FIRST NOTIONS
                                   OF
                                 LOGIC
                 (PREPARATORY TO THE STUDY OF GEOMETRY)


                                   BY

                          AUGUSTUS DE MORGAN,

                     OF TRINITY COLLEGE, CAMBRIDGE,
        PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.

  The root of all the mischief in the sciences, is this; that falsely
  magnifying and admiring the powers of the mind, we seek not its real
                             helps.—BACON.


                                LONDON:

                     PRINTED FOR TAYLOR AND WALTON,

           BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE.

                         28 UPPER GOWER STREET.

                             M.DCCC.XXXIX.




⁂ This Tract contains no more than the author has found, from
experience, to be much wanted by students who are commencing with
Euclid. It will ultimately form an Appendix to his Treatise on
Arithmetic.

The author would not, by any means, in presenting the minimum necessary
for a particular purpose, be held to imply that he has given enough of
the subject for all the ends of education. He has long regretted the
neglect of logic; a science, the study of which would shew many of its
opponents that the light esteem in which they hold it arises from those
habits of inference which thrive best in its absence. He strongly
recommends any student to whom this tract may be the first introduction
of the subject, to pursue it to a much greater extent.

_University College, Jan, 8, 1839._


                    LONDON:—PRINTED BY JAMES MOYES,
                    Castle Street, Leicester Square.




                             FIRST NOTIONS
                                   OF
                                 LOGIC.


What we here mean by Logic is the examination of that part of reasoning
which depends upon the manner in which inferences are formed, and the
investigation of general maxims and rules for constructing arguments, so
that the conclusion may contain no inaccuracy which was not previously
asserted in the premises. It has nothing to do with the truth of the
facts, opinions, or presumptions, from which an inference is derived;
but simply takes care that the inference shall certainly be true, if the
premises be true. Thus, when we say that all men will die, and that all
men are rational beings, and thence infer that some rational beings will
die, the _logical_ truth of this sentence is the same whether it be true
or false that men are mortal and rational. This logical truth depends
upon the structure of the sentence, and not on the particular matters
spoken of. Thus,

                 Instead of,                         Write,
              All men will die.                   Every A is B.
         All men are rational beings.             Every A is C.
   Therefore some rational beings will die. Therefore some Cs are Bs.

The second of these is the same proposition, logically considered, as
the first; the consequence in both is virtually contained in, and
rightly inferred from, the premises. Whether the premises be true or
false, is not a question of logic, but of morals, philosophy, history,
or any other knowledge to which their subject-matter belongs: the
question of logic is, does the conclusion certainly follow if the
premises be true?

Every act of reasoning must mainly consist in comparing together
different things, and either finding out, or recalling from previous
knowledge, the points in which they resemble or differ from each other.
That particular part of reasoning which is called _inference_, consists
in the comparison of several and different things with one and the same
other thing; and ascertaining the resemblances, or differences, of the
several things, by means of the points in which they resemble, or differ
from, the thing with which all are compared.

There must then be some propositions already obtained before any
inference can be drawn. All propositions are either assertions or
denials, and are thus divided into _affirmative_ and _negative_. Thus, A
is B, and A is not B, are the two forms to which all propositions may be
reduced. These are, for our present purpose, the most simple forms;
though it will frequently happen that much circumlocution is needed to
reduce propositions to them. Thus, suppose the following assertion, ‘If
he should come to-morrow, he will probably stay till Monday’; how is
this to be reduced to the form A is B? There is evidently something
spoken of, something said of it, and an affirmative connexion between
them. Something, if it happen, that is, the happening of something,
makes the happening of another something probable; or is one of the
things which render the happening of the second thing probable.

                 A                │is│               B


  The happening of his arrival    │  │an event from which it may be
    to-morrow                     │is│  inferred as probable that he
                                  │  │  will stay till Monday.

The forms of language will allow the manner of asserting to be varied in
a great number of ways; but the reduction to the preceding form is
always possible. Thus, ‘so he said’ is an affirmation, reducible as
follows:

  What you have just said (or     │is│the thing which he said.
    whatever else ‘so’ refers to) │  │

By changing ‘is’ into ‘is not,’ we make a negative proposition; but care
must always be taken to ascertain whether a proposition which appears
negative is really so. The principal danger is that of confounding a
proposition which is negative with another which is affirmative of
something requiring a negative to describe it. Thus ‘he resembles the
man who was not in the room,’ is affirmative, and must not be confounded
with ‘he does not resemble the man who was in the room.’ Again, ‘if he
should come to-morrow, it is probable he will not stay till Monday,’
does not mean the simple denial of the preceding proposition, but the
affirmation of the directly opposite proposition. It is,

                 A                │is│               B


  The happening of his arrival    │  │an event from which it may be
    to-morrow,                    │is│  inferred to be _im_probable
                                  │  │  that he will stay till Monday,

whereas the following,

 The happening of his arrival  │        │an event from which it may be
   to-morrow,                  │is _not_│  inferred as probable that he
                               │        │  will stay till Monday,

would be expressed thus: ‘If he should come to-morrow, that is no reason
why he should stay till Monday.’

Moreover, the negative words not, no, &c., have two kinds of meaning
which must be carefully distinguished. Sometimes they deny, and nothing
more: sometimes they are used to affirm the direct contrary. In cases
which offer but two alternatives, one of which is necessary, these
amount to the same thing, since the denial of one, and the affirmation
of the other, are obviously equivalent propositions. In many idioms of
conversation, the negative implies affirmation of the contrary in cases
which offer not only alternatives, but degrees of alternatives. Thus, to
the question, ‘Is he tall?’ the simple answer, ‘No,’ most frequently
means that he is the contrary of tall, or considerably under the
average. But it must be remembered, that, in all logical reasoning, the
negation is simply negation, and nothing more, never implying
affirmation of the contrary.

The common proposition that two negatives make an affirmative, is true
only upon the supposition that there are but two possible things, one of
which is denied. Grant that a man must be either able or unable to do a
particular thing, and then _not unable_ and able are the same things.
But if we suppose various degrees of performance, and therefore degrees
of ability, it is false, in the common sense of the words, that two
negatives make an affirmative. Thus, it would be erroneous to say, ‘John
is able to translate Virgil, and Thomas is not unable; therefore, what
John can do Thomas can do,’ for it is evident that the premises mean
that John is so near to the best sort of translation that an affirmation
of his ability may be made, while Thomas is considerably lower than
John, but not so near to absolute deficiency that his ability may be
altogether denied. It will generally be found that two negatives imply
an affirmative of a weaker degree than the positive affirmation.

Each of the propositions, ‘A is B,’ and ‘A is not B,’ may be subdivided
into two species: the _universal_, in which every possible case is
included; and the _particular_, in which it is not meant to be asserted
that the affirmation or negation is universal. The four species of
propositions are then as follows, each being marked with the letter by
which writers on logic have always distinguished it.

               A _Universal Affirmative_  Every A is    B
               E _Universal Negative_     No A is       B
               I _Particular Affirmative_ Some A is     B
               O _Particular Negative_    Some A is not B

In common conversation the affirmation of a part is meant to imply the
denial of the remainder. Thus, by ‘some of the apples are ripe,’ it is
always intended to signify that some are not ripe. This is not the case
in logical language, but every proposition is intended to make its
amount of affirmation or denial, and no more. When we say, ‘Some A is
B,’ or, more grammatically, ‘Some As are Bs,’ we do not mean to imply
that some are not: this may or may not be. Again, the word some means,
‘one or more, possibly all.’ The following table will shew the bearing
of each proposition on the rest.

 _Every A is B_ affirms and contains _Some A is B_ and│_No A is B_
   denies                                             │_Some A is not B_

 _No A is B_ affirms and contains _Some A is not B_   │_Every A is B_
   and denies                                         │_Some A is B_

 _Some A is B_ does not     │_Every A is B_   │but denies _No A is B_
   contradict               │_Some A is not B_│

 _Some A is not B_ does not │_No A is B_      │but denies _Every A is B_
   contradict               │_Some A is B_    │

_Contradictory_ propositions are those in which one denies _any thing_
that the other affirms; _contrary_ propositions are those in which one
denies _every thing_ which the other affirms, or affirms every thing
which the other denies. The following pair are contraries.

                       Every A is B and No A is B

and the following are contradictories,

                    Every A is B to Some A is not B
                    No A is B    to Some A is B

A contrary, therefore, is a complete and total contradictory; and a
little consideration will make it appear that the decisive distinction
between contraries and contradictories lies in this, that contraries may
both be false, but of contradictories, one must be true and the other
false. We may say, ‘Either P is true, or _something_ in contradiction of
it is true;’ but we cannot say, ‘Either P is true, or _every thing_ in
contradiction of it is true.’ It is a very common mistake to imagine
that the _denial_ of a proposition gives a right to _affirm_ the
contrary; whereas it should be, that the _affirmation_ of a proposition
gives a right to _deny_ the contrary. Thus, if we deny that Every A is
B, we do not affirm that No A is B, but only that Some A is not B;
while, if we affirm that Every A is B, we deny No A is B, and also Some
A is not B.

But, as to contradictories, affirmation of one is a denial of the other,
and denial of one is affirmation of the other. Thus, either Every A is
B, or Some A is not B: affirmation of either is denial of the other, and
_vice versá_.

Let the student now endeavour to satisfy himself of the following.
Taking the four preceding propositions, A, E, I, O, let the simple
letter signify the affirmation, the same letter in parentheses the
denial, and the absence of the letter, that there is neither affirmation
nor denial.

          From A follow│(E), I, (O)│From (A) follow         O
          From E       │(A), (I), O│From (E)                I
          From I       │(E)        │From (I)        (A), E, O
          From O       │(A)        │From (O)        A, (E), I

These may be thus summed up: The affirmation of a universal proposition,
and the denial of a particular one, enable us to affirm or deny all the
other three; but the denial of a universal proposition, and the
affirmation of a particular one, leave us unable to affirm or deny two
of the others.

In such propositions as ‘Every A is B,’ ‘Some A is not B,’ &c., A is
called the _subject_, and B the _predicate_, while the verb ‘is’ or ‘is
not,’ is called the _copula_. It is obvious that the words of the
proposition point out whether the subject is spoken of universally or
partially, but not so of the predicate, which it is therefore important
to examine. Logical writers generally give the name of _distributed_
subjects or predicates to those which are spoken of universally; but as
this word is rather technical, I shall say that a subject or predicate
enters wholly or partially, according as it is universally or
particularly spoken of.

1. In A, or ‘Every A is B,’ the subject enters wholly, but the predicate
only partially. For it obviously says, ‘Among the Bs are all the As,’
‘Every A is part of the collection of Bs, so that all the As make a part
of the Bs, the whole it _may_ be.’ Thus, ‘Every horse is an animal,’
does not speak of all animals, but states that all the horses make up a
portion of the animals.

2. In E, or ‘No A is B,’ both subject and predicate enter wholly. ‘No A
whatsoever is any one out of all the Bs;’ ‘search the whole collection
of Bs, and _every_ B shall be found to be something which is not A.’

3. In I, or ‘Some A is B,’ both subject and predicate enter partially.
‘Some of the As are found among the Bs, or make up a part (the whole
possibly, but not known from the preceding) of the Bs.’

4. In O, or ‘Some A is not B,’ the subject enters partially, and the
predicate wholly. ‘Some As are none of them any whatsoever of the Bs;
every B will be found to be no one out of a certain portion of the As.’

It appears then that,

In affirmatives, the predicate enters partially.

In negatives, the predicate enters wholly.

In contradictory propositions, both subject and predicate enter
differently in the two.

The _converse_ of a proposition is that which is made by interchanging
the subject and predicate, as follows:

                   The proposition.   Its converse.
                   A Every A is B    Every B is A
                   E No A is B       No B is A
                   I Some A is B     Some B is A
                   O Some A is not B Some B is not A

Now, it is a fundamental and self-evident proposition, that no
consequence must be allowed to assert more widely than its premises; so
that, for instance, an assertion which is only of some Bs can never lead
to a result which is true of all Bs. But if a proposition assert
agreement or disagreement, any other proposition which asserts the same,
to the same extent and no further, must be a legitimate consequence; or,
if you please, must amount to the whole, or part, of the original
assertion in another form. Thus, the converse of A is not true: for, in
‘Every A is B,’ the predicate enters partially; while in ‘Every B is A,’
the subject enters wholly. ‘All the As make up a part of the Bs, then a
part of the Bs are among the As, or some B is A.’ Hence, the only
_legitimate_ converse of ‘Every A is B’ is, ‘Some B is A.’ But in ‘No A
is B,’ both subject and predicate enter wholly, and ‘No B is A’ is, in
fact, the same proposition as ‘No A is B.’ And ‘Some A is B’ is also the
same as its converse ‘Some B is A;’ here both terms enter partially. But
‘Some A is not B’ admits of no converse whatever; it is perfectly
consistent with all assertions upon B and A in which B is the subject.
Thus neither of the four following lines is inconsistent with itself.

                 Some A is not B and Every B is A
                 Some A is not B and No    B is A
                 Some A is not B and Some  B is A
                 Some A is not B and Some  B is not A.

We find then, including converses, which are not identical with their
direct propositions, _six_ different ways of asserting or denying, with
respect to agreement or non-agreement, total or partial, between A and,
say X: these we write down, designating the additional assertions by U
and Y.

                     │Identical. │ Identical.  │
       A Every A is X│E No A is X│I Some A is X│O Some A is not X
       U Every X is A│„ No X is A│„ Some X is A│Y Some X is not A

We shall now repeat and extend the table of page 8 (A), &c., meaning, as
before, the denial of A, &c.

            From A or (O) follow  A,  (E),  I   (O)
            From E or (I)        (A),  E,  (I), O,  (U),  Y
            From I or (E)             (E)   I
            From O or (A)        (A),            O
            From U or (Y)             (E)   I,       U   (Y)
            From Y or (U)                           (U)   Y

Having thus discussed the principal points connected with the simple
assertion, we pass to the manner of making two assertions give a third.
Every instance of this is called a syllogism, the two assertions which
form the basis of the third are called premises, and the third itself
the conclusion.

If two things both agree with a third in any particular, they agree with
each other in the same; as, if A be of the same colour as X, and B of
the same colour as X, then A is of the same colour as B. Again, if A
differ from X in any particular in which B agrees with X, then A and B
differ in that particular. If A be not of the same colour as X, and B be
of the same colour as X, then A is not of the colour of B. But if A and
B both differ from X in any particular, nothing can be inferred; they
may either differ in the same way and to the same extent, or not. Thus,
if A and B be both of different colours from X, it neither follows that
they agree, nor differ, in their own colours.

The paragraph preceding contains the essential parts of all inference,
which consists in comparing two things with a third, and finding from
their agreement or difference with that third, their agreement or
difference with one another. Thus, Every A is X, every B is X, allows us
to infer that A and B have all those qualities in common which are
necessary to X. Again, from Every A is X, and ‘No B is X,’ we infer that
A and B differ from one another in all particulars which are essential
to X. The preceding forms, however, though they represent common
reasoning better than the ordinary syllogism, to which we are now
coming, do not constitute the ultimate forms of inference. Simple
_identity_ or _non-identity_ is the ultimate state to which every
assertion may be reduced; and we shall, therefore, first ask, from what
identities, &c., can other identities, &c., be produced? Again, since we
name objects in species, each species consisting of a number of
individuals, and since our assertion may include all or only part of a
species, it is further necessary to ask, in every instance, to what
extent the conclusion drawn is true, whether of all, or only of part?

Let us take the simple assertion, ‘Every living man respires;’ or, every
living man is one of the things (however varied they may be) which
respire. If we were to inclose all living men in a large triangle, and
all respiring objects in a large circle, the preceding assertion, if
true, would require that the whole of the triangle should be contained
in the circle. And in the same way we may reduce any assertion to the
expression of a coincidence, total or partial, between two figures.
Thus, a point in a circle may represent an individual of one species,
and a point in a triangle an individual of another species: and we may
express that the whole of one species is asserted to be contained or not
contained in the other by such forms as, ‘All the △ is in the ○’; ‘None
of the △ is in the ○’.

Any two assertions about A and B, each expressing agreement or
disagreement, total or partial, with or from X, and leading to a
conclusion with respect to A or B, is called a syllogism, of which X is
called the _middle term_. The plainest syllogism is the following:—

                   Every A is X│          All the △ is in the ○
                   Every X is B│          All the ○ is in the □
         Therefore Every A is B│Therefore All the △ is in the □

In order to find all the possible forms of syllogism, we must make a
table of all the elements of which they can consist; namely—

                       A and X           B and X
                   Every A is X    A Every B is X
                   No A is X       E No B is X
                   Some A is X     I Some B is X
                   Some A is not X O Some B is not X
                   Every X is A    U Every X is B
                   Some X is not A Y Some X is not B

Or their synonymes,

                △ and ○                         □ and ○
     All the △ is in the ○         A All the □ is in the ○
     None of the △ is in the ○     E None of the □ is in the ○
     Some of the △ is in the ○     I Some of the □ is in the ○
     Some of the △ is not in the ○ O Some of the □ is not in the ○
     All the ○ is in the △         U All the ○ is in the □
     Some of the ○ is not in the △ Y Some of the ○ is not in the □

Now, taking any one of the six relations between A and X, and combining
it with either of those between B and X, we have six pairs of premises,
and the same number repeated for every different relation of A and X. We
have then thirty-six forms to consider: but, thirty of these (namely,
all but (A, A) (E, E), &c.) are half of them repetitions of the other
half. Thus, ‘Every A is X, no B is X,’ and ‘Every B is X, no A is X,’
are of the same form, and only differ by changing A into B and B into A.
There are then only 15 + 6, or 21 distinct forms, some of which give a
necessary conclusion, while others do not. We shall select the former of
these, classifying them by their conclusions; that is, according as the
inference is of the form A, E, I, or O.

I. In what manner can a universal affirmative conclusion be drawn;
namely, that one figure is entirely contained in the other? This we can
only assert when we know that one figure is entirely contained in the
circle, which itself is entirely contained in the other figure. Thus,

                  Every A is X│  All the △ is in the ○ A
                  Every X is B│  All the ○ is in the □ A
                ∴ Every A is B│∴ All the △ is in the □ A

is the only way in which a universal affirmative conclusion can be
drawn.

II. In what manner can a universal negative conclusion be drawn; namely,
that one figure is entirely exterior to the other? Only when we are able
to assert that one figure is entirely within, and the other entirely
without, the circle. Thus,

                   Every A is X│All the △ is in the ○ A

                   No    B is X│None of the □ is in   E
                               │the ○

                 ∴ No    A is B│None of the △ is in   E
                               │the □

is the only way in which a universal negative conclusion can be drawn.

III. In what manner can a particular affirmative conclusion be drawn;
namely, that part or all of one figure is contained in the other? Only
when we are able to assert that the whole circle is part of one of the
figures, and that the whole, or part of the circle, is part of the other
figure. We have then two forms.

                  Every X is A│  All the ○ is in the △ A

                  Every X is B│  All the ○ is in the □ A

                ∴ Some  A is B│∴ Some of the △ is in   I
                              │  the □


                  Every X is A│  All the ○ is in the △ A

                  Some  X is B│  Some of the ○ is in   I
                              │  the □

                  Some  A is B│  Some of the △ is in   I
                              │  the □

The second of these contains all that is strictly necessary to the
conclusion, and the first may be omitted. That which follows when an
assertion can be made as to some, must follow when the same assertion
can be made of all.

IV. How can a particular negative proposition be inferred; namely, that
part, or all of one figure, is not contained in the other? It would seem
at first sight, whenever we are able to assert that part or all of one
figure is in the circle, and that part or all of the other figure is
not. The weakest syllogism from which such an inference can be drawn
would then seem to be as follows.

              Some A is X  │  Some of the △ is in the ○
            Some B is not X│  Some of the □ is not in the ○
          ∴ Some B is not A│∴ Some of the △ is not in the □

But here it will appear, on a little consideration, that the conclusion
is only thus far true; that those As which are Xs cannot be _those_ Bs
which are not Xs; but they may be _other_ Bs, about which nothing is
asserted when we say that _some_ Bs are not Xs. And further
consideration will make it evident, that a conclusion of this form can
only be arrived at when one of the figures is entirely within the
circle, and the whole or part of the other without; or else when the
whole of one of the figures is without the circle, and the whole or part
of the other within; or lastly, when the circle lies entirely within one
of the figures, and not entirely within the other. That is, the
following are the distinct forms which allow of a particular negative
conclusion, in which it should be remembered that a particular
proposition in the premises may always be changed into a universal one,
without affecting the conclusion. For that which necessarily follows
from “some,” follows from “all.”

             Every A is X  │  All the △ is in the ○         A
            Some B is not X│  Some of the □ is not in the ○ O
          ∴ Some B is not A│  Some of the □ is not in the △ O

               No A Is X   │  None of the △ is in the ○     E
              Some B is X  │  Some of the □ is in the ○     I
          ∴ Some B is not A│  Some of the □ is not in the △ O

             Every X is A  │  All the ○ is in the △         A
            Some X is not B│  Some of the ○ is not in the □ O
          ∴ Some A is not B│  Some of the △ is not in the □ O

It appears, then, that there are but six distinct syllogisms. All others
are made from them by strengthening one of the premises, or converting
one or both of the premises, where such conversion is allowable; or else
by first making the conversion, and then strengthening one of the
premises. And the following arrangement will shew that two of them are
universal, three of the others being derived from them by weakening one
of the premises in a manner which does not destroy, but only weakens,
the conclusion.

 1. Every A is X           3. Every A is X
    Every X is B              No    B is X                  .........
    ————————————              ————————————
    Every A is B              No    A is B
         │                           │
         │                 ┌─────────┴─────────┐
         │                 │                   │
 2. Some  A is X 4. Some A is X     5. Every A is X     6. Every X is A
    Every X is B    No   B is X        Some  B is not X    Some X is not B
    ————————————    ———————————————    ————————————————    ———————————————
    Some  A is B    Some A is not B    Some  B is not A    Some A is not B

We may see how it arises that one of the partial syllogisms is not
immediately derived, like the others, from a universal one. In the
preceding, AEE may be considered as derived from AAA, by changing the
term in which X enters universally into its contrary. If this be done
with the other term instead, we have

 No    A is X│from which universal premises we cannot deduce a universal
             │  conclusion, but only Some B is not A.

 Every X is B│                            „

If we weaken one and the other of these premises, as they stand, we
obtain

                  Some  A is not X     No   A is X
                  Every X is B     and Some X is B
                  ————————————————     ———————————————
                  No conclusion        Some B is not A

equivalent to the fourth of the preceding: but if we convert the first
premiss, and proceed in the same manner,

                 No    X is A               Some  X is not A
            From Every X is B     we obtain Every X is B
                 ————————————————           ————————————————
                 Some  B is not A           Some  B is not A

which is legitimate, and is the same as the last of the preceding list,
with A and B interchanged.

Before proceeding to shew that all the usual forms are contained in the
preceding, let the reader remark the following rules, which may be
proved either by collecting them from the preceding cases, or by
independent reasoning.

1. The middle term must enter universally into one or the other premiss.
If it were not so, the one premiss might speak of one part of the middle
term, and the other of the other; so that there would, in fact, be no
middle term. Thus, ‘Every A is X, Every B is X,’ gives no conclusion: it
may be thus stated;

                 All the As make up _a part_ of the Xs
                 All the Bs make up _a part_ of the Xs

And, before we can know that there is any common term of comparison at
all, we must have some means of shewing that the two parts are the same;
or the preceding premises by themselves are inconclusive.

2. No term must enter the conclusion more generally than it is found in
the premises; thus, if A be spoken of partially in the premises, it must
enter partially into the conclusion. This is obvious, since the
conclusion must assert no more than the premises imply.

3. From premises both negative no conclusion can be drawn. For it is
obvious, that the mere assertion of disagreement between each of two
things and a third, can be no reason for inferring either agreement or
disagreement between these two things. It will not be difficult to
reduce any case which falls under this rule to a breach of the first
rule: thus, No A is X, No B is X, gives

                 Every A is (something which is not X)
                 Every B is (something which is not X)

in which the middle term is not spoken of universally in either. Again,
‘No X is A, Some X is not B,’ may be converted into

                  Every A is (a thing which is not X)
                  Some (thing which is not B) is X

in which there is no middle term.

4. From premises both particular no conclusion can be drawn. This is
sufficiently obvious when the first or second rule is broken, as in
‘Some A is X, Some B is X.’ But it is not immediately obvious when the
middle term enters one of the premises universally. The following
reasoning will serve for exercise in the preceding results. Since both
premises are particular in form, the middle term can only enter one of
them universally by being the predicate of a negative proposition;
consequently (Rule 3) the other premiss must be affirmative, and, being
particular, neither of its terms is universal. Consequently both the
terms as to which the conclusion is to be drawn enter partially, and the
conclusion (Rule 2) can only be a particular _affirmative_ proposition.
But if one of the premises be negative, the conclusion must be
_negative_ (as we shall immediately see). This contradiction shews that
the supposition of particular premises producing a legitimate result is
inadmissible.

5. If one premiss be negative, the conclusion, if any, must be negative.
If one term agree with a second and disagree with a third, no agreement
can be inferred between the second and third.

6. If one premiss be particular, the conclusion must be particular. This
is not very obvious, since the middle term may be universally spoken of
in a particular proposition, as in Some B is not X. But this requires
one negative proposition, whence (Rule 3) the other must be affirmative.
Again, since the conclusion must be negative (Rule 5) its predicate is
spoken of universally, and, therefore, must enter universally; the other
term A must enter, then, in a universal affirmative proposition, which
is against the supposition.

In the preceding set of syllogisms we observe one form only which
produces A, or E, or I, but three which produce O.

Let an assertion be said to be weakened when it is reduced from
universal to particular, and strengthened in the contrary case. Thus,
‘Every A is B’ is called stronger than ‘Some A is B.’

Every form of syllogism which can give a legitimate result is either one
of the preceding six, or another formed from one of the six, either by
changing one of the assertions into its converse, if that be allowable,
or by strengthening one of the premises without altering the conclusion,
or both. Thus,

          Some A is X         may be written   Some X is A
          Every X is B              „          Every X is B

          What follows will still follow from  _Every_ X is A
                           „                   Every X is B

for all which is true when ‘Some X is A,’ is not less true when ‘Every X
is A.’

It would be possible also to form a legitimate syllogism by weakening
the conclusion, when it is universal, since that which is true of all is
true of some. Thus, ‘Every A is X, Every X is B,’ which yields ‘Every A
is B,’ also yields ‘Some A is B.’ But writers on logic have always
considered these syllogisms as useless, conceiving it better to draw
from any premises their strongest conclusion. In this they were
undoubtedly right; and the only question is, whether it would not have
been advisable to make the premises as weak as possible, and not to
admit any syllogisms in which more appeared than was absolutely
necessary to the conclusion. If such had been the practice, then

           Every X is A, Every X is B, therefore Some A is B

would have been considered as formed by a spurious and unnecessary
excess of assertion. The minimum of assertion would be contained in
either of the following,

           Every X is A, Some  X is B, therefore Some A is B
           Some  X is A, Every X is B, therefore Some A is B

In this tract, syllogisms have been divided into two classes: first,
those which prove a universal conclusion; secondly, those which prove a
partial conclusion, and which are (all but one) derived from the first
by weakening one of the premises, in such manner as to produce a
legitimate but weakened conclusion. Those of the first class are placed
in the first column, and the other in the second.

                   Universal.           Particular.

                A Every A is X        Some A is X     I
                A Every X is B ────── Every X is B    A
                  ————————————        —————————————
                A Every A is B        Some A is B     I

                                      Some A is X     I
                                      No X is B       E

                                    ┌ ———————————————
                A Every A is X      │ Some A is not B O
                E No X is B    ─────┼ Every A is X    A
                  ————————————      │
                E No A is B         │ Some B is not X O
                                    └ ———————————————
                                      Some B is not A O

                                      Every X is A    A
                     ......           Some X is not B O
                                      ———————————————
                                      Some A is not B O

In all works on logic, it is customary to write that premiss first which
contains the predicate of the conclusion. Thus,

          Every X is B                           Every A is X
          Every A is X would be written, and not Every X is B
          ————————————                           ————————————
          Every A is B                           Every A is B

The premises thus arranged are called major and minor; the predicate of
the conclusion being called the major term, and its subject the minor.
Again, in the preceding case we see the various subjects coming in the
order X, B; A, X; A, B: and the number of different orders which can
appear is four, namely—

                              XB BX XB BX
                              AX AX XA XA
                              —— —— —— ——
                              AB AB AB AB

which are called the four figures, and every kind of syllogism in each
figure is called a mood. I now put down the various moods of each
figure, the letters of which will be a guide to find out those of the
preceding list from which they are derived. Co means that a premiss of
the preceding list has been converted; + that it has been strengthened;
Co +, that both changes have taken place. Thus,

             A Every X is B         A Every X is B
             I Some  A is X becomes A Every X is A: (Co +)
               ————————————           ————————————
             I Some  A is B         I Some  A is B

And Co + abbreviates the following: If some A be X, then some X is A
(Co); and all that is true when Some X is A, is true when Every X is A
(+); therefore the second is legitimate, if the first be so.

                            _First Figure._

       A Every X is B                        A Every X is B
       A Every A is X                        I Some  A is X
         ————————————                          ——————————————
       A Every A is B                        I Some  A is B

       E No    X is B                        E No    X is B
       A Every A is X                        I Some  A is X
         ————————————                          ————————————————
       E No    A is B                        O Some  A is not B


                            _Second Figure._

       E No    B is X (Co)                   E No    B is X (Co)
       A Every A is X                        I Some  A is X
         ————————————                          —————————————————
       E No    A is B                        O Some  A is not B

       A Every B is X                        A Every B is X
       E No    A is X (Co)                   O Some  A is not X
         ————————————                          ————————————————
       E No    A is B                        O Some  A is not B


                            _Third Figure._

       A Every X is B                        E No    X is B
       A Every X is A (Co+)                  A Every X is A (Co+)
         ————————————                          ——————————————————
       I Some  A is B                        O Some  A is not B

       I Some  X is B (Co)                   O Some  X is not B
       A Every X is A                        A Every X is A
         ————————————                          ————————————————
       I Some  A is B                        O Some  A is not B

       A Every X is B                        E No    X is B
       I Some  X is A (Co)                   I Some  X is A (Co)
         ————————————                          —————————————————
       I Some  A is B                        O Some  A is not B


                            _Fourth Figure._

       A Every B is X (+)                    I Some  B is X
       A Every X is A                        A Every X is A
         ————————————                          ————————————
       I Some  A is B                        I Some  B is A

       A Every B is X                        E No    B is X (Co)
       E No    X is A                        A Every X is A (Co+)
         ————————————                          ——————————————————
       E No    A is B                        O Some  A is not B

                          E No   B is X (Co)
                          I Some X is A (Co)
                            ————————————————
                          O Some A is not B

The above is the ancient method of dividing syllogisms; but, for the
present purpose, it will be sufficient to consider the six from which
the rest can be obtained. And since some of the six have A in the
predicate of the conclusion, and not B, we shall join to them the six
other syllogisms which are found by transposing B and A. The complete
list, therefore, of syllogisms with the weakest premises and the
strongest conclusions, in which a comparison of A and B is obtained by
comparison of both with X, is as follows:

    Every A is X  Every B is X │Some  A is X      Some  B is X
    Every X is B  Every X is A │No    X is B      No    X is A
    ————————————  ———————————— │————————————————  ————————————————
    Every A is B  Every B is A │Some  A is not B  Some  B is not A
                               │
    Every A is X  Every B is X │Every A is X      Every B is X
    No    X is B  No    X is A │Some  B is not X  Some  A is not X
    ————————————  ———————————— │————————————————  ————————————————
    No    A is B  No    B is A │Some  B is not A  Some  A is not B
                               │
    Some  A is X  Some  B is X │Every X is A      Every X is B
    Every X is B  Every X is A │Some  X is not B  Some  X is not A
    ————————————  ———————————— │————————————————  ————————————————
    Some  A is B  Some  B is A │Some  A is not B  Some  B is not A

In the list of page 19, there was nothing but recapitulation of forms,
each form admitting a variation by interchanging A and B. This
interchange having been made, and the results collected as above, if we
take every case in which B is the predicate, or can be made the
predicate by allowable conversion, we have a collection of all possible
_weakest_ forms in which the result is one of the four ‘Every A is B,’
‘No A is B,’ ‘Some A is B,’ ‘Some A is not B’; as follows. The premises
are written in what appeared the most natural order, without distinction
of major or minor; and the letters prefixed are according to the forms
of the several premises, as in page 10.

                           A Every A is X
                           U Every X is B
                             ————————————
                           A Every A is B

                   I Some  A is X     I Some  B is X
                   U Every X is B     U Every X is A
                     ————————————       ————————————
                   I Some  A is B     I Some  A is B

                   A Every A is X     A Every B is X
                   E No    B is X     E No    A is X
                     ————————————       ————————————
                   E No    A is B     E No    A is B

        I Some  A is X     A Every B is X     U Every X is A
        E No    B is X     O Some  A is not X Y Some  X is not B
          ————————————————   ————————————————   ————————————————
        O Some  A is not B O Some  A is not B O Some  A is not B

Every assertion which can be made upon two things by comparison with any
third, that is, every simple inference, can be reduced to one of the
preceding forms. Generally speaking, one of the premises is omitted, as
obvious from the conclusion; that is, one premiss being named and the
conclusion, that premiss is implied which is necessary to make the
conclusion good. Thus, if I say, “That race must have possessed some of
the arts of life, for they came from Asia,” it is obviously meant to be
asserted, that all races coming from Asia must have possessed some of
the arts of life. The preceding is then a syllogism, as follows:

         ‘That race’ is ‘a race of Asiatic origin:’
         Every ‘race of Asiatic origin’ is ‘a race which must have
            possessed some of the arts of life:’
 Therefore, That race _is_ a race which must have possessed some of the
    arts of life.

A person who makes the preceding assertion either means to imply,
antecedently to the conclusion, that all Asiatic races must have
possessed arts, or he talks nonsense if he asserts the conclusion
positively. ‘A must be B, for it is X,’ can only be true when ‘Every X
is B.’ This latter proposition may be called the suppressed premiss; and
it is in such suppressed propositions that the greatest danger of error
lies. It is also in such propositions that men convey opinions which
they would not willingly express. Thus, the honest witness who said, ‘I
always thought him a respectable man—he kept his gig,’ would probably
not have admitted in direct terms, ‘Every man who keeps a gig must be
respectable.’


I shall now give a few detached illustrations of what precedes.

“His imbecility of character might have been inferred from his proneness
to favourites; for all weak princes have this failing.” The preceding
would stand very well in a history, and many would pass it over as
containing very good inference. Written, however, in the form of a
syllogism, it is,

                     All weak princes are prone to favourites
                            He        was prone to favourites
                     ———————————————— ———————————————————————
           Therefore        He        was a weak prince

which is palpably wrong. (Rule 1.) The writer of such a sentence as the
preceding might have meant to say, ‘for all who have this failing are
weak princes;’ in which case he would have inferred rightly. Every one
should be aware that there is much false inference arising out of
badness of style, which is just as injurious to the habits of the
untrained reader as if the errors were mistakes of logic in the mind of
the writer.

‘A is less than B; B is less than C: therefore A is less than C.’ This,
at first sight, appears to be a syllogism; but, on reducing it to the
usual form, we find it to be,

                          A is (a magnitude less than B)
                          B is (a magnitude less than C)
                Therefore A is (a magnitude less than C)

which is not a syllogism, since there is no middle term. Evident as the
preceding is, the following additional proposition must be formed before
it can be made explicitly logical. ‘If B be a magnitude less than C,
then every magnitude less than B is also less than C.’ There is, then,
before the preceding can be reduced to a syllogistic form, the necessity
of a deduction from the second premiss, and the substitution of the
result instead of that premiss. Thus,

                     A is less than B
           Less than B is less than C: following from B is less than C.
           ————————— —————————————————
 Therefore           A is less than C

But, if the additional argument be examined—namely, if B be less than C,
then that which is less than B is less than C—it will be found to
require precisely the same considerations repeated; for the original
inference was nothing more. In fact, it may easily be seen as follows,
that the proposition before us involves more than any simple syllogism
can express. When we say that A is less than B, we say that if A were
applied to B, every part of A would match a part of B, and there would
be parts of B remaining over. But when we say, ‘Every A is B,’ meaning
the premiss of a common syllogism, we say that every instance of A is an
instance of B, without saying any thing as to whether there are or are
not instances of B still left, after those which are also A are taken
away. If, then, we wish to write an ordinary syllogism in a manner which
shall correspond with ‘A is less than B, B is less than C, therefore A
is less than C,’ we must introduce a more definite amount of assertion
than was made in the preceding forms. Thus,

                 Every A is B, and there are Bs which are not As
                 Every B is C, and there are Cs which are not Bs
                 ———————————————————————————————————————————————
       Therefore Every A is C, and there are Cs which are not As

Or thus:

                  The Bs contain all the As, and more
                  The Cs contain all the Bs, and more
                  ———————————————————————————————————
                  The Cs contain all the As, and more

The most technical form, however, is,

                From    Every A is B; [Some B is not A]
                        Every B is C; [Some C is not B]
                Follows Every A is C; [Some C is not A]

This sort of argument is called _à fortiori_ argument, because the
premises are more than sufficient to prove the conclusion, and the
extent of the conclusion is thereby greater than its mere form would
indicate. Thus, ‘A is less than B, B is less than C, therefore, _à
fortiori_, A is less than C,’ means that the extent to which A is less
than C must be greater than that to which A is less than B, or B than C.
In the syllogism last written, either of the bracketed premises might be
struck out without destroying the conclusion; which last would, however,
be weakened. As it stands, then, the part of the conclusion, ‘Some C is
not A,’ follows it _à fortiori_.

The argument _à fortiori_, may then be defined as a universally
affirmative syllogism, in which both of the premises are shewn to be
less than the whole truth, or greater. Thus, in ‘Every A is X, Every X
is B, therefore Every A is B,’ we do not certainly imply that there are
more Xs than As, or more Bs than Xs, so that we do not know that there
are more Bs than As. But if we are at liberty to state the syllogism as
follows,

           All the As make up part (and part only) of the Xs
           Every X is B;

then we are certain that

           All the As make up part (and part only) of the Bs.

But if we are at liberty further to say that

           All the As make up part (and part only) of the Xs
           All the Xs make up part (and part only) of the Bs

then we conclude that

           All the As make up _part of part_ (only) of the Bs

and the words in Italics mark that quality of the conclusion from which
the argument is called _à fortiori_.

Most syllogisms which give an affirmative conclusion are generally meant
to imply _à fortiori_ arguments, except only in mathematics. It is
seldom, except in the exact sciences, that we meet with a proposition,
‘Every A is B,’ which we cannot immediately couple with ‘Some Bs are not
As.’

When an argument is completely established, with the exception of one
assertion only, so that the inference may be drawn as soon as that one
assertion is established, the result is stated in a form which bears the
name of an _hypothetical_ syllogism. The word hypothesis means nothing
but supposition; and the species of syllogism just mentioned first lays
down the assertion that a consequence will be true if a certain
condition be fulfilled, and then either asserts the fulfilment of the
condition, and thence the consequence, or else denies the consequence,
and thence denies the fulfilment of the condition. Thus, if we know that

                  When A is B, it follows that P is Q;

then, as soon as we can ascertain that A is B, we can conclude that P is
Q; or, if we can shew that P is not Q, we know that A is not B. But if
we find that A is not B, we can infer nothing; for the preceding does
not assert that P is Q _only_ when A is B. And if we find out that P is
Q, we can infer nothing. This conditional syllogism may be converted
into an ordinary syllogism, as follows. Let K be any ‘case in which A is
B,’ and Z a ‘case in which P is Q’; then the preceding assertion amounts
to ‘Every K is Z.’ Let L be a particular instance, the A of which may or
may not be B. If A be B in the instance under discussion, or if A be not
B, we have, in the one case and the other,

                        Every K is Z   Every K is Z
                              L is a K       L is not a K
                              ————————       —————————————
              Therefore       L is a Z       No conclusion

Similarly, according as a particular case (M) is or is not Z, we have

                   Every     K is Z   Every K is Z
                             M is a Z       M is not a Z
                   —————————————            ————————————
                   No conclusion            M is not a K

That is to say: The assertion of an hypothesis is the assertion of its
necessary consequence, and the denial of the necessary consequence is
the denial of the hypothesis; but the assertion of the necessary
consequence gives no right to assert the hypothesis, nor does the denial
of the hypothesis give any right to deny the truth of that which would
(were the hypothesis true) be its necessary consequence.


Demonstration is of two kinds: which arises from this, that every
proposition has a contradictory; and of these two, one must be true and
the other must be false. We may then either prove a proposition to be
true, or its contradictory to be false. ‘It is true that Every A is B,’
and, ‘it is false that there are some As which are not Bs,’ are the same
proposition; and the proof of either is called the indirect proof of the
other.

But how is any proposition to be proved false, except by proving a
contradiction to be true? By proving a necessary consequence of the
proposition to be false. But this is not a complete answer, since it
involves the necessity of doing the same thing; or, so far as this
answer goes, one proposition cannot be proved false unless by proving
another to be false. But it may happen, that a necessary consequence can
be obtained which is obviously and self-evidently false, in which case
no further proof of the falsehood of the hypothesis is necessary. Thus
the proof which Euclid gives that all equiangular triangles are
equilateral is of the following structure, logically considered.

(1.) If there be an equiangular triangle not equilateral, it follows
that a whole can be found which is not greater than its part.[1]

Footnote 1:

  This is the proposition in proof of which nearly the whole of the
  demonstration of Euclid is spent.

(2.) It is false that there can be any whole which is not greater than
its part (self evident).

(3.) Therefore it is false that there is any equiangular triangle which
is not equilateral; or all equiangular triangles are equilateral.

When a proposition is established by proving the truth of the matters it
contains, the demonstration is called _direct_; when by proving the
falsehood of every contradictory proposition, it is called _indirect_.
The latter species of demonstration is as logical as the former, but not
of so simple a kind; whence it is desirable to use the former whenever
it can be obtained.

The use of indirect demonstration in the Elements of Euclid is almost
entirely confined to those propositions in which the converses of simple
propositions are proved. It frequently happens that an established
assertion of the form

                 Every A is B                     (1)

may be easily made the means of deducing,

                 Every (thing not A) is not B     (2)

which last gives

                 Every B is A                     (3)

The conversion of the second proposition into the third is usually made
by an indirect demonstration, in the following manner. If possible, let
there be one B which is not A, (2) being true. Then there is one thing
which is not A and is B; but every thing not A is not B; therefore there
is one thing which is B and is not B: which is absurd. It is then absurd
that there should be one single B which is not A; or, Every B is A.

The following proposition contains a method which is of frequent use.

HYPOTHESIS.—Let there be any number of propositions or assertions,—three
for instance, A, B, and C,—of which it is the property that one or the
other must be true, _and one only_. Let there be three other
propositions, P, Q, and R, of which it is also the property that one,
and one only, must be true. Let it also be a connexion of those
assertions, that

                       When A is true, P is true
                       When B is true, Q is true
                       When C is true, R is true

CONSEQUENCE: then it follows that

                       When P is true, A is true
                       When Q is true, B is true
                       When R is true, C is true

For, when P is true, then Q and R must be false; consequently, neither B
nor C can be true, for then Q or R would be true. But either A, B, or C
must be true, therefore A must be true; or, when P is true, A is true.
In a similar way the remaining assertions may be proved.

      Case 1. If   │When P is Q,              A is B
                   │When P is not Q,          A is not B
    It follows that│When A is B,              P is Q
                   │When A is not B,          P is not Q

      Case 2. If   │When A is greater than B, P is greater than Q
           „       │When A is equal to B,     P is equal to Q
           „       │When A is less than B,    P is less than Q

    It follows that│When P is greater than Q, A is greater than B
           „       │When P is equal to Q,     A is equal to B
           „       │When P is less than Q,    A is less than B

                  *       *       *       *       *

We have hitherto supposed that the premises are actually true; and, in
such a case, the logical conclusion is as certain as the premises. It
remains to say a few words upon the case in which the premises are
probably, but not certainly, true.

The probability of an event being about to happen, and that of an
argument being true, may be so connected that the usual method of
measuring the first may be made to give an easy method of expressing the
second. Suppose an urn, or lottery, with a large number of balls, black
or white; then, if there be twelve white balls to one black, we say it
is twelve to one that a white ball will be drawn, or that a white ball
is twelve times as probable as a black one. A certain assertion may be
in the same condition as to the force of probability with which it
strikes the mind: that is, the questions

                      Is the assertion true?
                      Will a white ball be drawn?

may be such that the answer, ‘most probably,’ expresses the same degree
of likelihood in both cases.

We have before explained that logic has nothing to do with the truth or
falsehood of assertions, but only professes, supposing them true, to
collect and classify the legitimate methods of drawing inferences.
Similarly, in this part of the subject, we do not trouble ourselves with
the question, How are we to find the probability due to premises? but we
ask: Supposing (happen how it may) that we _have_ found the probability
of the premises, required the probability of the conclusion. When the
odds in favour of a conclusion are, say 6 to 1, there are, out of every
7 possible chances, 6 in favour of the conclusion, and 1 against it.
Hence ⁶⁄₇ and ⅐ will represent the proportions, for and against, of all
the possible cases which exist.

Thus we have the succession of such results as in the following table:—

     Odds in favour of an event Probability for Probability against

               1 to 1                  ½                 ½
               2 to 1                  ⅔                 ⅓
               3 to 1                  ¾                 ¼
               3 to 2                  ⅗                 ⅖
               4 to 1                  ⅘                 ⅕
               4 to 3                 ⁴⁄₇               ³⁄₇
               5 to 1                  ⅚                 ⅙
                &c.                   &c.               &c.

Let the probability of a conclusion, as derived from the premises (that
is on the supposition that it was never imagined to be possible till the
argument was heard), be called the _intrinsic probability_ of the
argument. This is found by multiplying together the probabilities of all
the assertions which are necessary to the argument. Thus, suppose that a
conclusion was held to be impossible until an argument of a single
syllogism was produced, the premises of which have severally five to one
and eight to one in their favour. Then ⅚ × ⁸⁄₉, or ⁴⁰⁄₅₄, is the
intrinsic probability of the argument, and the odds in its favour are 40
to 14, or 20 to 7.

But this intrinsic probability is not always that of the conclusion; the
latter, of course, depending in some degree on the likelihood which the
conclusion was supposed to have before the argument was produced. A
syllogism of 20 to 7 in its favour, advanced in favour of a conclusion
which was beforehand as likely as not, produces a much more probable
result than if the conclusion had been thought absolutely false until
the argument produced a certain belief in the possibility of its being
true. The change made in the probability of a conclusion by the
introduction of an argument (or of a new argument, if some have already
preceded) is found by the following rule.

From the sum of the existing probability of the conclusion and the
intrinsic probability of the new argument, take their product; the
remainder is the probability of the conclusion, as reinforced by the
argument. Thus, _a + b − ab_ is the probability of the truth of a
conclusion after the introduction of an argument of the intrinsic
probability _b_, the previous probability of the said conclusion having
been _a_.

Thus, a conclusion which has at present the chance ⅔ in its favour, when
reinforced by an argument whose intrinsic probability is ¾, acquires the
probability ⅔ + ¾ − ⅔ × ¾ or, ⅔ + ¾ − ½, or ¹¹⁄₁₂; or, having 2 to 1 in
its favour before, it has 11 to 1 in its favour after, the argument.

When the conclusion was neither likely nor unlikely beforehand (or had
the probability ½), the shortest way of applying the preceding rule (in
which _a + b − ab_ becomes ½ + ½_b_) is to divide the sum of the
numerator and denominator of the intrinsic probability of the argument
by twice the denominator. Thus, an argument of which the intrinsic
probability is ¾, gives to a conclusion on which no bias previously
existed, the probability ⅞ or (3 + 4)/(2 × 4).


                                THE END.


                    LONDON:—PRINTED BY JAMES MOYES,
                    Castle Street, Leicester Square.

------------------------------------------------------------------------




                          TRANSCRIBER’S NOTES


 1. Silently corrected obvious typographical errors and variations in
      spelling.
 2. Retained archaic, non-standard, and uncertain spellings as printed.
 3. Enclosed italics font in _underscores_.