Bernoulli V

Part Four.
The use and Application of the Preceding Doctrine in Civil, Moral, and Economic Matters.
Chapter 1 of Part 4 covers his definitions of key terms and concepts that he uses; including but not limited to: certainty, probability, more probable / doubtful, morally certain, necessary, and contingent, which we will cover here.
Certainty : He describes certainty as having an objective form and a subjective form. Objectively, he sets forth that anything that is or was or will be is certain. It is the truth of the existence of a thing; something that will not change except by divine intervention. Subjectively, certainty is the amount to which we have knowledge of a subject concerning the things which exist. So for us, certainty increases or decreases depending on our persuasion of the proportion of probabilities of a thing’s past, present or future existence.
Probability: This is a portion or “degree” of certainty. This degree of certainty is measured by the ratio of arguments for or against the existence of some outcome. The example is given that if complete certainty is represented by 1, and this certainty is divided up into five parts where 3 parts argue for and 2 parts against the existence of some outcome, then the totally probability would be the ratio 3/5 of certainty.
More probable / Doubtful: If one thing has a large portion of certainty than another, it is said to be more probably; this also means it must be more than ½ of certainty. If something has ½ or less of certainty it is said to be doubtful or undecided. If something has even a very small portion of certainty, it may still be said to be possible; however if the portion is too small it is said to be impossible. This bring us to Morally certain.
Morally Certain: “Something is morally certain if it comes so close to complete certainty that the difference cannot be perceived.” This means that if a thing has 999/1000 ratio of probability it is morally certain, while by the same token, a thing is morally impossible if its certainty is lacking by that amount of 999/1000. Namely, if morally certain is said to be 999/1000, then 1/1000 is said to be morally impossible.
Necessary: Something is necessary if it is impossible for it to not exist in either the past, present of future. Necessity can be physical, hypothetical, or contractual. He gives the example that fire necessarily burns and that a triangle necessarily has 3 sides. Hypothetically, if something is known to exist or have existed, it cannot not exist or have existed. Contractually, if players of a game of dice or some other game, agree before hand that a certain throw causes a player to win, and a player produces that throw, he is said to have won necessarily.
Contingent: “A thing that can now, in the future, or in the past not exist is contingent (either free depending on the will of a rational creature, or fortuitous and haphazard depending on accident or fortune.)”
Many other definitions are given of various concepts, but it is interesting to see his interpretation of these basics as he uses them repeatedly in the fourth section as he explains this “Art” of conjecturing. He presents “pertinent general axioms” and instructs on the “Various kids of arguments and how to assess their weights for computing probabilities of things.” Here we see 3 forms of arguments: those that exist necessarily and indicate contingently, those that exist contingently and indicate necessarily, and those where both exist contingently and indicate contingently. He gives the example using his “brother.” His brother has not written to him for a long time. The three arguments to be discussed are whether this has occurred because of his brother’s laziness, business, or death. Since he knows his brother to be lazy, this is an argument by hypothetical necessity. It is also a contingent argument because laziness may not be the true reason. The second argument indicates necessity, since he couldn’t not write if he were dead, but also exists contingently, because his brother may, in fact, be alive. As for the third argument, since the brother may or may not have business it exists contingently, but it also indicates contingently because even if the brother does have business the amount of business may or may not be great enough to keep him from writing. Another example given is that of a dice player. Though the player my win by contractual necessity if he throws a 7, his hope of winning exists contingently since there are possible outcomes of the dice other than 7.

Here I have presented some of the basic concepts with which everyone is familiar. However, Bernoulli’s The Art of Conjecturing covers many, many topics of probability at increasing levels of difficulty. Because of its clear nature and the details behind the concepts being presented up-front, this book should be required reading for all statistics students and even those in Modern Algebra! It is a very interesting, as well as challenging read.