The Bottom Line

There are two sealed boxes up for auc­tion, box A and box B. One and only one of these boxes con­tains a valuable di­a­mond. There are all man­ner of signs and por­tents in­di­cat­ing whether a box con­tains a di­a­mond; but I have no sign which I know to be perfectly re­li­able. There is a blue stamp on one box, for ex­am­ple, and I know that boxes which con­tain di­a­monds are more likely than empty boxes to show a blue stamp. Or one box has a shiny sur­face, and I have a sus­pi­cion—I am not sure—that no di­a­mond-con­tain­ing box is ever shiny.

Now sup­pose there is a clever ar­guer, hold­ing a sheet of pa­per, and they say to the own­ers of box A and box B: “Bid for my ser­vices, and who­ever wins my ser­vices, I shall ar­gue that their box con­tains the di­a­mond, so that the box will re­ceive a higher price.” So the box-own­ers bid, and box B’s owner bids higher, win­ning the ser­vices of the clever ar­guer.

The clever ar­guer be­gins to or­ga­nize their thoughts. First, they write, “And there­fore, box B con­tains the di­a­mond!” at the bot­tom of their sheet of pa­per. Then, at the top of the pa­per, the clever ar­guer writes, “Box B shows a blue stamp,” and be­neath it, “Box A is shiny,” and then, “Box B is lighter than box A,” and so on through many signs and por­tents; yet the clever ar­guer ne­glects all those signs which might ar­gue in fa­vor of box A. And then the clever ar­guer comes to me and re­cites from their sheet of pa­per: “Box B shows a blue stamp, and box A is shiny,” and so on, un­til they reach: “and there­fore, box B con­tains the di­a­mond.”

But con­sider: At the mo­ment when the clever ar­guer wrote down their con­clu­sion, at the mo­ment they put ink on their sheet of pa­per, the ev­i­den­tial en­tan­gle­ment of that phys­i­cal ink with the phys­i­cal boxes be­came fixed.

It may help to vi­su­al­ize a col­lec­tion of wor­lds—Everett branches or Teg­mark du­pli­cates—within which there is some ob­jec­tive fre­quency at which box A or box B con­tains a di­a­mond.1

The ink on pa­per is formed into odd shapes and curves, which look like this text: “And there­fore, box B con­tains the di­a­mond.” If you hap­pened to be a liter­ate English speaker, you might be­come con­fused, and think that this shaped ink some­how meant that box B con­tained the di­a­mond. Sub­jects in­structed to say the color of printed pic­tures and shown the word Green in red ink of­ten say “green” in­stead of “red.” It helps to be illiter­ate, so that you are not con­fused by the shape of the ink.

To us, the true im­port of a thing is its en­tan­gle­ment with other things. Con­sider again the col­lec­tion of wor­lds, Everett branches or Teg­mark du­pli­cates. At the mo­ment when all clever ar­guers in all wor­lds put ink to the bot­tom line of their pa­per—let us sup­pose this is a sin­gle mo­ment—it fixed the cor­re­la­tion of the ink with the boxes. The clever ar­guer writes in non-erasable pen; the ink will not change. The boxes will not change. Within the sub­set of wor­lds where the ink says “And there­fore, box B con­tains the di­a­mond,” there is already some fixed per­centage of wor­lds where box A con­tains the di­a­mond. This will not change re­gard­less of what is writ­ten in on the blank lines above.

So the ev­i­den­tial en­tan­gle­ment of the ink is fixed, and I leave to you to de­cide what it might be. Per­haps box own­ers who be­lieve a bet­ter case can be made for them are more li­able to hire ad­ver­tisers; per­haps box own­ers who fear their own defi­cien­cies bid higher. If the box own­ers do not them­selves un­der­stand the signs and por­tents, then the ink will be com­pletely un­en­tan­gled with the boxes’ con­tents, though it may tell you some­thing about the own­ers’ fi­nances and bid­ding habits.

Now sup­pose an­other per­son pre­sent is gen­uinely cu­ri­ous, and they first write down all the dis­t­in­guish­ing signs of both boxes on a sheet of pa­per, and then ap­ply their knowl­edge and the laws of prob­a­bil­ity and write down at the bot­tom: “There­fore, I es­ti­mate an 85% prob­a­bil­ity that box B con­tains the di­a­mond.” Of what is this hand­writ­ing ev­i­dence? Ex­am­in­ing the chain of cause and effect lead­ing to this phys­i­cal ink on phys­i­cal pa­per, I find that the chain of causal­ity wends its way through all the signs and por­tents of the boxes, and is de­pen­dent on these signs; for in wor­lds with differ­ent por­tents, a differ­ent prob­a­bil­ity is writ­ten at the bot­tom.

So the hand­writ­ing of the cu­ri­ous in­quirer is en­tan­gled with the signs and por­tents and the con­tents of the boxes, whereas the hand­writ­ing of the clever ar­guer is ev­i­dence only of which owner paid the higher bid. There is a great differ­ence in the in­di­ca­tions of ink, though one who fool­ishly read aloud the ink-shapes might think the English words sounded similar.

Your effec­tive­ness as a ra­tio­nal­ist is de­ter­mined by whichever al­gorithm ac­tu­ally writes the bot­tom line of your thoughts. If your car makes metal­lic squeal­ing noises when you brake, and you aren’t will­ing to face up to the fi­nan­cial cost of get­ting your brakes re­placed, you can de­cide to look for rea­sons why your car might not need fix­ing. But the ac­tual per­centage of you that sur­vive in Everett branches or Teg­mark wor­lds—which we will take to de­scribe your effec­tive­ness as a ra­tio­nal­ist—is de­ter­mined by the al­gorithm that de­cided which con­clu­sion you would seek ar­gu­ments for. In this case, the real al­gorithm is “Never re­pair any­thing ex­pen­sive.” If this is a good al­gorithm, fine; if this is a bad al­gorithm, oh well. The ar­gu­ments you write af­ter­ward, above the bot­tom line, will not change any­thing ei­ther way.

This is in­tended as a cau­tion for your own think­ing, not a Fully Gen­eral Coun­ter­ar­gu­ment against con­clu­sions you don’t like. For it is in­deed a clever ar­gu­ment to say “My op­po­nent is a clever ar­guer,” if you are pay­ing your­self to re­tain what­ever be­liefs you had at the start. The world’s clever­est ar­guer may point out that the Sun is shin­ing, and yet it is still prob­a­bly day­time.

1Max Teg­mark, “Par­allel Uni­verses,” in Science and Ul­ti­mate Real­ity: Quan­tum The­ory, Cos­mol­ogy, and Com­plex­ity, ed. John D. Bar­row, Paul C. W. Davies, and Charles L. Harper Jr. (New York: Cam­bridge Univer­sity Press, 2004), 459–491, http://​​arxiv.org/​​abs/​​as­tro-ph/​​0302131.