quantitative conditions. It was by means of quantitative measurement that Newton discovered the qualitative truth of the universal action of gravity; and some qualitative tests in chemistry can only be successfully made by adding the substances to each other in proper proportions. A qualitative truth is not one of degree; it is absolute. In a qualitative sense, a thing must either be or not be; but the idea of accuracy is a quantitative one, and accuracy may exist in all degrees from nothing to perfection. We cannot verify the exactness of the results obtained by means of a more exact method, by employing a less accurate one, because the range of uncertainty of the latter is greater than that of the former. Thus it is of no use to weigh in a coarse balance a light substance which has already been weighed in a delicate one. It is a good plan to indicate the degree of dubious accuracy by decimal numbers. CHAPTER XII. PROBABILITY IN MATTERS OF SCIENCE. PROBABILITY is a quantitative idea, and may exist in all degrees from nothing to infinity; and as in this treatise science is considered only in a qualitative aspect, I shall confine myself to but a few remarks upon it. Probability may be defined as likelihood based upon our intellectual perception of proper and sufficient evidence. It is an idea which is dependent for its existence upon the finite action of all our senses and mental powers; for if those powers were infinite, all our ideas would be as certain as truth itself, and that of probability would not exist. In other words, as all the existences and operations DEGREES OF PROBABILITY IN SCIENCE. 151 in nature are results of unerring laws, probability, uncertainty, or chance has no existence in them; and if all our capacities for perceiving and understanding nature were coextensive with nature itself, all our scientific beliefs would be certain, and we should not require to express any natural event in terms of proportionate truth. The degree of probability with which we regard an event depends largely upon our knowledge and experience; we are apt to miscalculate the true degree of probability of that of which we have no experience or knowledge. Our estimate of probability, like that of accuracy, is therefore a thing of degree, and depends upon the sufficiency of the evidence, and our capacity for receiving and understanding it. Probability is a very important and extensive subject in science; and, according to Butler, it is the very guide of life.' As our knowledge of nature is extremely imperfect, the element of probability enters into nearly all our thoughts and acts: and as the data from which we reason are very rarely certain, our inferences are often only probable, and we require to know the degree of probability of statements before we reason upon them. In this way a knowledge of probability is a necessary basis of inference and a guide of conduct. Our inferences of future events are all of them probable only, because of our finite experience and knowledge; but the degree of probability of an event is often sufficient to remove all serious degree of doubt. That which is very nearly certain, we accept as certain, in order to save us further labour; and as it is sufficiently certain for nearly all practical purposes, we usually act upon it and incur the risk. Scientific hypotheses are often only probably true, and that in a small degree; but we must not reject or even disregard them on that account. New truths are often but slowly perceived, and many great discoveries have arisen from what appeared at the outset to be but slightly probable hypotheses; Avogadro's law, for example.1 Many false beliefs also have at first been doubted, as being only probably false, but finally proved to be so; the theory of phlogiston, for instance. Any statement which does not actually contradict the fundamental laws of nature is not impossible, even though we are unable to prove it; and a belief is not necessarily a false or dubious one because we have not sufficient data upon which to base it, or no good argument to support it. With nothing to prove and nothing to disprove a scientific statement, the probability of its truth or falsity are equal. A weak argument in support of a scientific conclusion does not disprove the conclusion; and a strong argument, based upon many uncertain data, is itself extremely uncertain. To an unscientific mind, statements which are not provable often produce a stronger conviction of probability and truth than those which can be verified; and this, in some cases, arises partly from the fact that in scientific matters, external appearances are often the opposite of reality. In drawing conclusions as to the degree of probability of a given scientific statement, it is necessary to value, as far as possible, each point of evidence according to its own intrinsic degree of likelihood; and in forming a general scheme of science, we also require to quantify our ideas, and to value different scientific principles according to their relative probable degrees of intrinsic importance. 1 See p. 180. CRITERIA OF SCIENTIFIC TRUTH. 153 CHAPTER XIII. ON THE CRITERIA OF SCIENTIFIC TRUTH. Truth, like a perfect picture, filled in every part, WHAT is truth? and what are the complete criteria of truth? are questions which have occupied the minds of men in all ages, and can probably be fully solved only by means of perfect and infinite knowledge. Truth is universal consistency, unity in diversity; all truth is one by possessing these essential attributes. But although all truth is consistent, consistency is not necessarily truth, because there may be a limited consistency of imaginary existences, or a self-consistent limited system of error. Immediate consciousness also cannot be a criterion of truth, because all our senses are apt to deceive us and sometimes contradict each other; nor can our intuitive mental tendencies, because they are very misleading. According to Sir John Herschel, 'the grand and indeed only character of truth is its capability of enduring the test of universal experience, and coming unchanged out of every possible form of fair discussion." According to Archbishop Thomson, evidence is the sole means of establishing, and therefore the sole standard for testing, the truth of any proposition.''Four principal criteria of truth have been in different forms advocated by logicians, viz. :— 1 Discourse on Natural Philosophy, p. 10. '1st Criterion. The Principle of Contradiction.-The same attribute cannot at the same time be affirmed and denied of the same subject, or, the same subject cannot have two contradictory attributes. 2nd Criterion. The Principle of Identity.—Conceptions which agree can be affirmed of the same subject at the same time. This principle is the complement of the former. 6 3rd Criterion. The Principle of the Middle being Excluded. Either a given judgment must be true or its contradictory; there is no middle course. '4th Criterion. The Principle of Sufficient Reason.—Whatever exists, or is true, must have a sufficient reason why the thing or proposition should be as it is, and not otherwise. From this law are deduced such applications as these-1. Granting the reason, we must grant what follows from it. On this depends syllogistic inference. 2. If we reject the consequent, we must reject the reason.' The four criteria in question are useful in securing formal truth,' that is, in keeping our thoughts in harmony with each other; but for the discovery of material truth, for giving us thoughts that are true representatives of facts, they are either useless, or only useful as principles subordinate to the higher criterion-that every proposition must rest on sufficient evidence.' 'Viewed as instruments for judging of material truth, they' (i.e. the criteria) 'sink into mere rules for the reception of evidence. The first is a caution against receiving into our notion of a subject any attribute that is irreconcileable with some other, already proved upon evidence which we cannot doubt. The second is a permission to receive attributes that are not thus mutually opposed, or a hint to seek for such only. The third would compel us to reconsider the evidence of any proposition, when other evidence threatened |