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judgment arifes the divifion of propofitions into univerfal and particular. An univerfal propofition is that wherein the fubject is fome general term taken in its full latitude; infomuch that the predicate agrees to all the individuals comprehended under it, if it denotes a proper fpecies; and to all the feveral fpecies, and their individuals, if it marks an idea of a higher order. The words All, every, no, none, &c. are the proper figns of this univerfality; and as they feldom fail to accompany general truths, fo they are the most obvious criterion whereby to diftinguish them. All animals have a power of beginning motion. This is an univerfal propofition; as we know from the word all prefixed to the fubject animals, which denotes that it must be taken in its full extent. Hence the power of beginning motion may be affirmed of all the feveral species of animals.

copula. For as the copula, when placed by it-
felf, between the subject and the predicate, mani-
feftly binds them together; it is evident, that to
render a propofition negative, the particles of ne
gation must enter it in fuch a manner as to de-
ftroy this union. In a word, then only are two
ideas disjoined in a propofition, when the nega-
tive particle may be fo referred to the copula, as
to break the affirmation included in it, and undo
that connection it would otherwise establish.
When we fay, for inftance, No man is perfect;
take away the negation, and the copula of itself
plainly unites the ideas in the propofition. But as
this is the very reverse of what is intended, a nega-
tive mark is added, to show that this union does
not here take place. The negation, therefore, by
deftroying the effect of the copula, changes the
very nature of the propofition, infomuch that, in-
ftead of binding two ideas together, it denotes
their feparation. On the contrary, in this fen-
tence, The man who departs not from an upright
behaviour is beloved of God, the predicate beloved
of God is evidently affirmed of the fubject an up-
right man; so that, notwithstanding the negative
particle, the propofition is still affirmative. The
reafon is plain; the negation here affects not the
copula; but making properly a part of the fub-
ject ferves, with other terms in the fentence, to
form one complex idea, of which the predicate
beloved of God is directly affirmed.
SECT. III. Of UNIVERSAL and PARTICULAR

PROPOSITIONS.

I. THE next confiderable divifion of propofitions is into univerfal and particular. Our ideas, according to what has been already obferved in the Firft Part, are all fingular as they enter the mind, and reprefent individual objects. But as by abstraction we can render them univerfal, fo as to comprehend a whole clafs of things, and fometimes feveral claffes at once; hence the terms expreffing these ideas must be in like manner univerfal. If therefore we fuppofe any general term to become the subject of a propofition, 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 exterds. Thus, when we fay, Men are mortal; we confider mortality, not as confined to one or any number of particular men, but as what may be affirmed without reftriction of the whole fpecies. Thus the propofition becomes as general as the idea which makes the fubject of it; and indeed derives its univerfality entirely from that idea, being more or lefs fo according as this may be extended to more or fewer individuals. But it is further to be obferved of thefe general terms, that they fometimes enter a propofition in their full latitude, as in the example given above; and fometimes appear with a mark of limitation. In this laft cafe we are given to understand, that the predicate agrees not to the whole univerfal idea, but only to a part of it; as in the propofition, Some men are wife: For here wisdom is not affirmed of every particular man, but restrained to a few of the human species.

II. From this different appearance of the general idea, that conftitutes the fubject of any

III. A particular propofition has in like manner fome general term for its fubject; but with a mark of limitation added, to denote, that the predicate agrees only to fome of the individuals comprehended under a fpecies, or to one or more of the fpecies belonging to any genus, and not to the whole univerfal idea. Thus, Some ftones are heavier than iron; Some men have an uncommon Share of prudence. In the last of these propofitions, the fubject fome men implies only a certain number of individuals, comprehended under a fingle fpecies. In the former, where the fubject is a genus that extends to a great variety of diftinct claffes, fome flones may not only imply any num ber of particular stones, but alfo feveral whole fpecies of ftones, inafmuch as there may be not a few with the property there described. Hence we fee, that a propofition does not ceafe to be particular by the predicate's agreeing to a whole fpecies, unlefs that fpecies, fingly and diftinétly confidered, makes also the fubject of which we affirm or deny.

IV. There is still one fpecies of propofitions that remains to be defcribed, and which the more deferves our notice, as it is not yet agreed among logicians to which of the two claffes mentioned above they ought to be referred; namely, fingular propofitions, or thofe where the fubject is an individual. Of this nature are the following: Sir ISAAC NEWTON was the inventor of fluxions; This book contains many useful truths. There is fome difficulty as to the proper rank of these propofitions, becaufe the fubject being taken according to the whole of its extenfion, they fometimes have the fame effect in reafoning as univerfals. But if it be confidered that they are in truth the moft li mited kind of particular propofitions, and that no propofition can with any propriety be called univerfal but where the fubject is fome univerfal idea; we shall not be long in determining to which clafs they ought to be referred. When we fay, Some books contain useful truths; the propofition is particular, because the general term appears with a mark of reftriction. If therefore we fay, This book contains useful truths; it is evident that the propofition must be ftill more particular, as the limitation implied in the word this is of a more confined nature than in the former cafe.

V. We fee, therefore, that all propofitions are either affirmative or negative; nor is it lefs evi

dent,

dent, that in both cafes they may be univerfal or particular. Hence arifes that celebrated fourfold divifion of them into univerfil affirmative and univerfal negative, particular affirmative, and particular negative, which comprehends indeed all their varieties. The ufe of this method of diftinguishing them will appear more fully afterwards, when we come to treat of reasoning and fyllogifm.

SECT. IV. Of ABSOLUTE and CONDITIONAL PROPOSITIONS.

I. THE objects about which we are chiefly converfant 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 fmall part of our knowledge to distinguish rightly these variations, and trace the reasons upon which they depend. For it is obfervable, that amidft all the viciffitude of nature, fome things remain conftant and invariable; nor even are the changes, to which we fee others liable, effected but in confequence of uniform and fteady laws, which, when known, are fufficient to direct us in our judgments about them. Hence philosophers, in diftinguishing the objects of our perception into various claffes, have been very careful to note, that fome properties belong effentially to the general idea, fo as not to be feparable from it but by deftroying its very nature; while others are only accidental, and may be affirmed or denied of it in different circumftances. Thus folidity, a yellow colour, and great weight, are confidered as effential qualities of gold; but whether it fhall exift as an uniform conjoined mass, is not alike neceffary. We fee that by proper menftruum it may be reduced to a fine powder, and that an intenfe heat will bring it into a state of fufion.

II. From this diverfity in the feveral qualities of things arifes a confiderable difference as to the manner of our judging about them. For all fuch properties as are infeparable from objects when confidered as belonging to any genus or fpecies, are affirmed abfolutely and without referve of that general idea. Thus we fay, Gold is very weighty; Aftone is hard; Animals have a power of self-motion. But in the cafe of mutual or accidental quali ties, as they depend upon fome other confideration diftinct from the general idea, that alfo must be taken into the account, in order to form an accurate judgment. Should we affirm, for inftance, of fome ftones, that they are very fufceptible of a rolling motion; the propofition, while it remains in this general form, cannot with any advantage be introduced into our reafonings. An aptnefs to receive that mode of motion flows from the figure of the ftone; which, as it may vary infinitely, our judgment then only becomes applicable and determinate, when the particular figure, of which volubility is a confequence, is alfo taken into the account. Let us, then, bring in this other confideration, and the propofition will run as follows: Stones of a spherical form are eafily put into a rolling motion. Here we fee the condition upon which the predicate is affirmed, and therefore know in what particular cafes the propofition may be applied.

III. This confideration of propofitions, refpect

ing the manner in which the predicate is affirmed of the fubject, gives rife to the divifion of them into abfolute and conditional. Abfolute propofitions are thofe wherein we affirm fome property infeparable from the idea of the fubject, and which therefore belongs to it in all poffible cafes: as, GOD is infinitely wife; Virtue tends to the ultimate happiness of man. But where the predicate is not neceffarily connected with the idea of the fubject, idea, there the propofition is called conditional. unless upon fome confideration diftinct from that The reafon is from the supposition annexed, which is of the nature of a condition, and may be expreffed as fuch, thus: If a flone is expofed to the rays of the fun, it will contract fome degree of beats If a river runs in a very declining channel, its rapidity will conftantly increafe.

IV. There is not any thing of greater importance in philofophy than a due attention to this divifion of propofitions. If we are careful never to affirm things abfolutely but where the ideas are infeparably conjoined; and if in our other judgments we diftinctly 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 hu man life. It is owing to the exact observance of this rule that mathematicians have been fo happy in their discoveries, and that what they demonftrate of magnitude in general may be appli ed with eafe in all obvious occurrences.

V. The truth of it is, particular propofitions are then known to be true, when we can trace their connection with univerfals; and it is accordingly the great bufinefs of science to find out general truths that may be applied with fafety in all obvious inftances. Now the great advantage arifing from determining with care the conditions upon which one idea may be affirmed or denied of another is this: that thereby particular propofitions really become universal, may be introduced with certainty into our reasonings, and ferve as ftandards to conduct and regulate our judgments. To illuftrate this by a familiar inftance: if we fay, Some water acts very forcibly; the propofition is particular: and as the conditions on which this forcible action depends are not mentioned, it is as yet uncertain in what cafes it may be applied. Let us then supply these conditions, and the propofition will run thus: Water conveyed in fufficient quantity along a steep defcent acts very forcibly. Here we have an univerfal judgment, inasmuch as the predicate forci ble action may be afcribed to all water under the circumftances mentioned. Nor is it lefs evident that the propofition in this new form is of easy application; and in fact we find that men do apply it in inftances where the forcible action of water is required; as in corn mills and many other works of art.

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SECT. V. Of SIMPLE and COMPOUND PROPO

SITIONS.

I. HITHERTO We have treated of propofitions, where only two ideas are compared together. These are in the general called SIMPLE; because, having but one fubject and one predicate, they are the effect of a fimple judgment that admits of

no

no fubdivifion. But if it fo happens that feveral ideas offer themselves to our thoughts at once, whereby we are led to affirm the fame thing of different objects, or different things of the fame object; the propofitions expreffing thefe judgments are called COMPOUND; because they may be refolved into as many others as there are fub. jects or predicates in the whole complex deter. mination on the mind. Thus, GOD is infinitely wife and infinitely powerful. Here there are two predicates infinite wisdom and infinite power, both affirmed of the fame fubject; and accordingly the propofition may be refolved into two others, affirming thefe predicates feverally. In like manner, in the propofition, Neither kings nor people are exempt from death; the predicate is denied of both fubjects, and may therefore be separated from them in diftinct propofitions. Nor is it lefs evident, that if a complex judgment confifts of feveral fubjects and predicates, it may be refolved into as many fimple propofitions as are the number of different ideas compared together. Riches and honours are apt to elate the mind, and increase the number of our defires. In this judgment there are two fubjects and two predicates, and it is at the fame time apparent that it may be refolved into four diftinct propofitions. Riches are apt to elate the mind. Riches are apt to encrease the number of our defires. And so of honours.

II. Logicians have divided these compound propofitions into many different claffes; but not with a due regard to their proper definition. Thus conditionals, caufals, relatives, &c. are mentioned as fo many diftin&t species of this kind, though in fact they are no more than fimple propofitions. To give an inftance of a conditional; If a ftone is expofed to the rays of the fun, it will contract fome degree of heat. Here we have but one fubject and one predicate; for the complex expreffion, A ftone expofed to the rays of the fun, conftitutes the proper fubject of this propofition, and is no more than one determined idea. The fame thing happens in caufals. Rehoboam was unhappy be caufe be folloaved evil counsel. There is here an appearance of two propofitions arifing from the complexity of the expreffion; but when we come to confider the matter more nearly, it is evident that we have but a single subject and predicate. The purfuit of evil counfel brought mifery upon Rehoboam It is not enough, therefore, to render a propofition compound, that the subject and predicate are complex notions, requiring fometimes a whole sentence to exprefs them: for in this cafe the comparison is ftill confined to two ideas, and conftitutes what we call a fimple judgment. But where there are several subjects or predicates, or both, as the affirmation or negation may be alike extended to them all, the propofition expreffing fuch a judgment is truly a collection of as many fimple ones as there are different ideas compared. Confining ourselves therefore to this more ftrict and just notion of compound propofitions, they are all reducible to two kinds, viz. copulatives and disjunctives.

III. A COPULATIVE propofition is, where the fubjects and predicates are fo linked together, that they may be all feverally affirmed or denied one of another. Of this nature are the examples of

compound propofitions given above. "Riches and honours are apt to elate the mind, and increafe the number of our defires."-" Neither kings nor people are exempt from death." In the firft of thefe the two predicates may be affirmed severally of each subject, whence we have 4 diftinct propofitions. The other furnishes an example of the negative kind, where the fame predicate, being disjoined from both fubjects, may be alfo denied of them in feparate propofitions.

IV. The other species of compound propofitions are thofe called DISJUNCTIVES; in which, comparing feveral predicates with the same subject, we affirm that one of them neceffarily belongs to it, but leave the particular predicate undetermined. If any one, for example, fays, "This world either exists of itself, or is the work of fome all-wife and powerful caufe;" it is evident that one of the two predicates must belong to the world; but as the propofition determines not which, it is therefore of the kind we call disjunctive. Such too are the following: "The fun either moves round the earth, or is the centre about which the earth revolves."-" Friendfhip finds men equal, or makes them fo." It is the nature of all propofitions of this class, suppofing them to be exact in point of form, that, upon determining the particular predicate, the reft are of course to be removed; or if all the predicates but one are removed, that one neceffarily takes place. Thus, in the example given above, if we allow the world to be the work of fome wife and powerful caufe, we of course deny it to be felf-exiftent; or if we deny it to be felfexiftent, we muft neceffarily admit that it was produced by fome wife and powerful caufe. Now this particular manner of linking the predicates together, fo that the establishing one displaces all the reft; or the excluding all but one neceffarily eftablishes that one; cannot otherwise be effected than by means of disjunctive particles. And hence propofitions of this clafs take their names from thefe particles which make so neceffary a part of them, and indeed conftitute their very nature confidered as a diftinct species.

SECT. VI. Of the DIVISION of PROPOSITIONS into SELF-EVIDENT and DEMONSTRABLE.

I. WHEN any propofition is offered to the view of the mind, if the terms in which it is expreffed be understood, upon comparing the ideas together, the agreement or difagreement afferted is either immediately perceived, or found to lie beyond the prefent reach of the understanding. In the firft cafe, the proportion is faid to be SELFEVIDENT, and admits not of any proof, because a bare attention to the ideas themselves produces full conviction and certainty; nor is it poffible to call any thing more evident by way of confirmation. But where the connection or repugnance comes not fo readily under the infpection of the mind, there we must have recourfe to reasoning; and if by a clear feries of proofs we can make out the truth propofed, infomuch that felf-evidence fhall accompany every step of the procedure, we are then able to demonftrate what we affert, and the propofition itself is faid to be DEMONSTRABLE. When we affirm, for inftance, that "it is impos

ible for the fame thing to be and not to be;" who ever understands the terms made ufe of perceives at first glance the truth of what is afferted, nor can be by any efforts bring himself to believe the contrary. The propofition therefore is felf-evi dent, and fuch that it is impoffible by reafoning to make it plainer; because there is no truth more obvious or better known, from which as a confequence it may be deduced. But if we fay, This world had a beginning; the affertion is indeed equally true, but fhines not forth with the fame degree of evidence. We find great difficulty in conceiving how the world could be made out of nothing and are not brought to a free and full confent, until by reafoning we arrive at a clear view of the abfurdity involved in the contrary fuppofition. Hence this propofition is of the kind we call demonftrable, inafmuch 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 conféquence.

II. From what has been faid, it appears, that reafoning is employed only about demonftrable propofitions, and that our intuitive and felf-evident perceptions are the ultimate foundation on which it refts.

III. Self-evident propofitions furnish the first principles of reasoning; and it is certain, that if in our researches we employ only fuch principles as have this character of felf-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 exprefs model here mentioned, are an inconteftible proof of the firmnefs and ftability of human knowledge, when built upon fo fure a foundation. For not only have the propofitions of this science stood the teft of ages, but are found attended with that invincible evidence, as forces the affent of all who duly confider the proofs upon which they are eftablished. Since the mathematicians are univerfally allowed to have hit upon the right method of arriving at unknown truths, fince they have been the happiest in the choice as well as the application of their principles, it may not be amifs to explain here their method of stating self-evident propofitions, and applying them to the purpofes of demonstration.

IV. First then it is to be obferved, that they have been very careful in afcertaining their ideas and fixing the fignification of their terms. For this purpofe they begin with DEFINITIONS, in which the meaning of their words is fo diftinctly explained, that they cannot fail to excite in the mind of an attentive reader the very fame ideas as are annexed to them by the writer. And indeed the clearness and irrefiftible 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 juftly traced, it will not be difficult for another to comprehend that relation, if in setting himself to difcover it he brings the very fame ideas into comparison. But if, on the contrary, he affixes to his words ideas different from thofe that were

in the mind of him who first advanced the demonftration; it is evident, that as the fame ideas are not compared, the fame relation cannot fubfift, infomuch that a propofition will be rejected as falfe, which, had the terms been rightly underftood, muft have appeared inconteftibly true. A fquare, for inftance, 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 fides; and many properties demonftrated of the fquare flow entirely from its being a rectangular figure. If therefore we fuppofe a man, who has formed a partial notion of a fquare, comprehending only the equality of its fides, without regard to the angles, reading fome demonftration that implies alfo this latter confideration; it is plain he would reject it as not univerfally true, inafmuch as it could not be applied where the fides were joined together at equal angles. For this laft figure, anfwering ftill to bis idea of a fquare, would be yet found without the property affigned to it in the propofition. But if he comes afterwards to correct his notion, and render his idea complete, he will then readily own the truth and juftness of the demonftration.

V. We fee, therefore, that nothing contributes fo much to the improvement and certainty of human knowledge as the having determinate ideas, and keeping them steady and invariable in all our difcourfes and reafonings about them. On this account mathematicians always begin by defining their terms, and diftinctly unfolding the notions they are intended to exprefs. Hence fuch as apply themselves to thefe ftudies have exactly the fame views of things; and, bringing always the very fame ideas into comparifon, readily difcern the relation between them. It is likewise of importance, in every demonstration, to express the fame idea invariably by the fame word. From this practice mathematicians never deviate; and if it be neceffary in their demonftrations, where the reader's comprehenfion is aided by a diagram, it is much more fo in all reasonings about moral or intellectual truths where the ideas cannot be reprefented by a diagram. The obfervation of this rule may fometimes be productive of ill-founding periods; but when truth is the object, found ought to be difregarded.

VI. When the mathematicians have taken this first step, and made known the ideas whose relations they intend to inveftigate; their next care is, to lay down fome felf-evident truths, which may ferve as a foundation for their future reasonings. And here indeed they proceed with remarkable circumfpection, admitting no principles but what flow immediately from their definitions, and neceffarily force themselves upon a mind in any degree attentive to its ideas. Thus a circle is a figure formed by a right line moving round fome fixed point in the fame plane. The fixed point round which the line is fuppofed 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 firft fet out, describes a curve running into itself, and termed the circumference. All right lines drawn from the centre

to

to the circumference are called radii. From thefe definitions compared, geometricians derive this felf-evident truth, "that the radii of the fame circle are all equal to one another."

VII. We now observe, that, in all propofitions, we either affirm or deny fome property of the idea that conftitutes the fubject of our judgment, or we maintain that fomething may be done or effected. The firft fort are called speculative propofitions, as in the example mentioned above, the radii of the fame circle are all equal, one to another." The others are called practical, for a reafon quite obvious; thus, that a right line may be drawn from one point to another is a practical propofition, inafmuch as it expreffes that fomething may be done.

VIII. From this twofold confideration of propofitions arifes the twofold divifion of mathematical principles into axioms and poftulates. By an axiom they understand any self-evident fpeculative truth; as, "That the whole is greater than its parts:""That things equal to one and the fame thing are equal to one another." But a felf evident practical propofition is what they call a poftulate. Such are thofe of Euclid: "That a finite right line may be continued directly forwards;" ;" "That a circle may be described about any centre with any diftance." And here we are to observe, that as in an axiom the agreement or

there demonstrated in fo obvious a manner as to difcover their dependence upon the propofition whence they are deduced, almost as foon as propofed. 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 corrollary, "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 firft fight; becaufe in all cafes they are equal to two right ones, and things equal to one and the fame thing are equal to one another.

XI. The SCHOLIA of mathematicians are indifferently annexed to definitions, propofitions, or corollaries; and answer the fame purposes as annotations upon a claffic author. For in them occafion is taken to explain whatever may appear intricate and obfcure in a train of reasoning; to answer objections; to teach the application and uses of propofitions; to lay open the original and hiftory of the feveral difcoveries made in the science; and, in a word, to acquaint us with all fuch particulars as deferve to be known, whether confidered as points of curiofity or profit.

PART III.
OF REASONING.

PARTS of which it CONSISTS.

disagreement between the fubject and predicate SECT. I. Of REASONING in GENERAL, and the must come under the immediate infpection of the mind; fo in a poftulate, not only the poffibility of the thing afferted must be evident at firft view, but alfo the manner in which it may be effected. For where this manner is not of itself apparent, the propofition comes under the notion of the demonftrable kind, and is treated as such by geometrical writers. Thus, "to draw a right line from one point to another," is affumed by Euclid as a poftulate, because the manner of doing it is obvious, as to require no previous teaching. But then it is not equally evident, how we are to conftruct an equilateral triangle. For this reafon he advances it as a demonftrable propofition, lays down rules for the exact performance, and at the fame time proves, that if these rules are followed, the figure will be justly described.

IX. This leads us to take notice, that as felfevident truths are distinguished into different kinds, according as they are fpeculative or practical; fo is it alfo with demonftrable propofitions. A demonftrable fpeculative propofition is by mathematicians called a THEOREM. Such is the famous 47th propofition of the firft book of the elements, known by the name of the Pythagoric theorem, from its fuppofed inventor Pythagoras, viz. " that in every right-angled triangle, the fquare defcribed upon the fide fubtending the right angle is equal to both the fquares defcribed upon the fides containing the right-angle." On the other hand, a demonftrable practical propofition is called a problem; as where Euclid teaches us to defcribe a square upon a given right line.

X. Befides the 4 kinds of propofitions already mentioned, mathematicians have also a 5th, known by the name of corollaries. Thefe are ufually fubjoined to theorems or problems, and differ from them only in this, that they flow from what is

IT often happens, in comparing ideas together, that their agreement or difagreement cannot be difcerned at firft view, especially if they are of such a nature as not to admit of an exact application one to another. When, for inftance, we compare two figures of a different make, in order to judge of their equality or inequality, it is plain, that by barely confidering the figures themselves, we cannot arrive at an exact determination; because, by reason of their difagreeing forms, it is impoffible fo to put them together, as that their several parts fhall mutually coincide. Here then it becomes neceffary to look out for fome third idea that will admit of fuch an application as the present cafe requires; wherein if we fucceed, all difficulties vanish, and the relation we are in queft of may be traced with ease. Thus right-lined figures are all reduced to fquares, by means of which we can measure their areas, and determine exactly their agreement or disagreement in point of magnitude.

II. But how can any third idea ferve to difcover a relation between two others? The answer is, By being compared feverally with these others; for fuch a comparifon enables us to fee 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 fquare whofe area is known, and find the one exactly equal to it, and the other lefs by a square inch, we immediately conclude that the area of the firft figure is a fquare inch greater than that of the second. The manner of determining the relation between any two ideas, by the intervention of some third with which they may be com

pared,

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