_ON THE CONCEPTION AND TREATMENT OF MODALITY._
1. The reader who knows anything of the scholastic
Logic will have perceived before now that we have been
touching in a variety of places upon that most thorny and
repulsive of districts in the logical territory;--modality. It
will be advisable, however, to put together, somewhat more
definitely, what has to be said upon the subject. I propose,
therefore, to devote this chapter to a brief account of the
principal varieties of treatment which the modals have received
at the hands of professed logicians.
It must be remarked at the outset that the sense in
which modality and modal propositions have been at various
times understood, is by no means fixed and invariably the
same. This diversity of view has arisen partly from corresponding
differences in the view taken of the province and
nature of logic, and partly from differences in the philosophical
and scientific opinions entertained as to the constitution
and order of nature. In later times, moreover,
another very powerful agent in bringing about a change in
the treatment of the subject must be recognized in the
gradual and steady growth of the theory of Probability, as
worked out by the mathematicians from their own point of
view.
2. In spite, however, of these differences of treatment,
there has always been some community of subject-matter in
the discussions upon this topic. There has almost always
been some reference to quantity of belief; enough perhaps
to justify De Morgan's[1] remark, that Probability was "the
unknown God whom the schoolmen ignorantly worshipped
when they so dealt with this species of enunciation, that it
was said to be beyond human determination whether they
most tortured the modals, or the modals them." But this
reference to quantity of belief has sometimes been direct and
immediate, sometimes indirect and arising out of the nature
of the subject-matter of the proposition. The fact is, that
that distinction between the purely subjective and purely objective
views of logic, which I have endeavoured to bring out
into prominence in the eleventh chapter, was not by any
means clearly recognized in early times, nor indeed before
the time of Kant, and the view to be taken of modality
naturally shared in the consequent confusion. This will, I
hope, be made clear in the course of the following chapter,
which is intended to give a brief sketch of the principal
different ways in which the modality of propositions has
been treated in logic. As it is not proposed to give anything
like a regular history of the subject, there will be no
necessity to adhere to any strict sequence of time, or to
discuss the opinions of any writers, except those who may be
taken as representative of tolerably distinct views. The
outcome of such investigation will be, I hope, to convince
the reader (if, indeed, he had not come to that conviction
before), that the logicians, after having had a long and fair
trial, have failed to make anything satisfactory out of this
subject of the modals by their methods of enquiry and treatment;
and that it ought, therefore, to be banished entirely
from that science, and relegated to Probability.
3. From the earliest study of the syllogistic process
it was seen that, complete as that process is within its own
domain, the domain, at any rate under its simplest treatment,
is a very limited one. Propositions of the pure form,--All
(or some) A is (or is not) B,--are found in practice to form
but a small portion even of our categorical statements. We
are perpetually meeting with others which express the relation
of B to A with various degrees of necessity or probability;
e.g. A must be B, A may be B; or the effect of
such facts upon our judgment, e.g. I am perfectly certain
that A is B, I think that A may be B; with many others of
a more or less similar type. The question at once arises,
How are such propositions to be treated? It does not seem
to have occurred to the old logicians, as to some of their successors
in modern times, simply to reject all consideration of
this topic. Their faith in the truth and completeness of
their system of inference was far too firm for them to suppose
it possible that forms of proposition universally recognized
as significant in popular speech, and forms of inference
universally recognized there as valid, were to be omitted
because they were inconvenient or complicated.
4. One very simple plan suggests itself, and has indeed
been repeatedly advocated, viz. just to transfer all that
is characteristic of such propositions into that convenient receptacle
for what is troublesome elsewhere, the predicate.[2]
Has not another so-called modality been thus got rid of?[3]
and has it not been attempted by the same device to abolish
the distinctive characteristic of negative propositions, viz. by
shifting the negative particle into the predicate? It must
be admitted that, up to a certain point, something may be
done in this way. Given the reasoning, 'Those who take
arsenic will probably die; A has taken it, therefore he will
probably die;' it is easy to convert this into an ordinary
syllogism of the pure type, by simply wording the major,
'Those who take arsenic are people-who-will-probably-die,'
when the conclusion follows in the same form, 'A is one
who-will-probably-die.' But this device will only carry us
a very little way. Suppose that the minor premise also is
of the same modal description, e.g. 'A has probably taken
arsenic,' and it will be seen that we cannot relegate the
modality here also to the predicate without being brought to
a stop by finding that there are four terms in the syllogism.
But even if there were not this particular objection, it
does not appear that anything is to be gained in the way of
intelligibility or method by such a device as the above. For
what is meant by a modal predicate, by the predicate
'probably mortal,' for instance, in the proposition 'All poisonings
by arsenic are probably mortal'? If the analogy with
ordinary pure propositions is to hold good, it must be a
predicate referring to the _whole_ of the subject, for the subject
is distributed. But then we are at once launched into
the difficulties discussed in a former chapter (Ch. VI. §§19-25),
when we attempt to justify or verify the application of
the predicate. We have to enquire (at least on the view
adopted in this work) whether the application of the predicate
'probably mortal' to the _whole_ of the subject, really
means at bottom anything else than that the predicate
'mortal' is to be applied to a _portion_ (more than half) of the
members denoted by the subject. When the transference of
the modality to the predicate raises such intricate questions
as to the sense in which the predicate is to be interpreted,
there is surely nothing gained by the step.
5. A second, and more summary way of shelving all
difficulties of the subject, so far at least as logic, or the
writers upon logic, are concerned, is found by simply denying
that modality has any connection whatever with logic. This
is the course adopted by many modern writers, for instance,
by Hamilton and Mansel, in reference to whom one cannot
help remarking that an unduly large portion of their logical
writings seems occupied with telling us what does _not_ belong
to logic. They justify their rejection on the ground that
the mode belongs to the matter, and must be determined by
a consideration of the matter, and therefore is extralogical.
To a certain extent I agree with their grounds of rejection,
for (as explained in Chapter VI.) it is not easy to see how
the degree of modality of any proposition, whether premise
or conclusion, can be justified without appeal to the matter.
But then questions of justification, in any adequate sense of
the term, belong to a range of considerations somewhat alien
to Hamilton's and Mansel's way of regarding the science.
The complete justification of our inferences is a matter
which involves their truth or falsehood, a point with which
these writers do not much concern themselves, being only
occupied with the consistency of our reasonings, not with
their conformity with fact. Were I speaking as a Hamiltonian
I should say that modality _is_ formal rather than
material, for though we cannot justify the degree of our
belief of a proposition without appeal to the matter, we can
to a moderate degree of accuracy estimate it without any
such appeal; and this would seem to be quite enough to
warrant its being regarded as formal.
It must be admitted that Hamilton's account of the
matter when he is recommending the rejection of the modals,
is not by any means clear and consistent. He not only fails,
as already remarked, to distinguish between the formal and
the material (in other words, the true and the false) modality;
but when treating of the former he fails to distinguish
between the extremely diverse aspects of modality when
viewed from the Aristotelian and the Kantian stand-points.
Of the amount and significance of this difference we shall
speak presently, but it may be just pointed out here that
Hamilton begins (Vol. I. p. 257) by rejecting the modals on
the ground that the distinctions between the necessary, the
contingent, the possible, and the impossible, must be wholly
rested on an appeal to the matter of the propositions, in
which he is, I think, quite correct. But then a little further
on (p. 260), in explaining 'the meaning of three terms which
are used in relation to pure and modal propositions,' he gives
the widely different Kantian, or three-fold division into the
apodeictic, the assertory, and the problematic. He does not
take the precaution of pointing out to his hearers the very
different general views of logic from which these two accounts
of modality spring.[4]
6. There is one kind of modal syllogism which it
would seem unreasonable to reject on the ground of its not
being formal, and which we may notice in passing. The
premise 'Any A is probably B,' is equivalent to 'Most A are B.'
Now it is obvious that from two such premises as 'Most A
are B,' 'Most A are C,' we can deduce the consequence,
'Some C are B.' Since this holds good whatever may be
the nature of A, B, and C, it is, according to ordinary usage
of the term, a formal syllogism. Mansel, however, refuses to
admit that any such syllogisms belong to formal logic. His
reasons are given in a rather elaborate review[5] and criticism
of some of the logical works of De Morgan, to whom the
introduction of 'numerically definite syllogisms' is mainly
due. Mansel does not take the particular example given
above, as he is discussing a somewhat more comprehensive
algebraic form. He examines it in a special numerical
example:[6]--18 out of 21 Ys are X; 15 out of 21 Ys are Z;
the conclusion that 12 Zs are X is rejected from formal logic
on the ground that the arithmetical judgment involved is
synthetical, not analytical, and rests upon an intuition of
quantity. We cannot enter upon any examination of these
reasons here; but it may merely be remarked that his
criticism demands the acceptance of the Kantian doctrines
as to the nature of arithmetical judgments, and that it would
be better to base the rejection not on the ground that the
syllogism is not _formal_, but on the ground that it is not
_analytical_.
7. There is another and practical way of getting rid
of the perplexities of modal reasoning which must be noticed
here. It is the resource of ordinary reasoners rather than the
decision of professed logicians,[7] and, like the first method of
evasion already pointed out in this chapter, is of very partial
application. It consists in treating the premises, during the
process of reasoning, as if they were pure, and then reintroducing
the modality into the conclusion, as a sort of
qualification of its full certainty. When each of the premises
is nearly certain, or when from any cause we are not
concerned with the extent of their departure from full certainty,
this rough expedient will answer well enough. It is,
I apprehend, the process which passes through the minds of
most persons in such cases, in so far as they reason consciously.
They would, presumably, in such an example as that previously
given (§4), proceed as if the premises that 'those
who take arsenic will die,' and that 'the man in question
has taken it,' were quite true, instead of being only probably
true, and they would consequently draw the conclusion that
'he would die.' But bearing in mind that the premises are
not certain, they would remember that the conclusion was
only to be held with a qualified assent. This they would
express quite correctly, if the mere nature and not the
degree of that assent is taken into account, by saying that
'he is likely to die.' In this case the modality is rejected
temporarily from the premises to be reintroduced into the
conclusion.
It is obvious that such a process as this is of a very
rough and imperfect kind. It does, in fact, omit from accurate
consideration just the one point now under discussion.
It takes no account of the varying shades of expression by
which the degree of departure from perfect conviction is
indicated, which is of course the very thing with which
modality is intended to occupy itself. At best, therefore, it
could only claim to be an extremely rude way of deciding
questions, the accurate and scientific methods of treating
which are demanded of us.
8. In any employment of applied logic we have of
course to go through such a process as that just mentioned.
Outside of pure mathematics it can hardly ever be the case
that the premises from which we reason are held with absolute
conviction. Hence there must be a lapse from absolute
conviction in the conclusion. But we reason on the hypothesis
that the premises are true, and any trifling defection
from certainty, of which we may be conscious, is mentally
reserved as a qualification to the conclusion. But such considerations
as these belong rather to ordinary applied logic;
they amount to nothing more than a caution or hint to be
borne in mind when the rules of the syllogism, or of induction,
are applied in practice. When, however, we are
treating of modality, the extent of the defection from full
certainty is supposed to be sufficiently great for our language
to indicate and appreciate it. What we then want is of
course a scientific discussion of the principles in accordance
with which this departure is to be measured and expressed,
both in our premises and in our conclusion. Such a plan
therefore for treating modality, as the one under discussion,
is just as much a banishment of it from the field of real
logical enquiry, as if we had determined avowedly to reject it
from consideration.
9. Before proceeding to a discussion of the various
ways in which modality may be treated by those who admit
it into logic, something must be said to clear up a possible
source of confusion in this part of the subject. In the
cases with which we have hitherto been mostly concerned,
in the earlier chapters of this work, the characteristic of
modality (for in this chapter we may with propriety use this
logical term) has generally been found in singular and particular
propositions. It presented itself when we had to
judge of individual cases from a knowledge of the average,
and was an expression of the fact that the proposition relating
to these individuals referred to a portion only of the
whole class from which the average was taken. Given that
of men of fifty-five, three out of five will die in the course of
twenty years, we have had to do with propositions of the
vague form, 'It is probable that AB (of that age) will die,'
or of the more precise form, 'It is three to two that AB will
die,' within the specified time. Here the modal proposition
naturally presents itself in the form of a singular or particular
proposition.
10. But when we turn to ordinary logic we may find
_universal_ propositions spoken of as modal. This must mostly
be the case with those which are termed necessary or impossible,
but it may also be the case with the probable. We
may meet with the form 'All X is probably Y.' Adopting
the same explanation here as has been throughout adopted in
analogous cases, we must say that what is meant by the
modality of such a proposition is the proportional number of
times in which the universal proposition would be correctly
made. And in this there is, so far, no difficulty. The only
difference is that whereas the justification of the former, viz.
the particular or individual kind of modal, was obtainable
within the limits of the universal proposition which included
it, the justification of the modality of a universal proposition
has to be sought in a group or succession of other propositions.
The proposition has to be referred to some group of
similar ones and we have to consider the proportion of cases
in which it will be true. But this distinction is not at all
fundamental.
It is quite true that universal propositions from their
nature are much less likely than individual ones to be justified,
in practice, by such appeal. But, as has been already
frequently pointed out, we are not concerned with the way
in which our propositions are practically obtained, nor with
the way in which men might find it most natural to test
them; but with that ultimate justification to which we appeal
in the last resort, and which has been abundantly shown
to be of a statistical character. When, therefore, we say that
'it is probable that all X is Y,' what we mean is, that in
more than half the cases we come across we should be right
in so judging, and in less than half the cases we should be
wrong.
11. It is at this step that the possible ambiguity is
encountered. When we talk of the chance that All X is Y,
we contemplate or imply the complementary chance that it is
not so. Now this latter alternative is not free from ambiguity.
It might happen, for instance, in the cases of failure,
that no X is Y, or it might happen that some X, only, is not Y;
for both of these suppositions contradict the original proposition,
and are therefore instances of its failure. In practice,
no doubt, we should have various recognized rules and
inductions to fall back upon in order to decide between these
alternatives, though, of course, the appeal to them would be
in strictness extralogical. But the mere existence of such an
ambiguity, and the fact that it can only be cleared up by
appeal to the subject-matter, are in themselves no real
difficulty in the application of the conception of modality to
universal propositions as well as to individual ones.
12. Having noticed some of the ways in which the
introduction of modality into logic has been evaded or rejected,
we must now enter into a brief account of its treatment
by those who have more or less deliberately admitted
its claims to acceptance.
The first enquiry will be, What opinions have been held
as to the nature of modality? that is, Is it primarily an affection
of the matter of the proposition, and, if not, what is it
exactly? In reference to this enquiry it appears to me, as
already remarked, that amongst the earlier logicians no such
clear and consistent distinction between the subjective and
objective views of logic as is now commonly maintained, can
be detected.[8] The result of this appears in their treatment
of modality. This always had some reference to the subjective
side of the proposition, viz. in this case to the nature
or quantity of the belief with which it was entertained; but
it is equally clear that this characteristic was not estimated
at first hand, so to say, and in itself, but rather from a consideration
of the matter determining what it should be. The
commonly accepted scholastic or Aristotelian division, for
instance, is into the necessary, the contingent, the possible,
and the impossible. This is clearly a division according to
the matter almost entirely, for on the purely mental side the
necessary and the impossible would be just the same; one
implying full conviction of the truth of a proposition, and the
other of that of its contradictory. So too, on the same side, it
would not be easy to distinguish between the contingent and
the possible. On the view in question, therefore, the modality
of a proposition was determined by a reference to the nature
of the subject-matter. In some propositions the nature of
the subject-matter decided that the predicate was necessarily
joined to the subject; in others that it was impossible that
they should be joined; and so on.
13. The artificial character of such a four-fold division
will be too obvious to modern minds for it to be necessary to
criticize it. A very slight study of nature and consequent
appreciation of inductive evidence suffice to convince us that
those uniformities upon which all connections of phenomena,
whether called necessary or contingent, depend, demand extremely
profound and extensive enquiry; that they admit of
no such simple division into clearly marked groups; and
that, therefore, the pure logician had better not meddle with
them.[9]
The following extract from Grote's _Aristotle_ (Vol. I. p. 192)
will serve to show the origin of this four-fold division,
its conformity with the science of the day, and consequently
its utter want of conformity with that of our own time:--"The
distinction of Problematical and Necessary Propositions
corresponds, in the mind of Aristotle, to that capital and
characteristic doctrine of his Ontology and Physics, already
touched on in this chapter. He thought, as we have seen,
that in the vast circumferential region of the Kosmos, from
the outer sidereal sphere down to the lunar sphere, celestial
substance was a necessary existence and energy, sempiternal
and uniform in its rotations and influence; and that through
its beneficent influence, pervading the concavity between the
lunar sphere and the terrestrial centre (which included the
four elements with their compounds) there prevailed a regularizing
tendency called Nature; modified, however, and
partly counteracted by independent and irregular forces called
Spontaneity and Chance, essentially unknowable and unpredictable.
The irregular sequences thus named by Aristotle
were the objective correlate of the Problematical Proposition
in Logic. In these sublunary sequences, as to future time,
_may or may not_, was all that could be attained, even by the
highest knowledge; certainty, either of affirmation or negation,
was out of the question. On the other hand, the necessary
and uniform energies of the celestial substance, formed
the objective correlate of the Necessary Proposition in Logic;
this substance was not merely an existence, but an existence
necessary and unchangeable... he considers the Problematical
Proposition in Logic to be not purely subjective, as an
expression of the speaker's ignorance, but something more,
namely, to correlate with an objective essentially unknowable
to all."
14. Even after this philosophy began to pass away,
the divisions of modality originally founded upon it might
have proved, as De Morgan has remarked,[10] of considerable
service in mediæval times. As he says, people were much
more frequently required to decide in one way or the other
upon a single testimony, without there being a sufficiency of
specific knowledge to test the statements made. The old
logician "did not know but that any day of the week might
bring from Cathay or Tartary an account of men who ran on
four wheels of flesh and blood, or grew planted in the ground,
like Polydorus in the Æneid, as well evidenced as a great
many nearly as marvellous stories." Hence, in default of
better inductions, it might have been convenient to make
rough classifications of the facts which were and which were
not to be accepted on testimony (the necessary, the impossible, &c.),
and to employ these provisional inductions (which
is all we should now regard them) as testing the stories
which reached him. Propositions belonging to the class of
the impossible might be regarded as having an antecedent
presumption against them so great as to prevail over almost
any testimony worth taking account of, and so on.
15. But this old four-fold division of modals continued
to be accepted and perpetuated by the logicians long
after all philosophical justification for it had passed away.
So far as I have been able to ascertain, scarcely any logician
of repute or popularity before Kant, was bold enough to
make any important change in the way of regarding them.[11]
Even the Port-Royal Logic, founded as it is on Cartesianism,
repeats the traditional statements, though with extreme
brevity. This adherence to the old forms led, it need not be
remarked, to considerable inconsistency and confusion in
many cases. These forms were founded, as we have seen, on
an objective view of the province of logic, and this view was
by no means rigidly carried out in many cases. In fact it
was beginning to be abandoned, to an extent and in directions
which we have not opportunity here to discuss, before
the influence of Kant was felt. Many, for instance, added to
the list of the four, by including the true and the false;
occasionally also the probable, the supposed, and the certain
were added. This seems to show some tendency towards
abandoning the objective for the subjective view, or at least
indicates a hesitation between them.
16. With Kant's view of modality almost every one is
familiar. He divides judgments, under this head, into the
apodeictic, the assertory, and the problematic. We shall have
to say something about the number and mutual relations of
these divisions presently; we are now only concerned with
the general view which they carry out. In this respect it will
be obvious at once what a complete change of position has
been reached. The 'necessary' and the 'impossible' demanded
an appeal to the matter of a proposition in order to
recognize them; the 'apodeictic' and the 'assertory', on the
other hand, may be true of almost any matter, for they
demand nothing but an appeal to our consciousness in order
to distinguish between them. Moreover, the distinction
between the assertory and the problematic is so entirely
subjective and personal, that it may vary not only between
one person and another, but in the case of the same person
at different times. What one man knows to be true, another
may happen to be in doubt about. The apodeictic judgment
is one which we not only accept, but which we find ourselves
unable to reverse in thought; the assertory is simply accepted;
the problematic is one about which we feel in
doubt.
This way of looking at the matter is the necessary outcome
of the conceptualist or Kantian view of logic. It has
been followed by many logicians, not only by those who may
be called followers of Kant, but by almost all who have felt
his influence. Ueberweg, for instance, who is altogether at
issue with Kant on some fundamental points, adopts it.
17. The next question to be discussed is, How many
subdivisions of modality are to be recognized? The Aristotelian
or scholastic logicians, as we have seen, adopted a four-fold
division. The exact relations of some of these to each
other, especially the possible and the contingent, is an extremely
obscure point, and one about which the commentators
are by no means agreed. As, however, it seems tolerably
clear that it was not consciously intended by the use of these
four terms to exhibit a graduated scale of intensity of conviction,
their correspondence with the province of modern
probability is but slight, and the discussion of them, therefore,
becomes rather a matter of special or antiquarian interest.
De Morgan, indeed (_Formal Logic_, p. 232), says that
the schoolmen understood by contingent more likely than
not, and by possible less likely than not. I do not know
on what authority this statement rests, but it credits them
with a much nearer approach to the modern views of probability
than one would have expected, and decidedly nearer than
that of most of their successors.[12] The general conclusion at
which I have arrived, after a reasonable amount of investigation,
is that there were two prevalent views on the subject.
Some (e.g. Burgersdyck, Bk. I. ch. 32) admitted that there
were at bottom only two kinds of modality; the contingent
and the possible being equipollent, as also the necessary and
the impossible, provided the one asserts and the other
denies. This is the view to which those would naturally
be led who looked mainly to the nature of the subject-matter.
On the other hand, those who looked mainly at the form of
expression, would be led by the analogy of the four forms of
proposition, and the necessity that each of them should stand
in definite opposition to each other, to insist upon a distinction
between the four modals.[13] They, therefore, endeavoured
to introduce a distinction by maintaining (e.g. Crackanthorpe,
Bk. III. ch. 11) that the contingent is that which now is but
may not be, and the possible that which now is not but may
be. A few appear to have made the distinction correspondent
to that between the physically and the logically
possible.
18. When we get to the Kantian division we have
reached much clearer ground. The meaning of each of these
terms is quite explicit, and it is also beyond doubt that they
have a more definite tendency in the direction of assigning a
graduated scale of conviction. So long as they are regarded
from a metaphysical rather than a logical standing point,
there is much to be said in their favour. If we use introspection
merely, confining ourselves to a study of the judgments
themselves, to the exclusion of the grounds on which
they rest, there certainly does seem a clear and well-marked
distinction between judgments which we cannot even conceive
to be reversed in thought; those which we could
reverse, but which we accept as true; and those which we
merely entertain as possible.
Regarded, however, as a logical division, Kant's arrangement
seems to me of very little service. For such logical
purposes indeed, as we are now concerned with, it really
seems to resolve itself into a _two_-fold division. The distinction
between the apodeictic and the assertory will be
admitted, I presume, even by those who accept the metaphysical
or psychological theory upon which it rests, to be a
difference which concerns, not the quantity of belief with
which the judgments are entertained, but rather the violence
which would have to be done to the mind by the attempt to
upset them. Each is fully believed, but the one can, and
the other cannot, be controverted. The belief with which an
assertory judgment is entertained is full belief, else it would
not differ from the problematic; and therefore in regard to
the quantity of belief, as distinguished from the quality or
character of it, there is no difference between it and the apodeictic.
It is as though, to offer an illustration, the index
had been already moved to the top of the scale in the assertory
judgment, and all that was done to convert this into
an apodeictic one, was to _clamp_ it there. The only logical
difference which then remains is that between problematic
and assertory, the former comprehending all the judgments
as to the truth of which we have any degree of doubt, and
the latter those of which we have no doubt. The whole
range of the former, therefore, with which Probability is
appropriately occupied, is thrown undivided into a single
compartment. We can hardly speak of a 'division' where
one class includes everything up to the boundary line, and
the other is confined to that boundary line. Practically,
therefore, on this view, modality, as the mathematical student
of Probability would expect to find it, as completely
disappears as if it were intended to reject it.
19. By less consistent and systematic thinkers, and
by those in whom ingenuity was an over prominent feature,
a variety of other arrangements have been accepted or proposed.
There is, of course, some justification for such attempts
in the laudable desire to bring our logical forms into better
harmony with ordinary thought and language. In practice,
as was pointed out in an earlier chapter, every one recognizes
a great variety of modal forms, such as 'likely,' 'very
likely,' 'almost certainly,' and so on almost without limit in
each direction. It was doubtless supposed that, by neglecting
to make use of technical equivalents for some of these
forms, we should lose our logical control over certain possible
kinds of inference, and so far fall short even of the precision
of ordinary thought.
With regard to such additional forms, it appears to me
that all those which have been introduced by writers who
were uninfluenced by the Theory of Probability, have done
little else than create additional confusion, as such writers do
not attempt to marshal their terms in order, or to ascertain
their mutual relations. Omitting, of course, forms obviously
of material modality, we have already mentioned the true
and the false; the probable, the supposed, and the certain.
These subdivisions seem to have reached their climax at a
very early stage in Occam (Prantl, III. 380), who held that a
proposition might be modally affected by being 'vera, scita,
falsa, ignota, scripta, prolata, concepta, credita, opinata, dubitata.'
20. Since the growth of the science of Probability,
logicians have had better opportunities of knowing what
they had to aim at; and, though it cannot be said that their
attempts have been really successful, these are at any rate a
decided improvement upon those of their predecessors. Dr
Thomson,[14] for instance, gives a nine-fold division. He says
that, arranging the degrees of modality in an ascending
scale, we find that a judgment may be either possible,
doubtful, probable, morally certain for the thinker himself,
morally certain for a class or school, morally certain for all,
physically certain with a limit, physically certain without
limitation, and mathematically certain. Many other divisions
might doubtless be mentioned, but, as every mathematician
will recognize, the attempt to secure any general
agreement in such a matter of arrangement is quite hopeless.
It is here that the beneficial influence of the mathematical
theory of Probability is to be gratefully acknowledged. As
soon as this came to be studied it must have been perceived
that in attempting to mark off clearly from one another
certain gradations of belief, we should be seeking for breaches
in a continuous magnitude. In the advance from a slight
presumption to a strong presumption, and from that to moral
certainty, we are making a gradual ascent, in the course of
which there are no natural halting-places. The proof of this
continuity need not be entered upon here, for the materials
for it will have been gathered from almost every chapter of
this work. The reader need merely be reminded that the
grounds of our belief, in all cases which admit of number and
measurement, are clearly seen to be of this description; and
that therefore unless the belief itself is to be divorced from
the grounds on which it rests, what thus holds as to their
characteristics must hold also as to its own.
It follows, therefore, that modality in the old sense of the
word, wherein an attempt was made to obtain certain natural
divisions in the scale of conviction, must be finally abandoned.
All that it endeavoured to do can now be done incomparably
better by the theory of Probability, with its numerical scale
which admits of indefinite subdivision. None of the old systems
of division can be regarded as a really natural one;
those which admit but few divisions being found to leave the
whole range of the probable in one unbroken class, and those
which adopt many divisions lapsing into unavoidable vagueness
and uncertainty.
21. Corresponding to the distinction between pure
and modal propositions, but even more complicated and
unsatisfactory in its treatment, was that between pure and
modal syllogisms. The thing discussed in the case of the
latter was, of course, the effect produced upon the conclusion
in respect of modality, by the modal affection of one or both
premises. It is only when we reach such considerations as
these that we are at all getting on to the ground appropriate
to Probability; but it is obvious that very little could be
done with such rude materials, and the inherent clumsiness
and complication of the whole modal system come out very
clearly here. It was in reference probably to this complication
that some of the bitter sayings[15] of the schoolmen and
others which have been recorded, were uttered.
Aristotle has given an intricate investigation of this subject,
and his followers naturally were led along a similar
track. It would be quite foreign to my purpose in the slight
sketch in this chapter to attempt to give any account of
these enquiries, even were I competent to do so; for, as has
been pointed out, the connection between the Aristotelian
modals and the modern view of the nature of Probability,
though real, is exceedingly slight. It need only be remarked
that what was complicated enough with four modals to be
taken account of, grows intricate beyond all endurance when
such as the 'probable' and the 'true' and the 'false' have
also to be assigned a place in the list. The following examples[16]
will show the kind of discussions with which the logicians
exercised themselves. 'Whether, with one premise
certain, and the other probable, a certain conclusion may be
inferred': 'Whether, from the impossible, the necessary can
be inferred'; 'Whether, with one premise necessary and the
other _de inesse_, the conclusion is necessary', and so on,
endlessly.
22. On the Kantian view of modality the discussion
of such kinds of syllogisms becomes at once decidedly more
simple (for here but three modes are recognized), and also
somewhat more closely connected with strict Probability, (for
the modes are more nearly of the nature of gradations of
conviction). But, on the other hand, there is less justification
for their introduction, as logicians might really be expected
to know that what they are aiming to effect by their clumsy
contrivances is the very thing which Probability can carry
out to the highest desired degree of accuracy. The former
methods are as coarse and inaccurate, compared with the
latter, as were the roughest measurements of Babylonian
night-watchers compared with the refined calculations of the
modern astronomer. It is indeed only some of the general
adherents of the Kantian Logic who enter upon any such
considerations as these; some, such as Hamilton and Mansel,
entirely reject them, as we have seen. By those who do
treat of the subject, such conclusions as the following are laid
down; that when both premises are apodeictic the conclusion
will be the same; so when both are assertory or problematic.
If one is apodeictic and the other assertory, the latter, or
'weaker,' is all that is to be admitted for the conclusion;
and so on. The English reader will find some account of
these rules in Ueberweg's _Logic_.[17]
23. But although those modals, regarded as instruments
of accurate thought, have been thus superseded by the
precise arithmetical expressions of Probability, the question
still remains whether what may be termed our popular modal
expressions could not be improved and adapted to more
accurate use. It is true that the attempt to separate them
from one another by any fundamental distinctions is futile,
for the magnitude of which they take cognizance is, as we
have remarked, continuous; but considering the enormous
importance of accurate terminology, and of recognizing
numerical distinctions wherever possible, it would be a real
advance if any agreement could be arrived at with regard to
the use of modal expressions. We have already noticed (Ch. II. §16)
some suggestions by Mr Galton as to the possibility
of a natural system of classification, resting upon the regularity
with which most kinds of magnitudes tend to group
themselves about a mean. It might be proposed, for instance,
that we should agree to apply the term 'good'
to the first quarter, measuring from the best downwards;
'indifferent' to the middle half, and 'bad' to the last quarter.
There seems no reason why a similarly improved terminology
should not some day be introduced into the ordinary modal
language of common life. It might be agreed, for instance,
that 'very improbable' should as far as possible be confined
to those events which had odds of (say) more than 99 to 1
against them; and so on, with other similar expressions.
There would, no doubt, be difficulties in the way, for in all
applications of classification we have to surmount the two-fold
obstacles which lie in the way, firstly (to use Kant's
expression) of the faculty of making rules, and secondly of
that of subsumption under rules. That is to say, even if we
had agreed upon our classes, there would still be much doubt
and dispute, in the case of things which did not readily lend
themselves to be counted or measured, as to whether the
odds were more or less than the assigned quantity.
It is true that when we know the odds for or against an
event, we can always state them explicitly without the necessity
of first agreeing as to the usage of terms which shall
imply them. But there would often be circumlocution and
pedantry in so doing, and as long as modal terms are in
practical use it would seem that there could be no harm, and
might be great good, in arriving at some agreement as to the
degree of probability which they should be generally understood
to indicate. Bentham, as is well known, in despair of
ever obtaining anything accurate out of the language of common
life on this subject, was in favour of a direct appeal to
the numerical standard. He proposed the employment, in
judicial trials, of an instrument, graduated from 0 to 10, on
which scale the witness was to be asked to indicate the degree
of his belief of the facts to which he testified: similarly
the judge might express the force with which he held his
conclusion. The use of such a numerical scale, however, was
to be optional only, not compulsory, as Bentham admitted
that many persons might feel at a loss thus to measure the
degree of their belief. (_Rationale of Judicial Evidence_,
Bk. I., Ch. VI.)
24. Throughout this chapter we have regarded the
modals as the nearest counterpart to modern Probability
which was afforded by the old systems of logic. The reason
for so regarding them is, that they represented some slight
attempt, rude as it was, to recognize and measure certain
gradations in the degree of our conviction, and to examine
the bearing of such considerations upon our logical inferences.
But although it is amongst the modals that the germs of
the methods of Probability are thus to be sought; the true
subject-matter of our science, that is, the classes of objects
with which it is most appropriately concerned, are rather
represented by another part of the scholastic logic. This
was the branch commonly called Dialectic, in the old sense
of that term. Dialectic, according to Aristotle, seems to
have been a sort of sister art to Rhetoric. It was concerned
with syllogisms differing in no way from demonstrative syllogisms,
except that their premises were probable instead of
certain. Premises of this kind he termed topics, and the
syllogisms which dealt with them enthymemes. They were
said to start from 'signs and likelihoods' rather than from
axioms.[18]
25. The terms in which such reasonings are commonly
described sound very much like those applicable to
Probability, as we now understand it. When we hear of
likelihood, and of probable syllogisms, our first impression
might be that the inferences involved would be of a similar
character.[19] This, however, would be erroneous. In the
first place the province of this Dialectic was much too wide,
for it covered in addition the whole field of what we should
now term Scientific or Material Induction. The distinctive
characteristic of the dialectic premises was their want of
certainty, and of such uncertain premises Probability (as I
have frequently insisted) takes account of one class only,
Induction concerning itself with another class. Again, not the
slightest attempt was made to enter upon the enquiry, _How_
uncertain are the premises? It is only when this is attempted
that we can be considered to enter upon the field of
Probability, and it is because, after a rude fashion, the modals
attempted to grapple with this problem, that we have regarded
them as in any way occupied with our special subject-matter.
26. Amongst the older logics with which I have made
any acquaintance, that of Crackanthorpe gives the fullest discussion
upon this subject. He divides his treatment of the
syllogism into two parts, occupied respectively with the 'demonstrative'
and the 'probable' syllogism. To the latter a
whole book is devoted. In this the nature and consequences
of thirteen different 'loci'[20] are investigated, though it is not
very clear in what sense they can every one of them be regarded
as being 'probable.'
It is doubtless true, that if the old logicians had been
in possession of such premises as modern Probability is concerned
with, and had adhered to their own way of treating
them, they would have had to place them amongst such _loci_,
and thus to make the consideration of them a part of their
Dialectic. But inasmuch as there does not seem to have
been the slightest attempt on their part to do more here
than recognize the _fact_ of the premises being probable; that
is, since it was not attempted to _measure_ their probability
and that of the conclusion, I cannot but regard this part of
Logic as having only the very slightest relation to Probability
as now conceived. It seems to me little more than
one of the ways (described at the commencement of this
chapter) by which the problem of Modality is not indeed rejected,
but practically evaded.
27. As Logic is not the only science which is directly
and prominently occupied with questions about belief and
evidence, so the difficulties which have arisen there have
been by no means unknown elsewhere. In respect of the
modals, this seems to have been manifestly the case in Jurisprudence.
Some remarks, therefore, may be conveniently
made here upon this application of the subject, though of
course with the brevity suitable on the part of a layman who
has to touch upon professional topics.
Recall for a moment what are the essentials of modality.
These I understand to be the attempt to mark off from one
another, without any resort to numerical notation, varying
degrees of conviction or belief, and to determine the consequent
effect of premises, thus affected, upon our conclusions.
Moreover, as we cannot construct or retain a scale of any
kind without employing a standard from and by which to
measure it, the attainment and recognition of a standard of
certainty, or of one of the other degrees of conviction, is
almost inseparably involved in the same enquiry. In this
sense of the term, modal difficulties have certainly shown
themselves in the department of Law. There have been
similar attempts here, encountered by similar difficulties, to
come to some definite agreement as to a scale of arrangement
of the degrees of our assent. It is of course much
more practicable to secure such agreement in the case of a
special science, confined more or less to the experts, than in
subjects into which all classes of outsiders have almost equal
right of entry. The range of application under the former
circumstances is narrower, and the professional experts have
acquired habits and traditions by which the standards may
be retained in considerable integrity. It does not appear,
however, according to all accounts, as if any very striking
success had been attained in this direction by the lawyers.
28. The difficulty in its scientific, or strictly jurisprudential
shape, seems to have shown itself principally in the
attempt to arrange legal evidence into classes in respect of
the degree of its cogency. This, I understand, was the case
in the Roman law, and in some of the continental systems of
jurisprudence which took their rise from the Roman law.
"The direct evidence of so many witnesses was _plena probatio_.
Then came _minus plena probatio_, then _semiplenâ
major_ and _semiplenâ minor_; and by adding together a
certain number of half-proofs--for instance, by the production
of a tradesman's account-books, _plus_ his supplementary
oath--full proof might be made out. It was on
this principle that torture was employed to obtain a confession.
The confession was evidence suppletory to the circumstances
which were held to justify its employment."[21]
According to Bentham,[22] the corresponding scale in the
English school was:--Positive proof, Violent presumption.
Probable presumption, Light or Rash presumption. Though
admitted by Blackstone and others, I understand that these
divisions are not at all generally accepted at the present
day.
29. In the above we are reminded rather of modal
syllogisms. The principal practical form in which the difficulty
underlying the simple modal propositions presents
itself, is in the attempt to obtain some criterion of judicial
certainty. By 'certainty' here we mean, of course, not what
the metaphysicians term apodeictic,[23] for that can seldom or
never be secured in practical affairs, but such a degree of
conviction, short of this, as every reasonable person will feel
to be sufficient for all his wants. Here again, one would
think, the quest must appear, to accurate thinkers, an utterly
hopeless one; an effort to discover natural breaks in a continuous
magnitude. There cannot indeed be the least doubt
that, amongst limited classes of keen and practised intellects,
a standard of certainty, as of everything else, might be retained
and handed down with considerable accuracy: this is
possible in matters of taste and opinion where personal peculiarities
of judgment are far more liable to cause disagreement
and confusion. But then such a consensus is almost entirely
an affair of tact and custom; whereas what is wanted in the
case in question is some criterion to which the comparatively
uninitiated may be able to appeal. The standard, therefore,
must not merely be retained by recollection, but be generally
recognizable by its characteristics. If such a criterion could
be secured, its importance could hardly be overrated. But
so far as one may judge from the speeches of counsel, the
charges of judges, and the verdicts of juries, nothing really
deserving the name is ever attained.
30. The nearest approach, perhaps, to a recognized
standard is to be found in the frequent assurance that juries
are not bound to convict only in case they have _no_ doubt of
the guilt of the accused; for the absolute exclusion of all
doubt, the utter impossibility of suggesting any counter
hypothesis which this assumes, is unattainable in human
affairs. But, it is frequently said, they are to convict if
they have no 'reasonable doubt,' no such doubt, that is,
as would be 'a hindrance to acting in the important affairs
of life.' As a caution against seeking after unattainable
certainty, such advice may be very useful; but it need
hardly be remarked that the certainty upon which we act in
the important affairs of life is no fixed standard, but varies
exceedingly according to the nature of those affairs. The
greater the reward at stake, the greater the risk we are
prepared to run, and conversely. Hardly any degree of certainty
can exist, upon the security of which we should not
be prepared to act under appropriate circumstances.[24]
Some writers indeed altogether deny that any standard,
in the common sense of the word, either is, or ought to be,
aimed at in legal proceedings. For instance, Sir J. F.
Stephen, in his work on English Criminal Law,[25] after
noticing and rejecting such standards as that last indicated,
comes to the conclusion that the only standard recognized
by our law is that which induces juries to convict:--"What
is judicial proof? That which being permitted by
law to be given in evidence, induces twelve men, chosen
according to the Jury Act, to say that, having heard it, their
minds are satisfied of the truth of the proposition which it
affirms. They may be prejudiced, they may be timid, they
may be rash, they may be ignorant; but the oath, the
number, and the property qualification, are intended, as far
as possible, to neutralize these disadvantages, and answer
precisely to the conditions imposed upon standards of value
or length." (p. 263.)
To admit this is much about the same thing as to abandon
such a standard as unattainable. Evidence which induces
a jury to convict may doubtless be a standard to me and
others of what we ought to consider 'reasonably certain,'
provided of course that the various juries are tolerably uniform
in their conclusions. But it clearly cannot be proposed
as a standard to the juries themselves; if their decisions are
to be consistent and uniform, they want some external indication
to guide them. When a man is asking, _How_ certain
ought I to feel? to give such an answer as the above is,
surely, merely telling him that he is to be as certain as
he is. If, indeed, juries composed a close profession, they
might, as was said above, retain a traditional standard. But
being, as they are, a selection from the ordinary lay public,
their own decisions in the past can hardly be held up to
them as a direction what they are to do in future.
31. It would appear therefore that we may fairly say
that the English law, at any rate, definitely rejects the main
assumption upon which the logical doctrine of modality and
its legal counterpart are based: the assumption, namely, that
different grades of conviction can be marked off from one
another with sufficient accuracy for us to be able to refer
individual cases to their corresponding classes. And that
with regard to the collateral question of fixing a standard of
certainty, it will go no further than pronouncing, or implying,
that we are to be content with nothing short of, but
need not go beyond, 'reasonable certainty.'
This is a statement of the standard, with which the
logician and scientific man can easily quarrel; and they
may with much reason maintain that it has not the slightest
claim to accuracy, even if it had one to strict intelligibility.
If a man wishes to know whether his present degree of certainty
_is_ reasonable, whither is he to appeal? He can
scarcely compare his mental state with that which is experienced
in 'the important affairs of life,' for these, as
already remarked, would indicate no fixed value. At the
same time, one cannot suppose that such an expression is
destitute of all signification. People would not continue to
use language, especially in matters of paramount importance
and interest, without meaning something by it. We are
driven therefore to conclude that 'reasonable certainty' does
in a rude sort of way represent a traditional standard to
which it is attempted to adhere. As already remarked, this
is perfectly practicable in the case of any class of professional
men, and therefore not altogether impossible in the case of
those who are often and closely brought into connection with
such a class. Though it is hard to believe that any such
expressions, when used for purposes of ordinary life, attain
at all near enough to any conventional standard to be worth
discussion; yet in the special case of a jury, acting under
the direct influence of a judge, it seems quite possible that
their deliberate assertion that they are 'fully convinced'
may reach somewhat more nearly to a tolerably fixed standard
than ordinary outsiders would at first think likely.
32. Are there then any means by which we could
ascertain what this standard is; in other words, by which we
could determine what is the real worth, in respect of accuracy,
of this 'reasonable certainty' which the juries are supposed
to secure? In the absence of authoritative declarations
upon the subject, the student of Logic and Probability
would naturally resort to two means, with a momentary
notice of which we will conclude this enquiry.
The first of these would aim at determining the standard
of judicial certainty indirectly, by simply determining the
statistical frequency with which the decisions (say) of a jury
were found to be correct. This may seem to be a hopeless
task; and so indeed it is, but not so much on any theoretic
insufficiency of the determining elements as on account of
the numerous arbitrary assumptions which attach to most
of the problems which deal with the probability of testimony
and judgments. It is not necessary for this purpose that we
should have an infallible superior court which revised the
decisions of the one under consideration;[26] it is sufficient if a
large number of ordinary representative cases are submitted
to a court consisting even of exactly similar materials to the
one whose decisions we wish to test. Provided always that we
make the monstrous assumption that the judgments of men
about matters which deeply affect them are 'independent'
in the sense in which the tosses of pence are independent,
then the statistics of mere agreement and disagreement will
serve our purpose. We might be able to say, for instance,
that a jury of a given number, deciding by a given majority,
were right nine times out of ten in their verdict. Conclusions
of this kind, in reference to the French courts, are
what Poisson has attempted at the end of his great work on
the Probability of Judgments; though I do not suppose that
he attached much numerical accuracy to his results.
A scarcely more hopeful means would be found by a
reference to certain cases of legal 'presumptions.' A 'conclusive
presumption' is defined as follows:--"Conclusive, or
as they are elsewhere termed imperative or absolute presumptions
of law, are rules determining the quantity of evidence
requisite for the support of any particular averment
which is not permitted to be overcome by any proof that the
fact is otherwise."[27] A large number of such presumptions
will be found described in the text-books, but they seem to
refer to matters far too vague, for the most part, to admit of
any reduction to statistical frequency of occurrence. It is
indeed maintained by some authorities that any assignment
of degree of Probability is not their present object, but that
they are simply meant to exclude the troublesome delays
that would ensue if everything were considered open to
doubt and question. Moreover, even if they did assign a
degree of certainty this would rather be an indication of
what legislators or judges thought reasonable than of what
was so considered by the juries themselves.
There are indeed presumptions as to the time after which
a man, if not heard of, is supposed to be dead (capable of
disproof, of course, by his reappearance). If this time varied
with the age of the man in question, we should at once have
some such standard as we desire, for a reference to the Life
tables would fix his probable duration of life, and so determine
indirectly the measure of probability which satisfied
the law. But this is not the case; the period chosen is
entirely irrespective of age. The nearest case in point (and
that does not amount to much) which I have been able to
ascertain is that of the age after which it has been presumed
that a woman was incapable of bearing children.
This was the age of 53. A certain approach to a statistical
assignment of the chances in this case is to be found in
Quetelet's _Physique Sociale_ (Vol. I. p. 184, note). According
to the authorities which he there quotes it would seem that
in about one birth in 5500 the mother was of the age of 50
or upwards. This does not quite assign the degree of what
may be called the _à priori_ chance against the occurrence of a
birth at that age, because the fact of having commenced a
family at an early age represents some diminution of the
probability of continuing it into later life. But it serves to
give some indication of what may be called the odds against
such an event.
It need not be remarked that any such clues as these to
the measure of judicial certainty are far too slight to be of
any real value. They only deserve passing notice as a possible
logical solution of the problem in question, or rather as
an indication of the mode in which, in theory, such a solution
would have to be sought, were the English law, on those
subjects, a perfectly consistent scheme of scientific evidence.
This is the mode in which one would, under those circumstances,
attempt to extract from its proceedings an admission
of the exact measure of that standard of certainty which it
adopted, but which it declined openly to enunciate.
1. _Formal Logic_, p. 232.
2. This appears to be the purport of some statements in a very
confused passage in Whately's _Logic_ (Bk. II., ch. IV. §1). "A
modal proposition may be stated as a pure one by attaching the mode
to one of the terms, and the proposition will in all respects fall
under the foregoing rules;... 'It is probable that all knowledge is
useful;' 'probably useful' is here the predicate." He draws
apparently no such distinction as that between the true and false
modality referred to in the next note. What is really surprising is
that even Hamilton puts the two (the true and the false modality)
upon the same footing. "In regard to these [the former] the case is
precisely the same; the mode is merely a part of the predicate."
_Logic_, I. 257.
3. I allude of course to such examples as 'A killed B unjustly,' in
which the killing of B by A was sometimes said to be asserted not
simply but with a modification. (Hamilton's _Logic_, I. 256.) It is
obvious that the modification in such cases is by rights merely a
part of the predicate, there being no formal distinction between 'A
is the killer of B' and 'A is the unjust killer of B.' Indeed some
logicians who were too conservative to reject the generic name of
modality in this application adopted the common expedient of
introducing a specific distinction which did away with its meaning,
terming the spurious kind 'material modality' and the genuine kind
'formal modality'. The former included all the cases in which the
modification belonged by right either to the predicate or to the
subject; the latter was reserved for the cases in which the
modification affected the real conjunction of the predicate with the
subject. (Keckermann, _Systema Logicæ_, Lib. II. ch. 3.) It was, I
believe, a common scholastic distinction.
For some account of the dispute as to whether the negative particle
was to be considered to belong to the copula or to the predicate,
see Hamilton's _Logic_, I. 253.
4. He has also given a short discussion of the subject elsewhere
(_Discussions_, Ed. II. p. 702), in which a somewhat different view
is taken. The modes are indeed here admitted into logic, but only
in so far as they fall by subdivision under the relation of genus
and species, which is of course tantamount to their entire
rejection; for they then differ in no essential way from any other
examples of that relation.
5. _Letters, Lectures and Reviews_, p. 61. Elsewhere in the review
(p. 45) he gives what appears to me a somewhat different decision.
6. It must be remembered that this is not one of the proportional
propositions with which we have been concerned in previous chapters:
it is meant that there are _exactly_ 21 Ys, of which just 18 are X,
not that on the average 18 out of 21 may be so regarded.
7. I consider however, as I have said further on (p. 320), that the
treatment in the older logics of Probable syllogisms, and Dialectic
syllogisms, came to somewhat the same thing as this, though they
looked at the matter from a different point of view, and expressed
it in very different language.
8. The distinction is however by no means entirely neglected. Thus
Smiglecius, when discussing the modal affections of certainty and
necessity, says, "certitudo ad cognitionem spectat: necessitas vero
est in re" (_Disputationes;_ Disp. XIII., Quæst. XII.).
9. It may be remarked that Whately (_Logic_, Bk. II. ch. II. §2)
speaks of necessary, impossible and contingent matter, without any
apparent suspicion that they belong entirely to an obsolete point of
view.
10. _Formal Logic_, p. 233.
11. The subject was sometimes altogether omitted, as by Wolf. He says
a good deal however about probable propositions and syllogisms, and,
like Leibnitz before him, looked forward to a "logica probabilium"
as something new and desirable. I imagine that he had been
influenced by the writers on Chances, as of the few who had already
treated that subject nearly all the most important are referred to
in one passage (_Philosophia Rationalis sive Logica_, §593).
Lambert stands quite apart. In this respect, as in most others where
mathematical conceptions and symbols are involved, his logical
attitude is thoroughly unconventional. See, for instance, his
chapter 'Von dem Wahrscheinlichen', in his _Neues Organon_.
12. I cannot find the slightest authority for the statement in the
elaborate history of Logic by Prantl.
13. "Hi quatuor modi magnam censeri solent analogiam habere cum
quadruplici propositionum in quantitate et qualitate varietate"
(Wallis's _Instit. Logic._ Bk. II. ch. 8).
14. _Laws of Thought_, §118.
15. "Haud scio magis ne doctrinam modalium scholastici exercuerint,
quam ea illos vexarit. Certe usque adeo sudatum hic fuit, ut
dicterio locus sit datus; _De modalibus non gustabit asinus_."
Keckermann, _Syst. Log._ Bk. II. ch. 3.
16. _Smiglecii Disputationes_, Ingolstadt, 1618.
See also Prantl's _Geschichte der Logik_ (under _Occam_ and
_Buridan_) for accounts of the excessive complication which the
subtlety of those learned schoolmen evolved out of such suitable
materials.
17. Translation by T. M. Lindsay, p. 439.
18. "The εἰκòς and σημεῖον themselves are propositions; the former
stating a _general probability_, the latter a fact, which is known
to be an indication, more or less certain, of the truth of some
further statement, whether of a single fact, or of a general
belief. The former is a general proposition, nearly, though not
quite, universal; as 'most men who envy hate'; the latter is a
_singular_ proposition, which however is not regarded as a sign,
except relatively to some other proposition, which it is supposed
may by inferred from it." (Mansel's Aldrich; Appendix F, where an
account will be found of the Aristotelian enthymeme, and dialectic
syllogism. Also, of course, Grote's Aristotle, _Topics_ and
elsewhere.)
19. "Nam in hoc etiam differt demonstratio, sen demonstrativa
argumentatio, à probabili, quia in illâ tam conclusio quam præmissæ
necessariæ sunt; in probabili autem argumentatione sicut conclusio
ut probabilis infertur ita præmissæ ut probabiles afferuntur"
(Crackanthorpe, Bk. V., Ch. 1); almost the words with which De
Morgan distinguishes between logic and probability in a passage
already cited (see Ch. VI. §3).
Perhaps it was a development of some such view as this that Leibnitz
looked forward to. "J'ai dit plus d'une fois qu'il faudrait une
nouvelle espèce de Logique, qui traiteroit des degrés de
Probabilité, puisqu'Aristote dans ses Topiques n'a rien moins fait
que cela" (_Nouveaux essais_, Lib. IV. ch. XVI). It is possible,
indeed, that he had in his mind more what we now understand by the
mathematical theory of Probability, but in the infancy of a science
it is of course hard to say whether any particular subject is
definitely contemplated or not. Leibnitz (as Todhunter has shown in
his history) took the greatest interest in such chance problems as
had yet been discussed.
20. By _loci_ were understood certain general classes of
premises. They stood, in fact, to the major premise in somewhat the
same relation that the Category or Predicament did to the
term. Crackanthorpe says of them, "sed duci a _loco probabiliter
arguendi_, hoc vere proprium est Argumentationis probabilis; et in
hoc a Demonstratione differt, quia Demonstrator utitur solummodo
quatuor Locis eisque necessariis.... Præter hos autem, ex quibus
quoque probabiliter arguere licet, sunt multo plures Loci arguendi
probabiliter; ut a Genere, a Specie, ab Adjuncto, ab Oppositis, et
similia" (_Logica_, Lib. V., ch. II.).
21. Stephen's _General View of the Criminal Law of England_, p. 241.
22. _Rationale of Judicial Evidence_; Bk. I. ch. VI.
23. Though this is claimed by some Kantian logicians;--Nie darf an
einem angeblichen Verbrecher die gesetzliche Strafe vollzogen
werden, bevor er nicht selbst das Verbrechen eingestanden. Denn wenn
auch alle Zeugnisse und die übrigen Anzeigen wider ihn wären, so
bleibt doch das Gegentheil immer möglich" (Krug, _Denklehre_, §131).
24. As Mr C. J. Monro puts it: "Suppose that a man is suspected of
murdering his daughter. Evidence which would not convict him before
an ordinary jury might make a grand jury find a true bill; evidence
which would not do this might make a coroner's jury bring in a
verdict against him; evidence which would not do this would very
often prevent a Chancery judge from appointing the man guardian to a
ward of the court; evidence which would not affect the judge's mind
might make a father think twice on his death-bed before he appointed
the man guardian to _his_ daughter."
25. The portions of this work which treat of the nature of proof in
general, and of judicial proof in particular, are well worth reading
by every logical student. It appears to me, however, that the author
goes much too far in the direction of regarding proof as subjective,
that is as what _does_ satisfy people, rather than as what _should_
satisfy them. He compares the legislative standard of certainty with
that of value; this latter is declared to be a certain weight of
gold, irrespective of the rarity or commonness of that metal. So
with certainty; if people grow more credulous the intrinsic value of
the standard will vary.
26. The question will be more fully discussed in a future chapter, but
a few words may be inserted here by way of indication. Reduce the
case to the simplest possible elements by supposing only two judges
or courts, of the same average correctness of decision. Let this be
indicated by x. Then the chance of their agreeing is
x^{2} + (1 - x)^{2}, for they agree if both are right or both wrong.
If the statistical frequency of this agreement is known, that is, the
frequency with which the first judgment is confirmed by the second,
we have the means of determining x.
27. Taylor on Evidence: the latter part of the extract does not seem
very clear.