Guesswork: Enough is enough, but how much is enough?
Three weeks ago I recounted a lot of tough problems raised over the ages about how cause and effect really work. One of the toughest was that if cause and effect works like we normally think it doesâ€”thing A bumps into thing B and moves it in a precisely predictable directionâ€”then the whole universe should be precisely predictable as well. The universe would be like a giant computer churning out a precisely predictable sequence of events that was implicit in its initial state. Two weeks ago I went on to examine parallels and contrasts between the ways minds and computers work and I identified some of the ways in which we both dread and hope that minds are like computers.
Last week I detailed deduction, the deterministic process by which computers work. Deduction, it turns out, is always based on premises that canâ€™t have been deduced, but instead have to be the product of guesswork.
This week I want to say more about that guesswork.
Letâ€™s go back to deduction for a moment. Youâ€™ll remember from last week that this is the form a deduction takes:
General rule: If A then B.
If [human] then [mortal].
If you then apply this to a certain thing X, the statement becomes:
If [X is A] then [X is B].
If [Socrates is a man] then [Socrates is mortal].
Deduction is reasoning from the general law to a specific case, for example, from whatâ€™s true of all men to whatâ€™s true of one man. But where do these general laws come from and how do we know theyâ€™re right?
The general laws come from experience. The process by which experience leads us to general laws is called induction, building not from general laws to particular cases but the reverse, building from particular cases to general laws.
Induction is a kind of guesswork. All the men weâ€™ve ever checked up on have proven mortal, but most men each of us knows have yet to dieâ€”and for all we know, maybe one wonâ€™t. And what about future generations? Maybe theyâ€™ll be immortal. We donâ€™t know for sure that they wonâ€™t.
So how many men do you need to check before youâ€™re absolutely sure youâ€™ve got a true general law? Until youâ€™ve checked all past, present, and future men you canâ€™t say absolutely. No one can check that many, and besides, you need the general rule now, so you have to guess as to when to assume that the statement â€œall men are mortalâ€ is true. You might guess wrong, jumping to a false assumption too soon. Unless and until youâ€™ve checked all men, you canâ€™t be certain youâ€™ve checked enough men.
So hereâ€™s one disappointment for anyone dreaming of a determinate universe. Our general laws about how it works have to be derived from an indefinite number of specific instances. To come up with any general laws at all weâ€™ve got to guess. Thereâ€™s no way to simply deduce general laws from nothing. Any general law we come up with ultimately rests on a bed of unsettled guesswork.
Induction is one of two areas of guesswork. Hereâ€™s the other:
How do we know that the general law applies to the specific case? Sure, â€œSocrates is a manâ€ feels right and â€œRobbie the Robot is a manâ€ feels wrong, but how do we know that? Not by deduction or induction but by a kind of guesswork called abduction.
Socrates is a man and Robbie the Robot isnâ€™t because Socrates has all sorts of manlike qualities that Robby the Robot lacks. Socrates has hair growing on his toes. Heâ€™s covered in skin. He swills beer. Robby the Robot doesnâ€™t even have toes, let alone hair growing on them, heâ€™s covered in metal not skin and he doesnâ€™t swill beer.
Abduction is how we decide whether a thing fits a category. We use comparative traits. But this is guesswork too. How many manly traits does a man have? Potentially an infinite number. How many distinctive ones before we can call an entity a man? We can only guess. Maybe you think youâ€™re dealing with a man because you see so many manly traits, and then you discover to your shock one feature that youâ€™re sure no man can have.
For deduction it may not matter that much if you guess wrong about your abduction. So you decide that Robbie the Robot is a man and therefore mortal it may not be a big problem. But a wrong abduction can lead to a wrong induction and therefore a wrong general rule. If you make a mistake and assume that Robbie the Robot is a man you could end up deciding that not all men are mortal after all.
Both induction and abduction look like determinate processes at first glance. Take induction. If you wanted to know if all the bills in your wallet were twenties, a simple and decisive algorithm will let you find out: Check every bill. Bill A is a twenty, bill B is a twenty, on and on until you get to the end of your wallet. And then you would have a yes/no answer to your question. Likewise with abduction, if you wanted to know whether a particular car X is a black four-door hatchback, a simple algorithm will tell you: Check each trait. Car X is black, Car X has four doors, Car X is a hatchback. Check all three specifications and youâ€™ve got your yes/no answer.
The number of bills in your wallet or traits on a short list is finite. But in application you often deal with open-ended inductions or abductions. You wonâ€™t have a finite number of bills or car traits to check.
Will the sun rise tomorrow morning? Well, your experience of sunrises says it will, but then tomorrow is a new day and you canâ€™t be certain the sun will rise because you havenâ€™t checked every last sunrise ever the way you checked every bill in your wallet.
And the same is true for abduction. There were only three qualities to check regarding the black four-door hatchback, but most things have an infinite number of qualities. The guesswork resides in this open-endedness. Since the algorithms could go on forever, youâ€™ve got to decide when youâ€™re done deciding. Indeed, the man who brought us the computer, Alan Turing, made this point. Itâ€™s called â€œTuringâ€™s halting problemâ€: If you give a computer an open-ended abduction, the computer wonâ€™t know when to stop evaluating similarities.
Thereâ€™s something fundamentally circular about the three kinds of logic-deduction, induction, and abduction. Let me walk you through it but going backwards. We started with deduction:
All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.
But how did we come up with that first statement? By the guesswork of induction:
Bill is a man.
Bill is mortal.
Fred is a man.
Fred is mortal.
Joe is a man.
Joe is mortal.
Therefore all men are mortal.
Enough guys? Who knows? Thatâ€™s the guesswork in induction.
And how do we know that each of these guys is a man? For this we relied on the guesswork of abduction:
Men have hairy toes.
Bill has hairy toes.
Men have deep voices.
Bill has a deep voice.
Men swill beer.
Bill swills beer.
Enough manly qualities? Who knows? Thatâ€™s the guesswork of abduction.
Have you given your job, girlfriend, education, boyfriend, investment, business, child, sibling, mother, student, employee, boss, teacher enough time to prove his, her, or its worth? It depends on how you define enough. Have they given you enough time to prove your worth?
Perhaps this is enough (more than enough) on logic for a while. Next week, Iâ€™ll talk about a very personal application of Turingâ€™s halting problem to the question of forgiveness. Later, Iâ€™ll return to the logical mystery tour to draw a direct link from the guesswork of induction and abduction to the guesswork that makes life different from physics.