The Logic Alphabet

April 17, 2006 11:06 PM

"In 1953, while working a hotel switchboard, a college graduate named Shea Zellweger began a journey of wonder and obsession that would eventually lead to the invention of a radically new notation for logic. From a basement in Ohio, guided literally by his dreams and his innate love of pattern, Zellweger developed an extraordinary visual system - called the “Logic Alphabet” - in which a group of specially designed letter-shapes can be manipulated like puzzles to reveal the geometrical patterns underpinning logic."
posted by vacapinta (30 comments total)

Fascinating! Great post.
posted by Operation Afterglow at 11:19 PM on April 17, 2006

Great article from an excellent site. Check out other Cabinet topics, they are all fascinatin'
posted by owalt1 at 11:45 PM on April 17, 2006

Of course theres good old &&, ||, == and !=...
posted by Artw at 12:05 AM on April 18, 2006

will this be on the predicate logic exam?
posted by wumpus at 12:13 AM on April 18, 2006

What is the point? Zellweger admits that logicians are not impressed with his work. I thought in looking through all the links I might see some examples of how the notation is useful in some application but found nothing. He sure is not using "geometry" in the usual mathematical sense.

Is he just saying that the set of all 16 binary boolean functions in standard logic form a category which in itself is a boolean lattice?
posted by MonkeySaltedNuts at 12:16 AM on April 18, 2006

What is the point? Zellweger admits that logicians are not impressed with his work.

Um...everybody needs a hobby? Especially those tending towards the idiot-savant / obsessed genius end of the mental spectrum?

(Disclaimer: incorrect use of "idiot-savant". Just has a bit of that flavour for me...)
posted by UbuRoivas at 12:30 AM on April 18, 2006

This is an interesting mind game, but like many such games it does not seem to be very useful.
posted by b1tr0t at 1:15 AM on April 18, 2006

like many such games it does not seem to be very useful.

or even profound if you have a post high school math education.
posted by MonkeySaltedNuts at 4:44 AM on April 18, 2006

Interesting. I think MonkeySaltedNuts has the truth of it, but the fact that the set of functions is a boolean lattice with these relationships seems to be overlooked, unless diagrammed in a manner like Zellweger's.

As an aside, this diagram is really startlingly similar to the Sephirot. I'm not saying there's any deliberate connection, it's just a cute coincidence.
posted by aeschenkarnos at 5:15 AM on April 18, 2006

is there a simple, concise explanation of what the symbols mean anywhere? i can see how this could be useful, but i can't (quickly) find a good explanation of how it's done in practice.

i see that pierce did something similar - he's a pretty interesting chap.
posted by andrew cooke at 5:20 AM on April 18, 2006

You have to take a certain view of a site that considers Tony Smith to be a "Useful Link".
posted by Wolfdog at 5:22 AM on April 18, 2006

I stopped reading the interview after the word computor.
posted by emelenjr at 5:28 AM on April 18, 2006

aeschenkarnos: there is a closer correspondance than at first blush, although I don't know enough qabala to tell you if it is coincidental or not. Many computer science concepts developed out of medieval mystical symbols, like Ramon Lull's combinatory wheels. (Sort of like how chemistry developed out of alchemy, not directly but in a rather orthogonal fashion.)

In some qabalistic systems, after some "work", Malkuth (the bottom one) is mysteriously raised up to the position of Da'ath, the "hidden" sephiroth, making a more symmetrical design.

The "perfected" tree forms two overlapping cubes just like in the lower left corner of this picture, one on top and one on bottom. The lower one in this system is called "small face" and the upper cube is called "vast face."

Knuth has explored the 16 boolean bivalent operators, too.
posted by sonofsamiam at 5:39 AM on April 18, 2006

Before I spend to much effort understanding this, one question: Is this real or kinda Timecube?
posted by Toecutter at 5:47 AM on April 18, 2006

Oh, it's real. It just may not be super-deep.
posted by sonofsamiam at 5:54 AM on April 18, 2006

It's "real", but the content is not much more than arranging real things in pretty patterns.
posted by Wolfdog at 5:56 AM on April 18, 2006

i was assuming that things like de morgan's transform turn out to be mirror reflections. is that not the case? because if that *is* the csae, it seems useful to me.
posted by andrew cooke at 6:01 AM on April 18, 2006

There's a guy in my neighborhood who lives on the street and sticks diagrams that look very much like that onto trash cans, using Scotch tape.

The rest of the time he's sitting barefooted in doorways, surrounded by bags and trash, talking to himself.

It's hard to tell with people like that; for all I know, his diagrams might be saying something of world-shaking consequence. But then again, he might just be nuts.
posted by Nicholas West at 6:19 AM on April 18, 2006

The lack of coherent explanation anywhere of what the symbols mean is annoying. After staring at them for a bit, though, I think I see what's going on. Each symbol represents one of the 16 possible two-place logical connectives. They're divided into groups by the number of assignments to A and B that make A * B true (where * is the connective). The "o" is never true; the "x" is always true. The flipstick shows the groupings.

The visiaul gimmick is that you can imagine these symbols overlaid on a 2x2 grid. A is along the top and B along the side. True is up and to the right. (Illustration here.) The connective is true whenever A and B are assigned a value that occupy a square in which the symbol has an endpoint. Thus, for example, the "d" has one endpoint, in the upper-right. That's true crossed with true, so the "d" represents logical AND.

All the rest of the system is just pretty pictures laying out the symmetries involved.
posted by grimmelm at 6:44 AM on April 18, 2006

If you read the interview with Zellweger, he seems to be an extremely intelligent fellow and one who obviously takes enormous pleasure in knowledge for its own sake.

I think what he's trying to do is discover the over-arching patterns present in the constant stream of the smaller everyday logical patterns that we interact with. Kind of like the other arcane studies going on at present that are trying to find the larger symmetrical patterns in chaos.
posted by Nicholas West at 7:02 AM on April 18, 2006

is there a simple, concise explanation of what the symbols mean anywhere?

Here, and for comparison, here . Note that these are in different order, and it's Zellweger's fault that this is so. He's disordered the truth tables to fit his flip-chart's order.

A "truth table", a 4-digit binary number, is a compressed way of representing the answers the operation gives to the inputs F & F, F & T, T & F, T & T, which is logically equivalent to the binary 00, 01, 10, 11. So, a truth table of 0010 uncompresses to:

A B Result
0 0 0
0 1 0
1 0 1
1 1 0

which is the "but not" operator. "If A, but not B".

A major flaw in Zellweger's argument is that most logical operators are actually unnecessary, because they can be expressed solely out of combinations of NANDs, or NORs, or NOTs and ANDs, or a number of other smaller subsets of this 16-member operator set.
posted by aeschenkarnos at 7:03 AM on April 18, 2006

That's what I was going to say....
posted by Nicholas West at 7:09 AM on April 18, 2006

Nifty post.
posted by Smedleyman at 9:11 AM on April 18, 2006

See also WFF'N'PROOF
posted by dhartung at 9:31 AM on April 18, 2006

Karnaugh mapping.
posted by user92371 at 10:14 AM on April 18, 2006

I think what is interesting is that, without the formal education or understanding of say aeschenkarnos above, he intuitively imagined it anyway, despite it being slightly off. I, for one, am impressed. Great post vacapinta!
posted by sluglicker at 10:25 AM on April 18, 2006

Good post. The "Let the Mirrors Do the Thinking" article seems to offer more explanation and examples.
posted by StarForce5 at 2:03 PM on April 18, 2006

This is great. I have a similar obsession myself.
posted by leibniz at 3:11 PM on April 18, 2006

If anyone's still reading... :)

A major flaw in Zellweger's argument is that most logical operators are actually unnecessary, because they can be expressed solely out of combinations of NANDs, or NORs, or NOTs and ANDs, or a number of other smaller subsets of this 16-member operator set.

From the article:

"The standard notations are analogous to the Roman numerals and mine is like the Arabic system, because there are 16 logical connectives in all, but most notations only use three or four, cobbling the others together from combinations of these few. In my notation all 16 are given equal prominence, and each has its own symbol, just as each of the units in the base-ten Arabic system has its own symbol."

And also:

"The Arabs (with help from India) settled on a consistent use of base ten, gave each number within this base its own symbol (the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and then imposed a macro-structure in which the units, tens, hundreds, and thousands were each given their own distinct collumns or places. With this system you can more easily do arithmetic and uncover the patterns inherent in numbers."

I think Zellweger's trying to say that restructuring the notation of logical operators will enhance people's ability to develop proficiency with them. If so, then we will be able to have more people looking at various problems with higher degrees of logical sophistication than we currently do. Also people, especially children, will be able to attain proficiency at greater rates than they currently do. This can only help innovation.

It is too bad the trained logicians here are saying that the system is not very deep. It would be nice to have something which enhances facility but is deep at the same time. But if we can only have one, it is still a good thing.
posted by halonine at 10:12 AM on April 19, 2006

It is too bad the trained logicians here are saying that the system is not very deep. It would be nice to have something which enhances facility but is deep at the same time. But if we can only have one, it is still a good thing.

Not at all. The system is both shallow and overly complicated. The "geometrical patterns underpinning logic" appear to have little mathematical utility. They make nice pictures, but they do not appear to be mathematically useful.

Unfortunately, many people who are not well trained in mathematics seem to think that math is supposed to be complicated and esoteric. If you take the time to learn a subset of math, you will discover that the opposite is true. In order for a mathematical system to be useful, it should be simple and efficient. It may take some time to learn the notation and basic theorems, but most of the math that non-mathematicians are exposed to already has very good notation.

The system proposed by Zellweger might make for interesting art installations. I hope no one attempts to confuse young people by attempting to convince them that Zellweger's system is easier or more useful than ordinary boolean logic.
posted by b1tr0t at 9:54 PM on April 19, 2006