[288] Hum. Nat., c. 6.
Reasoning. 129. Reasoning is the addition or subtraction of parcels. “In whatever matter there is room for addition and subtraction, there is room for reason; and where these have no place, then reason has nothing at all to do.”[289] This is neither as perspicuously expressed, nor as satisfactorily illustrated, as is usual with Hobbes; but it is true that all syllogistic reasoning is dependent upon quantity alone, and consequently upon that which is capable of addition and subtraction. This seems not to have been clearly perceived by some writers of the old Aristotelian school, or perhaps by some others, who, as far as I can judge, have a notion that the relation of a genus to a species, or a predicate to its subject, considered merely as to syllogism or deductive reasoning, is something different from that of a whole to its parts; which would deprive that logic of its chief boast, axiomatic evidence. But, as this would appear too dry to some readers, I shall pursue it farther in a note.[290]
[289] Lev. c. 5.
[290] Dugald Stewart (Elements of Philosophy, &c. vol. ii., ch. ii., sect. 2) has treated this theory of Hobbes on reasoning, as well as that of Condillac, which seems much the same, with great scorn, as “too puerile to admit of (i.e. require) refutation.” I do not myself think the language of Hobbes either here, or as quoted by Stewart from his Latin treatise on Logic, so perspicuous as usual. But I cannot help being of opinion that he is substantially right. For surely, when we assert that A is B, we assert that all things which fall under the class B, taken collectively, comprehend A, or that B = A + X: B being here put, it is to be observed, not for the res prædicata itself, but for the concrete, de quibus prædicandum est. I mention this, because this elliptical use of the word predicate seems to have occasioned some confusion in writers on logic. The predicate strictly taken, being an attribute or quality, cannot be said to include or contain the subject. But to return, when we say B = A + X, or B - X = A, since we do not compare, in such a proposition, as is here supposed, A with X, we only mean that A = A, or that a certain part of B is the same as itself. Again, in a particular affirmative, Some A is B, we assert that part of A, or A - Y is contained in B, or that B may be expressed by [A - Y] + X. So also when we say, Some A is not B, we equally divide the class or genus B into A - Y and X, or assert that B = [A - Y] + X; but, in this case, the subject is no longer A - Y, but the remainder, or other part of A, namely, Y; and this is not found in either term of the predicate. Finally, in the universal negative, No A (neither [A - Y] nor Y) is B, the [A - Y] of the predicate vanishes or has no value, and B becomes equal to X, which is incapable of measurement with A, and consequently with either A - Y or Y, which make up A. Now if we combine this with another proposition, in order to form a syllogism, and say that C is A, we find, as before, that A = C + Z; and substituting this value of A in the former proposition, it appears that B = C + Z + X. Then, in the conclusion, we have, C is B; that is C is a part of C + Z + X. And the same in the three other cases or moods of the figure. This seems to be, in plainer terms, what Hobbes means by addition or subtraction of parcels, and what Condillac means by rather a lax expression, that equations and propositions are at bottom the same, or, as he phrases it better, “l’evidence de raison consiste uniquement dans l’identité.” If we add to this, as he probably intended, non-identity, as a condition of all negative conclusions, it seems to be no more than is necessarily involved in the fundamental principle of syllogism, the dictum de omni et nullo; which may be thus reduced to its shortest terms; “Whatever can be divided into parts, includes all those parts, add nothing else.” This is not limited to mathematical quantity, but includes everything which admits of more or less. Hobbes has a good passage in his Logic on this; Non putandum est computationi, id est, ratiocinationi in numeris tantum locum esse, tanquam homo a cæteris animantibus, quod censuisse narratur Pythagoras, sola numerandi facultate distinctus esset; nam et magnitudo magnitudini, corpus corpori, motus motui, tempus tempori, gradus qualitatis gradui, actio actioni, conceptus conceptui, proportio proportioni, oratio orationi, nomen nomini, in quibus omne philosophiæ genus continetur, adjici adimique potest.
But it does not follow by any means that we should assent to the strange passages quoted by Stewart from Condillac and Diderot, which reduce all knowledge to identical propositions. Even in geometry, where the objects are strictly magnitudes, the countless variety in which their relations may be exhibited constitutes the riches of that inexhaustible science; and in moral or physical propositions, the relation of quantity between the subject and predicate, as concretes, which enables them to be compared, though it is the sole foundation of all general deductive reasoning or syllogism, has nothing to do with the other properties or relations, of which we obtain a knowledge by means of that comparison. In mathematical reasoning, we infer as to quantity through the medium of quantity; in other reasoning, we use the same medium, but our inference is as to truths which do not lie within that category. Thus, in the hackneyed instance, All men are mortal; that is, mortal creatures include men and something more, it is absurd to assert, that we only know that men are men. It is true that our knowledge of the truth of the proposition comes by the help of this comparison of men in the subject with men in the predicate; but the very nature of the proposition discovers a constant relation between the individuals of the human species and that mortality which is predicated of them along with others; and it is to this, not in an identical equation, as Diderot seems to have thought, that our knowledge consists.
The remarks of Stewart’s friend. M. Prevost of Geneva, on the principle of identity as the basis of mathematical science, and which the former has candidly subjoined to his own volume, appear to me very satisfactory. Stewart comes to admit that the dispute is nearly verbal; but we cannot say that he originally treated it as such; and the principle itself, both as applied to geometry and to logic, is, in my opinion, of some importance to the clearness of our conceptions as to those sciences. It may be added, that Stewart’s objection to the principle of identity as the basis of geometrical reasoning is less forcible in its application to syllogism. He is willing to admit that magnitudes capable of coincidence by immediate superposition may be reckoned identical, but scruples to apply such a word to those which are dissimilar in figure, as the rectangles of the means and extremes of four proportional lines. Neither one nor the other are, in fact, identical as real quantities, the former being necessarily conceived to differ from each other by position in space, as much as the latter; so that the expression he quotes from Aristotle, εν τουτοις ἡ ισοτης ἑνοτης, or any similar one of modern mathematicians, can only refer to the abstract magnitude of their areas, which being divisible into the same number of equal parts, they are called the same. And there seems no real difference in this respect between two circles of equal radii and two such rectangles as are supposed above, the identity of their magnitudes being a distinct truth, independent of any consideration either of their figure or their position. But however this may be, the identity of the subject with part of the predicate in an affirmative proposition is never fictitious, but real. It means that the persons or things in the one are strictly the same beings with the persons or things to which they are compared in the other, though, through some difference of relations, or other circumstance, they are expressed in different language. It is needless to give examples, as all those who can read this note at all will know how to find them.
I will here take the liberty to remark, though not closely connected with the present subject, that Archbishop Whately seems not quite right in saying (Elements of Logic, p. 46), that in affirmative propositions the predicate is never distributed. Besides the numerous instances where this is, in point of fact, the case, all which he excludes, there are many in which it is involved in the very form of the proposition. Such are all those which assert identity or equality, and such also are all those particular affirmations which have previously been converted from universals. Of the first sort are all the theorems in geometry, asserting an equality of magnitude or ratios, in which the subject and predicate may always change places. It is true that in the instance given in the work quoted, that equilateral triangles are equiangular, the converse requires a separate proof, and so in many similar cases. But in these the predicate is not distributed by the form of the proposition; they assert no quality of magnitude.
The position, that where such equality is affirmed, the predicate is not logically distributed, would lead to the consequence that it can only be converted into a particular affirmation. Thus, after proving that the square of the hypothenuse, in all right-angled triangles, is equal to those of the sides, we could only infer that the squares of the sides are sometimes equal to that of the hypothenuse, which could not be maintained without rendering the rules, of logic ridiculous. The most general mode of considering the question, is to say, as we have done above, that, in an universal affirmative, the predicate B (that is, the class of which B is predicated), is composed of A the subject, and X, an unknown remainder. But if, by the very nature of the proposition, we perceive that X is nothing, or has no value, it is plain that the subject measures the entire predicate, and vice versâ, the predicate measures the subject; in other words, each is taken universally, or distributed.
False reasoning. 130. A man may reckon without the use of words in particular things, as in conjecturing from the sight of anything what is likely to follow; and if he reckons wrong, it is error. But in reasoning on general words, to fall on a false inference is not error, though often so called, but absurdity.[291] “If a man should talk to me of a round quadrangle, or accidents of bread in cheese, or immaterial substances, or of a free subject, a free will, or any free, but free from being hindered by opposition, I should not say he were in error, but that his words were without meaning, that is to say, absurd;” Some of these propositions, it will occur, are intelligible in a reasonable sense, and not contradictory, except by means of an arbitrary definition which he who employs them does not admit. It will be observed here, as we have done before, that Hobbes does not confine reckoning, or reasoning, to universals, or even to words.
[291] Lev. c. 5.