two tablespoons of bile

Two or three glasses of this tincture can give you a lifetime of self-satisfaction.

At some time you will probably be forced to invite people to your home and they may expect a drink. My advice is to offer them the cheapest tipple you can find; my local off-licence does a ghastly Mosel at 70p a bottle. I've never cared for even the best wines, and this should guarantee those poncing off you neither ask for top-ups nor stay long, thereby leaving you more money and time for the pub.

It is well known that only the very dullest of petit-bourgeois minds fail to over-imbibe on a daily basis, so I regard hangovers as a price worth paying for my brilliance. That said, I have found ways of coping with this metaphysical malaise. The first is to fuck someone; preferably somebody else's wife, but if your own is the only one around then she will do. The second is to read a book by that little shit Mart; it will either remind you you're not that bad a writer or give you some sleep.

The one downside to drinking is that it can make you fat. This is remedied by cutting out food entirely and drinking all spirits without mixers. My weight has gone down to 19st with this diet. There isn't much more to say, but as I'm being paid by the column I'd better repeat myself. And now that I'm dead, there's no harm in Bloomsbury repackaging the same material several times in the same collection.

I don't really like wine. Gin is for pansies, though a snifter with water doesn't go amiss. Liqueurs are best left to patent-shoed Wops. Or Americans. Champagne is an overrated girl's drink, though it can be drunk with any food; as such, it's a perfect breakfast drink because a scotch before 10am is very non-U.

I loathe pubs with loud music, but my utmost detestation is reserved for sanctimonious ex-topers. There's nothing worse than a man who doesn't drink. I once tried not drinking for several hours and my wives and mistresses said how dull it was that I was conscious and they were spared removing my soiled trousers from my bloated legs.

Whisky is my favourite tipple, though I recommend never giving it to a Welshman as it's wasted on someone with an IQ of less than 80. Have I mentioned that I'm partial to a Macallan? Gosh is that the time? Hilly's coming to change my IV drip before I fall unconscious again. The publisher can bloody well pad out the rest of the book with a pointless quiz without me.

Q: Who will buy this?

A: No one.

The digested read digested: The old pub bore.

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Declining genre spells gloom for publishers
Letter: Adam Phillips's analysis of Cinderella is interesting and, up to a point, plausible.