Hacker News Reader

Comment by Sharlin | Hacker News Reader

Boole didn’t introduce propositional logic; what he did was come up with an algebra that encodes propositional logic.

Abstract algebra was a new snd developing thing back then, the idea that you can generalize from numbers and addition and multiplication to other structures that have something like numbers and addition and multiplication.

Boole found that if you take the two-element set {0, 1} and choose saturating addition as the addition-like operation and normal multiplication as the multiplication-like operation, you get an algebra (specifically a ring) that is isomorphic to propositional logic with its AND and OR operations.

So the idea that the number 1 can represent true and the number 0 false was Boole’s insight and the foundation of modern digital circuits.

View on HN

Boole figured out Boolean Algebra, but nobody paid attention until Claude Shannon realized we could use Boolean Algebra to design digital circuits. In his masters thesis.

Very few masters theses have changed the world, but Claude Shannon's was one of them.

Note: That's not actually a ring. If you use XOR as addition as it's a ring. You could say it's a semiring, but that's probably not the most useful description of it. A more-fitting structure would be an appropriate sort of distributive lattice. Or really... a Boolean algebra! Except I also have to point out that you only included AND and OR; you left out NOT, which is also an important part of the structure.

Yes, good points!

Interesting. Thanks for the comment.

Unfortunately, I'm having trouble understanding though. The furthest I got with math is calculus. I'd love to hear a more ELI5 description of what you said. Something that doesn't require prior knowledge of abstract algebra. (I recognize that's a big/difficult ask. No obligation or anything ofc.)

Aristotle described propositional logic.

No. He described term logic.

And actually Boole is only dealing with two distinguishable states. Any two distinguishable physical states can be used to define binary 1 and 0. But in general, any N distinguishable states can be used to form an algebra. N=2 is just the simplest, most elegant, and what most real computers are based on. I've heard rumors that an N=3 computer exists, aka 'ternary' (EDITed) but I have my doubts. On the math side you can turn the integers into a discreet N-part thing with modulus.

The word is ternary, and Ternary Computer is a Wikipedia page you could read if you're interested.

No, Boole was dealing with probabilities. The first half of his investigation on the laws of thought is all ones and zeroes, but the second half admits any value in between.

Ternary based computers are far from a rumor.

They are flipping cool tbh. I wish more was done with the idea. Balanced ternary is beautiful.

>I've heard rumors that an N=3 computer exists, aka 'ternary' (EDITed) but I have my doubts

Rumours? This is the internet. Rumour to research is a few clicks away.

They did exist: https://en.wikipedia.org/wiki/Setun

Reply on news.ycombinator.com