Logic is the art of thinking and reasoning justly; or, it may be defined the science or history of the human mind, inasmuch as it traces the progress of our knowledge from our first and most simple conceptions through all their different combinations, and all those numerous deductions that result from variously comparing them one with another.

The precise business of logic therefore is, To explain the nature of the human mind, and the proper manner of conducting its several powers, in order to the attainment Part I.

ment of truth and knowledge. It lays open those errors Perception, and mistakes we are apt, through inattention, to run into; and teaches us how to distinguish between truth, and what only carries the appearance of it. By these means we grow acquainted with the nature and force of the understanding; see what things lie within its reach; where we may attain certainty and demonstra- tion; and when we must be contented with probability.

This science is generally divided into four parts, viz. Perception, Judgement, Reasoning, and Method. This division comprehends the whole history of the sensations and operations of the human mind.

PART I. OF PERCEPTION.

WE find ourselves surrounded with a variety of ob- jects, which acting differently upon our senses, con- vey distinct impressions into the mind, and thereby rouse the attention and notice of the understanding. By reflecting too on what passes within us, we become conscious of the operations of our own minds, and at- tend to them as a new set of impressions. But in all this there is only bare consciousness. The mind, with- out proceeding any farther, takes notice of the im- pressions that are made upon it, and views things in order, as they present themselves one after another. This attention of the understanding to the object act- ing upon it, whereby it becomes sensible of the im- pressions they make, is called by logicians perception; and the notices themselves, as they exist in the mind, are there treasured up to be the materials of think- ing and knowledge, are distinguished by the name of ideas. In the article Metaphysics it shall be shown at large, how the mind, being furnished with ideas, contrives to diversify and enlarge its stock: we have here chiefly to consider the means of making known our thoughts to others; that we may not only understand how knowledge is acquired, but also in what manner it may be communicated with the greatest certainty and advantage.

CHAP. I. Of Words, considered as the signs of our Ideas.

I. Our ideas, though manifold and various, are ne- vertheless all within our own breasts, invisible to oth- ers, nor can of themselves be made appear. But God, designing us for society, and to have fellowship with those of our kind, has provided us with organs fitted to frame articulate sounds, and given us also a capacity of using those sounds as signs of internal con- ceptions. Hence spring words and language: for, having once pitched upon any sound to stand as the mark of an idea in the mind, custom by degrees estab- lishes such a connexion between them, that the ap- pearance of the idea in the understanding always brings to our remembrance the sound or name by which it is expressed; as in like manner the hearing of the sound never fails to excite the idea for which it is made to stand. And thus it is easy to conceive how a man may record his own thoughts, and bring them again into view in any succeeding period of life. For this connexion being once settled, as the same sounds will always serve to excite the same ideas; if he can but contrive to register his words in the order and dispo- sition in which the present train of his thoughts pre- sent themselves to his imagination, it is evident he will be able to recall these thoughts at pleasure, and that too in the very manner of their first appearance. Ac- cordingly we find, that the inventions of writing and printing, by enabling us to fix and perpetuate such perishable things as sounds, have also furnished us with the means of giving a kind of permanency to the transactions of the mind, insomuch that they may be in the same manner subjected to our review as any other objects of nature.

II. But besides the ability of recording our own thoughts, there is this farther advantage in the use of mutual external signs, that they enable us to communicate com- munication of our thoughts to others, and also to receive information and knowledge of what passes in their breasts. For any number of men, from one having agreed to establish the same sounds as signs of man to ano- the same ideas, it is apparent that the repetition of other these sounds must excite the like perceptions in each, and create a perfect correspondence of thoughts. When, for instance, any train of ideas succeed one another in my mind, if the names by which I am wont to express them have been annexed by those with whom I converse to the very same set of ideas, nothing is more evident, than that, by repeating those names according to the tenor of my present concep- tions, I shall raise in their minds the same course of thought as has taken possession of my own. For by barely attending to what passes within themselves upon hearing the sounds which I repeat, they will al- so become acquainted with the ideas in my under- standing, and have them in a manner laid before their view. So that we here clearly perceive how a man may communicate his sentiments, knowledge, and dis- coveries to others, if the language in which he con- verses be extensive enough to mark all the ideas and transactions of his mind. But as this is not always the case, and men are often obliged to invent terms of their own to express new views and concep- tions of things; it may be asked, how in these cir- cumstances we can become acquainted with the thoughts of another, when he makes use of words, to which we have never annexed any ideas, and that of course can raise no perceptions in our minds? In order to un- veil this mystery, and give some little insight into the foundation, growth, and improvement of language, the following observations will be found of consider- able moment.

III. First, That no word can be to any man the sim- plest sign of an idea, till that idea comes to have a real ex- istence in his mind. For names, being only so far in- telligible as they denote known internal concep- tions, mind by where they have none such to answer them, there words, or they are plainly sounds without signification, and of a descrip- course convey no instruction or knowledge. But no- foonner are the ideas to which they belong raised in the understanding, than, finding it easy to connect them with the established names, we can join in any agree- ment of this kind made by others, and thereby enjoy the benefit of their discoveries. The first thing therefore to be considered is, how these ideas may be conveyed into the mind; that being there, we may learn to connect them with their appropriated sounds, and so become capable of understanding others when they make use of these sounds in laying open and communicating their thoughts. Now, to comprehend this distinctly, it will be necessary to attend to the division of our ideas into simple and complex, (see Metaphysics.) And first, as for our simple ideas; they can find no admission into the mind, but by the two original fountains of knowledge, sensation and reflection. If therefore any of these have as yet no being in the understanding, it is impossible by words or a description to excite them there. A man who had never felt the sensation of heat, could not be brought to comprehend that sensation by any thing we might say to explain it. If we would really produce the idea in him, it must be by applying the proper object to his senses, and bringing him within the influence of a hot body. When this is done, and experience has taught him the perception to which men have annexed the name heat, it then becomes to him the sign of that idea, and he thenceforth understands the meaning of the term, which, before, all the words in this world would not have been sufficient to convey into his mind. The case is the same in respect of light and colours. A man born blind, and thereby deprived of the only conveyance for the ideas of this class, can never be brought to understand the names by which they are expressed. The reason is plain: they stand for ideas that have no existence in his mind; and as the organ appropriated to their reception is wanting, all other contrivances are vain, nor can they by any force or description be raised in his imagination. But it is quite otherwise in our complex notions. For these being no more than certain combinations of simple ideas, put together in various forms; if the original ideas out of which the collections are made have already got admission into the understanding, and the names serving to express them are known; it will be easy, by enumerating the several ideas concerned in the composition, and marking the order and manner in which they are united, to raise any complex conception in the mind. Thus the idea answering to the word rainbow may be readily excited in the imagination of another who has never seen the appearance itself, by barely describing the figure, largeness, position, and order of colours; if we suppose these several simple ideas, with their names, sufficiently known to him.

IV. And this leads to a second observation upon this subject, namely, That words standing for complex ideas are all definable, but those by which we denote simple ideas are not; for simple ideas being secondary perceptions, which have no other entrance into the mind than by sensation or reflection, can only be got by experience, from the several objects of nature, proper to produce those perceptions in us. Words indeed may very well serve to remind us of them, if they have already found admission into the understanding, and their connexion with the established names is known; but they can never give them their original being and existence there. And hence it is, that when any one asks the meaning of a word denoting a simple idea, we pretend not to explain it to him by a definition, well knowing that to be impossible; but, supposing him already acquainted with the idea, and only ignorant of the name by which it is called, we either mention it to him by some other name with which we presume he knows its connexion, or appeal to the object where the idea itself is found. Thus, were any one to ask the meaning of the word white, we should tell him it stood for the same idea as albus in Latin, or blanc in French; or, if we thought him a stranger to these languages, we might appeal to an object producing the idea, by saying it denoted the colour we observe in snow or milk. But this is by no means a definition of the word, exciting a new idea in his understanding; but merely a contrivance to remind him of a known idea, and teach him its connexion with the established name. For if the ideas after which he inquires have never yet been raised in his mind; as suppose one who had seen no other colours than black and white, should ask the meaning of the word scarlet; it is easy to perceive, that it would be no more possible to make him comprehend it by words, or a definition, than to introduce the same perception into the imagination of a man born blind. The only method in this case is, to present some object, by looking at which the perception itself may be excited; and thus he will learn both the name and the idea together.

V. But how comes it to pass that men agree in the experience of their simple ideas, seeing they cannot view and observe the perceptions in one another's minds, nor make known their perceptions by words to others? The effect is produced by experience and observation. Thus finding, for instance, that the name of heat is annexed to names of that sensation which men feel when they approach the fire, I make it also the sign of the sensation excited in me by such an approach, nor have any doubt but it denotes the same perception in my mind as in theirs. For we are naturally led to imagine, that the same objects operate alike upon the organs of the human body, and produce an uniformity of sensations. No man fancies, that the idea raised in him by the taste of sugar, and which he calls sweetness, differs from that excited in another by the like means; or that wormwood, to whose relish he has given the epithet bitter, produces in another the sensation which he denotes by the word sweet. Presuming therefore upon this conformity of perceptions, when they arise from the same objects, we easily agree as to the names of our simple ideas; and if at any time, by a more narrow scrutiny into things, new ideas of this class come in our way, which we choose to express by terms of our own invention; these names are explained, not by a definition, but by referring to the objects whence the ideas themselves may be obtained.

VI. Being in this manner furnished with simple ideas, and the names by which they are expressed; the conveyance of meaning of terms that stand for complex ideas is easily got, because the ideas themselves answering to these terms may be conveyed into the mind by definitions, a combination of simple ideas. When therefore these terms are enumerated, and the manner in which they are united into one conception explained, nothing more is wanting to raise that conception in the understanding; and thus the term denoting it comes of course to be understood. And here it is worth while to reflect Part I.

Of a little upon the wise contrivance of nature, in thus furnishing us with the very aptest means of communicating our thoughts. For were it not so ordered, that we could thus convey our complex ideas from one to another by definitions, it would in many cases be impossible to make them known at all. This is apparent in those ideas which are the proper work of the mind. For as they exist only in the understanding, and have no real objects in nature in conformity to which they are framed; if we could not make them known by description, they must lie forever hid within our own breasts, and be confined to the narrow acquaintance of a single mind. All the fine scenes that arise from time to time in the poet's fancy, and by his lively painting give such entertainment to his readers, were he destitute of this faculty of laying them open to the view of others by words and description, could not extend their influence beyond his own imagination, or give joy to any but the original inventor.

VII. There is this farther advantage in the ability great avail we enjoy of communicating our complex notions by towards the definitions; that as these make by far the largest class of our ideas, and most frequently occur in the progress of knowledge, and improvement of knowledge, so they are by these means imparted with the greatest readiness, than which nothing would tend more to the increase and spreading of science; for a definition is soon perused; and if the terms of it are well understood, the idea itself finds an easy admission into the mind. Whereas, in simple perceptions, where we are referred to the objects producing them, if these cannot be come at, as is sometimes the case, the names by which they are expressed must remain empty sounds. But new ideas of this class occurring very rarely in the sciences, seldom create any great obstruction. It is otherwise with our complex notions; for every step we take leading us into new combinations and views of things, it becomes necessary to explain these to others, before they can be made acquainted with our discoveries: and as the manner of definitions is easy, requiring no apparatus but that of words, which are always ready, and at hand; hence we can with the least difficulty remove such obstacles as might arise from terms of our own invention, when they are made to stand for new complex ideas suggested to the mind by some present train of thinking. And thus at last we are let into the mystery hinted at in the beginning of this chapter, viz. how we may become acquainted with the thoughts of another, when he makes use of words to which we have as yet joined no ideas. The answer is obvious from what has been already said. If the terms denote simple perceptions, he must refer us to these objects of nature whence the perceptions themselves are to be obtained; but, if they stand for complex ideas, their meaning may be explained by a definition.

CHAP. II. Of Definition.

I. A Definition is the unfolding of some conception of the mind, answering to the word or term made use of as the sign of it. Now as, in exhibiting any idea to another, it is necessary that the description be such as may excite that precise idea in his mind; hence it is plain that definitions, properly speaking, are not arbitrary, but confined to the representing of certain determinate settled notions, such namely as are annexed by the speaker or writer to the words he uses. As nevertheless it is universally allowed that the signification of words is perfectly voluntary, and not the effect of any natural and necessary connexion between them and the ideas for which they stand; some may perhaps wonder why definitions are not so too. In order therefore to unravel this difficulty, and show distinctly what is and what is not arbitrary in speech, we must carefully distinguish between the connexion of our words and ideas, and the unfolding of the ideas themselves.

II. First, as to the connexion of our words and ideas; The connexion, it is plain, is a purely arbitrary institution. When, for instance, we have in our minds the idea of any particular species of metals, the calling it by the name ideas, a person gold is an effect of the voluntary choice of men speaking in the same language, and not of any peculiar aptness; but in that found to express that idea. Other nations we find make use of different sounds, and with the same effect. Thus aurum denotes that idea in Latin, and or in French; and even the word gold itself would have as well served to express the idea of that metal which we call silver, had custom in the beginning established it.

III. But although we are thus entirely at liberty in connecting any idea with any sound, yet it is quite otherwise in unfolding the ideas themselves. For every idea having a precise appearance of its own, by ed to which it is distinguished from every other idea: it is represented manifestly, that in laying it open to others, we must study such a description as shall exhibit that peculiar appearance. When we have formed to ourselves the idea of a figure bounded by four equal sides, joined and differing together at right angles, we are at liberty to express that idea by any sound, and call it either a square or a triangle. But whichever of these names we use, so long as the idea is the same, the description by which we would signify it to another must be so too. Let it be called square or triangle, it is still a figure having four equal sides, and all its angles right ones. Hence we clearly see what is and what is not arbitrary in the use of words. The establishing any sound as the mark of some determinate idea in the mind, is the effect of free choice, and a voluntary combination among men; and as different nations make use of different sounds to denote the same ideas, hence proceeds all that variety of languages which we meet with in the world. But when a connexion between our ideas and words is once settled, the unfolding of the idea answering to any word, which properly constitutes a definition, is by no means an arbitrary thing: for here we are bound to exhibit that precise conception which either the use of language, or our own particular choice, hath annexed to the term we use.

IV. And thus it appears, that definitions, considered as descriptions of ideas in the mind, are steady and invariable, being bounded to the representation of the idea that has precise ideas. But then, in the application of definitions to particular names, we are altogether left to our own free choice. Because as the connecting of any definition with any sound is a perfectly arbitrary institution, the applying the description of that idea to that sound must be so too. When therefore logicians tell... us that the definition of the name is arbitrary, they mean no more than this; that as different ideas may be connected with any term, according to the good pleasure of him that uses it; in like manner may different descriptions be applied to the term, suitable to the ideas so connected. But this connexion being fettered, and the term considered as the sign of some fixed idea in the understanding, we are no longer left to arbitrary explications, but must study such a description as corresponds with that precise idea. Now this alone, according to what has been before laid down, ought to be accounted a definition. What seems to have occasioned no small confusion in this matter, is, that many explanations of words, where no idea is unfolded, but merely the connexion between some word and idea alluded, have yet been dignified with the name of definitions. Thus, when we say that a clock is an instrument by which we measure time; that is by some called a definition; and yet it is plain that we are beforehand supposed to have an idea of this instrument, and only taught that the word clock serves in common language to denote that idea. By this rule all explications of words in our dictionaries will be definitions, nay, the names of even simple ideas may be thus defined. White, we may say, is the colour we observe in snow or milk; heat the sensation produced by approaching the fire; and so in innumerable other instances. But these, and all others of the like kind, are by no means definitions, exciting new ideas in the understanding, but merely contrivances to remind us of known ideas, and teach their connexion with the established names.

V. But now in definitions properly so called, we first consider the term we use, as the sign of some inward conception, either annexed to it by custom, or of our own free choice: and then the business of the definition is to unfold and explicate that idea. As therefore the whole art lies in giving just and true copies of our ideas; a definition is then said to be made perfect, when it serves distinctly to excite the idea described in the mind of another, even supposing him before wholly unacquainted with it. This point settled, let us next inquire what those ideas are which are capable of being thus unfolded? and in the first place it is evident, that all our simple ideas are necessarily excluded. We have seen already that experience alone is to be consulted here, inasmuch that if either the objects whence they are derived come not in our way, or the avenues appointed by nature for their reception are wanting, no description is sufficient to convey them into the mind. But where the understanding is already supplied with these original and primitive conceptions, as they may be united together in an infinity of different forms; so may all their several combinations be distinctly laid open, by enumerating the simple ideas concerned in the various collections, and tracing the order and manner in which they are linked one to another. Now these combinations of simple notices constitute what we call our complex notions, whence it is evident, that complex ideas, and those alone, admit of that kind of description which goes by the name of a definition.

VI. Definitions, then, are pictures or representations of our ideas; and as these representations are then only possible when the ideas themselves are complex, it is obvious to remark, that definitions cannot have place, but where we make use of terms standing for such complex ideas. But our complex ideas being, as we have said, nothing more than different combinations of simple ideas; we then know and comprehend them perfectly, when we know the several simple ideas of which they consist, and can so put them together in our minds as may be necessary towards the framing of that peculiar connexion which gives every idea its distinct and proper appearance.

VII. Two things are therefore required in every definition: first, That all the original ideas, out of which the complex one is formed, be distinctly enumerated; and, secondly, That the order and manner of combining them into one conception be clearly explained. Where a definition has these requisites, no explanation is wanting to its perfection; because every one who reads it and understands the terms, seeing at once their combination, can at pleasure form in his own mind the complex conception answering to the term defined. Let us, for instance, suppose the word square to stand for that idea by which we represent to ourselves a figure whose sides subtend quadrants of a circumscribed circle. The parts of this idea are the sides bounding the figure. These must be four in number, and all equal among themselves, because they are each to subtend a fourth part of the same circle. But, besides these component parts, we must also take notice of the manner of putting them together, if we would exhibit the precise idea for which the word square here stands. For four equal right lines, anyhow joined, will not subtend quadrants of a circumscribed circle. A figure with this property must have its sides standing also at right angles. Taking in therefore this last consideration respecting the manner of combining the parts, the idea is fully described, and the definition thereby rendered complete. For a figure bounded by four equal sides, joined together at right angles, has the property required; and is moreover the only right-angled figure to which that property belongs.

VIII. It will now be obvious to every one, in what manner we ought to proceed, in order to arrive at just and adequate definitions. First, We are to take an exact view of the idea to be described, trace it to its original principles, and mark the several simple perceptions that enter into the composition of it. Secondly, We are to consider the particular manner in which these elementary ideas are combined, in order to the forming of that precise conception for which the term we make use of stands. When this is done, and the idea wholly unravelled, we have nothing more to do than fairly transcribe the appearance it makes to our own minds. Such a description, by distinctly exhibiting the order and number of our primitive conceptions, cannot fail to excite at the same time in the mind of every one that reads it, the complex idea resulting from them; and therefore attains the true and proper end of a definition.

CHAP. III. Of the Composition and Resolutions of our Ideas, and the Rules of Definition thence arising.

I. The rule laid down in the foregoing chapter is general, extending to all possible cases; and is indeed that that to which alone we can have recourse, where any doubt or difficulty arises. It is not, however, necessary that we should practise it in every particular instance. Many of our ideas are extremely complicated, inasmuch that to enumerate all the simple perceptions out of which they are formed, would be a very troublesome and tedious work. For this reason logicians have established certain compendious rules of defining, of which it may not be amiss here to give some account. But in order to the better understanding of what follows, it will be necessary to observe, that there is a certain gradation in the composition of our ideas. The mind of man is very limited in its views, and cannot take in a great number of objects at once. We are therefore fain to proceed by steps, and make our first advances subservient to those which follow. Thus, in forming our complex notions, we begin at first with but a few simple ideas, such as we can manage with ease, and unite them together into one conception. When we are provided with a sufficient stock of these, and have by habit and use rendered them familiar to our minds, they become the component parts of other ideas still more complicated, and form what we may call a second order of compound notions. This process, as is evident, may be continued to any degree of composition we please, mounting from one stage to another, and enlarging the number of combinations.

II. But now in a series of this kind, whoever would acquaint himself perfectly with the last and highest order of ideas, finds it much the most expedient method to proceed gradually through all the intermediate steps. For, were he to take any very compound idea to pieces, and, without regard to the several classes of simple perceptions that have already been formed into distinct combinations, break it at once into its original principles, the number would be so great as perfectly to confound the imagination, and overcome the utmost reach and capacity of the mind. When we see a prodigious multitude of men jumbled together in crowds, without order or any regular position, we find it impossible to arrive at an exact knowledge of their number. But if they are formed into separate battalions, and so stationed as to fall within the leisure survey of the eye; by viewing them successively and in order, we come to an easy and certain determination. It is the same in our complex ideas. When the original perceptions, out of which they are framed, are very numerous, it is not enough that we take a view of them in loose and scattered bodies; we must form them into distinct classes, and unite these classes in a just and orderly manner, before we can arrive at a true knowledge of the compound notices resulting from them.

III. This gradual progress of the mind to its compound notions, through a variety of intermediate steps, plainly points out the manner of conducting the definitions by which these notions are conveyed into the minds of others. For as the series begins with simple and easy combinations, and advances through a succession of different orders, rising one above another in the degree of composition, it is evident, that, in a train of definitions expressing these ideas, a like gradation is to be observed. Thus the complex ideas of the lowest order can no otherwise be described than by enumerating the simple ideas out of which they are made, and explaining the manner of their union. But Perception, then in the second, or any other succeeding order, as they are formed out of those gradual combinations, and constitute the inferior classes, it is not necessary, in describing them, to mention one by one all the simple ideas of which they consist. They may be more distinctly and briefly unfolded, by enumerating the compound ideas of a lower order, from whose union they result, and which are all supposed to be already known in consequence of previous definitions. Here then it is that the logical method of defining takes place; which, that it may be the better understood, we shall explain somewhat more particularly the several steps and gradations of the mind in compounding its ideas, and thence deduce that peculiar form of a definition which logicians have thought fit to establish.

IV. All the ideas we receive from the several objects of nature that surround us, represent distinct in-by which individuals. These individuals, when compared together, are found in certain particulars to resemble each other. Hence, by collecting the resembling particulars into one conception, we form the notion of a species, general ideas. And here let it be observed, that this last idea is less complicated than that by which we represent any of the particular objects contained under it. For the idea of the species excludes the peculiarities of the several individuals, and retains only such properties as are common to them all. Again, By comparing several species together, and observing their resemblance, we form the idea of a genus; where, in the same manner as before, the composition is lessened, because we leave out what is peculiar to the several species compared, and retain only the particulars wherein they agree. It is easy to conceive the mind proceeding thus from one step to another, and advancing through its several classes of general notions, until at last it comes to the highest genus of all, denoted by the word being, where the bare idea of existence is only concerned.

V. In this procedure we see the mind unravelling the complex idea, and tracing it in the ascending scale, down to the greatest or least degrees of composition, until it terminates in one simple perception. If now we take its contrary way, and, beginning with the ideas, as it last or highest genus, carry our view downwards, advances through all the inferior genera and species, quite through the individuals, we shall thereby arrive at a distinct apprehension of the conduct of the understanding in compounding its ideas. For, in the several classes of our perceptions, the highest in the scale is for the most part made up of but a few simple ideas, such as the mind can take in and survey with ease. This first general notion, when branched out into the different subdivisions contained under it, has in every one of them something peculiar, by which they are distinguished among themselves; inasmuch that, in descending from the genus to the species, we always superadd some new idea, and thereby increase the degree of composition. Thus the idea denoted by the word figure is of a very general nature, and composed of but few simple perceptions, as implying no more than space everywhere bounded. But if we descend farther, and consider the boundaries of this space, as that they may be either lines or surface, we... fall into the several species of figure. For where the space is bounded by one or more surfaces, we give it the name of a solid figure; but where the boundaries are lines, it is called a plain figure (A).

VI. In this view of things it is evident, that the species is formed by superadding a new idea to the genus. Here, for instance, the genus is circumfribbed space. If now to this we superadd the idea of a circumscription by lines, we frame the notion of that species of figures which are called plain; but if we conceive the circumscription to be by surfaces, we have the species of solid figures. This superadded idea is called the specific difference, not only as it serves to divide the species from the genus, but because, being different in all the several subdivisions, we thereby also distinguish the species one from another. And as it is likewise that conception, which, by being joined to the general idea, completes the notion of the species: hence it is plain, that the genus and specific difference are to be considered as the proper and constituent parts of the species. If we trace the progress of the mind still farther, and observe it advancing through

(A) This account of the composition and resolution of our ideas is agreeable to the common doctrine of logicians on the subject. Into the truth of the doctrine itself we shall inquire afterwards under the article Metaphysics: but to prevent mistakes, it may be proper to observe here, that though every writer of logic has treated largely of general and specific ideas, there is in reality nothing general in the matter but the terms of language. When we utter, for instance, the word triangle, that general term does not, as has been often said, suggest to the mind the general idea of a triangle, which is neither oblique nor rectangle, neither equilateral nor scalene, &c., for such a triangle, as it cannot exist in nature, cannot be conceived in idea. In like manner, the general term Virtue does not excite a general idea of virtue, which is neither prudence, nor temperance, nor fortitude, nor justice, nor charity, &c., for that which is distinct from all these is not virtue. What then is the import of such general terms? The answer is obvious: they denote classes of objects; and are never used without some word of limitation, but when something that has no dependence upon the particular qualities, which distinguish the individuals from each other, is affirmed or denied of the whole class. Thus we may affirm, that the three angles of a plain triangle are equal to two right angles: and this proposition is demonstrably true, not of a triangle, which is neither oblique nor rectangle, neither equilateral nor scalene, for such a triangle never was conceived; but of all those triangles equally, as the truth of the proposition and the progress of the demonstration has no dependence upon the peculiarities which distinguish these triangles from one another. Again, when we say that a man of virtue will be rewarded by God, we do not mean by the word virtue a general idea making part of each of the complex and more particular ideas of prudence, fortitude, justice, &c., and at the same time different from them all; but we affirm, that the man who practises any or all of these virtues, according as he has opportunity, will be rewarded by God.

The history of our ideas is shortly this:—That act of the mind, if it may be called an act, which makes known an external object, is termed perception. That act of the mind which makes known an internal object, is termed consciousness. Objects once perceived may be recalled to the mind by the power of memory; and when they are so recalled, we have a perception of them in all respects similar to the original perception, only less distinct; we fancy ourselves in the same place, and the object perceived attended by the same circumstances. This indistinct secondary perception of an object is termed an idea; and therefore the precise and accurate definition of an idea, in contradistinction to an original perception, is "that perception of a real object which is raised in the mind by the power of memory." Now all our original perceptions being of particular objects, it is obvious that our ideas, which are only those perceptions recalled, must be of particular objects likewise, and that no man can have an idea of a thing of which the real existence is contradictory and impossible. But the general and specific ideas of logicians, are ideas of nothing which exists, or which can possibly exist. They are acquired, we are told, by abstraction, in the following manner. Among a number of individuals we perceive certain qualities the same in all, whilst in each individual there are other qualities which have nothing similar to them in any other individual: now the mind, it is said, has a power of abstracting the particular qualities of each individual from those which are common to the whole, and of these last forming a general idea of the whole class. Thus all men have nearly the same form; and they have each some stature and some colour, though there are not perhaps two individuals who have precisely the same stature and the same colour. Now, say the advocates for general ideas, if we abstract what is peculiar to each individual, and retain what is common to the whole race, we have the general idea signified by the word man. That is, if we conceive a being in human shape, which is of stature and colour, but neither tall nor short, neither white nor black, nor red nor brown, nor any other colour which we ever saw, we have the general idea of humanity, and understand the meaning of the word man! Surely no person who is not the slave of prejudice will pretend that he can frame such an idea as this—the idea of an object which cannot possibly exist in nature.

By this we do not mean to affirm, that we cannot frame ideas of such objects as have no real existence; for it is easy to imagine a man with ten heads as with one, because there is nothing contradictory between ten heads and one body. But figure, which is said to be space bounded neither by lines nor superficies; colour, which is neither red nor white, nor blue nor black, &c.; and animal, which is neither man, beast, bird, nor insect; are impossible in nature, and inconceivable in idea. There is, however, no harm in still retaining the phrase general idea, provided he who uses it takes care to let it be known, that by these words he means not any abstract and contradictory idea, but merely a class of real objects. The phrase may at times prevent much circumlocution; for which reason we have retained the use of it in the text. through the inferior species, we shall find its manner of proceeding to be always the same. For every lower species is formed by superadding some new idea to the species next above it; inasmuch that in this descending scale of our perceptions, the understanding passes through different orders of complex notions, which become more and more complicated at every step it takes. Let us resume here, for instance, the species of plain figures. They imply no more than space bounded by lines. But if we take in an additional consideration of the nature of these lines, as whether they are right or curves, we fall into the subdivisions of plain figure, distinguished by the names of rectilinear, curvilinear, and mixtilinear.

VII. And here we are to observe, that though plain figures, when considered as one of those branches that come under the notions of figure in general, take the name of a species; yet compared with the classes of curvilinear, rectilinear, and mixtilinear, into which they themselves may be divided, they really become a genus, of which the beforementioned subdivisions constitute the several species. These species, in the same manner as in the case of plane and solid figures, consist of the genus and specific difference as their constituent parts. For in the curvilinear kind, the curvity of the lines bounding the figure makes what is called the specific difference; to which if we join the genus, which here is a plain figure or space circumscribed by lines, we have all that is necessary towards completing the notion of this species. We are only to take notice, that this last subdivision, having two genera above it, viz. plain figure, and figure in general; the genus joined with the specific difference, in order to constitute the species of curvilinears, is that which lies nearest to the said species. It is the notion of plain figure, and not of figure in general, that joined with the idea of curvity, makes up the complex conception of curve-lined figures. For in this descending scale of our ideas, figure in general, plain figures, curved-lined figures, the two first are considered as genera in respect of the third; and the second in order, or that which stands next to the third, is called the nearest genus. But now as it is this second idea, which, joined with the notion of curvity, forms the species of curve-lined figures; it is plain, that the third or last idea in the series is made up of the nearest genus and specific difference. This rule holds invariably, however far the series is continued; because, in a train of ideas thus succeeding one another, all that precede the last are considered as so many genera in respect of that last; and the last itself is always formed by superadding the specific difference to the genus next it.

VIII. Here then we have an universal description, applicable to all our ideas of whatever kind, from the highest genus to the lowest species. For, taking them in order downwards from the said general idea, they everywhere consist of the genus proximum, and differentia specifica, as logicians love to express themselves. But when we come to the lowest species of all, comprehending under it only individuals, the superadded idea, by which these individuals are distinguished one from another, no longer takes the name of the specific difference. For here it serves not to denote distinct species, but merely a variety of individuals, each of which, having a particular existence of its own, is therefore numerically different from every other of the same kind. And hence it is, that in this last case, logicians choose to call the superadded idea by the name of the numerical difference; inasmuch that, as the idea of a species is made up of the nearest genus and specific difference, so the idea of an individual consists of the lowest species and numeric difference. Thus the circle is a species of curve-lined figures, and what we call the lowest species, as comprehending under it only individuals. Circles in particular are distinguished from one another by the length and position of their diameters. The length therefore and position of the diameter of a circle form what logicians call the numerical difference; because, there being given, the circle itself may be described, and an individual thereby constituted.

IX. Thus the mind, in compounding its ideas, begins, we see, with the most general notions, which, following one another consisting of but a few simple notices, are easily combined and brought together into one conception, and pass Thence it proceeds to the species comprehended under this general idea; and these are formed by joining together the genus and specific difference. And as it often happens, that these species may be still farther subdivided, and run on in a long series of continued gradations, producing various orders of compound perceptions; so all these several orders are regularly and successively formed by annexing in every step the specific difference to the nearest genus. When by this method of procedure we are come to the lowest order of all, by joining the species and numeric difference, we frame the ideas of individuals. And here the series necessarily terminates, because it is impossible any farther to bound or limit our conceptions. This view of the composition of our ideas, representing their constituent parts in every step of the progression, naturally points out the true and genuine form of a definition. For as definitions are no more than descriptions of the ideas for which the terms defined stand; and as ideas are then described, when we enumerate distinctly and in order the parts of which they consist; it is plain, that by making our definitions follow one another according to the natural train of our conceptions, they will be subject to the same rules, and keep pace with the ideas they describe.

X. As therefore the first order of our compound notions, or the ideas that constitute the highest genus in all the different scales of perception, are formed by uniting together a certain number of simple notices; so the terms expressing these genera are defined by conception, enumerating the simple notices so combined. And as the species comprehended under any genus, or the complex ideas of the second order, arise from superadding the specific difference to the said general idea; so the definition of the names of the species is abolished, in a detail of the ideas of the specific difference, connected with the term of the genus. For the genus having been before defined, the term by which it is expressed stands for a known idea, and may therefore be introduced into all subsequent definitions, in the same manner as the names of simple perceptions. It will now be sufficiently obvious, that the definitions of all the succeeding orders of compound notions will every- PART II. OF JUDGEMENT.

CHAP. I. Of the Grounds of Human Judgement.

THE mind being furnished with ideas, its next step in the way to knowledge is, the comparing these ideas together, in order to judge of their agreement or disagreement. In this joint view of our ideas, if the relation is such as to be immediately discoverable by the bare inspection of the mind, the judgements thence obtained are called intuitive, from a word that denotes to look at; for in this case, a mere attention to the ideas compared suffices to let us see now far they are connected or disjoined. Thus, that the Whole is greater than any of its Parts, is an intuitive judgement; nothing more being required to convince us of its truth, than an attention to the ideas of whole and part. And this too is the reason why we call the act of the mind forming these judgements intuition; as it is indeed no more than an immediate perception of the agreement or disagreement of any two ideas.

II. But here it is to be observed, that our knowledge of this kind respects only our ideas, and the relations between them; and therefore can serve only as a foundation to such reasonings as are employed in investigating those relations. Now it so happens, that many of our judgements are conversant about facts, and the real existence of things which cannot be traced by the bare contemplation of our ideas. It does not follow, because I have the idea of a circle in my mind, that therefore a figure answering to that idea has a real existence in nature. I can form to myself the notion of a centaur or golden mountain, but never imagine on that account that either of them exists. What then are the grounds of our judgement in relation to facts? experience and testimony. By experience we are informed of the existence of the several objects which surround us, and operate upon our senses. Testimony is of a wider extent, and reaches not only to objects beyond the present sphere of our observation, but also to facts and transactions, which being now past, and having no longer any existence, could not without this conveyance have fallen under our cognizance.

III. Here we have three foundations of human judgement, from which the whole system of our knowledge may with ease and advantage be derived. First, Intuition, which respects our ideas themselves, and their relations; and is the foundation of that species of reasoning which we call demonstration. For whatever is deduced from our intuitive perceptions, by a clear and connected series of proofs, is said to be demonstrated, and produces absolute certainty in the mind. Hence the knowledge obtained in this manner is what we properly term science; because in every step of the procedure it carries its own evidence along with it, and leaves no room for doubt or hesitation. And what is highly worthy of notice; as the truths of this class express the relations between our ideas, and the same relations must ever and invariably subsist between the same ideas, our deductions in the way of science constitute what we call eternal, necessary, and immutable truths. If it be true that the whole is equal to all its parts, it must be so unchangeably; because the relation of equality being attached to the ideas themselves, must ever intervene where the same ideas are compared. Of this nature all the truths of natural religion, morality, and mathematics, and in general whatever may be gathered from the bare view and consideration of our ideas.

IV. The second ground of human judgement is experience; from which we infer the existence of those objects that surround us, and fall under the immediate notice of our senses. When we see the sun, or caitlidge of our eyes towards a building, we not only have per the perceptions of these objects within ourselves, but ascribe and qualify to them a real existence out of the mind. It is also by the information of the senses that we judge of the qualities of bodies; as when we say that snow is white, fire hot, or steel hard. For as we are wholly unacquainted with the internal structure and constitution of the bodies that produce these sensations in us, nay, and are unable to trace any connexion between that structure and the sensations themselves, it is evident, that we build our judgements altogether upon observation, ascribing to bodies such qualities as are answerable to the perceptions they excite in us. Not that we ever suppose the qualities of bodies to be things of the same nature with our perceptions; for there is nothing in fire similar to our sensation of heat, or in a sword similar to pain; but that when different bodies excite in our minds similar perceptions, we necessarily ascribe to these bodies not only an existence independent of us, but likewise similar qualities, of which it is the nature to produce similar perceptions in the human mind. But this is not the only advantage derived from experience; for to that too we are indebted for all our knowledge regarding the co-existence of sensible qualities in objects, and the operations of bodies one upon another. Ivory, for instance, is hard and elastic; this we know by experience, and indeed by that alone. For, being altogether strangers to the true nature both of elasticity and hardness, we cannot by the bare contemplation of our ideas determine how far the one necessarily implies the other, or whether there may not be a repugnance between them. But when we observe them to exist both in the same object, we are then assured from experience that they are not incompatible; and when we also find, that a stone is hard and not elastic, and that air though ela- Part II.

Of Judgement.

Our affekt rests, and whereto we appeal when the highest degree of evidence is required.

VII. But there are many facts that will not allow of an appeal to the senses; and in this case testimony is the true and only foundation of our judgements. All human actions of whatever kind, when considered ground of already past, are of the nature here described; because having now no longer any existence, both the facts themselves, and the circumstances attending them, can be known only from the relations of such as had sufficient opportunities of arriving at the truth. Testimony therefore is justly accounted a third ground of human judgement; and as from the other two we have deduced scientific and natural knowledge, so we may from this derive historical; by which we mean, not merely a knowledge of the civil transactions of states and kingdoms, but of all facts whatsoever, where testimony is the ultimate foundation of our belief.

CHAP. II. Of Affirmative and Negative Propositions.

I. While the comparing of our ideas is considered merely as an act of the mind, assembling them together, and joining or disjoining them according to the result of its perceptions, we call it judgement; but when explained, our judgements are put into words, they then bear the name of propositions. A proposition therefore is a sentence expressing some judgement of the mind, whereby two or more ideas are affirmed to agree or disagree. Now, as our judgements include at least two ideas, one of which is affirmed or denied of the other, so must a proposition have terms answering to these ideas. The idea of which we affirm or deny, and of course the term expressing that idea, is called the subject of the proposition. The idea affirmed or denied, as also the term answering it, is called the predicate. Thus in the proposition, God is omnipotent: God is the subject, it being of him that we affirm omnipotence; and omnipotent is the predicate, because we affirm the idea expressed by that word to belong to God.

II. But as, in propositions, ideas are either joined or disjoined; it is not enough to have terms expressing those ideas, unless we have also some words to denote their agreement or disagreement. That word in a proposition, which connects two ideas together, is called the copula; and if a negative particle be annexed, we thereby understand that the ideas are disjoined. The substantive verb is commonly made use of for the copula: as in the above mentioned proposition, God is omnipotent; where is represents the copula, and signifies the agreement of the ideas of God and omnipotence. But if we mean to separate two ideas; then, besides the substantive verb, we must also use some particle of negation, to express this repugnance. The proposition, man is not perfect, may serve as an example of this kind; where the notion of perfection being removed from the idea of man, the negative particle not is inserted after the copula, to signify the disagreement between the subject and predicate.

III. Every proposition necessarily consists of three parts; but then it is not alike needless that they times express all severally expressed in words; because the copula prefixed by is often included in the term of the predicate, as when a single word. we say, he fits; which imports the same as he is fitting.

In the Latin language, a single word has often the force of a whole sentence. Thus ambulat is the same as ille est ambulans; amo, as ego sum amans; and so in innumerable other instances; by which it appears, that we are not so much to regard the number of words in a sentence, as the ideas they represent, and the manner in which they are put together. For wherever two ideas are joined or disjoined in an expression, though only but a single word; it is evident that we have a subject, predicate, and copula, and of consequence a complete proposition.

IV. When the mind joins two ideas, we call it an affirmative judgement; when it separates them, a negative; and as any two ideas compared together must necessarily either agree or not agree, it is evident that all our judgements fall under these two divisions. Hence likewise the propositions expressing these judgements are all either affirmative or negative. An affirmative proposition connects the predicate with the subject, as a stone is heavy; a negative proposition separates them, as God is not the author of evil. Affirmation therefore is the same as joining two ideas together; and this is done by means of the copula. Negation, on the contrary, marks the repugnance between the ideas compared; in which case a negative particle must be called in, to show that the connexion included in the copula does not take place.

V. Hence we see the reason of the rule commonly laid down by logicians, That in all negative propositions the negation ought to affect the copula. For as the copula, when placed by itself, between the subject and the predicate, manifestly binds them together; it is evident, that in order to render a proposition negative, the particles of negation must enter it in such a manner as to destroy this union. In a word, then only are two ideas disjoined in a proposition, when the negative particle may be referred to the copula, as to break the affirmation included in it, and undo that connexion it would otherwise establish. When we say, for instance, No man is perfect; take away the negation, and the copula of itself plainly unites the ideas in the proposition. But as this is the very reverse of what is intended, a negative mark is added, to show that this union does not here take place. The negation, therefore, by destroying the effect of the copula, changes the very nature of the proposition, inasmuch that, instead of binding two ideas together, it denotes their separation. On the contrary, in this sentence, The man who departs not from an upright behaviour is beloved of God, the predicate beloved of God is evidently affirmed of the subject an upright man; so that, notwithstanding the negative particle, the proposition is still affirmative. The reason is plain: the negation here affects not the copula; but, making properly a part of the subject, serves, with other terms in the sentence, to form one complex idea, of which the predicate beloved of God is directly affirmed.

CHAP. III. Of Universal and Particular Propositions.

I. The next considerable division of propositions is Division into universal and particular. Our ideas, according to proposition which has been already observed in the First Part, are divided into all singular as they enter the mind, and represent individual objects. But as by abstraction we can render them universal, so as to comprehend a whole class of things, and sometimes several classes at once; hence the terms expressing these ideas must be in like manner universal. If therefore we suppose any general term to become the subject of a proposition, it is evident, that whatever is affirmed of the abstract idea belonging to that term, may be affirmed of all the individuals to which that idea extends. Thus, when we say, Men are mortal; we consider mortality, not as confined to one or any number of particular men, but as what may be affirmed without restriction of the whole species. By this means the proposition becomes as general as the idea which makes the subject of it; and indeed derives its universality entirely from that idea, being more or less so according as this may be extended to more or fewer individuals. But it is further to be observed of these general terms, that they sometimes enter a proposition in their full latitude, as in the example given above; and sometimes appear with a mark of limitation. In this last case we are given to understand, that the predicate agrees not to the whole universal idea, but only to a part of it; as in the proposition, Some men are wise: For here wisdom is not affirmed of every particular man, but restrained to a few of the human species (u).

II. Now from this different appearance of the general idea that constitutes the subject of any proposition, arises the division of propositions into universal and particular. An universal proposition is that where the subject is some general term taken in its full latitude; insomuch that the predicate agrees to all the individuals comprehended under it, if it denotes a proper species; and to all the several species, and their individuals, if it marks an idea of a higher order. The words all, every, no, none, &c. are the proper signs of this universality; and as they seldom fail to accompany general truths, so they are the most obvious criterion whereby to distinguish them. All animals have a power of beginning motion. This is an universal proposition; as we know from the word all prefixed to the subject animals, which denotes that it must be taken in its full extent. Hence the power of beginning motion may be affirmed of all the several species of animals.

III. A particular proposition has in like manner some general term for its subject; but with a mark of particular where limitation some universal subject appears with a mark of limitation.

(B) See the preceding note, where it is demonstrated that the terms alone, and not the ideas, are in reality general. The term man is equally applicable to every individual of the human race; and therefore, what is affirmed or denied of men in general, is affirmed or denied of all the individuals, without regard to their discriminating qualities. Some is a definitive word (see Grammar), which, prefixed to the word man, limits the signification of that general term; and therefore what is affirmed of some men, is affirmed only of part of the race, but that part itself is not ascertained. Part II.

Of limitation added, to denote, that the predicate agrees only to one or the individuals comprehended under a species, or to one or more of the species belonging to any genus, and not to the whole universal idea. Thus, Some stones are heavier than iron; Some men have an uncommon share of prudence. In the last of these propositions, the subject some men implies only a certain number of individuals, comprehended under a single species. In the former, where the subject is a genus that extends to a great variety of distinct classes, some stones may not only imply any number of particular stones, but also several whole species of stones, inasmuch as there may be not a few with the property there described. Hence we see, that a proposition does not cease to be particular by the predicate's agreeing to a whole species, unless that species, singly and distinctly considered, makes also the subject of which we affirm or deny.

IV. There is still one species of propositions that remains to be described, and which the more deserves our notice, as it is not yet agreed among logicians to which of the two classes mentioned above they ought to be referred; namely, singular propositions, or those where the subject is an individual. Of this nature are the following: Sir Isaac Newton was the inventor of fluxions; This book contains many useful truths. What occasions some difficulty as to the proper rank of these propositions is, that, the subject being taken according to the whole of its extension, they sometimes have the same effect in reasoning as universals. But if it be considered that they are in truth the most limited kind of particular propositions, and that no proposition can with any propriety be called universal but where the subject is some universal idea; it shall not be long in determining to which class they ought to be referred. When we say, Some books contain useful truths; the proposition is particular, because the general term appears with a mark of restriction. If therefore we say, This book contains useful truths; it is evident that the proposition must be still more particular, as the limitation implied in the word this, is of a more confined nature than in the former case.

V. We see, therefore, that all propositions are either affirmative or negative; nor is it less evident, that in both cases they may be universal or particular. Hence arises that celebrated fourfold division of them into universal affirmative and universal negative, particular affirmative and particular negative, which comprehends indeed all their varieties. The use of this method of distinguishing them will appear more fully afterwards, when we come to treat of reasoning and syllogism.

CHAP. IV. Of Absolute and Conditional Propositions.

I. The objects about which we are chiefly conversant in this world, are all of a nature liable to change. What may be affirmed of them at one time, cannot often at another; and it makes no small part of our knowledge to distinguish rightly these variations, and trace the reasons upon which they depend. For it is observable, that amidst all the vicissitudes of nature, some things remain constant and invariable; nor even are the changes, to which we see others liable, effected but in consequence of uniform and steady laws, which, when known, are sufficient to direct us in our judgements about them. Hence philosophers, in distinguishing the objects of our perception into various classes, have been very careful to note, that some properties belong essentially to the general idea, so as not to be separable from it but by destroying its very nature; while others are only accidental, and may be affirmed or denied of it in different circumstances. Thus solidity, a yellow colour, and great weight, are considered as essential qualities of gold; but whether it shall exist as an uniform conjoined mass, is not alike necessary. We see that by a proper menstrum it may be reduced to a fine powder, and that an intense heat will bring it into a state of fusion.

II. From this diversity in the several qualities of things arises a considerable difference as to the manner of our judging about them. For all such properties as are inseparable from objects when considered as belonging to any genus or species, are affirmed absolutely and without reserve of that general idea. Thus we say, Gold is very weighty; A stone is hard; Animals have a power of self-motion. But in the case of mutual or accidental qualities, as they depend upon some other consideration distinct from the general idea; that also must be taken into the account, in order to form an accurate judgement. Should we affirm, for instance, of some stones, that they are very susceptible of a rolling motion; the proposition, while it remains in this general form, cannot with any advantage be introduced into our reasonings. An attempt to receive that mode of motion flows from the figure of the stone; which, as it may vary infinitely, our judgement then only becomes applicable and determinate, when the particular figure, of which volatility is a consequence, is also taken into the account. Let us then bring in this other consideration, and the proposition will run as follows: Stones of a spherical form are easily put into a rolling motion. Here we see the condition upon which the predicate is affirmed, and therefore know in what particular cases the proposition may be applied.

III. This consideration of propositions respecting the manner in which the predicate is affirmed of the subject gives rise to the division of them into absolute and conditional. Absolute propositions are those wherein we affirm some property inseparable from the idea of the subject, and which therefore belongs to it in all and possible cases; as, God is infinitely wise; Virtue tends to the ultimate happiness of man. But where the predicate is not necessarily connected with the idea of the subject, unless upon some consideration distinct from that idea, there the proposition is called conditional. The reason of the name is taken from the supposition annexed, which is of the nature of a condition, and may be expressed as such, thus: If a stone is exposed to the rays of the sun, it will contract some degree of heat; If a river runs in a very declining channel, its rapidity will constantly increase.

IV. There is not anything of greater importance in philosophy than a due attention to this division of propositions. If we are careful never to affirm things absolutely but where the ideas are inseparably connected; and if in our other judgements we distinctly mark the conditions which determine the predicate to belong to the subject; we shall be the less liable to mistake. mistake in applying general truths to the particular concerns of human life. It is owing to the exact observance of this rule that mathematicians have been so happy in their discoveries, and that what they demonstrate of magnitude in general may be applied with ease in all obvious occurrences.

V. The truth of it is, particular propositions are then known to be true, when we can trace their connexion with universals; and it is accordingly the great business of science to find out general truths that may be applied with safety in all obvious instances. Now the great advantage arising from determining with care the conditions upon which one idea may be affirmed or denied of another is this: that thereby particular propositions really become universal, may be introduced with certainty into our reasonings, and serve as standards to conduct and regulate our judgments. To illustrate this by a familiar instance: if we say, Some water acts very forcibly; the proposition is particular: and as the conditions on which this forcible action depends are not mentioned, it is as yet uncertain in what cases it may be applied. Let us then supply these conditions, and the proposition will run thus: Water conveyed in sufficient quantity along a steep descent acts very forcibly. Here we have an universal judgement, inasmuch as the predicate forcible action may be ascribed to all water under the circumstances mentioned. Nor is it less evident that the proposition in this new form is of easy application; and in fact we find that men do apply it in instances where the forcible action of water is required; as in corn-mills and many other works of art.

CHAP. V. Of Simple and Compound Propositions.

I. HITHERTO we have treated of propositions, where only two ideas are compared together. These are in the general called simple; because, having but one subject and one predicate, they are the effect of a simple judgement that admits of no subdivision. But if it happens that several ideas offer themselves to our thoughts at once, whereby we are led to affirm the same thing of different objects, or different things of the same object; the propositions expressing these judgements are called compound: because they may be resolved into as many others as there are subjects or predicates in the whole complex determination on the mind. Thus, God is infinitely wise and infinitely powerful. Here there are two predicates, infinite wisdom and infinite power, both affirmed of the same subject; and accordingly the proposition may be resolved into two others; affirming these predicates severally. In like manner in the proposition, Neither kings nor people are exempt from death; the predicate is denied of both subjects, and may therefore be separated from them in distinct propositions. Nor is it less evident, that if a complex judgement consists of several subjects and predicates, it may be resolved into as many simple propositions as are the number of different ideas compared together. Riches and honours are apt to elate the mind, and increase the number of our desires. In this judgement there are two subjects and two predicates, and it is at the same time apparent that it may be resolved into four distinct propositions. Riches are apt to elate the mind. Riches are apt to increase the number of our desires. And so of honours.

II. Logicians have divided these compound propositions into a great many different classes; but, in our opinion, not with a due regard to their proper definition of a nation. Thus conditionals, causals, relatives, &c., are compound mentioned as so many distinct species of this kind, though in fact they are no more than simple propositions. To give an instance of a conditional: If a stone is exposed to the rays of the sun, it will contract some degree of heat. Here we have but one subject and one predicate; for the complex expression, A stone exposed to the rays of the sun, constitutes the proper subject of this proposition, and is no more than one determined idea. The same thing happens in causals. Rehoboam was unhappy because he followed evil counsel. There is here an appearance of two propositions arising from the complexity of the expression; but when we come to consider the matter more nearly, it is evident that we have but a single subject and predicate. The pursuit of evil counsel brought misery upon Rehoboam. It is not enough, therefore, to render a proposition compound, that the subject and predicate are complex notions, requiring sometimes a whole sentence to express them: for in this case the composition is still confined to two ideas, and constitutes what we call a simple judgement. But where there are several subjects or predicates, or both, as the affirmation or negation may be alike extended to them all, the proposition expressing such a judgement is truly a collection of as many simple ones as there are different ideas compared. Confining ourselves therefore to this more strict and just notion of compound propositions, they are all reducible to two kinds, viz. copulatives and disjunctives.

III. A copulative proposition is, where the subjects and predicates are so linked together, that they may be all severally affirmed or denied one of another. Of these, either this nature are the examples of compound propositions given above. Riches and honours are apt to elate the mind, and increase the number of our desires. Neither kings nor people are exempt from death. In the first of these the two predicates may be affirmed severally of each subject, whence we have four distinct propositions. The other furnishes an example of the negative kind, where the same predicate, being disjoined from both subjects, may also be denied of them in separate propositions.

IV. The other species of compound propositions are or disjunctive those called disjunctives; in which, comparing several predicates with the same subject, we affirm that one of them necessarily belongs to it, but leave the particular predicate undetermined. If any one, for example, says, This world either exists of itself, or is the work of some all-wise and powerful cause, it is evident that one of the two predicates must belong to the world; but as the proposition determines not which, it is therefore of the kind we call disjunctive. Such too are the following: The sun either moves round the earth, or is the centre about which the earth revolves. Friendship finds men equal, or makes them so. It is the nature of all propositions of this class, supposing them to be exact in point of form, that upon determining the particular predicate, the rest are of course to be removed: or if all all the predicates but one are removed, that one necessarily takes place. Thus, in the example given above; if we allow the world to be the work of some wise and powerful cause, we of course deny it to be self-existent; or if we deny it to be self-existent, we must necessarily admit that it was produced by some wise and powerful cause. Now this particular manner of linking the predicates together, so that the establishing one displaces all the rest; or the excluding all but one necessarily establishes that one; cannot otherwise be effected than by means of disjunctive particles. And hence it is that propositions of this class take their names from these particles which make so necessary a part of them, and indeed constitute their very nature considered as a distinct species.

CHAP. VI. Of the Division of Propositions into Self-evident and Demonstrable.

I. When any proposition is offered to the view of the mind, if the terms in which it is expressed be understood; upon comparing the ideas together, the agreement or disagreement asserted is either immediately perceived, or found to lie beyond the present reach of the understanding. In the first case the proposition is said to be self-evident, and admits not of any proof, because a bare attention to the ideas themselves produces full conviction and certainty; nor is it possible to call in anything more evident by way of confirmation. But where the connexion or repugnance comes not to readily under the inspection of the mind, there we must have recourse to reasoning; and if by a clear series of proofs we can make out the truth proposed, inasmuch that self-evidence shall accompany every step of the procedure, we are then able to demonstrate what we assert, and the proposition itself is said to be demonstrable. When we affirm, for instance, that it is impossible for the same thing to be and not to be; whoever understands the terms made use of perceives at first glance the truth of what is asserted, nor can he by any efforts bring himself to believe the contrary. The proposition therefore is self-evident, and such that it is impossible by reasoning to make it plainer; because there is no truth more obvious or better known, from which as a consequence it may be deduced. But if we say, This world had a beginning; the assertion is indeed equally true, but shines not forth with the same degree of evidence. We find great difficulty in conceiving how the world could be made out of nothing; and are not brought to a free and full content, until by reasoning we arrive at a clear view of the absurdity involved in the contrary supposition. Hence this proposition is of the kind we call demonstrable, inasmuch as its truth is not immediately perceived by the mind, but yet may be made appear by means of others more known and obvious, whence it follows as an unavoidable consequence.

II. From what has been said, it appears, that reasoning is employed only about demonstrable propositions, and that our intuitive and self-evident perceptions are the ultimate foundation on which it rests.

III. Self-evident propositions furnish the first principles of reasoning; and it is certain, that if in our researches we employ only such principles as have this character of self-evidence, and apply them according to the rules to be afterwards explained, we shall be in no danger of error in advancing from one discovery to another. For this we may appeal to the writings of the mathematicians, which, being conducted by the express model here mentioned, are an incontrovertible proof of the firmness and stability of human knowledge, when built upon so sure a foundation. For not only have the propositions of this science stood the test of ages; but are found attended with that invincible evidence, as forces the assent of all who duly consider the proofs upon which they are established. Since the mathematicians are universally allowed to have hit upon the right method of arriving at unknown truths, since they have been the happiest in the choice as well as the application of their principles, it may not be amiss to explain here their method of stating self-evident propositions, and applying them to the purposes of demonstration.

IV. First then it is to be observed, that they have been very careful in ascertaining their ideas, and fixing a great signification of their terms. For this purpose help to begin with definitions, in which the meaning of their words is so distinctly explained, that they can scarcely fail to excite in the mind of an attentive reader knowledge of the very same ideas as are annexed to them by the writer. And indeed the clearness and irresistible evidence of mathematical knowledge is owing to nothing so much as this care in laying the foundation. Where the relation between any two ideas is accurately and justly traced, it will not be difficult for another to comprehend that relation, if in setting himself to discover it he brings the very same ideas into comparison. But if, on the contrary, he affixes to his words ideas different from those that were in the mind of him who first advanced the demonstration; it is evident that as the same ideas are not compared, the same relation cannot subsist, inasmuch that a proposition will be rejected as false, which, had the terms been rightly understood, must have appeared incontrovertibly true. A square, for instance, is a figure bounded by four equal right lines, joined together at right angles. Here the nature of the angles makes no less a part of the idea than the equality of the sides: and many properties demonstrated of the square flow entirely from its being a rectangular figure. If therefore we suppose a man, who has formed a partial notion of a square, comprehending only the equality of its sides, without regard to the angles, reading some demonstration that implies also this latter consideration; it is plain he would reject it as not universally true, inasmuch as it could not be applied where the sides were joined together at equal angles. For this last figure, answering still to his idea of a square, would be yet found without the property assigned to it in the proposition. But if he comes afterwards to correct his notion, and render his idea complete, he will then readily own the truth and justness of the demonstration.

V. We see, therefore, that nothing contributes so much to the improvement and certainty of human knowledge, as the having determinate ideas, and procure a keeping them steady and invariable in all our discourse and reasonings about them. And on this account it is, that mathematicians, as was before observed, always advance... always begin by defining their terms, and distinctly unfolding the notions they are intended to express. Hence such as apply themselves to these studies have exactly the same views of things; and, bringing always the very same ideas into comparison, readily discern the relations between them. It is likewise of importance, in every demonstration, to express the same idea invariably by the same word. From this practice mathematicians never deviate; and if it be necessary in their demonstrations, where the reader's comprehension is aided by a diagram, it is much more so in all reasonings about moral or intellectual truths where the ideas cannot be represented by a diagram. The observation of this rule may sometimes be productive of ill-founded periods; but when truth is the object, sound ought to be despised.

VI. When the mathematicians have taken this first step, and made known the ideas whose relations they intend to investigate; their next care is, to lay down some self-evident truths, which may serve as a foundation for their future reasonings. And here indeed they proceed with remarkable circumspection, admitting no principles but what flow immediately from their definitions, and necessarily force themselves upon a mind in any degree attentive to its ideas. Thus a circle is a figure formed by a right line moving round some fixed point in the same plane. The fixed point round which the line is supposed to move, and where one of its extremities terminates, is called the centre of the circle. The other extremity, which is conceived to be carried round until it returns to the point whence it first let out, describes a curve running into itself, and termed the circumference. All right lines drawn from the centre to the circumference are called radii. From these definitions compared, geometricians derive this self-evident truth; that the radii of the same circle are all equal to one another.

VII. We now observe, that in all propositions we either affirm or deny some property of the idea that constitutes the subject of our judgement, or we maintain that something may be done or effected. The first sort are called speculative propositions, as in the example mentioned above, the radius of the same circle are all equal one to another. The others are called practical, for a reason too obvious to be mentioned; thus, that a right line may be drawn from one point to another is a practical proposition; inasmuch as it expresses that something may be done.

VIII. From this twofold consideration of propositions arises the twofold division of mathematical principles into axioms and postulates. By an axiom they understand any self-evident speculative truth; as, That the whole is greater than its parts; That things equal to one and the same thing are equal to one another. But a self-evident practical proposition is what they call a postulate. Such are those of Euclid; that a finite right line may be continued directly forwards; that a circle may be described about any centre with any distance. And here we are to observe, that as in an axiom the agreement or disagreement between the subject and predicate must come under the immediate inspection of the mind; so in a postulate, not only the possibility of the thing asserted must be evident at first view, but also the manner in which it may be effected. For where this manner is not of itself apparent, the proposition comes under the notion of the demonstrable kind and is treated as such by geometrical writers. Thus, to draw a right line from one point to another, is assumed by Euclid as a postulate, because the manner of doing it is so obvious, as to require no previous teaching. But then it is not equally evident, how we are to construct an equilateral triangle. For this reason he advances it as a demonstrable proposition, lays down rules for the exact performance, and at the same time proves, that if these rules are followed, the figure will be justly described.

IX. This leads us to take notice, that as self-evident and demonstrable truths are distinguished into different kinds, according as they are speculative or practical; so is it also with demonstrable propositions. A demonstrable speculative proposition is by mathematicians called a theorem. Such is the famous 47th proposition of the first book of the Elements, known by the name of the Pythagorean theorem, from its supposed inventor Pythagoras, viz. "that in every right-angled triangle, the square described upon the side subtending the right angle is equal to both the squares described upon the sides containing the right angle." On the other hand, a demonstrable practical proposition is called a problem; as where Euclid teaches us to describe a square upon a given right line.

X. It may not be amiss to add, that, besides the Corollaries four kinds of propositions already mentioned, mathematicians have also a fifth, known by the name of corollaries. These are usually subjoined to theorems or problems, and differ from them only in this: that they flow from what is there demonstrated in so obvious a manner as to discover their dependence upon the proposition whence they are deduced, almost as soon as propounded. Thus Euclid having demonstrated, "that in every right-lined triangle all the three angles taken together are equal to two right angles;" adds by way of corollary, "that all the three angles of any one triangle taken together are equal to all the three angles of any other triangle taken together:" which is evident at first sight; because in all cases they are equal to two right ones, and things equal to one and the same thing are equal to one another.

XI. The scholia of mathematicians are indifferently annexed to definitions, propositions, or corollaries; serve the same purpose as annotations upon a work of a classic author. For in them occasion is taken to explain whatever may appear intricate or obscure in a train of reasoning; to answer objections; to teach the application and uses of propositions; to lay open the original and history of the several discoveries made in the science; and, in a word, to acquaint us with all such particulars as deserve to be known, whether considered as points of curiosity or profit. PART III. OF REASONING.

CHAP. I. Of Reasoning in general, and the Parts of which it consists.

IT often happens in comparing ideas together, that their agreement or disagreement cannot be discerned at first view, especially if they are of such a nature as not to admit of an exact application one to another.

When, for instance, we compare two figures of a different make, in order to judge of their equality or inequality, it is plain, that by barely considering the figures themselves, we cannot arrive at an exact determination; because, by reason of their disagreeing forms, it is impossible so to put them together, as that their several parts shall mutually coincide. Here then it becomes necessary to look out for some third idea that will admit of such an application as the present case requires; wherein if we succeed, all difficulties vanish, and the relation we are in quest of may be traced with ease. Thus, right-lined figures are all reduced to squares, by means of which we can measure their areas, and determine exactly their agreement or disagreement in point of magnitude.

II. But how can any third idea serve to discover a relation between two others? The answer is, By being compared severally with these others; for such a comparison enables us to see how far the ideas with which this third is compared are connected or disjoined between themselves. In the example mentioned above of two right-lined figures, if we compare each of them with some square whose area is known, and find the one exactly equal to it, and the other less by a square inch, we immediately conclude that the area of the first figure is a square inch greater than that of the second. This manner of determining the relation between any two ideas, by the intervention of some third with which they may be compared, is that which we call reasoning; and is indeed the chief instrument by which we push on our discoveries, and enlarge our knowledge. The great art lies in finding out such intermediate ideas, as when compared with the others in the question, will furnish evident and known truths; because, as will afterwards appear, it is only by means of them that we arrive at the knowledge of what is hidden and remote.

III. Hence it appears, that every act of reasoning that constitutes an act wherein the ideas whose relation we want to discover are severally compared with the middle idea, and a third wherein they are themselves connected or disjoined, according to the result of that comparison. Now, as in the second part of logic, our judgements, when put into words, were called propositions, so here in the third part the expressions of our reasonings are termed syllogisms. And hence it follows, that as every act of reasoning implies three several judgements, so every syllogism must include three distinct propositions. When a reasoning is thus put into words, and appears in form of a syllogism, the intermediate idea made use of, to discover the agreement or disagreement, we search for, is called the middle term; and the two ideas themselves with which this third is compared, go by the name of the extremes.

VI. But as these things are best illustrated by examples; let us, for instance, set ourselves to inquire whether men are accountable for their actions. As the account-relation between the ideas of man and accountability, comes not within the immediate view of the mind, our first care must be to find out some third idea that will enable us the more easily to discover and trace it. A very small measure of reflection is sufficient to inform us, that no creature can be accountable for his actions, unless we suppose him capable of distinguishing the good from the bad; that is, unless we suppose him possessed of reason. Nor is this alone sufficient. For what would it avail him to know good from bad actions, if he had no freedom of choice, nor could avoid the one and pursue the other? hence it becomes necessary to take in both considerations in the present case. It is at the same time equally apparent, that wherever there is ability of distinguishing good from bad actions, and of pursuing the one and avoiding the other, there also a creature is accountable. We have then got a third idea, with which accountability is inseparably connected, viz. reason and liberty; which are here to be considered as making up one complex conception. Let us now take this middle idea, and compare it with the other term in the question, viz. man, and we all know by experience that it may be affirmed of him. Having thus by means of the intermediate idea formed two several judgements, viz. that man is possessed of reason and liberty; and that reason and liberty imply accountability; a third obviously and necessarily follows, viz. that man is accountable for his actions. Here then we have a complete act of reasoning, in which, according to what has been already observed, there are three distinct judgements: two that may be styled previous, inasmuch as they lead to the other, and arise from comparing the middle idea with the two ideas in the question: the third is a consequence of these previous acts, and flows from combining the extreme ideas between themselves. If now we put this reasoning into words, it exhibits what logicians term a syllogism; and, when proposed in due form, runs thus:

"Every creature possessed of reason and liberty is accountable for his actions. Man is a creature possessed of reason and liberty: Therefore man is accountable for his actions."

V. In this syllogism we may observe, that there are premises, three several propositions expressing the three judge-conclusions implied in the act of reasoning; and so disposed, as to represent distinctly what passes within the mind in tracing the more distant relations of its ideas. The two first propositions answer the two previous judgements in reasoning, and are called the premises, because they are placed before the other. The third is termed the conclusion, as being gained in consequence of what was asserted in the premises. We are also to remember, that the terms expressing the two ideas whose relations we inquire after, as here man and accountability, are in general called the extremes; and that the intermediate idea, by means of which the relation is traced, viz., a creature possessed of reason and liberty, takes the name of the middle term. Hence it follows, that by the premises of a syllogism we are always to understand the two propositions where the middle term is severally compared with the extremes; for these constitute the previous judgements, whence the truth we are in quest of is by reasoning deduced. The conclusion is that other proposition, in which the extremes themselves are joined or separated agreeably to what appears upon the above comparison.

VI. The conclusion is made up of the extreme terms of the syllogism: and the extreme, which serves as the predicate of the conclusion, goes by the name of the major term: the other extreme, which makes the subject in the same proposition, is called the minor term. From this distinction of the extremes arises also a distinction between the premises, where these extremes are severally compared with the middle term. That proposition which compares the greater extreme, or the predicate of the conclusion, with the middle term, is called the major proposition: the other, wherein the same middle term is compared with the subject of the conclusion or lesser extreme, is called the minor proposition. All this is obvious from the syllogism already given, where the conclusion is, Man is accountable for his actions. For here the predicate accountable for his actions being connected with the middle term in the first of the two premises, every creature possessed of reason and liberty is accountable for his actions, gives what we call the major proposition. In the second of the premises, man is a creature possessed of reason and liberty, we find the lesser extreme, or subject of the conclusion, viz., man, connected with the same middle term, whence it is known to be the minor proposition. When a syllogism is proposed in due form, the major proposition is always placed first, the minor next, and the conclusion last.

VII. These things premised, we may in the general define reasoning to be an act or operation of the mind, deducing some unknown proposition from other previous ones that are evident and known. These previous propositions, in a simple act of reasoning, are only two in number; and it is always required that they be of themselves apparent to the understanding, inasmuch that we attend to and perceive the truth of them as soon as proposed. In the syllogism given above, the premises are supposed to be self-evident truths; otherwise the conclusion could not be inferred by a single act of reasoning. If, for instance, in the major, every creature possessed of reason and liberty is accountable for his actions, the connexion between the subject and predicate could not be perceived by a bare attention to the ideas themselves; it is evident that this proposition would not itself require a proof than the conclusion deduced from it. In this case a new middle term must be sought for, to trace the connexion here supposed; and this of course furnishes another syllogism, by which having established the proposition in question, we are then, and not before, at liberty to use it in any succeeding train of reasoning. And should it so happen, that in this second essay there was still some previous proposition whose truth did not appear at first sight, we must then have recourse to a third syllogism, in order to lay open that truth to the mind: because so long as the premises remain uncertain, the conclusion built upon them must be so too. When, by conducting our thoughts in this manner, we at last arrive at some syllogism where the previous propositions are intuitive truths; the mind then rests in full security, as perceiving that the several conclusions it has passed through stand upon the immoveable foundation of self-evidence, and when traced to their source terminate in it.

VIII. We see, therefore, that in order to infer a conclusion by a single act of reasoning, the premises in the highest must be intuitive propositions. Where they are not, yet exercise previous syllogisms are required; in which case reasoning becomes a complicated act, taking in a variety of steps of successive steps. This frequently happens in tracing the more remote relation of our ideas; where, many middle terms being called in, the conclusion cannot be made out but in consequence of a series of syllogisms following one another in train. But although in this concatenation of propositions, those that form the premises of the last syllogism are often considerably removed from self-evidence; yet if we trace the reasoning backwards, we shall find them the conclusions of previous syllogisms, whose premises approach nearer and nearer to intuition in proportion as we advance, and are found at last to terminate in it. And if, after having thus unravelled a demonstration, we take it the contrary way; and observe how the mind, setting out with intuitive perceptions, couples them together to form a conclusion: how, by introducing this conclusion into another syllogism, it still advances one step farther; and so proceeds, making every new discovery subservient to its future progress; we shall then perceive clearly, that reasoning, in the highest sense of that faculty, is no more than an orderly combination of those simple acts which we have already fully explained.

IX. Thus we see, that reasoning, beginning with first principles, rises gradually from one judgement to another, and connects them in such manner, that every stage of the progression brings intuitive certainty along of the process with it. And now at length we may clearly understand the definition given above of this distinguishing faculty of the human mind. Reason, we have said, is the ability of deducing unknown truths from principles or propositions that are already known. This evidently appears by the foregoing account, where we see that no proposition is admitted into a syllogism, to serve as one of the previous judgements on which the conclusion rests, unless it is itself a known and established truth, whose connexion with self-evident principles has been already traced.

CHAP. II. Of the several kinds of Reasoning: and firstly, of that by which we determine the Genera and Species of Things.

I. All the aims of human reason may in the general Reasoning be reduced to these two: 1. To rank things under twofold, those universal ideas to which they truly belong; and, 2. To ascribe to them their several attributes and properties in consequence of that distribution. II. One great aim of human reason is to determine the genera and species of things. We have seen in the First Part of this treatise, how the mind proceeds in framing general ideas*. We have also seen in the Second Part, how by means of these general ideas we come by universal propositions. Now as in these universal propositions we affirm some property of a genus or species, it is plain that we cannot apply this property to particular objects till we have first determined whether they are comprehended under that general idea of which the property is affirmed. Thus there are certain properties belonging to all even numbers, which nevertheless cannot be applied to any particular number, until we have first discovered it to be of the species expressed by that natural name. Hence reasoning begins with referring things to their several divisions and classes in the scale of our ideas; and as these divisions are all distinguished by particular names, we hereby learn to apply the terms expressing general conceptions to such particular objects as come under our immediate observation.

III. Now, in order to arrive at these conclusions, by which the several objects of perception are brought under general names, two things are manifestly necessary. First, That we take a view of the idea itself denoted by that general name, and carefully attend to the distinguishing marks which serve to characterize it. Secondly, That we compare this idea with the object under consideration, observing diligently wherever they agree or differ. If the idea is found to correspond with the particular object, we then without hesitation apply the general name; but if no such correspondence intervenes, the conclusion must necessarily take a contrary turn. Let us, for instance, take the number eight, and consider by what steps we are led to pronounce it an even number. First then, we call to mind the idea signified by the expression an even number, viz. that it is a number divisible into two equal parts. We then compare this idea with the number eight, and finding them manifestly to agree, see at once the necessity of admitting the conclusion. These several judgements therefore transferred into language, and reduced to the form of a syllogism, appear thus:

"Every number that may be divided into two equal parts is an even number: "The number eight may be divided into two equal parts; "Therefore the number eight is an even number."

IV. Here it may be observed, that where the general idea, to which particular objects are referred, is very familiar to the mind, and frequently in view; this reference, and the application of the general name, seem to be made without any apparatus of reasoning. When we see a horse in the fields, or a dog in the street, we readily apply the name of the species; habit, and a familiar acquaintance with the general idea, suggesting it instantaneously to the mind. We are not however to imagine on this account that the understanding departs from the usual rules of just thinking. A frequent repetition of acts begets a habit; and habits are attended with a certain promptness of execution, that prevents our observing the several steps and gradations by which any course of action is accomplished. But in other instances, where we judge not by precontracted habits, as when the general idea is very complex, or less familiar to the mind, we always proceed according to the form of reasoning established above. A goldsmith, for instance, who is in doubt as to any piece of metal, whether it be of the species called gold, first examines its properties, and then comparing them with the general idea signified by that name, if he finds a perfect correspondence, no longer hesitates under what class of metals to rank it.

V. Nor let it be imagined that our researches here, the great because in appearance bounded to the imposing of general names upon particular objects, are therefore of little consequence. Some of the most considerable debates among mankind, and such too as nearly regard their lives, interests, and happiness, turn wholly upon this article. Is it not the chief employment of our several courts of judicature to determine in particular instances, what is law, justice, and equity? Of what importance is it in many cases to decide aright whether an action shall be termed murder or manslaughter? We see then that no less than the lives and fortunes of men depend often upon these decisions. The reason is plain. Actions, when once referred to a general idea, draw after them all that may be affirmed of that idea; inasmuch that the determining the species of actions is all one with determining what proportion of praise or dispraise, commendation or blame, &c. ought to follow them. For as it is allowed that murder deserves death; by bringing any particular action under the head of murder, we of course decide the punishment due to it.

VI. But the great importance of this branch of reasoning, and the necessity of care and circumspection in referring particular objects to general ideas, is still farther evident from the practice of the mathematicians. Every one who has read Euclid, knows, that cians frequently requires us to draw lines through certain points, and according to such and such directions. The figures thence resulting are often squares, parallelograms, or rectangles. Yet Euclid never supposes this from their bare appearance, but always demonstrates it upon the strictest principles of geometry. Nor is the method he takes in anything different from that described above. Thus, for instance, having defined a square to be a figure bounded by four equal sides joined together at right angles; when such a figure arises in any construction previous to the demonstration of a proposition, yet he never calls it by that name until he has shown that its sides are equal, and all its angles right ones. Now this is apparently the same form of reasoning we have before exhibited in proving eight to be an even number.

VII. Having thus explained the rules by which we are to conduct ourselves in ranking particular objects under general ideas, and shown their conformity to a steady practice and manner of the mathematicians; it remains only to observe, that the true way of rendering names, rendering this part of knowledge both easy and certain, is by habituating ourselves to clear and determinate ideas, and keeping them steadily annexed to their respective words. For as all our aim is to apply general words and certain ideas, if these words stand for invariable ideas that are perfectly known to the mind, and can be readily distinguished upon occasion, there will be little danger of mistake or error in our reasonings. Let us suppose that, by examining any object, and carrying our attention successively from one part to another, we have acquainted ourselves with the several particulars observable in it. If among these we find such as constitute some general idea, framed and settled beforehand by the understanding, and distinguished by a particular name, the resemblance thus known and perceived necessarily determines the species of the object, and thereby gives it a right to the name by which that species is called. Thus four equal sides, joined together at right angles, make up the notion of a square. As this is a fixed and invariable idea, without which the general name cannot be applied; we never call any particular figure a square until it appears to have these several conditions; and contrarily, wherever a figure is found with these conditions, it necessarily takes the name of a square. The same will be found to hold in all our other reasonings of this kind, where nothing can create any difficulty but the want of settled ideas. If, for instance, we have not determined within ourselves the precise notion denoted by the word manlaughter, it will be impossible for us to decide whether any particular action ought to bear that name: because, however nicely we examine the action itself, yet, being strangers to the general idea with which it is to be compared, we are utterly unable to judge of their agreement or disagreement. But if we take care to remove this obstacle, and distinctly trace the two ideas under consideration, all difficulties vanish, and the resolution becomes both easy and certain.

VIII. Thus we see of what importance it is towards the improvement and certainty of human knowledge, that we accustom ourselves to clear and determinate ideas, and a steady application of words.

CHAP. III. Of Reasoning, as it regards the powers and Properties of Things, and the Relations of our general Ideas.

I. We now come to the second great end which men have in view in their reasonings; namely, the discovering and ascribing to things their several attributes and properties. And here it will be necessary to distinguish between reasoning, as it regards the sciences, and as it concerns common life. In the sciences, our reason is employed chiefly about universal truths, it being by them alone that the bounds of human knowledge are enlarged. Hence the division of things into various classes, called otherwise genera and species. For these universal ideas being set up as the representatives of many particular things, whatever is affirmed of them may be also affirmed of all the individuals to which they belong. Murder, for instance, is a general idea, representing a certain species of human actions. Reason tells us that the punishment due to it is death. Hence every particular action, coming under the notion of murder, has the punishment of death allotted to it. Here then we apply the general truth to some obvious instance; and this is what properly constitutes the reasoning of common life. For men, in their ordinary transactions and intercourse one with another, have, for the most part, to do only with particular objects. Our friends and relations, their characters and behaviour, the constitution of the several bodies that surround us, and the uses to which they may be applied, are what chiefly engage our attention. In all these, we reason about particular things; and the whole result of our reasoning is, the applying the general truths of the sciences in the ordinary transactions of human life. When we see a viper, we avoid it. Wherever we have occasion for the forcible action of water to move a body that makes considerable resistance, we take care to convey it in such a manner that it shall fall upon the object with impetuosity. Now all this happens in consequence of our familiar and ready application of these two general truths. The bite of a viper is mortal. Water falling upon a body with impetuosity, acts very forcibly towards setting it in motion. In like manner, if we set ourselves to consider any particular character, in order to determine the share of praise or dispraise that belongs to it, our great concern is to ascertain exactly the proportion of virtue and vice. The reason is obvious. A just determination, in all cases of this kind, depends entirely upon an application of these general maxims of morality: Virtuous actions deserve praise; vicious actions deserve blame.

II. Hence it appears that reasoning, as it regards the common life, is no more than the ascribing the general properties of things to those several objects with which we are more immediately concerned according as they are found to be of that particular division or common class to which the proprieties belong. The steps then by which we proceed are manifestly these. First, We refer the object under consideration to some general idea or class of things. We then recollect the several attributes of that general idea. And, lastly, Ascribe all those attributes to the present object. Thus, in considering the character of Sempronius, if we find it to be of the kind called virtuous, when we at the same time reflect that a virtuous character is deserving of esteem, it naturally and obviously follows that Sempronius is so too. These thoughts put into a syllogism, in order to exhibit the form of reasoning here required, run thus:

"Every virtuous man is worthy of esteem. "Sempronius is a virtuous man: "Therefore Sempronius is worthy of esteem."

III. By this syllogism it appears, that before we affirm any thing of a particular object, that object must next be referred to some general idea. Sempronius is pronounced worthy of esteem only in consequence of his being a virtuous man, or coming under that general branch of notion. Hence we see the necessary connexion of the reasoning various parts of reasoning, and the dependence they one upon another. The determining the genera and species of things is, as we have said, one exercise of human reason; and here we find that this exercise is the first in order, and previous to the other, which consists in ascribing to them their powers, properties, and relations. But when we have taken this previous step, and brought particular objects under general names; as the properties we ascribe to them are no other than those of the general idea, it is plain that, in order to a successful progress in this part of knowledge, we must thoroughly acquaint ourselves with the several relations and attributes of these our general ideas. Part III.

When this is done, the other part will be easy, and requires scarce any labour or thought, as being no more than an application of the general form of reasoning represented in the foregoing syllogism. Now, as we have already sufficiently shown how we are to proceed in determining the genera and species of things, which, as we have said, is the previous step to this second branch of human knowledge; all that is farther wanting towards a due explanation of it is, to offer some considerations as to the manner of investigating the general relations of our ideas. This is the highest exercise of the powers of the understanding, and that by means whereof we arrive at the discovery of universal truths; insomuch that our deductions in this way constitute that particular species of reasoning which we have before laid regards principally the sciences.

IV. But that we may conduct our thoughts with some order and method, we shall begin with observing, that the relations of our general ideas are of two kinds: either such as immediately discover themselves, upon comparing the ideas one with another; or such as, being more remote and distant, require art and contrivance to bring them into view. The relations of the first kind furnish us with intuitive and self-evident truths; those of the second are traced by reasoning, and a due application of intermediate ideas. It is of this last kind that we are to speak here, having depauperated what was necessary with regard to the other in the Second Part. As, therefore, in tracing the more distant relations of things, we must always have recourse to intervening ideas, and are more or less successful in our researches according to our acquaintance with these ideas, and ability of applying them; and it is evident, that to make a good reasoner, two things are principally required. First, An extensive knowledge of those intermediate ideas, by means of which things may be compared one with another. Secondly, The skill and talent of applying them happily in all particular instances that come under consideration.

V. In order to our successful progress in reasoning, we must have an extensive knowledge of those intermediate ideas by means of which things may be compared one with another. For as it is not every idea that will answer the purpose of our inquiries, but such only as are peculiarly related to the objects about which we reason, so as, by a comparison with them to furnish evident and known truths; nothing is more apparent than that the greater variety of conceptions we can call into view, the more likely we are to find some among them that will help us to the truths here required. And, indeed, it is found to hold in experience, that in proportion as we enlarge our views of things, and grow acquainted with a multitude of different objects, the reasoning faculty gathers strength; for, by extending our sphere of knowledge, the mind acquires a certain force and penetration, as being accustomed to examine the several appearances of its ideas, and observe what light they cast one upon another.

VI. This is the reason why, in order to excel remarkably in any one branch of learning, it is necessary to have at least a general acquaintance with the whole circle of arts and sciences. The truth of it is, all the various divisions of human knowledge are very nearly related among themselves, and, in innumerable instances, serve to illustrate and set off each other.

And although it is not to be denied that, by an ob- To excel in flinate application to one branch of study, a man may anyone make considerable progress, and acquire some degree branch of of eminence in it; yet his views will be always narrow and contracted, and he will want that matterly in general discernment which not only enables us to pursue our acquaintance discoveries with ease, but also, in laying them open with the others, to spread a certain brightness around them. But when our reasoning regards a particular science, and science it is farther necessary that we more nearly acquaint ourselves with whatever relates to that science. A general knowledge is a good preparation, and enables us to proceed with ease and expedition in whatever branch of learning we apply to. But then, in the minute and intricate questions of any science, we are by no means qualified to reason with advantage until we have perfectly mastered the science to which they belong.

VII. We come now to the second thing required, namely, the skill and talent of applying intermediate ideas happily in all particular instances that come under consideration. And here, rules and precepts are pily in particular instances. Use and experience are the best instructors. For, whatever logicians may boast of being able to form perfect reasoners by book and rule, we find by experience, that the study of their precepts does not always add any great degree of strength to the understanding. In short, it is the habit alone of reasoning that makes a reasoner. And therefore the true way to acquire this talent is, by being much conversant in those sciences where the art of reasoning is allowed to reign in the greatest perfection. Hence it was that the ancients, who so well understood the manner of forming the mind, always began with mathematics, as the foundation of their philosophical studies. Here the understanding is by degrees habituated to truth, contracts infinitely a certain fondness for it, and learns never to yield its affe- fect to any proposition but where the evidence is sufficient to produce full conviction. For this reason Plato has called mathematical demonstrations the cathartic or purgatives of the soul, as being the proper means to cleanse it from error, and restore that natural exercise of its faculties in which just thinking consists.

VIII. If therefore we would form our minds to a The study habit of reasoning closely and in train, we cannot take any more certain method than the exercising our- selves in mathematical demonstrations, so as to contract a kind of familiarity with them. Not that we look great avail upon it as necessary that all men should be deep ma, in this re- thematicians; but that, having got the way of reason- ing which that study necessarily brings the mind to, they may be able to transfer it to other parts of knowl- ledge, as they shall have occasion.

IX. But although the study of mathematics be of all others the most useful to form the mind and give it other sub- an early relish of truth, yet ought not other parts of objects, as are distinguished philosophy to be neglected. For there also we meet for with many opportunities of exercising the powers of strength the understanding; and the variety of subjects nature justifies rally of reasoning. rally leads us to observe all those different turns of thinking that are peculiarly adapted to the several ideas we examine, and the truth we search after. A mind thus trained acquires a certain mastery over its own thoughts, inasmuch that it can range and model them at pleasure, and call such into view as best suit its present designs. Now in this the whole art of reasoning consists; from among a great variety of different ideas to single out those that are most proper for the business in hand, and to lay them together in such order, that from plain and easy beginnings, by gentle degrees, and a continued train of evident truths, we may be insensibly led on to such discoveries, as at our first setting out appeared beyond the reach of human understanding. For this purpose, besides the study of mathematics before recommended, we ought to apply ourselves diligently to the reading of such authors as have distinguished themselves for strength of reasoning, and a just and accurate manner of thinking. For it is observable, that a mind exercised and seasoned to truth, seldom rests satisfied in a bare contemplation of the arguments offered by others; but will be frequently assaying its own strength, and pursuing its discoveries upon the plan it is most accustomed to. Thus we insensibly contract a habit of tracing truth from one stage to another, and of investigating those general relations and properties which we afterwards ascribe to particular things, according as we find them comprehended under the abstract ideas to which the properties belong.

CHAP. IV. Of the Forms of Syllogisms.

I. HITHERTO we have contented ourselves with a general notion of syllogisms, and of the parts of which they consist. It is now time to enter a little more particularly into the subject, to examine their various forms, and lay open the rules of argumentation proper to each. In the syllogisms mentioned in the foregoing chapters, we may observe, that the middle term is the subject of the major proposition, and the predicate of the minor. This disposition, though the most natural and obvious, is not however necessary; it frequently happens, that the middle term is the subject in both the premises, or the predicate in both; and sometimes, directly contrary to its disposition in the foregoing chapters, the predicate in the major, and the subject in the minor. Hence the division of syllogisms into various kinds, called figures by logicians. For figure, according to their use of the word, is nothing else but the order and disposition of the middle term in any syllogism. And as this disposition is, we see, fourfold, so the figures of syllogisms thence arising are four in number. When the middle term is the subject of the major proposition, and the predicate of the minor, we have what is called the first figure; As,

"No work of God is bad: "The natural passions and appetites of men are "the work of God: "Therefore none of them is bad."

If, on the other hand, it is the predicate of both the premises, the syllogism is said to be the second figure: As,

"Whatever is bad is not the work of God: "All the natural passions and appetites of men "are the work of God: "Therefore the natural passions and appetites of "men are not bad."

Again, In the third figure, the middle term is the subject of the two premises: As,

"All Africans are black: "All Africans are men: "Therefore some men are black."

And lastly, By making it the predicate of the major, and subject of the minor, we obtain syllogisms in the fourth figure: As,

"The only Being who ought to be worshipped is "the Creator and Governor of the world: "The Creator and Governor of the world is "God: "Therefore God is the only Being who ought to "be worshipped."

II. But, besides this fourfold distinction of syllogisms, there is also a farther subdivision of them in every figure, arising from the quantity and quality, as they are called, of the propositions. By quantity we mean the consideration of propositions, as universal or particular; by quality, as affirmative or negative.

Now as, in all the several dispositions of the middle term, the propositions of which a syllogism consists may be either universal or particular, affirmative or negative; the due determination of these, and so putting them together as the laws of argumentation require, constitute what logicians call the moods of syllogisms. Of these moods there is a determinate number to every figure, including all the possible ways in which propositions differing in quantity or quality can be combined, according to any disposition of the middle term, in order to arrive at a just conclusion.

The first figure has only four legitimate moods. The major proposition in this figure must be universal, and the minor affirmative; and it has this property, that it yields conclusions of all kinds, affirmative and negative, universal and particular.

The second figure has also four legitimate moods. Its major proposition must be universal, and one of the premises must be negative. It yields conclusions both universal and particular, but all negative.

The third figure has six legitimate moods. Its minor must always be affirmative; and it yields conclusions both affirmative and negative, but all particular.

These are all the figures which were admitted by the inventor of syllogisms, and of which, so far as we know, the number of legitimate moods has been ascertained, and severally demonstrated. In every figure it will be found upon trial, that there are sixty-four different moods of syllogism; and he who thinks it worth while to construct so many in the fourth figure, always remembering that the middle term in each must be the predicate of the major and the subject of the minor proposition, will easily discern what number of these moods are legitimate, and give true conclusions.

Besides the rules that are proper to each figure, Aristotle has given some that are common to all, by which the legitimacy of syllogisms may be tried. Part III.

There may be reduced to five:

1. There must be only three terms in a syllogism: As each term occurs in two of the propositions, it must be precisely the same in both; if it be not, the syllogism is said to have four terms, which makes a vicious syllogism.

2. The middle term must be taken universally in one of the premises.

3. Both premises must not be particular propositions, nor both negative.

4. The conclusion must be particular, if either of the premises be particular; and negative, if either of the premises be negative.

5. No term can be taken universally in the conclusion, if it be not taken universally in the premises.

For understanding the second and fifth of these rules, it is necessary to observe, that a term is said to be taken universally, not only when it is the subject of a universal proposition, but also when it is the predicate of a negative proposition. On the other hand, a term is said to be taken particularly, when it is either the subject of a particular or the predicate of an affirmative proposition.

III. The division of syllogisms according to mood and figure reflects those especially which are known by the name of plain simple syllogisms; that is, which are bounded to three propositions, all simple, and where the extremes and middle term are connected, according to the rules laid down above. But as the mind is not tied down to any one precise form of reasoning, but sometimes makes use of more, sometimes of fewer premises, and often takes in compound and conditional propositions, it may not be amiss to take notice of the different forms derived from this source, and explain the rules by which the mind conducts itself in the use of them.

IV. When in any syllogism the major is a conditional proposition, the syllogism itself is termed conditional. Thus:

"If there is a God, he ought to be worshipped: "But there is a God: "Therefore he ought to be worshipped."

In this example, the major, or first proposition, is, we see, conditional, and therefore the syllogism itself is also of the kind called by that name. And here we are to observe, that all conditional propositions are made of two distinct parts: one expressing the condition upon which the predicate agrees or disagrees with the subject, as in this now before us, "if there is a God;" the other joining or disjoining the said predicate and subject, as here, "he ought to be worshipped." The first of these parts, or that which implies the condition, is called the antecedent; the second, where we join or disjoin the predicate and subject, has the name of the consequent.

V. In all propositions of this kind, supposing them to be exact in point of form, the relation between the antecedent and consequent must ever be true and real; that is, the antecedent must always contain some certain and genuine condition, which necessarily implies the consequent; for otherwise the proposition itself will be false, and therefore ought not to be admitted into our reasonings. Hence it follows, that when any conditional proposition is assumed, if we admit the antecedent of that proposition, we must at the same time necessarily admit the consequent; but if we reject the consequent, we are in like manner bound to reject the antecedent. For as the antecedent always expresses some condition which necessarily implies the truth of the consequent; by admitting the antecedent, we allow of that condition, and therefore ought also to admit the consequent. In like manner, if it appears that the consequent ought to be rejected, the antecedent evidently must be so too: because, as was just now demonstrated, the admitting of the antecedent would necessarily imply the admission also of the consequent.

VI. There are two ways of arguing in hypothetical syllogisms, which lead to a certain and unavoidable conclusion. For as the major is always a conditional proposition, consisting of an antecedent and a consequent; if the minor admits the antecedent, it is plain that the conclusion must admit the consequent. This is called arguing from the admission of the antecedent to the admission of the consequent, and constitutes that mood or species of hypothetical syllogisms which is distinguished in the schools by the name of the modus ponens; inasmuch as by it the whole conditional proposition, both antecedent and consequent, is established. Thus:

"If God is infinitely wise, and acts with perfect freedom, he does nothing but what is best: "But God is infinitely wise, and acts with perfect freedom: "Therefore he does nothing but what is best."

Here we see the antecedent or first part of the conditional proposition is established in the minor, and the consequent or second part in the conclusion; whence the syllogism itself is an example of the modus ponens. But if now we on the contrary suppose that the minor reject the consequent, then it is apparent that the conclusion must also reject the antecedent. In this case we are said to argue from the removal of the consequent to the removal of the antecedent, and the particular mood or species of syllogisms thence arising is called by logicians the modus tollens; because in it both antecedent and consequent are rejected or taken away, as appears by the following example:

"If God were not a Being of infinite goodness, neither would he consult the happiness of his creatures: "But God does consult the happiness of his creatures: "Therefore he is a Being of infinite goodness."

VII. These two species take in the whole class of They include all the possible ways of arguing that lead to a legitimate conclusion; because we cannot here proceed by a contrary process of arguing, that is, from the removal of the antecedent to the removal of the consequent, or from the establishing of the consequent to the establishing of the antecedent. For although the antecedent always expresses some real condition, which, once admitted, necessarily implies the consequent, yet it does not follow that there is therefore no other condition; and if so, then, after removing the antecedent, the consequent may still hold, because of some other determination that infers it. When we say, If a stone is exposed some time to the rays of the sun, it will contract a certain degree of heat; the proposition is certainly true; and, admitting the antecedent, we must also admit... admit the consequent. But as there are other ways by which a stone may gather heat, it will not follow, from the ceasing of the before-mentioned condition, that therefore the consequent cannot take place. In other words, we cannot argue: *But the stone has not been exposed to the rays of the sun; therefore neither has it any degree of heat:* Inasmuch as there are a great many other ways by which heat might have been communicated to it. And if we cannot argue from the removal of the antecedent to the removal of the consequent, no more can we from the admission of the consequent to the admission of the antecedent: because, as the consequent may flow from a great variety of different suppositions, the allowing of it does not determine the precise supposition, but only that some one of them must take place. Thus in the foregoing proposition, *If a stone is exposed some time to the rays of the sun, it will contract a certain degree of heat;* admitting the consequent, viz. that it has contracted a certain degree of heat, we are not therefore bound to admit the antecedent, that it has been some time exposed to the rays of the sun: because there are many other causes whence that heat may have proceeded. These two ways of arguing, therefore, hold not in conditional syllogisms.

VIII. As from the major's being a conditional proposition, we obtain the species of conditional syllogisms: so, where it is a disjunctive proposition, the syllogism to which it belongs is also called disjunctive, as in the following example:

"The world is either self-existent, or the work of some finite, or of some infinite Being: "But it is not self-existent, nor the work of a finite being: "Therefore it is the work of an infinite Being."

Now, a disjunctive proposition is that, where of several predicates, we affirm one necessarily to belong to the subject, to the exclusion of all the rest, but leave that particular one undetermined. Hence it follows, that as soon as we determine the particular predicate, all the rest are of course to be rejected; or if we reject all the predicates but one, that one necessarily takes place. When, therefore, in a disjunctive syllogism, the several predicates are enumerated in the major; if the minor establishes any one of these predicates, the conclusion ought to remove all the rest; or if, in the minor, all the predicates but one are removed, the conclusion must necessarily establish that one. Thus, in the disjunctive syllogism given above, the major affirms one of the three predicates to belong to the earth, viz. self-existence, or that it is the work of a finite, or that it is the work of an infinite Being. Two of these predicates are removed in the minor, viz. self-existence, and the work of a finite being. Hence the conclusion necessarily attributes to it the third predicate, and affirms that it is the work of an infinite Being. If now we give the syllogism another turn, inasmuch that the minor may establish one of the predicates, by affirming the earth to be the production of an infinite Being: then the conclusion must remove the other two, asserting it to be neither self-existent, nor the work of a finite being. These are the forms of reasoning in these species of syllogisms, the justness of which appears at first sight: and that there can be no other, is evident from the very nature of a disjunctive proposition.

IX. In the several kinds of syllogisms hitherto mentioned, we may observe that the parts are complete; imperfect that is, the three propositions of which they consist are or mutually represented in form. But it often happens, that some syllogism one of the premises is not only an evident truth, but also familiar and in the minds of all men; in which case it is usually omitted, whereby we have an imperfect syllogism, that seems to be made up of only two propositions. Should we, for instance, argue in this manner:

"Every man is mortal: "Therefore every king is mortal:

the syllogism appears to be imperfect, as consisting but of two propositions. Yet it is really complete; only the minor [*every king is a man*] is omitted: and left to the reader to supply, as being a proposition so familiar and evident that it cannot escape him.

X. These seemingly imperfect syllogisms are called enthymemes; and occur very frequently in reasoning, especially where it makes a part of common conversation. Nay, there is a particular elegance in them, because, not displaying the argument in all its parts, they leave somewhat to the exercise and invention of the mind. By this means we are put upon exerting ourselves, and seem to share in the discovery of what is propounded to us. Now this is the great secret of fine writing, so to frame and put together our thoughts, as to give full play to the reader's imagination, and draw him infallibly into our very views and course of reasoning. This gives a pleasure not unlike to that which the author himself feels in composing. It befits shortens discourse, and adds a certain force and liveliness to our arguments, when the words in which they are conveyed favour the natural quickness of the mind in its operations, and a single expression is left to exhibit a whole train of thoughts.

XI. But there is another species of reasoning with ground of two propositions, which seems to be complete in itself, reasoning in immediate and where we admit the conclusion without supposing intermediate any tacit or suppressed judgement in the mind, from which it follows syllogistically. This happens between propositions, where the connexion is such, that the admission of the one necessarily and at the first sight implies the admission also of the other. For if it so falls out, that the proposition on which the other depends is self-evident, we content ourselves with barely affirming it, and infer that other by a direct conclusion. Thus, by admitting an universal proposition, we are forced also to admit of all the particular propositions comprehended under it, this being the very condition that constitutes a proposition universal. If then that universal proposition chances to be self-evident, the particular ones follow of course, without any farther train of reasoning. Whoever allows, for instance, that things equal to one and the same thing are equal to one another, must at the same time allow, that two triangles, each equal to a square whose side is three inches, are also equal between themselves. This argument therefore,

"Things equal to one and the same thing, are equal to one another: "Therefore," " Therefore these two triangles, each equal to the square of a line of three inches, are equal between themselves"—is complete in its kind, and contains all that is necessary towards a just and legitimate conclusion. For the first or universal proposition is self-evident, and therefore requires no farther proof. And as the truth of the particular is inseparably connected with that of the universal, it follows from it by an obvious and unavoidable consequence.

XII. Now, in all cases of this kind, where propositions are deduced one from another, on account of a known and evident connexion, we are said to reason by immediate consequence. Such a coherence of propositions manifest at first sight, and forcing itself upon the mind, frequently occurs in reasoning. Logicians have explained at some length the several suppositions upon which it takes place, and allow of all immediate consequences that follow in conformity to them. It is however observable, that these arguments, though seemingly complete, because the conclusion follows necessarily from the single proposition that goes before, may yet be considered as real enthymemes, whose major, which is a conditional proposition, is wanting. The syllogism but just mentioned, when represented according to this view, will run as follows:

"If things equal to one and the same thing, are equal to one another; these two triangles, each equal to a square whose side is three inches, are also equal between themselves.

But things equal to one and the same thing, are equal to one another:

Therefore also these triangles, &c. are equal between themselves."

This observation will be found to hold in all immediate consequences whatsoever, infomuch, that they are in fact no more than enthymemes of hypothetical syllogisms. But then it is particular to them, that the ground on which the conclusion rests, namely its coherence with the minor, is of itself apparent, and seen immediately to flow from the rules and reasons of logic.

XIII. The next species of reasoning we shall take notice of here is what is commonly known by the name of a fortis. This is a way of arguing, in which a great number of propositions are so linked together, that the predicate of one becomes continually the subject of the next following, until at last a conclusion is formed, by bringing together the subject of the first proposition, and the predicate of the last. Of this kind is the following argument:

"God is omnipotent: An omnipotent Being can do every thing possible: He that can do every thing possible, can do whatever involves not a contradiction: Therefore God can do whatever involves not a contradiction."

This particular combination of propositions may be continued to any length we please without in the least weakening the ground upon which the conclusion rests. The reason is, because the fortis itself may be resolved into as many simple syllogisms as there are middle terms in it; where this is found universally to hold, that when such a resolution is made, and the syllogisms are placed in train, the conclusion of the last in the series is also the conclusion of the fortis. This kind of argument, therefore, as it serves to unite several syllogisms into one, must stand upon the same foundation with the syllogisms of which it consists, and is indeed, properly speaking, no other than a compendious way of reasoning syllogistically.

XIV. What is here said of plain simple propositions may be as well applied to those that are conditional; that is, any number of them may be so joined together in a series, that the consequent of one shall become continually the antecedent of the next following; in which case, by establishing the antecedent of the first proposition, we establish the consequent of the last, or by removing the last consequent remove also the first antecedent. This way of reasoning is exemplified in the following argument:

"If we love any person, all emotions of hatred towards him cease: If all emotions of hatred towards a person cease, we cannot rejoice in his misfortunes: If we rejoice not in his misfortunes, we certainly wish him no injury: Therefore, if we love a person, we wish him no injury."

It is evident that this fortis, as well as the last, may be resolved into a series of distinct syllogisms, with this only difference, that here the syllogisms are all conditional.

XV. The last species of syllogism we shall take notice of in this chapter is that commonly distinguished of argued by the name of a dilemma. A dilemma is an argument by which we endeavour to prove the absurdity or falsehood of some assertion. In order to this, we assume a conditional proposition, the antecedent of which is the assertion to be disproved, and the consequent a disjunctive proposition, enumerating all the possible suppositions upon which that assertion can take place. If then it appears, that all these several suppositions ought to be rejected, it is plain, that the antecedent or assertion itself must be so too. When therefore such a proposition as that before mentioned is made the major of any syllogism; if the minor rejects all the suppositions contained in the consequent, it follows necessarily, that the conclusion ought to reject the antecedent, which, as we have said, is the very assertion to be disproved. This particular way of arguing is that which logicians call a dilemma; and from the account here given of it, it appears that we may in the general define it to be a hypothetical syllogism, where the consequent of the major is a disjunctive proposition, which is wholly taken away or removed in the minor. Of this kind is the following:

"If God did not create the world perfect in its kind, it must either proceed from want of inclination, or from want of power: But it could not proceed either from want of inclination, or from want of power: Therefore, he created the world perfect in its kind."

Part III.

Of Reasoning.

"kind." Or, which is the same thing: "It is absurd to say that he did not create the world perfect in its kind."

An universal description of it.

XVI. The nature then of a dilemma is universally this. The major is a conditional proposition, whose consequent contains all the several suppositions upon which the antecedent can take place. As therefore these suppositions are wholly removed in the minor, it is evident that the antecedent must be so too; in much that we here always argue from the removal of the consequent to the removal of the antecedent. That is, a dilemma is an argument in the modus tollens of hypothetical syllogisms, as logicians love to speak. Hence it is plain, that if the antecedent of the major is an affirmative proposition, the conclusion of the dilemma will be negative; but if it is a negative proposition, the conclusion will be affirmative.

CHAP. V. Of Induction.

I. All reasoning proceeds ultimately from first truths, either self-evident or taken for granted; and the first truths of syllogistic reasoning are general propositions. But except in the mathematics, and such other sciences as, being conversant about mere ideas, have no immediate relation to things without the mind, we cannot assume as truths propositions which are general. The mathematician indeed may be considered as taking his ideas from the beginning in their general form. Every proposition composed of such ideas is therefore general; and those which are theoretic are reducible to two parts or terms, a predicate and a subject, with a copula generally affirmative. If the agreement or the relation between the two terms be not immediate and self-evident, he has recourse to an axiom, which is a proposition still more general, and which supplies him with a third or middle term. This he compares first with the predicate, and then with the subject, or vice versa. These two comparisons, when drawn out in form, make two propositions, which are called the premises; and if they happen to be immediate and self-evident, the conclusion, consisting of the terms of the question proposed, is said to be demonstrated. This method of reasoning is conducted exactly in the syllogistic form explained in the preceding chapter.

II. But in sciences which treat of things external to the mind, we cannot assume as first principles the most general propositions, and from them infer others less and less general till we descend to particulars. The reason is obvious. Everything in the universe, whether of mind or body, presents itself to our observation in its individual state; so that perception and judgement employed in the investigation of truth, whether physical, metaphysical, moral, or historical, have in the first place to encounter with particulars. With these reason begins, or should begin, its operations. It observes, tries, canvasses, examines, and compares them together, and judges of them by some of those native evidences and original lights, which, as they are the first and indispensable inlets of knowledge to the mind, have been called the primary principles of truth." See Metaphysics.

III. "By such acts of observation and judgement, diligently practised and frequently repeated, on many in which, by individuals of the same class or of a similar nature, not attending their agreements, marking their differences however general or ever minute, and rejecting all instances which, however similar in appearance, are not in effect the same, reason, with much labour and attention, extracts some general laws respecting the powers, properties, qualities, actions, passions, virtues, and relations of real things. This is no hasty, premature, notional abstraction of the mind, by which images and ideas are formed that have no archetypes in nature: it is a rational, operative, experimental process, instituted and executed upon the constitution of beings, which in part compose the universe. By this process reason advances from particulars to generals, from less general to more general, till by a series of slow progression, and by regular degrees, it arrive at the most general notions, called forms or formal causes (c). And by affirming or denying a genus of a species, or an accident of a substance or class of substances, through all the stages of the gradation, we form conclusions, which, if logically drawn, are axioms (d), or general propositions ranged one above another,

(c) Qui formas novit, is, quae adhuc non facta sunt, qualia nec naturae vicissitudines, nec experimentales industrice unquam in actum produxissent, nec cogitationem humanam subiure sufflent, detegit et educit. Baconi Nov. Org.

(d) The word axiom, ἀξίωμα, literally signifies dignity: Hence it is used metaphorically to denote a general truth or maxim, and sometimes any truth that is self-evident, which is called a dignity on account of its importance in a process of reasoning. The axioms of Euclid are propositions extremely general; and so are the axioms of the Newtonian philosophy. But these two kinds of axioms have very different origins. The former appear true upon a bare contemplation of our ideas; whereas the latter are the result of the most laborious induction. Lord Bacon therefore strenuously contends that they should never be taken upon conjecture, or even upon the authority of the learned; but that, as they are the general principles and grounds of all learning, they should be canvassed and examined with the most scrupulous attention, "ut axiomatum corrigatur iniquitas, quae plerumque in exemplis vulgatis fundamentum habent;" De Augm. Sc. lib. ii. cap. 2. "Atque illa ipsa putativa principia ad rationes reddendas compellare decrevimus, quoique plane constat:" Distr. Operis.—Dr Tatham makes a distinction between axioms intuitive and axioms self-evident. Intuitive axioms, according to him, pass through the first inlets of knowledge, and flash direct conviction on the minds, as external objects do on the senses, of all men. Other axioms, though not intuitive, may be properly said to be self-evident; because, in their formation, reason judges by single comparisons without the help of a third idea or middle term; so that they have their evidence in themselves, and though inductively framed they cannot be syllogistically proved. If this distinction be just, and we think it is, only particular truths can be intuitive axioms. IV. "Thus, for instance, the evidence of the external senses is obviously the primary principle from which all physical knowledge is derived. But, whereas nature begins with causes, which, after a variety of changes, produce effects, the senses open upon the effects, and from them, through the slow and painful road of experiment and observation, ascend to causes. By experiments and observations skilfully chosen, artfully conducted, and judiciously applied, the philosopher advances from one stage of inquiry to another in the rational investigation of the general causes of physical truth. From different experiments and observations made on the same individual subject, and from the same experiments and observations made on different subjects of the same kind, by comparing and judging, he discovers some qualities, causes, or phenomena, which, after carefully distinguishing and rejecting all contradictory instances that occur, he finds common to many. Thus from many collateral comparisons and judgments formed upon particulars, he ascends to generals; and by a repetition of the same industrious process and laborious investigation, he advances from general to more general, till at last he is enabled to form a few of the most general, with their attributes and operations, into axioms or secondary principles, which are the well-founded laws enacted and enforced by the God of nature.—This is that just and philosophic method of reasoning which found logic procribes in this as well as in other parts of learning; by which, through the slow but certain road of experiment and observation, the mind ascends from appearances to qualities, from effects to causes; and from experiments upon many particular subjects forms general propositions concerning the powers and properties of physical body.

V. "Axioms to investigated and established are applicable to all parts of learning, and are the indispensable, and indeed the wonderful expedients, by which, in every branch of knowledge, reason pushes on its inquiries in the particular pursuit of truth: and the method of reasoning by which they are formed, is that of true and legitimate induction; which is therefore by Lord Bacon, the best and soundest of logicians, called the key of interpretation.

VI. "Instead of taking his axioms arbitrarily out of the great families of the categories (see Category), and erecting them by his own sophistical invention into the principles upon which his disputation was to be employed, had the analytical genius of Aristotle presented us with the laws of the true inductive logic, by which axioms are philosophically formed, and had he with his usual sagacity given us an example of it in a single branch of science; he would have brought to the temple of truth, an offering more valuable than he has done by the aggregate of all his logic and philosophical productions.

VII. "In all sciences, except the mathematics, it is only after the inductive process has been industriously pursued and successfully performed, that definition may be logically and usefully introduced, by beginning with the genus, passing through all the graduate and subordinate stages, and marking the specific difference as it descends, till it arrive at the individual, which is the reasoning subject of the question. And by adding an affirmation or negation of the attribute of the genus or the species or individual, or of a general accident on the particular substance so defined, making the definition a proposition, the truth of the question will be logically solved without any farther process. So that instead of being the first, as employed by the logic in common use, definition may be the last act of reason in the search of truth in general.

VIII. "These axioms or general propositions, thus and by inductively established, become another species of principles, which may be properly called secondary, and which lay the foundation of the syllogistic method of reasoning. When these are formed, but not before, we may safely admit the maxim with which logicians set out in the exercise of their art, as the great hinge on which their reasoning and disputation turn: From truths that are already known, to derive other which are not known. Or, to state it more comprehensively, so as to apply to probable as well as to scientific reasoning—From truths which are better known, to derive others which are less known. Philosophically speaking, syllogistic reasoning is, under general propositions to reduce others which are less general or which are particular; for the inferior ones are known to be true, only as we trace their connexion with the superior. Logically speaking, it is, To predicate a genus of a species or individual comprehended under it, or an accident of the substance in which it is inherent.

IX. "Thus induction and syllogism are the two methods of direct reasoning corresponding to the two kinds of principles, primary and secondary, on which syllogism is founded, and by which they are respectively different. In both methods, indeed, reason proceeds by judging and comparing, but the process is different throughout; and though it may have the sanction of Aristotle, an inductive syllogism is a fallacy.

X. "Till general truths are ascertained by induction, the third or middle terms by which syllogisms are the foundation are nowhere safely to be found. So that another position of the Stagyrite, that syllogism is naturally prior in order to induction, is equally unfounded; for induction does not only naturally but necessarily precede syllogism; and, except in mathematics, is in every respect indispensable to its existence; since, till generals are established, there can be neither definition, proposition, nor axiom, and of course no syllogism. And as induction is the first, so is it the most essential and fundamental instrument of reasoning: for as syllogism cannot produce its own principles, it must have them from induction; and if the general propositions or secondary principles be imperfectly or infirmly established, and much more if they be taken at hazard, upon authority, or by arbitrary assumption like those of Aristotle, all the syllogizing in the world is a vain and useless logicality, only instrumental to the multiplication of false learning, and to the invention and confirmation of error. The truth of syllogisms depends ultimately on the truth of axioms, and the truth of axioms on the foundations of inductions (E).”—But though induction is

(E) This chapter is almost wholly taken from Tatham's Chart and Scale of Truth; a work which, notwithstanding prior in order, as well as superior in utility, to syllogism; we have thought it expedient to treat of it last; both because syllogism is an easier exercise of the reasoning faculty than induction, and because it is the method of mathematics, the first science of reason in which the student is commonly initiated.

Chap. VI. Of Demonstration.

I. Having dispatched what seemed necessary to be said with regard to the two methods of direct reasoning, induction and syllogism; we now proceed to consider the laws of demonstration. And here it must be acknowledged, that in strict demonstration, which removes from the mind all possibility of doubt or error, the inductive method of reasoning can have no place. When the experiments and observations from which the general conclusion is drawn are numerous and extensive, the result of this mode of reasoning is moral certainty; and could the induction be made complete, it would be absolute certainty, equally convincing with mathematical demonstration. But however numerous and extensive the observations and experiments may be upon which an inductive conclusion is established, they must of necessity come short of the number and extent of nature; which, in some cases, by its immensity, will defeat all possibility of their co-extension; and in others, by its distance, lies out of the reach of their immediate application. Though truth does not appear in all other departments of learning with that bold and resolute conviction with which it presides in the mathematical science, it thines through them all, if not interrupted by prejudice or perverted by error, with a clear and useful, though inferior strength. And as it is not necessary for the general safety or convenience of a traveller, that he should always enjoy the heat and splendor of a mid-day sun, whilst he can with more ease pursue his journey under the weaker influence of a morning or an evening ray; so it is not requisite, for the various concerns and purposes of life, that men should be led by truth of the most redundant brightness. Such truth is to be had only in those sciences which are conversant about ideas and their various relations; where every thing being certainly what it appears to be, definitions and axioms arise from mere intuition. Here syllogism takes up the process from the beginning; and by a sublime intellectual motion advances from the simplest axioms to the most complicated speculations, and exhibits truth springing out of its first and purest elements, and spreading on all sides into a system of science. As each step in the progress of syllogistic, we shall endeavour to explain the use and application of syllogisms in this species of reasoning.

We have seen, that in all the different appearances they put on, we still arrive at a just and legitimate conclusion; now it often happens, that the conclusion of one syllogism becomes a previous proposition in another; by which means great numbers of them are sometimes linked together in a series, and truths are made to follow one another in train. And as in such a concatenation of syllogisms all the various ways of reasoning that are truly conclusive may be with safety introduced; hence it is plain, that in deducing any truth from its first principles, especially where it lies at a considerable distance from them, we are at liberty to combine all the several kinds of syllogisms above explained, according as they are found best to suit the end and purpose of our inquiries. When a proposition is thus, by means of syllogisms, collected from others more evident and known, it is said to be proved; so that we may in the general define the proof of a proposition to be a syllogism, or series of syllogisms, collecting that proposition from known and evident truths. But more particularly, if the syllogisms of which the proofs consist admit of no premises but definitions, self-evident truths, and propositions already established, then is the argument so constituted called a demonstration; whereby it appears that demonstrations are ultimately founded on definitions and self-evident propositions.

II. All syllogisms whatsoever, whether compound, all syllogism, or defective, are reducible to plain simple syllogisms whatever syllogisms in some one of the four figures. But this is reducible to all syllogisms of the first figure, in particular to the first figure, that is, any proposition whatsoever, whether an universal affirmation or universal negative, a particular affirmative or particular negative, which fourfold division embraces all their varieties; any one of these may be inferred by virtue of some syllogism in the first figure. By this means it happens that the syllogisms of all the other figures are reducible also to syllogisms of the first figure, and may be considered as standing on the same foundation with them. We cannot here demonstrate and explain the manner of this reduction, because it would too much swell the bulk of this treatise. It is enough to take notice that the thing is universally known and allowed among logicians, to whose writings we refer such as desire farther satisfaction in this matter. This then being laid down, it is plain that any demonstration whatsoever may be considered as composed of a series of syllogisms, all in the first figure. For, since all the syllogisms that enter the demonstration are reducible to syllogisms of some one of the four figures; and since the syllogisms of all the other figures are farther reducible to syllogisms of the first figure, it is evident, that the whole demonstration may be resolved into a series of these last syllogisms. Let us now, if possible, discover the ground upon which the conclusion rests in syllogisms of the first figure; because, by so doing, we shall come at an universal principle of certainty, whence the evidence of all demonstrations in all their parts may be ultimately derived.

III. The rules then of the first figure are briefly these. The middle term is the subject of the major proposition, and the predicate of the minor. The major is always an universal proposition and the minor always affirmative. Let us now see what effect these rules will have in reasoning. The major is an universal proposition of which the middle term is the subject. Of subject, and the predicate of the conclusion the predicate. Hence it appears, that in the major the predicate of the conclusion is always affirmed or denied universally of the middle term. Again, The minor is an affirmative proposition, whereof the subject of the conclusion is the subject, and the middle term the predicate. Here then the middle term is affirmed of the subject of the conclusion; that is, the subject of the conclusion is affirmed to be comprehended under, or to make a part of, the middle term. Thus then we see what is done in the premises of a syllogism of the first figure. The predicate of the conclusion is universally affirmed or denied of some idea. The subject of the conclusion is affirmed to be or to make a part of that idea. Hence it naturally and unavoidably follows, that the predicate of the conclusion ought to be affirmed or denied of the subject. To illustrate this by an example, we shall resume one of the syllogisms of the first chapter.

"Every creature possessed of reason and liberty is accountable for his actions: "Man is a creature possessed of reason and liberty: "Therefore man is accountable for his actions."

Here, in the first proposition, the predicate of the conclusion, accountability, is affirmed of all creatures that have reason and liberty. Again, In the second proposition, man, the subject of the conclusion, is affirmed to be or to make a part of this class of creatures. Hence the conclusion necessarily and unavoidably follows, viz. that man is accountable for his actions; because, if reason and liberty be that which constitutes a creature accountable, and man has reason and liberty, it is plain he has that which constitutes him accountable. In like manner, where the major is a negative proposition, or denies the predicate of the conclusion universally of the middle term, as the minor always affirms the subject of the conclusion to be or make a part of that middle term, it is no less evident that the predicate of the conclusion ought in this case to be denied of the subject. So that the ground of reasoning, in all syllogisms of the first figure, is manifestly this: "Whatever may be affirmed universally of any idea, may be affirmed of every or any number of particulars comprehended under that idea."

And again: "Whatever may be denied universally of any idea, may be in like manner denied of every or any number of its individuals. These two propositions are called by logicians the dictum de omni, and dictum de nullo; and are indeed the great principles of syllogistic reasoning, inasmuch as all conclusions whatsoever rest immediately upon them, or upon propositions deduced from them. But what adds greatly to their value is, that they are really self-evident truths, and such as we cannot gainsay without running into an express contradiction. To affirm, for instance, that no man is perfect, and yet argue that some men are perfect; or to say that all men are mortal, and yet that some men are not mortal, is to assert a thing to be and not to be at the same time.

IV. And now we may affirm, that, in all syllogisms of the first figure, if the premises are true, the conclusion must needs be true. If it be true that the predicate of the conclusion, whether affirmative or negative, agree universally to some idea; and if it be also true that the subject of the conclusion is a part of or comprehended under that idea; then it necessarily follows, that the predicate of the conclusion agrees also to the subject. For to assert the contrary, would be to run counter to some one of the two principles before established; that is, it would be to maintain an evident contradiction. And thus we are come at last to the point we have been all along endeavouring to establish; namely, that every proposition which can be demonstrated is necessarily true. For as every demonstration may be resolved into a series of syllogisms all in the first figure; and as in any one of these syllogisms, if the premises are true, the conclusion must needs be so too; it evidently follows, that if all the several premises are true, all the several conclusions are so, and consequently the conclusion also of the last syllogism, which is always the proposition to be demonstrated. Now that all the premises of a demonstration are true, will easily appear from the very nature and definition of that form of reasoning. A demonstration, as we have said, is a series of syllogisms, all whose premises are either definitions, self-evident truths, or propositions, already established. Definitions are identical propositions, wherein we connect the description of an idea with the name by which we choose to have that idea called, and therefore as to their truth there can be no dispute. Self-evident propositions appear true of themselves, and leave no doubt or uncertainty in the mind. Propositions, before established, are no other than conclusions gained by one or more steps from definitions and self-evident principles, that is, from true premises, and therefore must needs be true. Whence all the previous propositions of a demonstration being, we see, manifestly true; the last conclusion, or proposition to be demonstrated, must be so too. So that demonstration not only leads to certain truth, but we have here also a clear view of the ground and foundation of that certainty. For as, in demonstrating, we may be said to do nothing more than combine a series of syllogisms together, all resting on the same bottom; it is plain that one uniform ground of certainty runs through the whole, and that the conclusions are everywhere built upon some one of the two principles before established, as the foundation of all our reasoning. These two principles are easily reduced into one, and may be expressed thus: "Whatever predicate, whether affirmative or negative, agrees universally to any idea; the same must needs agree to every or any number of individuals comprehended under that idea." And thus at length we have, according to our first design, reduced the certainty of demonstration to one simple and universal principle; which carries its own evidence along with it, and which is indeed the ultimate foundation of all syllogistic reasoning.

V. Demonstration therefore serving as an infallible guide to truth and certainty. evident truths, or propositions previously established. To judge therefore of the validity of a demonstration, we must be able to distinguish whether the definitions that enter it are genuine, and truly descriptive of the ideas they are meant to exhibit; whether the propositions assumed without proofs as intuitive truths have really that self-evidence to which they lay claim; whether the syllogisms are drawn up in due form, and agreeable to the laws of argumentation; in fine, whether they are combined together in a just and orderly manner, so that no demonstrable propositions serve anywhere as premises unless they are conclusions of previous syllogisms. Now, it is the business of logic, in explaining the several operations of the mind, fully to instruct us in all these points. It teaches the nature and end of definitions, and lays down the rules by which they ought to be framed. It unfolds the several species of propositions, and distinguishes the self-evident from the demonstrable. It delineates also the different forms of syllogisms, and explains the laws of argumentation proper to each. In fine, it describes the manner of combining syllogisms, so that they may form a train of reasoning, and lead to the successive discovery of truth. The precepts of logic, therefore, as they enable us to judge with certainty when a proposition is duly demonstrated, furnish a sure criterion for the distinguishing between truth and falsehood.

VI. Perhaps it may be objected, that demonstration is a thing very rare and uncommon, as being the prerogative of but a few sciences, and therefore the criterion here given can be of no great use. But wherever, by the bare contemplation of our ideas, truth is discoverable, there also demonstration may be attained. Now that is an abundantly sufficient criterion which enables us to judge with certainty, in all cases where the knowledge of truth comes within our reach; for with discoveries, that lie beyond the limits of the human mind, we have, properly, no business or concernment. When a proposition is demonstrated, we are certain of its truth. When, on the contrary, our ideas are such as have no visible connection or repugnance, and therefore furnish not the proper means of tracing their agreement or disagreement, there we are sure that scientific knowledge is not attainable. But where there is some foundation of reasoning, which yet amounts not to the full evidence of demonstration, there the precepts of logic, by teaching us to determine aught of the degree of proof, and of what is still wanting to render it full and complete, enable us to make a due estimate of the measures of probability, and to proportion our assent to the grounds on which the proposition stands. And this is all we can possibly arrive at, or even so much as hope for, in the exercise of faculties so imperfect and limited as ours.

VII. Before we conclude this chapter, it may not be improper to take notice of the distinction of demonstration into direct and indirect. A direct demonstration is, when, beginning with definitions, self-evident propositions, or known and allowed truths, we form a train of syllogisms, and combine them in an orderly manner, continuing the series through a variety of successive steps, until at last we arrive at a syllogism whose conclusion is the proposition to be demonstrated. Proofs of this kind leave no doubt or uncertainty behind them; because, all the several premises being true, the conclusions must be so too, and of course the very last conclusion or proposition to be proved. The other species of demonstration is the indirect, or, as it is sometimes called, the apagogical. The manner of proceeding here is, by assuming a proposition which directly contradicts that we mean to demonstrate; and thence, by a continued train of reasoning, in the way of a direct demonstration, deducing some absurdity or manifest untruth. For hereupon we conclude, that the proposition assumed was false; and thence again, by an immediate consequence, that the proposition to be demonstrated is true. Thus Euclid, in his third book, being to demonstrate that circles which touch one another inwardly have not the same centre, affirms the direct contrary to this, viz. that they have the same centre; and thence, by an evident train of reasoning, proves that a part is equal to the whole. The supposition therefore leading to this absurdity he concludes to be false, viz. that circles touching one another inwardly have the same centre; and thence again immediately infers, that they have not the same centre.

VIII. Now, because this manner of demonstration is accounted by some not altogether so clear and satisfactory; we shall therefore endeavour to show, that it equally with the other leads to truth and certainty. Two propositions are said to be contradictory one of another, when that which is affirmed to be in the one is affirmed not to be in the other. Thus the propositions, Circles that touch one another inwardly have the same centre, and Circles that touch one another inwardly have not the same centre, are contradictory, because the second affirms the direct contrary of what is affirmed in the first. Now, in all contradictory propositions, this holds universally, That one of them is necessarily true, and the other necessarily false. For if it be true, that circles which touch one another inwardly have not the same centre; it is unavoidably false that they have the same centre. On the other hand, if it be false that they have the same centre, it is necessarily true that they have not the same centre. Since therefore it is impossible for them to be both true or both false at the same time; it unavoidably follows, that one is necessarily true, and the other necessarily false. This then being allowed, which is indeed self-evident; if any two contradictory propositions are assumed, and one of them can by a clear train of reasoning be demonstrated to be false, it necessarily follows that the other is true. For as the one is necessarily true, and the other necessarily false; when we come to discover which is the false proposition, we thereby also know the other to be true.

IX. Now this is precisely the manner of an indirect demonstration, as is evident from the account given of it above. For there we assume a proposition which directly contradicts that we mean to demonstrate; and having by a continued series of proofs shown it to be false, thence infer that it is contradictory, or the proposition to be demonstrated, is true. As, therefore, this last conclusion is certain and unavoidable; let us next inquire after what manner we come to be satisfied of the falsehood of the assumed proposition, that so no possible doubt may remain as to the force and validity of demonstrations of this kind. The manner then is plainly this: Beginning with the assumed proposition, Part IV.

Of Method: we, by the help of definitions, self-evident truths, or propositions already established, continue a series of reasoning, in the way of a direct demonstration, until at length we arrive at some absurdity or known fallacy. Thus Euclid, in the example before mentioned, from the supposition that circles touching one another inwardly have the same centre, deduces that a part is equal to the whole. Since, therefore, by a due and orderly process of reasoning, we come at last to a false conclusion; it is manifest, that all the premises cannot be true: for, were all the premises true, the last conclusion must be so too, by what has been before demonstrated. Now, as to all the other premises made use of in the course of reasoning, they are manifest and known truths by supposition, as being either definitions, self-evident propositions, or truths previously established. The assumed proposition is that only as to which any doubt or uncertainty remains. That alone, therefore, can be false; and indeed, from what has been already shown, must unavoidably be so. And thus we see, that in indirect demonstrations, two contradictory propositions being laid down, one of which is demonstrated to be false, the other, which is always the proposition to be proved, must necessarily be true; so that here, as well as in the direct way of proof, we arrive at a clear and satisfactory knowledge of truth.

X. This is universally the method of reasoning in all apagogical or direct demonstrations. But if any proposition is assumed, from which, in a direct train of reasoning, we can deduce its contradictory; the proposition so assumed is false, and the contradictory one true. For if we suppose the assumed proposition to be true, then, since all the other premises that enter the demonstration are also true, we shall have a series of reasoning consisting wholly of true premises; whence the last conclusion or contradictory of the assumed proposition must be true likewise: so that by this means we should have two contradictory propositions both true at the same time, which is manifestly impossible. The assumed proposition, therefore, whence this absurdity flows, must necessarily be false; and consequently its contradictory, which is here the proposition deduced from it, must be true. If then any proposition is supposed to be demonstrated, and we assume the contradictory of that proposition, and thence directly infer the proposition to be demonstrated; by this very means we know that the proposition so inferred is true. For, since from an assumed proposition we have deduced its contradictory, we are thereby certain that the assumed proposition is false; and if so, then its contradictory, or that deduced from it, which in this case is the same with the proposition to be demonstrated, must be true.

XI. We have a curious instance of this in the twelfth proposition of the ninth book of the Elements. Euclid there proposes to demonstrate, that in any series of numbers, rising from unity in geometrical progression, all the prime numbers that measure the last term of the progression will also measure the next after unity. In order to this, he assumes the contradictory of the proposition to be demonstrated; namely, that some prime number will not measure the last term in the series does not make us measure the next after unity; and thence, by a continued train of reasoning, proves that it actually does measure it. Hereupon he concludes the assumed proposition to be false; and that which is deduced from it, or its contradictory, which is the very proposition he proposed to demonstrate, to be true. Now that this is a just and conclusive way of reasoning, is abundantly manifest from what we have so clearly established above. Whence it appears, how necessary some knowledge of the rules of logic is, to enable us to judge of the force, justness, and validity, of demonstrations. For, though it is readily allowed, that by the mere strength of our natural faculties we can at once discern, that of two contradictory propositions, the one is necessarily true, and the other necessarily false; yet when they are so linked together in a demonstration, as that the one serves as a previous proposition whence the other is deduced, it does not so immediately appear, without some knowledge of the principles of logic, why that alone, which is collected by reasoning, ought to be embraced as true, and the other, whence it is collected, to be rejected as false.

XII. Having thus sufficiently evinced the certainty of and of its demonstration in all its branches, and thrown the rules by which we ought to proceed, in order to arrive at a just conclusion, according to the various ways of arguing, it is needless to enter upon a particular and false consideration of those several species of false reasoning—reasoning—which logicians distinguish by the name of sophisms. He that thoroughly understands the form and structure of a good argument, will of himself readily discern every deviation from it. And although sophisms have been divided into many classes, which are all called by sounding names, that therefore carry in them much appearance of learning; yet are the errors themselves so very palpable and obvious, that it would be lost labour to write for a man capable of being misled by them. Here, therefore, we choose to conclude this part of logic; and shall in the next give some account of Method, which, though inseparable from reasoning, is nevertheless always considered by logicians as a distinct operation of the mind; because its influence is not confined to the mere exercise of the reasoning faculty, but extends in some degree to all the transactions of the understanding.

PART IV. OF METHOD.

WE have now done with the three first operations of the mind, whose office it is to search after truth, and enlarge the bounds of human knowledge. There is yet a fourth, which regards the disposal and arrangement of our thoughts, when we endeavour to put them together as that their mutual connexion and dependence may be clearly seen. This is what logicians call Method, and place always the last in order in explaining the powers of the understanding; because it necessarily supposes a previous exercise of our other faculties, and some progress made in knowledge before we can exert it in any extensive degree. II. In this view, it is plain that we must be beforehand well acquainted with the truths we are to combine together; otherwise, how could we discern their several connexions and relations, or so dispose of them as their mutual dependence may require? But it often happens, that the understanding is employed, not in the arrangement and composition of known truths, but in the search and discovery of such as are unknown. And here the manner of proceeding is very different. We assemble at once our whole stock of knowledge relating to any subject, and, after a general survey of things, begin with examining them separately and by parts. Hence it comes to pass, that whereas, at our first setting out, we were acquainted only with some of the grand strokes and outlines of truth; by thus pursuing her through her several windings and recesses, we gradually discover those more inward and finer touches whence she derives all her strength, symmetry, and beauty. And here it is, that when, by a narrow scrutiny into things, we have unravelled any part of knowledge, and traced it to its first and original principles, infomuch that the whole frame and contexture of it lies open to the view of the mind; here it is, that, taking it the contrary way, and beginning with these principles, we can adjust and put together the parts as the order and method of science requires.

III. But as these things are best understood when illustrated by the multitude of examples, let us suppose any machine, for instance a watch, presented to us, whose structure and composition we are as yet unacquainted with, but want, if possible, to discover. The manner of proceeding, in this case, is, by taking the whole to pieces, and examining the parts separately, one after another. When, by such a scrutiny, we have thoroughly informed ourselves of the frame and contexture of each, we then compare them together, in order to judge of their mutual action and influence. By this means we gradually trace out the inward make and composition of the whole, and come at length to discern how parts of such a form, and so put together as we found in unravelling and taking them asunder, constitute that particular machine called a watch, and contribute to all the several motions and phenomena observable in it. This discovery being made, we can take things the contrary way, and, beginning with the parts, so dispose and connect them as their several uses and structures require, until at length we arrive at the whole itself, from the unravelling of which those parts resulted.

IV. And as it is in tracing and examining the works of human knowledge: for the relations and mutual habitudes of things do not always immediately appear upon comparing them one with another. Hence we have recourse to intermediate ideas; and, by means of them, are furnished with those previous propositions that lead to the conclusion we are in quest of. And if it so happens that the previous propositions themselves are not sufficiently evident, we endeavour, by new middle terms, to ascertain their truth; still tracing things backward, in a continual series, until at length we arrive at some syllogism where the premises are first and self-evident principles. This done, we become perfectly satisfied as to the truth of all the conclusions we have passed through, insomuch as they are now seen to stand upon the firm and immovable foundation of our intuitive perceptions. And as we arrived at this certainty by tracing things backward to the original principles whence they flow; so may we at any time renew it by a direct contrary process, if, beginning with these principles, we carry the train of our thoughts forward until they lead us, by a connected chain of proofs, to the very last conclusion of the series.

V. Hence it appears, that, in disposing and putting together our thoughts, either for our own use, that the method discoveries we have made may at all times lie open to analytic review of the mind, or where we mean to communicate and synthesize and unfold the discoveries to others, there are two ways of proceeding equally within our choice: for we may propose the truths relating to any part of knowledge, as they presented themselves to the mind in the manner of investigation; carrying on the series of proofs, in a reverse order, until they at last terminate in first principles; or, beginning with these principles, we may take the contrary way, and from them deduce, by a direct train of reasoning, all the several propositions we want to establish. This diversity in the manner of arranging our thoughts gives rise to the twofold division of method established among logicians; for method, according to their use of the word, is nothing else but the order and disposition of our thoughts relating to any subject. When truths are so proposed and put together as they were or might have been discovered, this is called the analytic method, or the method of resolution; insomuch as it traces things backward to their source, and resolves knowledge into its first and original principles. When, on the other hand, they are deduced from these principles, and connected according to their mutual dependence, infomuch that the truths first in order tend always to the demonstration of those that follow; this constitutes what we call the synthetic method or method of composition. For here we proceed by gathering together the several scattered parts of knowledge, and combining them into one whole or system, in such manner that the understanding is enabled distinctly to follow truth through all her different stages and gradations.

VI. There is this farther to be taken notice of, in calling attention to these two species of method; that the first therewith has also obtained the name of the method of invention, the method because it observes the order in which our thoughts succeed one another in the invention or discovery of the method truth. The other, again, is often denominated the doctrine, method of doctrine or instruction; insomuch as, in laying our thoughts before others, we generally choose to proceed in the synthetic manner, deducing them from their first principles. For we are to observe, that although there is great pleasure in pursuing truth in the method of investigation, because it places us in the condition of the inventor, and shows the particular train and process of thinking by which he arrived at his discoveries; yet it is not so well accommodated to the purposes of evidence and conviction. For, at our first setting out, we are commonly unable to divine where the analysis will lead us; infomuch that our researches are for some time little better than a mere groping in the dark. And even after light begins to break in upon us, we are still obliged to many reviews, and Part IV.

Of Method, and a frequent comparison of the several steps of the investigation among themselves. Nay, when we have unravelled the whole, and reached the very foundation on which our discoveries stand, all our certainty, in regard to their truth, will be found in a great measure to arise from that connexion we are now able to discern between them and first principles, taken in the order of composition. But in the synthetic manner of disposing our thoughts, the case is quite different: for as we here begin with the intuitive truths, and advance by regular deductions from them, every step of the procedure brings evidence and conviction along with it; so that, in our progress from one part of knowledge to another, we have always a clear perception of the ground on which our afflent rests. In communicating therefore our discoveries to others, this method is apparently to be chosen, as it wonderfully improves and enlightens the understanding, and leads to an immediate perception of truth.

VII. The logic which for so many ages kept possession of the schools, and was deemed the most important of the sciences, has long been condemned as a mere art of wrangling, of very little use in the pursuit of truth. Attempts have been made to restore it to credit, but without success; and of late years little or no attention whatever has been paid to the art of reasoning in the course of what is called a liberal education. As both extremes may be faulty, it should seem that we cannot conclude this short treatise more properly than with the following

REFLECTIONS ON THE UTILITY OF LOGIC.

If Aristotle was not the inventor of logic, he was certainly the prince of logicians. The whole theory of syllogisms he claims as his own, and as the fruit of much time and labour; and it is universally known, that the later writers on the art have borrowed their materials almost entirely from his Organon and Porphyry's Introduction. But after men had laboured near 2000 years in search of truth by the help of syllogisms, Lord Bacon proposed the method of induction, as a more effectual engine for that purpose; and since his days the art of logic has gradually fallen into disrepute.

To this consequence many causes contributed. The art of syllogism is admirably calculated for wrangling; and by the schoolmen it was employed with too much success, to keep in countenance the absurdities of the Romish church. Under their management it produced numberless disputes, and numberless sects, who fought against each other with much animosity without gaining or losing ground; but it did nothing considerable for the benefit of human life, whilst the method of induction has improved arts and increased knowledge. It is no wonder, therefore, that the excessive admiration of Aristotle, which continued for so many ages, should end in an undue contempt: and that the high esteem of logic, as the grand engine of science, should at last make way for too unfavourable an opinion, which seems now prevalent, of its being unworthy of a place in a liberal education. Men rarely leave one extreme without running into the contrary: Those who think according to the fashion, will be as prone to go into the present extreme as their grandfathers were to go into the former; and even they who in general think for themselves, when they are offended at the abuse of any thing, are too apt to entertain prejudices against the thing itself. "In practice (says the learned Warburton*), logic is more a trick than a science, formed rather to amuse than to instruct." And in some sort we may apply to the art of syllogism what a man of wit says of rhetoric, that it only tells us how to name those tools which nature had before put into our hands. In the service of chicane, indeed, it is a mere juggler's knot, now fast, now loose; and the schools where this legerdemain was exercised in great perfection are full of the stories of its wonders." The authority of Warburton is great; but it may be counterbalanced by another, which, on subjects of this nature, is confessedly greater.

"Laying aside prejudice, whether fashionable or unfashionable, let us consider (says Dr Reid†) whether logic is or may be made subservient to any good purpose. Its professed end is, to teach men to think, to judge, and to reason, with precision and accuracy. No man will say this is a matter of little importance; and the only thing therefore that can admit of doubt is whether it can be taught?

"To resolve this doubt, it may be observed, that our rational faculty is the gift of God, given to men in very different measures: Some have a large portion, some a less; and where there is a remarkable defect of the natural power, it cannot be supplied by any culture. But this natural power, even where it is the strongest, may lie dead for want of the means of improvement. Many a savage may have been born with as good faculties as a Newton, a Bacon, or an Aristotle; but their talents were buried by having never been put to use, whilst those of the philosophers were cultivated to the best advantage. It may likewise be observed, that the chief mean of improving our rational power, is the vigorous exercise of it in various ways and on different subjects, by which the habit is acquired of exercising it properly. Without such exercise, and good sense over and above, a man who has studied logic all his life may be only a petulant wrangler, without true judgement or skill of reasoning in any science."

This must have been Locke's meaning, when in his Thoughts on Education, he says, "If you would have your son to reason well, let him read Chillingworth." The state of things is much altered since Locke wrote; Logic has been much improved chiefly by his writings; and yet much less stress is laid upon it, and less time consumed in its study. His counsel, therefore, was judicious and feasible; to wit, That the improvement of our reasoning power is to be expected much more from an intimate acquaintance with the authors who reason best, than from studying voluminous systems of school logic. But if he had meant, that the study of logic was of no use, nor deserved any attention, he surely would not have taken the pains to make so considerable an addition to it, by his Essay on the Human Understanding, and by his Thoughts on the Conduct of the Understanding; nor would he have remitted his pupil to Chillingworth, the acutest logician as well as the best reasoner of his age."

There is no study better fitted to exercise and strengthen the reasoning powers than that of the mathematical sciences; because there is no other branch Of Method of science which gives such scope to long and accurate trains of reasoning, or in which there is so little room for authority or prejudice of any kind to give a false bias to the judgement. When a youth of moderate parts begins to study Euclid, every thing is new to him: His apprehension is unsteady; his judgement is feeble; and rests partly upon the evidence of the thing, and partly upon the authority of his teacher. But every time he goes over the definitions, the axioms, the elementary propositions, more light breaks in upon him; and as he advances, the road of demonstration becomes smooth and easy; he can walk in it firmly, and take wider steps, till at last he acquires the habit not only of understanding a demonstration, but of discovering and demonstrating mathematical truths.

It must indeed be confessed, that a man without the rules of logic may acquire a habit of reasoning justly in mathematics, and perhaps in any other science. Good sense, good examples, and affiduous exercise, may bring a man to reason justly and acutely in his own profession without rules. But whoever thinks, that from this conception he may infer the inutility of logic, betrays by this inference a great want of that art; for he might as well infer, because a man may go from Edinburgh to London by the way of Paris, that therefore any other road is useless.

There is perhaps no art which may not be acquired, in a very considerable degree, by example and practice, without reducing it to rules. But practice joined with rules may carry a man forward in his art farther and more quickly than practice without rules.—Every ingenious artist knows the utility of having his art reduced to rules, and thereby made a science. By rules he is enlightened in his practice, and works with more assurance. They enable him sometimes to correct his own errors, and often to detect the errors of others; and he finds them of great use to confirm his judgements, to justify what is right, and to condemn what is wrong.

Now mathematics are the noblest praxis of logic. Through them we may perceive how the stated forms of syllogism are exemplified in one subject, namely the predicament of quantity; and by marking the force of these forms, as they are there applied, we may be enabled to apply them of ourselves elsewhere. Whoever, therefore, will study mathematics with this view, will become not only by mathematics a more expert logician, and by logic a more rational mathematician, but a wiser philosopher, and an acute reasoner, in all the possible subjects either of science or deliberation. But when mathematics, instead of being applied to this excellent purpose, are used not to exemplify logic, but to supply its place; no wonder if logic fall into contempt, and if mathematics, instead of furthering science, become in fact an obstacle. For when men, knowing nothing of that reasoning which is universal, come to attach themselves for years to a single species, a species wholly involved in lines and numbers, the mind becomes incapacitated for reasoning at large, and especially in the search of moral truth. The object of mathematics is demonstration; and whatever in that science is not demonstration, is nothing, or at least below the sublime inquirer's regard. Probability, through its almost infinite degrees, from simple ignorance up to absolute certainty, is the terra incognita of the mathematician. And yet here it is that the great business of the human mind is carried on in the search and discovery of all the important truths which concern us as reasonable beings. And here too it is that all its vigour is exerted: for to proportion the assent to the probability accompanying every varying degree of moral evidence, requires the most enlarged and sovereign exercise of reason.

In reasonings of this kind, will any man pretend that it is of no use to be well acquainted with the various powers of the mind by which we reason? Is it of no use to resolve the various kinds of reasoning into their simple elements; and to discover, as far as we are able, the rules by which these elements are combined in judging and in reasoning? Is it of no use to mark the various fallacies in reasoning, by which even the most ingenious men have been led into error? It must surely betray great want of understanding, to think these things useless or unimportant. Now these are the things which logicians have attempted; and which they have executed—not indeed so completely as to leave no room for improvement, but in such a manner as to give very considerable aid to our reasoning powers. That the principles they have laid down with regard to definition and division, with regard to the conversion and opposition of propositions, and the general rules of reasoning, are not without use, is sufficiently apparent from the blunders committed daily by those who disdain any acquaintance with them.

Although the art of categorical syllogism is confessedly little fitted for the discovery of unknown truth, it may yet be employed to excellent purposes, as it is perhaps the most compendious method of detecting a fallacy. A man in quest of unknown truths must generally proceed by the way of induction, from effects to causes; but he, who as a teacher is to inculcate any system upon others, begins with one or more self-evident truths, and proceeds in the way of demonstration, to the conclusion which he wishes to establish. Now every demonstration, as has been already observed, may be resolved into a series of syllogisms, of which the conclusion of the preceding always enters into the premises of that which follows: and if the first principles be clear and evident, and every syllogism in some legitimate mode and figure, the conclusion of the whole must infallibly be admitted. But when the demonstration is thus broken into parts; if we find that the conclusion of one syllogism will not, without altering the meaning of the terms, enter legitimately into the premises of that which should immediately follow; or, supposing it to make one of the premises of a new syllogism, if we find that the conclusion, resulting from the whole series thus obtained, is different from that of the demonstration; we may, in either of these cases, rest assured that the author's reasoning is fallacious, and leads to error; and that if it carried an appearance of conviction before it was thus resolved into its elementary parts, it must have been owing to the inability of the mind to comprehend at once a long train of arguments. Whoever wishes to see the syllogistic art employed for this purpose, and to be convinced of the truth of what we have said respecting its utility, may consult the excellent writer recommended by Locke, who, in places innumerable of his incomparable book, has, without pedantry, even in that pedantic age, made the happiest application of the rules of... LOGISTÆ, certain officers at Athen, in number ten, whose busines consulted in receiving and passing the accounts of magistrates when they went out of office. The logifter were elected by lot, and had ten euthyni or auditors of accounts under them.