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 described will be an equilateral triangle.

9. This leads us to take notice, that as self-evident truths are distinguishable 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 forty-seventh 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 the sum of 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.

10. It may not be amiss to add, that, besides the four kinds of propositions already mentioned, mathematicians have also a fifth, known by the name of corollaries. These are usually subjoined in 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 proposed. 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 angles, and things which are equal to one another and the same thing are equal to one another.

11. The scholia of mathematicians are indifferently annexed to definitions, propositions, or corollaries, and answer the same purposes as annotations upon a classic author. For in them occasion is taken to explain whatever may appear intricate and obscure in a train of reasoning; to answer objections; to teach the application and uses of propositions; to lay open the origin and history of the several discoveries which have been 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 of profit.

III.—OF REASONING.

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

It often happens, in comparing together ideas, 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 form, in order to judge of their equality or inequality, relations 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 to put them together that their several parts shall mutually coincide. Here, then, it becomes necessary to look out for some third idea which will admit of such an application as the 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.

2. 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 discern how far the ideas with which this third idea 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 one with which they may be compared, is that which we call reasoning, and is indeed the chief instrument by which we extend our discoveries and enlarge our knowledge. The great art consists in finding out such intermediate ideas as, when compared with others in the question, will furnish evident and known truths; because, as will afterwards appear, it is only by means of these that we arrive at the knowledge of what is hidden and remote.

3. Hence it appears that every act of reasoning necessarily includes three distinct judgments; two in which the ideas whose relation we want to discover are compared with the middle idea, and a third in which they are themselves connected or disjoined, according to the result of that comparison. Now, as in the second part of logic, our judgments, 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 judgments, so every syllogism must include three distinct propositions. When reasoning is thus put into words, and appears in the 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 are denominated the extremes.

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It is proper here to caution the reader against being led into any misconception as to the real nature of axioms, which, as above stated, have commonly been understood to mean "self-evident speculative truths." That they are "self-evident" is undeniable; but it is not so easy to see how they can be called "speculative truths," since in all cases where they predicate equality they are identical propositions, and, where they do not, they are mere truisms, if we may use such a term, or propositions of such a kind that to suppose the contrary would involve an obvious absurdity. When we say, That the whole is equal to all its parts put together, or, That things which are equal to the same thing are equal to one another, we merely express two identical propositions, or in effect just affirm that \(a = a\), and \(y = y\); affirmations which can add nothing either to the evidence or the truth of any demonstration. Such propositions amount only to a sort of jeu de mots. Besides, they are illogical, inasmuch as the "self-evident" truth which is predicated in the second is invariably assumed in the first branch of the proposition. The "whole" necessarily includes all its parts, and "things equal to one and the same thing" involve mutual equality. If we were to repeat the axiom, "That whatever is, for an age, or for ages, we would not be in any respect wiser, or nearer to the discovery of any truth not self-evident, than we were before. Again, when we say, That the whole is greater than any of its parts, we merely affirm a truism, since the bare idea of a "whole" necessarily excludes that of the equality of any of its parts. It is, therefore, a mistake to suppose that axioms add anything to the clearness or force of mathematical evidence. No conclusion whatever can be deduced from an identical proposition, in which the thing assumed is also the thing predicated. The whole fabric of geometry is built, not upon axioms, but upon definitions, expressing simple facts, as, for instance, the definition of a circle; and the real character of geometrical evidence consists in this, that it is founded, not upon identical propositions or truisms, but upon an appeal to observation, though of the simplest and most elementary kind. 4. But as these things are best illustrated by examples, let us, for instance, set ourselves to inquire "Whether men be accountable for their actions." As the relation between the ideas of man and accountableness 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 good from evil; that is, unless we suppose him possessed of reason. Nor is this alone sufficient. For what would it avail him to know good from evil actions if he had no freedom of choice, and could not avoid the one or 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 the 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 accountableness is inseparably connected, viz. reason and liberty, which are here to be considered as forming 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 judgments, viz. that man is possessed of reason and liberty, and that reason and liberty imply accountableness, 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 judgments; two which may be styled previous, insomuch as they lead to the other, and arise from comparing the middle idea with the two ideas contained 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."

5. In this syllogism we may observe, that there are three several propositions expressing the three judgments 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 first two propositions answer the two previous judgments 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 the relations of which we inquire into, as man and accountableness, 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 judgments, 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.

6. 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, in which 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 stated, 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.

7. These things being premised, we may in general define reasoning to be an act or operation of the mind deducing some unknown proposition from other previous ones which 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, in somuch that we assent to and perceive the truth of them as soon as proposed. In the syllogism above given, 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 connection between the subject and predicate could not be perceived by a bare attention to the ideas themselves, it is evident that this proposition would no less require a proof than the conclusion deduced from it. In this case a new middle term must be sought for to trace the connection 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 still existed 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, as long as the premises remain uncertain, the conclusion built upon them must be so likewise. 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.

8. We see, therefore, that in order to infer a conclusion by a single act of reasoning, the premises must be only a concatenation of intuitive propositions. Where they are not, previous syllogisms are required; in which case reasoning becomes a complicated act, taking in a variety 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 which 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 so fully explained.

9. Thus we see, that reasoning, beginning with first principles, rises gradually from one judgment to another, and connects them in such manner that every stage of the progression brings intuitive certainty along 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 which are already known. This appears evident by the foregoing account, where we see that no proposition is admitted into a syllogism to serve as one of the previous judgments on which the conclusion rests, unless it be itself a known and established truth, the connection of which with self-evident principles has already been traced.

2. Of the several kinds of Reasoning; and, first, of that by which we determine the Genera and Species of Things.

All the aims of human reason may in the general be reduced to these two: First, to rank things under those universal ideas to which they truly belong; and, secondly, to ascribe to them their several attributes and properties in consequence of that distribution.

2. 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 thereby learn to apply the terms expressing general conceptions to such particular objects as come under our immediate observation.

3. 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 should take a view of the idea itself, which is denoted by that general name, and carefully attend to the distinguishing marks which serve to characterize it. Secondly, that we should compare this idea with the object under consideration, observing diligently wherein they agree or differ. If the idea be found to correspond with the particular object, we then without hesitation apply the general name; but if no such correspondence intervene, 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 judgments, 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."

4. Here it may be observed, that where the general idea, those steps to which particular objects are referred, is perfectly familiar always to the mind, and frequently in view, this reference, and the followed 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, which 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 find a perfect correspondence, no longer hesitates under what class of metals to rank it.

5. Nor let it be imagined that our researches here, because in appearance bounded to the imposing of general names upon particular objects, are therefore trivial and of little consequence. Some of the most considerable debates amongst mankind, and such too as nearly regard their lives, interest, 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 perceive 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; insomuch that the determining the species of actions is all one with determining what proportion of praise or dispraise, commendation or blame, 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.

6. 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 further evident from the practice of the mathematicians. Every one who has read Euclid, knows that he frequently requires us to draw lines through certain points, and according to such directions; and that 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, he never calls it by that name until he has shown that its sides are equal, and all its angles right angles. Now this is apparently the same form of reasoning we have before exhibited in proving eight to be an even number.

7. Having thus explained the rules by which we are to conduct ourselves in ranking particular objects under general ideas, and shown their conformity to the practice and ideas of the mathematicians, it remains only to observe, that the true way of rendering this part of knowledge both easy and certain is, by habituating ourselves to clear and determinate ideas, and by keeping these steadily annexed to their respective names. For as all our aim is to apply general words aright, if these words stand for invariable ideas which 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 of it to another, we have made ourselves acquainted with the several particulars observable in it. If amongst 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, constitute 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 to possess 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 manslaughter, it will be impossible for us to decide whether any particular action ought to bear that name; because, however nicely we may 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.

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

3. Of Reasoning, as it regards the Powers and Properties of Things, and the Relations of our general Ideas.

1. 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, because by these alone are the bounds of human knowledge 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 with one 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. Whenever we have occasion for the forcible action of water to remove a body which 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 any body with impetus, 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 which 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."

2. Hence it appears that reasoning, as it regards common life, is no more than the ascribing the general properties of things to those several objects with which we seem more immediately concerned, according as they are found to be of that particular division or class to which the properties 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, we ascribe all those attributes to the actual 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 an estimable man. 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?"

3. By this syllogism it appears, that before we affirm anything of a particular object, that object must be referred to some general idea. Sempronius is pronounced deservedly worthy of esteem only in consequence of his being a virtuous man, or coming under that general notion. Hence we see the necessary connection of the various parts of reasoning, and the dependence which they have 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. When this is done, the other part will be easy, and require scarcely any labour or thought, being no more than an application of the general form of reasoning represented in the foregoing syllogism. Now, as it has already been sufficiently shown how we are to proceed in determining the genera and species of things, which, as we have stated, is the previous step to this second branch of human knowledge, all that is further 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 of which 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 said, regards principally the sciences.

4. But, that we may conduct our thoughts with some degree of 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 the latter kind that we are to speak here, having despatched 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 in applying them; so 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; and, secondly, The skill and talent of applying them happily in all particular instances which come under consideration.

5. 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 only such 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 are we to find some amongst them which 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 throw upon one another.

6. 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 the arts and sciences. The truth is, that all the various divisions of human knowledge are very nearly related amongst themselves, and, in innumerable instances, serve to illustrate and set off one another. And although it is not to be denied that, by an obstinate application to one branch of study, a man may make considerable progress, and acquire some degree of eminence in it, yet his views will always be narrow and contracted, and he will want that masterly discernment which not only enables us to pursue our discoveries with ease, but also, in laying them open to others, to spread a certain brightness around them. But when our reasoning regards a particular science, it is further necessary that we should 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.

7. We come now to the second thing required in order to a successful progress in reasoning; namely, the skill and talent of applying intermediate ideas happily in all particular instances that come under consideration. And here rules and precepts are of little service. Use and experience are the best instructors. For, however 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 requires to be exercised 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. In this pursuit the understanding is by degrees habituated to truth, it contracts insensibly a certain fondness for it, and learns never to yield its assent to any proposition but where the evidence is sufficient to produce full conviction. For this reason Plato has called mathematical demonstrations the cathartics or purgatives of the soul, as being the proper means of cleansing it from error, and restoring that natural exercise of its faculties in which just thinking consists.

8. If, therefore, we would form our minds to a habit of Value of reasoning closely and in train, we cannot take any more certain method than that of exercising ourselves in mathematical demonstrations, so as to contract a kind of familiarity with them. Not that we look upon it as necessary that all men should be deep mathematicians; but that, having mastered the way of reasoning which that study necessarily brings the mind to, they may be able to transfer it to other parts of knowledge, as they shall afterwards have occasion.

9. But though the study of mathematics be of all others the most useful to form the mind, and to give it an early relish for truth, yet other parts of philosophy ought not to be neglected. For there also we meet with many opportunities of exercising the powers of the understanding; and the variety of subjects naturally leads us to observe all the different turns of thinking which 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, insomuch that it can arrange and model them at pleasure, and call such into view as best suit its actual designs. Now in this consists the whole art of reasoning; from amongst a great variety of different ideas to single out those which 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 would appear beyond the reach of human understanding. For this purpose, besides the study of mathematics already recommended, we should 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 with the bare contemplation of the arguments offered by others, but will be frequently essaying its own strength, and pursuing its discoveries upon the plan which 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.

4. Of the Forms of Syllogisms.

1. Hitherto we have contented ourselves with giving a Figures of general notion of syllogisms, and of the parts of which they syllogisms consist. It is now time to enter a little more particularly into the subject, to examine their various forms, and to lay down 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, as 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, it is the predicate in the major, and the subject in the minor. Hence the distinction 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, as we have shown, 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."

2. But, besides this fourfold distinction of syllogisms, there is also a further 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 the 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 evolve true conclusions.

Besides the rules which are proper to each figure, Aristotle has given some that are common to all, by which the legitimacy of syllogisms may be tried. These may be reduced to five, viz. 1. There must be only three terms in a syllogism; and as each term occurs in two of the propositions, it must be precisely the same in both; for 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; and, 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.

3. The division of syllogisms according to moods and figures respects those especially which are known by the name of plain simple syllogisms; that is, which are limited to three propositions, all of them simple, and where the extremes and middle term are connected, according to the rules above laid down. 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 employs 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.

4. When in any syllogism the major is a conditional proposition, the syllogism itself is termed conditional. Thus: "If there be 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 conditional. And here we must observe, that all conditional propositions are composed 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 be a God; and 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, receives the name of the consequent.

5. 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 appear that the consequent ought to be rejected, the antecedent must evidently be so too; because, as has just been demonstrated, the admission of the antecedent would necessarily imply the admission likewise of the consequent.

6. 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 admit 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 that 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, on the contrary, we suppose that the minor rejects 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. This will appear 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."

7. These two species include the whole class of conditional syllogisms, and all the possible ways of arguing which lead to a legitimate conclusion; because we cannot here proceed by a contrary process of reasoning; 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 which infers it. When we say, "If a stone be 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 the consequent. But as there are various other ways by which a stone may become heated, 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 argue 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 be 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 sources whence that heat may have proceeded. These two ways of arguing, therefore, hold not in conditional syllogisms.

8. As from the major 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 Manner of 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 becomes effectual. 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 be 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 world, 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, insomuch 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 such 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.

9. In the several kinds of syllogisms hitherto mentioned, we may observe that the parts are complete; that is, or mutilated, the three propositions of which they consist are represented in form. But it often happens that some 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, which 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.

10. These seemingly imperfect syllogisms are called *Enthymemes*, and occur very frequently in reasoning, especially where it forms 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 even seem to share in the discovery of what is proposed 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 insensibly into our very views and course of reasoning. This gives a pleasure not unlike to that which the author himself feels in composing. Besides, it 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

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1 Respecting the prevalent mistake relative to the *Enthymeme*, and Aristotle's doctrine concerning it, see *Edinburgh Review*, vol. Ixii. p. 221, et seq. Of its operations, and a single expression is left to exhibit a Reasoning whole train of thoughts.

11. But there is another species of reasoning with two propositions, which seems to be complete in itself; and where we admit the conclusion without supposing any tacit or suppressed judgment in the mind, from which it follows syllogistically. This happens between propositions where the connection 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 happens that the proposition upon which the other depends is self-evident, we content ourselves with barely affirming it, and infer the 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 which constitutes a proposition universal. If then that universal proposition chance to be self-evident, the particular ones follow of course, without any further 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 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 further 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.

12. Now, in all cases of this kind, where propositions are deduced one from another, on account of a known and evident connection, 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 propositions in which it takes place, and allow of all immediate consequences which follow in conformity to them. It is however observable, that these arguments, though seemingly complete, because the conclusion follows necessarily from the single proposition which 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, insomuch that they are in fact no more than the 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 observed to flow immediately from the rules and reasons of logic.

13. The next species of reasoning we shall take notice of here is what is commonly known by the name of a sorites. 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 that is possible: He that can do every thing that is 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 sorites itself may be resolved into as many simple syllogisms as there are middle terms in it; and 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 sorites. 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.

14. What is here said of plain simple propositions may be as well applied to those that are conditional; that is, of hypotheses of which the consequent of one shall become continually the antecedent of the next following; in which case, by establishing the consequent of the first proposition, we also 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 sorites, as well as the last, may be resolved into a series of distinct syllogisms, with only this difference, that here the syllogisms are all conditional.

15. The last species of syllogism which we shall take notice of under this head is that commonly distinguished 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 generally 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: Or, which is the same thing, It is absurd to say that he did not create the world perfect in its kind."

16. The nature, then, of a dilemma is universally this: Universal 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, insomuch 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 express it. Hence it is plain, that if the antecedent of the major be an affirmative proposition, the conclusion of the dilemma will be negative; but if it be a negative proposition, the conclusion will be affirmative.

5. Of Induction.

1. All reasoning is ultimately resolvable into first truths, which are either self-evident or taken for granted; and the first truths of syllogistic reasonings 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 theoretical 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, constitute 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.

2. 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. Every thing in the universe, whether of mind or body, presents itself to our observation in its individual state; so that perception and judgment employed in the investigation of truth, whether physical, metaphysical, moral, or historical, have in the first place to deal 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, so they have been called the primary principles of truth. See Metaphysics.

3. By such acts of observation and judgment, diligently practised and frequently repeated, upon many individuals of the same class or of a similar nature, observing their agreements, marking their differences however 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 abstraction of the mind, by which images and ideas are formed which 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 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. 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, or general propositions ranged one above another, till they terminate in those that are universal.

4. 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 sound logic prescribes 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.

5. Axioms so investigated and established are applicable to all parts of learning, and constitute the indispensable, applicable and indeed the wonderful expedients, by which, in every to all parts branch of knowledge, reason pushes on its inquiries in the particular pursuit of truth; and the method of reasoning is only particular truths can be intuitive axioms.

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On the distinction of logical and real inductions, a distinction hitherto wholly overlooked, see Edinburgh Review, vol. lvii. p. 224, et seq.

"Qui formae novit, is, quae adhae non facta sunt, qualia nec nature vielssitudines, nec experimentales industriae unquam in actum priduxissent, nec cognitionem humanam subiure fulsisset, detegit et educit." (Bacon Novum Organum.)

"The word axiom, ἀξιώματος, 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 corrigerat inquitas, qua plerumque in exemplis vulgatis fundamentum habent." (De Augen, Scient. lib. ii. cap. 2.) "Atque illa ipsa putativa principia ad rationes reddendas compellare decreverimus, quoquo plane constant." (Distrib. Operis.) Dr Tatham makes a distinction between axioms intuitives and axioms self-evident. Intuitive axioms, according to him, pass through the first inlets of knowledge, and flash direct conviction on the minds of 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 intuitively framed, they cannot be syllogistically proved. If this distinction be just, and we think it is, only particular truths can be intuitive axioms. Of by which they are formed is that of true and legitimate Reasoning. Induction, which is therefore by Lord Bacon, the best and soundest of logicians, called the key of interpretation.

6. Instead of taking his axioms arbitrarily out of the great families of the categories, 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 logical and philosophical productions.

7. In all sciences, excepting 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 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 further 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.

8. These axioms or general propositions, thus 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 deduce others 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 deduce 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 connection 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.

9. Thus induction and syllogism are the two methods of direct reasoning corresponding to the two kinds of principles, primary and secondary, on which they are founded, and by which they are respectively conducted. 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 solecism.

10. Till general truths are ascertained by induction, the third or middle terms by which syllogisms are made are nowhere safely to be found. So that another position of the Stagirite, 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 more 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 logomachy, and only instrumental in the multiplication of false learning, and the invention and confirmation of error. The truth of syllogisms depends ultimately on the truth of axioms, and the truth of axioms on the soundness of inductions. But though induction is 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.

6. Of Demonstration.

1. Having concluded all that seemed necessary to be said in regard to the two methods of direct reasoning, the syllogistic and inductive, 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 if the induction could 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 fall short of the number and extent of nature; which, in some cases, by its immensity, will defeat all possibility of their co-extension, whilst, in others, by its distance, it lies out of the reach of their immediate application. Though truth does not appear in all other departments of learning with that bold and restless conviction with which it presides in the mathematical sciences, it shines through them all, if not interrupted by prejudice or perverted by error, with a clear and an useful though inferior certainty. And as it is not necessary for the general safety or convenience of a traveller, that he should always enjoy the heat and splendour of a mid-day sun, whilst he can with more ease pursue his journey under the milder influence of a morning or an evening sky; 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 abstract ideas of space and number, 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

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1 This is a sad misrepresentation of Aristotle's doctrine. He makes the deductive syllogism prior in the order of nature to the inductive syllogism, as he makes the state in the same order prior to the family, and the family prior to the individual. But he expressly declares that the highest principles from which, in the syllogism proper, the deduction is made, are all the results of a foregoing induction. The error arises from ignorance of the difference of a priority in the order of nature and intention, and priority in the order of time and execution.

2 This chapter is almost wholly taken from Tatham's Chart and Scale of Truth; a work which, notwithstanding the ruggedness of its style, has so much real merit as a system of logic, that it cannot be too diligently studied by the young inquirer who wishes to travel by the straight road to the temple of science. purest elements, and spreading upon all sides into a system of science. As each step in the progress is 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 assume, 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 a train. And as in such a concatenation of syllogisms all the various modes of reasoning which 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 different kinds of syllogisms above explained, according as they are found best suited to 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, in general, we may 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 proof consists 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 upon definitions and self-evident propositions.

2. All syllogisms whatsoever, whether compound, uniform, or defective, are reducible to plain simple syllogisms in some one of the four figures. But this is not all. Syllogisms of the first figure, in particular, admit of all possible conclusions; that is, in any propositions whatsoever, whether universal affirmatives or universal negatives, particular affirmatives or particular negatives, which fourfold division embraces all their varieties, the conclusion 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 article. It is sufficient to notice, that the thing is universally known and allowed amongst logicians, to whose writings we refer such as desire further 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 further 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, then, 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 demonstrations in all their parts may be ultimately deduced.

3. 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, 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, of which 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 form 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 form 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 figure. "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, accountableness, is affirmed of all creatures possessed of reason and liberty. Again, in the second proposition, man, the subject of the conclusion, is affirmed to be or to form 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 asserts the subject of the conclusion to be or form 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 in 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 they 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 to 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.

4. And now we may affirm, that, in all syllogisms of the Demonstration an first figure, if the premises be true, the conclusion also must needs be true. If it be true that the predicate of the conclusion, whether affirmative or negative, agrees 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 have 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 likewise, it evidently follows, that if all the several premises are true, all the several conclusions are so, and consequently also the conclusion 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, in which 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. Hence all the previous propositions of a demonstration being thus manifestly true, the last conclusion, or proposition to be demonstrated, must likewise be so; 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 basis, 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 reducible 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.

5. Demonstration, therefore, serving as an infallible guide to truth, upon so sure and unalterable a basis, we may now venture to assert, that the rules of logic furnish a sufficient criterion for the distinguishing between truth and falsehood. For since every proposition that can be demonstrated is necessarily true, he is able to distinguish truth from falsehood who can with certainty judge when a proposition is truly demonstrated. Now, a demonstration, as we have said, is nothing more than a concatenation of syllogisms, all the premises of which are definitions, self-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 into 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 agreeably 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 the 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 these 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.

6. 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 sort of business or concern. 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 aright the degree of proof, and discern 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 upon 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.

7. Before we conclude this branch of the subject, it may distinctly 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 admitted 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 the conclusion of which 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 likewise be so, 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 which we mean to demonstrate; and thence, by a continued train of reasoning, in the way of a direct demonstration, deducing from it 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, having to demonstrate that circles which touch one another inwardly have not the same centre, assumes the direct contrary of 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.

8. Now, because this manner of demonstration is accounted by some not altogether so clear and satisfactory, reasoning we shall therefore endeavour to show that it equally with indirect demonstration leads to truth and certainty. Two propositions are said to be contradictory of one another, when that which is asserted to be in the one is asserted 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 contradictories, because the second asserts directly the contrary of that which is asserted 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 both to be 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 indeed is 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 know the other to be true.

9. Now this is precisely the manner of an indirect demonstration, as is evident from the account which has been given of it above. For there we assume a proposition directly contradictory of that which we mean to demonstrate; and, having by a continued series of proofs shown it to be false, we thence infer that it is contradictory, or that 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, 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 falsehood. Thus Euclid, in the example before mentioned, from the supposition that circles touching one another inwardly have the same centre, deduces the inference 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 likewise be so, by what has been before demonstrated. Now, as to all the other premises made use of in the course of reasoning, they are by supposition manifest and known truths, 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, it 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 mode of proof, we arrive at a clear and satisfactory knowledge of truth.

10. This is universally the method of reasoning in all apagogical or indirect 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; and hence 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 proposed 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 must be 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.

11. We have a curious instance of this in the twelfth proposition of the ninth book of the Elements. Euclid proposes to demonstrate, that in any series of numbers, rising from unity in geometrical progression, all the prime numbers that measure the last term in the series 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 measuring the last term in the series does not measure the next after unity; and, thence by a continued train of reasoning, proves that it actually does measure it. Hereupon he concludes that the assumed proposition is false; and that that which is deduced from it, or its contradictory, which is the very proposition he proposed to demonstrate, is true. Now that this is a just and conclusive way of reasoning, seems abundantly manifest from what has been 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, should be rejected as false.

12. Having thus sufficiently evinced the certainty of demonstration in all its branches, and shown the rules by which we ought to proceed in order to arrive at a just conclusion according to the various ways of arguing made use of, it is needless to enter upon a particular consideration of those several species of false reasoning which logicians distinguish by the name of sophisms. He who 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 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.

IV.—OF METHOD.

We have now done with the three first operations of General Method, 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 so to put them together that their mutual connection and dependence may be clearly seen. This is what logicians called 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 that some progress has been made in knowledge, before we can exert it in any extensive degree.

2. In this view it is plain that we must be beforehand well acquainted with the truths which we are to combine together, otherwise, how could we discern their several connections 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 although at our first setting out we were only acquainted with some of the grander 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, by a narrow scrutiny into things, we have unravelled any part of knowledge, and traced it to its first and original principles, insomuch 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 so adjust and put together the parts as the order and method of science requires.

3. But as these things are best understood when illustrated by examples, let us suppose any machine, for instance a watch, presented to us, the structure and composition of which 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.

4. And as it is in tracing and examining the works of art, so is it in a great measure in unfolding any part 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 which lead to the conclusion we are in quest of. And if it so happen that the previous propositions themselves are not sufficiently evident, we endeavour by new middle terms to ascertain their truth, still tracing things backwards in a continual series, until at length we arrive at some syllogism where the premises are first and self-evident principles. This being 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 immoveable foundation of our intuitive perceptions. And as we arrived at this certainty by tracing things backwards to the original principles whence they flow, so may we at any time renew it by a directly 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.

5. Hence it appears, that in disposing and putting together our thoughts, either for our own use, that the discoveries we have made may at all times lie open to the review of the mind, or where we mean to communicate and unfold these 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, the several propositions which we want to establish. This diversity in the manner of arranging our thoughts gives rise to the twofold division of method established amongst 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 proposed and put together as they were or might have been discovered, this is called the analytical method, or the method of resolution; insomuch as it traces things backwards to their source, and resolves knowledge into its first and elementary principles. When, on the other hand, they are deduced from these principles, and connected according to their mutual dependence, insomuch that the truths first in order tend always to the demonstration of those which follow, this constitutes what we call the synthetical method, or the 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.

6. There is this further to be taken notice of in relation to these two species of method, that the first has also obtained the name of the method of invention, because it observes the order in which our thoughts succeed one another in the invention or discovery of truth. The other, science again, is often denominated the 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 have 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. At our first setting out, we are commonly unable to divine where the analysis will lead us, insomuch, indeed, 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 make many reviews, and a frequent comparison of the several steps of the investigation amongst themselves. Nay, when we have unravelled the whole, and reached the very foundation upon which our discoveries stand, all our certainty, in regard to their truth, will be found in a great measure to arise from that connection which we are now able to discern between them and first principles, taken in the order of composition. But in the synthetical 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 upon which our ascent rests. In communicating our discoveries to others, therefore, this method is apparently to be chosen, as it wonderfully improves and enlightens the understanding, and leads to an immediate perception of truth.

7. 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 search after truth. Attempts have been made to restore it to credit, but without success; and for a considerable time little or no attention whatever has been paid to the science of logic in the course of what is called a liberal education. As both extremes may be faulty, it should seem that we cannot conclude this V.—OF 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 property, 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 nearly two thousand 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 science of logic has gradually and undeservedly fallen into disrepute.

To this effect a variety of causes have contributed. The syllogistic art is admirably calculated for wrangling; and by the schoolmen it was employed with but too much success to keep in countenance the absurdities of the scholastic philosophy. 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 the too unfavourable opinion which now seems prevalent, of its being unworthy of a place in a liberal education. Men rarely leave one extreme without running into the opposite. They who think according to the fashion of the time, 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 Bishop Warburton, "logic is more a trick than a science, formed rather to amuse than 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 chicanery, 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; 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 Utility of 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 judgment or skill of reasoning in any science."

This must have been Mr 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 much less time consumed in its study, than formerly. His counsel, therefore, was judicious and seasonable, viz. That the improvement of our reasoning powers is to be expected much more from an intimate acquaintance with the authors who reason best, than from studying voluminous systems of scholastic 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 concerning 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.

It must indeed be confessed, that a man without knowing the rules of logic may acquire a habit of reasoning justly in mathematics, and perhaps in any other science. Good sense, good examples, and assiduous exercise, may bring a man to reason justly and acutely in his own profession without rules. But whoever thinks, that from this concession he may infer the inutility of logic, betrays by this inference a great want of sense; for he might as well infer that, because a man may go from Edinburgh to London by the way of Paris, 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 conjoined 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 rendered 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 judgment, to justify what is right, and to condemn what is wrong. Now mathematics are the 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, shall study mathematics with this view, will become not only by mathematics a more expert logician, but by logic a more rational mathematician, but a wise philosopher, and an acuter reasoner on 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 should fall into contempt, and if mathematics, instead of furthering science, should become in fact an obstacle. For when men, knowing nothing of that reasoning which is universal, come to attach themselves for a series of 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 Utility of that science is not demonstration, is nothing, or at least below the regard of the geometrician. 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 whatever 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 simplest 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 also 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, the conversion and opposition of propositions, and the general rules of reasoning, are not without use, is sufficiently apparent from the blunders daily committed by those who disdain to cultivate any acquaintance with them.

Although the art of categorical syllogism is confessedly little fitted for the discovery of unknown truths, 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 method of induction, from effects to causes; but he who as a teacher has 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, this 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 Mr 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 logic for unravelling the sophistry of his Jesuitical antagonist.

Upon the whole, then, although we readily acknowledge that much time was wasted by our forefathers in syllogistic wrangling, and what might with little impropriety be termed the mechanical part of logic, yet the art of form-

The preceding article was written at a period when some views in regard to the object and the end of logic were entertained in this country, which, in the present state of the science, would be regarded as incorrect. The influence of these views upon this treatise does not, however, affect its utility. They are observable more in the definition than in the treatment of the subject. It is, accordingly, sufficient to correct the former, by showing what is the true scope and peculiar province of logic, and to point out to the reader the more accessible sources to which he may resort.

The common error, in which Bacon participated, of identifying logical (formal) and philosophical (real) induction; and the errors of Locke in supposing that logic supplied any peculiar method of reasoning; and that syllogism was ever intended as an instrument of discovery; had afforded to these philosophers a ground on which they found no difficulty in demonstrating the comparative insufficiency of the whole logical system of Aristotle. Nor was the fallacy of their refutation soon exposed by detecting the defect in its foundation. On the contrary, in this country, and in some others, it was long admitted to be conclusive; and this admission in a great measure determined the mode in which the subject was treated by the greater number of those British authors who subsequently presented to the public what they denominated systems of logic. Instead of showing that this science was only undervalued for not effecting what it really did not propose to accomplish; instead of showing that the end which it does propose is of high importance, and is fully accomplished by this science, and by this science alone; and, instead of rigorously excluding from their systems all not necessarily conducive to this end; they, on the contrary, proposed to logic an end which no one science could possibly accomplish; and in attempting this, they transmuted the most certain, definite, and independent of the philosophical sciences, into a precarious assemblage of foreign, incoherent, and insecure materials. Logic was regarded as the doctrine which taught the art of using rightly our intellectual faculties in the acquisition of knowledge; but as the promise here necessarily stood in melancholy contrast with the performance, this false and exaggerated estimate of the science only contributed to sink still lower the humble opinion now entertained of its importance.

Logic may be defined the science of the laws of thought, considered as thought. This is the central notion towards which the various views of the science from Aristotle downwards gravitate; it is the one definition, in which others, apparently the most opposite, find their complement and reconciliation. This definition, however, requires some elucidation.

In the first place, logic is said to be conversant about thought. The term thought properly denotes a self-active or spontaneous operation of the mind upon its objects, and in opposition to those passive determinations by which these objects are originally received into the mind, or subsequently suggested to its view. Logic is thus confined to the discursive agency of mind in its three degrees of Apprehension, Judgment, and Reasoning, to the exclusion of the subordinate operations of perception, imagination, and memory. Those who limit the definition of logic to Reasoning, are incorrect, inasmuch as, though the laws of reasoning are the principal, they are not the adequate object of the science. Those also are wrong who make logic proximately conversant about language, thus confounding it with grammar. We may, however, with the schools, il- L O G I C.

In the second place, logic is conversant about thought considered as thought. In every act of thinking we distinguish by reflection two things: 1. the object of thought (the matter); 2. the mode of thinking it (the form). For example, when I think "This pen is bad," the matter of the thought is the bad pen; the form in which the matter is contained is a judgment. Again, the word "animal;" in so far as it is viewed as expressive of a certain conceived multitude of living existences, is material; in so far as it is considered as only significant of a general notion, it is formal. This distinction of material and formal in thought was also expressed by the schoolmen in the contrast of entia realia and entia rationis, of notiones (intellecta, conceptus, intentiones) prima, and notiones secunda. This distinction is indeed only a mental abstraction for the sake of science, as the matter and the form of thought are in reality inseparable; for we are conscious of no object excepting under some form of thought, and of no form of thought without a reference to some determinate matter which constitutes the object of thought. Of these, the matter is infinite in variety and number; the form, of few and determinate modifications. Logic, therefore, can only consider the form of thought. For if it took the matter into account, it must either take all objects indifferently, or only some. On the former principle, logic would be a treatise upon the omne scibile, a complement of all the sciences, which is impossible and absurd; and upon the latter, no reason can be shown why certain objects should be selected for consideration, to the exclusion of all others. But if we can neither make all nor some objects of thought the object-matter of logic, it only remains that we propose as its object-matter the form of thought itself; that is, that we consider in logic thought as thinking, to the neglect of the particulars about which it thinks. Nor is this view of the province of logic any novelty. In the schools, to say nothing of more recent philosophers, logic was universally viewed as a rational not a real, as a formal not a material science; in other words, as conversant, not about things as conceived or known, but about the mode of conceiving and knowing them. Its object-matter was thus by the schoolmen limited to entia rationis, in contrast to entia realia; to notiones secunda, in contrast to notiones prima; to things in general, in so far as they stand under second notions, &c.; expressions which, when rightly understood, manifest that those persons who employed them held views, in regard to the scope and domain of the science, far more accurate than those latterly prevalent, at least in this country.

In the third place, logic is the science of the laws of thought. Logic, as we have seen, is solely conversant about the faculty of thought, considered in its formal relation. But the forms of thought can be made the object of consideration from two points of view. We can either contemplate them as objects of mere experience, that is, delineate and arrange the special modifications of the thinking principle, as facts of observation and history; or we may regard them as objects of speculation, that is, by means of reflection and abstraction seek out and determine what, as necessary in each act of thinking, affords the condition of the possibility of thought in general. The consideration of the human mind, as an object of experience, belongs not to logic, but to psychology. A knowledge, indeed, of the phenomena of mind, as facts of experience, must chronologically precede their consideration as objects of speculation; the philosophy of the human mind, as an empirical science, forms therefore the natural preliminary of logic as a speculative science. Logic in itself, however, constitutes not the less an independent and exclusive system. For where the knowledge of our intellectual nature founded upon observation and induction ends, and where, by an abstraction from all that experience affords of particular and contingent, we ascend by reflection to the universal, and necessary, and a priori conditions it involves, there the domain of logic commences, and that of psychology, as a doctrine explanatory of mere facts, terminates. But if logic be conversant about the universal and necessary in thought, it is thereby conversant about the laws of thought. For the universal and necessary is only conceived as what is determined by laws; and again, the very conception of a law involves the universality and necessity of its application.

Logic is thus conversant, not about thought considered in itself, but about thought viewed as subject to laws. Those therefore who, without qualification, make the operations or processes of the discursive faculty the object of logic, are wrong; for logic considers these only as governed by laws. Nor is this any modern observation. "To treat of thoughts as thoughts," says Simplicius, "is not the function of logic, but of psychology." Those schoolmen who held entia rationis or notiones secunda to be the object of logic, did not propose that these should be considered in themselves, but only in their application to entia realia or notiones prima, as the instruments and regulators of thought; whilst those again who proposed all or some of the three intellectual operations for this object, did so only in so far as these were dirigible or determinable by laws.

Thus the same correct view of the science may be detected under forms of expression which at first sight appear to be the most contradictory; and what we have shown in regard to the schoolmen may be more easily proved in regard to the Leibnitzian and Kantian logicians. But though a right conception of the scope and limits of logic must be allowed to have long and generally prevailed, it must be acknowledged that the speculative notion did not in practice regulate the contents and treatment of the science. From Aristotle downwards (the example and authority of the Stagirite having mainly determined the practice of his successors), the purity of the science has been, and is still, contaminated by foreign infusions; and it is only by slow and painful efforts that one extraneous doctrine after another has been banished from its domain. Thus, though it has always been speculatively allowed that logic is exclusively conversant with the form of thought; yet a consideration of the material modality of propositions and syllogisms (a subject as useless as it is irksome) remained, though often overlooked, an integral portion of the science, until it was proved to be intrusive in the article on logic in the Edinburgh Review (vol. lvii. p. 220, et seq.), and has accordingly, by subsequent writers on the science, been formally expunged, as paralogical, from their systems. In this country there is, however, no work on logic which displays the science in a pure and perfect form; which evolves its doctrines from fundamental principles, and displays its parts in that necessary connection and dependence, which constitutes the elegance of a scientific system. Whately's Elements of Logic is the English book which is best deserving of attention. Though somewhat vague and vacillating in its views of the science, and though its doctrines are neither developed from the primary laws of thought, nor combined together as the essential parts of one necessary whole, yet it displays a talent which shows that the author, with a more comprehensive knowledge of his subject, might have produced a work not easily to be superseded. There are also several minor works recently published on logic in this country, not unworthy of consideration, a detailed estimate of which may be seen in the Review previously referred to. LOGISTÆ, certain officers at Athens, ten in number, whose business consisted in receiving and passing the accounts of magistrates when they went out of office. The logistæ were elected by lot, and had under them ten euthymii or auditors of accounts.