ORDER OF TERMS, EULER'S DIAGRAMS, LOGICAL EQUATIONS, EXISTENTIAL IMPORT
OF PROPOSITIONS
Sec. 1. Of the terms of a proposition which is the Subject and which the
Predicate? In most of the exemplary propositions cited by Logicians it
will be found that the subject is a substantive and the predicate an
adjective, as in _Men are mortal_. This is the relation of Substance and
Attribute which we saw (chap. i. Sec. 5) to be the central type of
relations of coinherence; and on this model other predications may be
formed in which the subject is not a substance, but is treated as if it
were, and could therefore be the ground of attributes; as _Fame is
treacherous, The weather is changeable_. But, in literature, sentences
in which the adjective comes first are not uncommon, as _Loud was the
applause, Dark is the fate of man, Blessed are the peacemakers_, and so
on. Here, then, 'loud,' 'dark' and 'blessed' occupy the place of the
logical subject. Are they really the subject, or must we alter the order
of such sentences into _The applause was loud_, etc.? If we do, and then
proceed to convert, we get _Loud was the applause_, or (more
scrupulously) _Some loud noise was the applause_. The last form, it is
true, gives the subject a substantive word, but 'applause' has become
the predicate; and if the substantive 'noise' was not implied in the
first form, _Loud is the applause_, by what right is it now inserted?
The recognition of Conversion, in fact, requires us to admit that,
formally, in a logical proposition, the term preceding the copula is
subject and the one following is predicate. And, of course, materially
considered, the mere order of terms in a proposition can make no
difference in the method of proving it, nor in the inferences that can
be drawn from it.
Still, if the question is, how we may best cast a literary sentence into
logical form, good grounds for a definite answer may perhaps be found.
We must not try to stand upon the naturalness of expression, for _Dark
is the fate of man_ is quite as natural as _Man is mortal_. When the
purpose is not merely to state a fact, but also to express our feelings
about it, to place the grammatical predicate first may be perfectly
natural and most effective. But the grounds of a logical order of
statement must be found in its adaptation to the purposes of proof and
inference. Now general propositions are those from which most inferences
can be drawn, which, therefore, it is most important to establish, if
true; and they are also the easiest to disprove, if false; since a
single negative instance suffices to establish the contradictory. It
follows that, in re-casting a literary or colloquial sentence for
logical purposes, we should try to obtain a form in which the subject is
distributed--is either a singular term or a general term predesignate as
'All' or 'No.' Seeing, then, that most adjectives connote a single
attribute, whilst most substantives connote more than one attribute; and
that therefore the denotation of adjectives is usually wider than that
of substantives; in any proposition, one term of which is an adjective
and the other a substantive, if either can be distributed in relation to
the other, it is nearly sure to be the substantive; so that to take the
substantive term for subject is our best chance of obtaining an
universal proposition. These considerations seem to justify the practice
of Logicians in selecting their examples.
For similar reasons, if both terms of a proposition are substantive, the
one with the lesser denotation is (at least in affirmative
propositions) the more suitable subject, as _Cats are carnivores_. And
if one term is abstract, that is the more suitable subject; for, as we
have seen, an abstract term may be interpreted by a corresponding
concrete one distributed, as _Kindness is infectious_; that is, _All
kind actions suggest imitation_.
If, however, a controvertist has no other object in view than to refute
some general proposition laid down by an opponent, a particular
proposition is all that he need disentangle from any statement that
serves his purpose.
Sec. 2. Toward understanding clearly the relations of the terms of a
proposition, it is often found useful to employ diagrams; and the
diagrams most in use are the circles of Euler.
These circles represent the denotation of the terms. Suppose the
proposition to be _All hollow-horned animals ruminate_: then, if we
could collect all ruminants upon a prairie, and enclose them with a
circular palisade; and segregate from amongst them all the hollow-horned
beasts, and enclose them with another ring-fence inside the other; one
way of interpreting the proposition (namely, in denotation) would be
figured to us thus:
[Illustration: FIG. 1.]
An Universal Affirmative may also state a relation between two terms
whose denotation is co-extensive. A definition always does this, as _Man
is a rational animal_; and this, of course, we cannot represent by two
distinct circles, but at best by one with a thick circumference, to
suggest that two coincide, thus:
[Illustration: FIG. 2.]
The Particular Affirmative Proposition may be represented in several
ways. In the first place, bearing in mind that 'Some' means 'some at
least, it may be all,' an I. proposition may be represented by Figs. 1
and 2; for it is true that _Some horned animals ruminate_, and that
_Some men are rational_. Secondly, there is the case in which the 'Some
things' of which a predication is made are, in fact, not all; whilst the
predicate, though not given as distributed, yet might be so given if we
wished to state the whole truth; as if we say _Some men are Chinese_.
This case is also represented by Fig. 1, the outside circle representing
'Men,' and the inside one 'Chinese.' Thirdly, the predicate may
appertain to some only of the subject, but to a great many other things,
as in _Some horned beasts are domestic_; for it is true that some are
not, and that certain other kinds of animals are, domestic. This case,
therefore, must be illustrated by overlapping circles, thus:
[Illustration: FIG. 3.]
The Universal Negative is sufficiently represented by a single Fig. (4):
two circles mutually exclusive, thus:
[Illustration: FIG. 4.]
That is, _No horned beasts are carnivorous_.
Lastly, the Particular Negative may be represented by any of the Figs.
1, 3, and 4; for it is true that _Some ruminants are not hollow-horned_,
that _Some horned animals are not domestic_, and that _Some horned
beasts are not carnivorous_.
Besides their use in illustrating the denotative force of propositions,
these circles may be employed to verify the results of Obversion,
Conversion, and the secondary modes of Immediate Inference. Thus the
Obverse of A. is clear enough on glancing at Figs. 1 and 2; for if we
agree that whatever term's denotation is represented by a given circle,
the denotation of the contradictory term shall be represented by the
space outside that circle; then if it is true that _All hollow horned
animals are ruminants_, it is at the same time true that _No
hollow-horned animals are not-ruminants_; since none of the
hollow-horned are found outside the palisade that encloses the
ruminants. The Obverse of I., E. or O. may be verified in a similar
manner.
As to the Converse, a Definition is of course susceptible of Simple
Conversion, and this is shown by Fig. 2: 'Men are rational animals' and
'Rational animals are men.' But any other A. proposition is presumably
convertible only by limitation, and this is shown by Fig. 1; where _All
hollow-horned animals are ruminants_, but we can only say that _Some
ruminants are hollow-horned_.
That I. may be simply converted may be seen in Fig. 3, which represents
the least that an I. proposition can mean; and that E. may be simply
converted is manifest in Fig. 4.
As for O., we know that it cannot be converted, and this is made plain
enough by glancing at Fig. 1; for that represents the O., _Some
ruminants are not hollow-horned_, but also shows this to be compatible
with _All hollow-horned animals are ruminants_ (A.). Now in conversion
there is (by definition) no change of quality. The Converse, then, of
_Some ruminants are not hollow-horned_ must be a negative proposition,
having 'hollow-horned' for its subject, either in E. or O.; but these
would be respectively the contrary and contradictory of _All
hollow-horned animals are ruminants_; and, therefore, if this be true,
they must both be false.
But (referring still to Fig. 1) the legitimacy of contrapositing O. is
equally clear; for if _Some ruminants are not hollow-horned_, _Some
animals that are not hollow-horned are ruminants_, namely, all the
animals between the two ring-fences. Similar inferences may be
illustrated from Figs. 3 and 4. And the Contraposition of A. may be
verified by Figs. 1 and 2, and the Contraposition of E. by Fig. 4.
Lastly, the Inverse of A. is plain from Fig. 1--_Some things that are
not hollow-horned are not ruminants_, namely, things that lie outside
the outer circle and are neither 'ruminants' nor 'hollow-horned.' And
the Inverse of E may be studied in Fig. 4--_Some things that are
not-horned beasts are carnivorous_.
Notwithstanding the facility and clearness of the demonstrations thus
obtained, it may be said that a diagrammatic method, representing
denotations, is not properly logical. Fundamentally, the relation
asserted (or denied) to exist between the terms of a proposition, is a
relation between the terms as determined by their attributes or
connotation; whether we take Mill's view, that a proposition asserts
that the connotation of the subject is a mark of the connotation of the
predicate; or Dr. Venn's view, that things denoted by the subject (as
having its connotation) have (or have not) the attribute connoted by the
predicate; or, the Conceptualist view, that a judgment is a relation of
concepts (that is, of connotations). With a few exceptions artificially
framed (such as 'kings now reigning in Europe'), the denotation of a
term is never directly and exhaustively known, but consists merely in
'all things that have the connotation.' If the value of logical training
depends very much upon our habituating ourselves to construe
propositions, and to realise the force of inferences from them,
according to the connotation of their terms, we shall do well not to
turn too hastily to the circles, but rather to regard them as means of
verifying in denotation the conclusions that we have already learnt to
recognise as necessary in connotation.
Sec. 3. The equational treatment of propositions is closely connected with
the diagrammatic. Hamilton thought it a great merit of his plan of
quantifying the predicate, that thereby every proposition is reduced to
its true form--an equation. According to this doctrine, the proposition
_All X is all Y_ (U.) equates X and Y; the proposition _All X is some Y_
(A.) equates X with some part of Y; and similarly with the other
affirmatives (Y. and I.). And so far it is easy to follow his meaning:
the Xs are identical with some or all the Ys. But, coming to the
negatives, the equational interpretation is certainly less obvious. The
proposition _No X is Y_ (E.) cannot be said in any sense to equate X and
Y; though, if we obvert it into _All X is some not-Y_, we have (in the
same sense, of course, as in the above affirmative forms) X equated with
part at least of 'not-Y.'
But what is that sense? Clearly not the same as that in which
mathematical terms are equated, namely, in respect of some mode of
quantity. For if we may say _Some X is some Y_, these Xs that are also
Ys are not merely the same in number, or mass, or figure; they are the
same in every respect, both quantitative and qualitative, have the same
positions in time and place, are in fact identical. The proposition
2+2=4 means that any two things added to any other two are, _in respect
of number_, equal to any three things added to one other thing; and this
is true of all things that can be counted, however much they may differ
in other ways. But _All X is all Y_ means that Xs and Ys are the same
things, although they have different names when viewed in different
aspects or relations. Thus all equilateral triangles are equiangular
triangles; but in one case they are named from the equality of their
angles, and in the other from the equality of their sides. Similarly,
'British subjects' and 'subjects of King George V' are the same people,
named in one case from the person of the Crown, and in the other from
the Imperial Government. These logical equations, then, are in truth
identities of denotation; and they are fully illustrated by the
relations of circles described in the previous section.
When we are told that logical propositions are to be considered as
equations, we naturally expect to be shown some interesting developments
of method in analogy with the equations of Mathematics; but from
Hamilton's innovations no such thing results. This cannot be said,
however, of the equations of Symbolic Logic; which are the
starting-point of very remarkable processes of ratiocination. As the
subject of Symbolic Logic, as a whole, lies beyond the compass of this
work, it will be enough to give Dr. Venn's equations corresponding with
the four propositional forms of common Logic.
According to this system, universal propositions are to be regarded as
not necessarily implying the existence of their terms; and therefore,
instead of giving them a positive form, they are translated into symbols
that express what they deny. For example, the proposition _All devils
are ugly_ need not imply that any such things as 'devils' really exist;
but it certainly does imply that _Devils that are not ugly do not
exist_. Similarly, the proposition _No angels are ugly_ implies that
_Angels that are ugly do not exist_. Therefore, writing _x_ for
'devils,' _y_ for 'ugly,' and _[y]_ for 'not-ugly,' we may express A.,
the universal affirmative, thus:
A. _x[y]_ = 0.
That is, _x that is not y is nothing_; or, _Devils that are not-ugly do
not exist_. And, similarly, writing _x_ for 'angels' and _y_ for 'ugly,'
we may express E., the universal negative, thus:
E. _xy_ = 0.
That is, _x that is y is nothing_; or, _Angels that are ugly do not
exist_.
On the other hand, particular propositions are regarded as implying the
existence of their terms, and the corresponding equations are so framed
as to express existence. With this end in view, the symbol v is adopted
to represent 'something,' or indeterminate reality, or more than
nothing. Then, taking any particular affirmative, such as _Some
metaphysicians are obscure_, and writing _x_ for 'metaphysicians,' and
_y_ for 'obscure,' we may express it thus:
I. _xy_ = v.
That is, _x that is y is something_; or, _Metaphysicians that are
obscure do occur in experience_ (however few they may be, or whether
they all be obscure). And, similarly, taking any particular negative,
such as _Some giants are not cruel_, and writing _x_ for 'giants' and
_y_ for 'not-cruel,' we may express it thus:
O. _x[y]_ = v.
That is, _x that is not y is something_; or, _giants that are not-cruel
do occur_--in romances, if nowhere else.
Clearly, these equations are, like Hamilton's, concerned with
denotation. A. and E. affirm that the compound terms x[y] and xy have no
denotation; and I. and O. declare that x[y] and xy have denotation, or
stand for something. Here, however, the resemblance to Hamilton's system
ceases; for the Symbolic Logic, by operating upon more than two terms
simultaneously, by adopting the algebraic signs of operations, +,-, x, /
(with a special signification), and manipulating the symbols by
quasi-algebraic processes, obtains results which the common Logic
reaches (if at all) with much greater difficulty. If, indeed, the value
of logical systems were to be judged of by the results obtainable,
formal deductive Logic would probably be superseded. And, as a mental
discipline, there is much to be said in favour of the symbolic method.
But, as an introduction to philosophy, the common Logic must hold its
ground. (Venn: _Symbolic Logic_, c. 7.)
Sec. 4. Does Formal Logic involve any general assumption as to the real
existence of the terms of propositions?
In the first place, Logic treats primarily of the _relations_ implied in
propositions. This follows from its being the science of proof for all
sorts of (qualitative) propositions; since all sorts of propositions
have nothing in common except the relations they express.
But, secondly, relations without terms of some sort are not to be
thought of; and, hence, even the most formal illustrations of logical
doctrines comprise such terms as S and P, X and Y, or x and y, in a
symbolic or representative character. Terms, therefore, of some sort are
assumed to exist (together with their negatives or contradictories) _for
the purposes of logical manipulation_.
Thirdly, however, that Formal Logic cannot as such directly involve the
existence of any particular concrete terms, such as 'man' or 'mountain,'
used by way of illustration, is implied in the word 'formal,' that is,
'confined to what is common or abstract'; since the only thing common to
all terms is to be related in some way to other terms. The actual
existence of any concrete thing can only be known by experience, as with
'man' or 'mountain'; or by methodically justifiable inference from
experience, as with 'atom' or 'ether.' If 'man' or 'mountain,' or
'Cuzco' be used to illustrate logical forms, they bring with them an
existential import derived from experience; but this is the import of
language, not of the logical forms. 'Centaur' and 'El Dorado' signify to
us the non-existent; but they serve as well as 'man' and 'London' to
illustrate Formal Logic.
Nevertheless, fourthly, the existence or non-existence of particular
terms may come to be implied: namely, wherever the very fact of
existence, or of some condition of existence, is an hypothesis or datum.
Thus, given the proposition _All S is P_, to be P is made a condition of
the existence of S: whence it follows that an S that is not P does not
exist (_x[y]_ = 0). On the further hypothesis that S exists, it follows
that P exists. On the hypothesis that S does not exist, the existence of
P is problematic; but, then, if P does exist we cannot convert the
proposition; since _Some P is S_ (P existing) would involve the
existence of S; which is contrary to the hypothesis.
Assuming that Universals _do not_, whilst Particulars _do_, imply the
existence of their subjects, we cannot infer the subalternate (I. or O.)
from the subalternans (A. or E.), for that is to ground the actual on
the problematic; and for the same reason we cannot convert A. _per
accidens_.
Assuming, again, a certain _suppositio_ or universe, to which in a given
discussion every argument shall refer, then, any propositions whose
terms lie outside that _suppositio_ are irrelevant, and for the purposes
of that discussion are sometimes called "false"; though it seems better
to call them irrelevant or meaningless, seeing that to call them false
implies that they might in the same case be true. Thus propositions
which, according to the doctrine of Opposition, appear to be
Contradictories, may then cease to be so; for of Contradictories one is
true and the other false; but, in the case supposed, both are
meaningless. If the subject of discussion be Zoology, all propositions
about centaurs or unicorns are absurd; and such specious
Contradictories as _No centaurs play the lyre--Some centaurs do play the
lyre_; or _All unicorns fight with lions--Some unicorns do not fight
with lions_, are both meaningless, because in Zoology there are no
centaurs nor unicorns; and, therefore, in this reference, the
propositions are not really contradictory. But if the subject of
discussion or _suppositio_ be Mythology or Heraldry, such propositions
as the above are to the purpose, and form legitimate pairs of
Contradictories.
In Formal Logic, in short, we may make at discretion any assumption
whatever as to the existence, or as to any condition of the existence of
any particular term or terms; and then certain implications and
conclusions follow in consistency with that hypothesis or datum. Still,
our conclusions will themselves be only hypothetical, depending on the
truth of the datum; and, of course, until this is empirically
ascertained, we are as far as ever from empirical reality. (Venn:
_Symbolic Logic_, c. 6; Keynes: _Formal Logic_, Part II. c. 7: _cf._
Wolf: _Studies in Logic_.)