LOGICAL DIVISION AND CLASSIFICATION.
=1. NATURE OF LOGICAL DIVISION.=
The term _genus_ is used for any class name which stands for two or
more subordinate classes while the term _species_ is made to stand for
any one of the subordinate classes.
The proximate genus of any species is the _next_ class above. For
example the proximate genus of man is biped, not animal.
_Logical division is the process of separating a proximate genus into
its co-ordinate species._
ILLUSTRATIONS:
_Genus_ _Species_
{ Fixed stars
{ Planets
(1) Heavenly bodies { Satellites
{ Comets
{ Meteors
{ Nebulae
{ Leptocardians
{ Fishes
(2) Vertebrates { Amphibians
{ Reptiles
{ Birds
{ Mammals
{ Caucasian
{ Mongolian
(3) Man { Malay
{ Ethiopian
{ American Indian
{ Monarchy
(4) Government { Aristocracy
{ Democracy
=2. LOGICAL DIVISION DISTINGUISHED FROM ENUMERATION.=
When the genus is separated at once into _individual objects_ the
process is not logical division, but simple enumeration. Logical
division implies a separating into smaller class terms, each term being
a genus of still smaller subdivisions. This process may be continued
till the last division gives individuals as species. Enumeration takes
place when the first subdivision results in a list of individuals. To
illustrate:
_Logical Division._
{ Science teacher
Teacher { Mathematics teacher
{ English teacher
{ Modern language teacher
_Enumeration._
{ John J. Brown
Teacher { H. G. White
{ Mary Jones
{ Alice Smith
=3. LOGICAL DIVISION AS PARTITION.=
_Partition is the process of separating an individual thing into its
parts._
The partition is quantitative or mathematical when the separation is
in terms of space or time, but when otherwise the partition becomes
qualitative or logical. Or to put it in another way, the partition
is mathematical when the separation gives parts and logical when the
separation gives ingredients.
To illustrate:
{ { branches
{ quantitative { leaves
{ or { roots
{ (mathematical) { trunk
(1) Tree {
{ qualitative { woody fibre
{ or { capillary attraction
{ (logical) { sap
{ { chlorophyll
{ quantitative { roof
{ or { frame-work
{ (mathematical) { foundation
(2) House {
{ qualitative { wood
{ or { iron
{ (logical) { stone
{ { plaster
An easy way to determine that the separation involves logical division
proper and not partition is to affirm the connection between a class
and a sub-class. To wit: A man is a biped; a square is a rectangle;
a Caucasian is a man, etc. If such an affirmation cannot be made then
the separation involved is not properly logical division but probably
partition. For example it cannot be said that a _roof is a house_, or
that _sap is a tree_. It is seen, then, that a logical division of any
genus may be summarized in the form of a series of judgments of which
a species is the subject and the genus is the predicate. For example,
by a logical division quadrilaterals may be divided into trapeziums,
trapezoids and parallelograms; this process may then be summarized in
a series of three judgments: (1) A trapezium is a quadrilateral; (2) A
trapezoid is a quadrilateral; (3) A parallelogram is a quadrilateral.
=4. RULES OF LOGICAL DIVISION.=
When the logical division of a genus is under consideration there are
four rules which should be observed.
FIRST RULE. _There must be but one principle of division (fundamentum
divisionis)._ To divide mankind into white man, Australian, yellow man,
African and red man is a violation of this rule as the _two_ principles
of color and geographical location are involved. A division in which
more than one principle is used is sometimes referred to as _cross
division_ because the various species cross each other. For example in
the foregoing there are many white men who are Australians.
This rule applies only to one division. Where there is a series
of divisions a new principle may be employed in each division. For
example, in dividing triangles into scalene, isosceles and equilateral,
the equality of sides is the principle involved, but, in subdividing
isosceles triangles into right angled and oblique angled, the principle
employed concerns the nature of the angle.
SECOND RULE. _The co-ordinate species must be mutually exclusive._
There must be no overlapping. The illustration given in the first
rule is likewise a violation of this rule. Another example in which
this second rule is not obeyed may be found in most geometries where
triangles are divided into scalene, isosceles and equilateral. Here
the second and third classes are not mutually exclusive since all
equilateral triangles are isosceles according to the usual definition,
“An isosceles triangle is a triangle having two equal sides.” All
equilateral triangles have _two_ equal sides.
THIRD RULE. _The division must be exhaustive._ That is, the species
taken together must equal the whole genus. The sum of the species must
be co-extensive with the genus.
Dividing man into Caucasian, Ethiopian and Mongolian would be a
violation of this rule, as there are at least two other species of man,
Malay and American Indian.
A distinction should be made between an exhaustive division and
a complete division as the latter is not a logical requirement.
To divide government into monarchy, aristocracy and democracy is
exhaustive but incomplete. Exhaustive because there is no other kind
of government, all the species are included; but incomplete in that
monarchy may be divided into absolute and limited; democracy into pure
and representative.
FOURTH RULE. _The division must proceed from the proximate genus to the
immediate species._ There should be no sudden jumps from a high genus
to a low species. The division must be gradual and continuous; step by
step. To divide government into limited monarchy, absolute monarchy,
pure democracy and representative democracy would be a violation of
this rule, as government is the proximate genus of monarchy, not of
limited monarchy, therefore one step has been omitted. Such an omission
involves a step from grandfather to grandchild, so to speak, the
generation of father having been left out.
A violation of this rule is most insidious when some of the species
of a subdivision are immediate while others are not. To wit: dividing
government into monarchy, aristocracy, pure democracy and republic,
or dividing quadrilaterals into trapeziums, trapezoids, rectangles,
squares, rhomboids and rhombs.
=5. DICHOTOMY.=
Dichotomy comes from the Greek, meaning _to cut in two_. _Dichotomy
is a continual division of a genus into two species which are
contradictory in nature._
Contradictory terms are such as admit of no middle ground. They divide
the whole universe of thought into two classes. For example, honest and
not-honest, pure and impure, perfect and imperfect, are contradictory
terms. Dichotomy thus affords an easy opportunity for an exhaustive
division as in the use of contradictories nothing in the universe need
be omitted.
An historical illustration of dichotomy is the “Tree of Porphyry” named
after Porphyrius, a Neo-Platonic philosopher of the third century.
_Tree of Porphyry._
Substance.
/ \
/ \
Corporeal Incorporeal
\
\
Body
/ \
/ \
Animate Inanimate
\
\
Living Being
/ \
/ \
Sensible Insensible
\
\
Animal
/ \
/ \
Rational Irrational
\
\
Man
/ | \
/ | \
/ | \
Socrates Plato Other Men
This kind of division is not altogether satisfactory as the negative
side is too indefinite. On the other hand, if both subdivisions
are made positive then there is danger of making the opposing terms
contrary rather than contradictory. This, of course, would be a
serious logical fallacy, as contrary terms admit of middle ground while
contradictory terms give no choice, it is either the one or the other.
The use of dichotomy becomes evident in situations where new
and unexpected discoveries may be made. Without disturbing the
classification the new species may be appended to the negative side
of the division. The following illustrates:
Vertebrates
┌───────┴──────────┐
│ │
Leptocardians Not-leptocardians
┌──────────┴───┐
│ │
Fish Not-fish
┌───────────┴────┐
│ │
Amphibians Not-amphibians
┌──────────────┴────┐
│ │
Reptiles Not-reptiles
┌────────────────┴───┐
│ │
Birds Not-birds
┌──────────────────┴──┐
│ │
Mammals Not-mammals
│
_The New Species_
=6. CLASSIFICATION――COMPARED WITH DIVISION.=
_Classification is the process of grouping notions according to their
resemblances or connections._
So far as results are concerned there is no difference between
logical division and classification. Both processes may give us the
same orderly scheme of heads and subheads. The difference lies in
the process itself. Division is _deductive_ in nature as it proceeds
from the more general genus to the less general species. While
classification is _inductive_ as it groups the less general species
under the more general genus. Division differentiates unity into
multiplicity, while classification reduces multiplicity to unity. It
follows that the one is the inverse of the other. The difference in the
mode of procedure may be illustrated by using the common classification
or division of triangles. For example:
Without any knowledge of the kinds of triangles the student discovers
by examining the various shapes of many triangles that there is a
group in which _none_ of the sides are equal. For the lack of a better
name he terms these non-equilateral (scalene). Further observation
discloses another group in which two of the sides are equal. These he
names bi-equilateral (isosceles). Finally a third group is designated
as tri-equilateral (equilateral). This process is classification.
Division would consist in separating the genus triangle into the three
kinds――scalene, isosceles, equilateral.
=7. KINDS OF CLASSIFICATION――ARTIFICIAL AND NATURAL.=
_An artificial classification is one in which the grouping is made on
the basis of some arbitrary connection._ Cataloguing alphabetically the
books in a library illustrates this kind of classification. Likewise
the arrangement of the names in a directory or a telephone book. The
connecting mark being the initial letter of the title or name. The
reason why Mills and Meyers are put in the same group is that both
names happen to commence with the letter _M_.
Artificial classifications are resorted to for some special purpose,
designed by man, not by nature. Consequently artificial classifications
are sometimes called _special_ or _working_ classifications.
_A natural classification is one in which the grouping is made on the
basis of some inherent mark of resemblance._
Classifications in animal and plant life are the best illustrations
of this kind. Such classifications are suggested by nature and not by
man, and may, therefore, be called _general_ or scientific. The main
aim of natural classification is to derive general truths and arrange
knowledge so that it may be easily remembered.
=8. TWO RULES OF CLASSIFICATION.=
The rules of logical division are applicable in the making of a logical
classification. In addition to these an artificial classification
should be made to conform to the one rule: _The classification must be
appropriate to the purpose in hand._ Likewise a natural classification
should be made to conform to the rule: _Every classification should
afford opportunity for the greatest possible number of general
assertions._
=9. USE OF DIVISION AND CLASSIFICATION IN THE SCHOOL ROOM.=
It has been stated that classification and division aim at the
same result. Classification reduces multiplicity to unity while
division differentiates unity into multiplicity. In short, division is
_deductive_ while classification is _inductive_ in mode of procedure.
Therefore, classification should be used in those situations which
call for induction and division in cases where deduction is the better
method.
Speaking generally, classification should be used with small children
when the essential thing is to present the concrete facts with a view
of leading the children to discover for themselves the truths contained
therein.
With older pupils division may be used, if the purpose is to set in
order facts which are already known.
=10. TOPICAL OUTLINE.=
LOGICAL DIVISION AND CLASSIFICATION.
(1) Nature of Logical Division.
Genus――species.
Illustrations.
(2) Logical Division Distinguished from Enumeration.
Illustrations.
(3) Logical Division and Partition.
Quantitative――qualitative.
How summarized.
(4) Four Rules of Logical Division.
(1) One principle――cross division.
(2) Mutually exclusive.
(3) Exhaustive――complete.
(4) Immediate species.
(5) Dichotomy.
Contradictory terms.
Tree of Porphyry.
Use illustrated.
(6) Classification Compared with Division.
(7) Kinds. Artificial――Natural.
(8) Two Rules of Classification.
(1) Appropriate.
(2) Afford many Assertions.
(9) Use of Division and Classification.
=11. SUMMARY.=
(1) Logical division is the process of separating a proximate genus
into its co-ordinate species.
(2) The first subdivision in a logical division gives class terms,
while the first subdivision in an enumeration gives individual objects.
(3) Partition is the process of separating an individual thing into
its parts. These parts may be either quantitative or qualitative.
A logical division of any genus may be summarized in a series of
judgments of which a species is the subject and the genus is the
predicate.
(4) The four rules of logical division are: the division must (1) be
based on one principle, (2) have species mutually exclusive, (3) be
exhaustive and (4) proceed from proximate genus to immediate species.
A violation of the first rule gives a cross division.
Exhaustive division is easily confused with a complete or finished
division.
(5) Dichotomy is a continual division of a genus into two species which
are contradictory in nature.
An historical illustration of dichotomy is the Tree of Porphyry.
Dichotomy is of service in the field of new and unexpected discoveries.
(6) Classification is the process of grouping notions according to
their resemblances or connections.
Classification is inductive in nature, division deductive.
Classification unifies, division differentiates.
(7) An artificial classification is made on the basis of some
arbitrary connection; a natural classification, on some inherent mark
of resemblance.
(8) The rules of logical division are applicable in any classification.
In addition to these a classification should (1) be appropriate and
(2) afford opportunity for the greatest possible number of assertions.
(9) Classification should be the mode of procedure in the lower grades,
division in the higher grades.
=12. REVIEW QUESTIONS.=
(1) Define and illustrate logical division.
(2) What is the meaning of proximate genus?
(3) How does logical division differ from enumeration? Illustrate.
(4) Distinguish between logical division, and physical division or
partition.
(5) Illustrate a quantitative partition; a qualitative partition.
(6) Illustrate how a logical division may be summarized in the form
of a series of judgments.
(7) State and explain the rules of logical division.
(8) State the rules violated in the following divisions, explaining
in full:
{ Primary
{ Secondary
(1) Education { Collegiate
{ Technical
{ Scientific
{ Professional
{ Infancy
(2) Life { Childhood
{ Youth
{ Old age
{ Caucasian
{ Ethiopian
(3) Man { Malay
{ Mongolian
{ American
{ Cement
{ Frame
(4) Buildings { Stone
{ Dwellings
{ Barns
{ Churches
(9) Show the difference between contradictory and opposite terms.
(10) Define dichotomy.
(11) Illustrate the Tree of Porphyry and indicate its use to
scientists.
(12) Illustrate the difference between classification and division.
(13) Why should classification be the mode of procedure when dealing
with immature minds?
(14) Illustrate the difference between an artificial and a natural
classification.
(15) State and explain the two rules of classification.
(16) Show which of the following divisions are logical and which are
not:
(1) The manifestations of the mind into knowing, thinking and
feeling.
(2) Books into mathematical and non-mathematical.
(3) Students into those who are industrious, athletic and
shiftless.
(4) Flowers into roses, carnations and lilies.
(5) Planets into those which are larger than the earth and
those which are smaller.
=13. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=
(1) Analyze by mathematical partition the terms world, plant, book.
(2) Analyze by logical partition the terms granite, water, air.
(3) What rule is violated if the logical division is applied to the
genus rather than the proximate genus?
(4) Divide logically the following terms: school, religion, book,
vegetable, life.
(5) “Each new subdivision may adopt a new principle of division.”
Illustrate this.
(6) Explain and illustrate the meaning of the terms super-ordinate,
sub-ordinate and co-ordinate.
(7) Define and illustrate metaphysical division and verbal division.
(8) Give a definition of an isosceles triangle which will make
logical the division of triangles into scalene, isosceles and
equilateral.
(9) “The evolution of all truth develops progressively through three
stages.” “The first is the thesis; the second is the antithesis;
the third is the synthesis.” Explain this in terms of trichotomy.
(10) Illustrate the difference between a complete division and an
exhaustive division.
(11) Show in what ways, if any, the following divisions violate the
rules of logical division.
{ 1 Memory (History)
Human Learning { 2 Imagination (Poetry)
(by Bacon) { 3 Reason (Philosophy)
{ or (the Sciences)
{ 1 Mathematics
{ 2 Astronomy
Sciences { 3 Physics
(by Comte) { 4 Chemistry
{ 5 Biology
{ 6 Sociology
{ 7 Morals