Arnauld: Logic (1662), Part IV, chs. 13–16

19th-century portrait of Arnauld, the main author of the Logic.

The last four chapters of the Port-Royal Logic deal with reasoning under uncertainty. They touch briefly on issues of evidential support, balanced odds, and fair games.

In retrospect, these remarks are difficult not to read as a precursors of later probability theory. Many authors have pointed this out, including Ian Hacking and Lorraine Daston.

I'm reading the 1996 translation by Jill Buroker, published by Cambridge UP.

The content of the four chapters on probability, or plausibility, are:

• Chapter 13, "Some Rules for directing reason well in beliefs about events that depend on human faith," argues that the standard of geometric proof and mathematical certainty only applies to matters of "immutable essence," and that judgment about human affairs should be made by meditating on the available evidence and testimonies.
• Chapter 14, "Application of the preceding rule to the beliefs about miracles," claims that this means that there are several instances in which it is reasonable to believe miracles took place, even if this cannot be proven beyond all reasonable doubt.
• Chapter 15, "Another remark on the same subject of beliefs about events," adds that this explains why certain deeds are attested by two notaries, and that the technique of weighing the evidence also has applications to authorship attribution for ancient (religious) manuscripts.
• Chapter 16, "The Judgments we ought to make concerning future events," argues that the doctrine not only applies to reasoning about the past, but also about the prediction of the future; a number of fair and unfair games are described as examples, and a version of Pascal's wager is then put forward.

Quotes follow below.

Don't Hold Your Breath

In chapter 13, we are told that the methodology of geometric proof works for geometry,

But if we try to use the same rules for beliefs about human events, we will always judge them falsely, except by chance, and we will make a thousand fallacious inferences about them. (p. 263)

Instead, we have to look for circumstantial evidence:

In order to decide the truth about an event and to determine whether or not to believe in it, we must not consider it nakedly and in itself, as we would a proposition of geometry. But we must pay attention to all the accompanying circumstances, internal as well as external. I call those circumstances internal that belong to the fact itself, and those external that concern the persons whose testimony leads us to believe in it. Given this attention, if all the circumstances are such that it never or only rarely happens that similar circumstances are consistent with the falsity of the belief, the mind is naturally led to think that it is true. Moreover, it is right to do so, above all in the conduct of life, which does not require greater certainty than moral certainty, and which even ought to be satisfied in many cases with the greatest probability.

But if, on the contrary, these circumstances are such that they are often consistent with the falsity of the belief, reason would require either that we remain in suspense, or that we view as false whatever we are told when its truth does not look likely, even if it does not look completely impossible. (p. 264)

Cloister of the Hôpital Cochin, inhabiting the former site of the Port-Royal abbey.

 

A Perversion of Reason

This idea is echoed in chapter 15:

Since we should be satisfied with moral certainty in matters not susceptible of metaphysical certainty, so too when we cannot have complete moral certainty, the best we can do when we are committed to taking sides is to embrace the most probable, since it would be a perversion of reason to embrace the less probable. (p. 270)

We also hear that negative evidence "weaken or destroy in the mind the grounds for belief" (p. 270).

A couple of things worth noting about these quotes:

• The process described here is a mental therapy, not a decision calculus; you pay close attention, and your "mind is naturally led" to a certain belief.
• The focus is on human affairs, that is, on practical matters.
• As in frequentist statistics, the contrast is not between true and false, but between confirmed and unconfirmed; we thus "remain in suspense" if the evidence is insufficient.

Set the Record Straight

Chapter 16 pours scorn on "many people" for entertaining a certain "illusion":

This is that they consider only the greatness and importance of the benefit they desire or he disadvantage they fear, without considering in any way the likelihood or probability that this benefit or disadvantage will or will not come about. (p. 273)

That is, thinking only of utility while ignoring probability. Moreover,

This is what attracts so many people to lotteries: Is it not highly advantageous, they say, to win twenty thousand crowns for one crown? Each person thinks he will be the happy person who will win the jackpot. No one reflects that if it is, for example, twenty thousand crows, it may be thirty thousand times more probable for each individual to lose rather than to win it.

The flaw in this reasoning is that in order to decide what we ought to do to obtain some good or avoid some harm, it is necessary to consider not only the good or harm itself, but also the probability that it will or will not occur, and to view geometrically the proportion all these things have when taken together. This can be clarified by the following example.

There are game in which, if ten persons each put in a crown, only one wins the whole pot and all the others lose. This each person risks losing only a crown and may win nine. If we consider only the gain and loss in themselves, it would appear that each person has the advantage. But we must consider in addition that if each could win nine crowns and risks losing only one, it is also nine times more probability for each person to lose one crown and not to win nine. Hence each has nine crowns to hope for himself, one crown to lose, nine degrees of probability of losing a crown, and only one of winning the nine crowns. This puts the matter at perfect equality. (pp. 273–74)

This argument is then pushed a little further to deal with some more extreme bets; to the fear of lightning, which is allegedly irrational and has to be "set straight" (p. 275); and finally to a version of Pascal's wager.