\documentclass{article}
\def\overcomma{\mathaccent"7027}
\begin{document}
\title{The Calculus of Logic}
\author{George Boole}
\date{\ }
\maketitle
\vspace{-24pt}
[\emph{Cambridge and Dublin Mathematical Journal},
Vol.~III (1848), pp.~183--98]
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% D. R. Wilkins, School of Mathematics, Trinity College, Dublin
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\bigskip
In a work lately published\footnote{\emph{The Mathematical Analysis of
Logic, being an Essay towards a Calculus of Deductive Reasoning}.
Cambridge, Macmillan; London, G.~Bell.}
I have exhibited the application of a new and peculiar form of
Mathematics to the expression of the operations of the mind in
reasoning. In the present essay I design to offer such an account of
a portion of this treatise as may furnish a correct view of the
nature of the system developed. I shall endeavour to state distinctly
those positions in which its characteristic distinctions consist, and
shall offer a more particular illustration of some features which are
less prominently displayed in the original work. The part of the
system to which I shall confine my observations is that which treats
of categorical propositions, and the positions which, under this
limitation, I design to illustrate, are the following:
\medskip
(1) That the business of Logic is with the relations of classes, and
with the modes in which the mind contemplates those relations.
\medskip
(2) That antecedently to our recognition of the existence of
propositions, there are laws to which the conception of a class is
subject,---laws which are dependent upon the constitution of the
intellect, and which determine the character and form of the
reasoning process.
\medskip
(3) That those laws are capable of mathematical expression, and that
they thus constitute the basis of an interpretable calculus.
\medskip
(4) That those laws are, furthermore, such, that all equations which
are formed in subjection to them, even though expressed under
functional signs, admit of perfect solution, so that every problem in
logic can be solved by reference to a general theorem.
\medskip
(5) That the forms under which propositions are actually exhibited, in
accordance with the principles of this calculus, are analogous with
those of a philosophical language.
\medskip
(6) That although the symbols of the calculus do not depend for their
interpretation upon the idea of quantity, they nevertheless, in their
particular application to syllogism, conduct us to the quantitative
conditions of inference.
\medskip
It is specially of the two last of these positions that I here desire
to offer illustration, they having been but partially exemplified in
the work referred to. Other points will, however, be made the
subjects of incidental discussion. It will be necessary to premise
the following notation.
\medskip
The universe of conceivable objects is represented by 1 or unity.
This I assume as the primary and subject conception. All subordinate
conceptions of class are understood to be formed from it by
limitation, according to the following scheme.
Suppose that we have the conception of any group of objects consisting
of $\mathrm{X}$s, $\mathrm{Y}$s, and others, and that $x$, which we
shall call an elective symbol, represents the mental operation of
selecting from that group all the $\mathrm{X}$s which it contains, or
of fixing the attention upon the $\mathrm{X}$s to the exclusion of all
which are not $\mathrm{X}$s, $y$ the mental operation of selecting the
$\mathrm{Y}$s, and so on; then, 1 or the universe being the subject
conception, we shall have
\begin{eqnarray*}
x \, 1 \mbox{ or } x &=& \mbox{the class $\mathrm{X}$,}\\
y \, 1 \mbox{ or } y &=& \mbox{the class $\mathrm{Y}$,}\\
xy 1 \mbox{ or } xy &=& \mbox{the class each member of which is
both $\mathrm{X}$ and $\mathrm{Y}$,}
\end{eqnarray*}
and so on.
In like manner we shall have
\begin{eqnarray*}
1 - x &=& \mbox{the class not-$\mathrm{X}$,}\\
1 - y &=& \mbox{the class not-$\mathrm{Y}$,}\\
x(1 - y) &=& \mbox{the class whose members are $\mathrm{X}$s but
not-$\mathrm{Y}$s,}\\
(1 - x)(1 - y) &=& \mbox{the class whose members are neither $\mathrm{X}$s
nor $\mathrm{Y}$s,}\\
\&c.&&
\end{eqnarray*}
Furthermore, from consideration of the nature of the mental operation
involved, it will appear that the following laws are satisfied.
Representing by $x$, $y$, $z$ any elective symbols whatever,
\begin{eqnarray}
x(y+z) &=& xy + xz, \label{eqn-1}\\
xy &=& yx, \,\,\&c., \label{eqn-2}\\
x^n &=& x, \,\,\&c. \label{eqn-3}
\end{eqnarray}
From the first of these it is seen that elective symbols are
distributive in their operation; from the second that they are
\emph{commutative}. The third I have termed the index law; it is
peculiar to elective symbols.
The truth of these laws does not at all depend upon the nature, or the
number, or the mutual relations, of the individuals included in the
different classes. There may be but one individual in a class, or
there may be a thousand. There may be individuals common to different
classes, or the classes may be mutually exclusive. All elective
symbols are distributive, and commutative, and all elective symbols
satisfy the law expressed by (\ref{eqn-3}).
These laws are in fact embodied in every spoken or written language.
The equivalence of the expressions ``good wise man'' and ``wise good
man,'' is not a mere truism, but an assertion of the law of
commutation exhibited in (\ref{eqn-2}). And there are similar
illustrations of the other laws.
With these laws there is connected a general axiom. We have seen that
algebraic operations performed with elective symbols represent mental
processes. Thus the connexion of two symbols by the sign~$+$
represents the aggregation of two classes into a single class, the
connexion of two symbols $xy$ as in multiplication, represents the
mental operation of selecting from a class $\mathrm{Y}$ those members
which belong also to another class $\mathrm{X}$, and so on. By such
operations the conception of a class is modified. But beside this the
mind has the power of perceiving relations of equality among classes.
The axiom in question, then, is that \emph{if a relation of equality
is perceived between two classes, that relation remains unaffected
when both subjects are equally modified by the operations above
described}. (A). This axiom, and not ``Aristotle's dictum,'' is the
real foundation of all reasoning, the form and character of the
process being, however, determined by the three laws already stated.
It is not only true that every elective symbol representing a class
satisfies the index law (\ref{eqn-3}), but it may be rigorously
demonstrated that any combination of elective symbols $\phi(xyz.\,.)$,
which satisfies the law $\phi(xyz.\,.)^n = \phi(xyz.\,.)$, represents
an intelligible conception,---a group or class defined by a greater or
less number of properties and consisting of a greater or less number
of parts.
The four categorical propositions upon which the doctrine of ordinary
syllogism is founded, are
\begin{quote}
\begin{tabular}{ll}
All $\mathrm{Y}$s are $\mathrm{X}$s. &A,\\
No $\mathrm{Y}$s are $\mathrm{X}$s. &E,\\
Some $\mathrm{Y}$s are $\mathrm{X}$s. &I,\\
Some $\mathrm{Y}$s are not $\mathrm{X}$s. &O.\\
\end{tabular}
\end{quote}
We shall consider these with reference to the classes among which
relation is expressed.
\medskip
A. The expression All~$\mathrm{Y}$s represents the class $\mathrm{Y}$
and will therefore be expressed by $y$, the copula are by the
sign~$=$, the indefinite term, $\mathrm{X}$s, is equivalent to
Some~$\mathrm{X}$s. It is a convention of language, that the word
Some is expressed in the subject, but not in the predicate of a
proposition. The term Some~$\mathrm{X}$s will be expressed by $vx$,
in which $v$ is an elective symbol appropriate to a
class~$\mathrm{V}$, some members of which are $\mathrm{X}$s, but which
is in other respects arbitrary. Thus the proposition $\mathrm{A}$
will be expressed by the equation
\begin{equation}
y = vx. \label{eqn-4}
\end{equation}
\medskip
E. In the proposition, No $\mathrm{Y}$s are $\mathrm{X}$s, the
negative particle appears to be attached to the subject instead of to
the predicate to which it manifestly belongs.\footnote{There are but
two ways in which the proposition, No $\mathrm{X}$s are $\mathrm{Y}$s,
can be understood. 1st, In the sense of All $\mathrm{X}$s are
not-$\mathrm{Y}$. 2nd, In the sense of It is not true that any
$\mathrm{X}$s are $\mathrm{Y}$s, \emph{i.e.} the proposition ``Some
$\mathrm{X}$s are $\mathrm{Y}$s'' is false. The former of these is a
single categorical proposition. The latter is \emph{an assertion
respecting a proposition}, and its expression belongs to a distinct
part of the elective system. It appears to me that it is the latter
sense, which is really adopted by those who refer the negative,
\emph{not}, to the copula. To refer it to the predicate is not a
useless refinement, but a necessary step, in order to make the
proposition truly a \emph{relation between classes}. I believe it
will be found that this step is really taken in the attempts to
demonstrate the Aristotelian rules of distribution.
The transposition of the negative is a very common feature of
language. Habit renders us almost insensible to it in our own
language, but when in another language the same principle is
differently exhibited, as in the Greek,
$o\overcomma{\upsilon}$ $\phi\eta\mu\grave{\iota}$ for
$\phi\eta\mu\grave{\iota}$ $o\overcomma{\upsilon}$, it claims attention.}
We do not intend to say that those things which are not-$\mathrm{Y}$s
are $\mathrm{X}$s, but that things which are $\mathrm{Y}$s are
not-$\mathrm{X}$s. Now the class not-$\mathrm{X}$s is expressed by
$1 - x$; hence the proposition No $\mathrm{Y}$s are $\mathrm{X}$s, or
rather All $\mathrm{Y}$s are not-$\mathrm{X}$s, will be expressed by
\begin{equation}
y = v(1 - x). \label{eqn-5}
\end{equation}
\medskip
I. In the proposition Some $\mathrm{Y}$s are $\mathrm{X}$s, or Some
$\mathrm{Y}$s are Some $\mathrm{X}$s, we might regard the Some in the
subject and the Some in the predicate as having reference to the same
arbitrary class $\mathrm{V}$, and so write
\[ vy = vx, \]
but it is less of an assumption to refrain from doing this. Thus we
should write
\begin{equation}
vy = v'x, \label{eqn-6}
\end{equation}
$v'$ referring to another arbitrary class $\mathrm{V}'$.
\medskip
O. Similarly, the proposition Some $\mathrm{Y}$s are
not-$\mathrm{X}$s, will be expressed by the equation
\begin{equation}
vy = v'(1 - x). \label{eqn-7}
\end{equation}
\medskip
It will be seen from the above that the forms under which the four
categorical propositions A, E, I, O are exhibited in the notation of
elective symbols are analogous with those of pure language,
\emph{i.e.} with the forms which human speech would assume, were its
rules entirely constructed upon a scientific basis. In a vast
majority of the propositions which can be conceived by the mind, the
laws of expression have not been modified by usage, and the analogy
becomes more apparent, \emph{e.g.} the interpretation of the equation
\[ z = x(1 - y) + y(1 - x), \]
is, the class $\mathrm{Z}$ consists of all $\mathrm{X}$s which are
not-$\mathrm{Y}$s and of all $\mathrm{Y}$s which are
not-$\mathrm{X}$s.
\subsection*{General Theorems relating to Elective Functions.}
We have now arrived at this step,---that we are in possession of a
class of symbols $x$, $y$, $z$, \&c.\ satisfying certain laws, and
applicable to the rigorous expression of any categorical proposition
whatever. It will be our next business to exhibit a few of the
general theorems of the calculus which rest upon the basis of those
laws, and these theorems we shall afterwards apply to the discussion
of particular examples.
Of the general theorems I shall only exhibit two sets: those which
relate to the development of functions, and those which relate to the
solution of equations.
\subsection*{Theorems of Development.}
(1) If $x$ be any elective symbol, then
\begin{equation}
\phi(x) = \phi(1) x + \phi(0) (1 - x), \label{eqn-8}
\end{equation}
the coefficients $\phi(1)$, $\phi(0)$, which are quantitative or
common algebraic functions, are called the moduli, and $x$ and $1 - x$
the constituents.
\medskip
(2) For a function of two elective symbols we have
\begin{equation}
\phi(xy) = \phi(11)xy + \phi(10) x(1 - y)
+ \phi(01)(1 - x)y + \phi(00)(1 - x)(1 - y), \label{eqn-9}
\end{equation}
in which $\phi(11)$, $\phi(10)$, \&c.\ are quantitative, and are called
the moduli, and $xy$, $x(1 - y)$, \&c.\ the constituents.
\medskip
(3) Functions of three symbols:
\begin{eqnarray}
\phi(xyz) &=& \phi(111)xyz + \phi(110) xy(1 - z) \nonumber\\
&&+ \phi(101)x(1 - y)z + \phi(100) x(1 - y)(1 - z) \nonumber\\
&&+ \phi(011)(1 - x)yz + \phi(010)(1 - x)y(1 - z) \nonumber\\
&&+ \phi(001)xy(1 - z) + \phi(000)(1 - x)(1 - y)(1 - z), \label{eqn-10}
\end{eqnarray}
in which $\phi(111)$, $\phi(110)$, \&c.\ are the moduli, and $xyz$,
$xy(1 - z)$, \&c.\ the constituents.
From these examples the general law of development is obvious. And I
desire it to be noted that this law is a mere consequence of the
primary laws which have been expressed in (\ref{eqn-1}),
(\ref{eqn-2}), (\ref{eqn-3}).
\medbreak
\textsc{Theorem.} \emph{If we have any equation $\phi(xyz.\,.) = 0$,
and fully expand the first member, then every constituent whose
modulus does not vanish may be equated to 0.}
\medbreak
This enables us to interpret any equation by a general rule.
\medbreak
\textsc{Rule.} \emph{Bring all the terms to the first side, expand
this in terms of all the elective symbols involved in it, and equate
to 0 every constituent whose modulus does not vanish.}
\medbreak
For the demonstration of these and many other results, I must refer to
the original work. It must be noted that on p.~66, $z$ has been,
through mistake, substituted for $w$, and that the reference on p.~80
should be to Prop.~2.
As an example, let us take the equation
\begin{equation}
x + 2y - 3xy = 0. \label{eqn-11a}
\end{equation}
Here $\phi(xy) = x + 2y - 3xy$, whence the values of the moduli are
\[ \phi(11) = 0,\quad \phi(10) = 1,\quad
\phi(01) = 2,\quad \phi(00) = 0, \]
so that the expansion (\ref{eqn-9}) gives
\[ x(1 - y) + 2y(1 - x) = 0, \]
which is in fact only another form of (\ref{eqn-11a}). We have, then,
by the Rule
\addtocounter{equation}{-1}% Boole's numbering
\begin{eqnarray}
x(1 - y) &=& 0, \label{eqn-11b}\\
y(1 - x) &=& 0; \label{eqn-12}
\end{eqnarray}
the former implies that there are no $\mathrm{X}$s which are
not-$\mathrm{Y}$s, the latter that there are no $\mathrm{Y}$s which
are not-$\mathrm{X}$s, these together expressing the full significance
of the original equation.
We can, however, often recombine the constituents with a gain of
simplicity. In the present instance, subtracting (\ref{eqn-12}) from
(\ref{eqn-11b}), we have $x - y = 0$, or $x = y$, that is, the
class~$\mathrm{X}$ is identical with the class~$\mathrm{Y}$. This
proposition is equivalent to the two former ones.
All equations are thus of equal significance which give, on expansion,
the same series of constituent equations, and \emph{all are
interpretable}.
\subsection*{General Solution of Elective Equations.}
(1) The general solution of the equation $\phi(xy) = 0$, in which two
elective symbols only are involved, $y$ being the one whose value is
sought, is
\begin{equation}
y = \frac{\phi(10)}{\phi(10) - \phi(11)} x
+ \frac{\phi(00)}{\phi(00) - \phi(01)} (1 - x). \label{eqn-13}
\end{equation}
The coefficients
\[ \frac{\phi(10)}{\phi(10) - \phi(11)},\qquad
\frac{\phi(00)}{\phi(00) - \phi(01)} \]
are here the moduli.
\medskip
(2) The general solution of the equation $\phi(xyz) = 0$, $z$ being
the symbol whose value is to be determined, is
\begin{eqnarray}
z &=& \frac{\phi(110)}{\phi(110) - \phi(111)} xy
+ \frac{\phi(100)}{\phi(100) - \phi(101)} x(1 - y) \nonumber\\
& & + \frac{\phi(010)}{\phi(010) - \phi(011)} (1 - x)y
+ \frac{\phi(000)}{\phi(000) - \phi(001)} (1 - x)(1 - y),
\nonumber\\
\label{eqn-14}
\end{eqnarray}
the coefficients of which we shall still term the moduli. The law of
their formation will readily be seen, so that the general theorems
which have been given for the solution of elective equations of two
and three symbols, may be regarded as examples of a more general
theorem applicable to all elective equations whatever. in applying
these results it is to be observed, that if a modulus assume the form
$\frac{0}{0}$ it is to be replaced by an arbitrary elective
symbol~$w$, and that if a modulus assume any numerical value except
$0$ or $1$, the constituent of which it is a factor must be separately
equated to $0$. Although these conditions are deduced solely from the
laws to which the symbols are obedient, and without any reference to
interpretation, they nevertheless render the solution of every
equation interpretable in logic. To such formulae also \emph{every
question upon the relations of classes may be referred}. One or two
very simple illustrations may suffice.
\medskip
(1) Given $yx = yz + x(1 - z)$.\hfill (\textit{a})\break
The $\mathrm{Y}$s which are $\mathrm{X}$s consist of the $\mathrm{Y}$s
which are $\mathrm{Z}$s and the $\mathrm{X}$s which are
not-$\mathrm{Z}$s. Required the class $\mathrm{Z}$.
Here $\phi(xyz) = yx - yz - x(1 - z)$,
\[ \phi(111) = 0,\quad \phi(110) = 0,\quad \phi(101) = 0, \]
\[ \phi(100) = -1,\quad \phi(011) = -1,\quad \phi(010) = 0, \]
\[ \phi(001) = 0,\quad \phi(000) = 0; \]
and substituting in (\ref{eqn-14}), we have
\begin{eqnarray}
z &=& \frac{0}{0} xy + x(1 - y) + \frac{0}{0}(1 - x)(1 - y) \nonumber\\
&=& x(1 - y) + wxy + w'(1 - x)(1 - y). \label{eqn-15}
\end{eqnarray}
Hence the class~$\mathrm{Z}$ includes all $\mathrm{X}$s which are
not-$\mathrm{Y}$s, an indefinite number of $\mathrm{X}$s which are
$\mathrm{Y}$s, and an indefinite number of individuals which are
neither $\mathrm{X}$s nor $\mathrm{Y}$s. The classes $w$ and $w'$
being quite arbitrary, the indefinite remainder is equally so; it may
vanish or not.\footnote{This conclusion may be illustrated and
verified by considering an example such as the following.
\begin{tabular}{ll}
Let &$x$ denote all steamers, or steam-vessels,\\
&$y$ denote all armed vessels,\\
&$z$ denote all vessels of the Mediterranean.
\end{tabular}\\
Equation (\textit{a}) would then express that \emph{armed steamers
consist of the armed vessels of the Mediterranean and the
steam-vessels not of the Mediterranean}. From this it follows---
(1) That there are no armed vessels except steamers in the
Mediterranean.
(2) That all unarmed steamers are in the Mediterranean (since the
steam-vessels not of the Mediterranean are armed). Hence we infer
that \emph{the vessels of the Mediterranean consist of all unarmed
steamers; any number of armed steamers; and any number of unarmed
vessels without steam}. This, expressed symbolically, is equation
(\ref{eqn-15}).}
Since $1 - z$ represents a class, not-$\mathrm{Z}$, and satisfies the
index law
\[ (1 - z)^n = 1 - z, \]
as is evident on trial, we may, if we choose, determine the value of
this element just as we should determine that of $z$.
Let us take, in illustration of this principle, the equation $y = vx$,
(All $\mathrm{Y}$s are $\mathrm{X}$s), and seek the value of $1 - x$,
the class not-$\mathrm{X}$.
Put $1 - x = z$ then $y = v(1 - z)$, and if we write this in the form
$y - v(1 - z) = 0$ and represent the first member by $\phi(vyz)$, $v$
here taking the place of $x$, in (14), we shall have
\[ \phi(111) = 1,\quad \phi(110) = 0,\quad
\phi(101) = 0,\quad \phi(100) = -1, \]
\[ \phi(011) = 1,\quad \phi(010) = 1,\quad
\phi(001) = 0,\quad \phi(000) = 0; \]
the solution will thus assume the form
\[ z = \frac{0}{0 - 1}vy + \frac{-1}{-1 - 0}v(1 - y)
+ \frac{1}{1 - 1}(1 - v)y + \frac{0}{0 - 0}(1 - v)(1 - y),\]
or
\begin{equation}
1 - x = v(1 - y) + \frac{1}{0}(1 - v)y + \frac{0}{0}(1 - v)(1 - y).
\label{eqn-16}
\end{equation}
The infinite coefficient of the second term in the second member
permits us to write
\begin{equation}
y(1 - v) = 0, \label{eqn-17}
\end{equation}
the coefficient $\frac{0}{0}$ being then replaced by $w$, an
arbitrary elective symbol, we have
\[ 1 - x = v(1 - y) + w(1 - v)(1 - y), \]
or
\begin{equation}
1 - x = \{v + w(1 - v)\} (1 - y). \label{eqn-18}
\end{equation}
We may remark upon this result that the coefficient $v + w(1 - v)$ in
the second member satisfies the condition
\[ \{v + w(1 - v) \}^n = v + w(1 - v), \]
as is evident on squaring it. It therefore represents a \emph{class}.
We may replace it by an elective symbol~$u$, we have then
\begin{equation}
1 - x = u(1 - y), \label{eqn-19}
\end{equation}
the interpretation of which is
\begin{quote}
All not-$\mathrm{X}$s are not-$\mathrm{Y}$s.
\end{quote}
This is a known transformation in logic, and is called conversion by
contraposition, or negative conversion. But it is far from exhausting
the solution we have obtained. Logicians have overlooked the fact,
that when we convert the proposition All $\mathrm{Y}$s are (some)
$\mathrm{X}$s into All not-$\mathrm{X}$s are (some) not-$\mathrm{Y}$s
there is a relation between the two (\emph{somes}), understood in the
predicates. The equation (\ref{eqn-18}) shews that whatever may be
that condition which limits the $\mathrm{X}$s in the original
proposition,---the not-$\mathrm{Y}$s in the converted proposition
consist of all which are subject to the same condition, and of an
arbitrary remainder which are not subject to that condition. The
equation (\ref{eqn-17}) further shews that there are no $\mathrm{Y}$s
which are not subject to that condition.
We can similarly reduce the equation $y = v(1 - x)$, No $\mathrm{Y}$s
are $\mathrm{X}$s, to the form $x = v'(1 - y)$ No $\mathrm{X}$s are
$\mathrm{Y}$s, with a like relation between $v$ and $v'$. If we solve
the equation $y = vx$ All $\mathrm{Y}$s are $\mathrm{X}$s, with
reference to $v$, we obtain the subsidiary relation $y(1 - x) = 0$
No $\mathrm{Y}$s are not-$\mathrm{X}$s, and similarly from the
equation $y = v(1 - x)$ (No $\mathrm{Y}$s are $\mathrm{X}$s) we get
$xy = 0$. These equations, which may also be obtained in other ways,
I have employed in the original treatise. All equations whose
interpretations are connected are similarly connected themselves, by
solution or development.
\subsection*{On Syllogism.}
The forms of categorical propositions already deduced are
\begin{quote}
\begin{tabular}{ll}
$\phantom{v}y = vx$,
&All $\mathrm{Y}$s are $\mathrm{X}$s,\\
$\phantom{v}y = v(1 - x)$,
&No $\mathrm{Y}$s are $\mathrm{X}$s,\\
$vy = v'x$,
&Some $\mathrm{Y}$s are $\mathrm{X}$s,\\
$vy = v'(1 - x)$,
&Some $\mathrm{Y}$s are not-$\mathrm{X}$s,
\end{tabular}
\end{quote}
whereof the two first give, by solution, $1 - x = v'(1 - y)$. All
not-$\mathrm{X}$s are not-$\mathrm{Y}$s, $x = v'(1 - y)$, No
$\mathrm{X}$s are $\mathrm{Y}$s. To the above scheme, which is that
of Aristotle, we might annex the four categorical propositions
\begin{quote}
\begin{tabular}{ll}
$\phantom{v()}1 - y = vx$,
&All not-$\mathrm{Y}$s are $\mathrm{X}$s,\\
$\phantom{v()}1 - y = v(1 - x)$,
&All not-$\mathrm{Y}$s are not-$\mathrm{X}$s,\\
$v(1 - y) = v'x$,
&Some not-$\mathrm{Y}$s are $\mathrm{X}$s,\\
$v(1 - y) = v'(1 - x)$,
&Some not-$\mathrm{Y}$s are not-$\mathrm{X}$s,\\
\end{tabular}
\end{quote}
the first two of which are similarly convertible into
\begin{quote}
\begin{tabular}{lll}
$1 - x = v'y$, &&All not-$\mathrm{X}$s are $\mathrm{Y}$s,\\
$\phantom{1 - {}}x = v'y$, &&All $\mathrm{X}$s are $\mathrm{Y}$s,\\
&or &No not-$\mathrm{X}$s are $\mathrm{Y}$s,
\end{tabular}
\end{quote}
If now the two premises of any syllogism are expressed by equations of
the above forms, the elimination of the common symbol~$y$ will lead us
to an equation expressive of the conclusion.
\medskip
\begin{quote}
\begin{tabular}{lll}
Ex.~1.&All $\mathrm{Y}$s are $\mathrm{X}$s, &$y = vx$,\\
&All $\mathrm{Z}$s are $\mathrm{Y}$s, &$z = v'y$,
\end{tabular}
\end{quote}
the elimination of $y$ gives
\[ z = vv'x, \]
the interpretation of which is
\begin{quote}
All $\mathrm{Z}$s are $\mathrm{X}$s,
\end{quote}
the form of the coefficient $vv'$ indicates that the predicate of the
conclusion is limited by both the conditions which separately limit
the predicates of the premises.
\medskip
\begin{quote}
\begin{tabular}{lll}
Ex.~2.&All $\mathrm{Y}$s are $\mathrm{X}$s, &$y = vx$,\\
&All $\mathrm{Y}$s are $\mathrm{Z}$s, &$y = v'z$.
\end{tabular}
\end{quote}
The elimination of $y$ gives
\[ v'z = vx, \]
which is interpretable into Some $\mathrm{Z}$s are $\mathrm{X}$s. It
is always necessary that one term of the conclusion should be
interpretable by means of the equations of the premises. In the above
case both are so.
\medskip
\begin{quote}
\begin{tabular}{lll}
Ex.~3.&All $\mathrm{X}$s are $\mathrm{Y}$s, &$x = vy$,\\
&No $\mathrm{Z}$s are $\mathrm{Y}$s, &$z = v'(1 - y)$.
\end{tabular}
\end{quote}
Instead of directly eliminating $y$ let either equation be transformed
by solution as in (\ref{eqn-19}). The first gives
\[ 1 - y = u(1 - x), \]
$u$ being equivalent to $v + w(1 - v)$, in which $w$ is arbitrary.
Eliminating $1 - y$ between this and the second equation of the
system, we get
\[ z = v'u(1 - x), \]
the interpretation of which is
\begin{quote}
No $\mathrm{Z}$s are $\mathrm{X}$s.
\end{quote}
Had we directly eliminated $y$, we should have had
\[ vz = v'(v - x), \]
the reduced solution of which is
\[ z = v'\{v + (1 - v)\}(1 - x), \]
in which $w$ is an arbitrary elective symbol. This exactly agrees
with the former result.
These examples may suffice to illustrate the employment of the method
in particular instances. But its applicability to the demonstration
of general theorems is here, as in other cases, a more important
feature. I subjoin the results of a recent investigation of the Laws
of Syllogism. While those results are characterized by great
simplicity and bear, indeed, little trace of their mathematical
origin, it would, I conceive, have been very difficult to arrive at
them by the examination and comparison of particular cases.
\subsection*{Laws of Syllogism deduced from the Elective Calculus.}
We shall take into account all propositions which can be made out of
the classes $\mathrm{X}$, $\mathrm{Y}$, $\mathrm{Z}$, and referred to
any of the forms embraced in the following system,
\begin{quote}
\begin{tabular}{ll}
A, All $\mathrm{X}$s are $\mathrm{Z}$s.
&A${}'$, All not-$\mathrm{X}$s are $\mathrm{Z}$s.\\[3pt]
E, No $\mathrm{X}$s are $\mathrm{Z}$s.
&E${}'$, $\left\{ \vcenter{%
\hbox{No not-$\mathrm{X}$s are $\mathrm{Z}$s, or}
\hbox{(All not-$\mathrm{X}$s are not-$\mathrm{Z}$s.)}}
\right.$\\[3pt]
I, Some $\mathrm{X}$s are $\mathrm{Z}$s.
&I${}'$, Some not-$\mathrm{X}$s are $\mathrm{Z}$s.\\[3pt]
O, Some $\mathrm{X}$s are not-$\mathrm{Z}$s.
&O${}'$, Some not-$\mathrm{X}$s are not-$\mathrm{Z}$s.
\end{tabular}
\end{quote}
It is necessary to recapitulate that quantity (universal and
particular) and quality (affirmative and negative) are understood to
belong to the \emph{terms} of propositions which is indeed the correct
view.\footnote{When \emph{propositions} are said to be affected with
quantity and quality, the quality is really that of the
\emph{predicate}, which expresses the \emph{nature} of the assertion,
and the quantity that of the \emph{subject}, which shews its extent.}
Thus, in the proposition All $\mathrm{X}$s are $\mathrm{Y}$s, the
subject All $\mathrm{X}$s is universal-affirmative, the predicate
(some) $\mathrm{Y}$s particular-affirmative.
In the proposition, Some $\mathrm{X}$s are $\mathrm{Z}$s, both terms
are particular-affirmative.
The proposition No $\mathrm{X}$s are $\mathrm{Z}$s would in
philosophical language be written in the form All $\mathrm{X}$s are
not-$\mathrm{Z}$s. The subject is universal-affirmative, the
predicate particular-negative.
In the proposition Some $\mathrm{X}$s are not-$\mathrm{Z}$s, the
subject is particular-affirmative, the predicate particular-negative.
In the proposition All not-$\mathrm{X}$s are $\mathrm{Y}$s the subject
is universal-negative, the predicate particular-affirmative, and so
on.
In a pair of premises there are four terms, viz. two subjects and two
predicates; two of these terms, viz. those involving the $\mathrm{Y}$
or not-$\mathrm{Y}$ may be called the middle terms, the two others the
extremes, one of these involving $\mathrm{X}$ or not-$\mathrm{X}$, the
other $\mathrm{Z}$ or not-$\mathrm{Z}$.
The following are then the conditions and the rules of inference.
\medskip
Case 1st. The middle terms of like quality.
Condition of Inference. One middle term universal.
Rule. Equate the extremes.
\medskip
Case 2nd. The middle terms of opposite qualities.
1st. Condition of Inference. One extreme universal.
Rule. Change the quantity and quality of that extreme, and equate the
result to the other extreme.
2nd. Condition of inference. Two universal middle terms.
Rule. Change the quantity and quality of either extreme, and equate
the result to the other extreme.
\medskip
I add a few examples,
\medskip
\begin{tabular}{ll}
1st&All $\mathrm{Y}$s are $\mathrm{X}$s.\\
&All $\mathrm{Z}$s are $\mathrm{Y}$s.
\end{tabular}
\medskip
This belongs to Case~1. All $\mathrm{Y}$s is the universal middle
term. The extremes equated give All $\mathrm{Z}$s are $\mathrm{X}$s,
the stronger term becoming the subject.
\medskip
\vbox to 0pt{2nd\par\vss}
\leavevmode\phantom{2nd}\quad $\left. \vcenter{%
\hbox{All $\mathrm{X}$s are $\mathrm{Y}$s}
\hbox{No $\mathrm{Z}$s are $\mathrm{Y}$s}}\right\}
= \left\{ \vcenter{%
\hbox{All $\mathrm{X}$s are $\mathrm{Y}$s}
\hbox{No $\mathrm{Z}$s are not-$\mathrm{Y}$s}}\right.$
\medskip
This belongs to Case~2, and satisfies the first condition. The middle
term is particular-affirmative in the first premise,
particular-negative in the second. Taking All-$\mathrm{Z}$s as the
universal extreme, we have, on changing its quantity and quality, Some
not-$\mathrm{Z}$s, and this equated to the other extreme gives
\begin{quote}
All $\mathrm{X}$s are (some) not-$\mathrm{Z}$s
= No $\mathrm{X}$s are $\mathrm{Z}$s.
\end{quote}
If we take All~$\mathrm{X}$s as the universal extreme we get
\begin{quote}
No $\mathrm{Z}$s are $\mathrm{X}$s.
\end{quote}
\medskip
\begin{tabular}{ll}
3rd&All $\mathrm{X}$s are $\mathrm{Y}$s.\\
&Some $\mathrm{Z}$s are not-$\mathrm{Y}$s.
\end{tabular}
\medskip
This also belongs to Case~2, and satisfies the first condition. The
universal extreme All $\mathrm{X}$s becomes, some not-$\mathrm{X}$s,
whence
\begin{quote}
Some $\mathrm{Z}$s are not-$\mathrm{X}$s.
\end{quote}
\medskip
\begin{tabular}{ll}
4th&All $\mathrm{Y}$s are $\mathrm{X}$s.\\
&All not-$\mathrm{Y}$s are $\mathrm{Z}$s.
\end{tabular}
\medskip
This belongs to Case~2, and satisfies the second condition. The
extreme Some $\mathrm{X}$s becomes All not-$\mathrm{X}$s,
\begin{quote}
$\mathrel{.\,\raise1ex\hbox{.}\,.}$
All not-$\mathrm{X}$s are $\mathrm{Z}$s.
\end{quote}
The other extreme treated in the same way would give
\begin{quote}
All not-$\mathrm{Z}$s are $\mathrm{X}$s,
\end{quote}
which is an equivalent result.
\penalty-1000
If we confine ourselves to the Aristotelian premises A, E, I, O, the
second condition of inference in Case~2 is not needed. The conclusion
will not necessarily be confined to the Aristotelian system.
\medskip
\vbox to 0pt{Ex.\par\vss}
\leavevmode\phantom{Ex.}\quad $\left. \vcenter{%
\hbox{Some $\mathrm{Y}$s are not-$\mathrm{X}$s}
\hbox{No $\mathrm{Z}$s are $\mathrm{Y}$s}}\right\}
= \left\{ \vcenter{%
\hbox{Some $\mathrm{Y}$s are not-$\mathrm{X}$s}
\hbox{All $\mathrm{Z}$s are not-$\mathrm{Y}$s}}\right.$
\medskip
This belongs to Case~2, and satisfies the first condition. The result
is
\begin{quote}
Some not-$\mathrm{Z}$s are not-$\mathrm{X}$s.
\end{quote}
These appear to me to be the ultimate laws of syllogistic inference.
They apply to every case, and they completely abolish the distinction
of figure, the necessity of conversion, the arbitrary and
partial\footnote{Partial, because they have reference only to the
quantity of the $\mathrm{X}$, even when the proposition relates to the
not-$\mathrm{X}$. It would be possible to construct an exact
counterpart to the Aristotelian rules of syllogism, by quantifying
only the not-$\mathrm{X}$. The system in the text is
\emph{symmetrical} because it is complete.}
rules of distribution, \&c. If all logic were reducible to the
syllogism these might claim to be regarded as the rules of logic. But
logic, considered as the science of the relations of classes has been
shewn to be of far greater extent. Syllogistic inference, in the
elective system, corresponds to elimination. But this is not the
highest in the order of its processes. All questions of elimination
may in that system be regarded as subsidiary to the more general
problem of the solution of elective equations. To this problem all
questions of logic and of reasoning, without exception, may be
referred. For the fuller illustrations of this principle I must
however refer to the original work. The theory of hypothetical
propositions, the analysis of the positive and negative elements, into
which all propositions are ultimately resolvable, and other similar
topics are also there discussed.
Undoubtedly the final aim of speculative logic is to assign the
conditions which render reasoning possible, and the laws which
determine its character and expression. The general axiom (A) and the
laws (\ref{eqn-1}), (\ref{eqn-2}), (\ref{eqn-3}), appear to convey the
most definite solution that can at present be given to this question.
When we pass to the consideration of hypothetical propositions, the
same laws and the same general axiom which ought perhaps also to be
regarded as a law, continue to prevail; the only difference being that
the subjects of thought are no longer classes of objects, but cases of
the coexistent truth or falsehood of propositions. Those relations
which logicians designate by the terms conditional, disjunctive, \&c.,
are referred by Kant to distinct conditions of thought. But it is a
very remarkable fact, that the expressions of such relations can be
deduced the one from the other by mere analytical process. From the
equation $y = vx$, which expresses the \emph{conditional} proposition,
``If the proposition $\mathrm{Y}$ is true the proposition~$\mathrm{X}$
is true,'' we can deduce
\[ yx + (1 - y)x + (1 - y)(1 - x) = 1, \]
which expresses the \emph{disjunctive} proposition, ``Either
$\mathrm{Y}$ and $\mathrm{X}$ are together true, or $\mathrm{X}$ is
true and $\mathrm{Y}$ is false, or they are both false,'' and again
the equation $y(1 - x) = 0$, which expresses a relation of
coexistence, \emph{viz.} that the truth of $\mathrm{Y}$ and the
falsehood of $\mathrm{X}$ do not coexist. The distinction in the
mental regard, which has the best title to be regarded as fundamental,
is, I conceive, that of the affirmative and the negative. From this
we deduce the direct and the inverse in operations, the true and the
false in propositions, and the opposition of qualities in their
terms.
The view which these enquiries present of the nature of language is a
very interesting one. They exhibit it not as a mere collection of
signs, but as a system of expression, the elements of which are
subject to the laws of the thought which they represent. That those
laws are as rigorously mathematical as are the laws which govern the
purely quantitative conceptions of space and time, of number and
magnitude, is a conclusion which I do not hesitate to submit to the
exactest scrutiny.
\end{document}