The Parable of Hemlock

“All men are mor­tal. Socrates is a man. There­fore Socrates is mor­tal.”
— Aris­to­tle(?)

Socrates raised the glass of hem­lock to his lips…
“Do you sup­pose,” asked one of the on­look­ers, “that even hem­lock will not be enough to kill so wise and good a man?”
“No,” replied an­other by­stan­der, a stu­dent of philos­o­phy; “all men are mor­tal, and Socrates is a man; and if a mor­tal drink hem­lock, surely he dies.”
“Well,” said the on­looker, “what if it hap­pens that Socrates isn’t mor­tal?”
“Non­sense,” replied the stu­dent, a lit­tle sharply; “all men are mor­tal by defi­ni­tion; it is part of what we mean by the word ‘man’. All men are mor­tal, Socrates is a man, there­fore Socrates is mor­tal. It is not merely a guess, but a log­i­cal cer­tainty.
“I sup­pose that’s right...” said the on­looker. “Oh, look, Socrates already drank the hem­lock while we were talk­ing.”
“Yes, he should be keel­ing over any minute now,” said the stu­dent.
And they waited, and they waited, and they waited…
“Socrates ap­pears not to be mor­tal,” said the on­looker.
“Then Socrates must not be a man,” replied the stu­dent. “All men are mor­tal, Socrates is not mor­tal, there­fore Socrates is not a man. And that is not merely a guess, but a log­i­cal cer­tainty.

The fun­da­men­tal prob­lem with ar­gu­ing that things are true “by defi­ni­tion” is that you can’t make re­al­ity go a differ­ent way by choos­ing a differ­ent defi­ni­tion.

You could rea­son, per­haps, as fol­lows: “All things I have ob­served which wear cloth­ing, speak lan­guage, and use tools, have also shared cer­tain other prop­er­ties as well, such as breath­ing air and pump­ing red blood. The last thirty ‘hu­mans’ be­long­ing to this cluster, whom I ob­served to drink hem­lock, soon fell over and stopped mov­ing. Socrates wears a toga, speaks fluent an­cient Greek, and drank hem­lock from a cup. So I pre­dict that Socrates will keel over in the next five min­utes.”

But that would be mere guess­ing. It wouldn’t be, y’know, ab­solutely and eter­nally cer­tain. The Greek philoso­phers—like most pre­scien­tific philoso­phers—were rather fond of cer­tainty.

Luck­ily the Greek philoso­phers have a crush­ing re­join­der to your ques­tion­ing. You have mi­s­un­der­stood the mean­ing of “All hu­mans are mor­tal,” they say. It is not a mere ob­ser­va­tion. It is part of the defi­ni­tion of the word “hu­man”. Mor­tal­ity is one of sev­eral prop­er­ties that are in­di­vi­d­u­ally nec­es­sary, and to­gether suffi­cient, to de­ter­mine mem­ber­ship in the class “hu­man”. The state­ment “All hu­mans are mor­tal” is a log­i­cally valid truth, ab­solutely un­ques­tion­able. And if Socrates is hu­man, he must be mor­tal: it is a log­i­cal de­duc­tion, as cer­tain as cer­tain can be.

But then we can never know for cer­tain that Socrates is a “hu­man” un­til af­ter Socrates has been ob­served to be mor­tal. It does no good to ob­serve that Socrates speaks fluent Greek, or that Socrates has red blood, or even that Socrates has hu­man DNA. None of these char­ac­ter­is­tics are log­i­cally equiv­a­lent to mor­tal­ity. You have to see him die be­fore you can con­clude that he was hu­man.

(And even then it’s not in­finitely cer­tain. What if Socrates rises from the grave a night af­ter you see him die? Or more re­al­is­ti­cally, what if Socrates is signed up for cry­on­ics? If mor­tal­ity is defined to mean finite lifes­pan, then you can never re­ally know if some­one was hu­man, un­til you’ve ob­served to the end of eter­nity—just to make sure they don’t come back. Or you could think you saw Socrates keel over, but it could be an illu­sion pro­jected onto your eyes with a reti­nal scan­ner. Or maybe you just hal­lu­ci­nated the whole thing...)

The prob­lem with syl­l­o­gisms is that they’re always valid. “All hu­mans are mor­tal; Socrates is hu­man; there­fore Socrates is mor­tal” is—if you treat it as a log­i­cal syl­l­o­gism—log­i­cally valid within our own uni­verse. It’s also log­i­cally valid within neigh­bor­ing Everett branches in which, due to a slightly differ­ent evolved bio­chem­istry, hem­lock is a deli­cious treat rather than a poi­son. And it’s log­i­cally valid even in uni­verses where Socrates never ex­isted, or for that mat­ter, where hu­mans never ex­isted.

The Bayesian defi­ni­tion of ev­i­dence fa­vor­ing a hy­poth­e­sis is ev­i­dence which we are more likely to see if the hy­poth­e­sis is true than if it is false. Ob­serv­ing that a syl­l­o­gism is log­i­cally valid can never be ev­i­dence fa­vor­ing any em­piri­cal propo­si­tion, be­cause the syl­l­o­gism will be log­i­cally valid whether that propo­si­tion is true or false.

Syl­l­o­gisms are valid in all pos­si­ble wor­lds, and there­fore, ob­serv­ing their val­idity never tells us any­thing about which pos­si­ble world we ac­tu­ally live in.

This doesn’t mean that logic is use­less—just that logic can only tell us that which, in some sense, we already know. But we do not always be­lieve what we know. Is the num­ber 29384209 prime? By virtue of how I define my dec­i­mal sys­tem and my ax­ioms of ar­ith­metic, I have already de­ter­mined my an­swer to this ques­tion—but I do not know what my an­swer is yet, and I must do some logic to find out.

Similarly, if I form the un­cer­tain em­piri­cal gen­er­al­iza­tion “Hu­mans are vuln­er­a­ble to hem­lock”, and the un­cer­tain em­piri­cal guess “Socrates is hu­man”, logic can tell me that my pre­vi­ous guesses are pre­dict­ing that Socrates will be vuln­er­a­ble to hem­lock.

It’s been sug­gested that we can view log­i­cal rea­son­ing as re­solv­ing our un­cer­tainty about im­pos­si­ble pos­si­ble wor­lds—elimi­nat­ing prob­a­bil­ity mass in log­i­cally im­pos­si­ble wor­lds which we did not know to be log­i­cally im­pos­si­ble. In this sense, log­i­cal ar­gu­ment can be treated as ob­ser­va­tion.

But when you talk about an em­piri­cal pre­dic­tion like “Socrates is go­ing to keel over and stop breath­ing” or “Socrates is go­ing to do fifty jump­ing jacks and then com­pete in the Olympics next year”, that is a mat­ter of pos­si­ble wor­lds, not im­pos­si­ble pos­si­ble wor­lds.

Logic can tell us which hy­pothe­ses match up to which ob­ser­va­tions, and it can tell us what these hy­pothe­ses pre­dict for the fu­ture—it can bring old ob­ser­va­tions and pre­vi­ous guesses to bear on a new prob­lem. But logic never flatly says, “Socrates will stop breath­ing now.” Logic never dic­tates any em­piri­cal ques­tion; it never set­tles any real-world query which could, by any stretch of the imag­i­na­tion, go ei­ther way.

Just re­mem­ber the Li­tany Against Logic:

Logic stays true, wher­ever you may go,
So logic never tells you where you live.