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The Calculus of Logic
In a work lately published (1) That the business of Logic is with the relations of classes, and with the modes in which the mind contemplates those relations. (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. (3) That those laws are capable of mathematical expression, and that they thus constitute the basis of an interpretable calculus. (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. (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. (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. 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. 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 Xs, Ys, and others, and that and so on. In like manner we shall have 1 - Furthermore, from consideration of the nature of the mental operation involved, it will appear that the following laws are satisfied.
Representing by From the first of these it is seen that elective symbols are distributive in their operation; from the second that they are
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 (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 (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
It is not only true that every elective symbol representing a class
satisfies the index law (3), but it may be rigorously
demonstrated that any combination of elective symbols
φ( 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 We shall consider these with reference to the classes among which relation is expressed.
A. The expression All Ys represents the class Y
and will therefore be expressed by
E. In the proposition, No Ys are Xs, the
negative particle appears to be attached to the subject instead of to
the predicate to which it manifestly belongs.
We do not intend to say that those things which are not-Ys
are Xs, but that things which are Ys are
not-Xs. Now the class not-Xs is expressed by
1 -
I. In the proposition Some Ys are Xs, or Some Ys are Some Xs, we might regard the Some in the subject and the Some in the predicate as having reference to the same arbitrary class V, and so write but it is less of an assumption to refrain from doing this. Thus we should write
v`'` referring to another arbitrary class
V`'` .
O. Similarly, the proposition Some Ys are not-Xs, will be expressed by the equation
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,
is, the class Z consists of all Xs which are not-Ys and of all Ys which are not-Xs. ## General Theorems relating to Elective Functions.
We have now arrived at this step, - that we are in possession of a
class of symbols 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. ## Theorems of Development.
(1) If φ(the coefficients φ(1), φ(0), which are quantitative or common algebraic functions, are called the moduli, and x and 1 - x the constituents.
(2) For a function of two elective symbols we have φ(in which φ(11), φ(10), &c. are quantitative, and are called the moduli, and xy,
x(1 - y), &c. the constituents.
(3) Functions of three symbols: φ(in which φ(111), φ(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 (1), (2), (3).
THEOREM. This enables us to interpret any equation by a general rule.
RULE.
For the demonstration of these and many other results, I must refer to
the original work. It must be noted that on p. 66, As an example, let us take the equation Here φ( xy)
= x + 2y - 3xy,
whence the values of the moduli are
φ(11) = 0, φ(10) = 1, φ(01) = 2, φ(00) = 0,so that the expansion (9) gives which is in fact only another form of (11). We have, then, by the Rule the former implies that there are no Xs which are not-Ys, the latter that there are no Ys which are not-Xs, 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 (12) from
(11), we have
All equations are thus of equal significance which give, on expansion,
the same series of constituent equations, and ## General Solution of Elective Equations.
(1) The general solution of the equation
φ( y = φ(10) / (φ(10) - φ(11)) x + φ(00) / (φ(00) - φ(01)) (1-x) ........(13)The coefficients φ(10) / (φ(10) - φ(11)), φ(00) / (φ(00) - φ(01))are here the moduli.
(2) The general solution of the equation
φ( z =φ(110) / (φ(110) + φ(111)) xy + φ(100) / (φ(100) - φ(101) x(1-y) + φ(010) / (φ(010) - φ(011) (1-x)y + φ(000) / (φ(000) - φ(001) (1-x)(1-y) ........(14)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 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 every
question upon the relations of classes may be referred. One or two
very simple illustrations may suffice.
(1) Given
Here
φ( φ(111) = 0, φ(110) = 0, φ(101) = 0,and substituting in (14), we have Hence the class Z includes all Xs which are not-Ys, an indefinite number of Xs which are Ys, and an indefinite number of individuals which are neither Xs nor Ys. The classes w and w`'`
being quite arbitrary, the indefinite remainder is equally so; it may
vanish or not.
Since 1 - (1 -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
Put 1 - φ(111) = 1, φ(110) = 0, φ(101) = 0, φ(100) = -1,the solution will thus assume the form z = 0/(0 - 1) vy + (-1)/(-1 - 0) v(1 - y) + 1/(1 - 1) (1 - v)y + 0/(0 - 0) (1 - v)(1 - y),or 1 -The infinite coefficient of the second term in the second member permits us to write the coefficient 0/0 being then replaced by w, an
arbitrary elective symbol, we have
1 -or 1 -
We may remark upon this result that the coefficient
{as is evident on squaring it. It therefore represents a class.
We may replace it by an elective symbol u, we have then
1 -the interpretation of which is All not-Xs are not-Ys.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 Ys are (some) Xs into All not-Xs are (some) not-Ys there is a relation between the two ( somes), understood in the
predicates. The equation (18) shews that whatever may be
that condition which limits the Xs in the original
proposition, - the not-Ys 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 (17) further shews that there are no Ys
which are not subject to that condition.
We can similarly reduce the equation ## On Syllogism.The forms of categorical propositions already deduced are whereof the two first give, by solution, 1 - x = v`'` (1 - y).
All not-Xs are not-Ys,
x = v`'` (1 - y),
No Xs are Ys. To the above scheme, which is that
of Aristotle, we might annex the four categorical propositions
the first two of which are similarly convertible into
If now the two premises of any syllogism are expressed by equations of
the above forms, the elimination of the common symbol
the elimination of y gives
the interpretation of which is All Zs are Xs,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.
The elimination of y gives
which is interpretable into Some Zs are Xs. 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.
Instead of directly eliminating y let either equation be transformed
by solution as in (19). The first gives
1 - 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
the interpretation of which is No Zs are Xs.Had we directly eliminated y, we should have had
the reduced solution of which is 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. ## Laws of Syllogism deduced from the Elective Calculus.We shall take into account all propositions which can be made out of the classes X, Y, Z, and referred to any of the forms embraced in the following system,
It is necessary to recapitulate that quantity (universal and
particular) and quality (affirmative and negative) are understood to
belong to the Thus, in the proposition All Xs are Ys, the subject All Xs is universal-affirmative, the predicate (some) Ys particular-affirmative. In the proposition, Some Xs are Zs, both terms are particular-affirmative. The proposition No Xs are Zs would in philosophical language be written in the form All Xs are not-Zs. The subject is universal-affirmative, the predicate particular-negative. In the proposition Some Xs are not-Zs, the subject is particular-affirmative, the predicate particular-negative. In the proposition All not-Xs are Ys 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 Y or not-Y may be called the middle terms, the two others the extremes, one of these involving X or not-X, the other Z or not-Z. The following are then the conditions and the rules of inference. Case 1st. The middle terms of like quality. Condition of Inference. One middle term universal. Rule. Equate the extremes. 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. I add a few examples,
This belongs to Case 1. All Ys is the universal middle term. The extremes equated give All Zs are Xs, the stronger term becoming the subject.
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-Zs as the universal extreme, we have, on changing its quantity and quality, Some not-Zs, and this equated to the other extreme gives All Xs are (some) not-Zs = No Xs are Zs.If we take All Xs as the universal extreme we get No Zs are Xs.
This also belongs to Case 2, and satisfies the first condition. The universal extreme All Xs becomes, some not-Xs, whence Some Zs are not-Xs.
This belongs to Case 2, and satisfies the second condition. The extreme Some Xs becomes All not-Xs, .·. All not-Xs are Zs.The other extreme treated in the same way would give All not-Zs are Xs,which is an equivalent result. 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.
This belongs to Case 2, and satisfies the first condition. The result is Some not-Zs are not-Xs. 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 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 (1), (2), (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 which expresses the disjunctive proposition, ``Either
Y and X are together true, or X is
true and Y is false, or they are both false,'' and again
the equation y(1 - x) = 0, which expresses a relation of
coexistence, viz. that the truth of Y and the
falsehood of 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. 1.Cambridge and Dublin Mathematical Journal, Vol. III (1848), pp. 183-98
Originally transcribed by
D.R. Wilkins
(dwilkins@maths.tcd.ie) |