Guesswork

“ Finding the minor premise is a matter of observation, and if you fail to observe the significant fact about the problem, the really useful major premise may lie dormant, known and retained but not recalled, while false clues suggest inapplicable major premises and give birth to plenty of reasoning but all to no purpose. Some persons with abundant knowledge are ineffective reasoners from lack of a sense for probability. The efficient reasoner must be a good guesser. ”

“ His manner was wholly facetious, but I professed to take his question seriously, and replied—
"By a guess, I mean a conclusion arrived at without data."
"Impossible!" he exclaimed, with mock sternness. "Nobody but an utter fool arrives at a conclusion without data."
"Then I must revise my definition instantly," I rejoined. "Let us say that a guess is a conclusion drawn from insufficient facts."
"That is better," said he; "but perhaps it would be better still to say that a guess is a particular and definite conclusion deduced from facts which properly yield only a general and indefinite one. ”

“ If Logic were confined to strict demonstration, it would cover a narrow field. The greater part of our conclusions can only be more or less probable. It may, indeed, be maintained, not unreasonably, that no judgments concerning matters of fact can be more than probable. Some say that all scientific results should be considered as giving the average of cases, from which deviations are to be expected. Many matters can only be treated statistically and by the methods of Probability. Our ordinary beliefs are adopted without any methodical examination. ”

“ It would place us in that position of more advanced knowledge, in which the probabilities, to us, would be different from what they were before; and in estimating these new probabilities we should have to proceed on a totally different set of data, furnished no longer by mere counting of possible suppositions, but by specific knowledge of facts. Such data it should always be our endeavor to obtain; and in all inquiries, unless on subjects equally beyond the range of our means of knowledge and our practical uses, they may be obtained, if not good, at least better than none at all. ”

“ Why should we want to know the probability that the fact will accord with our conclusion? That implies that we are interested in all possible worlds, and not merely the one in which we find ourselves placed. ”

“ Those philosophers, who have divided human reason into knowledge and probability, and have defin'd the first to be that evidence, which arises from the comparison of ideas, are oblig'd to comprehend all our arguments from causes or effects under the general term of probability. ”

“ From the preceding principles it is easy to deduce the demonstration of that theorem of the doctrine of probabilities, which is the foundation of its application to inquiries for ascertaining the occurrence of a given event, or the reality of an individual fact. The signs or evidences by which a fact is usually proved, are some of its consequences: and the inquiry hinges upon determining what cause is most likely to have produced a given effect. ”

“ In a word, instead of taking known events and deducing from them what has occurred, we must build up a fanciful explanation if it will only be consistent with known events. We can then test this explanation by any fresh facts which may arise. If they all fit into their places, the probability is that we are upon the right track, and with each fresh fact this probability increases in a geometrical progression until the evidence becomes final and convincing. ”

“ At least that is what it would be like if your guesses were accurate, not only in their statement, but also in their proportion, and also in their order. It is millions to one that your guesses are nothing of the kind. ”

“ To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary about it. In the choice of this convention we can be guided only by the principle of sufficient reason. ”

“ As we teach children we should encourage the intelligent guess. We would not, of course, encourage mere random guessing, which may be engaged in by children to have something to say or to blind the teacher. A child who offers a guess or hypothesis should be asked to give his ground for the inference, should show that his guess has grown out of his consideration of the data in hand. It is fallacious to suppose that this kind of thinking is beyond the power of children. ”

“ All the cases must be known for a perfect induction; there is no unknown to argue to. This, then, is only a short statement of many individual truths, and has but little of value. Induction that is imperfect is more valuable; for with many cases the probability becomes so strong that it is a practical certainty. It is the method of science.
133 More valuable for literature is another division of arguments into arguments from cause, arguments from sign, and arguments from example. ”

“ I observe, then, that there are many investigations in every subject-matter which only lead us a certain way towards truth, and not the whole way: either leading us, for instance, to a strong probability, not to a certainty, or again, proving only some things out of the whole number which are true. ”

“ Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth.
If one considers a series of objects of the same nature one perceives among them and in their changes ratios which manifest themselves more and more in proportion as the series is prolonged, and which, extending and generalizing continually, lead finally to the principle from which they were derived. ”

“ At any rate we can reason quite as well with hypotheses and probabilities as with absolute truths, and this is in fact what we usually do, whether or not we are aware that our premisses are conditional and hypothetical. ”

“ According to one view the ground is subjective: probability depends, it is said, upon the quantity of our Belief in the happening of a certain event, or in its happening in a particular way. According to the other view the ground is objective, and, in fact, is nothing else than experience, which is most trustworthy when carefully expressed in statistics. ”

“ Since, therefore, all knowledge resolves itself into probability, and becomes at last of the same nature with that evidence which we employ in common life, we must now examine this latter species of reasoning, and see on what foundation it stands. ”

“ And here, Jardine, I would draw your attention to the interesting way in which, when a long train of hypothetical reasoning has at length elicited an actual, demonstrable truth, that truth instantly reacts on the hypothesis, lifting it as a whole on to an entirely different plane of probability. ”

“ And here we laid down the canon, "that the value of any probability, in its last analysis, is determined by the number, the importance, and the definiteness of the relations known, as compared with those of the relations unknown;" and, consequently, that in cases where the unknown relations are more numerous, more important, or more indefinite than are the known relations, the value of our inference varies inversely as the difference in these respects between the relations compared. ”

“ Such a use of probability-theory is a good example of the way in which mathematics may be used to cover the false assumptions which have to be made in order to make a mathematical treatment of certain sorts of subject-matter initially plausible as description of concrete fact. ”

“ Let us avoid confounding what is in the domain of accurate reasoning with what is the object of a more or less happy guess; what is rational with what is empirical. It is enough to have to guard against errors in logic on the first score ”

“ As we go on drawing inference after inference of the given kind, during the first ten or hundred cases the ratio of successes may be expected to show considerable fluctuations; but when we come into the thousands and millions, these fluctuations become less and less; and if we continue long enough, the ratio will approximate toward a fixed limit. We may, therefore, define the probability of a mode of argument as the proportion of cases in which it carries truth with it. ”

“ Strictly speaking it may even be said that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, the principal means for ascertaining truth—induction and analogy—are based on probabilities; so that the entire system of human knowledge is connected with the theory set forth in this essay. ”

“ It is true that when our knowledge is very precise, when we have made many drawings from the bag, or, as in most of the examples in the books, when the total contents of the bag are absolutely known, the number which expresses the uncertainty of the assumed probability and its liability to be changed by further experience may become insignificant, or utterly vanish. ”

“ My second reflection is founded on those large probabilities, which the mind can judge of, and the minute differences it can observe betwixt them. When the chances or experiments on one side amount to ten thousand, and on the other to ten thousand and one, the judgment gives the preference to the latter, upon account of that superiority; tho’ ‘tis plainly impossible for the mind to run over every particular view, and distinguish the superior vivacity of the image arising from the superior number, where the difference is so inconsiderable. ”

“ These probabilities rise so near to certainty, that they govern our thoughts as absolutely, and influence all our actions as fully, as the most evident demonstration ; and in what concerns us we make little or no difference between them and certain knowledge. ”

“ But so also, for aught we knew, might there be more red balls than black and white, or more black balls than white and red, and in such case [pg 381] the effect of additional knowledge would be to prove to us that our bet was more advantageous than we had supposed it to be. There is in the existing state of our knowledge a rational probability of two to one against white; a probability fit to be made a basis of conduct. ”

“ What we have found once to follow from any object, we conclude will for ever follow from it; and if this maxim be not always built upon as certain, ‘tis not for want of a sufficient number of experiments, but because we frequently meet with instances to the contrary; which leads us to the second species of probability, where there is a contrariety in our experience and observation. ”

“ We may arrive at them only by multiplying the observations or the experiences, so that the strange effects finally destroy reciprocally each other, the mean results putting in evidence those phenomena and their divers elements. The more numerous the number of observations and the less they vary among themselves the more their results approach the truth. We fulfil this last condition by the choice of the methods of observations, by the precision of the instruments, and by the care which we take to observe closely ”

“ As to the ventures at conjecture which are sparingly put forth, let them be rated at their seeming worth, after the facts have been carefully studied and the opinions weighed, which have called out these ventures. They are confessedly only entitled to a claim for a certain degree, higher or lower, of probability. ”

“ When therefore we see other creatures, in millions of instances, perform like actions, and direct them to like ends, all our principles of reason and probability carry us with an invincible force to believe the existence of a like cause. ”

“ In judging how often one large number is contained in another, a first and rough guess may be made by striking off the same number of figures from both, and using the results instead of the numbers themselves. ”

“ His invention connects the utmost stretch of possibility with the most plausible degree of probability, in a manner that equally surprises our fancy, persuades our judgment, and affects our heart. His composition always hastens to the most necessary point as its centre, and from that disseminates, to that leads back as rays, all secondary ones. Group, form, and contrast are subordinate to the event, and common-place ever excluded. ”

“ Therefore, if the chances for the truth of a theory, in consequence of its explaining a certain set of details, be three to one, and because of its explaining another set,—for the purposes of argument unrelated to the first,—four to one, then the chances in its favor from its explaining both sets are not seven to one but twelve to one. ”

“ The every-day thought and reading of common life have a certain resemblance to the thought and reading demanded by the learned faculties. The research, the balance of evidence, the estimation of probabilities, which are used in a question of medicine, are closely akin in character, however different the matter of application, to those which serve a merchant to draw his conclusions about the markets. But the mathematicians have methods of their own, to which nothing in common life bears close analogy, as to the nature of the results or the character of the conclusions. ”

“ If our intention, therefore, be to consider the proportions of contrary events in a great number of instances, the images presented by our past experience must remain in their first form, and preserve their first proportions. ”

“ As demonstration is subject to the controul of probability, so is probability liable to a new correction by a reflex act of the mind, wherein the nature of our understanding, and our reasoning from the first probability become our objects. ”