A story about fickle memory, pirates, unnecessary details and boolean logic.
"Hey Lenny, it's me, Teddy!"
Leonard looks blankly at the stranger for what seems like minutes. He then pulls a bunch of yellowed Polaroids from his pockets and checks them out. Eventually he finds a photo of this man, the photo reads "Teddy".
"Hi, Teddy", he says, "did I tell you about..."
"Yes, yes", Teddy cuts him off, "memory fades, never trust memory, memory's unreliable, I know the drill"
"Look, Lenny, I need a favor", Teddy continues, "I'd do it myself, but you know how it is, very busy and all"
"What do you need?" asks Leonard.
"I need you to help me reduce a logical expression!", shouts Teddy excitedly.
If Leonard could remember, he'd get the feeling Teddy always has an agenda, alas, he can't.
"Uhm", replies Leonard, "is that the whole business with the ands and the ors and the trues and the falses?"
"Yes! That's exactly it... except this might be a bit more complex than your typical run-of-the-mill expression", Teddy looks amused, "Are you familiar with De Morgan's laws?"
Leonard looks blankly for a couple of seconds, "I can't say I am"
Teddy saw this coming so he continues, "OK, bring a chair, I'll tell you a story. I'm pretty sure De Morgan wasn't a pirate but for the purpose of this story he might as well be one..."
Back in the day where pirates were the rage, there was one pirate captain called De Morgan. Now De Morgan's crew was very well known in the seven seas as a bunch of weirdos.
They were very proficient in all their piraty skills, all but one: their pirate lingo.
They were obsessed with negations. You couldn't get a straight answer from them even if you bribed them with grog. You couldn't even begin to disbelieve the lengths they'd go not to avoid saying things in a way not simple.
Needless to say they got on De Morgan's nerves. Commanding this crew was proving more and more difficult. He needed to find a way to make this work.
He though long and hard, while the crew plundered, pillaged and had the most outrageous philosophical discussions on deck.
Eventually Captain De Morgan created a bunch of rules to help him understand his crew:
1) When they say "Nay (A or B)" that is the same as "(Nay A) and (Nay B)"
2) When they say "Nay (A and B)" that is the same as "(Nay A) or (Nay B)"
And because back then a Captain's word was law, these became known as De Morgan's laws:
not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B)
(Teddy concludes his story and waits for it to sink in)
Leonard thinks about this for a while and asks Teddy an unexpected question, "How can you be sure those laws actually work in practice? How can you trust them?"
The question catches Teddy off guard but he manages to come up with an answer, "I can tell you, but first I'll need to tell you about..."
(Teddy pulls out a piece of paper and starts another monologue)
Take a very simple expression, for instance:
A and B
We all know the values for
A can be either
False, and the same goes for
We could put all those values right after the expression above and we'd get something like this:
A and B ------- T T # both A and B are true T F # A is true, B is false F T # A is false, B is true F F # both A and B are false
Now, if we also compute the value of the whole expression and put that under the
and operator we get:
A and B ------- T (T) T T (F) F F (F) T F (F) F
But if we negate the expression above, we get
not (A and B), and we can also compute the values of this new expression for every possible combination of values for
not (A and B) ------------- [F] (T (T) T) # `` is the the opposite of `()` [T] (T (F) F) [T] (F (F) T) [T] (F (F) F)
So in other words...
value of A | value of B | not (A and B) -----------+------------+--------------- T | T | F T | F | T F | T | T F | F | T
We'll get back to this in a second, but lets think about something else.
What would be the values of
A or B?
A or B ------- T (T) T T (T) F F (T) T F (F) F
And, if we negate both
(not A) or (not B) ------------------ ([F] T) (F) ([F] T) ([F] T) (T) ([T] F) ([T] F) (T) ([F] T) ([T] F) (T) ([T] F)
In other words...
value of A | value of B | not (A and B) | (not A) or (not B) -----------+------------+---------------+-------------------- T | T | F | F T | F | T | T F | T | T | T F | F | T | T
As you can see, for every possible combination of values of
B, both expressions are equivalent!
That's what De Morgan saw. You can do the same analysis for the other law.
These little table things are called Truth Tables, by the way.
(Just when Teddy was about to sourly comment on the futility of explaining this whole process, Leonard surprises him yet again...)
Leonard takes out his camera, takes a picture of De Morgan's laws and writes below the picture, "You can trust these, they work, you've seen the demonstration".
"What are you doing?" asks Teddy curiously.
I have a method, because of my condition, you know. If every time I need to use these laws I have to do the same demonstration, I wouldn't be able to get anything done.
This way I can trust that things just work and build upon them. I don't need to see the gory details of everything every time.
I call this method...
(...Leonard's explanation continues...)
The annotated Polaroid acts as an abstraction interface.
Only things that I trust go in the Polaroids.
Because I trust these things, I can just build upon that knowledge without being bothered by the implementation details.
If every time I use something I first need to stop and think about how that something is implemented, my time would run out before getting anything done.
The implementation details are important in context. Looking in, drilling down, they make sense. Looking up, abstracting away, they don't.
(Leonard stays quiet and after a while he asks...)
"Yes, yes, Sammy Jankis, I know" interrupts Teddy.
"No, no, I mean, did I ever tell you about Ireneo Funes?" Leonard continues, "I read about him somewhere"
(Leonard tells the story of Funes)
It turns out this Funes person could remember everything in excruciating detail. He suffered in some ways from the opposite of my condition.
He was so obsessed with the details that he was almost incapable of creating new thoughts.
Contemplation prevented composition.
"To think is to forget the differences, to generalize, to abstract" or so I read somewhere.
(Leonard trails off his face blank once again)
Teddy seizes this lapse of distraction to slip another Polaroid into Leonard's pocket. The photo shows:
if (A) then (B) = (not A) or (B)
"Ok, so anyway about my expression, I need you to help me reduce this" Teddy says as he hands Leonard a piece of paper.
The paper reads:
(C) and (not D) and (not J) (not C) and (D) and (not J) (not C) and (not D) and (J) if A and ((B or ((C or D) and E))) then F if A and (not (B or ((C or D) and E))) and (G and ((H and (not I)) or (not H))) then F if A and (not (B or ((C or D) and E))) and (not (G and ((H and (not I)) or (not H)))) and H then L if A and (not (B or ((C or D) and E))) and (not (G and ((H and (not I)) or (not H)))) and (not H) then M if (not A) and J then K if (not A) and (not J) and (C or D) and ((not E) and (not B)) then K if (not A) and (not J) and (C or D) and (not ((not E) and (not B))) then F if (not A) and (not J) and (not (C or D)) then F
Leonard looks at the paper confused but when he looks up Teddy is gone.
(Some time later)
"Hey Lenny, it's me, Teddy!"
Leonard looks blankly at the stranger for what seems like minutes. He then pulls a bunch of yellowed Polaroids from his pockets and checks them out. Eventually he finds a photo of this man, the photo reads "Teddy, check your left pocket".
Leonard checks his left pocket and pulls out a crumpled piece of paper, he reads it carefully and then addresses the stranger in front him.
"Hi, Teddy", he says, "you must be here about your expression..."
"Yes, yes!" says Teddy excitedly, "did you manage to reduce it?!?!"
"Oh yes I did", answers Leonard.
"Ok, what is it, Leonard, this is very important, what's the answer?" asks Teddy almost losing his patience.
Leonard looks blank for a second and then answers...
"I can't remember"