The Project Gutenberg EBook of The Algebra of Logic, by Louis Couturat
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net
Title: The Algebra of Logic
Author: Louis Couturat
Release Date: January 26, 2004 [EBook #10836]
Language: English
Character set encoding: TeX
*** START OF THIS PROJECT GUTENBERG EBOOK THE ALGEBRA OF LOGIC ***
Produced by David Starner, Arno Peters, Susan Skinner
and the Online Distributed Proofreading Team.
THE ALGEBRA OF LOGIC
BY
LOUIS COUTURAT
AUTHORIZED ENGLISH TRANSLATION
BY
LYDIA GILLINGHAM ROBINSON, B. A.
With a Preface by PHILIP E. B. JOURDAIN. M. A. (Cantab.)
Preface
Mathematical Logic is a necessary preliminary to logical Mathematics. "Math-
ematical Logic" is the name given by Peano to what is also known (after
Venn ) as "Symbolic Logic"; and Symbolic Logic is, in essentials, the Logic
of Aristotle, given new life and power by being dressed up in the wonderful —
almost magical — armour and accoutrements of Algebra. In less than seventy
years, logic, to use an expression of De Morgan's, has so thriven upon sym-
bols and, in consequence, so grown and altered that the ancient logicians would
not recognize it, and many old-fashioned logicians will not recognize it. The
metaphor is not quite correct: Logic has neither grown nor altered, but we now
see more of it and more into it.
The primary significance of a symbolic calculus seems to lie in the econ-
omy of mental effort which it brings about, and to this is due the characteristic
power and rapid development of mathematical knowledge. Attempts to treat
the operations of formal logic in an analogous way had been made not infre-
quently by some of the more philosophical mathematicians, such as Leibniz
and Lambert ; but their labors remained little known, and it was Boole
and De Morgan, about the middle of the nineteenth century, to whom a
mathematical — though of course non- quantitative — way of regarding logic was
due. By this, not only was the traditional or Aristotelian doctrine of logic
reformed and completed, but out of it has developed, in course of time, an
instrument which deals in a sure manner with the task of investigating the fun-
damental concepts of mathematics — a task which philosophers have repeatedly
taken in hand, and in which they have as repeatedly failed.
First of all, it is necessary to glance at the growth of symbolism in mathe-
matics; where alone it first reached perfection. There have been three stages in
the development of mathematical doctrines: first came propositions with par-
ticular numbers, like the one expressed, with signs subsequently invented, by
"2 + 3 = 5"; then came more general laws holding for all numbers and expressed
by letters, such as
lastly came the knowledge of more general laws of functions and the formation
of the conception and expression "function". The origin of the symbols for par-
ticular whole numbers is very ancient, while the symbols now in use for the
operations and relations of arithmetic mostly date from the sixteenth and sev-
enteenth centuries; and these "constant" symbols together with the letters first
used systematically by ViETE (1540-1603) and Descartes (1596-1650),
serve, by themselves, to express many propositions. It is not, then, surprising
that Descartes, who was both a mathematician and a philosopher, should
have had the idea of keeping the method of algebra while going beyond the
material of traditional mathematics and embracing the general science of what
thought finds, so that philosophy should become a kind of Universal Mathemat-
ics. This sort of generalization of the use of symbols for analogous theories is a
characteristic of mathematics, and seems to be a reason lying deeper than the
erroneous idea, arising from a simple confusion of thought, that algebraical sym-
bols necessarily imply something quantitative, for the antagonism there used to
be and is on the part of those logicians who were not and are not mathemati-
cians, to symbolic logic. This idea of a universal mathematics was cultivated
especially by Gottfried Wilhelm Leibniz (1646-1716).
Though modern logic is really due to Boole and De Morgan, Leibniz
was the first to have a really distinct plan of a system of mathematical logic.
That this is so appears from research — much of which is quite recent — into
Leibniz's unpublished work.
The principles of the logic of Leibniz, and consequently of his whole philos-
ophy, reduce to two^ : (1) All our ideas are compounded of a very small number of
simple ideas which form the "alphabet of human thoughts"; (2) Complex ideas
proceed from these simple ideas by a uniform and symmetrical combination
which is analogous to arithmetical multiplication. With regard to the first prin-
ciple, the number of simple ideas is much greater than Leibniz thought; and,
with regard to the second principle, logic considers three operations — which we
shall meet with in the following book under the names of logical multiplication,
logical addition and negation — instead of only one.
"Characters" were, with Leibniz, any written signs, and "real" characters
were those which — as in the Chinese ideography — represent ideas directly, and
not the words for them. Among real characters, some simply serve to represent
ideas, and some serve for reasoning. Egyptian and Chinese hieroglyphics and the
symbols of astronomers and chemists belong to the first category, but Leibniz
declared them to be imperfect, and desired the second category of characters
for what he called his "universal characteristic".^ It was not in the form of an
algebra that Leibniz first conceived his characteristic, probably because he
was then a novice in mathematics, but in the form of a universal language or
script.^ It was in 1676 that he first dreamed of a kind of algebra of thought,^ and
it was the algebraic notation which then served as model for the characteristic.^
Leibniz attached so much importance to the invention of proper symbols
that he attributed to this alone the whole of his discoveries in mathematics.^
And, in fact, his infinitesimal calculus affords a most brilliant example of the
importance of, and Leibniz' s skill in devising, a suitable notation.^
Now, it must be remembered that what is usually understood by the name
"symbolic logic", and which — though not its name — is chiefiy due to Boole, is
what Leibniz called a Calculus ratiocinator, and is only a part of the Universal
Characteristic. In symbolic logic Leibniz enunciated the principal properties
of what we now call logical multiplication, addition, negation, identity, class-
inclusion, and the null-class; but the aim of Leibniz's researches was, as he
^ CouTURAT, La Logique de Leibniz d^apres des documents inedits, Paris, 1901, pp. 431-
432, 48.
"^Ibid., p. 81.
^Ibid., pp. 51, 78
"^Ibid., p. 61.
^Ibid., p. 83.
^Ibid., p. 84.
^Ibid., p. 84-87.
said, to create "a kind of general system of notation in which ah the truths
of reason should be reduced to a calculus. This could be, at the same time,
a kind of universal written language, very different from all those which have
been projected hitherto; for the characters and even the words would direct
the reason, and the errors — excepting those of fact — would only be errors of
calculation. It would be very difficult to invent this language or characteristic,
but very easy to learn it without any dictionaries". He fixed the time necessary
to form it: "I think that some chosen men could finish the matter within five
years"; and finally remarked: "And so I repeat, what I have often said, that a
man who is neither a prophet nor a prince can never undertake any thing more
conducive to the good of the human race and the glory of God".
In his last letters he remarked: "If I had been less busy, or if I were younger
or helped by well-intentioned young people, I would have hoped to have evolved
a characteristic of this kind"; and: "I have spoken of my general characteristic
to the Marquis de I'Hopital and others; but they paid no more attention than if
I had been telling them a dream. It would be necessary to support it by some
obvious use; but, for this purpose, it would be necessary to construct a part at
least of my characteristic; — and this is not easy, above all to one situated as I
am".
Leibniz thus formed projects of both what he called a characteristica uni-
versalis, and what he called a calculus ratiocinator; it is not hard to see that
these projects are interconnected, since a perfect universal characteristic would
comprise, it seems, a logical calculus. Leibniz did not publish the incomplete
results which he had obtained, and consequently his ideas had no continuators,
with the exception of Lambert and some others, up to the time when Boole,
De Morgan, Schroder, MacColl, and others rediscovered his theorems.
But when the investigations of the principles of mathematics became the chief
task of logical symbolism, the aspect of symbolic logic as a calculus ceased to be
of such importance, as we see in the work of Frege and Russell. Frege's
symbolism, though far better for logical analysis than Boole's or the more
modern Peano's, for instance, is far inferior to Peano's — a symbolism
in which the merits of internationality and power of expressing mathematical
theorems are very satisfactorily attained — in practical convenience. Russell,
especially in his later works, has used the ideas of Frege, many of which he
discovered subsequently to, but independently of, Frege, and modified the
symbolism of Peano as little as possible. Still, the complications thus intro-
duced take away that simple character which seems necessary to a calculus, and
which Boole and others reached by passing over certain distinctions which a
subtler logic has shown us must ultimately be made.
Let us dwell a little longer on the distinction pointed out by Leibniz be-
tween a calculus ratiocinator and a characteristica universalis or lingua char-
acteristica. The ambiguities of ordinary language are too well known for it to
be necessary for us to give instances. The objects of a complete logical sym-
bolism are: firstly, to avoid this disadvantage by providing an ideography, in
which the signs represent ideas and the relations between them directly (with-
out the intermediary of words), and secondly, so to manage that, from given
m
premises, we can, in this ideography, draw all the logical conclusions which they
imply by means of rules of transformation of formulas analogous to those of
algebra, — in fact, in which we can replace reasoning by the almost mechanical
process of calculation. This second requirement is the requirement of a calculus
ratiocinator. It is essential that the ideography should be complete, that only
symbols with a well-defined meaning should be used — to avoid the same sort of
ambiguities that words have — and, consequently, — that no suppositions should
be introduced implicitly, as is commonly the case if the meaning of signs is not
well defined. Whatever premises are necessary and sufficient for a conclusion
should be stated explicitly.
Besides this, it is of practical importance, — though it is theoretically irrelevant,-
that the ideography should be concise, so that it is a sort of stenography.
The merits of such an ideography are obvious: rigor of reasoning is ensured
by the calculus character; we are sure of not introducing unintentionally any
premise; and we can see exactly on what propositions any demonstration de-
pends.
We can shortly, but very fairly accurately, characterize the dual development
of the theory of symbolic logic during the last sixty years as follows: The calculus
ratiocinator aspect of symbolic logic was developed by Boole, De Morgan,
Jevons, Venn, C. S. Peirce, Schroder, Mrs. Ladd-Franklin and
others; the lingua characteristica aspect was developed by Frege, Peano
and Russell. Of course there is no hard and fast boundary-line between the
domains of these two parties. Thus Peirce and Schroder early began to
work at the foundations of arithmetic with the help of the calculus of relations;
and thus they did not consider the logical calculus merely as an interesting
branch of algebra. Then Peano paid particular attention to the calculative
aspect of his symbolism. Frege has remarked that his own symbolism is
meant to be a calculus ratiocinator as well as a lingua characteristica, but the
using of Frege 's symbolism as a calculus would be rather like using a three-
legged stand-camera for what is called "snap-shot" photography, and one of the
outwardly most noticeable things about Russell's work is his combination of
the symbolisms of Frege and Peano in such a way as to preserve nearly all
of the merits of each.
The present work is concerned with the calculus ratiocinator aspect, and
shows, in an admirably succinct form, the beauty, symmetry and simplicity of
the calculus of logic regarded as an algebra. In fact, it can hardly be doubted
that some such form as the one in which Schroder left it is by far the best
for exhibiting it from this point of view.^ The content of the present volume
corresponds to the two first volumes of Schroder's great but rather prolix
treatise.^ Principally owing to the infiuence of C. S. Peirce, Schroder
^Cf. A.N. Whitehead, A Treatise on Universal Algebra with Applications, Cambridge,
1898.
^ Vorlesungen iiber die Algebra der Logik, VoL L, Leipsic, 1890; VoL II, 1891 and 1905. We
may mention that a much shorter Abriss of the work has been prepared by Eugen Muller.
VoL III (1895) of Schroder's work is on the logic of relatives founded by De Morgan
and C. S. Peirce, — a branch of Logic that is only mentioned in the concluding sentences
IV
departed from the custom of Boole, Jevons, and himself (1877), which
consisted in the making fundamental of the notion of equality, and adopted the
notion of subordination or inclusion as a primitive notion. A more orthodox
BOOLIAN exposition is that of Venn, ^^ which also contains many valuable
historical notes.
We will finally make two remarks.
When Boole (cf. §0.2 below) spoke of propositions determining a class of
moments at which they are true, he really (as did MacColl ) used the word
"proposition" for what we now call a "propositional function". A "proposition"
is a thing expressed by such a phrase as "twice two are four" or "twice two are
five", and is always true or always false. But we might seem to be stating a
proposition when we say: "Mr. William Jennings Bryan is Candidate for
the Presidency of the United States", a statement which is sometimes true and
sometimes false. But such a statement is like a mathematical function in so far
as it depends on a variable — the time. Functions of this kind are conveniently
distinguished from such entities as that expressed by the phrase "twice two
are four" by calling the latter entities "propositions" and the former entities
"propositional functions": when the variable in a propositional function is fixed,
the function becomes a proposition. There is, of course, no sort of necessity
why these special names should be used; the use of them is merely a question
of convenience and convention.
In the second place, it must be carefully observed that, in §0.13, and 1 are
not defined by expressions whose principal copulas are relations of inclusion. A
definition is simply the convention that, for the sake of brevity or some other
convenience, a certain new sign is to be used instead of a group of signs whose
meaning is already known. Thus, it is the sign of equality that forms the princi-
pal copula. The theory of definition has been most minutely studied, in modern
times by Frege and Peano.
Philip E. B. Jourdain.
Girton, Cambridge. England.
of this volume.
lOSymbolic Logic, London, 1881; 2nd ed., 1894.
Contents
Preface i
Bibliography viii
0.1 Introduction 2
0.2 The Two Interpretations of the Logical Calculus 2
0.3 Relation of Inclusion 3
0.4 Definition of Equality 4
0.5 Principle of Identity 5
0.6 Principle of the Syllogism 6
0.7 Multiplication and Addition 6
0.8 Principles of Simplification and Composition 8
0.9 The Laws of Tautology and of Absorption 9
0.10 Theorems on Multiplication and Addition 10
0.11 The First Formula for Transforming Inclusions into Equalities . . 11
0.12 The Distributive Law 13
0.13 Definition of and 1 14
0.14 The Law of Duality 16
0.15 Definition of Negation 17
0.16 The Principles of Contradiction and of Excluded Middle 19
0.17 Law of Double Negation 19
0.18 Second Formulas for Transforming Inclusions into Equalities ... 20
0.19 The Law of Contraposition 21
0.20 Postulate of Existence 22
0.21 The Development of and of 1 23
0.22 Properties of the Constituents 23
0.23 Logical Functions 24
0.24 The Law of Development 24
0.25 The Formulas of De Morgan 26
0.26 Disjunctive Sums 27
0.27 Properties of Developed Functions 28
0.28 The Limits of a Function 30
0.29 Formula of Poretsky 32
0.30 Schroder's Theorem 33
0.31 The Resultant of Elimination 34
0.32 The Case of Indetermination 36
0.33 Sums and Products of Functions 36
VI
0.34 The Expression of an Inclusion by Means of an Indeterminate . . 39
0.35 The Expression of a Double Inclusion by Means of an Indeterminate 40
0.36 Solution of an Equation Involving One Unknown Quantity .... 42
0.37 Elimination of Several Unknown Quantities 45
0.38 Theorem Concerning the Values of a Function 47
0.39 Conditions of Impossibility and Indetermination 48
0.40 Solution of Equations Containing Several Unknown Quantities . 49
0.41 The Problem of Boole 51
0.42 The Method of Poretsky 52
0.43 The Law of Forms 53
0.44 The Law of Consequences 54
0.45 The Law of Causes 56
0.46 Forms of Consequences and Causes 59
0.47 Example: Venn's Problem 60
0.48 The Geometrical Diagrams of Venn 62
0.49 The Logical Machine of Jevons 64
0.50 Table of Consequences 64
0.51 Table of Causes 65
0.52 The Number of Possible Assertions 67
0.53 Particular Propositions 67
0.54 Solution of an Inequation with One Unknown 69
0.55 System of an Equation and an Inequation 70
0.56 Formulas Peculiar to the Calculus of Propositions 71
0.57 Equivalence of an Implication and an Alternative 72
0.58 Law of Importation and Exportation 74
0.59 Reduction of Inequalities to Equalities 76
0.60 Conclusion 77
1 PROJECT GUTENBERG "SMALL PRINT"
vn
Bibliography
11
George Boole. The Mathematical Analysis of Logic (Cambridge and Lon-
don, 1847).
— An Investigation of the Laws of Thought (London and Cambridge, 1854).
W. Stanley Jevons. Pure Logic (London, 1864).
— "On the Mechanical Performance of Logical Inference" {Philosophical Trans-
actions^ 1870).
Ernst Schroder. Der Operationskreis des Logikkalkuls (Leipsic, 1877).
— Vorlesungen iiher die Algebra der Logik, Vol. I (1890), Vol. II (1891), Vol. Ill:
Algebra und Logik der Relative (1895) (Leipsic). ^^
Alexander MacFarlane. Principles of the Algebra of Logic, with Exam-
ples (Edinburgh, 1879).
John Venn. Symbolic Logic, 1881; 2nd. ed., 1894 (London). ^^ Studies in
Logic by members of the Johns Hopkins University (Boston, 1883): par-
ticularly Mrs. Ladd-Franklin, O. Mitchell and C. S. Peirce.
A. N. Whitehead. A Treatise on Universal Algebra. Vol. I (Cambridge,
1898).
— "Memoir on the Algebra of Symbolic Logic" {American Journal of Mathe-
matics, Vol. XXIII, 1901).
EuGEN MtJLLER. Ubcr die Algebra der Logik: I. Die Grundlagen des Ge-
bietekalkuls; II. Das Eliminationsproblem und die Syllogistik; Programs of
the Grand Ducal Gymnasium of Tauberbischofsheim (Baden), 1900, 1901
(Leipsic).
W. E. Johnson. "Sur la theorie des egalites logiques" {Bibliotheque du Con-
gres international de Philosophic. Vol. Ill, Logique et Histoire des Sci-
ences; Paris, 1901).
Platon Poretsky. Sept Lois fondamentales de la theorie des egalites logiques
(Kazan, 1899).
— Quelques lois ulterieures de la theorie des egalites logiques (Kazan, 1902).
— "Expose elementaire de la theorie des egalites logiques a deux termes" {Revue
de Metaphysique et de Morale. Vol. VIII, 1900.)
^^This list contains only the works relating to the system of Boole and Schroder
explained in this work.
^^EuGEN MuLLER has prepared a part, and is preparing more, of the pubhcation of supple-
ments to Vols. II and III, from the papers left by Schroder.
^^A valuable work from the points of view of history and bibliography.
vni
— "Theorie des egalites logiques a trois termes" {Bibliotheque du Congres in-
ternational de Philosophie). Vol. III. {Logique et Histoire des Sciences).
(Paris, 1901, pp. 201-233).
— Theorie des non-egalites logiques (Kazan, 1904).
E. V. Huntington. "Sets of Independent Postulates for the Algebra of
Logic" {Transactions of the American Mathematical Society^ Vol. V, 1904).
IX
THE ALGEBRA OF LOGIC
0.1 Introduction
The algebra of logic was founded by George Boole (1815-1864); it was
developed and perfected by Ernst Schroder (1841-1902). The fundamental
laws of this calculus were devised to express the principles of reasoning, the
"laws of thought". But this calculus may be considered from the purely formal
point of view, which is that of mathematics, as an algebra based upon certain
principles arbitrarily laid down. It belongs to the realm of philosophy to decide
whether, and in what measure, this calculus corresponds to the actual operations
of the mind, and is adapted to translate or even to replace argument; we cannot
discuss this point here. The formal value of this calculus and its interest for the
mathematician are absolutely independent of the interpretation given it and of
the application which can be made of it to logical problems. In short, we shall
discuss it not as logic but as algebra.
0.2 The Two Interpretations of the Logical Cal-
culus
There is one circumstance of particular interest, namely, that the algebra in
question, like logic, is susceptible of two distinct interpretations, the parallelism
between them being almost perfect, according as the letters represent concepts
or propositions. Doubtless we can, with Boole and Schroder, reduce
the two interpretations to one, by considering the concepts on the one hand
and the propositions on the other as corresponding to assemblages or classes;
since a concept determines the class of objects to which it is applied (and which
in logic is called its extension), and a proposition determines the class of the
instances or moments of time in which it is true (and which by analogy can also
be called its extension). Accordingly the calculus of concepts and the calculus of
propositions become reduced to but one, the calculus of classes, or, as Leibniz
called it, the theory of the whole and part, of that which contains and that which
is contained. But as a matter of fact, the calculus of concepts and the calculus
of propositions present certain differences, as we shall see, which prevent their
complete identification from the formal point of view and consequently their
reduction to a single "calculus of classes".
Accordingly we have in reality three distinct calculi, or, in the part common
to all, three different interpretations of the same calculus. In any case the reader
must not forget that the logical value and the deductive sequence of the formulas
does not in the least depend upon the interpretations which may be given them,
and, in order to make this necessary abstraction easier, we shall take care to
place the symbols "C. I." {conceptual interpretation) and "P. I." {prepositional
interpretation) before all interpretative phrases. These interpretations shall
serve only to render the formulas intelligible, to give them clearness and to
make their meaning at once obvious, but never to justify them. They may be
omitted without destroying the logical rigidity of the system.
In order not to favor either interpretation we shall say that the letters repre-
sent terms; these terms may be either concepts or propositions according to the
case in hand. Hence we use the word term only in the logical sense. When we
wish to designate the "terms" of a sum we shall use the word summand in order
that the logical and mathematical meanings of the word may not be confused.
A term may therefore be either a factor or a summand.
0.3 Relation of Inclusion
Like all deductive theories, the algebra of logic may be established on various
systems of principles'^; we shall choose the one which most nearly approaches
the exposition of Schroder and current logical interpretation.
The fundamental relation of this calculus is the binary (two-termed) relation
which is called inclusion (for classes), suhsumption (for concepts), or implication
(for propositions) . We will adopt the first name as affecting alike the two logical
interpretations, and we will represent this relation by the sign < because it has
formal properties analogous to those of the mathematical relation < ("less than")
or more exactly <, especially the relation of not being symmetrical. Because of
this analogy Schroder represents this relation by the sign G which we shall
not employ because it is complex, whereas the relation of inclusion is a simple
one.
In the system of principles which we shall adopt, this relation is taken as a
primitive idea and is consequently indefinable. The explanations which follow
are not given for the purpose of defining it but only to indicate its meaning
according to each of the two interpretations.
C. I.: When a and b denote concepts, the relation a < b signifies that the
concept a is subsumed under the concept b; that is, it is a species with respect
to the genus b. From the extensive point of view, it denotes that the class of a's
is contained in the class of 6's or makes a part of it; or, more concisely, that "All
a's are 6's". From the comprehensive point of view it means that the concept
b is contained in the concept a or makes a part of it, so that consequently the
character a implies or involves the character b. Example: "All men are mortal";
"Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore
mortal".
P. I.: When a and b denote propositions, the relation a < b signifies that the
proposition a implies or involves the proposition 6, which is often expressed by
the hypothetical judgement, "If a is true, b is true"; or by "a implies 6"; or more
simply by "a, therefore 6". We see that in both interpretations the relation <
may be translated approximately by "therefore".
^^See Huntington, "Sets of Independent Postulates for the Algebra of Logic", Transactions
of the Am. Math. Soc, Vol. V, 1904, pp. 288-309. [Here he says: "Any set of consistent
postulates would give rise to a corresponding algebra, viz., the totality of propositions which
follow from these postulates by logical deductions. Every set of postulates should be free from
redundances, in other words, the postulates of each set should be independent, no one of them
deducible from the rest."!
Remark. — Such a relation as "a < 6" is a proposition, whatever may be the
interpretation of the terms a and b. Consequently, whenever a < relation has
two like relations (or even only one) for its members, it can receive only the
propositional interpretation, that is to say, it can only denote an implication.
A relation whose members are simple terms (letters) is called a primary
proposition; a relation whose members are primary propositions is called a sec-
ondary proposition, and so on.
From this it may be seen at once that the propositional interpretation is
more homogeneous than the conceptual, since it alone makes it possible to give
the same meaning to the copula < in both primary and secondary propositions.
0.4 Definition of Equality
There is a second copula that may be defined by means of the first; this is the
copular = ("equal to"). By definition we have
a = 6,
whenever
a < b and b < a
are true at the same time, and then only. In other words, the single relation
a = b 18 equivalent to the two simultaneous relations a <b and b < a.
In both interpretations the meaning of the copula = is determined by its
formal definition:
C. I.: a = b means, "All a's are 6's and all 6's are a's"; in other words, that
the classes a and b coincide, that they are identical. ^^
P. I.: a = b means that a implies b and b implies a; in other words, that the
propositions a and b are equivalent, that is to say, either true or false at the
same time.^^
Remark. — The relation of equality is symmetrical by very reason of its def-
inition: a = b is equivalent to b = a. But the relation of inclusion is not
symmetrical: a < b is not equivalent to 6 < a, nor does it imply it. We might
agree to consider the expression a > b equivalent to 6 < a, but we prefer for
the sake of clearness to preserve always the same sense for the copula <. How-
ever, we might translate verbally the same inclusion a < b sometimes by "a is
contained in 6", and sometimes by "6 contains a".
In order not to favor either interpretation, we will call the first member of
this relation the antecedent and the second the consequent.
CI.'. The antecedent is the subject and the consequent is the predicate of a
universal affirmative proposition.
^^This does not mean that the concepts a and h have the same meaning. Examples: "trian-
gle" and "trilateral", "equiangular triangle" and "equilateral triangle".
^^This does not mean that they have the same meaning. Example: "The triangle ABC has
two equal sides", and "The triangle ABC has two equal angles".
p. I.: The antecedent is the premise or the cause, and the consequent is the
consequence. When an imphcation is translated by a hypothetical (or condi-
tional) judgment the antecedent is cahed the hypothesis (or the condition) and
the consequent is cahed the thesis.
When we shah have to demonstrate an equahty we shah usuahy analyze it
into two converse inclusions and demonstrate them separately. This analysis is
sometimes made also when the equality is a datum (a premise).
When both members of the equality are propositions, it can be separated into
two implications, of which one is called a theorem and the other its reciprocal.
Thus whenever a theorem and its reciprocal are true we have an equality. A
simple theorem gives rise to an implication whose antecedent is the hypothesis
and whose consequent is the thesis of the theorem.
It is often said that the hypothesis is the sufficient condition of the thesis, and
the thesis the necessary condition of the hypothesis; that is to say, it is sufficient
that the hypothesis be true for the thesis to be true; while it is necessary that
the thesis be true for the hypothesis to be true also. When a theorem and its
reciprocal are true we say that its hypothesis is the necessary and sufficient
condition of the thesis; that is to say, that it is at the same time both cause and
consequence.
0.5 Principle of Identity
The first principle or axiom of the algebra of logic is the principle of identity,
which is formulated thus:
Ax. 1
a < a,
whatever the term a may be.
C. I.: "All a's are a's", i.e., any class whatsoever is contained in itself.
P. I.: "a implies a", i.e., any proposition whatsoever implies itself.
This is the primitive formula of the principle of identity. By means of the
definition of equality, we may deduce from it another formula which is often
wrongly taken as the expression of this principle:
a = a,
whatever a may be; for when we have
a < a,a < a,
we have as a direct result,
a = a.
C. I.: The class a is identical with itself.
P. I.: The proposition a is equivalent to itself.
0.6 Principle of the Syllogism
Another principle of the algebra of logic is the principle of the syllogism, which
may be formulated as follows:
Ax. 2
(a < b){b < c) < {a < c).
C. I.: "If all a's are 6's, and if all 6's are c's, then all a's are c's". This is the
principle of the categorical syllogism.
P. I.: "If a implies b, and if b implies c, a implies c." This is the principle of
the hypothetical syllogism.
We see that in this formula the principal copula has always the sense of
implication because the proposition is a secondary one.
By the definition of equality the consequences of the principle of the syllogism
may be stated in the following formulas^^:
(a <b) {b = c) < {a < c),
(a = b) {b < c) < {a < c),
(a = b) {b- c) < {a = c).
The conclusion is an equality only when both premises are equalities.
The preceding formulas can be generalized as follows:
(a <b) {b < c) {c< d) < {a < d),
{a = b) {b = c) {c = d)<{a = d).
Here we have the two chief formulas of the sorites. Many other combinations
may be easily imagined, but we can have an equality for a conclusion only when
all the premises are equalities. This statement is of great practical value. In
a succession of deductions we must pay close attention to see if the transition
from one proposition to the other takes place by means of an equivalence or only
of an implication. There is no equivalence between two extreme propositions
unless all intermediate deductions are equivalences; in other words, if there is
one single implication in the chain, the relation of the two extreme propositions
is only that of implication.
0.7 Multiplication and Addition
The algebra of logic admits of three operations, logical multiplication, logical
addition, and negation. The two former are binary operations, that is to say,
combinations of two terms having as a consequent a third term which may or
may not be different from each of them. The existence of the logical product
^^Strictly speaking, these formulas presuppose the laws of multiplication which will be
established further on; but it is fitting to cite them here in order to compare them with the
principle of the syllogism from which they are derived.
and logical sum of two terms must necessarily answer the purpose of a double
postulate, for simply to define an entity is not enough for it to exist. The two
postulates may be formulated thus:
Ax. 3 Given any two terms, a and b, then there is a term p such that
p < a,p < 6,
and that for every value of x for which
X < a, X < 6,
we have also
X < p.
Ax. 4 Given any two terms, a and b, there exists a term s such that
a < 5, 6 < 5,
we have also
s < X.
It is easily proved that the terms p and s determined by the given conditions
are unique, and accordingly we can define the product ab and the sum a + 6 as
being respectively the terms p and s.
C. I.: 1. The product of two classes is a class p which is contained in each
of them and which contains every (other) class contained in each of them;
2. The sum of two classes a and 6 is a class s which contains each of them
and which is contained in every (other) class which contains each of them.
Taking the words "less than" and "greater than" in a metaphorical sense
which the analogy of the relation < with the mathematical relation of inequality
suggests, it may be said that the product of two classes is the greatest class
contained in both, and the sum of two classes is the smallest class which contains
both.^^ Consequently the product of two classes is the part that is common to
each (the class of their common elements) and the sum of two classes is the class
of all the elements which belong to at least one of them.
P. I.: 1. The product of two propositions is a proposition which implies each
of them and which is implied by every proposition which implies both:
2. The sum of two propositions is the proposition which is implied by each
of them and which implies every proposition implied by both.
Therefore we can say that the product of two propositions is their weakest
common cause, and that their sum is their strongest common consequence,
strong and weak being used in a sense that every proposition which implies
^^According to another analogy Dedekind designated the logical sum and product by the
same signs as the least common multiple and greatest common divisor ( Was sind und was
sollen die Zahlen? Nos. 8 and 17, 1887. [Cf. English translation entitled Essays on Number
(Chicago, Open Court Publishing Co. 1901, pp. 46 and 48)] Georg Cantor originally gave
them the same designation (Mathematische Annalen, Vol. XVII, 1880).
another is stronger than the latter and the latter is weaker than the one which
implies it. Thus it is easily seen that the product of two propositions consists
in their simultaneous affirmation: "a and b are true", or simply "a and 6"; and
that their sum consists in their alternative affirmation, "either a or 6 is true", or
simply "a or U\
Remark. — Logical addition thus defined is not disjunctive;^^ that is to say,
it does not presuppose that the two summands have no element in common.
0.8 Principles of Simplification and Composition
The two preceding definitions, or rather the postulates which precede and justify
them, yield directly the following formulas:
(1) ab < a, ab < 6,
(2) {x < a){x <b) < {x < ab),
(3) a < a^b, b < a^b,
(4) (a < x){b < x) < {a^b < x).
Formulas (1) and (3) bear the name of the principle of simplification because
by means of them the premises of an argument may be simplified by deducing
therefrom weaker propositions, either by deducing one of the factors from a
product, or by deducing from a proposition a sum (alternative) of which it is a
summand.
Formulas (2) and (4) are called the principle of composition, because by
means of them two inclusions of the same antecedent or the same consequent
may be combined (composed) . In the first case we have the product of the
consequents, in the second, the sum of the antecedents.
The formulas of the principle of composition can be transformed into equal-
ities by means of the principles of the syllogism and of simplification. Thus we
have
1 (Syh.) {x < ab){ab < a) < {x < a),
(Syh.) {x < ab){ab <b) < {x <b).
Therefore
(Comp.) {x < ab) < {x < a){x <b).
2 (Syh.) {a < a^ b){a -^ b < x) < {a < x),
(Syh.) (6 < a + b){a ^b<x)<{b<x).
^^ Boole, closely following analogy with ordinary mathematics, premised, as a necessary
condition to the definition of "x + y", that x and y were mutually exclusive. Jevons, and
practically all mathematical logicians after him, advocated, on various grounds, the definition
of "logical addition" in a form which does not necessitate mutual exclusiveness.
Therefore
(Comp.) {a^b < x) < {a < x){b < x).
If we compare the new formulas with those preceding, which are their con-
verse propositions, we may write
{x < ah) = {x < a){x < b),
{a -\- b < x) = {a < x){b < x).
Thus, to say that x's contained in ab is equivalent to saying that it is con-
tained at the same time in both a and b; and to say that x contains a -\- b is
equivalent to saying that it contains at the same time both a and b.
0.9 The Laws of Tautology and of Absorption
Since the definitions of the logical sum and product do not imply any order
among the terms added or multiplied, logical addition and multiplication evi-
dently possess commutative and associative properties which may be expressed
in the formulas
ab = ba, a -\- b = b -\- a,
{ab)c = a{bc)^ {a ^b) ^ c = a ^ {b ^ c).
Moreover they possess a special property which is expressed in the law of
tautology:
a = aa. a = a -\- a.
Demonstration:
1 (Simpl.)
aa < a,
(Comp.)
(a < a){a < a) = {a < aa)
whence, by the definition of equality,
{aa < a){a < aa) = (a — aa)
In the same way:
2 (Simpl.) a < a-\- a^
(Comp.) (a < a){a < a) = {a -\- a < a),
whence
{a < a-\- a){a ^ a < a) = {a = a ^ a).
From this law it follows that the sum or product of any number whatever
of equal (identical) terms is equal to one single term. Therefore in the algebra
of logic there are neither multiples nor powers, in which respect it is very much
simpler than numerical algebra.
Finally, logical addition and multiplication posses a remarkable property
which also serves greatly to simplify calculations, and which is expressed by the
law of absorption:
a -\- ab = a, a{a -\- b) = a.
Demonstration:
1 (Comp.) (a < a){ab < a) < {a -\- ab < a),
(Simpl.) a < a -\- ab,
whence, by the definition of equality,
{a-\- ab < a){a < a -\- ab) = {a -\- ab = a).
In the same way:
1 (Comp.) {a < a){a < a -\- b) < [a < a{a -\- 6)],
(Simpl.) a{a + 6) < a,
whence
[a < a{a + b)][a{a + 6) < a] = [a{a + 6) = a].
Thus a term (a) absorbs a summand {ab) of which it is a factor, or a factor
(a + b) of which it is a summand.
0.10 Theorems on Multiplication and Addition
We can now establish two theorems with regard to the combination of inclusions
and equalities by addition and multiplication:
Th. 1
{a <b) < {ac < be), {a < b) < {a ^ c < b ^ c).
Demonstration:
1 (Simpl.) ac < c,
(Syh.) {ac < a){a < b) < {ac < b),
(Comp.) {ac < b){ac < c) < {ac < be).
2 (Simpl.) c<b^c,
(Syh.) {a < b){b <b^c) <{a<b^c).
(Comp.) (a < 6 + c){a < b ^ c) < {a ^ c < b ^ c).
10
This theorem may be easily extended to the case of equaUties:
{a = b) < {ac = bc)^ {a = b) < {a -\- c = b -\- c).
Th. 2
(a < b){c < d) < {ac < bd),
{a < b){c <d) <{a^c<b^d).
Demonstration:
1 (Syh.) {ac < a){a <b) < {ac < b),
(Syh.) {ac < c){c < d) < {ac < a),
(Comp.) {ac < b){ac < d) < {ac < bd).
2 (Syh.) {a < b){b <b^d)<{a<b^d),
(Syh.) (c < d){d <b^d)<{c<b^d),
(Comp.) (a < 6 + d){c <b ^ d) < {a ^ c <b ^ d).
This theorem may easily be extended to the case in which one of the two
inclusions is replaced by an equality:
{a = b){c <d) < {ac < bd),
{a = b){c <d) <{a^c<b^d).
When both are replaced by equalities the result is an equality:
{a = b){c = d) < {ac = bd),
{a = b){c = d) < {a^c = b^d).
To sum up, two or more inclusions or equalities can be added or multiplied
together member by member; the result will not be an equality unless all the
propositions combined are equalities.
0.11 The First Formula for Transforming Inclu-
sions into Equalities
We can now demonstrate an important formula by which an inclusion may be
transformed into an equality, or vice versa:
{a < b) = {a = ab) {a < b) = {a -\- b = b)
Demonstration:
1. {a <b) < {a = ab), {a < b) < {a ^ b = b).
11
For
(Comp.) (a < a){a < b) < {a < ab)^
{a < b){b <b) < {a^b<b).
On the other hand, we have
(Simpl.) ab < a^b < a -\- b,
(Def. =) (a < ab){ab < a) = {a = ab)
(a + 6 < b){b <a^b) = {a^b = b).
2. (a = ab) <{a <b), {a ^ b = b) < {a < b).
For
(a - ab){ab < b) < {a < b),
{a < a^ b){a -^ b = b) < {a < b).
Remark. — If we take the relation of equahty as a primitive idea (one not
defined) we shah be able to define the relation of inclusion by means of one
of the two preceding formulas. ^^ We shall then be able to demonstrate the
principle of the syllogism. ^^
From the preceding formulas may be derived an interesting result:
{a = b) = {ab = a -\-b).
For
1. [a = b) = {a < b){b < a),
{a < b) = {a = ab)^ {b < a) = {a -\- b = a),
(Syll.) (a = ab){a -\- b = a) < {ab = a -\- b).
2. {ab = a^b) < {a^b < ab),
(Comp.) {a^b < ab) = {a < ab){b < ab),
{a < ab){ab < a) = {a = ab) = {a < b),
{b < ab){ab < b) = {b = ab) = {b < a),
Hence
{ab = a^b) < {a < b){b < a) = {a = b).
^'-'See Huntington, op. cit., §??.
^■•^This can be demonstrated as follows: By definition we have {a < b) = (a = ab), and
(b < c) = {b = be). If in the first equahty we substitute for b its value derived from the second
equality, then a = abc. Substitute for a its equivalent ab, then ab = abc. This equality is
equivalent to the inclusion, ab < c. Conversely substitute a for ab; whence we have a < c.
Q.E.D.
12
0.12 The Distributive Law
The principles previously stated make it possible to demonstrate the converse
distributive law, both of multiplication with respect to addition, and of addition
with respect to multiplication,
ac-\- be < {a -\- b)c, ab -\- c < {a -\- c){b -\- c).
Demonstration:
{a < a^b) <[ac < {a^ b)c],
(6 < a + 6) < [6c < (a + b)c];
whence, by composition,
[ac < (a + b)c] [be < (a + b)c] < [ac -^ be < {a ^ b)c]
2. {ab < a) < {ab -\- c < a -\- c),
{ab < 6) < (a6 + c < 6 + c),
whence, by composition,
{ab^c< a^ c){ab -^ c < b ^ c) < [ab ^ c < {a ^ c){b + c)].
But these principles are not sufficient to demonstrate the direct distributive
law
{a -\- b)c < ac^ be, {a ^ c){b ^ c) < ab -\- c,
and we are obliged to postulate one of these formulas or some simpler one
from which they can be derived. For greater convenience we shall postulate the
formula
Ax. 5
{a -\- b)c < ac^ be.
This, combined with the converse formula, produces the equality
{a + b)c = ac-\- be
which we shall call briefly the distributive law.
From this may be directly deduced the formula
{a -\-b){c-\- d) = ac -\- be -\- ad -\- bd,
and consequently the second formula of the distributive law,
{a -\- c){b -\- c) = ab -\- c.
13
For
(a -\- c){b -\- c) = ab -\- ac -\- be -\- c,
and, by the law of absorption,
ac -\- be -\- c = c.
This second formula implies the inclusion cited above,
(a -\- c){b -\- c) < ab -\- c,
which thus is shown to be proved.
Corollary. — We have the equality
ab -\- ac -\- be = {a -\- b){a + c)(6 + c),
for
{a^b){a^c){b^c) = {a^bc){b^c) = ab^ac^bc.
It will be noted that the two members of this equality differ only in having
the signs of multiplication and addition transposed (compare §0.14).
0.13 Definition of and 1
We shall now define and introduce into the logical calculus two special terms
which we shall designate by and by 1, because of some formal analogies that
they present with the zero and unity of arithmetic. These two terms are formally
defined by the two following principles which affirm or postulate their existence.
Ax. 6 There is a term such that whatever value may be given to the term x,
we have
0<x.
Ax. 7 There is a term 1 such that whatever value may be given to the term x,
we have
X <1.
It may be shown that each of the terms thus defined is unique; that is to
say, if a second term possesses the same property it is equal to (identical with)
the first.
The two interpretations of these terms give rise to paradoxes which we shall
not stop to elucidate here, but which will be justified by the conclusions of the
theory.^^
C. I.: denotes the class contained in every class; hence it is the "null" or
"void" class which contains no element (Nothing or Naught), 1 denotes the class
which contains all classes; hence it is the totality of the elements which are
^^ Compare the author's Manuel de Logistique, Chap. I., §8, Paris, 1905 [This work, however,
did not appear].
14
contained within it. It is called, after Boole, the "universe of discourse" or
simply the "whole".
P. I.: denotes the proposition which implies every proposition; it is the
"false" or the "absurd", for it implies notably all pairs of contradictory proposi-
tions, 1 denotes the proposition which is implied in every proposition; it is the
"true", for the false may imply the true whereas the true can imply only the
true.
By definition we have the following inclusions
0<0, 0<1, 1< 1,
the first and last of which, moreover, result from the principle of identity. It is
important to bear the second in mind.
C. I.: The null class is contained in the whole?^
P. I.: The false implies the true.
By the definitions of and 1 we have the equivalences
(a < 0) = (a = 0), (1 < a) = (a = 1),
since we have
< a, a < 1
whatever the value of a.
Consequently the principle of composition gives rise to the two following
corollaries:
(a = 0)(6 = 0) = (a + 6 = 0),
(a = l){h= 1) = {ah= 1).
Thus we can combine two equalities having for a second member by adding
their first members, and two equalities having 1 for a second member by multi-
plying their first members.
Conversely, to say that a sum is "null" [zero] is to say that each of the
summands is null; to say that a product is equal to 1 is to say that each of its
factors is equal to 1.
Thus we have
(a + 6 = 0) < (a = 0),
{ah = 1) < (a = 1),
and more generally (by the principle of the syllogism)
(a<6)(6 = 0) < (a = 0),
{a<h){a = 1) < (6 = 1).
It will be noted that we can not conclude from these the equalities a6 =
and a + 6 = 1. And indeed in the conceptual interpretation the first equality
^The rendering "Nothing is everything" must be avoided.
15
denotes that the part common to the classes a and b is nuh; it by no means
fohows that either one or the other of these classes is null. The second denotes
that these two classes combined form the whole; it by no means follows that
either one or the other is equal to the whole.
The following formulas comprising the rules for the calculus of and 1, can
be demonstrated:
axO = 0, a + l = l,
a + = a, a X 1 = a.
For
(0 < a) = (0 = X a) = (a + = a),
(a < 1) = (a = a X 1) = (a + 1 = 1).
Accordingly it does not change a term to add to it or to multiply it by 1.
We express this fact by saying that is the modulus of addition and 1 the
modulus of multiplication.
On the other hand, the product of any term whatever by is and the sum
of any term whatever with 1 is 1.
These formulas justify the following interpretation of the two terms:
C. I.: The part common to any class whatever and to the null class is the
null class; the sum of any class whatever and of the whole is the whole. The
sum of the null class and of any class whatever is equal to the latter; the part
common to the whole and any class whatever is equal to the latter.
P. I.: The simultaneous affirmation of any proposition whatever and of a
false proposition is equivalent to the latter {i.e., it is false); while their alter-
native affirmation is equal to the former. The simultaneous affirmation of any
proposition whatever and of a true proposition is equivalent to the former; while
their alternative affirmation is equivalent to the latter {i.e., it is true).
Remark. — If we accept the four preceding formulas as axioms, because of
the proof afforded by the double interpretation, we may deduce from them the
paradoxical formulas
< X, and x < 1,
by means of the equivalences established above,
{a — ah) = {a < b) = {a -\- b = b).
0.14 The Law of Duality
We have proved that a perfect symmetry exists between the formulas relating to
multiplication and those relating to addition. We can pass from one class to the
other by interchanging the signs of addition and multiplication, on condition
that we also interchange the terms and 1 and reverse the meaning of the
sign < (or transpose the two members of an inclusion). This symmetry, or
16
duality as it is called, which exists in principles and definitions, must also exist
in all the formulas deduced from them as long as no principle or definition is
introduced which would overthrow them. Hence a true formula may be deduced
from another true formula by transforming it by the principle of duality; that
is, by following the rule given above. In its application the law of duality makes
it possible to replace two demonstrations by one. It is well to note that this
law is derived from the definitions of addition and multiplication (the formulas
for which are reciprocal by duality) and not, as is often thought^^, from the
laws of negation which have not yet been stated. We shall see that these laws
possess the same property and consequently preserve the duality, but they do
not originate it; and duality would exist even if the idea of negation were not
introduced. For instance, the equality (§0.12)
ab -\- ac -\- be = {a -\- b) {a -\- c){b -\- c)
is its own reciprocal by duality, for its two members are transformed into each
other by duality.
It is worth remarking that the law of duality is only applicable to primary
propositions. We call [after Boole ] those propositions primary which contain
but one copula (< or =). We call those propositions secondary of which both
members (connected by the copula < or =) are primary propositions, and so
on. For instance, the principle of identity and the principle of simplification are
primary propositions, while the principle of the syllogism and the principle of
composition are secondary propositions.
0.15 Definition of Negation
The introduction of the terms and 1 makes it possible for us to define negation.
This is a "uni-nary" operation which transforms a single term into another term
called its negative?^ The negative of a is called not-a and is written a' ?^ Its
formal definition implies the following postulate of existence^^:
^^ Boole thus derives it {Laws of Thought, London 1854, Chap. Ill, Prop. IV).
25 pj^ French] the same word negation denotes both the operation and its result, which
becomes equivocaL The result ought to be denoted by another word, like [the English] "nega-
tive". Some authors say, "supplementary" or "supplement", [e.g. Boole and Huntington
], Classical logic makes use of the term "contradictory" especially for propositions.
^^We adopt here the notation of MacColl; Schroder indicates not-a by ai which
prevents the use of indices and obliges us to express them as exponents. The notation a' has
the advantage of excluding neither indices nor exponents. The notation a employed by many
authors is inconvenient for typographical reasons. When the negative affects a proposition
written in an explicit form (with a copula) it is applied to the copula < or =) by a vertical
bar (^) or /). The accent can be considered as the indication of a vertical bar appHed to
letters.
^^ Boole follows Aristotle in usually calhng the law of duality the principle of contradiction
"which affirms that it is impossible for any being to possess a quality and at the same time
not to possess it". He writes it in the form of an equation of the second degree, x — x'^ = 0, or
x{l — cc) = in which 1 — x expresses the universe less x, or not x. Thus he regards the law
of duality as derived from negation as stated in note 24 above.
17
Ax. 8 Whatever the term a may be, there is also a term a' such that we have
at the same time
aa = 0, a -\- a = 1.
It can be proved by means of the fohowing lemma that if a term so defined
exists it is unique:
If at the same time
ac = be, a -\- c = b -\- c,
then
a = b.
Demonstration. — Multiplying both members of the second premise by a, we
have
a -\- ac = ab -\- ac.
Multiplying both members by b,
ab -\- be = b -\- be.
By the first premise,
ab -\- ae = ab -\- be.
Hence
a -\- ae = b -\- be,
which by the law of absorption may be reduced to
a = b.
Remark. — This demonstration rests upon the direct distributive law. This
law cannot, then, be demonstrated by means of negation, at least in the system
of principles which we are adopting, without reasoning in a circle.
This lemma being established, let us suppose that the same term a has two
negatives; in other words, let a[ and 02 be two terms each of which by itself
satisfies the conditions of the definition. We will prove that they are equal.
Since, by hypothesis,
aa[ =0, a -\- a[ = 1,
aa'2 = 0, a + a2 = 1,
we have
aa'i = aa'2 5 a + a 1 = a + ^2 5
whence we conclude, by the preceding lemma, that
We can now speak of the negative of a term as of a unique and well-defined
term.
The uniformity of the operation of negation may be expressed in the follow-
ing manner:
If a = 6, then also a' = b' . By this proposition, both members of an equality
in the logical calculus may be "denied".
18
0.16 The Principles of Contradiction and of Ex-
cluded Middle
By definition, a term and its negative verify the two formulas
aa = 0, a + a = 1,
which represent respectively the principle of contradiction and the principle of
excluded middle?^
C. I.: 1. The classes a and a' have nothing in common; in other words, no
element can be at the same time both a and not-a.
2. The classes a and a' combined form the whole; in other words, every
element is either a or not-a.
P. I.: 1. The simultaneous affirmation of the propositions a and not-a is
false; in other words, these two propositions cannot both be true at the same
time.
2. The alternative affirmation of the propositions a and not-a is true; in
other words, one of these two propositions must be true.
Two propositions are said to be contradictory when one is the negative of
the other; they cannot both be true or false at the same time. If one is true the
other is false; if one is false the other is true.
This is in agreement with the fact that the terms and 1 are the negatives
of each other; thus we have
0x1=0, + 1 = 1.
Generally speaking, we say that two terms are contradictory when one is the
negative of the other.
0.17 Law of Double Negation
Moreover this reciprocity is general: if a term h is the negative of the term a,
then the term a is the negative of the term h. These two statements are expressed
by the same formulas
a6 = 0, a + 6 = 1,
and, while they unequivocally determine h in terms of a, they likewise determine
a in terms of h. This is due to the symmetry of these relations, that is to say, to
the commutativity of multiplication and addition. This reciprocity is expressed
by the law of double negation
{a')' = a.
^^As Mrs. Ladd-Franklin has truly remarked ( Baldwin, Dictionary of Philosophy
and Psychology^ article "Laws of Thought"), the principle of contradiction is not sufficient to
define contradictories-^ the principle of excluded middle must be added which equally deserves
the name of principle of contradiction. This is why Mrs. Ladd-Franklin proposes to
call them respectively the principle of exclusion and the principle of exhaustion^ inasmuch
as, according to the first, two contradictory terms are exclusive (the one of the other); and,
according to the second, they are exhaustive (of the universe of discourse).
19
which may be formany proved as fohows: a' being by hypothesis the negative
of a, we have
aa = 0, a + a = 1.
On the other hand, let a" be the negative of a'] we have, in the same way,
a' a" = 0, a' ^a" = 1.
But, by the preceding lemma, these four equalities involve the equality
a = a .
Q. E. D.
This law may be expressed in the following manner:
lib = a' ^ we have a = h' ^ and conversely, by symmetry.
This proposition makes it possible, in calculations, to transpose the negative
from one member of an equality to the other.
The law of double negation makes it possible to conclude the equality of two
terms from the equality of their negatives (if a' = h' then a = b), and therefore
to cancel the negation of both members of an equality.
From the characteristic formulas of negation together with the fundamental
properties of and 1, it results that every product which contains two contra-
dictory factors is null, and that every sum which contains two contradictory
summands is equal to 1.
In particular, we have the following formulas:
a = ab -\- ab\ a = {a-\-b){a-\- b')^
which may be demonstrated as follows by means of the distributive law:
a = a X 1 = a{b + b') = ab -\- ab\
a = a + = a + 66' = (a + b){a + b').
These formulas indicate the principle of the method of development which
we shall explain in detail later (§§0.21 sqq.)
0.18 Second Formulas for Transforming Inclusions
into Equalities
We can now establish two very important equivalences between inclusions and
equalities:
{a<b) = {ab' = 0), {a<b) = {a' ^b = 1).
Demonstration. — 1. If we multiply the two members of the inclusion a < b
by b' we have
{ab' < bb') = {ab' < 0) = {ab' = 0).
20
2. Again, we know that
a = ab -\- ah' .
Now if a6' = 0,
a = a6 + = a6.
On the other hand: 1. Add a' to each of the two members of the inclusion
a < 6; we have
{a' ^a<a' ^h) = {l<a' ^h) = a' ^h = 1).
2. We know that
h= {a^h){a' ^h).
Now, if a' + 6= 1,
6 = (a + 6) X 1 = a + 6.
By the preceding formulas, an inclusion can be transformed at will into
an equality whose second member is either or 1. Any equality may also be
transformed into an equality of this form by means of the following formulas:
{a = h) = {ah' + a'h = 0), {a = h) = [{a + h'){a' + 6) = 1].
Demonstration:
{a = h) = {a< h){h <a) = {ah' = 0){a'h = 0) = {ah' + a'h = 0),
{a = h) = {a< h){h <a) = {a' ^h= l){a ^ h' = 1) = [{a' + h'){a' + 6) = 1].
Again, we have the two formulas
{a = h) = [{a + h){a' + h') = 0], {a = h) = {ah + a'h' = 1),
which can be deduced from the preceding formulas by performing the indicated
multiplications (or the indicated additions) by means of the distributive law.
0.19 The Law of Contraposition
We are now able to demonstrate the law of contraposition,
{a<h) = {h' < a').
Demonstration. — By the preceding formulas, we have
{a<h) = {ah = 0) = {h' <a').
Again, the law of contraposition may take the form
{a < h') = {h< a'),
21
which presupposes the law of double negation. It may be expressed verbally as
follows: "Two members of an inclusion may be interchanged on condition that
both are denied".
C. I.: "If all a is 6, then all not-b is not-a, and conversely".
P. I.: "If a implies 6, not-b implies not-a and conversely"; in other words, "If
a is true b is true", is equivalent to saying, "If b is false, a is false".
This equivalence is the principle of the reductio ad absurdum (see hypothet-
ical arguments, modus tollens, §0.58).
0.20 Postulate of Existence
One final axiom may be formulated here, which we will call the postulate of
existence:
Ax. 9
1 ^0
whence may be also deduced 1 7^ 0.
In the conceptual interpretation (C. I.) this axiom means that the universe
of discourse is not null, that is to say, that it contains some elements, at least
one. If it contains but one, there are only two classes possible, 1 and 0. But
even then they would be distinct, and the preceding axiom would be verified.
In the propositional interpretation (P. I.) this axiom signifies that the true
and false are distinct; in this case, it bears the mark of evidence and necessity.
The contrary proposition, 1=0, is, consequently, the type of absurdity (of the
formally false proposition) while the propositions = 0, and 1 = 1 are types of
identity (of the formally true proposition) . Accordingly we put
(1 = 0) =0, (0 = 0) = (/ = 1) = 1.
More generally, every equality of the form
X = X
is equivalent to one of the identity-types; for, if we reduce this equality so that
its second member will be or 1, we find
{xx' + xx' = 0) = (0 = 0), {xx + xV = 1) = (1 = 1).
On the other hand, every equality of the form
is equivalent to the absurdity-type, for we find by the same process,
{xx + xV = 0) = (1 = 0), {xx' + xx' = 1) = (0 = 1).
22
0.21 The Development of and of 1
Hitherto we have met only such formulas as directly express customary modes
of reasoning and consequently offer direct evidence.
We shall now expound theories and methods which depart from the usual
modes of thought and which constitute more particularly the algebra of logic
in so far as it is a formal and, so to speak, automatic method of an absolute
universality and an infallible certainty, replacing reasoning by calculation.
The fundamental process of this method is development. Given the terms
a,b,c. . . (to any finite number) , we can develop or 1 with respect to these terms
(and their negatives) by the following formulas derived from the distributive law:
= aa\
= aa' + bb' = (a + b){a + b'){a' + b){a' + b'),
= aa' + bb' -^ cc' = (a + 6 + c){a + 6 + c^){a + 6' + c)
X {a^b' ^c'){a' ^b^c)
X (a' + 6 + c'){a' + 6' + c){a' + 6' + c');
1 = a + a',
1 = (a + a'){b + b') = ab ^ ab' + a'b + a'b' ,
1 = (a + a'){b + b'){c + c') = abc + abc^ + ab^c + ab^c^
and so on. In general, for any number n of simple terms; will be developed in
a product containing 2'^ factors, and 1 in a sum containing 2'^ summands. The
factors of zero comprise all possible additive combinations, and the summands
of 1 all possible multiplicative combinations of the n given terms and their
negatives, each combination comprising n different terms and never containing
a term and its negative at the same time.
The summands of the development of 1 are what Boole called the con-
stituents (of the universe of discourse). We may equally well call them, with
PORETSKY, ^^ the minima of discourse, because they are the smallest classes
into which the universe of discourse is divided with reference to the n given
terms. In the same way we shall call the factors of the development of the
maxima of discourse, because they are the largest classes that can be determined
in the universe of discourse by means of the n given terms.
0.22 Properties of the Constituents
The constituents or minima of discourse possess two properties characteristic
of contradictory terms (of which they are a generalization); they are mutually
exclusive, i.e., the product of any two of them is 0; and they are collectively
exhaustive, i.e., the sum of all "exhausts" the universe of discourse. The latter
^See the Bibliography, page xiv.
23
property is evident from the preceding formulas. The other results from the
fact that any two constituents differ at least in the "sign" of one of the terms
which serve as factors, i.e., one contains this term as a factor and the other the
negative of this term. This is enough, as we know, to ensure that their product
be null.
The maxima of discourse possess analogous and correlative properties; their
combined product is equal to 0, as we have seen; and the sum of any two of
them is equal to 1, inasmuch as they differ in the sign of at least one of the
terms which enter into them as summands.
For the sake of simplicity, we shall confine ourselves, with Boole and
Schroder, to the study of the constituents or minima of discourse, i.e., the
developments of 1. We shall leave to the reader the task of finding and demon-
strating the corresponding theorems which concern the maxima of discourse or
the developments of 0.
0.23 Logical Functions
We shall call a logical function any term whose expression is complex, that is,
formed of letters which denote simple terms together with the signs of the three
logical operations. ^^
A logical function may be considered as a function of all the terms of dis-
course, or only of some of them which may be regarded as unknown or variable
and which in this case are denoted by the letters x,y,z. We shall represent a
function of the variables or unknown quantities, x,y,z, by the symbol f{x,y,z)
or by other analogous symbols, as in ordinary algebra. Once for all, a logical
function may be considered as a function of any term of the universe of discourse,
whether or not the term appears in the explicit expression of the function.
0.24 The Law of Development
This being established, we shall proceed to develop a function f{x) with respect
to X. Suppose the problem solved, and let
ax -\- hx'
be the development sought. By hypothesis we have the equality
f{x) = ax -\- hx'
for all possible values of x. Make x = 1 and consequently x' = 0. We have
/(!) = «•
^•-•in this algebra the logical function is analogous to the integral function of ordinary algebra,
except that it has no powers beyond the first.
24
Then put x = and x' = 1; we have
/(O) = b.
These two equahties determine the coefficients a and b of the development
which may then be written as fohows:
f{x) = mx+f{o)x',
in which /(I), /(O) represent the value of the function f{x) when we let x = 1
and X = respectively.
Corollary. — Multiplying both members of the preceding equalities by x and x^
in turn, we have the following pairs of equalities ( MacColl ):
xf{x) = ax x' f{x) = hx'
xf{x)=xf{l), x'f{x) = x'f{0).
Now let a function of two (or more) variables be developed with respect to
the two variables x and y. Developing f{x,y) ffist with respect to x, we find
f{x,y)=f{l,y)x + f{0,y)x'.
Then, developing the second member with respect to y, we have
/(x, y) = /(I, l)xy + /(I, 0)xy' + /(O, l)x'y + /(O, 0)xy
This result is symmetrical with respect to the two variables, and therefore
independent of the order in which the developments with respect to each of
them are performed.
In the same way we can obtain progressively the development of a function
of 3,4, , variables.
The general law of these developments is as follows:
To develop a function with respect to n variables, form all the constituents
of these n variables and multiply each of them by the value assumed by the
function when each of the simple factors of the corresponding constituent is
equated to 1 (which is the same thing as equating to those factors whose
negatives appear in the constituent).
When a variable with respect to which the development is made, y for in-
stance, does not appear explicitly in the function (/(x) for instance), we have,
according to the general law,
fix) = f{x)y + f{x)y'.
In particular, if a is a constant term, independent of the variables with re-
spect to which the development is made, we have for its successive developments,
a = ax ^ ax' ^
a = axy + axy' + ax'y + ax'y' ^
a = axyz + axyz' + axy' z + axy' z' + ax'yz + ax'yz' + ax'y' z
+ ax'y'z'.'^^
25
and so on. Moreover these formulas may be directly obtained by multiplying
by a both members of each development of 1.
Cor. 1. We have the equivalence
(a + x^){b -\- x) = ax -\- bx -\- ab = ax -\- bx' .
For, if we develop with respect to x, we have
ax + bx' + abx + abx' = (a + ab)x + (6 + ab)x' = ax + bx' .
Cor. 2. We have the equivalence
ax + bx' + c = (a + c)x + (6 + c)x' .
For if we develop the term c with respect to x, we find
ax + 6x' + ex + ex' = (a + c)x + (6 + c)x' .
Thus, when a function contains terms (whose sum is represented by c) inde-
pendent of X, we can always reduce it to the developed form ax^bx' by adding c
to the coefficients of both x and x' . Therefore we can always consider a function
to be reduced to this form.
In practice, we perform the development by multiplying each term which
does not contain a certain letter (x for instance) by (x + x') and by developing
the product according to the distributive law. Then, when desired, like terms
may be reduced to a single term.
0.25 The Formulas of De Morgan
In any development of 1, the sum of a certain number of constituents is the
negative of the sum of all the others.
For, by hypothesis, the sum of these two sums is equal to 1, and their product
is equal to 0, since the product of two different constituents is zero.
From this proposition may be deduced the formulas of De Morgan:
(a + b)' = a'b', (ab)' = a' ^ b' .
Demonstration. — Let us develop the sum {a-\-b):
a -\- b = ab -\- ab' -\- ab -\- a'b = ab -\- ab' -\- a'b.
Now the development of 1 with respect to a and b contains the three terms
of this development plus a fourth term a'b' . This fourth term, therefore, is the
negative of the sum of the other three.
We can demonstrate the second formula either by a correlative argument
{i.e., considering the development of by factors) or by observing that the
development of {a' -\- b'),
a'b + ab' + a'b',
^ These formulas express the method of classification by dichotomy.
26
differs from the development of 1 only by the summand ab.
How De Morgan's formulas may be generalized is now clear; for instance
we have for a sum of three terms,
This development differs from the development of 1 only by the term a^b^c\
Thus we can demonstrate the formulas
(a + 6 + c)' = a%^c\ {ahc)' = a' ^h' ^ c' ,
which are generalizations of De Morgan's formulas.
The formulas of De Morgan are in very frequent use in calculation,
for they make it possible to perform the negation of a sum or a product by
transferring the negation to the simple terms: the negative of a sum is the
product of the negatives of its summands; the negative of a product is the sum
of the negatives of its factors.
These formulas, again, make it possible to pass from a primary proposition
to its correlative proposition by duality, and to demonstrate their equivalence.
For this purpose it is only necessary to apply the law of contraposition to the
given proposition, and then to perform the negation of both members.
Example:
ab -\- ac -\- be = (a + b){a -\- c){b -\- c).
Demonstration:
{ab ^ac^ be)' = [{a + b){a + c){b + c)],
{abYiacYibcY = (a + b)' + (a + c)' + (6 + c)^
{a' + b'){a' + c'){b' + cO = a'b' + a'c' + b'c\
Since the simple terms, a, 6, c, may be any terms, we may suppress the sign
of negation by which they are affected, and obtain the given formula.
Thus De Morgan's formulas furnish a means by which to find or to
demonstrate the formula correlative to another; but, as we have said above
(§0.14), they are not the basis of this correlation.
0.26 Disjunctive Sums
By means of development we can transform any sum into a disjunctive sum,
i.e., one in which each product of its summands taken two by two is zero. For,
let (a + 6 + c) be a sum of which we do not know whether or not the three terms
are disjunctive; let us assume that they are not. Developing, we have:
a-\-b -\- c = abc -\- abc' + ab'c + ab'c' + a'bc + a'bc' + a'b'c.
Now, the first four terms of this development constitute the development of
a with respect to b and c; the two following are the development of a'b with
respect to c. The above sum, therefore, reduces to
a + a'b + a'b' c.
27
and the terms of this sum are disjunctive hke those of the preceding, as may be
verified. This process is general and, moreover, obvious. To enumerate without
repetition ah the a's, ah the 6's, and ah the c's, etc., it is clearly sufficient to
enumerate all the a's, then all the 6's which are not a's, and then all the c's
which are neither a's nor 6's, and so on.
It will be noted that the expression thus obtained is not symmetrical, since
it depends on the order assigned to the original summands. Thus the same sum
may be written:
b -\- ab^ -\- a'b'c^ c-\- ac^ -\- a'bc' ^ ....
Conversely, in order to simplify the expression of a sum, we may suppress as
factors in each of the summands (arranged in any suitable order) the negatives
of each preceding summand. Thus, we may find a symmetrical expression for a
sum. For instance,
a + a'b = b -\- ab^ = a -\- b.
0.27 Properties of Developed Functions
The practical utility of the process of development in the algebra of logic lies in
the fact that developed functions possess the following property:
The sum or the product of two functions developed with respect to the
same letters is obtained simply by finding the sum or the product of their
coefficients. The negative of a developed function is obtained simply by replacing
the coefficients of its development by their negatives.
We shall now demonstrate these propositions in the case of two variables;
this demonstration will of course be of universal application.
Let the developed functions be
dixy -\- bixy' + cix'y + dix'y' ^
a2xy + b2xy' + C2x'y + d2x'y' .
1. I say that their sum is
(ai + a2)xy + {bi + b2)xy' + (ci + C2)x'y + {di + d2)x'y' .
This result is derived directly from the distributive law.
2. I say that their product is
aia2xy + bib2xy' + ciC2x'y + did2x'y' ,
for if we find their product according to the general rule (by applying the dis-
tributive law), the products of two terms of different constituents will be zero;
therefore there will remain only the products of the terms of the same con-
stituent, and, as (by the law of tautology) the product of this constituent mul-
tiplied by itself is equal to itself, it is only necessary to obtain the product of
the coefficients.
28
3. Finally, I say that the negative of
axy + hxy' + cx'y + dx'y'
is
a'xy + h'xy' + c'x'y + d'x'y' .
In order to verify this statement, it is sufficient to prove that the product of
these two functions is zero and that their sum is equal to 1. Thus
{axy + hxy' + cx'y + dx'y'){a'xy + h'xy' + c'x'y + d'x'y')
= {aa'xy -\- bb'xy' -\- cc'x'y -\- dd'x'y')
= {0 ' xy ^ ' xy' ^ ' x'y ^ ' x'y') =
{axy -\- hxy' + cx'y + dx'y') + {a'xy + h'xy' + c'x'?/ + d'x'y')
= [{a + a')xy + (6 + ^O^?/' + (c + c^x'?/ + (<i + ^O^'^/T
= (Ix?/ + Ixy' -\- Ix'y -\- Ix'y') = 1.
Special Case. — We have the equalities:
{ah + a'h'Y = ah' + a'h,
{ah'^a'h'Y = ah^a'h',
which may easily be demonstrated in many ways; for instance, by observing that
the two sums {ah -\- a'h') and {ah' -\- a'h) combined form the development of 1;
or again by performing the negation {ah -\- a'h')' by means of De Morgan's
formulas (§0.25).
From these equalities we can deduce the following equality:
{ah' + a6 = 0) = {ah + a'h' = 1),
which result might also have been obtained in another way by observing that (§0.18)
{a = h) = {ah' + a'h = 0) = [{a + h'){a' + 6) = 1],
and by performing the multiplication indicated in the last equality.
Theorem. — We have the following equivalences:^'^
{a = he' + h'c) = {h = ac' + a'c) = (c = ah' + a'h).
For, reducing the ffist of these equalities so that its second member will be 0,
a{hc^h'c')^a'{hc' ^h'c) =0,
ahc + ah'c' + a'hc' + a'h'c = 0.
Now it is clear that the first member of this equality is symmetrical with
respect to the three terms a^h^c. We may therefore conclude that, if the two
W. Stanley Jevons, Pure Logic, 1864, p. 61.
29
other equalities which differ from the first only in the permutation of these three
letters be similarly transformed, the same result will be obtained, which proves
the proposed equivalence.
Corollary. — If we have at the same time the three inclusions:
a < bc^ -\- h' c^ b < ac^ -\- a'c^ c < ab^ -\- a'b.
we have also the converse inclusion, an therefore the corresponding equalities
a = bc^ -\- b'c^ b = ac^ -\- a'c^ c = ab^ -\- a'b.
For if we transform the given inclusions into equalities, we shall have
abc + ab'c' = 0, abc + a'bc' = 0, abc + a'b'c = 0,
whence, by combining them into a single equality,
abc + ab'c' + a'bc' + a'b'c = 0.
Now this equality, as we see, is equivalent to any one of the three equalities
to be demonstrated.
0.28 The Limits of a Function
A term x is said to be comprised between two given terms, a and b, when it
contains one and is contained in the other; that is to say, if we have, for instance,
a < X, X < b,
which we may write more briefiy as
a < X < b.
Such a formula is called a double inclusion. When the term x is variable and
always comprised between two constant terms a and b, these terms are called
the limits of x. The first (contained in x) is called inferior limit; the second
(which contains x) is called the superior limit.
Theorem. — A developed function is comprised between the sum and the
product of its coefficients.
We shall first demonstrate this theorem for a function of one variable,
ax -\- bx' .
We have, on the one hand,
{ab < a) < {abx < ax),
{ab <b) < {abx' < bx').
30
Therefore
abx -\- ahx' < ax + hx' ^
or
ah < ax ^ hx' .
On the other hand,
Therefore
or
To sum up,
{a < a ^h) < [ax < {a -\- b)x],
{b <a^b) < [bx' < (a + b)x'].
ax -\- bx' < {a-\-b){x -\- x')^
ax + bx' < a -\- b.
ab < ax -\- bx' < a -\- b.
Q. E. D.
Remark 1. This double inclusion may be expressed in the following form:^^
f{b) < fix) < f{a).
For
/(a) = aa -\- ba' = a -\- b,
f{b) =ab^ bb' = ab.
But this form, pertaining as it does to an equation of one unknown quantity,
does not appear susceptible of generalization, whereas the other one does so
appear, for it is readily seen that the former demonstration is of general appli-
cation. Whatever the number of variables n (and consequently the number of
constituents 2^) it may be demonstrated in exactly the same manner that the
function contains the product of its coefficients and is contained in their sum.
Hence the theorem is of general application.
Remark 2. — This theorem assumes that all the constituents appear in the
development, consequently those that are wanting must really be present with
the coefficient 0. In this case, the product of all the coefficients is evidently 0.
Likewise when one coefficient has the value 1, the sum of all the coefficients is
equal to 1.
It will be shown later (§0.38) that a function may reach both its limits, and
consequently that they are its extreme values. As yet, however, we know only
that it is always comprised between them.
EuGEN MuLLER, Aus dcv Algebra der Logik, Art. II.
31
0.29 Formula of Poretsky.^^
We have the equivalence
{x = ax -\- hx') = {h < X < a).
Demonstration. — First multiplying by x both members of the given equality
[which is the first member of the entire secondary equality], we have
which, as we know, is equivalent to the inclusion
X < a.
Now multiplying both members by x' ^ we have
= hx',
which, as we know, is equivalent to the inclusion
h < X.
Summing up, we have
{x = ax ^ hx') < {h < X < a).
Conversely,
{h < X < a) < {x = ax ^ hx').
For
{x < a) = {x = ax),
{h<x) = {hx' = 0).
Adding these two equalities member to member [the second members of the
two larger equalities],
{x = ax){o = hx) < {x = ax -\- hx').
Therefore
{h < X < a) < {x = ax -\- hx')
and thus the equivalence is proved.
^^ PoRETSKY, "Sur les methodes pour resoudre les egalites logiques". (Bull, de la Soc.
math, de Kazan, Vol. II, 1884).
32
0.30 Schroder's Theorem. ^^
The equality
ax -\- hx' =
signifies that x hes between a' and h.
Demonstration:
{ax + hx' = 0) = {ax = {)){hx' = 0),
{ax = 0) = (x < a'),
{hx' = {)) = {h<x).
{ax ^hx' = ^) = {h <x < a').
Hence
Comparing this theorem with the formula of Poretsky, we obtain at once
the equality
{ax + hx' = 0) = (x = a'x + hx')^
which may be directly proved by reducing the formula of Poretsky to an
equality whose second member is 0, thus:
{x = a'x + hx') = [x{ax + h'x') + x' {a'x + hx') = 0] = {ax + hx' = 0).
If we consider the given equality as an equation in which x is the unknown
quantity, Poretsky's formula wih be its solution.
From the double inclusion
h <x <a'
we conclude, by the principle of the syllogism, that
h<a'
This is a consequence of the given equality and is independent of the un-
known quantity x. It is called the resultant of the elimination of x in the given
equation. It is equivalent to the equality
a6 = 0.
Therefore we have the implication
{ax^hx' = 0) < (a6 = 0).
Taking this consequence into consideration, the solution may be simplified,
for
(a6 = 0) = {h = a'h).
Schroder, Operationskreis des Logikkalkiils (1877), Theorem 20.
33
Therefore
X = a'x + bx' = a'x + a'bx'
= a'bx -\- a'b'x + a'bx' = a'6 + a'6'x
This form of the solution conforms most closely to common sense: since x'
contains b and is contained in a', it is natural that x should be equal to the sum
of b and a part of a' (that is to say, the part common to a' and x) . The solution
is generally indeterminate (between the limits a' and 6); it is determinate only
when the limits are equal,
a' = b,
for then
X = b ^ a'x = b -\-bx = b = a\
Then the equation assumes the form
{ax -\- a'x' = 0) = {a' = x)
and is equivalent to the double inclusion
{a' < X < a') = {x = a').
0.31 The Resultant of Elimination
When ab is not zero, the equation is impossible (always false), because it has a
false consequence. It is for this reason that Schroder considers the resultant
of the elimination as a condition of the equation. But we must not be misled
by this equivocal word. The resultant of the elimination of x is not a cause
of the equation, it is a consequence of it; it is not a sufficient but a necessary
condition.
The same conclusion may be reached by observing that ab is the inferior limit
of the function ax-\-bx' , and that consequently the function can not vanish unless
this limit is 0.
{ab < ax -\- bx'){ax -\- bx' = 0) < {ab = 0).
We can express the resultant of elimination in other equivalent forms; for
instance, if we write the equation in the form
{a^x'){b^x) =0,
we observe that the resultant
ab =
is obtained simply by dropping the unknown quantity (by suppressing the
terms x and x'). Again the equation may be written:
a'x -\- b'x' = 1
34
and the resultant of elimination:
a' + 6' = l.
Here again it is obtained simply by dropping the unknown quantity.^^
Remark. If in the equation
ax -\-bx^ =
we substitute for the unknown quantity x its value derived from the equations,
X = a^x -\- bx\ x^ = ax -\- b^x\
we find
{abx -\- abx' = 0) = {ab = 0),
that is to say, the resultant of the elimination of x which, as we have seen, is
a consequence of the equation itself. Thus we are assured that the value of x
verifies this equation. Therefore we can, with VoiGT, define the solution of an
equation as that value which, when substituted for x in the equation, reduces
it to the resultant of the elimination of x.
Special Case. — When the equation contains a term independent of x, z.e.,
when it is of the form
ax + 6x' + c =
it is equivalent to
(a + c)x + (6 + c)x' = 0,
and the resultant of elimination is
{a -\- c){b -\- c) = a6 + c = 0,
whence we derive this practical rule: To obtain the resultant of the elimination
of X in this case, it is sufficient to equate to zero the product of the coefficients
of X and x', and add to them the term independent of x.
^^This is the method of ehmination of Mrs. Ladd-Franklin and Mr. Mitchell, but
this rule is deceptive in its apparent simpHcity, for it cannot be appHed to the same equation
when put in either of the forms
ax + bx' = 0, (a + x){b' + x) = 1.
Now, on the other hand, as we shall see (§0.54), for inequalities it may be applied to the
forms
ax + bx' / 0, (a' + x'){b' + x) / 1.
and not to the equivalent forms
{a-\-x){b-\- x) ^0, a'x^b'x +1.
Consequently, it has not the mnemonic property attributed to it, for, to use it correctly, it
is necessary to recall to which forms it is applicable.
35
0.32 The Case of Indetermination
Just as the resultant
ab =
corresponds to the case when the equation is possible, so the equality
corresponds to the case of absolute indetermination. For in this case the equation
both of whose coefficients are zero (a = 0), (6 = 0), is reduced to an identity
(0 = 0), and therefore is "identically" verified, whatever the value of x may be;
it does not determine the value of x at all, since the double inclusion
b < X < a'
then becomes
<x< 1
which does not limit in any way the variability of x. In this case we say that
the equation is indeterminate.
We shall reach the same conclusion if we observe that (a + b) is the superior
limit of the function ax-\-bx and that, if this limit is 0, the function is necessarily
zero for all values of x,
{ax + 6x' < a + b){a + 6 = 0) < (ax + bx^ = 0).
Special Case. — When the equation contains a term independent of x,
ax -\- bx^ -\- c = 0,
the condition of absolute indetermination takes the form
a + 6 + c = 0.
For
ax -\- bx' -\- c = (a + c)x -\- {b -\- c)x' ^
(a + c) + (6 + c) = a + 6 + c = 0.
0.33 Sums and Products of Functions
It is desirable at this point to introduce a notation borrowed from mathematics,
which is very useful in the algebra of logic. Let f{x) be an expression containing
one variable; suppose that the class of all the possible values of x is determined;
then the class of all the values which the function f{x) can assume in conse-
quence will also be determined. Their sum will be represented by ^^ f{x) and
their product by Y\x /(^) This is a new notation and not a new notion, for it is
merely the idea of sum and product applied to the values of a function.
36
When the symbols ^ and Yl ^^^ apphed to propositions, they assume an
interesting significance:
X
means that f{x) =0 is true for every value of x; and
X
that f{x) = is true for some value of x. For, in order that a product may
be equal to 1 {i.e., be true), all its factors must be equal to 1 {i.e., be true);
but, in order that a sum may be equal to 1 {i.e., be true), it is sufficient that
only one of its summands be equal to 1 {i.e., be true). Thus we have a means
of expressing universal and particular propositions when they are applied to
variables, especially those in the form: "For every value of x such and such a
proposition is true", and "For some value of x, such and such a proposition is
true", etc.
For instance, the equivalence
{a = b) = {ac = be) {a -\- c = b -\- c)
is somewhat paradoxical because the second member contains a term (c) which
does not appear in the first. This equivalence is independent of c, so that we
can write it as follows, considering c as a variable x
TT[(a = b) = {ax = bx){a -\- x = b -\- x)],
X
or, the first member being independent of x,
(a = 6) = 1 I [{ax = bx) {a -\- x = b -\- x)].
X
In general, when a proposition contains a variable term, great care is neces-
sary to distinguish the case in which it is true for every value of the variable,
from the case in which it is true only for some value of the variable. ^^ This is
the purpose that the symbols Yl ^^^ XI serve.
Thus when we say for instance that the equation
ax -
bx' =
is possible, we are stating that it can be verified by some value of x; that is to
say,
(ax -\- bx' = 0),
E(^
^^This is the same as the distinction made in mathematics between identities and equations,
except that an equation may not be verified by any value of the variable.
37
and, since the necessary and sufficient condition for this is that the resultant
{ab = 0) is true, we must write
2_^{cix -\- bx = 0) = {ab = 0),
although we have only the implication
{ax^bx = 0) < {ab = 0).
On the other hand, the necessary and sufficient condition for the equation
to be verified by every value of x is that
a + 6 = 0.
Demonstration. — 1. The condition is sufficient, for if
(a + 6 = 0) = (a = 0)(6 = 0),
we obviously have
ax -\-bx' =
whatever the value of x; that is to say,
Y[{ax^bx' = 0).
X
2. The condition is necessary, for if
W{ax + bx') = 0,
X
the equation is true, in particular, for the value x = a; hence
Therefore the equivalence
Y[{ax + 6x' = 0) = (a + 6 = 0)
X
is proved. ^^ In this instance, the equation reduces to an identity: its first
member is "identically" null.
^^ EUGEN MULLER, Op. Clt.
38
0.34 The Expression of an Inclusion by Means of
an Indeterminate
The foregoing notation is indispensable in almost every case where variables or
indeterminates occur in one member of an equivalence, which are not present in
the other. For instance, certain authors predicate the two following equivalences
{a < b) = {a = hu) = {a-\-v = b)^
in which u, v are two "indeterminates". Now, each of the two equalities has the
inclusion (a < b) as its consequence, as we may assure ourselves by eliminating
u and V respectively from the following equalities:
1. [a{b' + u') + a%u = 0] = [{ab' + a%)u + au' = 0].
Resultant:
[{ab' + a'b)a = 0] = {ab' = 0) = {a<b).
2. [(a + v)b' + a%v = 0] = [b'v + {ab' + a^y = 0].
Resultant:
[b\ab' + a%) = 0] = {ab' = 0) + (a < b).
But we cannot say, conversely, that the inclusion implies the two equalities
for any values of u and v; and, in fact, we restrict ourselves to the proof that
this implication holds for some value of u and v, namely for the particular values
u = a, b = v;
for we have
(a = ab) = {a < b) = {a -\- b = b).
But we cannot conclude, from the fact that the implication (and therefore
also the equivalence) is true for some value of the indeterminates, that it is true
for all; in particular, it is not true for the values
IX = 1, V = 0,
for then (a = bu) and {a-\-v = b) become {a = b), which obviously asserts more
than the given inclusion (a < b).^^
Therefore we can write only the equivalences
{a <b) = ^(a = bu) = ^(a -^v = b),
^^ Likewise if we make
u = 0, V = 1,
we obtain the equalities
(a = 0), (6=1),
which assert stih more than the given inclusion.
39
but the three expressions
are not equivalent. ^^
0.35 The Expression of a Double Inclusion by
Means of an Indeterminate
Theorem. The double inclusion
b < X < a
is equivalent to the equality x = au -\- bu' together with the condition (b < a), u
being a term absolutely indeterminate.
Demonstration. — Let us develop an equality in question,
x{a'u + b'u') + x'{au + bu') = 0,
{a'x + ax')u + {b'x + bx')u' = 0.
Eliminating u from it,
a'b'x + abx' = 0.
This equality is equivalent to the double inclusion
ab < X < a -\- b.
^'-' According to the remark in the preceding note, it is clear that we have
]J(a = bu) = {a = b = 0), J|(a + i; = b) = {a = b=l),
V V
since the equahties affected by the sign H ^^ay be Hkewise verified by the values
u = 0, u = 1 and v = 0, v = 1.
If we wish to know within what limits the indeterminates u and v are variable, it is sufficient
to solve with respect to them the equations
(a < b) = (a = bu), (a < b) = {a -\- v = b),
or
{ab' = a'bu + ab' + au , ab' = ab' + b' v + a'bv' ,
or
a'bu + abu' = 0, a'b'v + a'bv' = 0,
from which (by a formula to be demonstrated later on) we derive the solutions
u = ab -\- w{a + b'), v = a'b + w{a + 6),
or simply
u = ab + wb' , V = a'b + wa,
w being absolutely indeterminate. We would arrive at these solutions simply by asking: By
what term must we multiply b in order to obtain a? By a term which contains ab plus any
part of b' . What term must we add to a in order to obtain b? A term which contains a'b plus
any part of a. In short, u can vary between ab and a + b' , v between a'b and a + 6.
40
But, by hypothesis, we have
{b < a) = {ah = b) = {a -\- b = a).
The double inclusion is therefore reduced to
b < X < a.
So, whatever the value of u, the equality under consideration involves the
double inclusion. Conversely, the double inclusion involves the equality, what-
ever the value of x may be, for it is equivalent to
a'x -\- bx' = 0,
and then the equality is simplified and reduced to
ax'u + b'xu' = 0.
We can always derive from this the value of u in terms of x, for the resultant
{ab'xx' = 0) is identically verified. The solution is given by the double inclusion
b'x < u < a' -\- X.
Remark. — There is no contradiction between this result, which shows that
the value of u lies between certain limits, and the previous assertion that u is
absolutely indeterminate; for the latter assumes that x is any value that will
verify the double inclusion, while when we evaluate u in terms of x the value of
X is supposed to be determinate, and it is with respect to this particular value
of X that the value of u is subjected to limits. ^^
In order that the value of u should be completely determined, it is necessary
and sufficient that we should have
b'x = a' + X,
that is to say,
b'xax' + (6 + x'){a' + x) =
or
bx + a'x' = 0.
Now, by hypothesis, we already have
a'x + bx' = 0.
If we combine these two equalities, we find
(a + 6 = 0) = (a = l)(6 = 0).
^■•^ Moreover, if we substitute for x its inferior limit b in the inferior limit of u, this limit
becomes bb' = 0; and, if we substitute for x its superior limit a in the superior limit of u, this
limit becomes a + a' = 1.
41
This is the case when the value of x is absolutely indeterminate, since it lies
between the limits and 1.
In this case we have
u = b^x = a -\- X = X.
In order that the value of u be absolutely indeterminate, it is necessary and
sufficient that we have at the same time
b'x = 0, a' + x = l,
or
b'x -\- ax' = 0,
that is
a < X < b.
Now we already have, by hypothesis,
b < X < a;
so we may infer
b = X = a.
This is the case in which the value of x is completely determinate.
0.36 Solution of an Equation Involving One Un-
known Quantity
The solution of the equation
ax -\- bx' =
may be expressed in the form
X = a'u + bu' ^
u being an indeterminate, on condition that the resultant of the equation be
verified; for we can prove that this equality implies the equality
ab'x + a'bx' = 0,
which is equivalent to the double inclusion
a'b < X < a' -\- b.
Now, by hypothesis, we have
{ab = 0) = {a% = b) = {a' ^b = a').
Therefore, in this hypothesis, the proposed solution implies the double in-
clusion
b < X < a']
42
which is equivalent to the given equation.
Remark. — In the same hypothesis in which we have
{ab = 0) = {h<a'),
we can always put this solution in the simpler but less symmetrical forms
X = b-\- a'u, x = a\b-\-u).
For
1. We have identically
b = bu-\- bu' .
Now
{b < a') < {bu < a'u).
Therefore
{x = bu' + a'u) = (x = 6 + a'u).
2. Let us now demonstrate the formula
Now
a'b = b.
Therefore
X = b -\- a'u
which may be reduced to the preceding form.
Again, we can put the same solution in the form
X = a'b -\- u{ab + a'b')^
which follows from the equation put in the form
ab'x + a'bx' = 0,
if we note that
a' ^b = ab^a'b^ a'b'
and that
ua'b < a'b.
This last form is needlessly complicated, since, by hypothesis,
ab = 0.
Therefore there remains
X = a'b -\- ua'b'
which again is equivalent to
X = b -\- ua' ^
43
since
a'h = h and a' = a'h + a'h' .
Whatever form we give to the solution, the parameter u in it is absolutely
indeterminate, z.e., it can receive all possible values, including and 1; for when
1^ = we have
X = 6,
and when i^ = 1 we have
and these are the two extreme values of x.
Now we understand that x is determinate in the particular case in which
a' = 6, and that, on the other hand, it is absolutely indeterminate when
6 = 0, a' = 1, (ora = 0).
Summing up, the formula
X = a'u + hu'
replaces the "limited" variable x (lying between the limits a' and h) by the
"unlimited" variable u which can receive all possible values, including and 1.
Remark}'^ — The formula of solution
X = a'x + hx'
is indeed equivalent to the given equation, but not so the formula of solution
X = a'u + hu'
as a function of the indeterminate u. For if we develop the latter we find
ah'x + a'hx' + ah{xu + x'u') + a'h'{xu' + x'u) = 0,
and if we compare it with the developed equation
ah + ah'x + a'hx' = 0,
we ascertain that it contains, besides the solution, the equality
ah{xu' -\- x'u) = 0,
and lacks of the same solution the equality
a'h'{xu' ^x'u) = 0.
Moreover these two terms disappear if we make
PoRETSKY. Sept lois, Chaps. XXXIII and XXXIV.
44
and this reduces the formula to
X = a'x + hx' .
From this remark, Poretsky concluded that, in general, the solution of
an equation is neither a consequence nor a cause of the equation. It is a cause
of it in the particular case in which
ah = 0,
and it is a consequence of it in the particular case in which
{a'h' = {)) = {a^b = i).
But if ab is not equal to 0, the equation is unsolvable and the formula of
solution absurd, which fact explains the preceding paradox. If we have at the
same time
ab = and a -\- b = 1,
the solution is both consequence and cause at the same time, that is to say, it
is equivalent to the equation. For when a = b the equation is determinate and
has only the one solution
X = a' = b.
Thus, whenever an equation is solvable, its solution is one of its causes; and,
in fact, the problem consists in finding a value of x which will verify it, i.e.,
which is a cause of it.
To sum up, we have the following equivalence:
{ax -\- bx' = 0) = {ab = 0) /_^(^ = cl'u + bu')
u
which includes the following implications:
{ax^bx' = 0) < (a6 = 0),
{ax + bx' = 0) < /_^(^ = ci'u + bu'),
u
{ab = 0) ^(x = a'u + bu') < {ax + bx' = 0).
u
0.37 Elimination of Several Unknown Quantities
We shall now consider an equation involving several unknown quantities and
suppose it reduced to the normal form, i.e., its first member developed with
respect to the unknown quantities, and its second member zero. Let us first
concern ourselves with the problem of elimination. We can eliminate the un-
known quantities either one by one or all at once.
45
For instance, let
0(x, 7/, z) = axyz -\- hxyz' + cxy' z + dxy' z'
+ fx'yz + gx'yz' + hx'y' z + kx'y' z' =
be an equation involving three unknown quantities.
We can eliminate 2: by considering it as the only unknown quantity, and we
obtain as resultant
{axy + cxy' + fx'y + hx'y'){hxy + dxy' + 5'x'?/ + kx'y') =
or
(6) ahxy + C(ix7/' + fgx'y -\- hkx'y' = 0.
If equation (5) is possible, equation (6) is possible as well; that is, it is verified
by some values of x and y. Accordingly we can eliminate y from the equation
by considering it as the only unknown quantity, and we obtain as resultant
{abx -\- fgx'){cdx + hkx') =
or
(7) abcdx -\- fghkx' = 0.
If equation (5) is possible, equation (7) is also possible;, that is, it is verified
by some values of x. Hence we can eliminate x from it and obtain as the final
resultant,
abed • fghk =
which is a consequence of (5), independent of the unknown quantities. It is
evident, by the principle of symmetry, that the same resultant would be obtained
if we were to eliminate the unknown quantities in a different order. Moreover
this result might have been foreseen, for since we have (§0.28)
abcdfghk < (/)(x, ?/, z),
0(x, 7/, z) can vanish only if the product of its coefficients is zero:
[(l){x,y,z) = 0] < {abcdfghk = 0).
Hence we can eliminate all the unknown quantities at once by equating to
the product of the coefficients of the function developed with respect to all
these unknown quantities.
We can also eliminate some only of the unknown quantities at one time. To
do this, it is sufficient to develop the first member with respect to these unknown
quantities and to equate the product of the coefficients of this development to 0.
This product will generally contain the other unknown quantities. Thus the
resultant of the elimination of z alone, as we have seen, is
abxy -\- cdxy' -\- fgx'y -\- hkx'y' =
46
and the resultant of the ehmination of y and z is
ahcdx + fghkx' = 0.
These partial resultants can be obtained by means of the following practical
rule: Form the constituents relating to the unknown quantities to be retained;
give each of them, for a coefficient, the product of the coefficients of the con-
stituents of the general development of which it is a factor, and equate the sum
to 0.
0.38 Theorem Concerning the Values of a Func-
tion
All the values which can he assumed by a function of any number of variables
f{x^y^z...) are given by the formula
abc . . .k -\-u{a-\-b-\- c-\- . . . -\- k)^
in which u is absolutely indeterminate, and a^b^c. . . ,k are the coefficients of
the development of f.
Demonstration. — It is sufficient to prove that in the equality
/(x, y^z . . .) = abc . . .k -\- u{a + 6 + c + . . . + /c)
u can assume all possible values, that is to say, that this equality, considered as
an equation in terms of ix, is indeterminate.
In the first place, for the sake of greater homogeneity, we may put the second
member in the form
u'abc . . .k -\- u{a + 6 + c + . . . + /c),
for
abc . . .k = uabc . . .k -\- u'abc . . . /c,
and
uabc . . .k < u{a -\- b -\- c-\- . . . -\- k).
Reducing the second member to (assuming there are only three variables
x,y,z)
{axyz + bxyz' + cxy' z + . . . + kx'y' z')
X [ua'b'c' ...k' ^ u\a' + 6' + c' + . . . + k')]
+ {a'xyz + b'xyz' + c'xy' z + . . . + k'x'y' z')
X [u{a + 6 + c+... + A;)+ u'abc . . . /c] = 0,
or more simply
u{a + 6 + c + . . . + k) {a'xyz -\- b'xyz -\- c'xy' z + . . . + k'x'y' z')
-^ u' {a' ^b' ^ c' ^ . . . ^ k'){axyz -\- bxyz + . . . + kx'yz) = 0.
47
If we eliminate all the variables x, ?/, z, but not the indeterminate u, we get
the resultant
1^(0 + 6 + c + . . . + k)a'h'c' ...k'
+ u'{a' + 6' + c' + . . . + k')ahc ...k = 0.
Now the two coefficients of u and u' are identically zero; it follows that u is
absolutely indeterminate, which was to be proved. ^^
From this theorem follows the very important consequence that a function
of any number of variables can be changed into a function of a single variable
without diminishing or altering its "variability".
Corollary. — A function of any number of variables can become equal to either
of its limits.
For, if this function is expressed in the equivalent form
abc . . .k -\- u{a -\- b -\- c-\- . . . -\- k),
it will be equal to its minimum {abc. . . k) when i^ = 0, and to its maximum
(a + 6 + c + . . . + /c) when u = 1.
Moreover we can verify this proposition on the primitive form of the function
by giving suitable values to the variables.
Thus a function can assume all values comprised between its two limits,
including the limits themselves. Consequently, it is absolutely indeterminate
when
abc . . .k = and a -\- b -\- c-\- . . . -\- k = 1
at the same time, or
abc...k = = a%'c' ...k\
0.39 Conditions of Impossibility and Indetermi-
nation
The preceding theorem enables us to find the conditions under which an equation
of several unknown quantities is impossible or indeterminate. Let /(x, y^z . . .)
be the first member supposed to be developed, and a^b^c. . . ,k its coefficients.
The necessary and sufficient condition for the equation to be possible is
abc. . .k = 0.
For, (1) if / vanishes for some value of the unknowns, its inferior limit
abc. . . k must be zero; (2) if abc. . .k is zero, / may become equal to it, and
therefore may vanish for certain values of the unknowns.
The necessary and sufficient condition for the equation to be indeterminate
(identically verified) is
a + 6 + c... + A: = 0.
^^ Whitehead, Universal Algebra, Vol. I, §33 (4).
48
For, (1) if a + 6 + c + . . . + A: is zero, since it is the superior limit of /, this
function wih always and necessarily be zero; (2) if / is zero for all values of the
unknowns, a -\- b -\- c-\- . . . -\- k will be zero, for it is one of the values of /.
Summing up, therefore, we have the two equivalences
^[/(x,7/,z,...)=0] = (a6c.../c = 0).
l[[f{x, 7/, z, . . .) = 0] = (a + 6 + c. . . + /c = 0).
The equality abc ... /c = is, as we know, the resultant of the elimination of
all the unknowns; it is the consequence that can be derived from the equation
(assumed to be verified) independently of all the unknowns.
0.40 Solution of Equations Containing Several Un-
known Quantities
On the other hand, let us see how we can solve an equation with respect to its
various unknowns, and, to this end, we shall limit ourselves to the case of two
unknowns
axy -\- hxy' + cx'y + dx'y' = 0.
First solving with respect to x,
X = {a'y + h'y')x + {cy + dy')x' .
The resultant of the elimination of x is
acy + hdy' = 0.
If the given equation is true, this resultant is true.
Now it is an equation involving y only; solving it,
y = {a' ^c')y^hdy'.
Had we eliminated y first and then x, we would have obtained the solution
y = {a'x + c'x')y + {hx + dx')y'
and the equation in x
abx -\- cdx' = 0,
whence the solution
X = (a' + h')x + cdx' .
We see that the solution of an equation involving two unknown quantities
is not symmetrical with respect to these unknowns; according to the order in
which they were eliminated, we have the solution
X = {a'y + b'y')x -\- {cy -\- dy')x' ,
y = {a'^ c')y + hdy',
49
or the solution
X = {a' ^ h')x + cdx,
y = {a'x + c'x')y + {hx + dx')y' .
If we replace the terms x, ?/, in the second members by indeterminates u^ v,
one of the unknowns will depend on only one indeterminate, while the other
will depend on two. We shall have a symmetrical solution by combining the two
formulas,
X = {a^ -\- h')u + cdu\
y = {a'^ c')v + hdv',
but the two indeterminates u and v will no longer be independent of each other.
For if we bring these solutions into the given equation, it becomes
ahcd + ah' c'uv + a'hd'uv' + a'cd'u'v + h'c'du'v' =
or since, by hypothesis, the resultant ahcd = is verified,
ah' c'uv + a'hd'uv' + a'cdu'v + h'c'du'v' = 0.
This is an "equation of condition" which the indeterminates u and v must
verify; it can always be verified, since its resultant is identically true,
ah'c' • a'hd' • a'cd' • h'c'd = aa' • hh' • cc' • dd' = 0,
but it is not verified by any pair of values attributed to u and v.
Some general symmetrical solutions, i.e., symmetrical solutions in which
the unknowns are expressed in terms of several independent indeterminates,
can however be found. This problem has been treated by Schroder ^^, by
Whitehead ^^ and by Johnson. ^^
This investigation has only a purely technical interest; for, from the practical
point of view, we either wish to eliminate one or more unknown quantities (or
even all), or else we seek to solve the equation with respect to one particular
unknown. In the first case, we develop the first member with respect to the
unknowns to be eliminated and equate the product of its coefficients to 0. In
the second case we develop with respect to the unknown that is to be extricated
and apply the formula for the solution of the equation of one unknown quantity.
If it is desired to have the solution in terms of some unknown quantities or in
terms of the known only, the other unknowns (or all the unknowns) must first
be eliminated before performing the solution.
"^"^ Algebra der Logik, Vol. I, §24.
"^^ Universal Algebra, Vol. I, §§35-37.
^^"Sur la theorie des egalites logiques", Bibl. du Cong, intern, de Phil., Vol. Ill, p. 185
(Paris, 1901).
50
0.41 The Problem of Boole
According to Boole the most general problem of the algebra of logic is the
following^^ :
Given any equation (which is assumed to be possible)
f{x,y,z,...) = 0,
and, on the other hand, the expression of a term t in terms of the variables
contained in the preceding equation
to determine the expression of t in terms of the constants contained in / and in
Suppose / and Lp developed with respect to the variables x^y^z . . . and let
PiiP2^P3^ • • • be their constituents:
/(x, ?/, z, . . .) = Api + Bp2 + Cps + . . . ,
0(x, ?/, z, . . .) = api + 6p2 + cp3 + . . . .
Then reduce the equation which expresses t so that its second member will
be 0:
(t0' + t> = 0) = [(aVi + b'p2 + cVs + . . 0^
+ {api + bp2 + cp3 + . . 0^' = 0]-
Combining the two equations into a single equation and developing it with
respect to t:
[{A + a')pi + (5 + b')p2 + (C + c')ps + . . .]t
+ [{A + a)pi + (5 + b)p2 + (C + c)ps + . . .]t' = 0.
This is the equation which gives the desired expression of t. Eliminating t,
we obtain the resultant
Api + 5p2 + Cp3 + . . . = 0,
as we might expect. If, on the other hand, we wish to eliminate x, 7/, z, . . . {i.e.,
the constituents Pi,P2,P3 • • O? we put the equation in the form
{A + a't + at')pi + (5 + b't + bt')p2 + (C + c't + ct')ps + . . . = 0,
and the resultant will be
{A + a't + at'){B + b't + bt'){C + c't + ct') . . . = 0,
^ Laws of Thought, Chap. IX,
51
an equation that contains only the unknown quantity t and the constants of
the problem (the coefficients of / and of (f). From this may be derived the
expression of t in terms of these constants. Developing the first member of this
equation
{A + a'){B + b'){C + cO . . . X t + (A + a){B + b){C + c) . . . x t' = 0.
The solution is
t = (A + a)(5 + 6)(C + c) . . . + u{A'a + B% ^C'c^.. .).
The resultant is verified by hypothesis since it is
ABC . . . = 0,
which is the resultant of the given equation
f{x,y,z,...) = 0.
We can see how this equation contributes to restrict the variability of t.
Since t was defined only by the function (f, it was determined by the double
inclusion
abc ... < t < a-\-b -\- c-\-
Now that we take into account the condition / = 0, t is determined by the
double inclusion
(A + a)(5 + 6)(C + c) . . . < t < {A'a + B% + C'c + . . .).48
The inferior limit can only have increased and the superior limit diminished,
for
abc...< (A + a)(5 + 6)(C + c)...
and
A'a + 5'6 + C'c . . . < a + 6 + c . . . .
The limits do not change if A = B = C = . . . = 0, that is, if the equation
/ = is reduced to an identity, and this was evident a priori.
0.42 The Method of Poretsky
The method of Boole and Schroder which we have heretofore discussed
is clearly inspired by the example of ordinary algebra, and it is summed up in
two processes analogous to those of algebra, namely the solution of equations
with reference to unknown quantities and elimination of the unknowns. Of these
processes the second is much the more important from a logical point of view,
and Boole was even on the point of considering deduction as essentially con-
sisting in the elimination of middle terms. This notion, which is too restricted.
Whitehead, Universal Algebra, p. 63.
52
was suggested by the example of the synogism, in which the conclusion results
from the elimination of the middle term, and which for a long time was wrongly
considered as the only type of mediate deduction. ^^
However this may be, Boole and Schroder have exaggerated the anal-
ogy between the algebra of logic and ordinary algebra. In logic, the distinction
of known and unknown terms is artificial and almost useless. All the terms
are — in principle at least — known, and it is simply a question, certain relations
between them being given, of deducing new relations (unknown or not explic-
itly known) from these known relations. This is the purpose of Poretsky's
method which we shall now expound. It may be summed up in three laws, the
law of forms, the law of consequences and the law of causes.
0.43 The Law of Forms
This law answers the following problem: An equality being given, to find for
any term (simple or complex) a determination equivalent to this equality. In
other words, the question is to find all the forms equivalent to this equality, any
term at all being given as its first member.
We know that any equality can be reduced to a form in which the second
member is or 1; i.e., to one of the two equivalent forms
N = 0, N' = l.
The function N is what Poretsky calls the logical zero of the given
equality; N^ is its logical whole.^^
Let U be any term; then the determination of U:
U = N'U + NU'
is equivalent to the proposed equality; for we know it is equivalent to the equality
{NU ^NU' = {)) = {N = {)).
Let us recall the signification of the determination
U = N'U ^NU'.
^^In fact, the fundamental formula of elimination
{ax + hx' = 0) < {ah = 0)
is, as we have seen, only another form and a consequence of the principle of the syllogism
{b < X < a) < {b < a).
^•-•They are called "logical" to distinguish them from the identical zero and whole, i.e., to
indicate that these two terms are not equal to and 1 respectively except by virtue of the
data of the problem.
53
It denotes that the term U is contained in N' and contains N. This is easily
understood, since, by hypothesis, N is equal to and A^' to 1. Therefore we
can formulate the law of forms in the following way:
To obtain all the forms equivalent to a given equality, it is sufficient to express
that any term contains the logical zero of this equality and is contained in its
logical whole.
The number of forms of a given equality is unlimited; for any term gives rise
to a form, and to a form different from the others, since it has a different first
member. But if we are limited to the universe of discourse determined by n
simple terms, the number of forms becomes finite and determinate. For, in this
limited universe, there are 2'^ constituents. Now, all the terms in this universe
that can be conceived and defined are sums of some of these constituents. Their
number is, therefore, equal to the number of combinations that can be made
with 2^ constituents, namely 2^ (including 0, the combination of constituent,
and 1, the combination of all the constituents). This will also be the number of
different forms of any equality in the universe in question.
0.44 The Law of Consequences
We shall now pass to the law of consequences. Generalizing the conception
of Boole, who made deduction consist in the elimination of middle terms,
PORETSKY makes it consist in the elimination of known terms (connaissances) .
This conception is explained and justified as follows.
All problems in which the data are expressed by logical equalities or inclu-
sions can be reduced to a single logical equality by means of the formula^^
(A = 0)(5 = 0)(C = 0) . . . = (A + 5 + C . . . = 0).
In this logical equality, which sums up all the data of the problem, we develop
the first member with respect to all the simple terms which appear in it (and
not with respect to the unknown quantities). Let n be the number of simple
terms; then the number of the constituents of the development of 1 is 2'^. Let
m (< 2^) be the number of those constituents appearing in the first member of
the equality. All possible consequences of this equality (in the universe of the n
terms in question) may be obtained by forming all the additive combinations of
these m constituents, and equating them to 0; and this is done in virtue of the
formula
{A^B = 0)< {A = 0).
We see that we pass from the equality to any one of its consequences by
suppressing some of the constituents in its first member, which correspond to
as many elementary equalities (having for second member), i.e., as many as
there are data in the problem. This is what is meant by "eliminating the known
terms".
^^We employ capitals to denote complex terms (logical functions) in contrast to simple
terms denoted by small letters (a, b,c, . . .)
54
The number of consequences that can be derived from an equahty (in the
universe of n terms with respect to which it is developed) is equal to the number
of additive combinations that may be formed with its m constituents; i.e., T^ .
This number includes the combination of constituents, which gives rise to the
identity = 0, and the combination of the m constituents, which reproduces
the given equality.
Let us apply this method to the equation with one unknown quantity
ax + hx' = 0.
Developing it with respect to the three terms a, 6, x\
{abx -\- ah'x + ahx' + a'hx' = 0)
= [ah{x + x') + ah'x + a'hx' = 0]
= {ah = 0){ah'x = 0){a'hx' = 0).
Thus we find, on the one hand, the resultant ah = 0, and, on the other hand,
two equalities which may be transformed into the inclusions
X < a' -\- h, a'h < x.
But by the resultant which is equivalent to 6 < a', we have
a' ^h' = a', a'h = h.
This consequence may therefore be reduced to the double inclusion
X < a' , h < X,
that is, to the known solution.
Let us apply the same method to the premises of the syllogism
(a < h){h < c).
Reduce them to a single equality
{a<h) = {ah' = 0), {h < c) = {he' = 0), {ah' + he' = 0),
and seek all of its consequences.
Developing with respect to the three terms a, 6, c:
ahe' + ah'e + ah'e' + a'he' = 0.
The consequences of this equality, which contains four constituents, are 16
(2^) in number as follows:
1. {ahe' = 0) = {ah < c);
2. {ah'e = 0) = {ae < 6);
3. {ah'e' = 0) = (a<6 + c);
4. {a'he' = 0) = (6<a + c);
5. {ahe' + ah'e = 0) = (a < 6c + h'e');
6. {ahe' + ah'e' = 0) = {ae' = 0) = (a < c).
55
This is the traditional conclusion of the syllogism. ^^
7. {abc' + a%c' = 0) = {be' = 0) = (6 < c).
This is the second premise.
8. {ab'c + ab'c' = 0) = {ab' = 0) = {a<b).
This is the first premise.
9. {ab'c + a'bc' = 0) = (ac < 6 < a + c);
10. {ab'c' + a%c' = 0) = (a6' + a'6 < c);
11. (a6c' + ab'c + a6'c' = 0) = (a6' + ac' = 0) = (a < 6c);
12. {abc' + a6'c + a'6c' = 0) = {ab'c + 6c' = 0) = (ac < 6 < c);
13. {abc' + a6'c' + a' be' = 0) = (ac' + 6c' = 0) = (a + 6 < c);
14. {ab'c + a6'c' + a'6c' = 0) = {ab' + a'6c' = 0) = (a < 6 < a + c).
The last two consequences (15 and 16) are those obtained by combining
constituent and by combining all; the first is the identity
15. = 0,
which confirms the paradoxical proposition that the true (identity) is implied
by any proposition (is a consequence of it); the second is the given equality itself
16. ab' + be' = 0,
which is, in fact, its own consequence by virtue of the principle of identity. These
two consequences may be called the "extreme consequences" of the proposed
equality. If we wish to exclude them, we must say that the number of the
consequences properly so called of an equality of m constituents is 2^ — 2.
0.45 The Law of Causes
The method of finding the consequences of a given equality suggests directly
the method of finding its causes, namely, the propositions of which it is the
consequence. Since we pass from the cause to the consequence by eliminating
known terms, i.e., by suppressing constituents, we will pass conversely from the
consequence to the cause by adjoining known terms, i.e., by adding constituents
to the given equality. Now, the number of constituents that may be added to
^^It will be observed that this is the only consequence (except the two extreme consequences
[see the text below]) independent of b; therefore it is the resultant of the elimination of that
middle term.
56
it, i.e., that do not already appear in it, is 2^ — m. We will obtain all the
possible causes (in the universe of the n terms under consideration) by forming
all the additive combinations of these constituents, and adding them to the first
member of the equality in virtue of the general formula
(A + 5 = 0)<(A = 0),
which means that the equality {A = 0) has as its cause the equality {A-\-B = 0),
in which B is any term. The number of causes thus obtained will be equal to
the number of the aforesaid combinations, or 2'^^ — m.
This method may be applied to the investigation of the causes of the premises
of the syllogism
(a < h){h < c)
which, as we have seen, is equivalent to the developed equality
ahc' + ah'c + ah'c' + a'hc' = 0.
This equality contains four of the eight (2^) constituents of the universe of
three terms, the four others being
abc, a'hc, a'h'c, a'h'c' .
The number of their combinations is 16 (2^), this is also the number of the
57
causes sought, which are:
{abc + abc' + ab'c + ab'c' + a%c' = 0)
^^^ = (a + 6c' = 0) = (a = 0)(6 < c);
(a6c' + a6'c + ab'c' + a'6c + a'bc' = 0)
^ ^ = (a6c' + ab' + a'6 = 0) = {ab < c){a = b);
(abc' + ab'c + a6'c' + a%c' + a'6'c = 0)
^ ^ = {be' + 6'c + a6'c' = 0) = (6 = c)(a < 6 + c);
(a6c' + a6'c + ab'c' + a'6c' + a'b'c' = 0)
^^^^ = (c' + a6' = 0) = (c = l)(a < 6);
(a6c + a6c' + ab'c + a6'c' + a'bc + a'6c' = 0)
(12)
^ ^ =(a + 6 = 0) = (a = 0)(6 = 0);
(a6c + abc' + a6'c + ab'c' + a'6c' + a'6'c = 0)
fl3)
^ ^ = (a + 6c' + b'c = 0) = (a = 0)(6 = c);
{abc + a6c' + a6'c + ab'c' + a'6c' + a'b'c' = 0)
^^^^ = (a + c' = 0) = (a = 0)(c = l)^^
(a6c' + ab'c + a6'c' + a'6c + a'bc' + a'6'c = 0)
(15) = {ac' + a'c + a6'c + a'6c' = 0)
= (a = c)(ac <6<a + c) = (a = 6 = c);
{abc' + a6'c + a6'c' + a'bc + a' 6c' + a'b'c = 0)
^^^^ = {c' + a6' + a'6 = 0) = (c = l)(a = 6);
, , {abc' + a6'c + ab'c' + a'6c' + a'6'c + a'b'c' = 0)
(17)
^ ^ =(6' + c' = 0) = (6 = c=l).
Before going any further, it may be observed that when the sum of certain
constituents is equal to 0, the sum of the rest is equal to 1. Consequently, instead
of examining the sum of seven constituents obtained by ignoring one of the four
missing constituents, we can examine the equalities obtained by equating each
of these constituents to 1:
(18) (a'6V = 1) = (a + 6 + c = 0) =(a = 6 = c = 0);
(19) {a'b'c = 1) = (a + 6 + c' = 0) = {a = b = 0)(c = 1);
(20) {a'bc = 1) = (a + 6' + c' = 0) = (a = 0)(6 = c = 1);
(21) (a6c=l) =(a = 6 = c=l).
Note that the last four causes are based on the inclusion
0<1.
The last two causes (22 and 23) are obtained either by adding all the miss-
ing constituents or by not adding any. In the first case, the sum of all the
constituents being equal to 1, we find
(22) 1 = 0,
58
that is, absurdity, and this confirms the paradoxical proposition that the false
(the absurd) implies any proposition (is its cause). In the second case, we obtain
simply the given equality, which thus appears as one of its own causes (by the
principle of identity) :
(23) ab' + be' = 0.
If we disregard these two extreme causes, the number of causes properly so
called will be
0.46 Forms of Consequences and Causes
We can apply the law of forms to the consequences and causes of a given equality
so as to obtain all the forms possible to each of them. Since any equality is
equivalent to one of the two forms
N = 0, N' = l,
each of its consequences has the form^^
NX = 0, orA^' + X' = l,
and each of its causes has the form
A^ + X = 0, otN'X' = 1.
In fact, we have the following formal implications:
{N^X = 0) <{N = 0)< {NX = 0),
{N'X' = 1) < {N' = 1) = {N' + X' = 1).
Applying the law of forms, the formula of the consequences becomes
U = {N' ^X')U ^NXU',
and the formula of the causes
U = N'X'U ^{N ^X)U']
^^In §0.44 we said that a consequence is obtained by taking a part of the constituents of
the first member A^, and not by multiplying it by a term X; but it is easily seen that this
amounts to the same thing. For, suppose that X (like A^) be developed with respect to the
n terms of discourse. It will be composed of a certain number of constituents. To perform
the multiplication of A^ by X, it is sufficient to multiply all their constituents each by each.
Now, the product of two identical constituents is equal to each of them, and the product of
two different constituents is 0. Hence the product of A^ by X becomes reduced to the sum of
the constituents common to A^ and X, which is, of course, contained in A^. So, to multiply A^
by an arbitrary term is tantamount to taking a part of its constituents (or all, or none).
59
or, more generally, since X and X' are indeterminate terms, and consequently
are not necessarily the negatives of each other, the formula of the consequences
will be
U = {N' ^X)U^NYU',
and the formula of the causes
U = N'XU ^{N ^Y)U'.
The first denotes that U is contained in {N' + X) and contains NY] which
indeed results, a fortiori^ from the hypothesis that U is contained in N' and
contains N .
The second formula denotes that U is contained in N'X and contains N' ^Y
whence results, a fortiori^ that U is contained in N' and contains N .
We can express this rule verbally if we agree to call every class contained
in another a sub- class ^ and every class that contains another a super- class. We
then say: To obtain all the consequences of an equality (put in the form U =
N'U + NU')^ it is sufficient to substitute for its logical whole N' all its super-
classes, and, for its logical zero N, all its sub-classes. Conversely, to obtain all
the causes of the same equality, it is sufficient to substitute for its logical whole
all its sub-classes, and for its logical zero, all its super-classes.
0.47 Example: Venn's Problem
The members of the administrative council of a financial society are either bond-
holders or shareholders, but not both. Now, all the bondholders form a part of
the council. What conclusion must we draw?
Let a be the class of the members of the council; let b be the class of the
bondholders and c that of the shareholders. The data of the problem may be
expressed as follows:
a < bc^ -\- b' c^ b < a.
Reducing to a single developed equality,
a{bc = b'c') = 0, a'b = 0,
(24) abc + ab'c' + a'bc + a'bc' = 0.
This equality, which contains 4 of the constituents, is equivalent to the fol-
lowing, which contains the four others,
(25) abc' + ab'c + a' b'c + a' b'c' = 1.
This equality may be expressed in as many different forms as there are classes
in the universe of the three terms a, 6, c.
Ex. 1. a = abc' -\- ab'c -\- a' be -\- a'bc'^
60
that is,
Ex. 2.
Ex. 3.
b < a <bc^ ^ h'c,
b = abc^ -\- ab'c = ac']
ab'c + a'b'c + ab'c' + a' be'
that is,
ab' ^a'b < c < b' .
These are the solutions obtained by solving equation (24) with respect to a,
b, and c.
From equality (24) we can derive 16 consequences as follows :
1. abc ■■
2. {ab'c' = 0) :
3. {a'bc = 0) :
4. {a'bc = 0) :
5. {abc ^ ab'c' = 0)
6. {abc + a'bc = 0) :
7. {abc + a'6c' = 0) :
8. {ab'c' + a'bc = 0) :
9. {ab'c' + a'6c' = 0) :
10. {a'bc + a'6c' = 0) :
11. {abc + ab'c' + a'6c = 0) :
12. abc + ab'c + a'6c' :
13. {abc + a'6c + a'6c' = 0) :
14. ab'c' + a'bc + a'6c' :
0;
{a < b -\- c);
{be < a);
{b < a -\- c);
{a < be' + b'c) [1^* premise];
(6c = 0);
(6 < ac' -\- a'c);
{be < a < b -\- c);
{ab' -\- a'b < c);
{a'b = 0) [2^ premise];
{be ^ ab'c' = 0);
0;
{bc-
0.
a'6c0 = 0;
The last two consequences, as we know, are the identity (0 = 0) and the
equality (24) itself. Among the preceding consequences will be especially noted
the 6*^ {be = 0), the resultant of the elimination of a, and the 10*^ {a'b = 0),
the resultant of the elimination of c. When b is eliminated the resultant is the
identity
[{a' ^ c)ac' = 0] = (0 = 0).
Finally, we can deduce from the equality (24) or its equivalent (25) the
61
following 16 causes:
1. {abc' = 1) = (a = l){b = l)(c = 0)
2. {ab'c = 1) = (a = l){b = 0)(c = 1)
3. {a%'c = 1) = (a = 0)(6 = 0)(c = 1)
4. (a'6'c' = 1) = (a = 0){b = 0)(c = 0)
5. {abc' + a6'c = 1) = (a = l){b' = c);
6. (a6c' + a'b'c = 1) = (a = 6 = c');
7. (a6c' + a%'c' = 1) = (c = 0)(a = 6);
8. (a6'c + a%'c = 1) = (6 = 0)(c = 1);
9. {ab'c + a'6'c' = 1) = (6 = 0)(a = c);
10. {a'b'c + a'6'c' = 1) = (a = 0)(6 = 0);
11. {abc' + ab'c + a'6'c = 1) = (6 = cO(c' < a);
12. (a6c' + ab'c + a'6'c' = 1) = (6c = 0)(a = 6 + c);
13. {abc' + a'6'c + a'6'c' = 1) = {ac = 0){a = b);
14. (a6'c + a'b'c + a'6'c' = 1) = (6 = o)(a < c).
The last two causes, as we know, are the equality (24) itself and the absurdity
(1=0). It is evident that the cause independent of a is the 8*^ (6 = 0)(c = 1),
and the cause independent of c is the 10*^ (a = 0)(6 = 0). There is no cause,
properly speaking, independent of b. The most "natural" cause, the one which
may be at once divined simply by the exercise of common sense, is the 12*^:
(6c = 0)(a = 6 + c).
But other causes are just as possible; for instance the 9*^ {b = 0)(a = c), the
7*^ (c = 0)(a = 6), or the 13*^ {ac = 0){a = b).
We see that this method furnishes the complete enumeration of all possible
cases. In particular, it comprises, among the forms of an equality, the solutions
deducible therefrom with respect to such and such an "unknown quantity", and,
among the consequences of an equality, the resultants of the elimination of such
and such a term.
0.48 The Geometrical Diagrams of Venn
Poretsky's method may be looked upon as the perfection of the methods of
Stanley Jevons and Venn.
Conversely, it finds in them a geometrical and mechanical illustration, for
Venn's method is translated in geometrical diagrams which represent all the
constituents, so that, in order to obtain the result, we need only strike out
(by shading) those which are made to vanish by the data of the problem. For
instance, the universe of three terms a, 6, c, represented by the unbounded plane.
62
Figure 1:
is divided by three simple closed contours into eight regions which represent the
eight constituents (Fig. 1).
To represent geometrically the data of Venn's problem we must strike out
the regions abc, ah'c' ^ a' he and a'hc'-^ there will then remain the regions ahc' ^
ah'c^ a'h'c^ and a'h'c' which will constitute the universe relative to the problem^
being what Poretsky calls his logical whole (Fig. 2). Then every class will be
contained in this universe, which will give for each class the expression resulting
from the data of the problem. Thus, simply by inspecting the diagram, we see
that the region he does not exist (being struck out); that the region h is reduced
to ah'c' (hence to a6); that all a is 6 or c, and so on.
Figure 2:
This diagrammatic method has, however, serious inconveniences as a method
for solving logical problems. It does not show how the data are exhibited by
canceling certain constituents, nor does it show how to combine the remaining
constituents so as to obtain the consequences sought. In short, it serves only to
exhibit one single step in the argument, namely the equation of the problem;
it dispenses neither with the previous steps, z.e., "throwing of the problem into
63
an equation" and the transformation of the premises, nor with the subsequent
steps, i.e., the combinations that lead to the various consequences. Hence it is
of very httle use, inasmuch as the constituents can be represented by algebraic
symbols quite as well as by plane regions, and are much easier to deal with in
this form.
0.49 The Logical Machine of Jevons
In order to make his diagrams more tractable, Venn proposed a mechanical
device by which the plane regions to be struck out could be lowered and caused to
disappear. But Jevons invented a more complete mechanism, a sort of logical
piano. The keyboard of this instrument was composed of keys indicating the
various simple terms (a, 6, c, d), their negatives, and the signs + and =. Another
part of the instrument consisted of a panel with movable tablets on which were
written all the combinations of simple terms and their negatives; that is, all the
constituents of the universe of discourse. Instead of writing out the equalities
which represent the premises, they are "played" on a keyboard like that of a
typewriter. The result is that the constituents which vanish because of the
premises disappear from the panel. When all the premises have been "played",
the panel shows only those constituents whose sum is equal to 1, that is, forms
the universe with respect to the problem, its logical whole. This mechanical
method has the advantage over Venn's geometrical method of performing
automatically the "throwing into an equation", although the premises must first
be expressed in the form of equalities; but it throws no more light than the
geometrical method on the operations to be performed in order to draw the
consequences from the data displayed on the panel.
0.50 Table of Consequences
But Poretsky's method can be illustrated, better than by geometrical and
mechanical devices, by the construction of a table which will exhibit directly all
the consequences and all the causes of a given equality. (This table is relative to
this equality and each equality requires a different table). Each table comprises
the 2^ classes that can be defined and distinguished in the universe of discourse
of n terms. We know that an equality consists in the annulment of a certain
number of these classes, viz., of those which have for constituents some of the
constituents of its logical zero N. Let m be the number of these latter con-
stituents, then the number of the subclasses of N is 2^ which, therefore, is the
number of classes of the universe which vanish in consequence of the equality
considered. Arrange them in a column commencing with and ending with N
(the two extremes). On the other hand, given any class at all, any preceding
class may be added to it without altering its value, since by hypothesis they
are null (in the problem under consideration). Consequently, by the data of the
problem, each class is equal to 2^ classes (including itself). Thus, the assem-
64
blage of the 2^ classes of discourse is divided into 2^~"^ series of 2^ classes, each
series being constituted by the sums of a certain class and of the 2^ classes of
the first column (sub-classes of N). Hence we can arrange these 2^ sums in the
following columns by making them correspond horizontally to the classes of the
first column which gave rise to them. Let us take, for instance, the very simple
equality a = b, which is equivalent to
ab' + a'b = 0.
The logical zero (N) in this case is ab^ -\- a'b. It comprises two constituents
and consequently four sub-classes: 0, ab' ^ a'b^ and ab' ^a'b. These will compose
the first column. The other classes of discourse are a6, a'b' ^ ab-\-a'b', and those
obtained by adding to each of them the four classes of the first column. In this
way, the following table is obtained:
ab a'b' ab^a'b'
ab' a b' a ^ b'
a'b b a' a' ^ b
ab'^a'b a^b a'^b' 1
By construction, each class of this table is the sum of those at the head of
its row and of its column, and, by the data of the problem, it is equal to each
of those in the same column. Thus we have 64 different consequences for any
equality in the universe of discourse of 2 letters. They comprise 16 identities
(obtained by equating each class to itself) and 16 forms of the given equality,
obtained by equating the classes which correspond in each row to the classes
which are known to be equal to them, namely
= a6' + a'b, ab = a^b, a'b' = a' ^ b' ab ^ a'b' = 1
a = b, b' = a', ab' = a'b, a -\- b' = a' -\- b.
Each of these 8 equalities counts for two, according as it is considered as a
determination of one or the other of its members.
0.51 Table of Causes
The same table may serve to represent all the causes of the same equality in
accordance with the following theorem:
When the consequences of an equality A^ = are expressed in the form of
determinations of any class U, the causes of this equality are deduced from the
consequences of the opposite equality, A^ = 1, put in the same form, by changing
U to U' in one of the two members.
For we know that the consequences of the equality A^ = have the form
U = {N' ^ X)U ^ NYU' ,
and that the causes of the same equality have the form
U = N'XU^{N^Y)U'.
65
Now, if we change U into U^ in one of the members of this last formula, it
becomes
U = {N ^ X')U ^ N'Y'U' ,
and the accents of X and Y can be suppressed since these letters represent
indeterminate classes. But then we have the formula of the consequences of the
equality A^' = or A^ = 1.
This theorem being established, let us construct, for instance, the table of
causes of the equality a = b. This will be the table of the consequences of the
opposite equality a = h' ^ for the first is equivalent to
and the second to
ab' + a'b
= 0,
{ab +
a'b' = 0) =
{ab'
+ a'b=l).
ab'
a'b
ab' + a'b
ah
a
b
a + b
a'h'
b'
a'
a' + b'
ah ^ a'h' a^h' a' ^h 1
To derive the causes of the equality a = h from this table instead of the con-
sequences of the opposite equality a = 6', it is sufficient to equate the negative
of each class to each of the classes in the same column. Examples are:
a' ^y = 0, a' ^y = a'h', a' ^h' = ah^ a'h',
a' ^h = a, a' ^h = h', a' ^ h = a ^ h'; . . . .
Among the 64 causes of the equality under consideration there are 16 ab-
surdities (consisting in equating each class of the table to its negative); and 16
forms of the equality (the same, of course, as in the table of consequences, for
two equivalent equalities are at the same time both cause and consequence of
each other).
It will be noted that the table of causes differs from the table of consequences
only in the fact that it is symmetrical to the other table with respect to the
principal diagonal (0, 1); hence they can be made identical by substituting the
word "row" for the word "column" in the foregoing statement. And, indeed,
since the rule of the consequences concerns only classes of the same column, we
are at liberty so to arrange the classes in each column on the rows that the rule
of the causes will be verified by the classes in the same row.
It will be noted, moreover, that, by the method of construction adopted for
this table, the classes which are the negatives of each other occupy positions
symmetrical with respect to the center of the table. For this result, the sub-
classes of the class N' (the logical whole of the given equality or the logical zero
of the opposite equality) must be placed in the first row in their natural order
from to N'; then, in each division, must be placed the sum of the classes at
the head of its row and column.
66
With this precaution, we may sum up the two rules in the fohowing practical
statement:
To obtain every consequence of the given equality (to which the table relates)
it is sufficient to equate each class to every class in the same column; and, to
obtain every cause, it is sufficient to equate each class to every class in the row
occupied by its symmetrical class.
It is clear that the table relating to the equality A^ = can also serve for
the opposite equality A^ = 1, on condition that the words "row" and "column"
in the foregoing statement be interchanged.
Of course the construction of the table relating to a given equality is useful
and profitable only when we wish to enumerate all the consequences or the
causes of this equality. If we desire only one particular consequence or cause
relating to this or that class of the discourse, we make use of one of the formulas
given above.
0.52 The Number of Possible Assertions
If we regard logical functions and equations as developed with respect to all the
letters, we can calculate the number of assertions or different problems that may
be formulated about n simple terms. For all the functions thus developed can
contain only those constituents which have the coefficient 1 or the coefficient
(and in the latter case, they do not contain them). Hence they are additive
combinations of these constituents; and, since the number of the constituents
is 2"^, the number of possible functions is 2^ . From this must be deducted
the function in which all constituents are absent, which is identically 0, leaving
2^—1 possible equations (255 when n = 3). But these equations, in their turn,
may be combined by logical addition, i.e., by alternation; hence the number of
their combinations is 2^ ~^ — 1, excepting always the null combination. This is
the number of possible assertions affecting n terms. When n = 2, this number is
as high as 32767.^^ We must observe that only universal premises are admitted
in this calculus, as will be explained in the following section.
0.53 Particular Propositions
Hitherto we have only considered propositions with an affirmative copula {i.e.,
inclusions or equalities) corresponding to the universal propositions of classi-
^^ G. Peano, Calcolo geometrico (1888) p. x; Schroder, Algebra der Logik, Vol. II,
p. 144-148.
67
cal logic. ^^ It remains for us to study propositions with a negative copula (non
inclusions or inequalities), which translate particular propositions^^; but the cal-
culus of propositions having a negative copula results from laws already known,
especially from the formulas of De Morgan and the law of contraposition. We
shall enumerate the chief formulas derived from it.
The principle of composition gives rise to the following formulas:
(c ^ ab) = {c ^ a) -\- {c ^ 6),
{a-\-b ^ c) = {a ^ c) -\- {b ^ c)^
whence come the particular instances
(a5^1) = (a^l) + (6^1),
{a + b^O) = {a^O) + {c^O).
From these may be deduced the fohowing important imphcations:
{a^O)<{a + b^O),
{aj^l)< (abj^l).
From the principle of the syllogism, we deduce, by the law of transposition,
{a<b){a^O)< (b^O),
{a<b){bj^l) <{a^l).
The formulas for transforming inclusions and equalities give corresponding
formulas for the transformation of non-inclusions and inequalities,
{aftb) = {ab' ^0) = {a' + b^l),
{a^b) = {ab' + a'b ^Q) = {ab + a'b' + 1).
^^The universal affirmative, "All a's are 6's", may be expressed by the formulas
(a <b) = {a = ab) = (ab' = 0) = (a' + b = 1),
and the universal negative, "No a's are 6's", by the formulas
{a < b') = {a = ab') = {ab = 0) = {a' + b' = 1).
^^For the particular affirmative, "Some a's are 6's", being the negation of the universal
negative, is expressed by the formulas
{a ^ b') = (a / ab') = {ab / 0) = {a' + b' ^ 1),
and the particular negative, "Some a's are not 6's", being the negation of the universal affir-
mative, is expressed by the formulas
{a^b) = {a^ ab) = {ab' / 0) = (a' + 6 / 1).
68
0.54 Solution of an Inequation with One Unknown
If we consider the conditional inequality (inequation) with one unknown
ax -\- bx ^ 0,
we know that its first member is contained in the sum of its coefficients
ax -\- hx' < a -\- b.
From this we conclude that, if this inequation is verified, we have the in-
equality
a^b^O.
This is the necessary condition of the solvability of the inequation, and the
resultant of the elimination of the unknown x. For, since we have the equivalence
Y[{ax -^bx' = 0) = {a^b = 0),
X
we have also by contraposition the equivalence
^{ax -^bx' ^0) = {a^b^ 0).
X
Likewise, from the equivalence
^{ax + bx' = 0) = {ab = 0)
X
we can deduce the equivalence
Y[{ax^bx' ^0) = {ab^O),
X
which signifies that the necessary and sufficient condition for the inequation to
be always true is
(ab + 0);
and, indeed, we know that in this case the equation
[ax + bx' = 0)
is impossible (never true).
Since, moreover, we have the equivalence
{ax -\- bx' = 0) = (x = a'x + bx')^
we have also the equivalence
{ax + bx' 7^ 0) = (x 7^ a'x + bx').
69
Notice the significance of this solution:
{ax + bx' 7^ 0) = {ax 7^ 0) + {bx' ^ 0) = {x it a') + (6 ^ x).
"Either x is not contained in a\ or it does not contain U\ This is the negative
of the double inclusion
b < X < a.
Just as the product of several equalities is reduced to one single equality,
the sum (the alternative) of several inequalities may be reduced to a single
inequality. But neither several alternative equalities nor several simultaneous
inequalities can be reduced to one.
0.55 System of an Equation and an Inequation
We shall limit our study to the case of a simultaneous equality and inequality.
For instance, let the two premises be
{ax + bx^ = 0) {ex + dx' ^ 0).
To satisfy the former (the equation) its resultant a6 = must be verified.
The solution of this equation is
X = a! X + bx' .
Substituting this expression (which is equivalent to the equation) in the
inequation, the latter becomes
{a!c + ad)x + (be + b' d)x' ^ 0.
Its resultant (the condition of its solvability) is
{a!c ^ad^bc^ b'd 7^ 0) = [{a' + b)c + (a + b')d ^ 0],
which, taking into account the resultant of the equality,
{ab = 0) = (a' + 6 = a') = {a^b' = b')
may be reduced to
a'c + b'd ^ 0.
The same result may be reached by observing that the equality is equivalent
to the two inclusions
{x<a'){x' <b'),
and by multiplying both members of each by the same term
{ex < a' c){dx' < b'd) < {ex -\- dx' < a'c -\- b'd)
{ex + dx' 7^ 0) < {a'c + b'd ^ 0).
70
This resultant implies the resultant of the inequality taken alone
c + ^7^0,
so that we do not need to take the latter into account. It is therefore sufficient
to add to it the resultant of the equality to have the complete resultant of the
proposed system
{ab = 0){a'c^b'd^O).
The solution of the transformed inequality (which consequently involves the
solution of the equality) is
X 7^ {a'c' -\- ad')x + {he + h' d)x' .
0.56 Formulas Peculiar to the Calculus of Propo-
sitions.
All the formulas which we have hitherto noted are valid alike for propositions
and for concepts. We shall now establish a series of formulas which are valid
only for propositions, because all of them are derived from an axiom peculiar
to the calculus of propositions, which may be called the principle of assertion.
This axiom is as follows:
Ax. 10
(a = 1) = a.
P. I.: To say that a proposition a is true is to state the proposition itself. In
other words, to state a proposition is to affirm the truth of that proposition.^^
Corollary.
a' = (a' = 1) = (a = 0).
P. I.: The negative of a proposition a is equivalent to the affirmation that
this proposition is false.
By Ax. 9 (§0.20), we already have
(a = l)(a = 0) = 0,
"A proposition cannot be both true and false at the same time", for
(Syh.) (a = l)(a = 0) < (1 = 0) = 0.
But now, according to Ax. 10, we have
(a = 1) + (a = 0) = a + a' = 1.
^^We can see at once that this formula is not susceptible of a conceptual interpretation
(C. I.); for, if a is a concept, (a = 1) is a proposition, and we would then have a logical
equality (identity) between a concept and a proposition, which is absurd.
71
"A proposition is either true or false". From these two formulas combined
we deduce directly that the propositions (a = 1) and (a = 0) are contradictory,
{a^l) = {a = 0), {a^O) = {a = l).
From the point of view of calculation Ax. 10 makes it possible to reduce
to its first member every equality whose second member is 1, and to transform
inequalities into equalities. Of course these equalities and inequalities must have
propositions as their members. Nevertheless all the formulas of this section
are also valid for classes in the particular case where the universe of discourse
contains only one element, for then there are no classes but and 1. In short,
the special calculus of propositions is equivalent to the calculus of classes when
the classes can possess only the two values and 1.
0.57 Equivalence of an Implication and an Alter-
native
The fundamental equivalence
(a < 6) = (a' + 6 = 1)
gives rise, by Ax. 10, to the equivalence
{a<b) = (a' + b)
which is no less fundamental in the calculus of propositions. To say that a
implies b is the same as affirming "not-a or U\ i.e., "either a is false or b is true."
This equivalence is often employed in every day conversation.
Corollary. — For any equality, we have the equivalence
{a = b) = ab^ a'b' .
Demonstration:
{a = b) = {a< b){b < a) = {a' + b){b' -^ a) = ab ^ o!b'
"To affirm that two propositions are equal (equivalent) is the same as stating
that either both are true or both are false".
The fundamental equivalence established above has important consequences
which we shall enumerate.
In the first place, it makes it possible to reduce secondary, tertiary, etc.,
propositions to primary propositions, or even to sums (alternatives) of elemen-
tary propositions. For it makes it possible to suppress the copula of any proposi-
tion, and consequently to lower its order of complexity. An implication (A < 5),
in which A and B represent propositions more or less complex, is reduced to
the sum A' -\- B, in which only copulas within A and B appear, that is, propo-
sitions of an inferior order. Likewise an equality {A = B) is reduced to the sum
{AB -\- A'B') which is of a lower order.
72
We know that the principle of composition makes it possible to combine
several simultaneous inclusions or equalities, but we cannot combine alternative
inclusions or equalities, or at least the result is not equivalent to their alternative
but is only a consequence of it. In short, we have only the implications
{a <c)^{b <c) < {ah < c),
{c<a)^{c<b) < {c<a^ b),
which, in the special cases where c = and c = 1, become
(a = 0) + (6 = 0) < (a6 = 0),
(a = l) + (6=l)<(a + 6 = l).
In the calculus of classes, the converse implications are not valid, for, from
the statement that the class ah is null, we cannot conclude that one of the
classes a or 6 is null (they can be not-null and still not have any element in
common) ; and from the statement that the sum (a + b) is equal to 1 we cannot
conclude that either a or 6 is equal to 1 (these classes can together comprise all
the elements of the universe without any of them alone comprising all). But
these converse implications are true in the calculus of propositions
{ab < c) < (a < c) + (6 < c),
{c<a^b) < {c<a)^{c< b);
for they are deduced from the equivalence established above, or rather we may
deduce from it the corresponding equalities which imply them,
(1) {ab < c) = (a < c) + (6 < c),
(2) {c<a^b) = {c<a)^{c< b).
Demonstration:
(1) {ab <c) = a^ ^b^ ^c,
{a <c) ^{b <c) = {a' + c) + {b' -^ c) = a' ^ b' ^ c;
(2) {c<a^b) =c' ^a^b,
{c<a)^{c<b) = {c' + a) + (c' + 6) = c' + a + 6.
In the special cases where c = and c = 1 respectively, we find
(3) (a6 = 0) = (a = 0) + (6 = 0),
(4) (a + 6 = l) = (a = l) + (6 = l).
P. I.: (1) To say that two propositions united imply a third is to say that
one of them implies this third proposition.
(2) To say that a proposition implies the alternative of two others is to say
that it implies one of them.
73
(3) To say that two propositions combined are false is to say that one of
them is false.
(4) To say that the alternative of two propositions is true is to say that one
of them is true.
The paradoxical character of the first three of these statements will be noted
in contrast to the self-evident character of the fourth. These paradoxes are
explained, on the one hand, by the special axiom which states that a proposition
is either true or false; and, on the other hand, by the fact that the false implies
the true and that only the false is not implied by the true. For instance, if both
premises in the first statement are true, each of them implies the consequence,
and if one of them is false, it implies the consequence (true or false). In the
second, if the alternative is true, one of its terms must be true, and consequently
will, like the alternative, be implied by the premise (true or false). Finally, in
the third, the product of two propositions cannot be false unless one of them is
false, for, if both were true, their product would be true (equal to 1).
0.58 Law of Importation and Exportation
The fundamental equivalence {a < b) = a^ -\- b has many other interesting
consequences. One of the most important of these is the law of importation and
exportation, which is expressed by the following formula:
[a < {b < c)] = {ab < c).
"To say that if a is true b implies c, is to say that a and b imply c".
This equality involves two converse implications: If we infer the second mem-
ber from the first, we import into the implication {b < c) the hypothesis or
condition a; if we infer the first member from the second, we, on the contrary,
export from the implication {ab < c) the hypothesis a.
Demonstration:
[a < {b < c)] = a' + (6 < c) = a' + 6' + c,
{ab <c) = {ab)' ^c = a' ^b' ^c.
Cor. 1. — Obviously we have the equivalence
[a<{b < c)] = [b <{a< c)],
since both members are equal to {ab < c) , by the commutative law of multipli-
cation.
Cor. 2. — We have also
[a < {a < b)] = {a < b),
for, by the law of importation and exportation,
[a < {a < b)] = {aa < b) = {a < b).
74
If we apply the law of importation to the two following formulas, of which the
first results from the principle of identity and the second expresses the principle
of contraposition,
{a <b) < {a < h), {a < b) < {h' < a'),
we obtain the two formulas
(a < h)a < b), {a < h)h' < a' ,
which are the two types of hypothetical reasoning: "If a implies 6, and if a is
true, h is true" {modus ponens); "If a implies b, and if b is false, a is false" {modus
tollens).
Remark. These two formulas could be directly deduced by the principle of
assertion, from the following
{a<b){a = l) < (6 = 1),
{a<b){b = {)) < (a = 0),
which are not dependent on the law of importation and which result from the
principle of the syllogism.
From the same fundamental equivalence, we can deduce several paradoxical
formulas:
1. a < {b < a), a' < {a < b).
"If a is true, a is implied by any proposition 6; if a is false, a implies any
proposition 6". This agrees with the known properties of and 1.
2. a< [{a <b) < b], a' < [{b < a) < b'].
"If a is true, then 'a implies V implies 6; if a is false, then '6 implies a' implies
not-6."
These two formulas are other forms of hypothetical reasoning {modus ponens
and modus tollens).
3. [{a <b) <a]=a, ^^ [{b < a) < a'] = a\
"To say that, if a implies b, a is true, is the same as affirming a; to say that,
if b implies a, a is false, is the same as denying a".
Demonstration:
[{a < b) < a] = {a' ^ b < a) = ab' ^ a = a,
[{b < a) < a^] = {b' ^a < a') = a'b + a' = a' .
^^This formula is Bertrand Russell's "principle of reduction". See The Principles of
Mathematics, Vol. I, p. 17 (Cambridge, 1903).
75
In formulas (1) and (3), in which b is any term at ah, we might introduce
the sign Yl with respect to b. In the fohowing formula, it becomes necessary to
make use of this sign.
4. Y[ {[^ < {b < x)] < x} = ab.
X
Demonstration:
{[a < {b < x)] <x} = {[a^ + (6 < x)] < x}
= [(a' -\- b^ -\- x) < x] = abx^ -\- x = ab -\- x.
We must now form the product Ylx{cLb-\-x), where x can assume every value,
including and 1. Now, it is clear that the part common to all the terms of
the form {ab -\- x) can only be ab. For, (1) ab is contained in each of the sums
{ab-\-x) and therefore in the part common to all; (2) the part common to all the
sums {ab-\-x) must be contained in (a6 + 0), that is, in ab. Hence this common
part is equal to a6,^^ which proved the theorem.
0.59 Reduction of Inequalities to Equalities
As we have said, the principle of assertion enables us to reduce inequalities to
equalities by means of the following formulas:
(a^O) = (a = l), (a^l) = (a = 0),
{a^b) = {a = b').
For,
{a^b) = {ab' + a'b + 0) = {ab' + ab' = 1) = (a = b').
Consequently, we have the paradoxical formula
{a^b) = {a = b').
This is easily understood, for, whatever the proposition 6, either it is true
and its negative is false, or it is false and its negative is true. Now, whatever
the proposition a may be, it is true or false; hence it is necessarily equal either
to b or to b' . Thus to deny an equality (between propositions) is to affirm the
opposite equality.
^•-•This argument is general and from it we can deduce the formula
]^(a + x) = a,
X
whence may be derived the correlative formula
2J ax = a.
76
Thence it results that, in the calculus of propositions, we do not need to
take inequalities into consideration — a fact which greatly simplifies both theory
and practice. Moreover, just as we can combine alternative equalities, we can
also combine simultaneous inequalities, since they are reducible to equalities.
For, from the formulas previously established (§0.57)
we deduce by contraposition
{ab--
= 0) = (a =
= 0) + (& =
= 0),
(a + b--
= 1) = (a =
= !) + (& =
= 1),
)ition
(a^0)(5^0) =
= {ab ^ 0),
(a^:
l)(5^1) =
--{a + b^
!)•
These two formulas, moreover, according to what we have just said, are
equivalent to the known formulas
(a = l){b = l) = (a6 = l),
(a = 0)(6 = 0) = (a + 6 = 0).
Therefore, in the calculus of propositions, we can solve all simultaneous
systems of equalities or inequalities and all alternative systems of equalities or
inequalities, which is not possible in the calculus of classes. To this end, it is
necessary only to apply the following rule:
First reduce the inclusions to equalities and the non-inclusions to inequali-
ties; then reduce the equalities so that their second members will be 1, and the
inequalities so that their second members will be 0, and transform the latter
into equalities having 1 for a second member; finally, suppress the second mem-
bers 1 and the signs of equality, i.e., form the product of the first members of
the simultaneous equalities and the sum of the first members of the alternative
equalities, retaining the parentheses.
0.60 Conclusion
The foregoing exposition is far from being exhaustive; it does not pretend to be
a complete treatise on the algebra of logic, but only undertakes to make known
the elementary principles and theories of that science. The algebra of logic is
an algorithm with laws peculiar to itself. In some phases it is very analogous to
ordinary algebra, and in others it is very widely different. For instance, it does
not recognize the distinction of degrees; the laws of tautology and absorption
introduce into it great simplifications by excluding from it numerical coefficients.
It is a formal calculus which can give rise to all sorts of theories and problems,
and is susceptible of an almost infinite development.
But at the same time it is a restricted system, and it is important to bear
in mind that it is far from embracing all of logic. Properly speaking, it is only
the algebra of classical logic. Like this logic, it remains confined to the domain
77
circumscribed by Aristotle, namely, the domain of the relations of inclusion
between concepts and the relations of implication between propositions. It is
true that classical logic (even when shorn of its errors and superfluities) was
much more narrow than the algebra of logic. It is almost entirely contained
within the bounds of the theory of the syllogism whose limits to-day appear
very restricted and artiflcial. Nevertheless, the algebra of logic simply treats,
with much more breadth and universality, problems of the same order; it is at
bottom nothing else than the theory of classes or aggregates considered in their
relations of inclusion or identity. Now logic ought to study many other kinds of
concepts than generic concepts (concepts of classes) and many other relations
than the relation of inclusion (of subsumption) between such concepts. It ought,
in short, to develop into a logic of relations, which Leibniz foresaw, which
Peirce and Schroder founded, and which Peano and Russell seem to
have established on deflnite foundations.
While classical logic and the algebra of logic are of hardly any use to math-
ematics, mathematics, on the other hand, flnds in the logic of relations its
concepts and fundamental principles; the true logic of mathematics is the logic
of relations. The algebra of logic itself arises out of pure logic considered as
a particular mathematical theory, for it rests on principles which have been
implicitly postulated and which are not susceptible of algebraic or symbolic ex-
pression because they are the foundation of all symbolism and of all the logical
calculus. ^^ Accordingly, we can say that the algebra of logic is a mathematical
logic by its form and by its method, but it must not be mistaken for the logic
of mathematics.
^^The principle of deduction and the principle of substitution. See the author's Manuel
de Logistique, Chapter 1, §§ 2 and 3 [not published], and Les Principes des Mathematiques,
Chapter 1, A.
78
Index
Absorption, Law of
Absurdity, Type of
Addition, and multiplication. Logi-
cal
and multiplication. Modulus of
and multiplication. Theorems on
Logical, not disjunctive
Affirmative propositions
Algebra, of logic an algorithm
of logic compared to mathemat-
ical algebra
of thought
Algorithm, Algebra of logic an
Alphabet of human thought
Alternative
affirmation
Equivalence of an implication
and an
Antecedent
Aristotle
Assertion, Principle of
Assertions, Number of possible
Axioms
Baldwin
Boole
Problem of
Bryan, William Jennings
Calculus, Infinitesimal
Logical
ratiocinator
Cantor, Georg
Categorical syllogism
Cause
Causes, Forms of
Law of
Sixteen
Table of
Characters
Classes, Calculus of
Classification of dichotomy
Commutativity
Composition, Principle of
Concepts, Calculus of
Condition
Necessary and sufficient
Necessary but not sufficient
of impossibility and indetermi-
nation
Connaissances
Consequence
Consequences, Forms of
Law of
of the syllogism
Sixteen
Table of
Consequent
Constituents
Properties of
Contradiction, Principle of
Contradictory propositions
terms
Contraposition, Law of
Principle of
Council, Members of
Couturat, v
Dedekind
Deduction
Principle of
Definition, Theory of
De Morgan
Formulas of
79
Descartes
Development
Law of
of logical functions
of mathematics
of symbolic logic
Diagrams of Venn, Geometrical
Dichotomy, Classification of
Disjunctive, Logical addition not
sums
Distributive law
Double inclusion
expressed by an indeterminate
Negative of the
Double negation
Duality, Law of
Economy of mental effort
Elimination of known terms
of middle terms
of unknowns
Resultant of
Rule for resultant of
Equalities, Formulas for transform-
ing inclusions into
Reduction of inequalities to
Equality a primitive idea
Definition of
Notion of
Equation, and an inequation
Throwing into an
Equations, Solution of
Excluded middle. Principle of
Exclusion, Principle of
Exclusive, Mutually
Existence, Postulate of
Exhaustion, Principle of
Exhaustive, Collectively
Forms, Law of
of consequences and causes
Frege
Symbolism of
Functions
Development of logical
Integral
Limits of
Logical
of variables
Properties of developed
Propositional
Sums and products of
Values of
Hopital, Marquis de 1'
Huntington, E. V
Hypothesis
Hypothetical arguments
reasoning
syllogism
Ideas, Simple and complex
Identity
Principle of
Type of
Ideography
Implication
and an alternative. Equivalence
of an
Relations of
Importation and exportation. Law
of
Impossibility, Condition of
Inclusion
a primitive idea
Double
expressed by an indeterminate
Negative of the double
Relation of
Inclusions into equalities. Formulas
for transforming
Indeterminate
Inclusion expressed by an
Indetermination
Condition of
Inequalities, to equalities. Reduction
of
Transformation of non-inclusions
and
Inequation, Equation and an
Solution of an
Infinitesimal calculus
80
Integral function
Interpretations of the calculus
Jevons
Logical piano of
Johnson, W. E
Known terms (connaissances)
Ladd-Franklin, Mrs
Lambert
Leibniz
Limits of a function
MacCoh
MacFarlane, Alexander
Mathematical function
logic
Mathematics, Philosophy a univer-
sal
Maxima of discourse
Middle, Principle of excluded
terms. Elimination of
Minima of discourse
Mitcheh, O
Modulus of addition and multiplica-
tion
Modus ponens
Modus tollens
Miiller, Eugen
Multiplication. See s. v. "Addi-
tion."
Negation
defined
Double
Duality not derived from
Negative
of the double inclusion
propositions
Non-inclusions and inequalities. Trans-
formation of
Notation
Null-class
Number of possible assertions
One, Definition of.
Particular propositions,
Peano,
Peirce, C. S.,
Philosophy a universal mathemat-
ics.
Piano of Jevons, Logical,
Poretsky,
Formula of.
Method of.
Predicate,
Premise,
Primary proposition.
Primitive idea.
Equality a.
Inclusion a.
Product, Logical,
Propositions,
Calculus of.
Contradictory,
Formulas peculiar to the calcu-
lus of.
Implication between,
reduced to lower orders.
Universal and particular.
Reciprocal,
Reductio ad ahsurdum,
Reduction, Principle of.
Relations, Logic of.
Relatives, Logic of.
Resultant of elimination.
Rule for,
Russell, B.,
Schroder,
Theorem of.
Secondary proposition.
Simplification, Principle of.
Simultaneous affirmation.
Solution of equations,
of inequations.
Subject,
Substitution, Principle of,
Subsumption,
Summand,
Sums,
and products of functions.
81
Disjunctive,
Logical,
Syllogism, Principle of the.
Theory of the.
Symbolic logic.
Development of.
Symbolism in mathematics.
Symbols, Origin of.
Symmetry,
Tautology, Law of
Term,
Theorem,
Thesis,
Thought,
Algebra of.
Alphabet of human.
Economy of.
Transformation
of inclusions into equalities,
of inequalities into equalities,
of non-inclusions and inequali-
ties.
Universal
characteristic of Leibniz,
propositions.
Universe of discourse.
Unknowns, Elimination of.
Variables, Functions of,
Venn, John,
metrical diagrams of.
Mechanical device of.
Problem of,
Viete,
Voigt,
Whitehead, A. N.,
Whole, Logical,
Zero,
Definition of.
Logical,
82
Chapter 1
PROJECT GUTENBERG
"SMALL PRINT"
End of the Project Gutenberg EBook of The Algebra of Logic, by Louis Couturat
*** END OF THIS PROJECT GUTENBERG EBOOK THE ALGEBRA OF LOGIC ***
***** This file should be named 10836-pdf .pdf or 10836-pdf .zip *****
This and all associated files of various formats will be found in:
http : //www . gutenberg . net/1/0/8/3/10836/
Produced by David Starner, Arno Peters, Susan Skinner
and the Online Distributed Proofreading Team.
Updated editions will replace the previous one- -the old editions
will be renamed.
Creating the works from public domain print editions means that no
one owns a United States copyright in these works, so the Foundation
(and you!) can copy and distribute it in the United States without
permission and without paying copyright royalties. Special rules,
set forth in the General Terms of Use part of this license, apply to
copying and distributing Project Gutenberg-tm electronic works to
protect the PROJECT GUTENBERG-tm concept and trademark. Project
Gutenberg is a registered trademark, and may not be used if you
charge for the eBooks, unless you receive specific permission. If you
do not charge anything for copies of this eBook, complying with the
rules is very easy. You may use this eBook for nearly any purpose
such as creation of derivative works, reports, performances and
research. They may be modified and printed and given away- -you may do
practically ANYTHING with public domain eBooks. Redistribution is
subject to the trademark license, especially commercial
redistribution .
*** START: FULL LICENSE ***
THE FULL PROJECT GUTENBERG LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
To protect the Project Gutenberg-tm mission of promoting the free
distribution of electronic works, by using or distributing this work
(or any other work associated in any way with the phrase "Project
Gutenberg") , you agree to comply with all the terms of the Full Project
Gutenberg-tm License (available with this file or online at
http://gutenberg.net/license) .
Section 1. General Terms of Use and Redistributing Project Gutenberg-tm
electronic works
l.A. By reading or using any part of this Project Gutenberg-tm
electronic work, you indicate that you have read, understand, agree to
and accept all the terms of this license and intellectual property
(trademark/ copyright) agreement. If you do not agree to abide by all
the terms of this agreement, you must cease using and return or destroy
all copies of Project Gutenberg-tm electronic works in your possession.
If you paid a fee for obtaining a copy of or access to a Project
Gutenberg-tm electronic work and you do not agree to be bound by the
terms of this agreement, you may obtain a refund from the person or
entity to whom you paid the fee as set forth in paragraph I.E. 8.
l.B. "Project Gutenberg" is a registered trademark. It may only be
used on or associated in any way with an electronic work by people who
agree to be bound by the terms of this agreement. There are a few
things that you can do with most Project Gutenberg-tm electronic works
even without complying with the full terms of this agreement . See
paragraph l.C below. There are a lot of things you can do with Project
Gutenberg-tm electronic works if you follow the terms of this agreement
and help preserve free future access to Project Gutenberg-tm electronic
works. See paragraph l.E below.
l.C. The Project Gutenberg Literary Archive Foundation ("the Foundation"
or PGLAF) , owns a compilation copyright in the collection of Project
Gutenberg-tm electronic works. Nearly all the individual works in the
collection are in the public domain in the United States. If an
individual work is in the public domain in the United States and you are
located in the United States, we do not claim a right to prevent you from
copying, distributing, performing, displaying or creating derivative
works based on the work as long as all references to Project Gutenberg
are removed. Of course, we hope that you will support the Project
Gutenberg-tm mission of promoting free access to electronic works by
freely sharing Project Gutenberg-tm works in compliance with the terms of
this agreement for keeping the Project Gutenberg-tm name associated with
the work. You can easily comply with the terms of this agreement by
keeping this work in the same format with its attached full Project
Gutenberg-tm License when you share it without charge with others .
l.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are in
a constant state of change. If you are outside the United States, check
the laws of your country in addition to the terms of this agreement
before downloading, copying, displaying, performing, distributing or
creating derivative works based on this work or any other Project
Gutenberg-tm work. The Foundation makes no representations concerning
the copyright status of any work in any country outside the United
States .
I.E. Unless you have removed all references to Project Gutenberg:
l.E.l. The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg-tm License must appear prominently
whenever any copy of a Project Gutenberg-tm work (any work on which the
phrase "Project Gutenberg" appears, or with which the phrase "Project
Gutenberg" is associated) is accessed, displayed, performed, viewed,
copied or distributed:
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net
I.E. 2. If an individual Project Gutenberg-tm electronic work is derived
from the public domain (does not contain a notice indicating that it is
posted with permission of the copyright holder) , the work can be copied
and distributed to anyone in the United States without paying any fees
or charges. If you are redistributing or providing access to a work
with the phrase "Project Gutenberg" associated with or appearing on the
work, you must comply either with the requirements of paragraphs l.E.l
through I.E. 7 or obtain permission for the use of the work and the
Project Gutenberg-tm trademark as set forth in paragraphs I.E. 8 or
I.E. 9.
I.E. 3. If an individual Project Gutenberg-tm electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs l.E.l through I.E. 7 and any additional
terms imposed by the copyright holder. Additional terms will be linked
to the Project Gutenberg-tm License for all works posted with the
permission of the copyright holder found at the beginning of this work.
I.E. 4. Do not unlink or detach or remove the full Project Gutenberg-tm
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg-tm.
I.E. 5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph l.E.l with
active links or immediate access to the full terms of the Project
Gutenberg-tm License.
I.E. 6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including any
word processing or hypertext form. However, if you provide access to or
distribute copies of a Project Gutenberg-tm work in a format other than
"Plain Vanilla ASCII" or other format used in the official version
posted on the official Project Gutenberg-tm web site (www.gutenberg.net),
you must, at no additional cost, fee or expense to the user, provide a
copy, a means of exporting a copy, or a means of obtaining a copy upon
request, of the work in its original "Plain Vanilla ASCII" or other
form. Any alternate format must include the full Project Gutenberg-tm
License as specified in paragraph l.E.l.
I.E. 7. Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg-tm works
unless you comply with paragraph I.E. 8 or I.E. 9.
I.E. 8. You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg-tm electronic works provided
that
- You pay a royalty fee of 20'/, of the gross profits you derive from
the use of Project Gutenberg-tm works calculated using the method
you already use to calculate your applicable taxes . The fee is
owed to the owner of the Project Gutenberg-tm trademark, but he
has agreed to donate royalties under this paragraph to the
Project Gutenberg Literary Archive Foundation. Royalty payments
must be paid within 60 days following each date on which you
prepare (or are legally required to prepare) your periodic tax
returns. Royalty payments should be clearly marked as such and
sent to the Project Gutenberg Literary Archive Foundation at the
address specified in Section 4, "Information about donations to
the Project Gutenberg Literary Archive Foundation."
- You provide a full refund of any money paid by a user who notifies
you in writing (or by e-mail) within 30 days of receipt that s/he
does not agree to the terms of the full Project Gutenberg-tm
License. You must require such a user to return or
destroy all copies of the works possessed in a physical medium
and discontinue all use of and all access to other copies of
Project Gutenberg-tm works.
- You provide, in accordance with paragraph 1.F.3, a full refund of any
money paid for a work or a replacement copy, if a defect in the
electronic work is discovered and reported to you within 90 days
of receipt of the work.
- You comply with all other terms of this agreement for free
distribution of Project Gutenberg-tm works.
I.E. 9. If you wish to charge a fee or distribute a Project Gutenberg-tm
electronic work or group of works on different terms than are set
forth in this agreement, you must obtain permission in writing from
both the Project Gutenberg Literary Archive Foundation and Michael
Hart, the owner of the Project Gutenberg-tm trademark. Contact the
Foundation as set forth in Section 3 below.
l.F.
l.F.l. Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread
public domain works in creating the Project Gutenberg-tm
collection. Despite these efforts. Project Gutenberg-tm electronic
works, and the medium on which they may be stored, may contain
"Defects," such as, but not limited to, incomplete, inaccurate or
corrupt data, transcription errors, a copyright or other intellectual
property infringement, a defective or damaged disk or other medium, a
computer virus, or computer codes that damage or cannot be read by
your equipment .
l.F. 2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
of Replacement or Refund" described in paragraph l.F. 3, the Project
Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg-tm trademark, and any other party distributing a Project
Gutenberg-tm electronic work under this agreement, disclaim all
liability to you for damages, costs and expenses, including legal
fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.
l.F. 3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
defect in this electronic work within 90 days of receiving it, you can
receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from. If you
received the work on a physical medium, you must return the medium with
your written explanation. The person or entity that provided you with
the defective work may elect to provide a replacement copy in lieu of a
refund. If you received the work electronically, the person or entity
providing it to you may choose to give you a second opportunity to
receive the work electronically in lieu of a refund. If the second copy
is also defective, you may demand a refund in writing without further
opportunities to fix the problem.
l.F.4. Except for the limited right of replacement or refund set forth
in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO OTHER
WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.
l.F.5. Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of damages.
If any disclaimer or limitation set forth in this agreement violates the
law of the state applicable to this agreement, the agreement shall be
interpreted to make the maximum disclaimer or limitation permitted by
the applicable state law. The invalidity or unenforceability of any
provision of this agreement shall not void the remaining provisions.
l.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg-tm electronic works in accordance
with this agreement, and any volunteers associated with the production,
promotion and distribution of Project Gutenberg-tm electronic works,
harmless from all liability, costs and expenses, including legal fees,
that arise directly or indirectly from any of the following which you do
or cause to occur: (a) distribution of this or any Project Gutenberg-tm
work, (b) alteration, modification, or additions or deletions to any
Project Gutenberg-tm work, and (c) any Defect you cause.
Section 2. Information about the Mission of Project Gutenberg-tm
Project Gutenberg-tm is synonymous with the free distribution of
electronic works in formats readable by the widest variety of computers
including obsolete, old, middle-aged and new computers. It exists
because of the efforts of hundreds of volunteers and donations from
people in all walks of life.
Volunteers and financial support to provide volunteers with the
assistance they need, is critical to reaching Project Gutenberg-tm' s
goals and ensuring that the Project Gutenberg-tm collection will
remain freely available for generations to come. In 2001, the Project
Gutenberg Literary Archive Foundation was created to provide a secure
and permanent future for Project Gutenberg-tm and future generations.
To learn more about the Project Gutenberg Literary Archive Foundation
and how your efforts and donations can help, see Sections 3 and 4
and the Foundation web page at http://www.pglaf.org.
Section 3. Information about the Project Gutenberg Literary Archive
Foundation
The Project Gutenberg Literary Archive Foundation is a non profit
501(c)(3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service. The Foundation's EIN or federal tax identification
number is 64-6221541. Its 501(c)(3) letter is posted at
http://pglaf.org/fundraising. Contributions to the Project Gutenberg
Literary Archive Foundation are tax deductible to the full extent
permitted by U.S. federal laws and your state's laws.
The Foundation's principal office is located at 4557 Melan Dr. S.
Fairbanks, AK, 99712., but its volunteers and employees are scattered
throughout numerous locations. Its business office is located at
809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email
business@pglaf.org. Email contact links and up to date contact
information can be found at the Foundation's web site and official
page at http : //pglaf . org
For additional contact information:
Dr. Gregory B. Newby
Chief Executive and Director
gbnewby@pglaf . org
Section 4. Information about Donations to the Project Gutenberg
Literary Archive Foundation
Project Gutenberg-tm depends upon and cannot survive without wide
spread public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine readable form accessible by the widest
array of equipment including outdated equipment. Many small donations
($1 to $5,000) are particularly important to maintaining tax exempt
status with the IRS.
The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States. Compliance requirements are not uniform and it takes a
considerable effort, much paperwork and many fees to meet and keep up
with these requirements. We do not solicit donations in locations
where we have not received written confirmation of compliance. To
SEND DONATIONS or determine the status of compliance for any
particular state visit http://pglaf.org
While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.
International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States. U.S. laws alone swamp our small staff.
Please check the Project Gutenberg Web pages for current donation
methods and addresses. Donations are accepted in a number of other
ways including including checks, online payments and credit card
donations. To donate, please visit: http://pglaf.org/donate
Section 5. General Information About Project Gutenberg-tm electronic
works .
Professor Michael S. Hart is the originator of the Project Gutenberg-tm
concept of a library of electronic works that could be freely shared
with anyone. For thirty years, he produced and distributed Project
Gutenberg-tm eBooks with only a loose network of volunteer support .
Project Gutenberg-tm eBooks are often created from several printed
editions, all of which are confirmed as Public Domain in the U.S.
unless a copyright notice is included. Thus, we do not necessarily
keep eBooks in compliance with any particular paper edition.
Each eBook is in a subdirectory of the same number as the eBook's
eBook number, often in several formats including plain vanilla ASCII,
compressed (zipped), HTML and others.
Corrected EDITIONS of our eBooks replace the old file and take over
the old filename and etext number. The replaced older file is renamed.
VERSIONS based on separate sources are treated as new eBooks receiving
new filenames and etext numbers.
Most people start at our Web site which has the main PG search facility:
http : //www . gutenberg . net
This Web site includes information about Project Gutenberg-tm,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.
EBooks posted prior to November 2003, with eBook numbers BELOW #10000,
are filed in directories based on their release date. If you want to
download any of these eBooks directly, rather than using the regular
search system you may utilize the following addresses and just
download by the etext year. For example:
http : //www . gutenberg . net/etext06
(Or /etext 05, 04, 03, 02, 01, 00, 99,
98, 97, 96, 95, 94, 93, 92, 92, 91 or 90)
EBooks posted since November 2003, with etext numbers OVER #10000, are
filed in a different way. The year of a release date is no longer part
of the directory path. The path is based on the etext number (which is
identical to the filename) . The path to the file is made up of single
digits corresponding to all but the last digit in the filename. For
example an eBook of filename 10234 would be found at:
http : //www . gutenberg . net/1/0/2/3/10234
or filename 24689 would be found at:
http : //www . gutenberg . net/2/4/6/8/24689
An alternative method of locating eBooks:
http : //www . gutenberg . net/GUTINDEX . ALL
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11