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

The precise business of logic, therefore, is to explain the nature of the human mind, and the proper manner of conducting its several powers, in order to the attainment of truth and knowledge. It lays open those errors and mistakes we are apt, through inattention, to run into; and teaches us how to distinguish between truth, and what only carries the appearance of it. By this means we grow acquainted with the nature and force of the understanding; see what things lie within its reach; where we may attain certainty and demonstration; and when we must be contented with probability.

This science is generally divided into four parts, viz.: Perception, Judgment, Reasoning, and Method. This division comprehends the whole history of the sensations and operations of the human mind. But we must refer the reader for the first part, viz., Perception, and Ideas, to Metaphysics, where it will be more conveniently and fully treated, and confine ourselves in this place to the three last, viz., Judgment, Reasoning, and Method.

PART I. Of JUDGMENT.

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

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

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

The second ground of human judgment is experience; from which we infer the existence of those objects that surround us, and fall under the immediate notice of our senses. When we see the sun, or cast our eyes towards a building, we not only have ideas of these objects within ourselves, but ascribe to them a real existence out of the mind. It is also by the information of the senses, that we judge of the qualities of bodies; as when we say that snow is white, fire hot, or steel hard. For as we are wholly unacquainted with the internal structure and constitution of the bodies that produce these sensations in us, and are unable to trace any connection between that structure and the sensations themselves, it is evident, that we build our judgments altogether upon observation, ascribing to bodies such qualities, as are answerable to the perceptions they excite in us. But this is not the only advantage derived from experience; for we are likewise indebted to it for all our knowledge regarding the coexistence of sensible qualities in objects, and the operations of bodies one upon another. Ivory, for instance, is hard and elastic; this we know by experience, and indeed by that alone. For being altogether strangers to the true nature both of elasticity and hardness, we cannot by the bare contemplation of our ideas determine, how far the one necessarily implies the other, or whether there may not be a repugnance between them. But when we observe them to exist both in the same object, we are then assured from experience, that they are not incompatible; and when we also find that a stone is hard and not elastic, and that air though elastic is not hard, we also conclude upon the same foundation, that the ideas are not necessarily conjoined; but may exist separately in different objects. In like manner with regard

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to the operations of bodies one upon another, it is evident, that our knowledge this way, is all derived from observation. Aqua regia dissolves gold, as has been found by frequent trial, nor is there any other way of arriving at the discovery. Naturalists may tell us if they please, that the parts of aqua regia are of a texture apt to infuse between the corpuscles of gold, and thereby loosen and shake them asunder. If this is a true account of the matter, we believe it will notwithstanding be allowed, that our conjecture in regard to the conformation of these bodies is deduced from the experiment, and not the experiment from the conjecture. It was not from any previous knowledge of the intimate structure of aqua regia and gold, and the aptness of their parts to act or be acted upon, that we came by the conclusion above mentioned. The internal constitution of bodies is in a manner wholly unknown to us, and could we even surmount this difficulty, yet as the separation of the parts of gold, implies something like an active force in the menstruum, and we are unable to conceive how it comes to be possessed of this activity; the effect must be owned to be altogether beyond our comprehension. But when repeated trials had once confirmed it, insomuch that it was admitted as an established truth in natural knowledge, it was then easy for men, to spin out theories of their own invention, and contrive such a structure of parts, both for gold and aqua regia, as would best serve to explain the phenomenon, upon the principles of that system of philosophy they had adopted.

From what has been said it is evident, that as intuition is the foundation of what we call scientific knowledge, so is experience of natural. For this last, being wholly taken up with objects of sense, or those bodies that constitute the natural world; and their properties, as far as we can discover them, being to be traced only by a long and painful series of observations, it is apparent, that in order to improve this branch of knowledge, we must be take ourselves to the method of trial and experiment.

But though experience is what we may term the immediate foundation of natural knowledge, yet with respect to particular persons, its influence is very narrow and confined. The bodies that surround us are numerous, many of them lie at a great distance, and some quite beyond our reach. Life too is short, and so crowded with cares, that but little time is left for any single man to employ himself in unfolding the mysteries of nature. Hence it is necessary to admit many things upon the testimony of others, which by this means becomes the foundation of a great part of our knowledge of body. No man doubts of the power of aqua regia to dissolve gold, though perhaps he never himself made the experiment. In these therefore and such like cases, we judge of the facts and operations of nature, upon the mere ground of testimony. However, as we can always have recourse to experience, where any doubt or scruple arises, this is justly considered as the true foundation of natural philosophy; being indeed the ultimate support upon which our assent rests, and whereto we appeal, when the highest degree of evidence is required.

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

Of affirmative and negative propositions.

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

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

Every proposition necessarily consists of these three parts, but then it is not alike needful that they be all generally expressed in words; because the copula is often included in the term of the predicate, as when we say, He sits; which imports the same as he is sitting. In the Latin language, a single word has often the force of a whole sentence. Thus, ambulat is the same as ille est ambulans; amo, as ego sum amans, and so in innumerable other instances; by which it appears, that we are not so much to regard the number of words in a sentence, as the ideas they represent, and the manner in which they are put together. For where ever two ideas are joined or disjoined in an expression, though of but a single word, it is evident that we have a subject, predicate, and copula, and of consequence a complete proposition.

When the mind joins two ideas, we call it an affirmative judgment; when it separates them, a negative; and as any two ideas compared together, must necessarily either agree or not agree, it is evident, that all our judgments fall under these two divisions. Hence likewise, the propositions expressing these judgments, are all either affirmative or negative.

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

Of universal and particular propositions.

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

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

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

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

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

Of absolute and conditional propositions.

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

From this diversity in the several qualities of things, arises a considerable difference as to the manner of our judging about them. For all such properties as are inseparable from objects, when considered as belonging to any genus or species, are affirmed absolutely and without reserve of that general idea. Thus we say, Gold is very weighty, a stone is hard, animals have a power of self-motion. But in the case of mutable or accidental qualities, as they depend upon some other consideration, distinct from the general idea; that also must be taken into the account, in order to form an accurate judgment. Should we affirm, for instance, of some stones, that they are very susceptible of a rolling motion; the proposition while it remains in the general form, cannot with any advantage be introduced into our reasonings. An aptness to receive that mode of motion, flows from the figure of the stone; which, as it may vary infinitely, our judgment then only becomes applicable and determinate, when the particular figure, of which volatility is a consequence, is also taken into the account. Let us then bring in this other consideration, and the proposition will run as follows: Stones of a spherical form, are easily put into a rolling motion. Here we see the condition upon which the predicate is affirmed, and therefore know in what particular cases the proposition may be applied. This consideration of propositions, respecting the manner in which the predicate is affirmed of the subject, gives rise to the division of them into absolute and conditional. Absolute propositions are those wherein we affirm some property inseparable from the idea of the subject, and which therefore belongs to it in all possible cases; as, God is infinitely wise. Virtue tends to the ultimate happiness of man. But where the predicate is not necessarily connected with the idea of the subject, unless upon some consideration distinct from that idea, there the proposition is called conditional. The reason of the name is taken from the supposition annexed, which is of the nature of a condition, and may be expressed as such. Thus, If a stone is exposed to the rays of the sun, it will contract some degree of heat. If a river runs in a very declining channel, its rapidity will constantly increase.

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

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

Of simple and compound propositions.

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

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

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

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

Of the division of propositions into self-evident and demonstrable.

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

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

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

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

We see therefore, that nothing contributes so much to the improvement and certainty of human knowledge, as the having determinate ideas, and keeping them steady and invariable in all our discourses and reasonings about them. And on this account it is, that mathematicians always begin by defining their terms, and distinctly unfolding the notions they are intended to express. Hence such as apply themselves to these studies, having exactly the same views of things, and bringing always the very same ideas into comparison, readily discern the relations between them.

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

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

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

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

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 to theorems, or problems, and differ from them only in this; that they flow from what is there demonstrated, in so obvious a manner, as to discover their dependence upon the proposition whence they are deduced, almost as soon as 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 ones, and things equal to one and the same thing are equal to one another.

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 original and history of the several discoveries made in the science; and in a word, to acquaint us with all such particulars as deserve to be known, whether considered as points of curiosity or profit.

PART II. Of Reasoning.

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

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

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

Hence it appears, that every act of reasoning necessarily includes three distinct judgments; two wherein the ideas whose relation we want to discover are severally compared with the middle idea, and a third wherein they are themselves connected or disjoined according to the result of that comparison. Now, as, in the first part of logic, our judgments, when put into words, were called propositions; so here, in the second 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 a reasoning is thus put into words, and appears in form of a syllogism, the intermediate idea made use of to discover the agreement or disagreement we search for is called the middle term; and the two ideas themselves, with which this third is compared, go by the name of the extremes.

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

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.

In this syllogism 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 two first 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. The terms expressing the two ideas whose relation we inquire after, as here man and accountability, are in general called the extremes; and 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.

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. proposition; the other, wherein the same middle term is compared with the subject of the conclusion, or lesser extreme, is called the minor proposition. All this is obvious from the syllogism already given, where the conclusion is, Man is accountable for his actions. For here the predicate, accountable for his actions, being connected with the middle term in the first of the two premises, every creature possessed of reason and liberty is accountable for his actions, gives what we call the major proposition. In the second of the premises, Man is a creature possessed of reason and liberty, we find the lesser extreme, or subject of the conclusion, viz., man, connected with the same middle term, whence it is known to be the minor proposition. When a syllogism is propounded in due form, the major proposition is always placed first, the minor next, and the conclusion last.

These things premised, we may in the general define reasoning to be an act or operation of the mind, deducing some unknown proposition from other previous ones that are evident and known. These previous propositions, in a simple act of reasoning, are only two in number; and it is always required that they be of themselves apparent to the understanding, insomuch that we assent to and perceive the truth of them as soon as proposed. In the syllogism given above, the premises are supposed to be self-evident truths, otherwise the conclusion could not be inferred by a single act of reasoning. If, for instance, in the major, every creature possessed of reason and liberty is accountable for his actions, the 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 was still some previous proposition whose truth did not appear at first sight, we must then have recourse to a third syllogism in order to lay open that truth to the mind; because, so long as the premises remain uncertain, the conclusion built upon them must be so too. When by conducting our thoughts in this manner, we at last arrive at some syllogism, where the previous propositions are intuitive truths; the mind then rests in full security, as perceiving that the several conclusions it has passed through stand upon the immovable foundation of self-evidence, and when traced to their source terminate in it.

We see therefore, that in order to infer a conclusion by a single act of reasoning, the premises must be 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 relations 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 a train. But although in this concatenation of propositions, those that form the premises of the last syllogism are often considerably removed from self-evidence; yet if we trace the reasoning backwards, we shall find them the conclusions of previous syllogisms, whose premises approach nearer and nearer to intuition, in proportion as we advance, and are found at last to terminate in it. And if, after having thus unravelled a demonstration, we take it the contrary way; and observe how the mind, setting out with intuitive perceptions, couples them together to form a conclusion; how, by introducing this conclusion into another syllogism, it still advances one step farther; and so proceeds, making every new discovery subservient to its future progress; we shall then perceive clearly, that reasoning, in the highest exercise of that faculty, is no more than an orderly combination of those simple acts which we have already so full explained.

Thus we see, that reasoning, beginning with first principles, rises gradually from one judgment to another, and connects them in such a 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 is the ability of deducing unknown truths from principles or propositions that are already known. This evidently appears by the foregoing account, where we see, that no proposition is admitted into a syllogism, to serve as one of the previous judgments on which the conclusion rests, unless it is itself a known and established truth, whose connection with self-evident principles has been already traced.

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 be reduced to these two: 1. To rank things under those universal ideas to which they truly belong; and, 2. To ascribe to them their several attributes and properties in consequence of that distribution.

One great aim of human reason is, to determine the genera and species of things. Now, as in 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 general 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 peculiar names, we hereby learn to apply the terms expressing general conceptions to such particular objects as come under our immediate observation.

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

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

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

VOL. II. NO 68.

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

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

Of Reasoning, as it regards the powers and properties of things, and the relations of our general ideas.

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

In like manner, if we set ourselves to consider any particular character, in order to determine the share of praise or dispraise that belongs to it, our great concern is, to ascertain exactly the proportion of virtue and vice. The reason is obvious. A just determination in all cases of this kind depends entirely upon an application of these general maxims of morality: Virtuous actions deserve praise. Vicious actions deserve blame.

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 are more immediately concerned, according as they are found to be 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, ascribe all those attributes to the present object. Thus, in considering the character of Sempronius, if we find it to be of the kind called virtuous; when we at the same time reflect, that a virtuous character is deserving of esteem, it naturally and obviously follows that Sempronius is so too. These thoughts put into a syllogism, in order to exhibit the form of reasoning here required, run thus.

Every virtuous man is worthy of esteem.

Sempronius is a virtuous man.

Therefore Sempronius is worthy of esteem.

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 worthy of esteem, only in consequence of his being a virtuous man. Hence we see the necessary connection of the various parts of reasoning, and the dependence 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 scarce any labour of thought, as being no more than an application of the general form of reasoning represented in the foregoing syllogism. Now as we have already sufficiently shewn how we are to proceed in determining the genera and species of things, all that is farther wanting towards a due explanation of it is, to offer some considerations as to the manner of investigating the general relations of our ideas. This is the highest exercise of the powers of the understanding, and that by means whereof we arrive at the discovery of universal truths, insomuch that our deductions in this way constitute that particular species of reasoning which we have before said regards principally the sciences.

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

In order to our successful progress in reasoning, we must have an extensive knowledge of those intermediate ideas by means of which things may be compared one with another. For as it is not every idea that will answer the purpose of our inquiries, but such only as are peculiarly related to the objects about which we reason, so as, by a comparison with them, to furnish evident and known truths; nothing is more apparent, than that the greater variety of conceptions we can call into view, the more likely we are to find some among them that will help us to the truths here required. And indeed it is found to hold in experience, that in proportion as we enlarge our views of things, and grow acquainted with a multitude of different objects, the reasoning faculty gathers strength. For by extending our sphere of knowledge, the mind acquires a certain force and penetration, as being accustomed to examine the several appearances of its ideas, and observe what light they cast one upon another. This is the reason, why, in order to excel remarkably in any one branch of learning, it is necessary to have at least a general acquaintance with the whole circle of arts and sciences. The truth is, all the various divisions of human knowledge are very nearly related among themselves, and in innumerable instances serve to illustrate and set off each other. And although it is not to be denied, that, by an 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 narrow and contract, 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 farther necessary, that we more nearly acquaint ourselves with whatever relates to that science. A general knowledge is a good preparation, and enables us to proceed with ease and expedition, in whatever branch of learning we apply to. But then in the minute and intricate questions of any science we are by no means qualified to reason with advantage, until we have perfectly mastered the science to which they belong.

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. Use and exercise are the best instructors in the present case. And therefore the true way to acquire this talent is, by being much conversant in those sciences where the art of reasoning is allowed to reign in the greatest perfection. Hence it was that the ancients, who so well understood the manner of forming the mind, always began with mathematics, as the foundation of their philosophical studies. Here the understanding is by degrees habituated to truth, contracts 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 catharticks or purgatives of the soul, as being the proper means to cleanse it from error, and restore that natural exercise of its faculties, in which just thinking consists.

If therefore we would form our minds to a habit of reasoning closely and in train, we cannot take any more certain method, than the 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 got 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 have occasion.

But although the study of mathematics be of all others the most useful to form the mind, and give it an early relish of truth, yet ought not other parts of philosophy 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 those different turns of thinking that are peculiarly adapted to the several ideas we examine and the truths we search after. For this purpose, besides the study of mathematics, we ought to apply ourselves diligently to the reading of such authors as have distinguished themselves for strength of reasoning, and a just and accurate manner of thinking. For it is observable, that a mind exercised and seasoned to truth, seldom rests satisfied in a bare contemplation of the arguments offered by others, but will be frequently essaying its own strength, and pursuing its discoveries upon the plan it is most accustomed to. Thus we insensibly contract a habit of tracing truth from one stage to another, and of investigating those general relations and properties, which we afterwards ascribe to particular things, according as we find them comprehended under the abstract ideas to which the properties belong.

Of the forms of Syllogisms.

Hitherto we have contented ourselves with a general notion of syllogisms, and of the parts of which they consist. It is now time to enter a little more particularly into the subject, to examine their various forms, and lay open the rules of argumentation proper to each. In the syllogisms mentioned in, we may observe, that the middle term is the subject of the major proposition, and the predicate of the minor. This disposition, though the most natural and obvious, is not however necessary; it frequently happening, that the middle term is the subject in both the premises, or the predicate in both; and sometimes directly contrary, 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 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. If, on the other hand, it is the predicate of both the premises, the syllogism is said to be in the second figure. Again, in the third figure, the middle term is the subject of the two premises. And lastly, by making it the predicate of the major, and subject of the minor, we obtain syllogisms in the fourth figure.

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 proposition of which a syllogism consists may be either universal or particular, affirmative or negative; the due determination of these, and so putting them together as the laws of argumentation require, constitute what logicians call the moods of syllogisms. Of these moods there are 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 division of syllogisms according to mood and figure, gure, respects those especially which are known by the name of plain simple syllogisms; that is, which are bound- ed to three propositions, all simple, and where the ex- tremes and middle term are connected according to the rules laid down above. But as the mind is not tied down to any one precise form of reasoning, but sometimes makes use of more, sometimes of fewer premises, and often takes in compound and conditional propositions, it may not be amiss to take notice of the different forms derived from this source, and explain the rule by which the mind conducts itself in the use of them.

When, in any syllogism, the major is a conditional pro- position, the syllogism itself is termed conditional. Thus: If there is a God, he ought to be worshipped. But there is a God: Therefore he ought to be worshipped.

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

In all propositions of this kind, supposing them to be exact in point of form, the relation between the antec- endent and consequent must ever be true and real; that is, the antecedent must always contain some certain and ge- nuine 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 conse- quent; but if we reject the consequent, we are in like manner bound to reject also the antecedent. For as the antecedent always expresses some condition, which ne- cessarily implies the truth of the consequent; by admit- ting the antecedent we allow of that condition, and there- fore ought also to admit the consequent. In like manner if it appears that the consequent ought to be rejected, the antecedent evidently must be so too; because the ad- mitting of the antecedent would necessarily imply the admission also of the consequent.

There are two ways of arguing in hypothetical syllo- gisms, which lead to a certain and unavoidable conclusion. For as the major is always a conditional proposition, consisting of an antecedent and a consequent; if the minor admits the antecedent, it is plain that the conclu- sion 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, in as much as by it the whole conditional proposition both antecedent and consequent is established. Thus:

If God is infinitely wise, and acts with perfect free- dom, he does nothing but what is best.

But God is infinitely wise, and acts with perfect free- dom: Therefore he does nothing but what is best.

Here the antecedent or first part of the conditional pro- position 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 we, on the contrary, suppose, that the minor rejects the conse- quent; 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 syllo- gisms thence arising is called by logicians the modus tol- lens; because in it both antecedent and consequent are rejected or taken away, as appears by the following ex- ample.

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.

These two species take in the whole class of conditional syllogisms, and include all the possible ways of arguing that 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 ante- cedent 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 con- sequent may still hold, because of some other determina- tion that infers it. When we say: If a stone is exposed some time to the rays of the sun, it will contract a cer- tain degree of heat; the proposition is certainly true; and admitting the antecedent, we must also admit the conse- quent. But as there are other ways by which a stone may gather heat, it will not follow, from the ceasing of the before-mentioned condition, that therefore the con- sequent 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: in as much as there are a great many other ways by which heat might have been communicated to it. And if we cannot argue from the removal of the antecedent to the removal of the consequent, no more can we from the ad- mission of the consequent to the admission of the antece- dent; 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 propo- sition, If a stone is exposed some time to the rays of the sun, it will contract a certain degree of heat; admitting the consequent, viz. that it has contracted a certain degree of heat, we are not therefore bound to admit the antecedent, that it has been some time exposed to the rays of the sun, because there are many other causes whence that heat may have proceeded. These two ways of arguing, therefore, hold not in conditional syllogisms.

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

The world is either self-existent, or the work of some finite, or of some infinite being.

But it is not self-existent, nor the work of a finite being.

Therefore it is the work of an infinite being.

Now a disjunctive proposition is that where, of several predicates, we affirm one necessarily to belong to the subject, to the exclusion of all the rest, but leave that particular one undetermined. Hence it follows, that as soon as we determine the particular predicate, all the rest are of course to be rejected; or if we reject all the predicates but one, that one necessarily takes place. When therefore, in a disjunctive syllogism, the several predicates are enumerated in the major; if the minor establishes any one of these predicates, the conclusion ought to remove all the rest; or if, in the minor, all the predicates but one are removed, the conclusion must necessarily establish that one. Thus, in the disjunctive syllogism given above, the major affirms one of three predicates to belong to the earth, viz. self-existence, or that it is the work of a finite or that it is the work of an infinite being. Two of these predicates are removed in the minor, viz. self-existence, and the work of a finite being. Hence the conclusion necessarily attributes to it the third predicate, and affirms that it is the work of an infinite being.

If now we give the syllogism another turn, 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 this 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.

In the several kinds of syllogisms hitherto mentioned, the parts are complete, that is, 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, that seems to be made up of only two propositions. Should we, for instance, argue in this manner;

Every man is mortal;

Therefore every king is mortal;

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

These seemingly imperfect syllogisms are called enthymemes, and occur very frequently in reasoning, especially where it makes a part of common conversation. Nay, there is a particular elegance in them; because, not displaying the argument in all its parts, they leave somewhat to the exercise and invention of the mind. By this means we are put upon exerting ourselves, and seem to share in the discovery of what is 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. It besides shortens discourse, and adds a certain force and liveliness to our arguments, when the words in which they are conveyed favour the natural quickness of the mind in its operations, and a single expression is left to exhibit a whole train of thoughts.

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 falls out, that the proposition on which the other depends is self-evident, we content ourselves with barely affirming it, and infer that other by a direct conclusion. Thus, by admitting an universal proposition, we are forced also to admit of all the particular propositions comprehended under it, this being the very condition that constitutes a proposition universal. If then that universal proposition chances to be self-evident, the particular ones follow of course, without any farther train of reasoning. Whoever allows, for instance, that things equal to one and the same thing are equal to one another, must at the same time allow, that two triangles, each equal to a square whose side is three inches, are also equal between themselves. This argument therefore,

Things equal to one and the same thing are equal to one another;

Therefore these two triangles, each equal to the square of a line of three inches, are equal between themselves,

is complete in its kind, and contains all that is necessary towards a just and legitimate conclusion. For the first or universal proposition is self-evident, and therefore requires no farther proof. And as the truth of the particular is inseparably connected with that of the universal, it follows from it by an obvious and unavoidable consequence.

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 suppositions upon which it takes place, and allow of all immediate consequences that follow in conformity to them. It is however observable, that these arguments, though seemingly complete, because the conclusion follows necessarily from the single proposition that goes before, may yet be considered as real enthymemes, whose major, which is a conditional proposition, is wanting. The syllogism but just mentioned, when represented according to this view, will run as follows:

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

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

The next species of reasoning we shall take notice of is what is 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 possible. He that can do every thing possible, can do whatever involves not a contradiction. Therefore God can do whatever involves not a contradiction.

This particular combination of propositions, may be continued to any length we please, without in the least weakening the ground upon which the conclusion rests. The reason is, because the sorites itself may be resolved into as many simple syllogisms as there are middle terms in it; where this is found universally to hold, that when such a resolution is made, and the syllogisms are placed in train, the conclusion of the last in the series is also the conclusion of the 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.

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

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

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

We come now to that kind of argument which logicians called induction; in order to the right understanding of which, it will be necessary to observe, that our general ideas are for the most part capable of various subdivisions. Thus the idea of the lowest species may be subdivided into its several individuals, the idea of any genus into the different species it comprehends, and so of the rest. If then we suppose this distribution to be duly made, and so as to take in the whole extent of the idea to which it belongs; then it is plain, that all the subdivisions or parts of any idea together constitute that whole idea. Thus the several individuals of any species taken together constitute the whole species, and all the various species comprehended under any genus make up the whole genus. This being allowed, whatever may be affirmed of all the several subdivisions and classes of any idea, ought to be affirmed of the whole general idea to which these subdivisions belong. What may be affirmed of all the individuals of any species, may be affirmed of the whole species, and what may be affirmed of all the species of any genus may be also affirmed of the whole genus; because all the individuals taken together are the same with the species, and all the species taken together the same with the genus.

This way of arguing, where we infer universally concerning any idea what we had before affirmed or denied separately of all its several subdivisions and parts, is called reasoning by induction. Thus if we suppose the whole tribe of animals, subdivided into men, beasts, birds, insects, and fishes, and then reason concerning them after this manner: All men have a power of beginning motion; all beasts, birds, and insects, have a power of beginning motion; all fishes have a power of beginning motion; therefore all animals have a power of beginning motion; the argument is an induction. When the subdivisions are just, so as to take in the whole general idea, and the enumeration is perfect, that is, extends to all and every of the inferior classes or parts; there the induction is compleat, and the manner of reasoning by induction is apparently conclusive.

The last species of syllogisms we shall take notice of, 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 the several suppositions ought to be rejected, it is plain that the antecedent or assertion itself must be so too. When therefore such a proposition as that before-mentioned is made the major of any syllogism; if the minor rejects all the suppositions contained in the consequent, it follows necessarily, that the conclusion ought to reject the antecedent, which, as we have said, is the very assertion to be disproved. This particular way of arguing is that which logicians call a dilemma; and from the account here given of it, it appears, that we may in the general define it to be a hypothetical syllogism, where the consequent of the major is a disjunctive proposition, which is wholly taken away or removed in the minor. Of this kind is the following:

If God did not create the world perfect in its kind, it must 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.

The nature then of a dilemma is universally this. The major is a conditional proposition, whose consequent contains all the several suppositions upon which the antecedent can take place. As therefore these suppositions are wholly removed in the minor, it is evident that the antecedent must be so too; 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 speak. Hence it is plain, that if the antecedent of the major is an affirmative proposition, the conclusion of the dilemma will be negative; but if it is a negative proposition, the conclusion will be affirmative.

Of Demonstration.

Having dispatched what seemed necessary with regard to the forms of syllogisms, we shall now explain their use and application in reasoning. We have seen, that in all the different appearances they put on, we still arrive at a just and legitimate conclusion: now it often happens, that the conclusion of one syllogism becomes a previous proposition in another, by which means great numbers of them are sometimes linked together in a series, and truths are made to follow one another in train. And as in such a concatenation of syllogisms, all the various ways of reasoning that are truly conclusive may be with safety introduced; hence it is plain, that in deducing any truth from its first principles, especially where it lies at a considerable distance from them, we are at liberty to combine all the several kinds of arguments above explained, according as they are found best to suit the end and purpose of our inquiries. When a proposition is thus, by means of syllogisms, collected from others more evident and known, it is said to be proved; so that we may in the general define the proof of the 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 on definitions and self-evident propositions.

All syllogisms whatsoever, whether compound, multiform, 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, any proposition whatsoever, whether an universal affirmative, or universal negative, a particular affirmative, or particular negative, which fourfold division embraces all their varieties; any one of these may be inferred, by virtue of some syllogism in the first figure. By this means 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. It is enough to take notice, that the thing is universally known and allowed among logicians, to whose writings we refer such as desire farther satisfaction in this matter. This then being laid down, it is plain, that any demonstration whatsoever may be considered as composed of a series of syllogisms, all in the first figure. For since all the syllogisms that enter the demonstration are reducible to syllogisms of some one of the four figures, and since the syllogisms of all the other figures are farther reducible to syllogisms of the first figure, it is evident, that the whole demonstration may be resolved into a series of these last syllogisms. Let us now, if possible, discover the ground upon which the conclusion rests, in syllogisms of the first figure; because, by so doing, we shall come at an universal principle of certainty, whence the evidence of all demonstrations in all their parts may be ultimately derived.

The rules then of the first figure are 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, whereof the subject of the conclusion is the subject, and the middle term the predicate. Here then the middle term is affirmed of the subject of the conclusion; that is, the subject of the conclusion is affirmed to be comprehended under, or to make a part of the middle term. Thus then we see what is done in the premises of a syllogism of the first figure. The predicate of the conclusion is universally affirmed or denied of some idea. The subject of the conclusion is affirmed to be or to make a part of that idea. Hence it naturally and unavoidably follows, that the predicate of the conclusion ought to be affirmed or denied of the subject. To illustrate this by an example, we shall resume one of the former syllogisms.

Every creature possessed of reason and liberty is accountable for his actions.

Man is a creature possessed of reason and liberty.

Therefore man is accountable for his actions.

Here, in the first proposition, the predicate of the conclusion, accountability, is affirmed of all creatures that have reason and liberty. Again, in the second proposition, man, the subject of the conclusion, is affirmed to be or to make a part of the class of creatures. Hence the conclusion necessarily and unavoidably follows, viz. that man is accountable for his actions; because if reason and liberty be the which constitutes a creature accountable, and man has reason and liberty, it is plain he has that which constitutes him accountable. In like manner, where the major is a negative proposition, or denies the predicate of the conclusion universally of the middle term, as the minor always affirms the subject of the conclusion, to be or make a part of that middle term, it is no less evident, that the predicate of the conclusion ought in this case to be denied of the subject. So that the ground of reasoning in all syllogisms of the first figure is manifestly this. Whatever may be affirmed universally of any idea, may be affirmed of every or any number of particulars comprehended under that idea. And again: Whatever may be denied universally of any idea, may be in like manner denied of every or any number of its individuals. These two propositions are called by logicians the dictum de omni, and dictum de nullo, and are indeed the great principles of syllogistic reasoning, insomuch as all conclusions whatsoever either 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 gainlay without running into an express contradiction. To affirm, for instance, that no man is perfect, and yet argue that some men are perfect; or to say that all men are mortal, and yet that some men are not mortal, is to assert a thing to be and not to be at the same time.

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

Demonstration therefore serving as an infallible guide to truth, and standing on 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 duly demonstrated. Now a demonstration is nothing more than a concatenation of syllogisms, all whose premises 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 it are genuine, and truly descriptive of the ideas they are meant to exhibit; whether the propositions assumed without proof as intuitive truths have really that self-evidence to which they lay claim; whether the syllogisms are drawn up in due form, and agreeable to the laws of argumentation; in fine, whether they are combined together in a just and orderly manner, so that no demonstrable propositions serve anywhere as premises, unless they are conclusions of previous syllogisms. Now it is the business of logic, in explaining the several operations of the mind, fully to instruct us in all these points. It teaches the nature and end of definitions, and lays down the rules by which they ought to be framed. It unfolds the several species of propositions, and distinguishes the self-evident from the demonstrable. It delineates also the different forms of syllogisms, and explains the laws of argumentation proper to each. In fine, it describes the manner of combining syllogisms, so as 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.

But 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 where ever, by the bare contemplation of our ideas, truth is discoverable, there also demonstration may be attained. Now that is an abundantly sufficient criterion, which enables us to judge with certainty in all cases where the knowledge of truth comes within our reach; for with discoveries, that lie beyond the limits of the human mind, we have properly no business. When a proposition is demonstrated, we are certain of its truth, When, on the contrary, our ideas are such as have no visible connection nor repugnance, and therefore furnish not the proper means of tracing their agreement or disagreement, there we are sure that scientifical 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 of the degree of proof, and of what is still wanting to render it full and complete, enable us to make a due estimate of the measures of probability, and to proportion our assent to the grounds on which the proposition stands. And this is all we can possibly arrive at, or even so much as hope for, in the exercise of faculties so imperfect and limited as ours.

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

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

Now this is precisely the manner of an indirect demonstration. For there we assume a proposition, which directly contradicts that we mean to demonstrate, and having by a continued series of proofs shewn it to be false, thence infer that its contradictory, or the proposition to be demonstrated, is true. As therefore this last conclusion is certain and unavoidable, let us next inquire, after what manner we come to be satisfied of the falsehood of the assumed proposition, that so no possible doubt may remain as to the force and validity of demonstrations of this kind. The manner then is plainly this. Beginning with the assumed proposition, 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, from the supposition that circles touching one another inwardly have the same centre, deduces that a part is equal to the whole. Since therefore, by a due and orderly process of reasoning, we come at last to a false conclusion, it is manifest that all the premises cannot be true. For were all the premises true, the last conclusion must be so too. Now as to all the other premises made use of in the course of reasoning, they are manifest and known truths by supposition, as being either definitions, self-evident propositions, or truths previously established. The assumed proposition is that only as to which any doubt or uncertainty remains. That alone therefore can be false, and indeed, from what has been already shewn, must unavoidably be so. And thus we see, that, in indirect demonstrations, two contradictory propositions being laid down, one of which is demonstrated to be false, the other, which is always the proposition to be proved, must necessarily be true; so that here, as well as in the direct way of proof, we arrive at a clear and satisfactory knowledge of truth.

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; whence the last conclusion or contradictory of the assumed proposition must be true likewise. So that by this means we should have two contradictory propositions both true at the same time, which is manifestly impossible. The assumed proposition therefore, whence this absurdity flows, must necessarily be false, and consequently its contradictory, which is here the proposition deduced from it, must be true. If then any proposition is 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 is false; and if so, then its contradictory, or that deduced from it, which in this case is the same with the proposition to be demonstrated, must be true.

We have a curious instance of this in the twelfth proposition of the ninth book of the elements. Euclid there proposes to demonstrate, that in any series of numbers, rising from unity in geometrical progression, all the prime numbers that measure the last term 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 the assumed proposition to be false, and that which is deduced from it, or its contradictory, which is the very proposition he proposed to demonstrate, to be true. Now that this is a just and conclusive way of reasoning, is abundantly manifest from what we have so clearly established above.

Having thus sufficiently evinced the certainty of demonstration in all its branches, and shewn 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 that thoroughly understands the form and structure of a good argument, will of himself readily discern every deviation from it. And although sophisms have been divided into many classes, which are all called by sounding names, that therefore carry in them much appearance of learning; yet are the errors themselves so very palpable and obvious, that it is lost labour to write for a man capable of being misled by them. Here therefore we choose to conclude this second part of logic, and shall in the next part give some account of method, which, though inseparable from reasoning, is nevertheless always considered by logicians as a distinct operation of the mind; because its influence is not confined to the mere exercise of the reasoning faculty, but extends in some degree to all the transactions of the understanding.

Part III. Of Method.

Of method in general, and the division of it into analytic and synthetical.

We have now done with the two first operations of the mind, whose office it is to search after truth, and enlarge the bounds of human knowledge. There is yet a third, 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 call method, and place always the last in order in explaining the powers of the understanding; because it necessarily supposes a previous exercise of our other faculties, and some progress made in knowledge, before we can exert it in any extensive degree.

In this view it is plain, that we must be before-hand well acquainted with the truths 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 whereas, at our first setting out, we were acquainted only with some of the grand strokes and outlines of truth, by thus pursuing her through her several windings and recesses we gradually discover those more inward and finer touches whence she derives all her strength, symmetry, and beauty. And here it is, that when, by a narrow scrutiny into things, we have unravelled any part of knowledge, and traced it to its first and original principles, in somuch 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.

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

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 LOGIC

with those previous propositions, that 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 backward in a continued series, until at length we arrive at some syllogism where the premises are first and self-evident principles. This done, we become perfectly satisfied as to the truth of all the conclusions we have passed through, in as much as they are now seen to stand upon the firm and immovable foundation of our intuitive perceptions. And as we arrived at this certainty by tracing things backward to the original principles whence they flow, so may we at any time renew it by a direct contrary process; if, beginning with these principles, we carry the train of our thoughts forward, until they lead us by a connected chain of proofs to the very last conclusion of the series.

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

There is this farther 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 again is often denominated the method of doctrine or instruction; in as much as, in laying our thoughts before others, we generally choose to proceed in the synthetick manner, deducing them from their first principles. For we are to observe, that although there is great pleasure in pursuing truth in the method of investigation, because it places us in the condition of the inventor, and shews the particular train and process of thinking by which he arrived at his discoveries; yet it is not so well accommodated to the purposes of evidence and conviction. For at our first setting out, we are commonly unable to divine where the analysis will lead us, insomuch that our researches are for some time little better than a mere groping in the dark. And even after light begins to break upon us, we are still obliged to many reviews, and a frequent comparison of the several steps of the investigation among themselves. Nay, when we have unravelled the whole, and reached the very foundation on which our discoveries stand, all our certainty in regard to their truth will be found in a great measure to arise from that connection we are now able to discern between them and first principles taken in the order of composition. But in the synthetick manner of disposing our thoughts, the case is quite different. For as we here begin with intuitive truths, and advance by regular deductions from them, every step of the procedure brings evidence and conviction along with it; so that in our progress from one part of knowledge to another, we have always a clear perception of the ground on which our assent rests. In communicating therefore our discoveries to others, this method is apparently to be chosen, as it wonderfully improves and enlightens the understanding, and leads to an immediate perception of truth.

LOCH

LOHOCH, or Loch, in pharmacy, a composition of a middle consistence between a soft electuary and a syrup, principally used in disorders of the lungs.

There are several kinds of lohoshs, denominated from the principal ingredient that enters into their composition. The common lohocb is made thus: take of fresh drawn oil of sweet almonds, and of pectoral or balsamic syrup, one ounce; white sugar, two drams: mix, and make them into a lohocb.

LOINS; in anatomy, the two lateral parts of the umbilical region of the abdomen. See Anatomy.