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Six years later Leibniz returns to the subject in a letter to Jacob Bernoulli. Within the two year period of April 1703 and April 1705 Jacob Bernoulli and Leibniz exchanged a number of letters regarding topics relevant to the Ars Conjectandi.
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OF
PROBABILITY
§
1. Philalethes.
If demonstration
shows the connection
of ideas, probability is nothing else than the appearance of this
connection based
upon proofs in which immutable connection is not seen.
§ 2. There are several degrees of assent from assurance
down to conjecture,
doubt,
distrust. §
3. When there
is certainty, there is intuition in all parts of the reasoning which
show its connection; but what makes me believe
is something extraneous. §
4. Now
probability is grounded in its conformity with what we know, or in the
testimony of those who know.
Theophilus. I prefer to maintain that it is always grounded in likelihood (vraisemblance) or in conformity with the truth; and the testimony of another is also a thing which the truth has been wont to have for itself as regards the facts that are within reach. It may be said then that the similarity of the probable and the truth is taken either from the thing itself, or from some extraneous thing. The rhetoricians emply two kinds of arguments: the artificial, drawn from things by reasoning, and the non-artificial, based only upon the express testimony either of man or perhaps also of the thing itself. But there are mixed arguments also, for testimony may itself furnish a fact which serves to from an artificial argument. § 5. Ph. It is for lack of similarity to truth that we do not readily believe that which has nothing like that which we know. Thus when an ambassador told the king of Siam that with us the water was so hardened in winter that an elephant might walk thereon without breaking through, the king said to him: Hitherto I have believed you as a man of good faith; now I see that you lie. § 6. But if the testimony of others can render a fact probable, the opinion of others should not pass of itself as a true ground of probability. For there is more error than knowledge among men, and if the belief of those whom we know and esteem is a legitimate ground of assent, men have reason to be Heathen in Japan, Mahometans in Turkey, Papists in Spain, Calvinists in Holland, and Lutherans in Sweden. Th. The testimony of men is no doubt of more weight than their opinion, and in reason it is also the result of more reflection. But you know that the judge sometimes makes them take the oath de credulitate, as it is called; that in the examinations, we often ask witnesses not only what they have seen but also what they think, demanding of them at the same time the reasons of their judgment, and whether they have reflected thereupon to such an extent as behooves them. Judges also defer much to the views and opinions of experts in each profession; private individuals, in proportion as it is inconvenient for them to present themselves at the appropriate examination, are not less compelled to do this. Thus a child, or other human being whose condition is but little better in this respect, is obliged, whenever he finds himself in a certain situation, to follow the religion of the country, so long as he sees nothing bad therein, and so long as he is not in a condition to find out whether there is a better. A tutor of pages, whatever his sect, will compel them each to go to the church where those who profess the same belief as this young man go. The discussions between Nicole and others on the argument from the great number in a matter of faith may be consulted, in which sometimes one defers to it too much and another does not consider it enough. There are other similar prejudgments by which men would very easily exempt themselves from discussion. These are what Tertullian, in a special treatise, calls Prescriptiones, [De Praescriptione Haereticorum] availing himself of a term which the ancient jurisconsults (whose language was not known to him) intended for many kinds of exceptions or foreign and predisposing allegations, but which now means merely the temporal prescription when it is intended to repel the demand of another because not made within the time fixed by law. Thus there was reason for making known the legitimate prejudgments both on the side of the Roman Church and on that of the Protestants. It has been found that there are means of opposing novelty, for example, on the part of both in certain respects; as, for example, when the Protestants for the most part abandoned the ancient form of ordination of clergymen, and the Romanists changed the ancient canon of the Old Testament books of Holy Scripture, as I have clearly enough shown in a discussion I had in writing, and from time to time, with the bishop of Meaux, whom we have just lost, according to the news which came some days since. although it presents suspicion of error in these matters, is not a certain proof thereof. |
ON
THE DEGREE OF ASSENT
Th.
... The mathematicians of
our times have begun to calculate chances upon the occations of games.
Chevalier de Méré, whose "Agrémens"and
other works have been printed, a man of penetrating mind who was both a
player and a philosopher, gave them an opportunity by forming questions
regarding the profits in order to know how much the game would be
worth, if interrupted at such or such a stage. In this way he induced
Pascal, his friend, to examine these things a little. The question made
a stir and gave Huygens the opportunity to produce his treatise "de
Alea." Other learned men entered into the subject. Some principles were
established of
which the Pensioner De Witt also availed himself in a brief discourse
printed in Dutch on annuities. The foundation on which they have built
goes back to the prosthaphaeresis,
i.
e. the taking of an arithmetical
mean between several equally receivable suppositions. Our peasants also
have made use of it for a long time according to their natural
mathematics. For example,
when some inheritance or land is to be sold, they form three bodies of
appraisers; these bodies are called Schurzen
in Low Saxon, and each body makdes an estimate of the property in
question. Suppose, then, that the first estimates its value to be 1000
crowns, the second 1400, the third 1500; the sum of these three
estimates is taken, viz. 3900, and because there were three bodies, the
third, i.e.
1300, is taken as the mean value asked for; or rather, they take the
sum of the third part of each estiamte which is the same thing. This is
the axiom: aequalibus aequalia, equal suppositions must have equal
consideration. But when the suppositions are unequal they compare them
with each other. Suppose, for example, that with two dice, the one
ought to win if it makes 7 points, the other if it makes 9, the
question is asked what proportion obtains between their probabilities
of winning? I reply that the probability of the last is worth only
two-thirds of the probability of the first, for the first can make 7 in
three ways with two dice, viz.: by 1 and 6, or 2 and 5, or 3 and 4; and
the other can make 9 in two ways only, by throwing 3 and 6, or 4 and 5;
and all these methods are equally possible. Then the probabilities,
which are as the numbers of equal possiblilities, will be as 3 to 2, or
as 1 to 2/3. I have more than once said that a new kind of logic
would be required which would treat of the degrees of probability,
since Aristotle in his "Topics" has done nothing less than this, and
has contented himself with putting in a certain order certain popular
rules distributed according to the topics, which may be of use on some
occasion where the question concerns the amplification of the discourse
and the giving to it probability without putting it to the trouble of
furnishing us a necessary balance for weighing probabilities and
forming thereupon a solid judgment. It would be well for him who should
treat of this matter to pursue the examination of games of chance;
and in general I wish that some skillful mathematician would
produce
an ample work with full details and thoroughly reasoned upon all sorts
of games, which would be very useful in perfecting the art of
invention, the human mind appearing to better advantage in games than
in the most serious matters.
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