The Logic Diagram - MacSphere - McMaster University

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THE LOGIC DIAGRAM
THE LOGIC DIAGRAM
By
, ,
GAlLAND MAC QUEEN, B.A.
A Thesis
Submitted to the Faculty of Graduate Studies
in Partial Fulfilment of the Requirements
for the Degree:
Master of Arts;
McMaster University
(October) 1967
ACKNOWLEDGEMENTS
I should like to thank everyone who has assisted me in the preparation
of this paper:
Dr. J. Mayer, now at Brock University, for fostering my interest in the
logic diagram
t
Dr. D. D. Roberts of Waterloo University for the use of his doctoral thesis,
Dr. J. Noxon for approving the project,
Mrs. M. Parker and her assistants in the Inter-Library Loan office for
locating many obscure books,
Mrs.. D. Brown for typing the last half of the paper,
Mr. Wm. Newell and Miss Elaine Lively for assistance with translation,
Dr. A. F. Burghardt of the Geography Department and Dr. A. Shalom for
suggestions,
Mr. S. Ajzenstat for his careful reading,
Dr. J. Thomas whose kindness went weil beyond the requirements of a
thesis supervisor,
And my wife for typing part of the paper, her encouragement and
fo~
assistance, and especially for ,being herself. .
1i
TABLE OF CONTENTS
. .
ACKNOWLEDGEMENTS • .. " • .. • .. .. .. .. " • " .. " ' • • II • .. .. ii
.
TABLE OF CONTENTS .. .. .. .. • .. .. .. .. .. iii
" "
INTRODUCTION .. • 1
.
I. THE LIFE HISTORY OF THE LOGIC DIAGRAM .. • • • .0 ,. Ct ' .. . ..... ..•. .. .. . • 4
1. Its Conceptio.n .• .• .• .• .• .• • .• ,. •.. • ..• • ., . .. 4
\ ..
2. Its Birth •• .,..... •.•.•. •.•. 6
• e .•.
3 •. Its Adolescence ,. ... .• .• ," ... ,. ,. ,. .. ,• .• ,. .• ,. .• .•." .• .• .• 8
4 ... Its Maturity ......................., •.•, .0 11
0, e, •.•. e, e.
65• ;. Its Senesce.nG,e ., ...... ,• ." .• .• ... .• .• .. ,., ,• .•.• ,. " ... ,., .... , 15
Its Death • .., ...... Or .', e. 'IJ·~ o. :. Od ~I o~ O ••~ e ........., e.. ... • ., .' • 18
7. Post Mortem _ • • • • • • • • o • • • .' .. • .. .. .. .. • 19
II THE CLASSIFICATION JU'{D .,DESCRIPTION .OF. VARIQUS, S,YSTEMS..
OF' LOGIC DIAGRAHS • • • ,,". • " .. .'. .. • • • • " • .. •• • • 2l
"
1. Scheme of Claspification .. .. .. .. " .. " .. 21
$ • .. 0 • " o ..
2. Geometric Diagrams ~ " 0 0 ~ .. .. ~ .. 0 .. ~ Q .. ~ .. u .." 22
(a) Euler ..... 22
$" O.~ o. ill. o~ G.. f1... 0", o~ 0., 0·, •• O. ~!II
(b) Haass "i' 41
... .,IIJ ,.0 ....0 .. C)
(c) Venn '" • .... .,) 0 ... e. Q. 81 0, CI_ tP~ 8 ... 0 .. __ e. 0 e. lit .. (ijI", iII~ /I.) -, 45
(d) Marquand .. .. ~.. ~i ~"! 66
0T "" .... 0_ ~ "... "'.. 0'1 41, 41'. .... GIl.
((ef)) MCaacrfraorl1la "n,., e~ . .0, .~ .i ., 0. ..,0~ ", e , ..... 0. , 0 "Qt~ ..~ QI, "- .0~ o' ISS ,.1). ,_~ " ... c;v. ..! i,. _"_ .0 .~, ,(all ",$. 0 ,et .. ~ .Q 08 ',G" ':~1 .' Ii 7693
(g) Newlin 0 .. 0 ~ .. ",0 .... 0 ," _~ .,0 ",0 .. 8 ',e ~ ... ~ _iii ... $ .....e .... 0 1<1 ,Q .. Q 80
(((hji))) GLHeoowncksisei nth•g .....~ , •.. ~~ •"~ ""I 0t~t..,., : o.01~ ~ '•(Iit ),..~. 0eC t..,, 0•4 tr, •8Q 1.' .0Go.,.... ..(I.I,.. ..r.;, ,-: .0e. .. .,. .e&).~ .$~... .., .e6 ... . ....~ "9Ct .. •e0", •t~il . 998534
3. Linear Diagrams • • • • • • • • • • • • • • • • • .. • • • 97
(a) Leibniz ., •••.•, •.•., . .• .• •. .e, .e" e. _. e. .., .0 ... 97
(b) Lambert • • • • • " Goo .. .. • .. • • •.~ •.~ 98
.
..
4. Network Diagrams • • • • • • • • • • • • o o • !J 0 • 103
(a) Squares of Opposition. • .. .. .. .. .. .. , •. 103
0 .,. ... ,.
(b) The Pons Assinorum an,d .. the .Ars Magna ,. 105
,II .0 .0 , ••
(c) Frege •• .• .• ._ .• .• '.. .• ,. .• • • • .• 106
,0 ... • ," ,. ••
(d) Peirce • _. .• ,. _. .• .• .,. ,. .• 113
,0 ,. •• ... .,. ,. ,. ,0 •• .CO ,
(e) Gardner • • • • • • • • • • 130
e .. • .. .. • .. .. • •
iii
5. 'Unclassified Diagrams .141
• • • • • ~ 0 • • • • • • • • • •
(a) Hamilton. '. ....... '• ••• ,. ............. 141
(b) De Morgan • .. • • • • • • • • _ • .. • • • • • • 147
6. Conclusion .. .. .. .. .. 150
III THE LOGIC DIAGRAM TODAY .. .. • • 152
• • • • • • • • • • • • • 0 •
1. Vfuat is a Logic Diagram? • • 152
• e • • • • • • • • • 0 ~ •
(a) Peirce s Concept of L:anguage .. .. .. .. .. 152
I • Co • •
(b) The Weak Statement of ,the ,valu,e .
(c) Souf mthmea rLy.o g•i c • D•i ag..r a...m ·. . " ." •. ~ • • .. <> .. .. • .. .. .. •. " • ... • . 11 • ,$ • ~CII .. 115557
.
2. The Uses of the Logic Diagram • • • • .. • • .. .. • • .. 158
. .. .. . . .. ., .. .. .,
(a) In Logic .. .. • .. , ,. . .. . . .... . ,•.. . .. .. .., 158
((((ebdc)))) SIIInnun·m MBtmhuaaserit yhnS e~•csm isea... nt ic..c. ,,s e. •s, . . ......~ ,, . . ..•." •.•,, ."., ..... .,. • , .•e. . , •..., " •C•o ...., , • ••-~, . .... .,, . 0. _..". . . ., 4. 1 ...... ... ' .•". , •...,. ,c o ....... . 111166661653
3. The Evaluation of the Logic Di,agram • 167
II .. • .. .. .. .. ....
4. The Central Issues Raised by this Paper .. .. .. .. 173
0 .. " "
(a) Quantification .ot; the. Predicate ... ....... . •. •. 173
G, e, .. .
(b) Existence .. ~ s .. " .. .. .. .. .. .. ., .. " ... 176
5. Conclusion .. .. .. .. .. .. ... .. 179
Ii> " .. " .f> .. .. " .. .. .. .. .. .. ..
. . . .
APPENDIX I: DIAGRAMS .. .. .. • • • • • • • • • • • 0 " • • • 181
.... . . ....
APPENDIX II: INDEX .. .. .. • • • .. .. .. G • 0 • • • .. • 284
. . . . . . . .
BIBLIOGRAPHY .. • • • • • • • .. .. • • • • • .. .. • • 286
iv
INTRODUCTION
The logic diagram was introduc€ld in 17611 and was, from the
time of Hamiiton to the publishing of thE~ Principia Mathematica, a center-
of controversy among logicians. Some of the major logical problems of
our time are crystallized and clarified, not solved, in these
tho~gh
'2
diagrams. More important, the history ()f nineteenth-century logic, with
the important exception of Boole, can be traced in the development of
these diagrams.. Yet there has been little work done in our own time or
in the past on this fascinating branch of logico
Sir William Hamilton and John Venn both attempted to collect
what was known about diagrams in their own time but Ham.i lton's analysis is
,
prejudiced as we shall see, and Venn's is sketchy. C.I. Lewis introduced
only geometric diagrams in A Survey of Sirmbolic Logic and used them with
presumptions which must be examinedo Gardner collected several
MaJ~tin
systems of logic diagrams in Logic Machllles and Diagrams but treated them,
with the exception of Venn's, as interes1~ing curiositieso Some work has
3
been done on the diagrams of individual logicians but in the interest of
~or qualifications of this statement see chapter I section 1
and ,chapter' II section 3 part (a) and chapter II section 4 parts (al and
(b). Hereafter cross-references to this thesis will be abbreviated. The
above reference would be written thus: I 1, II 3 (a) and 4 (a) and (b).
2III 4 (a) and (b).'
~.g" D.'D; Roberts, The Existential Graphs of C.S. Peirce,
of
Urbana: University Illinois, 1963, unpublished thesis.
1
2
completeness this needs the context of the complete development of logic
diagrams
0
On the other hand, we need only pick up' a textbook on elementary
set theory, switching circuits or even arithmetic if it is the
logic~
"new math" to be faced with a wide variety of logic diagrams. Euler, Venn,
Marquand, Lambert and Carroll are all represented, sometimes in the same
booko That these systems are incompatible4 seems of little importance
to their users.
It would seem therefore, that there is a necessity for an exam in-
ation of logic diagrams, qua logic diagrams, which will endeavour to dis-
cover what they are and what characteristics they must possess if they
are to function as their users intend. Since the diagrams were first
introduced in logic,and since the mathematician and electrical engineer
can hardly be expected to perform such an analysis, and since, moreover,
,
Aristotelian and nineteenth-century symbolic logic wou,ld seem to be the
most adequate tools to be used in such an analysis, we may drop this work
in the logicians' 'lape The purpose of this paper is to lay the foundat-
ion for such an analysise It will, of course, be impossible to examine
any single problem extensively but will be within our purpose to locate
those areas in which problems of a logical or philosophical nature should
be raised. Our work will be divided into thIe~· parts, the purposes of
which will be 'as follows:
•
I. to give a chronological resume of the history of the diagram
in logic from 1761 to 1910 with emphasis on those logicians who contributed
4See the comparative, portions of II.
3
to its development and their relevant writings.
II. to establish a simple classification for such diagrams
and to describe and compare the various s:ystems of diagrams within this
classification.
III. in the ,light of the forego1ing, to describe the uses and
assess the worth of the logic diagram in our own time and to indicate
some relevant problems raised by the diagrams.
If the diagrams are to be used they ought to be used well.
This can only be done if they are examined ..
system~,tically
All of the diagrams mentioned in this work will be found in
numerical order in the first appendixe
I
THELI:F'E HISTORY OF THE LOGIC DIAGRAM (1763 - 1910)
1. Its Conception (prior to 1763)
The first diagrams used by logicians are in all probability now
lost. We have, nevertheless, many early Mediaeval diagrams which represent
the individual valid arguments of Aristotelian logic. These are not
actual logic diagrams: illustrate the argument after it has been
t~ey
solved and are not primarily intended as aids to reasoning as more recent
l
systems are. Hamilton, whose scholarship we shall have reason to question ,
traces these as far back the fifth century A.D. Giordano Bruno incor
~s
porates three of these Mediaeval diaerams in one diagram~'(Diagram I).
3
Gardner is fascinated by the life and works of Raymon Lull.
The Ars Ivlar.t:na was more mechanical than diagrammatic and more metaphysical
than logical. We may, therefore, safely and thankfully'ignore Lull's
incredibly obscure system for our purposes in this chapter and turn to
lHamilton's scholarship will be found to be questionable in his,
discussions of all prior logicians, particularly Euler and Maass. II 2 (a)
and (b).
2Hart~n Gardner, Logic Hachines and Diagrams, Toronto: McGraw-Hill,
1958, po 30.. I find these diagrams confusing and will not treat them in
this work. For more information see John 'Venn, SymboliC Logic, 2nd ed.,
London: Hacmillan, 1894, pp. 50.4 ff.
3Gardner, Ope cit., pp. 1 ff.
4
5
4
more modern sources.
Hamilton attributed the geometric diagram to Christian Weise in
Nucleous Logicae Weisiania (1712).5 This was a mistake for, as Venn
pointed Weise did not write this book. The author was, in fact,
out~
6
Johann Christian Lange. Hamilton did not seem to recognize that Lange
used the diagram to represent propositions but not syllogisms for he
equates Langeis diagrams with Euler's. Johann Christoph Strum in
Universalia Euclides (1661) and Leibniz both used circles to represent
7
propositions prior to Lange. This would seem to indicate further that
Hamilton's historical research was not as thorough as he thought.
In A Survey of Symbolic Logic, C. I. Lewis translates two brief
portions of Leibniz which indicate that Leibni~ understood the principles
8
of the linear diagram almost a century b,efore Lrunbert. These fragments
are taken from Gerhardt's text, Die Philosophischen Schriften von G. W.
Leibniz, Band VII, tlScientia Generalis. Characteristica,U XIX and XX.9
Hamilton, with his usual historical scholarship, attributes the linear
diagram to J. H. Alsted in his Logic (1614). Venn pointed out that there
4.An example of Lull's system will be described in II 4 (b)~
5This book was not available to the author of this thesis.
6
Venn, OPe cit., p. 509.
7See ·Gardner, Ope cit., p. 31, where he cites Church. The Strum
book was unavailable but we discuss Leibniz briefly, from what information
is available, in II 3 (a).
8See II 1 for a definition of "linear" and II 3 for descriptions
of linear systems.
9C• I. Lewis, A Survey of Symbolic Logic, New York: Dover, 1960,
pp. 291 ff.