The Complete Works of Boole George

The Complete Works of Boole George

Boole George

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ISBN 10: 599894598

ISBN 13: 9780599894594

The Complete Works of Boole George

This collection includes the following:

An Investigation of the Laws of Thought

The Mathematical Analysis of Logic Being an Essay Towards a Calculus of Deductive Reasoning

Accordingly, since by the logical reduction the solution of all questions in

the theory of probabilities is brought to a form in which, from the probabilities

of simple events, $s$, $t$, \&c. under a given condition, $V$, it is required to determine

the probability of some combination, $A$, of those events under the same condition,

the principle of the demonstration in Prop. IV. is really the following:--``The

probability of such combination $A$ under the condition $V$ must be calculated

as if the events $s$, $t$, \&c. were independent, and possessed of such probabilities

as would cause the derived probabilities of the said events under the same

condition $V$ to be such as are assigned to them in the data.'' This principle I

regard as axiomatic. At the same time it admits of indefinite verification, as

well directly as through the results of the method of which it forms the basis.

I think it right to add, that it was in the above form that the principle first presented

itself to my mind, and that it is thus that I have always understood it,

the error in the particular problem referred to having arisen from inadvertence

in the choice of a material illustration.

\mainmatter

\chapter[NATURE AND DESIGN OF THIS WORK]{\large NATURE AND DESIGN OF THIS WORK.}

1. The design of the following treatise is to investigate the

fundamental laws of those operations of the mind by which

reasoning is performed; to give expression to them in the symbolical

language of a Calculus, and upon this foundation to establish the

science of Logic and construct its method; to make that method

itself the basis of a general method for the application of the mathematical

doctrine of Probabilities; and, finally, to collect from

the various elements of truth brought to view in the course of

these inquiries some probable intimations concerning the nature

and constitution of the human mind.

2. That this design is not altogether a novel one it is almost

needless to remark, and it is well known that to its two main

practical divisions of Logic and Probabilities a very considerable

share of the attention of philosophers has been directed. In its

ancient and scholastic form, indeed, the subject of Logic stands

almost exclusively associated with the great name of Aristotle.

As it was presented to ancient Greece in the partly technical,

partly metaphysical disquisitions of the Organon, such, with

scarcely any essential change, it has continued to the present

day. The stream of original inquiry has rather been directed

towards questions of general philosophy, which, though they

have arisen among the disputes of the logicians, have outgrown

their origin, and given to successive ages of speculation their peculiar

bent and character. The eras of Porphyry and Proclus,

of Anselm and Abelard, of Ramus, and of Descartes, together

with the final protests of Bacon and Locke, rise up before the

mind as examples of the remoter influences of the study upon the

course of human thought, partly in suggesting topics fertile of

discussion, partly in provoking remonstrance against its own undue

pretensions. The history of the theory of Probabilities, on

the other hand, has presented far more of that character of steady

growth which belongs to science. In its origin the early genius

of Pascal,--in its maturer stages of development the most recondite

of all the mathematical speculations of Laplace,--were directed

to its improvement; to omit here the mention of other names

scarcely less distinguished than these. As the study of Logic has

been remarkable for the kindred questions of Metaphysics to

which it has given occasion, so that of Probabilities also has been

remarkable for the impulse which it has bestowed upon the

higher departments of mathematical science. Each of these subjects

has, moreover, been justly regarded as having relation to a

speculative as well as to a practical end. To enable us to deduce

correct inferences from given premises is not the only object of

Logic; nor is it the sole claim of the theory of Probabilities that

it teaches us how to establish the business of life assurance on a

secure basis; and how to condense whatever is valuable in the

records of innumerable observations in astronomy, in physics, or

in that field of social inquiry which is fast assuming a character

of great importance. Both these studies have also an interest

of another kind, derived from the light which they shed upon

the intellectual powers. They instruct us concerning the mode

in which language and number serve as instrumental aids to the

processes of reasoning; they reveal to us in some degree the

connexion between different powers of our common intellect;

they set before us what, in the two domains of demonstrative and

of probable knowledge, are the essential standards of truth and

correctness,--standards not derived from without, but deeply

founded in the constitution of the human faculties. These ends

of speculation yield neither in interest nor in dignity, nor yet, it

may be added, in importance, to the practical objects, with the

pursuit of which they have been historically associated. To unfold

the secret laws and relations of those high faculties of

thought by which all beyond the merely perceptive knowledge

of the world and of ourselves is attained or matured, is an object

which does not stand in need of commendation to a rational

mind.

3. But although certain parts of the design of this work have

been entertained by others, its general conception, its method,

and, to a considerable extent, its results, are believed to be original.

For this reason I shall offer, in the present chapter, some

preparatory statements and explanations, in order that the real

aim of this treatise may be understood, and the treatment of its

subject facilitated.

It is designed, in the first place, to investigate the fundamental

laws of those operations of the mind by which reasoning is

performed. It is unnecessary to enter here into any argument to

prove that the operations of the mind are in a certain real sense

subject to laws, and that a science of the mind is therefore {\it possible}.

If these are questions which admit of doubt, that doubt is not

to be met by an endeavour to settle the point of dispute \textit{\`{a} priori},

but by directing the attention of the objector to the evidence of

actual laws, by referring him to an actual science. And thus the

solution of that doubt would belong not to the introduction to

this treatise, but to the treatise itself. Let the assumption be

granted, that a science of the intellectual powers is possible, and

let us for a moment consider how the knowledge of it is to be

obtained.

4. Like all other sciences, that of the intellectual operations

must primarily rest upon observation,--the subject of such observation

being the very operations and processes of which we

desire to determine the laws. But while the necessity of a foundation

in experience is thus a condition common to all sciences,

there are some special differences between the modes in which

this principle becomes available for the determination of general

truths when the subject of inquiry is the mind, and when the

subject is external nature. To these it is necessary to direct

attention.

The general laws of Nature are not, for the most part, immediate

objects of perception. They are either inductive inferences

from a large body of facts, the common truth in which they express,

or, in their origin at least, physical hypotheses of a causal

nature serving to explain ph{\ae}nomena with undeviating precision,

and to enable us to predict new combinations of them. They

are in all cases, and in the strictest sense of the term, \textit{probable}

conclusions, approaching, indeed, ever and ever nearer to certainty,

as they receive more and more of the confirmation of experience.

But of the character of probability, in the strict and

proper sense of that term, they are never wholly divested. On the

other hand, the knowledge of the laws of the mind does not require

as its basis any extensive collection of observations. The general

truth is seen in the particular instance, and it is not confirmed

by the repetition of instances. We may illustrate this position

by an obvious example. It may be a question whether that formula

of reasoning, which is called the \textit{dictum} of Aristotle, \textit{de omni et nullo},

expresses a primary law of human reasoning or not; but

it is no question that it expresses a general truth in Logic. Now

that truth is made manifest in all its generality by reflection

upon a single instance of its application. And this is both an

evidence that the particular principle or formula in question is

founded upon some general law or laws of the mind, and an illustration

of the doctrine that the perception of such general truths

is not derived from an induction from many instances, but is involved

in the clear apprehension of a single instance. In connexion

with this truth is seen the not less important one that

our knowledge of the laws upon which the science of the intellectual

powers rests, whatever may be its extent or its deficiency, is

not probable knowledge. For we not only see in the particular

example the general truth, but we see it also as a certain truth,--a

truth, our confidence in which will not continue to increase

with increasing experience of its practical verifications.

5. But if the general truths of Logic are of such a nature that

when presented to the mind they at once command assent,

wherein consists the difficulty of constructing the Science of

Logic? Not, it may be answered, in collecting the materials of

knowledge, but in discriminating their nature, and determining

their mutual place and relation. All sciences consist of general

truths, but of those truths some only are primary and fundamental,

others are secondary and derived. The laws of elliptic motion,

discovered by Kepler, are general truths in astronomy, but

they are not its fundamental truths. And it is so also in the

purely mathematical sciences. An almost boundless diversity of

theorems, which are known, and an infinite possibility of others,

as yet unknown, rest together upon the foundation of a few simple

axioms; and yet these are all \textit{general} truths. It may be

added, that they are truths which to an intelligence sufficiently

refined would shine forth in their own unborrowed light, without

the need of those connecting links of thought, those steps

of wearisome and often painful deduction, by which the knowledge

of them is actually acquired. Let us define as fundamental

those laws and principles from which all other general truths of

science may be deduced, and into which they may all be again

resolved. Shall we then err in regarding that as the true science

of Logic which, laying down certain elementary laws, confirmed

by the very testimony of the mind, permits us thence to deduce,

by uniform processes, the entire chain of its secondary consequences,

and furnishes, for its practical applications, methods of

perfect generality? Let it be considered whether in any science,

viewed either as a system of truth or as the foundation of a practical

art, there can properly be any other test of the completeness

and the fundamental character of its laws, than the completeness

of its system of derived truths, and the generality of the methods

which it serves to establish. Other questions may indeed present

themselves. Convenience, prescription, individual preference,

may urge their claims and deserve attention. But as

respects the question of what constitutes science in its abstract

integrity, I apprehend that no other considerations than the

above are properly of any value.

6. It is designed, in the next place, to give expression in this

treatise to the fundamental laws of reasoning in the symbolical

language of a Calculus. Upon this head it will suffice to say, that

those laws are such as to suggest this mode of expression, and

to give to it a peculiar and exclusive fitness for the ends in view.

There is not only a close analogy between the operations of the

mind in general reasoning and its operations in the particular

science of Algebra, but there is to a considerable extent an exact

agreement in the laws by which the two classes of operations are

conducted. Of course the laws must in both cases be determined

independently; any formal agreement between them can only be

established \textit{\`{a} posteriori} by actual comparison. To borrow the

notation of the science of Number, and then assume that in its

new application the laws by which its use is governed will remain

unchanged, would be mere hypothesis. There exist, indeed,

certain general principles founded in the very nature of language,

by which the use of symbols, which are but the elements of

scientific language, is determined. To a certain extent these

elements are arbitrary. Their interpretation is purely conventional:

we are permitted to employ them in whatever sense we

please. But this permission is limited by two indispensable conditions,--first,

that from the sense once conventionally established

we never, in the same process of reasoning, depart; secondly,

that the laws by which the process is conducted be founded exclusively

upon the above fixed sense or meaning of the symbols

employed. In accordance with these principles, any agreement

which may be established between the laws of the symbols of

Logic and those of Algebra can but issue in an agreement of processes.

The two provinces of interpretation remain apart and

independent, each subject to its own laws and conditions.

Now the actual investigations of the following pages exhibit

Logic, in its practical aspect, as a system of processes carried on

by the aid of symbols having a definite interpretation, and subject

to laws founded upon that interpretation alone. But at the

same time they exhibit those laws as identical in form with the

laws of the general symbols of algebra, with this single addition,

viz., that the symbols of Logic are further subject to a special

law (Chap, II.), to which the symbols of quantity, as such, are

not subject. Upon the nature and the evidence of this law it is not

purposed here to dwell. These questions will be fully discussed

in a future page. But as constituting the essential ground of

difference between those forms of inference with which Logic is

conversant, and those which present themselves in the particular

science of Number, the law in question is deserving of more

than a passing notice. It may be said that it lies at the very

foundation of general reasoning,--that it governs those intellectual

acts of conception or of imagination which are preliminary to

the processes of logical deduction, and that it gives to the processes

themselves much of their actual form and expression. It

may hence be affirmed that this law constitutes the germ or seminal

principle, of which every approximation to a general method

in Logic is the more or less perfect development.

7. The principle has already been laid down (5) that the

sufficiency and truly fundamental character of any assumed system

of laws in the science of Logic must partly be seen in the

perfection of the methods to which they conduct us. It remains,

then, to consider what the requirements of a general method in

Logic are, and how far they are fulfilled in the system of the present

work.

Logic is conversant with two kinds of relations,--relations

among things, and relations among facts. But as facts are expressed

by propositions, the latter species of relation may, at

least for the purposes of Logic, be resolved into a relation among

propositions. The assertion that the fact or event $A$ is an invariable

consequent of the fact or event $B$ may, to this extent at

least, be regarded as equivalent to the assertion, that the truth

of the proposition affirming the occurrence of the event $B$ always

implies the truth of the proposition affirming the occurrence of

the event $A$. Instead, then, of saying that Logic is conversant

with relations among things and relations among facts, we are

permitted to say that it is concerned with relations among things

and relations among propositions. Of the former kind of relations

we have an example in the proposition--``All men are mortal;''

of the latter kind in the proposition--``If the sun is totally

eclipsed, the stars will become visible.'' The one expresses a relation

between ``men'' and ``mortal beings,'' the other between

the elementary propositions--``The sun is totally eclipsed;''

``The stars will become visible.'' Among such relations I suppose

to be included those which affirm or deny existence with

respect to things, and those which affirm or deny truth with respect

to propositions. Now let those things or those propositions

among which relation is expressed be termed the elements of

the propositions by which such relation is expressed. Proceeding

from this definition, we may then say that the \textit{premises} of any

logical argument express \textit{given} relations among certain elements,

and that the conclusion must express an \textit{implied} relation among

those elements, or among a part of them, i.e. a relation implied

by or inferentially involved in the premises.

8. Now this being premised, the requirements of a general

method in Logic seem to be the following:--

1st. As the conclusion must express a relation among the

whole or among a part of the elements involved in the premises,

it is requisite that we should possess the means of eliminating

those elements which we desire not to appear in the conclusion,

and of determining the whole amount of relation implied by the

premises among the elements which we wish to retain. Those

elements which do not present themselves in the conclusion are,

in the language of the common Logic, called middle terms; and

the species of elimination exemplified in treatises on Logic consists

in deducing from two propositions, containing a common element

or middle term, a conclusion connecting the two remaining terms.

But the problem of elimination, as contemplated in this work,

possesses a much wider scope. It proposes not merely the elimination

of one middle term from two propositions, but the elimination

generally of middle terms from propositions, without

regard to the number of either of them, or to the nature of their

connexion. To this object neither the processes of Logic nor

those of Algebra, in their actual state, present any strict parallel.

In the latter science the problem of elimination is known to be

limited in the following manner:--From two equations we can

eliminate one symbol of quantity; from three equations two

symbols; and, generally, from $n$ equations $n-1$ symbols. But

though this condition, necessary in Algebra, seems to prevail in

the existing Logic also, it has no essential place in Logic as a

science. There, no relation whatever can be proved to prevail

between the number of terms to be eliminated and the number

of propositions from which the elimination is to be effected.

From the equation representing a single proposition, any number

of symbols representing terms or elements in Logic may be

eliminated; and from any number of equations representing propositions,

one or any other number of symbols of this kind may

be eliminated in a similar manner. For such elimination there

exists one general process applicable to all cases. This is one of

the many remarkable consequences of that distinguishing law of

the symbols of Logic, to which attention has been already

directed.

2ndly. It should be within the province of a general method

in Logic to express the final relation among the elements of the

conclusion by any admissible \textit{kind} of proposition, or in any selected

\textit{order} of terms. Among varieties of kind we may reckon

those which logicians have designated by the terms categorical,

hypothetical, disjunctive, \&c. To a choice or selection in the

order of the terms, we may refer whatsoever is dependent upon

the appearance of particular elements in the subject or in the

predicate, in the antecedent or in the consequent, of that proposition

which forms the ``conclusion.'' But waiving the language

of the schools, let us consider what really distinct species of

problems may present themselves to our notice. We have seen

that the elements of the final or inferred relation may either be

\textit{things} or \textit{propositions}. Suppose the former case; then it might

be required to deduce from the premises a definition or description

of some one thing, or class of things, constituting an element of

the conclusion in terms of the other things involved in it. Or

we might form the conception of some thing or class of things,

involving more than one of the elements of the conclusion, and

require its expression in terms of the other elements. Again,

suppose the elements retained in the conclusion to be propositions,

we might desire to ascertain such points as the following,

viz., Whether, in virtue of the premises, any of those propositions,

taken singly, are true or false?--Whether particular

combinations of them are true or false?--Whether, assuming a

particular proposition to be true, any consequences will follow,

and if so, what consequences, with respect to the other

propositions?--Whether any particular condition being assumed with

reference to certain of the propositions, any consequences, and

what consequences, will follow with respect to the others? and

so on. I say that these are general questions, which it should

fall within the scope or province of a general method in Logic to

solve. Perhaps we might include them all under this one statement

of the final problem of practical Logic. Given a set of

premises expressing relations among certain elements, whether

things or propositions: required explicitly the whole relation

consequent among \textit{any} of those elements under any proposed

conditions, and in any proposed form. That this problem, under

all its aspects, is resolvable, will hereafter appear. But it is not

for the sake of noticing this fact, that the above inquiry into the

nature and the functions of a general method in Logic has been

introduced. It is necessary that the reader should apprehend

what are the specific ends of the investigation upon which we

are entering, as well as the principles which are to guide us to

the attainment of them.

9. Possibly it may here be said that the Logic of Aristotle,

in its rules of syllogism and conversion, sets forth the elementary

processes of which all reasoning consists, and that beyond these

there is neither scope nor occasion for a general method. I have

no desire to point out the defects of the common Logic, nor do I

wish to refer to it any further than is necessary, in order to place

in its true light the nature of the present treatise. With this

end alone in view, I would remark:--1st. That syllogism, conversion,

\&c., are not the ultimate processes of Logic. It will

be shown in this treatise that they are founded upon, and are resolvable

into, ulterior and more simple processes which constitute

the real elements of method in Logic. Nor is it true in fact that

all inference is reducible to the particular forms of syllogism and

conversion.--\textit{Vide} Chap. xv. 2ndly. If all inference were reducible

to these two processes (and it has been maintained that

it is reducible to syllogism alone), there would still exist the

same necessity for a general method. For it would still be requisite

to determine in what order the processes should succeed

each other, as well as their particular nature, in order that the

desired relation should be obtained. By the desired relation I

mean that full relation which, in virtue of the premises, connects

any elements selected out of the premises at will, and which,

moreover, expresses that relation in any desired form and order.

If we may judge from the mathematical sciences, which are the

most perfect examples of method known, this \textit{directive} function

of Method constitutes its chief office and distinction. The fundamental

processes of arithmetic, for instance, are in themselves

but the elements of a possible science. To assign their nature is

the first business of its method, but to arrange their succession

is its subsequent and higher function. In the more complex

examples of logical deduction, and especially in those which form

a basis for the solution of difficult questions in the theory of

Probabilities, the aid of a directive method, such as a Calculus

alone can supply, is indispensable.

10. Whence it is that the ultimate laws of Logic are mathematical

in their form; why they are, except in a single point,

identical with the general laws of Number; and why in that particular

point they differ;--are questions upon which it might not

be very remote from presumption to endeavour to pronounce a

positive judgment. Probably they lie beyond the reach of our

limited faculties. It may, perhaps, be permitted to the mind to

attain a knowledge of the laws to which it is itself subject, without

its being also given to it to understand their ground and

origin, or even, except in a very limited degree, to comprehend

their fitness for their end, as compared with other and conceivable

systems of law. Such knowledge is, indeed, unnecessary for the

ends of science, which properly concerns itself with what is, and

seeks not for grounds of preference or reasons of appointment.

These considerations furnish a sufficient answer to all protests

against the exhibition of Logic in the form of a Calculus. It is

not because we choose to assign to it such a mode of manifestation,

but because the ultimate laws of thought render that mode

possible, and prescribe its character, and forbid, as it would

seem, the perfect manifestation of the science in any other form,

that such a mode demands adoption. It is to be remembered

that it is the business of science not to create laws, but to discover

them. We do not originate the constitution of our own minds,

greatly as it may be in our power to modify their character.

And as the laws of the human intellect do not depend upon our

will, so the forms of the science, of which they constitute the basis,

are in all essential regards independent of individual choice.

11. Beside the general statement of the principles of the

above method, this treatise will exhibit its application to the

analysis of a considerable variety of propositions, and of trains of

propositions constituting the premises of demonstrative arguments.

These examples have been selected from various writers,

they differ greatly in complexity, and they embrace a wide range

of subjects. Though in this particular respect it may appear to

some that too great a latitude of choice has been exercised, I do

not deem it necessary to offer any apology upon this account.

That Logic, as a science, is susceptible of very wide applications

is admitted; but it is equally certain that its ultimate forms and

processes are mathematical. Any objection \textit{\`{a} priori} which may

therefore be supposed to lie against the adoption of such forms

and processes in the discussion of a problem of morals or of general

philosophy must be founded upon misapprehension or false

analogy. It is not of the essence of mathematics to be conversant

with the ideas of number and quantity. Whether as a general

habit of mind it would be desirable to apply symbolical processes

to moral argument, is another question. Possibly, as I have

elsewhere observed,\footnote{Mathematical Analysis of Logic. London : G. Bell. 1847.}

the perfection of the method of Logic may

be chiefly valuable as an evidence of the speculative truth of its

principles. To supersede the employment of common reasoning,

or to subject it to the rigour of technical forms, would be the last

desire of one who knows the value of that intellectual toil and

warfare which imparts to the mind an athletic vigour, and teaches

it to contend with difficulties, and to rely upon itself in emergencies.

Nevertheless, cases may arise in which the value of a

scientific procedure, even in those things which fall confessedly

under the ordinary dominion of the reason, may be felt and acknowledged.

Some examples of this kind will be found in the

present work.

12. The general doctrine and method of Logic above explained

form also the basis of a theory and corresponding method

of Probabilities. Accordingly, the development of such a theory

and method, upon the above principles, will constitute a distinct

object of the present treatise. Of the nature of this application

it may be desirable to give here some account, more especially as

regards the character of the solutions to which it leads. In connexion

with this object some further detail will be requisite concerning

the forms in which the results of the logical analysis are

presented.

The ground of this necessity of a prior method in Logic, as

the basis of a theory of Probabilities, may be stated in a few

words. Before we can determine the mode in which the expected

frequency of occurrence of a particular event is dependent upon

the known frequency of occurrence of any other events, we must be

acquainted with the mutual dependence of the events themselves.

Speaking technically, we must be able to express the event

whose probability is sought, as a function of the events whose

probabilities are given. Now this explicit determination belongs

in all instances to the department of Logic. Probability, however,

in its mathematical acceptation, admits of numerical measurement.

Hence the subject of Probabilities belongs equally to

the science of Number and to that of Logic. In recognising the

co-ordinate existence of both these elements, the present treatise

differs from all previous ones; and as this difference not only

affects the question of the possibility of the solution of problems

in a large number of instances, but also introduces new and important

elements into the solutions obtained, I deem it necessary

to state here, at some length, the peculiar consequences of the

theory developed in the following pages.

13. The measure of the probability of an event is usually

defined as a fraction, of which the numerator represents the number

of cases favourable to the event, and the denominator the

whole number of cases favourable and unfavourable; all cases

being supposed equally likely to happen. That definition is

adopted in the present work. At the same time it is shown that

there is another aspect of the subject (shortly to be referred to)

which might equally be regarded as fundamental, and which

would actually lead to the same system of methods and conclusions.

It may be added, that so far as the received conclusions

of the theory of Probabilities extend, and so far as they are consequences

of its fundamental definitions, they do not differ from

the results (supposed to be equally correct in inference) of the

method of this work.

Again, although questions in the theory of Probabilities

present themselves under various aspects, and may be variously

modified by algebraical and other conditions, there seems to be

one general type to which all such questions, or so much of each

of them as truly belongs to the theory of Probabilities, may be

referred. Considered with reference to the \textit{data} and the \textit{qu\ae{}situm},

that type may be described as follows:---1st. The data are

the probabilities of one or more given events, each probability

being either that of the absolute fulfilment of the event to which

it relates, or the probability of its fulfilment under given supposed

conditions. 2ndly. The \textit{qu\ae{}situm}, or object sought, is the

probability of the fulfilment, absolutely or conditionally, of some

other event differing in expression from those in the data, but

more or less involving the same elements. As concerns the data,

they are either \textit{causally given},---as when the probability of a particular

throw of a die is deduced from a knowledge of the constitution

of the piece,---or they are derived from observation of

repeated instances of the success or failure of events. In the

latter case the probability of an event may be defined as the

limit toward which the ratio of the favourable to the whole number

of observed cases approaches (the uniformity of nature being

presupposed) as the observations are indefinitely continued.

Lastly, as concerns the nature or relation of the events in question,

an important distinction remains. Those events are either

\textit{simple} or \textit{compound}. By a compound event is meant one of

which the expression in language, or the conception in thought,

depends upon the expression or the conception of other events,

which, in relation to it, may be regarded as \textit{simple} events. To

say ``it rains,'' or to say ``it thunders,'' is to express the occurrence

of a simple event; but to say ``it rains and thunders,'' or

to say ``it either rains or thunders,'' is to express that of a compound

event. For the expression of that event depends upon

the elementary expressions, ``it rains,'' ``it thunders.'' The criterion

of simple events is not, therefore, any supposed simplicity

in their nature. It is founded solely on the mode of their expression

in language or conception in thought.

14. Now one general problem, which the existing theory of

Probabilities enables us to solve, is the following, viz.:---Given

the probabilities of any simple events: required the probability of

a given compound event, i.e. of an event compounded in a given

manner out of the given simple events. The problem can also

be solved when the compound event, whose probability is required,

is subjected to given conditions, i.e. to conditions dependent

also in a given manner on the given simple events.

Beside this general problem, there exist also particular problems

of which the principle of solution is known. Various questions

relating to \textit{causes} and \textit{effects} can be solved by known methods

under the particular hypothesis that the causes are mutually exclusive,

but apparently not otherwise. Beyond this it is not

clear that any advance has been made toward the solution of

what may be regarded as the general problem of the science, viz.:

Given the probabilities of any events, simple or compound, conditioned

or unconditioned: required the probability of any other

event equally arbitrary in expression and conception. In the

statement of this question it is not even postulated that the

events whose probabilities are given, and the one whose probability

is sought, should involve some common elements, because

it is the office of a method to determine whether the data of a

problem are sufficient for the end in view, and to indicate, when

they are not so, wherein the deficiency consists.

This problem, in the most unrestricted form of its statement,

is resolvable by the method of the present treatise; or, to speak

more precisely, its theoretical solution is completely given, and

its practical solution is brought to depend only upon processes

purely mathematical, such as the resolution and analysis of equations.

The order and character of the general solution may be

thus described.

15. In the first place it is always possible, by the preliminary

method of the Calculus of Logic, to express the event whose

probability is sought as a logical function of the events whose

probabilities are given. The result is of the following character:

Suppose that $X$ represents the event whose probability is sought,

$A$, $B$, $C$, \&c. the events whose probabilities are given, those

events being either simple or compound. Then the \textit{whole} relation

of the event $X$ to the events $A$, $B$, $C$, \&c. is deduced in the

form of what mathematicians term a \textit{development}, consisting, in

the most general case, of four distinct classes of terms. By the

first class are expressed those combinations of the events $A$, $B$, $C$,

which both necessarily accompany and necessarily indicate the

occurrence of the event $X$; by the second class, those combinations

which necessarily accompany, but do not necessarily imply,

the occurrence of the event $X$; by the third class, those combinations

whose occurrence in connexion with the event $X$ is impossible,

but not otherwise impossible; by the fourth class,

those combinations whose occurrence is impossible under any circumstances.

I shall not dwell upon this statement of the result

of the logical analysis of the problem, further than to remark

that the elements which it presents are precisely those by which

the expectation of the event $X$, as dependent upon our knowledge

of the events $A$, $B$, $C$, is, or alone can be, affected. General

reasoning would verify this conclusion; but general reasoning

would not usually avail to disentangle the complicated web

events and circumstances from which the solution above described

must be evolved. The attainment of this object constitutes

the first step towards the complete solution of the question I

proposed. It is to be noted that thus far the process of solution

is logical, i. e. conducted by symbols of logical significance, and

resulting in an equation interpretable into a \textit{proposition}. Let this

result be termed the \textit{final logical equation}.

The second step of the process deserves attentive remark.

From the final logical equation to which the previous step has

conducted us, are deduced, by inspection, a series of algebraic

equations implicitly involving the complete solution of the problem

proposed. Of the mode in which this transition is effected

let it suffice to say, that there exists a definite relation between

the laws by which the probabilities of events are expressed as

algebraic functions of the probabilities of other events upon which

they depend, and the laws by which the logical connexion of

the events is itself expressed. This relation, like the other coincidences

of formal law which have been referred to, is not

founded upon hypothesis, but is made known to us by observation

(I.4), and reflection. If, however, its reality were assumed \textit{\`{a} priori}

as the basis of the very definition of Probability, strict deduction

would thence lead us to the received numerical definition as a

necessary consequence. The Theory of Probabilities stands, as

it has already been remarked (I.12), in equally close relation to

Logic and to Arithmetic; and it is indifferent, so far as results

are concerned, whether we regard it as springing out of the latter

of these sciences, or as founded in the mutual relations which

connect the two together.

16. There are some circumstances, interesting perhaps to the

mathematician, attending the general solutions deduced by the

above method, which it may be desirable to notice.

1st. As the method is independent of the number and the

nature of the data, it continues to be applicable when the latter

are insufficient to render determinate the value sought. When

such is the case, the final expression of the solution will contain

terms with arbitrary constant coefficients. To such terms there

will correspond terms in the final logical equation (I. 15), the

interpretation of which will inform us what new data are requisite

in order to determine the values of those constants, and

thus render the numerical solution complete. If such data are

not to be obtained, we can still, by giving to the constants their

limiting values $0$ and $1$, determine the limits within which the

probability sought must lie independently of all further experience.

When the event whose probability is sought is \textit{quite} independent

of those whose probabilities are given, the limits thus

obtained for its value will be $0$ and $1$, as it is evident that they

ought to be, and the interpretation of the constants will only

lead to a re-statement of the original problem.

2ndly. The expression of the final solution will in all cases

involve a particular element of quantity, determinable by the solution

of an algebraic equation. Now when that equation is of

an elevated degree, a difficulty may seem to arise as to the selection

of the proper root. There are, indeed, cases in which

both the elements given and the element sought are so obviously

restricted by necessary conditions that no choice remains. But

in complex instances the discovery of such conditions, by unassisted

force of reasoning, would be hopeless. A distinct method

is requisite for this end,---a method which might not

appropriately be termed the Calculus of Statistical Conditions,

into the nature of this method I shall not here further enter

than to say, that, like the previous method, it is based upon the

employment of the ``final logical equation,'' and that it definitely

assigns, 1st, the conditions which must be fulfilled among the

numerical elements of the data, in order that the problem may

be real, i.e. derived from a \textit{possible experience}; 2ndly, the numerical

limits, within which the probability sought must have

been confined, if, instead of being determined by theory, it had

been deduced directly by observation from the same system of

ph{\ae}nomena from which the data were derived. It is clear that

these limits will be actual limits of the probability sought.

Now, on supposing the data subject to the conditions above assigned

to them, it appears in every instance which I have examined

that there exists one root, and only one root, of the final

algebraic equation which is subject to the required limitations.

Every source of ambiguity is thus removed. It would even seem

that new truths relating to the theory of algebraic equations

are thus incidentally brought to light. It is remarkable that

the special element of quantity, to which the previous discussion

relates, depends only upon the \textit{data}, and not at all upon the

\textit{qu\ae{}situm} of the problem proposed. Hence the solution of each

particular problem unties the knot of difficulty for a system of

problems, viz., for that system of problems which is marked by

the possession of common data, independently of the nature of

their \textit{qu\ae{}sita}. This circumstance is important whenever from a

particular system of data it is required to deduce a series of connected

conclusions. And it further gives to the solutions of

particular problems that character of relationship, derived from

their dependence upon a central and fundamental unity, which

not unfrequently marks the application of general methods.

17. But though the above considerations, with others of a

like nature, justify the assertion that the method of this treatise,

for the solution of questions in the theory of Probabilities, is a

general method, it does not thence follow that we are relieved in

all cases from the necessity of recourse to hypothetical grounds.

It has been observed that a solution may consist entirely of terms

affected by arbitrary constant coefficients,---may, in fact, be

wholly indefinite. The application of the method of this work to

some of the most important questions within its range would--were

the data of experience alone employed--present results of

this character. To obtain a \textit{definite} solution it is necessary, in

such cases, to have recourse to hypotheses possessing more or less

of independent probability, but incapable of exact verification.

Generally speaking, such hypotheses will differ from the immediate

results of experience in partaking of a logical rather than of a

numerical character; in prescribing the conditions under which

ph{\ae}nomena occur, rather than assigning the relative frequency

of their occurrence. This circumstance is, however, unimportant.

Whatever their nature may be, the hypotheses assumed must

thenceforth be regarded as belonging to the actual data, although

tending, as is obvious, to give to the solution itself somewhat of

a hypothetical character. With this understanding as to the

possible sources of the data actually employed, the method is

perfectly general, but for the correctness of the hypothetical elements

introduced it is of course no more responsible than for the

correctness of the numerical data derived from experience.

In illustration of these remarks we may observe that the

theory of the reduction of astronomical observations\footnote{

The author designs to treat this subject either in a separate work or in a

future Appendix. In the present treatise he avoids the use of the integral

calculus.}

rests, in

part, upon hypothetical grounds. It assumes certain positions

as to the nature of error, the equal probabilities of its occurrence

in the form of excess or defect, \&c., without which it would be

impossible to obtain any \textit{definite} conclusions from a system of

conflicting observations. But granting such positions as the

above, the residue of the investigation falls strictly within the

province of the theory of Probabilities. Similar observations

apply to the important problem which proposes to deduce from

the records of the majorities of a deliberative assembly the mean

probability of correct judgment in one of its members. If the

method of this treatise be applied to the mere numerical data,

the solution obtained is of that wholly indefinite kind above described.

And to show in a more eminent degree the insufficiency

of those data by themselves, the interpretation of the arbitrary

constants (I. 16) which appear in the solution, merely produces

a re-statement of the original problem. Admitting, however,

the hypothesis of the independent formation of opinion in the

individual mind, either absolutely, as in the speculations of

Laplace and Poisson, or under limitations imposed by the actual

data, as will be seen in this treatise, Chap. XXI., the problem assumes

a far more definite character. It will be manifest that the

ulterior value of the theory of Probabilities must depend very

much upon the correct formation of such mediate hypotheses,

where the purely experimental data are insufficient for \textit{definite}

solution, and where that further experience indicated by the interpretation

of the final logical equation is unattainable. Upon

the other hand, an undue readiness to form hypotheses in subjects

which from their very nature are placed beyond human

ken, must re-act upon the credit of the theory of Probabilities,

and tend to throw doubt in the general mind over its most legitimate

conclusions.

18. It would, perhaps, be premature to speculate here upon

the question whether the methods of abstract science are likely at

any future day to render service in the investigation of social

problems at all commensurate with those which they have rendered

in various departments of physical inquiry. An attempt

to resolve this question upon pure \textit{\`{a} priori} grounds of reasoning

would be very likely to mislead us. For example, the consideration

of human free-agency would seem at first sight to preclude

the idea that the movements of the social system should ever manifest

that character of orderly evolution which we are prepared

to expect under the reign of a physical necessity. Yet already

do the researches of the statist reveal to us facts at variance with

such an anticipation. Thus the records of crime and pauperism

present a degree of regularity unknown in regions in which the

disturbing influence of human wants and passions is unfelt. On

the other hand, the distemperature of seasons, the eruption of

volcanoes, the spread of blight in the vegetable, or of epidemic

maladies in the animal kingdom, things apparently or chiefly the

product of natural causes, refuse to be submitted to regular and

apprehensible laws. ``Fickle as the wind,'' is a proverbial expression.

Reflection upon these points teaches us in some degree

to correct our earlier judgments. We learn that we are not to

expect, under the dominion of necessity, an order perceptible to

human observation, unless the play of its producing causes is

sufficiently simple; nor, on the other hand, to deem that free

agency in the individual is inconsistent with regularity in the

motions of the system of which he forms a component unit.

Human freedom stands out as an apparent fact of our consciousness,

while it is also, I conceive, a highly probable deduction of

analogy (Chap, XXII.) from the nature of that portion of the

mind whose scientific constitution we are able to investigate.

But whether accepted as a fact reposing on consciousness, or as

a conclusion sanctioned by the reason, it must be so interpreted

as not to conflict with an established result of observation, viz.:

that ph\ae{}nomena, in the production of which large masses of men

are concerned, do actually exhibit a very remarkable degree of

regularity, enabling us to collect in each succeeding age the elements

upon which the estimate of its state and progress, so far

as manifested in outward results, must depend. There is thus no

sound objection \textit{\`{a} priori} against the possibility of that species of

data which is requisite for the experimental foundation of a

science of social statistics. Again, whatever other object this

treatise may accomplish, it is presumed that it will leave no

doubt as to the existence of a system of abstract principles and of

methods founded upon those principles, by which any collective

body of social data may be made to yield, in an explicit form,

whatever information they implicitly involve. There may, where

the data are exceedingly complex, be very great difficulty in obtaining

this information,---difficulty due not to any imperfection

of the theory, but to the laborious character of the analytical

processes to which it points. It is quite conceivable that in many

instances that difficulty may be such as only united effort could