A few technical terms are useful. Suppose we want to establish inductively that there is some probability in favour of the generalisation: “Everything that has the property _F_ also has the property _f_”. We will call this generalisation _g_. Suppose we have observed a number of instances in which _F_ and _f_ go together, and no instances to the contrary. These instances may have other common properties as well; the sum-total of their common properties is called the _total positive analogy_, and the sum-total of their _known_ common qualities is called the _known positive analogy_. The properties belonging to some but not to all of the instances in question are called the _negative analogy_: all of them constitute the _total negative analogy_, all those that are _known_ constitute the _known negative analogy_. To strengthen an induction, we want to diminish the positive analogy to the utmost possible extent; this, according to Mr. Keynes, is why numerous instances are useful. On “pure” induction, where we rely solely upon number of instances, without _knowing_ how they affect the analogy, Mr. Keynes concludes (_Treatise in Probability_, p. 236): “We have shown that if each of the instances necessarily follows from the generalisation, then each additional instance increases the probability of the generalisation, so long as the new instance could not have been predicted with certainty from a knowledge of the former instances.... The common notion, that each successive verification of a doubtful principle strengthens it, is formally proved, therefore without any appeal to conceptions of law or of causality. _But we have not proved_ that this probability approaches certainty as a limit, or even that our conclusion becomes more likely than not, as the number of verifications or instances is indefinitely increased.” It is obvious that induction is not much use unless, with suitable care, its conclusions can be rendered more likely to be true than false. This problem therefore necessarily occupies Mr. Keynes. It is found that an induction will approach certainty as a limit if two conditions are fulfilled: (1) If the generalisation is false, the probability of its being true in a new instance when it has been found to be true in a certain number of instances, however great that number may be, falls short of certainty by a finite amount. (2) There is a finite _a priori_ probability in favour of our generalisation. Mr. Keynes uses “finite” here in a special sense. He holds that not all probabilities are numerically measurable; a “finite” probability is one which exceeds some numerically measurable probability however small. _E.g._ our generalisation has a finite _a priori_ probability if it is less unlikely than throwing heads a billion times running.