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Existential
Graphs - 4.418-529 |
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Chapter 4: On Existential
Graphs, Euler's Diagrams, and Logical Algebra |
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CHAPTER 4 |
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ON EXISTENTIAL GRAPHS,
EULER'S DIAGRAMS,
AND LOGICAL ALGEBRAP(*1) |
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§INTRODUCTION |
| 418. |
A diagram is a representamen
(*2)
which is predominantly an icon of
relations and is aided to be so by conventions.
Indices are also more or less
used. It should be carried out upon a perfectly
consistent system of
representation, founded upon a simple and easily
intelligible basic idea. |
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| 419. |
A graph is a superficial diagram composed of the sheet upon which
it is written or drawn, of spots or their equivalents, of lines of connection,
and (if need be) of enclosures. The type, which it is supposed more or less to
resemble, is the structural formula of the chemist. |
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| 420. |
A logical graph is a graph representing logical relations iconically, so as to be an aid to logical analysis. |
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| 421. |
An existential graph is a logical graph governed by a. system of
representation founded upon the idea that the sheet upon which it is written,
as well as every portion of that sheet, represents one recognized universe,
real or fictive, and that every graph drawn on that sheet, and not cut off from
the main body of it by an enclosure, represents some fact existing in that
universe, and represents it independently of the representation of another such
fact by any other graph written upon another part of the sheet, these graphs,
however, forming one composite graph. |
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| 422. |
No other system of existential graphs than that herein set forth
having hitherto been proposed, this one will need, for the present, no more
distinctive designation. Should such designation hereafter become desirable, I
desire that this system should be called the Existential System of 1897, in
which year I wrote an account of it and offered it for publication to the
Editor of The Monist, who declined it on the ground that it might later be
improved upon. No changes have been found desirable since that date, although
it has been under continual examination; but the exposition has been rendered
more formal. |
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| 423. |
The following exposition of this system will be arranged as
follows: Part I will explain the expression of ordinary forms of language in
graphs and the interpretation of the latter into the former in three sections,
as follows:
A will state all the fundamental conventions of the system, separating those which are essentially different, showing the need which each is designed to meet together with the reasons for meeting it by the particular convention chosen, so far as these can be given at this stage of the development. A complete discussion will be given in an Appendix
(*1) to this part. To aid the
understanding of all this, various logical analyses will be interspersed where they become pertinent.
B will enunciate other rules of interpretation whose validity will be demonstrated from the fundamental conventions as premisses. This section will also introduce certain modifications of some of the signs established in A, the modified signs being convenient, although good reasons forbid their being considered fundamental.
C will redescribe the system in a compact form, which, on account of its uniting into one many rules that had, in the first instance; to be considered separately, is more easily grasped and retained in the mind.
Part II will develop formal "rules," or permissions, by which one graph
may be transformed into another without danger of passing from truth to falsity
and without recurring to any interpretation of the graphs; such transformations
being of the nature of immediate inferences. The part will be divided into
sections corresponding to those of Part I.
A will prove the basic rules of transformation directly from the fundamental conventions of A of Part I.
B will deduce further rules of transformation from those of A, without further recourse to the principles of transformation.
C will restate the rules in more compact form.
Part III will show how the system may be made useful.(*1) |
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Part I. PRINCIPLES OF INTERPRETATION
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A. Fundamental Conventions
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§1. OF CONVENTIONS NOS. 1 AND 2(*1)
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| 424. |
In order to understand why this system of expression has the
construction it has, it is indispensable to grasp the precise purpose of it,
and not to confuse this with four other purposes, to wit:
- First, although the study of it and practice with it will be highly
useful in helping to train the mind to accurate thinking, still that
consideration has not had any influence in determining the characters of the
signs employed; and an exposition of it, which should have that aim, ought to
be based upon psychological researches of which it is impossible here to take
account.
- Second, this system is not intended to serve as a universal language
for mathematicians or other reasoners, like that of Peano.
- Third, this system is not intended as a calculus, or apparatus by which
conclusions can be reached and problems solved with greater facility than by
more familiar systems of expression. Although some writers
(*2) have studied the
logical algebras invented by me with that end apparently in view, in my own
opinion their structure, as well as that of the present system, is quite
antagonistic to much utility of that sort. The principal desideratum in a
calculus is that it should be able to pass with security at one bound over a
series of difficult inferential steps. What these abbreviated inferences may
best be, will depend upon the special nature of the subject under discussion.
But in my algebras and graphs, far from anything of that sort being attempted,
the whole effort has been to dissect the operations of inference into as many
distinct steps as possible.
- Fourth, although there is a certain fascination about these graphs, and
the way they work is pretty enough, yet the system is not intended for a
plaything, as logical algebra has sometimes been made, but has a very serious
purpose which I proceed to explain.
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| 425. |
Admirable as the work of research of the special sciences --
physical and psychical -- is, as a whole, the reasoning [employed in them] is
of an elementary kind except when it is mathematical, and it is not
infrequently loose. The philosophical sciences are greatly inferior to the
special sciences in their reasoning. Mathematicians alone reason with great
subtlety and great precision. But hitherto nobody has succeeded in giving a
thoroughly satisfactory logical analysis of the reasoning of mathematics. That
is to say, although every step of the reasoning is evidently such that the
collective premisses cannot be true and yet the conclusion false, and although
for each such step, A, we are able to draw up a self-evident general rule that
from a premiss of such and such a form such and such a form of conclusion will
necessarily follow, this rule covering the particular inferential step, A, yet
nobody has drawn up a complete list of such rules covering all mathematical
inferences. It is true that mathematics has its calculus which solves problems
by rules which are fully proved; but, in the first place, for some branches of
the calculus those proofs have not been reduced to self-evident rules, and in
the second place, it is only routine work which can be done by simply following
the rules of the calculus, and every considerable step in mathematics is
performed in other ways. |
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| 426. |
If we consult the ordinary treatises on logic for an account of
necessary reasoning, all the help that they afford is the rules of syllogism.
They pretend that ordinary syllogism explains the reasoning of mathematics; and
books have professed to exhibit considerable parts of the reasoning of the
first book of Euclid's Elements stated in the form of syllogisms. But if this
statement is examined, it will be found that it represents transformations of
statements to be made that are not reduced to strict syllogistic form; and on
examination it will be found that it is precisely in these transformations that
the whole gist of the reasoning lies. The nearest approach to a logical
analysis of mathematical reasoning that has ever been made was Schröder's
statement, with improvements, in a logical algebra of my invention, of
Dedekind's reasoning (itself in a sort of logical form) concerning the
foundations of arithmetic.(*1) But though this relates only to an exceptionally
simple kind of mathematics, my opinion -- quite against my natural leanings
toward my own creation -- is that the soul of the reasoning has even here not
been caught in the logical net. |
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| 427. |
No other book has, during the nineteenth century, been deeply
studied by so large a proportion of the strong intellects of the civilized
world as Kant's Critic of the Pure Reason; and the reason has undoubtedly been
that they have all been greatly struck by Kant's logical power. Yet Kant, for
all this unquestionable power, had paid so little attention to logic that he
makes it manifest that he supposed that ordinary syllogism explains
mathematical reasoning, and indeed [in] the simplest mood of syllogism,
Barbara. Now, at the very utmost, from n propositions only 1/4n2 conclusions
can be drawn by Barbara. In the thirteen books of Euclid's Elements there [are]
14 premisses (5 postulates and 9 axioms) excluding the definitions, which are
merely verbal. Therefore, even if these premisses were related to one another
in the most favorable way, which is far from being the case, there could only
be 49 conclusions from them. But Euclid draws over ten times that number (465
propositions, 27 corollaries, and 17 lemmas) besides which his editors have
inserted hundreds of corollaries. There are 48 propositions in the first book.
Moreover, in Barbara or any sorites, or complexus of such syllogisms, to
introduce the same premiss twice is idle. But throughout mathematics the same
premisses are used over and over again. Moreover a person of fairly good mind
and some logical training will instantly see the syllogistic conclusions from
any number of premisses. But this is far from being true of mathematical
inferences. |
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| 428. |
There is reason to believe that a thorough understanding of the
nature of mathematical reasoning would lead to great improvements in
mathematics. For when a new discovery is made in mathematics, the demonstration
first found is almost always replaced later by another much simpler. Now it may
be expected that, if the reasoning were thoroughly understood, the unnecessary
complications of the first proof would be eliminable at once. Indeed, one might
expect that the shortest route would be taken at the outset. Then again,
consider the state of topical geometry, or geometrical topics, otherwise called
topology. Here is a branch of geometry which not only leaves out of
consideration the proportions of the different dimensions of figures and the
magnitudes of angles (as does also graphics, or projective geometry --
perspective, etc.) but also leaves out of account the straightness or mode of
curvature of lines and the flatness or mode of bending of surfaces, and
confines itself entirely to the connexions of the parts of figures
(distinguishing, for example, a ring from a ball). Ordinary metric geometry
equally depends on the connections of parts; but it depends on much besides.
It, therefore, is a far more complicated subject, and can hardly fail to be of
its own nature much the more difficult. And yet geometrical topics stands idle
with problems to all appearance very simple staring it unsolved in the face,
merely because mathematicians have not found out how to reason about it. Now a
thorough understanding of mathematical reasoning must be a long stride toward
enabling us to find a method of reasoning about this subject as well, very
likely, as about other subjects that are not even recognized to be
mathematical. |
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| 429. |
This, then, is the purpose for which my logical algebras were
designed but which, in my opinion, they do not sufficiently fulfill. The
present system of existential graphs is far more perfect in that respect, and
has already taught me much about mathematical reasoning. Whether or not it will
explain all mathematical inferences is not yet known.
Our purpose, then, is to study the workings of necessary inference.
What we want, in order to do this, is a method of representing diagrammatically
any possible set of premisses, this diagram to be such that we can observe the
transformation of these premisses into the conclusion by a series of steps each
of the utmost possible simplicity. |
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| 430. |
What we have to do, therefore, is to form a perfectly consistent
method of expressing any assertion diagrammatically. The diagram must then
evidently be something that we can see and contemplate. Now what we see appears
spread out as upon a sheet. Consequently our diagram must be drawn upon a
sheet. We must appropriate a sheet to the purpose, and the diagram drawn or
written on the sheet is to express an assertion. We can, then, approximately
call this sheet our sheet of assertion. The entire graph, or all that is drawn
on the sheet, is to express a proposition, which the act of writing is to
assert. |
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| 431. |
But what are
our assertions to be about? The answer must be that they
are to be about an arbitrarily hypothetical universe, a
creation of a mind. For it is necessary
reasoning alone that we intend to study; and the
necessity of such reasoning consists in this, that not
only does the conclusion happen to be true of a
pre-determinate universe, but will be true, so
long as the premisses are true, howsoever the universe may subsequently turn out to be
determined. Thus, conformity to an existing, that is, entirely determinate,
universe does not make necessity, which consists in what always will be, that
is, what is determinately true of a universe not yet entirely determinate.
Physical necessity consists in the fact that whatever may happen will conform
to a law of nature; and logical necessity, which is what we have here to deal
with, consists of something being determinately true of a universe not entirely
determinate as to what is true, and thus not existent. In order to fix our
ideas, we may imagine that there are two persons, one of whom, called the grapheus, creates the universe by the continuous development of his idea of it,
every interval of time during the process adding some fact to the universe,
that is, affording justification for some assertion, although, the process
being continuous, these facts are not distinct from one another in their mode
of being, as the propositions, which state some of them, are. As fast as this
process in the mind of the grapheus takes place, that which is thought acquires
being, that is, perfect definiteness, in the sense that the effect of what, is
thought in any lapse of time, however short, is definitive and irrevocable; but
it is not until the whole operation of creation is complete that the universe
acquires existence, that is, entire determinateness, in the sense that nothing
remains undecided. The other of the two persons concerned, called the graphist,
is occupied during the process of creation in making successive modifications
(i.e., not by a continuous process, since each modification, unless it be
final, has another that follows next after it), of the entire graph.
Remembering that the entire graph is whatever is, at any time, expressed in
this system on the sheet of assertion, we may note that before anything has
been drawn on the sheet, the blank is, by that definition, a graph. It may be
considered as the expression of whatever must be well-understood between the graphist and the interpreter of the graph before the latter can understand what
to expect of the graph. There must be an interpreter, since the graph, like
every sign founded on convention, only has the sort of being that it has if it
is interpreted; for a conventional sign is neither a mass of ink on a piece of
paper or any other individual existence, nor is it an image present to
consciousness, but is a special habit or rule of interpretation and consists
precisely in the fact that certain sorts of ink spots -- which I call its
replicas -- will have certain effects on the conduct, mental and bodily, of the
interpreter. So, then, the blank of the blank sheet may be considered as
expressing that the universe, in process of creation by the grapheus, is
perfectly definite and entirely determinate, etc. Hence, even the first writing
of a graph on the sheet is a modification of the graph already written. The
business of the graphist is supposed to come to an end before the work of
creation is accomplished. He is supposed to be a mind-reader to such an extent
that he knows some (perhaps all) the creative work of the grapheus so far as it
has gone, but not what is to come. What he intends the graph to express
concerns the universe as it will be when it comes to exist. If he risks an
assertion for which he has no warrant in what the grapheus has yet thought, it
may or may not prove true. |
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| 432. |
The above
considerations constitute a sufficient reason for
adopting the following convention, which is hereby
adopted:
Convention No. 1. A certain sheet, called the sheet of assertion, is
appropriated to the drawing upon it of such graphs that whatever may be at any
time drawn upon it, called the entire graph, shall be regarded as expressing an
assertion by an imaginary person, called the graphist, concerning a universe,
perfectly definite and entirely determinate, but the arbitrary creation of an
imaginary mind, called the grapheus. |
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| 433. |
The convention which has next to be considered is the most
arbitrary of all. It is, nevertheless, founded on two good reasons. A diagram
ought to be as iconic as possible; that is, it should represent relations by
visible relations analogous to them. Now suppose the graphist finds himself
authorized to write each of two entire graphs. Say, for example, that he can
draw:
The pulp of some oranges is red;
and that he is equally authorized to draw:
To express oneself naturally is the last perfection
of a writer's art.
Each proposition is true independently of the other, and either may therefore
be expressed on the sheet of assertion. If both are written on different parts
of the sheet of assertion, the independent presence on the sheet of the two
expressions is analogous to the independent truth of the two propositions that
they would, when written separately, assert. It would, therefore, be a highly
iconic mode of representation to understand,
The pulp of some oranges is red.
To express oneself naturally is the last perfection of a writer's art.
where both are written on different parts of the sheet, as the assertion of
both propositions. |
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| 434. |
It is a subsidiary recommendation of a mode of diagrammatization,
but one which ought to be accorded some weight, that it is one that the nature
and habits of our minds will cause us at once to understand, without our being
put to the trouble of remembering a rule that has no relation to our natural
and habitual ways of expression. Certainly, no convention of representation
could possess this merit in a higher degree than the plan of writing both of
two assertions in order to express the truth of both. It is so very natural,
that all who have ever used letters or almost any method of graphic
communication have resorted to it. It seems almost unavoidable, although in my
first invented system of graphs, which I call entitative graphs,(*1)
propositions
written on the sheet together were not understood to be independently asserted
but to be alternatively asserted. The consequence was that a blank sheet
instead of expressing only what was taken for granted had to be interpreted as
an absurdity. One system seems to be about as good as the other, except that
unnaturalness and aniconicity haunt every part of the system of entitative
graphs, which is a curious example of how late a development simplicity is.
These two reasons will suffice to make every reader very willing to accede to
the following convention, which is hereby adopted. Convention No. 2. Graphs on different parts of the sheet, called
partial graphs, shall independently assert what they would severally assert,
were each the entire graph. |
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§2. OF CONVENTION NO. 3
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| 435. |
If a system of expression is to be adequate to the analysis of all
necessary consequences,(*P1) it is requisite that it should be able to express
that an expressed consequent, C, follows necessarily from an expressed
antecedent, A. The conventions hitherto adopted do not enable us to express
this. In order to form a new and reasonable convention for this purpose we must
get a perfectly distinct idea of what it means to say that a consequent follows
from an antecedent. It means that in adding to an assertion of the antecedent
an assertion of the consequent we shall be proceeding upon a general principle
whose application will never convert a true assertion into a false one. This,
of course, means that so it will be in the universe of which alone we are
speaking. But when we talk logic -- and people occasionally insert logical
remarks into ordinary discourse -- our universe is that universe which embraces
all others, namely The Truth, so that, in such a case, we mean that in no
universe whatever will the addition of the assertion of the consequent to the
assertion of the antecedent be a conversion of a true proposition into a false
one. But before we can express any proposition referring to a general
principle, or, as we say, to a "range of possibility," we must first find means
to express the simplest kind of conditional proposition, the conditional de inesse, in which "If A is true, C is true" means only that, principle or no
principle, the addition to an assertion of A of an assertion of C will not be a
conversion of a true assertion into a false one. That is, it asserts that the
graph of Fig. 69, anywhere on the sheet of assertion, might be transformed into
the graph of Fig. 70 without passing from truth to falsity. |
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Figure 69 |
Figure 70 |
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This conditional
de inesse has to be expressed as a graph in such a way as
distinctly to express in our system both a and c, and to exhibit their relation
to one another. To assert the graph thus expressing the conditional de inesse,
it must be drawn upon the sheet of assertion, and in this graph the expressions
of a and of c must appear; and yet neither a nor c must be drawn upon the sheet
of assertion. How is this to be managed? Let us draw a closed line which we may
call a sep (sæpes, a fence), which shall cut off its contents from the sheet of
assertion. Let this sep together with all that is within it, considered as a
whole, be called an enclosure, this close, being written on the sheet of
assertion, shall assert the conditional de inesse; but that which it encloses,
considered separately from the sep, shall not be considered as on the sheet of
assertion. Then, obviously, the antecedent and consequent must be in separate
compartments of the close. In order to make the representation of the relation
between them iconic, we must ask ourselves what spatial relation is analogous
to their relation. Now if it be true that "If a is true, b is true" and "If b
is true, c is true," then it is true that "If a is true, c is true." This is
analogous to the geometrical relation of inclusion. So naturally striking is
the analogy as to be (I believe) used in all languages to express the logical
relation; and even the modern mind, so dull about metaphors, employs this one
frequently. It is reasonable, therefore, that one of the two compartments
should be placed within the other. But which shall be made the inner one? Shall
we express the conditional de inesse by Fig. 71 or by Fig. 72? In order to
decide which is the more appropriate mode of representation, one should observe
that the consequent of a conditional proposition asserts what is true, not
throughout the whole universe of possibilities considered, but in a subordinate
universe marked off by the antecedent. This is not a fanciful notion, but a
truth. Now in Fig. 72, the consequent appears in a special part of the sheet
representing the universe, the space between the two lines containing the
definition of the sub-universe.
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Figure 71(*1) |
Figure 72 |
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There is no such expressiveness in Fig. 71 -- or, if there be, it is only of a
superficial and fanciful sort. Moreover, the necessity of using two kinds of
enclosing lines -- a necessity which, we shall find, does not exist in Fig. 72
-- is a defect of Fig. 71; and when we come to consider the question of
convenience, the superiority of Fig. 72 will appear still more strongly. This,
then, will be the method for us to adopt.
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| 436. |
The two seps
of Fig. 72, taken together, form a curve which I shall
call a scroll. The node is of no particular
significance. The scroll may equally well be drawn as in
Fig. 73. |
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Figure 73 |
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The only
essential feature is that there should be two seps, of which the inner, however drawn,
may be called the inloop. The node merely serves to aid the mind in the
interpretation, and will be used only when it can have this effect. The two
compartments will be called the inner, or second, close, and the
outer close,
the latter excluding the former. The outer close considered as containing the inloop will be called the
close. |
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| 437. |
Convention No. 3. An
enclosure shall be a graph consisting of a
scroll with its contents.
The scroll shall be a real curve of two closed branches, the one within
the other, called seps, and the inner specifically called the
loop; and these
branches may or may not be joined at a node.
The contents of the scroll shall consist of whatever is in the area
enclosed by the outer sep, this area being called the close and
consisting of the inner, or second,
close, which is the area enclosed by the loop,
and the outer, or first close,
which is the area outside the loop but inside the outer
sep.
When an enclosure is written on the sheet of assertion, although it is
asserted as a whole, its contents shall be cut off from the sheet, and shall
not be asserted in the assertion of the whole. But the enclosure shall assert
de inesse that if every graph in the outer close be true, then every graph in
the inner close is true. |
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§3. OF CONVENTIONS NOS. 4 TO 9(*1P) |
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438. |
Let a heavy dot or dash be used in place of a noun which has been
erased from a proposition. A blank form of proposition produced by such
erasures as can be filled, each with a proper name, to make a proposition
again, is called a rhema, or, relatively to the proposition of which it is
conceived to be a part, the predicate of that proposition. The following are
examples of rhemata: |
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Every proposition has one predicate and one only. But what that predicate is
considered to be depends upon how we choose to analyze it. Thus, the
proposition |
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may be considered as having for its predicate either of the following rhemata: |
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In the last case the entire proposition is considered as predicate. A rhema
which has one blank is called a monad; a rhema of two blanks, a dyad; a rhema
of three blanks, a triad; etc. A rhema with no blank is called a medad, and is
a complete proposition. A rhema of more than two blanks is a polyad. A rhema of
more than one blank is a relative. Every proposition has an ultimate predicate,
produced by putting a blank in every place where a blank can be placed, without
substituting for some word its definition. Were this done we should call it a
different proposition, as a matter of nomenclature. If on the other hand, we
transmute the proposition without making any difference as to what it leaves
unanalyzed, we say the expression only is different, as, if we say, |
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Each part of a proposition which might be replaced by a proper name, and still
leave the proposition a proposition is a subject of the proposition.(*P1) It is,
however, the rhema which we have just now to attend to. |
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| 439. |
A rhema is, of course, not a proposition. Supposing, however, that
it be written on the sheet of assertion, so that we have to adopt a meaning for
it as a proposition, what can it most reasonably be taken to mean? Take, for
example, Fig. 74. Shall this, since it represents the universe, be taken to
mean that "Something in the universe is beautiful," or that "Anything in the
universe is beautiful," or that "The universe, as a whole, is beautiful"? The
last interpretation may be rejected at once for the reason that we are
generally unable to assert anything of the universe not reducible to one of the
other forms except what is well-understood between graphist and interpreter. We
have, therefore, to choose between interpreting Fig. 74 to mean "Something is
beautiful" |
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Figure 74 |
Figure 75 |
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and to mean "Anything is beautiful." Each asserts the rhema of an individual;
but the former leaves that individual to be designated by the grapheus, while
the latter allows the rhema [interpreter q to fill the blank with any proper
name he likes. If Fig. 74 be taken to mean "Something is beautiful," then Fig.
75 will mean "Everything is beautiful"; while if Fig. 74 be taken to mean
"Everything is beautiful," then Fig. 75 will mean "Something is beautiful." In
either case, therefore, both propositions will be expressible, and the main
question is, which gives the most appropriate expressions? The question of
convenience is subordinate, as a general rule; but in this case the difference
is so vast in this respect as to give this consideration more than its usual
importance. |
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| 440. |
In order to decide the question of appropriateness, we must ask
which form of proposition, the universal or the particular, "Whatever
salamander there may be lives in fire," or "Some existing salamander lives in
fire," is more of the nature of a conditional proposition; for plainly, these
two propositions differ in form from "Everything is beautiful" and "Something
is beautiful" respectively, only in their being limited to a subsidiary
universe of salamanders. Now to say "Any salamander lives in fire" is merely to
say "If anything, X, is a salamander, X lives in fire." It differs from a
conditional, if at all, only in the identification of X which it involves. On
the other hand, there is nothing at all conditional in saying "There is a
salamander, and it lives in fire." Thus the interpretation of Fig. 74 to mean "Something is beautiful" is
decidedly the more appropriate; and since reasonable arrangements generally
prove to be the most convenient in the end, we shall not be surprised when we
come to find, as we shall, the same interpretation to be incomparably the
superior in that respect also.
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| 441. |
Convention No. 4. In this system, the unanalyzed expression of a rhema shall be called a
spot. A distinct place on its periphery shall be
appropriated to each blank, which place shall be called a hook. A spot with a
dot at each hook shall be a graph expressing the proposition which results from
filling every blank of the rhema with a separate sign of an indesignate
individual existing in the universe and belonging to some determinate category,
usually that of "things." |
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| 442. |
In many reasonings it becomes necessary to write a copulative
proposition in which two members relate to the same individual so as to
distinguish these members. Thus we have to write such a proposition as,
A is greater than something that is greater than B,
so as to exhibit the two partial graphs of Fig. 76. |
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Figure 76 |
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The proposition we wish to express adds to those of Fig. 76 the assertion of
the identity of the two "somethings." But this addition cannot be effected as
in Fig. 77. |
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Figure 77 |
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For the "somethings," being indesignate, cannot be described in general terms.
It is necessary that the signs of them should be connected in fact. No way of
doing this can be more perfectly iconic than that exemplified in Fig. 78. |
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Figure 78 |
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Any sign of such identification of individuals may be called a
connexus, and the particular sign here used, which we shall do well to adopt,
may be called a line of identity. |
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| 443. |
Convention No. 5. Two coincident points, not more, shall denote
the same individual. |
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| 444. |
Convention
No. 6. A heavy line, called a line of identity,
shall be a graph asserting the numerical identity of the
individuals denoted by its two extremities. |
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| 445. |
The next
convention to be laid down is so perfectly natural that
the reader may well have a difficulty in perceiving that
a separate convention is required for it. Namely, we may
make a line of identity branch to express the identity
of three individuals. Thus, Fig. 79 |
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 |
|
Figure 79 |
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will express
that some black bird is thievish. No doubt, it would
have been easy to draw up Convention No. 4 in such a
form as to cover this procedure. But it is not our
object in this section to find ingenious modes of
statement which, being borne in mind, may serve as rules
for as many different acts as possible. On the contrary,
what we are here concerned to do is to distinguish all
proceedings that are essentially different. Now it is
plain that no number of mere bi-terminal bonds, each
terminal occupying a spot's hook, can ever assert the
identity of three things, although |
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 |
|
Figure 80 |
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when we once
have a three-way branch, any higher number of terminals
can be produced from it, as in Fig. 80. |
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| 446. |
We ought to, and must, then, make a distinct convention to cover
this procedure, as follows: Convention No. 7. A branching line of identity shall express a triad
rhema signifying the identity of the three individuals, whose designations are
represented as filling the blanks of the rhema by coincidence with the three
terminals of the line. |
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| 447. |
Remark how peculiar a sign the line of identity is. A sign, or, to
use a more general and more definite term, a representamen, is of one or other
of three kinds:(*1) it is either an icon, an index, or a symbol. An icon is a representamen of what it represents and for the mind that interprets it as
such, by virtue of its being an immediate image, that is to say by virtue of
characters which belong to it in itself as a sensible object, and which it
would possess just the same were there no object in nature that it resembled,
and though it never were interpreted as a sign. It is of the nature of an
appearance, and as such, strictly speaking, exists only in consciousness,
although for convenience in ordinary parlance and when extreme precision is not
called for, we extend the term icon to the outward objects which excite in
consciousness the image itself. A geometrical diagram is a good example of an
icon. A pure icon can convey no positive or factual information; for it affords
no assurance that there is any such thing in nature. But it is of the utmost
value for enabling its interpreter to study what would be the character of such
an object in case any such did exist. Geometry sufficiently illustrates that.
Of a completely opposite nature is the kind of representamen termed an index.
This is a real thing or fact which is a sign of its object by virtue of being
connected with it as a matter of fact and by also forcibly intruding upon the
mind, quite regardless of its being interpreted as a sign. It may simply serve
to identify its object and assure us of its existence and presence. But very
often the nature of the factual connexion of the index with its object is such
as to excite in consciousness an image of some features of the object, and in
that way affords evidence from which positive assurance as to truth of fact may
be drawn. A photograph, for example, not only excites an image, has an
appearance, but, owing to its optical connexion with the object, is evidence
that that appearance corresponds to a reality. A symbol is a representamen
whose special significance or fitness to represent just what it does represent
lies in nothing but the very fact of there being a habit, disposition, or other
effective general rule that it will be so interpreted. Take, for example, the
word "man." These three letters are not in the least like a man; nor is the
sound with which they are associated. Neither is the word existentially
connected with any man as an index. It cannot be so, since the word is not an
existence at all. The word does not consist of three films of ink. If the word
"man" occurs hundreds of times in a book of which myriads of copies are
printed, all those millions of triplets of patches of ink are embodiments of
one and the same word. I call each of those embodiments a replica of the
symbol. This shows that the word is not a thing. What is its nature? It
consists in the really working general rule that three such patches seen by a
person who knows English will effect his conduct and thoughts according to a
rule. Thus the mode of being of the symbol is different from that of the icon
and from that of the index. An icon has such being as belongs to past
experience. It exists only as an image in the mind. An index has the being of
present experience. The being of a symbol consists in the real fact that
something surely will be experienced if certain conditions be satisfied.
Namely, it will influence the thought and conduct of its interpreter. Every
word is a symbol. Every sentence is a symbol. Every book is a symbol. Every
representamen depending upon conventions is a symbol. Just as a photograph is an
index having an icon incorporated into it, that is,
excited in the mind by its force, so a symbol may have
an icon or an index incorporated into it, that is, the
active law that it is may require its interpretation to
involve the calling up of an image, or a composite
photograph of many images of past experiences, as
ordinary common nouns and verbs do; or it may require
its interpretation to refer to the actual surrounding
circumstances of the occasion of its embodiment, like
such words as that, this, I, you, which, here, now,
yonder, etc. Or it may be pure symbol, neither iconic
nor indicative, like the words and, or, of, etc. Peirce: |
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| 448. |
The value of an icon consists in its exhibiting the features of a
state of things regarded as if it were purely imaginary. The value of an index
is that it assures us of positive fact. The value of a symbol is that it serves
to make thought and conduct rational and enables us to predict the future. It
is frequently desirable that a representamen should exercise one of those three
functions to the exclusion of the other two, or two of them to the exclusion of
the third; but the most perfect of signs are those in which the iconic,
indicative, and symbolic characters are blended as equally as possible. Of this
sort of signs the line of identity is an interesting example. As a conventional
sign, it is a symbol; and the symbolic character, when present in a sign, is of
its nature predominant over the others. The line of identity is not, however,
arbitrarily conventional nor purely conventional. Consider any portion of it
taken arbitrarily (with certain possible exceptions shortly to be considered)
and it is an ordinary graph for which Fig. 81 might perfectly well be
substituted. But when we consider the |
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Figure 81 |
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connexion of this portion with a next adjacent portion, although the two
together make up the same graph, yet the identification of the something, to
which the hook of the one refers, with the something, to which the hook of the
other refers, is beyond the power of any graph to effect, since a graph, as a
symbol, is of the nature of a law, and is therefore general, while here there
must be an identification of individuals. This identification is effected not
by the pure symbol, but by its replica which is a thing. The termination of one
portion and the beginning of the next portion denote the same individual by
virtue of a factual connexion, and that the closest possible; for both are
points, and they are one and the same point. In this respect, therefore, the
line of identity is of the nature of an index. To be sure, this does not affect
the ordinary parts of a line of identity, but so soon as it is even conceived,
[it is conceived] as composed of two portions, and it is only the factual
junction of the replicas of these portions that makes them refer to the same
individual. The line of identity is, moreover, in the highest degree iconic.
For it appears as nothing but a continuum of dots, and the fact of the identity
of a thing, seen under two aspects, consists merely in the continuity of being
in passing from one apparition to another. Thus uniting, as the line of
identity does, the natures of symbol, index, and icon, it is fitted for playing
an extraordinary part in this system of representation. |
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| 449. |
There is no difficulty in interpreting the line of identity until
it crosses a sep. To interpret it in that case, two new conventions will be
required. How shall we express the proposition "Every salamander lives in fire,"
or "If it be true that something is a salamander then it will always be true
that that something lives in fire"? If we omit the assertion of the identity of
the somethings, the expression is obviously given in Fig. 82. |
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Figure 82 |
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To that, we
wish to add the expression of individual identity. We
ought to use our line of identity for that. Then, we
must draw Fig. 83. |
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Figure 83 |
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It would be unreasonable, after having adopted the line of identity as
our instrument for the expression of individual identity, to hesitate to employ
it in this case. Yet to regularize such a mode of expression two new
conventions are required. For, in the first place, we have not hitherto had any
such sign as a line of identity crossing a sep. This part of the line of
identity is not a graph; for a graph must be either outside or inside of each
sep.*1 In order, therefore, to legitimate our interpretation of Fig. 83, we
must agree that a line of identity crossing a sep simply asserts the identity
of the individual denoted by its outer part and the individual denoted by its
inner part. But this agreement does not of itself necessitate our
interpretation of Fig. 83; since this might be understood to mean, "There is
something which, if it be a salamander, lives in fire," instead of meaning, "If
there be anything that is a salamander, it lives in fire." But although the
last interpretation but one would involve itself in no positive contradiction,
it would annul the convention that a line of identity crossing a sep still
asserts the identity of its extremities -- not, indeed, by conflict with that
convention, but by rendering it nugatory. What does it mean to assert de inesse
that there is something, which if it be a salamander, lives in fire? It
asserts, no doubt, that there is something. Now suppose that anything lives in
fire. Then of that it will be true de inesse that if it be a salamander, it
lives in fire; so that the proposition will then be true. Suppose that there is
anything that is not a salamander. Then, of that it will be true de inesse that
if it be a salamander, it lives in fire; and again the proposition will be
true. It is only false in case whatever there may be is a salamander while
nothing lives in fire. Consequently, Fig. 83 would be precisely equivalent to
Fig. 84 [Click here to view], |
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Figure 84 |
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and there
would be no need of any line of identity's crossing a
sep. It would then be impossible to express a universal
categorical analytically except by resorting to an unanalytic expression of
such a proposition or something substantially equivalent to that.(*P1)
Two conventions, then, are necessary. In stating them, it will be well
to avoid the idea of a graph's being cut through by a sep, and confine
ourselves to the effects of joining dots on the sep to dots outside and inside
of it.
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| 450. |
Convention No. 8. Points on a sep shall be considered to lie
outside the close of the sep so that the junction of such a point with any
other point outside the sep by a line of identity shall be interpreted as it
would be if the point on the sep were outside and away from the sep. |
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| 451. |
Convention No. 9. The junction by a line of identity of a point on
a sep to a point within the close of the sep shall assert of such individual as
is denoted by the point on the sep, according to the position of that point by
Convention No. 8, a hypothetical conditional identity, according to the
conventions applicable to graphs situated as is the portion of that line that
is in the close of the sep. |
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| 452. |
It will be well to illustrate these conventions by some examples.
Fig. 85 asserts that if it be true that something is good, then this assertion
is false. That is, the assertion is that nothing is good. But in Fig. 86, the
terminal of the line of identity on the outer sep asserts that something, X,
exists, and it is only of this existing individual, X, that it is asserted that
if that is good the assertion is false. It therefore means |
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 |
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Figure
85 |
Figure
86 |
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"Something is not good." On Fig. 87 and Fig. 88 the points on the seps are
marked with letters, for convenience of reference. Fig. 87 asserts that
something, A, is a woman; and that if there is an individual, X, that is a
catholic, and an individual, Y, that is identical with A, then X adores Y; that
is, some woman is adored by all catholics, if there are any. Fig. 88 asserts
that if there be an individual, X, and if X is a catholic, then X adores
somebody that is a woman. That is, whatever |
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 |
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Figure
87 |
Figure
88 |
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catholic there may be adores some woman or other. This does
not positively assert that any woman exists, but only that if
there is a catholic, then there is a woman whom he adores.
|
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| 453. |
A triad rhema gives twenty-six affirmative forms of
simple general propositions, as follows: |
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For a tetrad there are 150 such forms; for a pentad 1082; for a hexad 9366;
etc. |
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B. Derived Principles of InterpretationP |
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§1. OF THE PSEUDOGRAPH AND CONNECTED SIGNSP |
| 454. |
It is, as will soon appear, sometimes desirable to express a
proposition either absurd, contrary to the understanding between the graphist
and the interpreter, or at any rate well-known to be false. From any such
proposition, as antecedent, any proposition whatever follows as a consequent de
inesse. Hence, every such proposition may be regarded as implying that
everything is true; and consequently all such propositions are equivalent. The
expression of such a proposition may very well fill the entire close in which
it is, since nothing can be added to what it already implies. Hence we may
adopt the following secondary convention. Convention No. 10. The pseudograph, or expression in this system of a
proposition implying that every proposition is true, may be drawn as a black
spot entirely filling the close in which it is. |
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| 455. |
Since the size of signs has no significance, the blackened close
may be drawn invisibly small. Thus Fig. 97 [may be scribed] as in Fig. 98, or
even as in Fig. 99, Fig. 100, or lastly as in Fig. 101.(*1) |
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|
Figure 97 |
Figure 98 |
Figure 99 |
Figure 100 |
Figure 101 |
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| 456. |
Interpretational Corollary 1. A scroll with its contents having
the pseudograph in the inner close is equivalent to the precise denial of the
contents of the outer close.
For the assertion, as in Fig. 97, that de inesse if a is true
everything is true, is equivalent to the assertion that a is not true, since if
the conditional proposition de inesse be true a cannot be true, and if a is not
true the conditional proposition de inesse, having a for its antecedent, is
true. Hence the one is always true or false with the other, and they are
equivalent.
This corollary affords additional justification for writing Fig. 97 as
in Fig. 101, since the effect of the loop enclosing the pseudograph is to make
a precise denial of the absurd proposition; and to deny the absurd is
equivalent to asserting nothing. |
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| 457. |
Interpretational Corollary 2. A disjunctive proposition may be
expressed by placing its members in as many inloops of one sep. But this will
not exclude the simultaneous truth of several members or of all.
Thus, Fig. 102 will express that either
a or b or
c or d or e is true. For it will deny the simultaneous denial of all. |
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Figure 102 |
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| 458. |
Interpretational Corollary 3. A graph may be interpreted by
copulations and disjunctions. Namely, if a graph within an odd number of seps
be said to be oddly enclosed, and a graph within no sep or an even number of seps be said to be
evenly enclosed, then spots in the same compartment are
copulated when evenly enclosed, and disjunctively combined when oddly enclosed;
and any line of identity whose outermost part is evenly enclosed refers to
something, and any one whose outermost part is oddly enclosed refers to
anything there may be. And the interpretation must begin outside of all seps
and proceed inward. And spots evenly enclosed are to be taken affirmatively;
those oddly enclosed negatively.
For example, Fig. 83 may be read, Anything whatever is either not a
salamander or lives in fire. Fig. 87 may be read, Something, A, is a woman, and
whatever X may be, either X is not a catholic or X adores A. Fig. 88 may be
read, Whatever X may be, either X is not a catholic or there is something Y,
such that X adores Y and Y is a woman. Fig. 96 may be read, Whatever A may be,
there is something C, such that whatever B may be, A blames B to C. Fig. 103
may be read, Whatever X and Y may be, either X is not a
saint or Y is not a saint or X loves Y; that is, Every
saint there may be loves every saint. So Fig. 104 |
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Figure 103 |
Figure 104 |
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may be read,
Whatever X and Y may be, either X is not best or Y is
not best or X is identical with Y; that is, there are
not two bests. Fig. 105 [Click here to view] may be
read, Whatever X and Y may be, either X does not love Y
or Y does not love X; that is, no two love each other.
Fig. 106 [Click here to view] may be read, Whatever X
and Y may be either X does not love Y or there is
something L and X is not L but Y loves L; that is,
nobody loves anybody who does not love somebody else |
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|
Figure 105 |
Figure 106 |
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| 459. |
Interpretational Corollary 4. A sep which is vacant,
except for a line of identity traversing it, expresses
with its contents the non-identity of the extremities of
that line. |
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§2. SELECTIVES AND PROPER NAMESP |
| 460. |
It is sometimes impossible upon an ordinary surface to draw a
graph so that lines of identity will not cross one another. If, for example, we
express that x is a value that can result from raising z to the power whose
exponent is y, by means of Fig. 107, and express that u is a value that can
result from multiplying w by v, by Fig. 108, then in order to express that |
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|
Figure 107 |
Figure 108 |
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whatever values
x, y, and z may be, there is a value resulting from raising x
to a power whose exponent is a value of the product of z by y which same value
is also one of the values resulting from raising to the power z a value
resulting from raising x to the power y (this being one of the propositions
expressed by the equation x(yz) = (xy)z) we may draw Fig. 109 ; |
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but there is an unavoidable intersection of two lines of identity. In
such a case, and indeed in any case in which the lines of identity become too
intricate to be perspicuous, it is advantageous to replace some of them by
signs of a sort that in this system are called selectives. A selective is very
much of the same nature as a proper name; for it denotes an individual and its
outermost occurrence denotes a wholly indesignate individual of a certain
category (generally a thing) existing in the universe, just as a proper name,
on the first occasion of hearing it, conveys no more. But, just as on any
subsequent hearing of a proper name, the hearer identifies it with that
individual concerning which he has some information, so all occurrences of the
selective other than the outermost must be understood to denote that identical
individual. If, however, the outermost occurrence of any given selective is
oddly enclosed, then, on that first occurrence the selective will refer to any
individual whom the interpreter may choose, and in all other occurrences to the
same individual. If there be no one outermost occurrence, then any one of those
that are outermost may be considered as the outermost. The later capital
letters are used for selectives. For example, Fig. 109 is otherwise expressed
in Figs. 110 and 111. |
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|
Figure 110(z2) |
Figure 111 |
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Fig. 111 may
be read, "Either no value is designated as
U,
or no value is designated as V,
or no value is designated as W,
or else a value designated as
Y results from raising
W
to the V power, and a value
designated as Z
results from multiplying U
by V, and a value
designated as X results
from raising Y to the
U power, while this same value
X results from raising
W to the Z
power." |
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| 461. |
Convention No. 11. The capital letters of the alphabet shall be
used to denote single individuals of a well-understood category, the individual
existing in the universe, the early letters preferably as proper names of
well-known individuals, the later letters, called selectives, each on its first
occurrence, as the name of an individual (that is, an object existing in the
universe in a well-understood category; that is, having such a mode of being as
to be determinate in reference to every character as wholly possessing it or
else wholly wanting it), but an individual that is indesignate (that is, which
the interpreter receives no warrant for identifying); while in every occurrence
after the first, it shall denote that same individual. Of two occurrences of
the same selective, either one may be interpreted as the earlier, if and only
if, enclosed by no sep that does not enclose the other. A selective at its
first occurrence shall be asserted in the mode proper to the compartment in
which it occurs. If it be on that occurrence evenly enclosed, it is only
affirmed to exist under the same conditions under which any graph in the same
close is asserted; and it is then asserted, under those conditions, to be the
subject filling the rhema-blank corresponding to any hook against which it may
be placed. If, however, at its first occurrence, it be
oddly enclosed, then, in the disjunctive mode of
interpretation, it will be denied, subject to the
conditions proper to the close in which it occurs, so
that its existence being disjunctively denied, a
non-existence will be affirmed, and as a subject, it
will be universal (that is, freed from the
condition of wholly possessing or wholly wanting each
character) and at the same time designate (that is, the
interpreter will be warranted in identifying it with whatever the context may
allow), and it will be, subject to the conditions of the close, disjunctively
denied to be the subject filling the rhema-blank of the hook against which it
may be placed. In all subsequent occurrences it shall
denote the individual with which the interpreter may, on
its first occurrence, have identified it, and otherwise
will be interpreted as on its first occurrence. Resort must be had to the examples to trace out the sense of this long
abstract statement; and the line of identity will aid in explaining the
equivalent selectives. Fig. 112 may be read |
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Figure 112 |
Figure 113 |
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there exists
something that may be called X and it is good. Fig. 113,
the precise denial of Fig. 112, may be read "Either
there is not anything to be called X or whatever there
may be is not good," or "Anything you may choose to call
X is not good," or "all things are non-good." "Anything"
is not an individual subject, since the two
propositions, "Anything is good" and "Anything is bad,"
do not exhaust the possibilities. Both may be false. |
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| 462. |
Convention No. 12. The use of selectives may be avoided, where it
is desired to do so, by drawing parallels on both sides of the lines of
identity where they appear to cross.(*1) |
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§3. OF ABSTRACTION AND ENTIA RATIONIS
(*1)P |
| 463. |
The term abstraction bears two utterly different meanings in
philosophy. In one sense it is applied to a psychological act by which, for
example, on seeing a theatre, one is led to call up images of other theatres
which blend into a sort of composite in which the special features of each are
obliterated. Such obliteration is called precisive abstraction. We shall have
nothing to do with abstraction in that sense. But when that fabled old doctor,
being asked why opium put people to sleep, answered that it was because opium
has a dormative virtue, he performed this act of immediate inference:
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Opium causes people to sleep; |
| Hence, |
Opium possesses a power of causing sleep |
The peculiarity of such inference is that the conclusion relates to something
-- in this case, a power -- that the premiss says nothing about; and yet the
conclusion is necessary. Abstraction, in the sense in which it will here be
used, is a necessary inference whose conclusion refers to a subject not
referred to by the premiss; or it may be used to denote the characteristic of
such inference. But how can it be that a conclusion should necessarily follow
from a premiss which does not assert the existence of that whose existence is
affirmed by it, the conclusion itself? The reply must be that the new
individual spoken of is an ens rationis; that is, its being consists in some
other fact. Whether or not an ens rationis can exist or be real, is a question
not to be answered until existence and reality have been very distinctly
defined. But it may be noticed at once, that to deny every mode of being to
anything whose being consists in some other fact would be to deny every mode of
being to tables and chairs, since the being of a table depends on the being of
the atoms of which it is composed, and not vice versa. |
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| 464. |
Every symbol is an
ens rationis, because it consists in a habit,
in a regularity; now every regularity consists in the future conditional
occurrence of facts not themselves that regularity. Many important truths are
expressed by propositions which relate directly to symbols or to ideal objects
of symbols, not to realities. If we say that two walls collide, we express a
real relation between them, meaning by a real relation one which involves the
existence of its correlates. If we say that a ball is red, we express a
positive quality of feeling really connected with the ball. But if we say that
the ball is not blue, we simply express -- as far as the direct expression goes
-- a relation of inapplicability between the predicate blue, and the ball or
the sign of it. So it is with every negation. Now it has already been shown
that every universal proposition involves a negation, at least when it is
expressed as an existential graph. On the other hand, almost every graph
expressing a proposition not universal has a line of identity. But identity,
though expressed by the line as a dyadic relation, is not a relation between
two things, but between two representamens of the same thing. |
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| 465. |
Every rhema whose blanks may be filled by signs of ordinary
individuals, but which signifies only what is true of symbols of those
individuals, without any reference to qualities of sense, is termed a rhema of
second intention. For second intention is thought about thought as symbol.
Second intentions and certain entia rationis demand the special attention of
the logician. Avicenna defined logic as the science of second intentions, and
was followed in this view by some of the most acute logicians, such as Raymund
Lully, Duns Scotus, Walter Burleigh, and Armandus de Bello Visu; while the
celebrated Durandus ā Sancto Porciano, followed by Gratiadeus Esculanus, made
it relate exclusively to entia rationis, and quite rightly. |
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| 466. |
Interpretational Corollary 5. A blank, considered as a medad,
expresses what is well-understood between graphist and interpreter to be true;
considered as a monad, it expresses "_ exists" or "_
is true"; considered as a
dyad, it expresses " _ coexists with
_" or "and." |
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| 467. |
Interpretational Corollary 6. An empty sep with its surrounding
blank, as in Fig. 114, is the pseudograph. Whether it be taken as medad, monad,
or dyad, for which purpose it will be written as in Figs. 115, 116, it is the
denial of the blank. |
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Figure 114 |
Figure 115 |
Figure 116 |
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| 468. |
Interpretational Corollary 7. A line of identity traversing a sep
will signify non-identity. Thus Fig. 117 will express that
there are at least two men. |
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Figure 117 |
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| 469. |
Interpretational Corollary 8. A branching of a line of identity
enclosed in a sep, as in Fig. 118, will express that three
individuals are not all identical. |
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Figure 118 |
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We now come to another kind of graphs which may go under the general
head of second intentional graphs.(*1)
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| 470. |
Convention No. 13. The letters,
r0
r1,
r2,
r3,
etc., each
with a number of hooks greater by one than the subscript number, may be taken
as rhemata, signifying that the individuals joined to the hooks, other than the
one vertically above the {r}, taken in their order clockwise, are capable of
being asserted of the rhema indicated by the line of identity joined vertically
to the r. |
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Figure 119 |
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Thus, Fig. 119 expresses that there is a relation
in which every man stands to some woman to whom no other man stands in the same
relation; that is, there is a woman corresponding to every man or, in other
words, there are at least as many women as men. The dotted lines, between
which, in Fig. 119, the line of identity denoting the ens rationis is
placed, are by no means necessary. |
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| 471. |
Convention No. 14. The line of identity representing an
ens
rationis may be placed between two rows of dots, or it may be drawn in ink of
another colour, and any graph, which is to be spoken of as a thing, may be
enclosed in a dotted oval with a dotted line attached to it. Other entia
rationis may be treated in the same way, the patterns of the dotting being
varied for those of different category.
The graph of Fig. 120 is an example. It may be
read, as follows: "Euclid *2 enunciates it as a postulate that if two straight
lines are cut by a third straight line so that those angles the two make with
the third, these angles lying between the first two lines(τας
έντος γωνιας) and on the same side of the third, are less than two right angles, then that
those two lines shall meet on that same side; and in this enunciation, by a
side, μέρη, of the third line must be understood part of a plane that contains
that third line, which part is bounded by that line and by the infinitely
distant parts of the plane." . . . |
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Figure 120 |
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