Deductive Logic

S >> St. George Stock >> Deductive Logic

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374. We may distinguish therefore between two kinds of definition,
namely,

(1) Final.

(2) Provisional.

375. A distinction is also observed between Real and Nominal
Definitions. Both of these explain the meaning of a term: but a real
definition further assumes the actual existence of the thing
defined. Thus the explanation of the term 'Centaur' would be a
nominal, that of 'horse' a real definition.

It is useless to assert, as is often done, that a nominal definition
explains the meaning of a term and a real definition the nature of a
thing; for, as we have seen already, the meaning of a term is whatever
we know of the nature of a thing.

376. It now remains to lay down certain rules for correct
definition.

377. The first rule that is commonly given is that a definition
should state the essential attributes of the thing defined. But this
amounts merely to saying that a definition should be a definition;
since it is only by the aid of definition that we can distinguish
between essential and non-essential among the common attributes
exhibited by a class of things. The rule however may be retained as a
material test of the soundness of a definition, in the sense that he
who seeks to define anything should fix upon its most important
attributes. To define man as a mammiferous animal having two hands, or
as a featherless biped, we feel to be absurd and incongruous, since
there is no reference to the most salient characteristic of man,
namely, his rationality. Nevertheless we cannot quarrel with these
definitions on formal, but only on material grounds. Again, if anyone
chose to define logic as the art of thinking, all we could say is that
we differ from him in opinion, as we think logic is more properly to
be regarded as the science of the laws of thought. But here also it is
on material grounds that we dissent from the definition.

378. Confining ourselves therefore to the sphere with which we are
properly concerned, we lay down the following

_Rules for Definition._

(1) A definition must be co-extensive with the term defined.

(2) A definition must not state attributes which imply one another.

(3) A definition must not contain the name defined, either directly
or by implication.

(4) A definition must be clearer than the term defined.

(5) A definition must not be negative, if it can be affirmative.

Briefly, a definition must be adequate (1), terse (2), clear (4); and
must not be tautologous (3), or, if it can be avoided, negative (5).

379. It is worth while to notice a slight ambiguity in the term
'definition' itself. Sometimes it is applied to the whole proposition
which expounds the meaning of the term; at other times it is confined
to the predicate of this proposition. Thus in stating the first four
rules we have used the term in the latter sense, and in stating the
fifth in the former.

380. We will now illustrate the force of the above rules by giving
examples of their violation.

Rule 1. Violations. A triangle is a figure with three equal sides.

A square is a four-sided figure having all its sides equal.

In the first instance the definition is less extensive than the term
defined, since it applies only to equilateral triangles. This fault
may be amended by decreasing the intension, which we do by eliminating
the reference to the equality of the sides.

In the second instance the definition is more extensive than the term
defined. We must accordingly increase the intension by adding a new
attribute 'and all its angles right angles.'

Rule 2. Violation. A triangle is a figure with three sides and three
angles.

One of the chief merits of a definition is to be terse, and this
definition is redundant, since what has three sides cannot but have
three angles.

Rule 3. Violations. A citizen is a person both of whose parents were
citizens.

Man is a human being.

Rule 4. Violations. A net is a reticulated fabric, decussated at
regular intervals.

Life is the definite combination of heterogeneous changes, both
simultaneous and successive, in correspondence with external
co-existences and sequences.

Rule 5. Violations. A mineral is that which is neither animal nor
vegetable.

Virtue is the absence of vice.

381. The object of definition being to explain what a thing is, this
object is evidently defeated, if we confine ourselves to saying what
it is not. But sometimes this is impossible to be avoided. For there
are many terms which, though positive in form, are privative in force.
These terms serve as a kind of residual heads under which to throw
everything within a given sphere, which does not exhibit certain
positive attributes. Of this unavoidably negative nature was the
definition which we give of 'accident,' which amounted merely to
saying that it was any attribute which was neither a difference nor a
property.

382. The violation of Rule 3, which guards against defining a thing
by itself, is technically known as 'circulus in definiendo,' or
defining in a circle. This rule is often apparently violated, without
being really so. Thus Euclid defines an acute-angled triangle as one
which has three acute angles. This seems a glaring violation of the
rule, but is perfectly correct in its context; for it has already been
explained what is meant by the terms 'triangle' and 'acute angle,' and
all that is now required is to distinguish the acute-angled triangle
from its cognate species. He might have said that an acute-angled
triangle is one which has neither a right angle nor an obtuse angle:
but rightly preferred to throw the same statement into a positive
form.

383. The violation of Rule 4 is known as 'ignotum per ignotius' or
'per aeque ignotum.' This rule also may seemingly be violated when it
is not really so. For a definition may be correct enough from a
special point of view, which, apart from that particular context,
would appear ridiculous. From the point of view of conic sections, it
is correct enough to define a triangle as that section of a cone which
is formed by a plane passing through the vertex perpendicularly to the
base, but this could not be expected to make things clearer to a
person who was inquiring for the first time into the meaning of the
word triangle. But a real violation of the fourth rule may arise, not
only from obscurity, but from the employment of ambiguous language or
metaphor. To say that 'temperance is a harmony of the soul' or that
'bread is the staff of life,' throws no real light upon the nature of
the definiend.

384. The material correctness of a definition is, as we have already
seen, a matter extraneous to formal logic. An acquaintance with the
attributes which terms imply involves material knowledge quite as much
as an acquaintance with the things they apply to; knowledge of the
intension and of the extension of terms is alike acquired by
experience. No names are such that their meaning is rendered evident
by the very constitution of our mental faculties; yet nothing short of
this would suffice to bring the material content of definition within
the province of formal logic.

CHAPTER VIII.

_Of Division._

385. To divide a term is to unfold its extension, that is, to set
forth the things of which it is a name.

386. But as in definition we need not completely unfold the
intension of a term, so in division we must not completely unfold its
extension.

387. Completely to unfold the extension of a term would involve
stating all the individual objects to which the name applies, a thing
which would be impossible in the case of most common terms. When it is
done, it is called Enumeration. To reckon up all the months of the
year from January to December would be an enumeration, and not a
division, of the term 'month.'

388. Logical division always stops short at classes. It may be
defined as the statement of the various classes of things that can be
called by a common name. Technically we may say that it consists in
breaking up a genus into its component species.

389. Since division thus starts with a class and ends with classes,
it is clear that it is only common terms which admit of division, and
also that the members of the division must themselves be common terms.

390. An 'individual' is so called as not admitting of logical
division. We may divide the term 'cow' into classes, as Jersey,
Devonshire, &c., to which the name 'cow' will still be applicable, but
the parts of an individual cow are no longer called by the name of the
whole, but are known as beefsteaks, briskets, &c.

391. In dividing a term the first requisite is to fix upon some
point wherein certain members of the class differ from others. The
point thus selected is called the Fundamentum Divisionis or Basis of
the Division.

392. The basis of the division will of course differ according to
the purpose in hand, and the same term will admit of being divided on
a number of different principles. Thus we may divide the term 'man,'
on the basis of colour, into white, black, brown, red, and yellow; or,
on the basis of locality, into Europeans, Asiatics, Africans,
Americans, Australians, New Zealanders, and Polynesians; or again, on
a very different principle, into men of nervous, sanguine, bilious,
lymphatic and mixed temperaments.

393. The term required to be divided is known as the Totum Divisum
or Divided Whole. It might also be called the Dividend.

394. The classes into which the dividend is split up are called the
Membra Dividentia, or Dividing Members.

395. Only two rules need be given for division--

(1) The division must be conducted on a single basis.

(2) The dividing members must be coextensive with the divided whole.

396. More briefly, we may put the same thing thus--There must be no
cross-division (1) and the division must be exhaustive (2).

397. The rule, which is commonly given, that each dividing member
must be a common term, is already provided for under our definition of
the process.

398. The rule that the dividend must be predicable of each of the
dividing members is contained in our second rule; since, if there were
any term of which the dividend were not predicable, it would be
impossible for the dividing members to be exactly coextensive with it.
It would not do, for instance, to introduce mules and donkeys into a
division of the term horse.

399. Another rule, which is sometimes given, namely, that the
constituent species must exclude one another, is a consequence of our
first; for, if the division be conducted on a single principle, the
constituent species must exclude one another. The converse, however,
does not hold true. We may have a division consisting of mutually
exclusive members, which yet involves a mixture of different bases,
e.g. if we were to divide triangle into scalene, isosceles and
equiangular. This happens because two distinct attributes may be found
in invariable conjunction.

400. There is no better test, however, of the soundness of a
division than to try whether the species overlap, that is to say,
whether there are any individuals that would fall into two or more of
the classes. When this is found to be the case, we may be sure that we
have mixed two or more different fundamenta divisionis. If man, for
instance, were to be divided into European, American, Aryan, and
Semitic, the species would overlap; for both Europe and America
contain inhabitants of Aryan and Semitic origin. We have here members
of a division based on locality mixed up with members of another
division, which is based on race as indicated by language.

401. The classes which are arrived at by an act of division may
themselves be divided into smaller classes. This further process is
called Subdivision.

402. Let it be noticed that Rule 1 applies only to a single act of
division. The moment that we begin to subdivide we not only may, but
must, adopt a new basis of division; since the old one has, 'ex
hypothesi,' been exhausted. Thus, having divided men according to the
colour of their skins, if we wish to subdivide any of the classes, we
must look out for some fresh attribute wherein some men of the same
complexion differ from others, e.g. we might divide black men into
woolly-haired blacks, such as the Negroes, and straight-haired blacks,
like the natives of Australia.

403. We will now take an instance of division and
subdivision, with a view to illustrating some of the
technical terms which are used in connection with the
process. Keeping closely to our proper subject, we will
select as an instance a division of the products of thought,
which it is the province of logic to investigate.

Product of thought
_______________|____________________________
| | |
Term Proposition Inference
____|___ ______|_____ _____|______
| | | | | |
Singular Common Universal Particular Immediate Mediate
___|___ ___|___
| | | |
A E I O

Here we have first a threefold division of the products of thought
based on their comparative complexity. The first two of these, namely,
the term and the proposition, are then subdivided on the basis of
their respective quantities. In the case of inference the basis of the
division is again the degree of complexity. The subdivision of the
proposition is carried a step further than that of the others. Having
exhausted our old basis of quantity, we take a new attribute, namely,
quality, on which to found the next step of subdivision.

404. Now in such a scheme of division and subdivision as the
foregoing, the highest class taken is known as the Summum Genus. Thus
the summum genus is the same thing as the divided whole, viewed in a
different relation. The term which is called the divided whole with
reference to a single act of division, is called the summum genus
whenever subdivision has taken place.

405. The classes at which the division stops, that is, any which are
not subdivided, are known as the Infimae Species.

406. All classes intermediate between the summum genus and the
infimae species are called Subaltern Genera or Subaltern Species,
according to the way they are looked at, being genera in relation to
the classes below them and species in relation to the classes above
them.

407. Any classes which fall immediately under the same genus are
called Cognate Species, e.g. singular and common terms are cognate
species of term.

408. The classes under which any lower class successively falls are
called Cognate Genera. The relation of cognate species to one another
is like that of children of the same parents, whereas cognate genera
resemble a line of ancestry.

409. The Specific Difference of anything is the attribute or
attributes which distinguish it from its cognate species. Thus the
specific difference of a universal proposition is that the predicate
is known to apply to the whole of the subject. A specific difference
is said to constitute the species.

410. The specific difference of a higher class becomes a Generic
Difference with respect to the class below it. A generic difference
then may be said to be the distinguishing attribute of the whole class
to which a given species belongs. The generic difference is common to
species that are cognate to one another, whereas the specific
difference is peculiar to each. It is the generic difference of an A
proposition that it is universal, the specific difference that it is
affirmative.

411. The same distinction is observed between the specific and
generic properties of a thing. A Specific Property is an attribute
which flows from the difference of a thing itself; a Generic Property
is an attribute which flows from the difference of the genus to which
the thing belongs. It is a specific property of an E proposition that
its predicate is distributed, a generic property that its contrary
cannot be true along with it ( 465); for this last characteristic
flows from the nature of the universal proposition generally.

412. It now remains to say a few words as to the place in logic of
the process of division. Since the attributes in which members of the
same class differ from one another cannot possibly be indicated by
their common name, they must be sought for by the aid of experience;
or, to put the same thing in other words, since all the infimae
species are alike contained under the summum genus, their distinctive
attributes can be no more than separable accidents when viewed in
relation to the summum genus. Hence division, being always founded on
the possession or non-possession of accidental attributes, seems to
lie wholly outside the sphere of formal logic. This however is not
quite the case, for, in virtue of the Law of Excluded Middle, there is
always open to us, independently of experience, a hypothetical
division by dichotomy. By dichotomy is meant a division into two
classes by a pair of contradictory terms, e.g. a division of the
class, man, into white and not-white. Now we cannot know,
independently of experience, that any members of the class, man,
possess whiteness; but we may be quite sure, independently of all
experience, that men are either white or not. Hence division by
dichotomy comes strictly within the province of formal logic. Only it
must be noticed that both sides of the division must be hypothetical.
For experience alone can tell us, on the one hand, that there are any
men that are white, and on the other, that there are any but white
men.

413. What we call a division on a single basis is in reality the
compressed result of a scheme of division and subdivision by
dichotomy, in which a fresh principle has been introduced at every
step. Thus when we divide men, on the basis of colour, into white,
black, brown, red and yellow, we may be held to have first divided men
into white and not-white, and then to have subdivided the men that are
not-white into black and not-black, and so on. From the strictly
formal point of view this division can only be represented as
follows--

Men
___________________|_____
| |
White (if any) Not-white (if any)
_________________|_____
| |
Black (if any) Not-black (if any)
__________________|____
| |
Brown (if any) Not-brown (if any)
____________________|____
| |
Red (if any) Not-red (if any).

414. Formal correctness requires that the last term in such a series
should be negative. We have here to keep the term 'not-red' open, to
cover any blue or green men that might turn up. It is only experience
that enables us to substitute the positive term 'yellow' for
'not-red,' since we know as a matter of fact that there are no men but
those of the five colours given in the original division.

415. Any correct logical division always admits of being arrived at
by the longer process of division and subdivision by dichotomy. For
instance, the term quadrilateral, or four-sided rectilinear figure, is
correctly divided into square, oblong, rhombus, rhomboid and
trapezium. The steps of which this division consists are as follows--

Quadrilateral
__________|_________
| |
Parallelogram Trapezium
_____|_____________________
| |
Rectangle Non-rectangle
___|___ _____|_____
| | | |
Square Oblong Rhombus Rhomboid.

416. In reckoning up the infimae species in such a scheme, we must
of course be careful not to include any class which has been already
subdivided; but no harm is done by mixing an undivided class, like
trapezium, with the subdivisions of its cognate species.

417. The two processes of definition and division are intimately
connected with one another. Every definition suggests a division by
dichotomy, and every division supplies us at once with a complete
definition of all its members.

418. Definition itself, so far as concerns its content, is, as we
have already seen, extraneous to formal logic: but when once we have
elicited a genus and difference out of the material elements of
thought, we are enabled, without any further appeal to experience, to
base thereon a division by dichotomy. Thus when man has been defined
as a rational animal, we have at once suggested to us a division of
animal into rational and irrational.

419. Again, the addition of the attributes, rational and irrational
respectively, to the common genus, animal, ipso facto supplies us with
definitions of the species, man and brute. Similarly, when we
subdivided rectangle into square and oblong on the basis of the
equality or inequality of the adjacent sides, we were by so doing
supplied with a definition both of square and oblong--'A square is a
rectangle having all its sides equal,' and 'An oblong is a rectangle
which has only its opposite sides equal.'

420. The definition of a square just given amounts to the same thing
as Euclid's definition, but it complies with a rule which has value as
a matter of method, namely, that the definition should state the
Proximate Genus of the thing defined.

421. Since definition and division are concerned with the intension
and extension of terms, they are commonly treated of under the first
part of logic: but as the treatment of the subject implies a
familiarity with the Heads of Predicables, which in their turn imply
the proposition, it seems more desirable to deal with them under the
second.

422. We have already had occasion to distinguish division from
Enumeration. The latter is the statement of the individual things to
which a name applies. In enumeration, as in division, the wider term
is predicable of each of the narrower ones.

423. Partition is the mapping out of a physical whole into its
component parts, as when we say that a tree consists of roots, stem,
and branches. In a partition the name of the whole is not predicable
of each of the parts.

424. Distinction is the separation from one another of the various
meanings of an equivocal term. The term distinguished is predicable
indeed of each of the members, but of each in a different sense. An
equivocal term is in fact not one but several terms, as would quickly
appear, if we were to use definitions in place of names.

425. We have seen that a logical whole is a genus viewed in relation
to its underlying species. From this must be distinguished a
metaphysical whole, which is a substance viewed in relation to its
attributes, or a class regarded in the same way. Logically, man is a
part of the class, animal; metaphysically, animal is contained in
man. Thus a logical whole is a whole in extension, while a
metaphysical whole is a whole in intension. From the former point of
view species is contained in genus; from the latter genus is contained
in species.

PART III.--OF INFERENCES.

CHAPTER I.

_Of Inferences in General_.

426. To infer is to arrive at some truth, not by direct experience,
but as a consequence of some truth or truths already known. If we see
a charred circle on the grass, we infer that somebody has been
lighting a fire there, though we have not seen anyone do it. This
conclusion is arrived at in consequence of our previous experience of
the effects of fire.

427. The term Inference is used both for a process and for a product
of thought.

As a process inference may be defined as the passage of the mind from
one or more propositions to another.

As a product of thought inference may be loosely declared to be the
result of comparing propositions.

428. Every inference consists of two parts--

(1) the truth or truths already known;

(2) the truth which we arrive at therefrom.

The former is called the Antecedent, the latter the Consequent. But
this use of the terms 'antecedent' and 'consequent' must be carefully
distinguished from the use to which they were put previously, to
denote the two parts of a complex proposition.

429. Strictly speaking, the term inference, as applied to a product
of thought, includes both the antecedent and consequent: but it is
often used for the consequent to the exclusion of the
antecedent. Thus, when we have stated our premisses, we say quite
naturally, 'And the inference I draw is so and so.'

430. Inferences are either Inductive or Deductive. In induction we
proceed from the less to the more general; in deduction from the more
to the less general, or, at all events, to a truth of not greater
generality than the one from which we started. In the former we work
up to general principles; in the latter we work down from them, and
elicit the particulars which they contain.

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