George Boole

In 1847, George Boole published one of the founding documents of modern algebraic or symbolic logic, The Mathematical Analysis of Logic.

Augustus DeMorgan's Formal Logic, Or the Calculus of Inference appeared the same year. In the next few decades, Charles Sanders Peirce would develop many of Boole's ideas into modern quantification logic, but the core of the theory was already in Boole.

It appeared to me that, although Logic might be viewed with reference to the idea of quantity,* it had also another and a deeper system of relations. If it was lawful to regard it from without, as connecting itself through the medium of Number with the intuitions of Space and Time, it was lawful also to regard it from within, as based upon facts of another order which have their abode in the constitution of the Mind...

THEY who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation...

Boole's "universe" is the universe of discourse, the set of all true or false propositions
LET US employ the symbol 1, or unity, to represent the Universe, and let us understand it as comprehending every conceivable class of objects whether actually existing or not, it being premised that the same individual may be found in more than one class, inasmuch as it may possess more than one quality in common with other individuals. Let us employ the letters X, Y, Z, to represent the individual members of classes, X applying to every member of one class, as members of that particular class, and Y to every member of another class as members of such class, and so on, according to the received language of treatises on Logic.

Further let us conceive a class of symbols x, y, z, possessed of the following character.

The symbol x operating upon any subject comprehending individuals or classes, shall be supposed to select from that subject all the Xs which it contains. In like manner the symbol y, operating upon any subject, shall be supposed to select from it all individuals of the class Y which are comprised in it, and so on. When no subject is expressed, we shall suppose 1 (the Universe) to be the subject understood, so that we shall have

x = x           (1).

the meaning of either term being the selection from the Universe of all the Xs which it contains, and the result of the operation being in common language, the class X, i. e. the class of which each member is an X...

In virtue of the principle, that a Proposition is either true or false, every elective symbol employed in the expression of hypotheticals admits only of the values 0 and 1, which are the only quantitative forms of an elective symbol. It is in fact possible, setting out from the theory of Probabilities (which is purely quantitative), to arrive at a system of methods and processes for the treatment of hypotheticals exactly similar to those which have been given.

Boole is aware that contingency requires probability values between 0 and 1
The two systems of elective symbols and of quantity osculate, if I may use the expression, in the points 0 and 1. It seems to me to be implied by this, that unconditional truth (categoricals) and probable truth meet together in the constitution of contingent truth (hypotheticals). The general doctrine of elective symbols and all the more characteristic applications are quite independent of any quantitative origin.
Nature of the Work,
Chapter I, p.20-23 of The Laws of Thought

Boole's The Laws of Thought (on which are founded the mathematical theories of logic and probabilities) was an enormously influential work. Writing in the heyday of Quételet and Buckle, Boole concluded that

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. (p.20)

If we regard the intellect as free, and this is apparently the view most in accordance with the general spirit of these speculations, its freedom must be viewed as opposed to the dominion of necessity, not to the existence of a certain just supremacy of truth.

I would especially direct attention to that view of the constitution of the intellect which represents it as subject to laws determinate in their character, but not operating by the power of necessity; which exhibits it as redeemed from the dominion of fate, without being abandoned to the lawlessness of chance. (p.420)

Nature of the Work,
Chapter I, p.20-23 of The Laws of Thought
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 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 phaenomena, 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 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 overcome. But that we possess theoretically in all cases, and practically, so far as the requisite labour of calculation may be supplied, the means of evolving from statistical records the seeds of general truths which lie buried amid the mass of figures, is a position which may, I conceive, with perfect safety be affirmed.

19. But beyond these general positions I do not venture to speak in terms of assurance. Whether the results which might be expected from the application of scientific methods to statistical records, over and above those the discovery of which requires no such aid, would so far compensate for the labour involved as to render it worth while to institute such investigations upon a proper scale of magnitude, is a point which could, perhaps, only be determined by experience. It is to be desired, and it might without great presumption be expected, that in this, as in other instances, the abstract doctrines of' science should minister to more than intellectual gratification. Nor, viewing the apparent order in which the sciences have been evolved, and have successively contributed their aid to the service of mankind, does it seem very improbable that a day may arrive in which similar aid may accrue from departments of the field of knowledge yet more intimately allied with the elements of human welfare. Let the speculations of this treatise, however, rest at present simply upon their claim to be regarded as true.

20. I design, in the last place, to endeavour to educe from the scientific results of the previous inquiries some general intimations respecting the nature and constitution of the human mind. Into the grounds of the possibility of this species of inference it is not necessary to enter here. One or two general observations may serve to indicate the track which I shall endeavour to follow. It cannot but be admitted that our views of the science of Logic must materially influence, perhaps mainly determine, our opinions upon the nature of the intellectual faculties. For example, the question whether reasoning consists merely in the application of certain first or necessary truths, with which the mind has been originally imprinted, or whether the mind is itself a seat of law, whose operation is as manifest and as conclusive in the particular as in the general formula, or whether, as some not undistinguished writers seem to maintain, all reasoning is of particulars; this question, I say, is one which not merely affects the science of Logic, but also concerns the formation of just views of the constitution of the intellectual faculties. Again, if it is concluded that the mind is by original constitution a seat of law, the question of the nature of its subjection to this law, — whether, for instance, it is an obedience founded upon necessity, like that which sustains the revolutions of the heavens, and preserves the order of Nature, — or whether it is a subjection of some quite distinct kind, is also a matter of deep speculative interest. Further, if the mind is truly determined to be a subject of law, and if its laws also are truly assigned, the question of their probable or necessary influence upon the course of human thought in different ages is one invested with great importance, and well deserving a patient investigation, as matter both of philosophy and of history. These and other questions I propose, however imperfectly, to discuss in the concluding portion of the present work. They belong, perhaps, to the domain of probable or conjectural, rather than to that of positive, knowledge. But it may happen that where there is not sufficient warrant for the certainties of science, there may be grounds of analogy adequate for the suggestion of highly probable opinions. It has seemed to me better that this discussion should be entirely reserved for the sequel of the main business of this treatise, — which is the investigation of scientific truths and laws. Experience sufficiently instructs us that the proper order of advancement in all inquiries after truth is to proceed from the known to the unknown. There are parts, even of the philosophy and constitution of the human mind, which have been placed fully within the reach of our investigation. To make a due acquaintance with those portions of our nature the basis of all endeavours to penetrate amid the shadows and uncertainties of that conjectural realm which lies beyond and above them, is the course most accordant with the limitations of our present condition.

On the Theory of Probabilities,
Chapter XVI, p.243-246 of The Laws of Thought
1. BEFORE the expiration of another year just two centuries will have rolled away since Pascal solved the first known question in the theory of Probabilities, and laid, in its solution, the foundations of a science possessing no common share of the attraction which belongs to the more abstract of mathematical speculations. The problem which the Chevalier de Mere, a reputed gamester, proposed to the recluse of Port Royal (not yet withdrawn from the interests of science* by the more distracting contemplation of the " greatness and the misery of man"), was the first of a long series of problems, destined to call into existence new methods in mathematical analysis, and to render valuable service in the practical concerns of life. Nor does the interest of the subject centre merely in its mathematical connexion, or its associations of utility. The attention is repaid which is devoted to the theory of Probabilities as an independent object of speculation, — to the fundamental modes in which it has been conceived, — to the great secondary principles which, as in the contemporaneous science of Mechanics, have gradually been annexed to it, — and, lastly, to the estimate of the measure of perfection which has been actually attained. I speak here of that perfection which consists in unity of conception and harmony of processes. Some of these points it is designed very briefly to consider in the present chapter.

2. A distinguished writer (Poisson) has thus stated the fundamental definitions of the science:

"The probability of an event is the reason we have to believe that it has taken place, or that it will take place."

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible" (equally likely to happen).

From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand fora theory of probabilities.

3. Though our expectation of an event grows stronger with the increase of the ratio of the number of the known cases favourable to its occurrence to the whole number of equally possible cases, favourable or unfavourable, it would be unphilosophical to affirm that the strength of that expectation, viewed as an emotion of the mind, is capable of being referred to any numerical standard. The man of sanguine temperament builds high hopes where the timid despair, and the irresolute are lost in doubt. As subjects of scientific inquiry, there is some analogy between opinion and sensation. The thermometer and the carefully prepared photographic plate indicate, not the intensity of the sensations of heat and light, but certain physical circumstances which accompany the production of those sensations. So also the theory of probabilities contemplates the numerical measure of the circumstances upon which expectation is founded; and this object embraces the whole range of its legitimate applications. The rules which we employ in life-assurance, and in the other statistical applications of the theory of probabilities, are altogether independent of the mental phaenomena of expectation. They are founded upon the assumption that the future will bear a resemblance to the past; that under the same circumstances the same event will tend to recur with a definite numerical frequency; not upon any attempt to submit to calculation the strength of human hopes and fears.

Now experience actually testifies that events of a given species do, under given circumstances, tend to recur with definite frequency, whether their true causes be known to us or unknown. Of course this tendency is, in general, only manifested when the area of observation is sufficiently large. The judicial records of a great nation, its registries of births and deaths, in relation to age and sex, &c., present a remarkable uniformity from year to year. In a given language, or family of languages, the same sounds, and successions of sounds, and, if it be a written language, the same characters and successions of characters recur with determinate frequency. The key to the rude Ogham inscriptions, found in various parts of Ireland, and in which no distinction of words could at first be traced, was, by a strict application of this principle, recovered. The same method, it is understood, has been applied to the deciphering of the cuneiform records recently disentombed from the ruins of Nineveh by the enterprise of Mr. Layard.

4. Let us endeavour from the above statements and definitions to form a conception of the legitimate object of the theory of Probabilities.

Probability, it has been said, consists in the expectation founded upon a particular kind of knowledge, viz., the knowledge of the relative frequency of occurrence of events. Hence the probabilities of events, or of combinations of events, whether deduced from a knowledge of the particular constitution of things under which they happen, or derived from the long-continued observation of a past series of their occurrences and failures, constitute, in all cases, our data. The probability of some connected event, or combination of events, constitutes the corresponding quaesitum, or object sought. Now in the most general, yet strict meaning of the term "event," every combination of events constitutes also an event. The simultaneous occurrence of two or more events, or the occurrence of an event under given conditions, or in any conceivable connexion with other events, is still an event. Using the term in this liberty of application, the object of the theory of probabilities might be thus defined. Given the probabilities of any events, of whatever kind, to find the probability of some other event connected with them.

Constitution of the Intellect,
Chapter XXII, p.420-424 of The Laws of Thought
9. Refraining from the further prosecution of a train of thought which to some may appear to be of too speculative a character, let us briefly review the positive results to which we have been led. It has appeared that there exist in our nature faculties which enable us to ascend from the particular facts of experience to the general propositions which form the basis of Science; as well as faculties whose office it is to deduce from general propositions accepted as true the particular conclusions which they involve. It has been seen, that those faculties are subject in their operations to laws capable of precise scientific expression, but invested with an authority which, as contrasted with the authority of the laws of nature, is distinct, sui generis, and underived. Further, there has appeared to be a manifest fitness between the intellectual procedure thus made known to us, and the conditions of that system of things by which we are surrounded, — such conditions, I mean, as the existence of species connected by general resemblances, of facts associated under general laws; together with that union of permanency with order, which while it gives stability to acquired knowledge, lays a foundation for the hope of indefinite progression. Human nature, quite independently of its observed or manifested tendencies, is seen to be constituted in a certain relation to Truth; and this relation, considered as a subject of speculative knowledge, is as capable of being studied in its details, is, moreover, as worthy of being so studied, as are the several departments of physical science, considered in the same aspect. I would especially direct attention to that view of the constitution of the intellect which represents it as subject to laws determinate in their character, but not operating by the power of necessity; which exhibits it as redeemed from the dominion of fate, without being abandoned to the lawlessness of chance. We cannot embrace this view without accepting at least as probable the intimations which, upon the principle of analogy, it seems to furnish respecting another and a higher aspect of our nature, — its subjection in the sphere of duty as well as in that of knowledge to fixed laws whose authority does not consist in power, — its constitution with reference to an ideal standard and a final purpose. It has been thought, indeed, that scientific pursuits foster a disposition either to overlook the specific differences between the moral and the material world, or to regard the former as in no proper sense a subject for exact knowledge. Doubtless all exclusive pursuits tend to produce partial views, and it may be, that a mind long and deeply immersed in the contemplation of scenes over which the dominion of a physical necessity is unquestioned and supreme, may admit with difficulty the possibility of another order of things. But it is because of the exclusiveness of this devotion to a particular sphere of knowledge, that the prejudice in question takes possession, if at all, of the mind. The application of scientific methods to the study of the intellectual phaenomena, conducted in an impartial spirit of inquiry, and without overlooking those elements of error and disturbance which must be accepted as facts, though they cannot be regarded as laws, in the constitution of our nature, seems to furnish the materials of a juster analogy.

10. If it be asked to what practical end such inquiries as the above point, it may be replied, that there exist various objects, in relation to which the courses of men's actions are mainly determined by their speculative views of human nature. Education, considered in its largest sense, is one of those objects. The ultimate ground of all inquiry into its nature and its methods must be laid in some previous theory of what man is, what are the ends for which his several faculties were designed, what are the motives which have power to influence them to sustained action, and to elicit their most perfect and most stable results. It may be doubted, whether these questions have ever been considered fully, and at the same time impartially, in the relations here suggested. The highest cultivation of taste by the study of the pure models of antiquity, the largest acquaintance with the facts and theories of modern physical science, viewed from this larger aspect of our nature, can only appear as parts of a perfect intellectual discipline. Looking from the same point of view upon the means to be employed, we might be led to inquire, whether that all but exclusive appeal is made in the present day to the spirit of emulation or cupidity, does not tend to weaken the influence of those more enduring motives which seem to have been implanted in our nature for the immediate end in view. Upon these, and upon many other questions, the just limits of authority, the reconciliation of freedom of thought with discipline of feelings, habits, manners, and upon the whole moral aspect of the question,—what unfixedness of opinion, what diversity of practice, do we meet with ! Yet, in the sober view of reason, there is no object within the compass of human endeavours which is of more weight and moment than this, considered, as I have said, in its largest meaning. Now, whatsoever tends to make more exact and definite our view of human nature, in any of its real aspects, tends, in the same proportion, to reduce these questions into narrower compass, and restrict the limits of their possible solution. Thus may even speculative inquiries prove fruitful of the most important principles of action.

11. Perhaps the most obviously legitimate bearing of such speculations would be upon the question of the place of Mathematics in the system of human knowledge, and the nature and office of mathematical studies, as a means of intellectual discipline. No one who has attended to the course of recent discussions can think this question an unimportant one. Those who have maintained that the position of Mathematics is in both respects a fundamental one, have drawn one of their strongest arguments from the actual constitution of things. The material frame is subject in all its parts to the relations of number. All dynamical, chemical, electrical, thermal, actions, seem not only to be measurable in themselves, but to be connected with each other, even to the extent of mutual convertibility, by numerical relations of a perfectly definite kind. But the opinion in question seems to me to rest upon a deeper basis than this. The laws of thought, in all its processes of conception and of reasoning, in all those operations of which language is the expression or the instrument, are of the same kind as are the laws of the acknowledged processes of Mathematics. It is not contended that it is necessary for us to acquaint ourselves with those laws in order to think coherently, or, in the ordinary sense of the terms, to reason well. Men draw inferences without any consciousness of those elements upon which the entire procedure depends. Still less is it desired to exalt the reasoning faculty over the faculties of observation, of reflection, and of judgment. But upon the very ground that human thought, traced to its ultimate elements, reveals itself in mathematical forms, we have a presumption that the mathematical sciences occupy, by the constitution of our nature, a fundamental place in human knowledge, and that no system of mental culture can be complete or fundamental, which altogether neglects them.

But the very same class of considerations shows with equal force the error of those who regard the study of Mathematics, and of their applications, as a sufficient basis either of knowledge or of discipline. If the constitution of the material frame is mathematical, it is not merely so. If the mind, in its capacity of formal reasoning, obeys, whether consciously or unconsciously, mathematical laws, it claims through its other capacities of sentiment and action, through its perceptions of beauty and of moral fitness, through its deep springs of emotion and affection, to hold relation to a different order of things. There is, moreover, a breadth of intellectual vision, a power of sympathy with truth in all its forms and manifestations, which is not measured by the force and subtlety of the dialectic faculty. Even the revelation of the material universe in its boundless magnitude, and pervading order, and constancy of law, is not necessarily the most fully apprehended by him who has traced with minutest accuracy the steps of the great demonstration. And if we embrace in our survey the interests and duties of life, how little do any processes of mere ratiocination enable us to comprehend the weightier questions which they present ! As truly, therefore, as the cultivation of the mathematical or deductive faculty is a part of intellectual discipline, so truly is it only a part. The prejudice which would either banish or make supreme any one department of knowledge or faculty of mind, betrays not only error of judgment, but a defect of that intellectual modesty which is inseparable from a pure devotion to truth. It assumes the office of criticising a constitution of things which no human appointment has established, or can annul. It sets aside the ancient and just conception of truth as one though manifold. Much of this error, as actually existent among us, seems due to the special and isolated character of scientific teaching — which character it, in its turn, tends to foster. The study of philosophy, notwithstanding a few marked instances of exception, has failed to keep pace with the advance of the several departments of knowledge, whose mutual relations it is its province to determine. It is impossible, however, not to contemplate the particular evil in question as part of a larger system, and connect it with the too prevalent view of knowledge as a merely secular thing, and with the undue predominance, already adverted to, of those motives, legitimate within their proper limits, which are founded upon a regard to its secular advantages. In the extreme case it is not difficult to see that the continued operation of such motives, uncontrolled by any higher principles of action, uncorrected by the personal influence of superior minds, must tend to lower the standard of thought in reference to the objects of knowledge, and to render void and ineffectual whatsoever elements of a noble faith may still survive. And ever in proportion as these conditions are realized must the same effects follow. Hence, perhaps, it is that we sometimes find juster conceptions of the unity, the vital connexion, and the subordination to a moral purpose, of the different parts of Truth, among those who acknowledge nothing higher than the changing aspect of collective humanity, than among those who profess an intellectual allegiance to the Father of Lights. But these are questions which cannot further be pursued here. To some they will appear foreign to the professed design of this work. But the consideration of them has arisen naturally, either out of the speculations which that design involved, or in the course of reading and reflection which seemed necessary to its accomplishment.

References

"The Calculus of Logic", Cambridge and Dublin Mathematical Journal Vol. III (1848), pp. 183-98

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