L 376, December 6, 1911.  Houghton Library.

My dear Risteen:

A Diagrammatic Syntax.  I mentioned to you, while you were [here] last
year, that I have a diagrammatic syntax which analyzes the syllogism
into no less than six inferential steps.  I now describe its latest
state of development for the first time.  I am glad to think that my
account of it will have one such a reader as you.  C.S.P.

This syntax, which I have hitherto called the "system of Existential Graphs",
was suggested to me in reading the proof sheets of an article by me that
was published in the Monist of Jan. 1897; and I at once wrote a full
account of it for the same journal.  But Dr.  Carus would not print it.
I gave an oral account of it, soon after, to the National Academy of
Sciences; and in 1903 for my audience of a course of Lectures before the
Lowell Institute, I printed a brief account of it.  An account of
slightly further development of it was given in the Monist of Oct. 1906.
In this I made an attempt to make the syntax cover Modals; but it has
not satisfied me.  The description was, on the whole, as bad as it well
could be, in great contrast to the one Dr.  Carus rejected.  For
although the system itself is marked by extreme simplicity, the
description fills 55 pages, and defines over a hundred technical terms
applying to it.  The necessity for these was chiefly due to the lines
called "cuts" which simply appear in the present description as the
boundaries of shadings, or shaded parts of the sheet.  

The better
exposition of 1903 divided the system into three parts, distinguished as
the Alpha, the Beta, and the Gamma, parts; a division I shall here
adhere to, although I shall now have to add a Delta part in order to
deal with modals.  A cross division of the description which here, as in
that of 1903, is given precedence over the other is into the
Conventions, the Rules, and the working of the System.

The Conventions.  The ultimate purpose of contriving this diagrammatic
syntax, is to enable one with facility to divide any necessary, or 
mathematical, reasoning into its ultimate logical steps.  It is more
accurate to call such reasoning "necessary" than to call it
"mathematical"; but the latter designation will give a person who
recognizes that not all mathematical reasonings have to do with
quantity, but who has happily not received any of the sort of
instruction in Logic that is now usually given, a better notion of what
is meant than if the reasoning of the kind meant were described as
"necessary".  

For example, all sound reasoning in applying the doctrine
of chances belongs to the class of reasonings meant; and all such
reasoning is truly "necessary" reasoning,|that is to say, what it
concludes must necessarily (i.e. would always) be true, provided its
"premisses", or the hypotheses upon which it is based, be true.  But
because what it reasons about is probability,|concluding that such and
such an event would always, under such and such circumstances, have a
stated probability, a person unfamiliar with the theory of reasoning
might very naturally take it for probable reasoning.  But I usually
designate the kind of reasoning in question as Deductive Reasoning, or
Deduction, so giving this word a broader meaning than many logicians do.

If I am asked how a person can, in deductive reasoning, be absolutely
certain that his conclusion is true, if his premisses are so, I answer
that in point of fact he is not.  One can never be absolutely certain of
anything.  All men make mistakes in addition and multiplication, for
example; and they may repeat the same mistakes in going over the
computation a second time.  Consequently, however improbable it may be,
it is possible, strictly speaking, that the same mistake should be
repeated a million of million of times.  This may have
happened,|ridiculous as it would be really to suppose so,|every time
that anybody undertook to say how many twice two came too.  Absolute
accuracy is beyond human powers.  But a phrase often used in reference
to deductive reasoning expresses correctly the nature of such approach
to certainty as we can attain.  Namely, it is often said that the
reasoner must admit the deductive conclusion from his own premisses, or
"fall into self-contradiction".  That is precisely the essence of
Deduction:  the fact which its conclusion asserts, if it is a logically
perfect deduction, was already asserted in its premisses,|not, usually,
in either one, but in the totality of the premisses taken together.

By means of the accurate expression of the meanings of premisses and
conclusions in this diagrammatic syntax, it will be made clear whether
or not the nature of deductive [reasoning] really be such as I say it
is.

One obvious reason for doubting the theory is that it would seem to
render any valid deductive conclusion instantaneously evident upon an
examination of the premisses; so that mathematics would be almost too
simple a thing to be dignified by the name of a science, in strange 
conflict with the known reputation of that science as being the most
difficult of all the sciences.  But the theory ought not to be condemned
upon this ground alone, without careful analysis of the reasonings of
mathematics, since there are two reasons for believing that a science
where reasonings should be of the sort that the theory supposes might
have all the difficulty that mathematics has.  

The first of these reasons
is that if certain propositions of such a science were of such extreme
complexity, owing to the great number of individual objects almost
exactly alike of which the premisses might make assertions, these
assertions at the same time each of them dealing with many such objects,
the result might be, if there were many such premisses, that almost any
mind would become confused in considering them; so that it might readily
happen that it should become far from an easy matter for a person to
determine offhand whether a given state of things, only describable in a
similarly confusing way, had been asserted to exist or not. Now it
cannot be denied that one is frequently confronted in mathematics with
situations of the character of that which has just been described.
Whoever has read or tried to read Jordan's Traite des Substitutions, or
Gauss's six demonstrations of the fundamental theorem of algebra, or a
great deal of the Mecanique celeste must acknowledge it.  But this is
not all.  

My second reason is found in the peculiar character of
mathematical postulates.  These pronounce that certain things are
possible.  But these possibles are not, of course, single things, for a
single thing must be more or less than possible:  they embrace whole
infinite series of infinite series of objects in each postulate; and it is
upon the statement of the possibility of one single one of those objects
or single one for each set of certain others, that some essential part
of the conclusion is founded.  How many demonstrations, for example, and
very simple ones too, as mathematics goes, depend, each of them, upon
the possibility of a single straight line; while this possibility is
only asserted in the postulate that there is, or may be, a straight line
through any two points of space.  In that statement the possibility of
every single straight line in space is asserted, including the single
one whose existence is pertinent and concerning which a similar
postulate directly or mediately asserts something which is an essential
ingredient of the conclusion.

The Phemic Sheet.  Since the sole purpose of the Syntax I am describing
is to facilitate the anatomy, and thereby the physiology of deductive
reasonings, the reader will have anticipated the fact that no occasion
has been found for supplying it with any means of expressing mere
feelings or complexes of mere feelings, such as abound in the arts of
music and of painting.  Nor has any need been found for furnishing it
with means of expressing commands,|not even such as take the softened
forms of requests and inquiries.  We need only a mode of indicating that
what is "scribed",|i.e. is marked, whether by coveting or drawing or by
a mixture of these two arts, is meant, and is not scribed for some other
purpose, as, for example, to show how it might be asserted.  

Moreover,
however minutely we may analyze our assertions, there will never be the
slightest need of any such fragment of meaning as that of a noun or that
of an English verb.  The simplest part of speech which this syntax
contemplates, which, as scribed, I shall term a blot (a vocable I choose
because it is cognate with Greek [Gr.] floid, for which see L.& S. p.
1692, under [Gr.] flao, where many words containing this root are given;
and there are many others in all European languages] is itself an
assertion.  Ought it to be an affirmation or a denial?  A denial is
logically the simpler, because it implies merely that the utterer
recognizes, however vaguely, some discrepancy between the fact and the
speech, while an affirmation implies that he has examined all the
implications of the latter and finds no discrepancy with the fact.  This
is a circumstance to be borne in mind; but since the denial implies
recognition of the affirmation, while the affirmation is so far from
implying recognition of the denial, that one might imagine a paradisaic
state of innocence in which men never had the idea of falsity, and yet
might reason, we must admit that affirmation is psychically the simpler.
Now I think that upon this point we must prefer psychical to logical
simplicity.  I therefore make the blot an affirmation.  The subject of
it must for simplicity be completely indefinite.  The simplest blots,
which are not relative, are such as `it snows', `it thunders', etc.

All thought, which is the process of forming, under self-control, an
intellectual habit, requires two functionaries; an utterer and an
interpreter, and though these two functionaries may live in one brain,
they are nevertheless two.  In order to distinguish the actual
performance of an assertion, though it be altogether a mental act, from
a mere representation or appearance, the difference between a mere idea
jotted down on a bit of paper, from an affidavit made before a notary,
for which the utterer is substantially responsible,|I provide my system
with a phemic sheet, which is a surface upon which the utterer and
interpreter will, by force of a voluntary and actually contracted habit,
recognize that whatever is scribed upon it and is interpretable as an
assertion is to be recognized as an assertion, although it may refer to
a mere idea as its subject.  

If "snows" is scribed upon the Phemic
Sheet, it asserts that in the universe to which a special understanding
between utterer and interpreter has made the special part of the phemic
sheet on which it is scribed to relate, it sometime does snow.  For they
two may conceive that the "phemic sheet" embraces many papers, so that
one part of it is before the common attention at one time and another
part at another, and that actual conventions between them equivalent to
scribed graphs make some of those pieces relate to one subject and part
to another.  

Any visible form which, if it were scribed on the phemic
sheet would be an assertion is called a graph.  If it actually be so
scribed, it would be incorrect to say that the graph itself is put upon
the sheet.  For that would be an impossibility, since the graph itself
[is] a mere form, an abstraction, a "general", or as I call it a "might
be", i.e. something which might be if conditions were otherwise than
they are; and in that respect it [is] just like a "word", any word, say
camel.  (Not that it is right to pronounce a might-be to be a fiction, or
to deny its reality without other reason than that it is a mere
might-be.  For what do we mean by real?  The legal sense, that in which
we speak of "real" estate, is a good deal older than the ordinary sense.
This latter, though I have once or twice met with it in older
writings,|it is such a natural formation from res,|may be said with
substantial truth to have been introduced by Duns Scotus, who died, if I
recollect the exact date (it can only be a few years out) in 1309.  It
only came slowly into use, outside of the Scotists, because "in re" etc.
were used instead.  It would be no use quoting Scotus in order to fix the
meaning:  that would but render confusion worse confounded.  The best
way is to analyze our own meaning in some of the many common phrases in
which we use it.  What would a man have in his mind who should inquire
whether Sancho Panza was a real person?  Observe that his phrase may be
"Is Sancho Panza a real character".  He knows well enough that he cannot
now be living or existent.  But that does not affect his Reality.  Is it
really true, or, in other words, "is it a Real fact", that two brothers
founded Rome?  If the nature of it is such that its being depends upon
how ancient historians have thought, that comes pretty nearly to the
same thing as its not being "Real", or "in re", as the scholastics would
have said.)

Yet this would not hold in all cases; since it is certainly a Real fact
that I had a dream last night,|a pretty rare occurrence with me.  Yet
the truth of this consisted in my thinking, in a general sense (that is,
imagining) something.  On the other hand, Kant's holding that such
concepts as cause etc. are merely due to the nature of the human mind is
not considered by him, and certainly ought to be considered any reason
for pronouncing "cause" etc. not to be Real.  Most philosophers,
nowadays, would, on the contrary, think, as I certainly do, that if it
were absolutely certain that anything,|Time, for example,|were a mental
phenomenon of mental origin due to the structure of the human mind in
such a way that no man could ever by any possibility escape from the
idea of events owing in time, that would constitute the reality of time.
For the real universe in which man lives is the system of those ideas
which research, carried far enough, would ultimately establish.  If, on
the other hand, all that is true concerning a given object is due to how
an individual man, or an individual collection of men happened to think,
and that not about other things, as in the case of the dream, but how
they happened to think about the very fact or thing whose reality is in
question, then plainly the very being and truth of that object is a mere
accident of thought; and all we have to do is to stop thinking about it,
not only to cause the cessation of its being, but to cause it to cease
from ever having been; and that sort of thing we all call unreal.  Now a
great many people believe the laws of nature are little more than that.
They think that Sir Isaac Newton not merely discovered gravitation but
that he created it.  No less a person than Dr.  Karl Pearson says so in
plain terms in his Grammar of Science.  They do not think, and recognize
themselves that they do not believe, that there really are any laws, or
regularities in Nature.  I do not agree with them.  John Stuart Mill who
thought nearly as Pearson does, so far differed on that point that he
thought the whole truth of Inductive reasoning,|not the fact that there
are such truths, but that our observations afford any good reason for
believing Inductive conclusions, to be likely to be more or less
true,|is that it is a fact, whether the reasoner knows it or not, that
under exactly the same circumstances, the same thing always occurs.  I
think that is no sound reason for believing in induction as a sound way
of reasoning, because, in so far as we have any reason for such a
pronouncement (and as a general proposition nothing can be more false),
we only know it as an inductive conclusion.  It is as the veracity of a
witness were in question, and in his defence a lot of others were put on
the stand, and on cross-examination it were found that all they knew of
the suspected witness's veracity was what he himself had told them.  But
if I could have had an opportunity of tackling Mill, who had in him the
stuff to make a philosopher, but who had taken his philosophic creed
ready made from Bentham and from his father, James Mill, having no
reading in philosophy to speak of, and who, while he was

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