Is Boundary Logic interesting?

So does LoF.
You're still fifty years back, here.

They abandon BL for Boolean notation to pursue imaginary truth values?
Seems unlikely.

 

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Along with most logicians on the planet it seems.
Why, if LoF is the "answer", is it not being used? Why isn't it included in university courses?

Or have those questions been asked and answered?

Your response here indicates that you don't understand what Bricken or Kauffman "do".

Kauffman demonstrates that BL is "just" the closure of TL, as a knot is the closure of a braid.
But you know, "so what"?

 

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LoF is fifty years out of date, so the real question would be why the logic and notational innovations of BL are so missing.
A couple of speculations appear in my posts, as repeated and highlighted above. It is an interesting question, and I have never seen an informed and sufficient answer.

Of course not. I haven't been keeping up. But your apparent claim is that they abandoned BL notation when working with four valued logic - that seemed unlikely, to me. Did you intend to make that claim?

If they did, that would make a good starting point for considering why BL notation and so forth have not become more common and more often employed.

 

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There is more than one way to represent an algebra, as I'm sure you know.

But I have a problem specifically with the notion of an imaginary number being taken to the notion of an imaginary logical value. As I say, division is not defined in logic; division defines the relation between i and -i. Another problem is that meet (∧) isn't really like scalar multiplication.

To change the sign of a number, multiply by -1. In Boolean logic, to change the 'sign' you 'multiply' by ¬. It doesn't look the same, you see.
Further, would you need to restrict everything to the Gaussian integers modulo 2? That is, you want a map from the set {1, -1, i, -i} to 'imaginary' Booleans { ¬¬, ¬, √¬, ¬√¬}?

 

So don't do that. LoF does not do that, and there seems to be no obvious motive or advantage.

 

I think BL, or at least Spencer Brown's version of it, is interesting because it's connected to Temperley-Lieb algebra.

As Kauffman shows in the online book Knots and Applications:

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The arrow in the lower diagram is drawn in the only place it means anything. That is, it identifies where two capforms are joined together below the line.
The rule is that the identified boundary gets lifted above the arrow, and the arrow stays where it is so it identifies the next join/lift.

 

I'll add the parenthetical expression for the capforms above the line in the lower diagram: (( )) ( ) ((( ))).
This doesn't need an arrow anywhere because there's a topological invariant. The arrow represents a convenient symbol (or icon if you like), which isn't really there.

However, if you put the algebra in a computational context, the arrow looks like a sort-of instruction pointer.

 

If anyone's still interested you can try this exercise.

Start with the two upper diagrams, the one on the right is a pair of disconnected halves of the curve on the left.
The lower diagram is topologically equivalent to the reconnected halves, the reconnection is the identity element in \( TL_6 \).

The exercise is to redraw the lower curve so the disconnected halves form an element of \( TL_6 \) which is rank 0. That is, the disconnected halves in the upper diagram are inverted inward, and their (re-)connections are inverted outward.

 

So here's another question: how many ways can you close the identity in TLn (this is the diagram monoid, it's "unparameterised") with rank-0 elements of TLn?

What if n is odd? How does that coincide with Jordan's curve theorem? (answer, it doesn't, n is even because all rank-o elements are even, a consequence of the geometric fact that a straight line intersects a closed curve an even number of times if it passes completely through it and is not tangent anywhere).

Moreover, all the rank-0 elements are composed of pairs of "simple" loops, for example in \( TL_6 \) these are:

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​

These are all drawn below the six points, but if you take pairs you can alter this. They also represent BL forms; the first one is ( ) ( ) ( ).

 

Spencer Brown's Laws of Form "live" on a boundary which is artificial, it's just an identity in TLn, n > 0.

TL1 is not very interesting, it has one element, the identity as a vertical line between two points. This is topologically a single point; remember you cut a closed figure with a straight line, and "artificially" drag the halves apart, perhaps along with a rule that you do it continuously, no disconnections allowed, say.

So what looks like a pair of lines with points intersecting a closed curve is really two sides of the same line, the identity as a set of non-intersecting lines is "imaginary".

The two rules of BL in parenthetical form, ( ) ( ), and (( )) (equality is based on connectedness here), live in TLn for even n, hence you can find them in TL2 where the form (( )) is the trace closure of the only other element.

TLn is a diagram monoid, there's a nice introductory paper here, along with a sample:

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Oops I posted the wrong link. It should be this one https://digitalcommons.macalester.edu/cgi/viewcontent.cgi?article=1011&context=mathcs_honors

 

Just some more ideas that follow on from the OP, about what an algebra is. Boundary logic in terms of expressions using only "( )" as a single object, gets more interesting when you start replacing expressions or parts of them with different symbols.

There are plenty of sources quoting the idea that BL is a subset (a subalgebra in fact), of TL. But TL is a category of algebras over sets of points, TLn is that subset of TL that has n points going to n points. More general "algebras" mean n going to m points, say.

Braids have crossings in them, the simple idea is that moving towards or away from a crossing means an area, call this A, is to your left if you move towards the crossing along the undercrossing line. So A, and its inverse 1/A, are identifications of this movement. This means you can decompose a braid into vectors with coefficients in A, where the vectors are elements of the algebra of TL2 (after "going to" one less dimension, i.e 2, than a braid is in). So that for instance, \( TL_4 = TL_2 {\otimes}^2 \).

QM has been "recast" in terms of Category Theory, TL is a hot topic but doesn't seem to have a connection to classical logic although BL does (albeit, not all by itself).

The Yang-Baxter equation is something you can describe with all this diagram logic too.

 

No, it doesn't look interesting, in any way.
EB

 

Well, I'm glad you've managed to sort that out. How does it feel?

 

That in brackets must be performed first, the second bracket second, and the third bracket third. Thus (1) identifies the individual as having to be dealt with before all others.

 

Not like half a brain.
Hey, don't complain, I just answered the question you asked.
I'm sure it must be interesting for some people, like the rubik's cube is, but come on, what does it even mean in real life out there?
Your best hope is that someday someone finds it applies straightaway to quantum physics or something. Serendipity, it's called.
I would encourage you to pursue. Someone has to and it won't be me.
EB

 

Not really, but thanks for trying.

What does an algebraic object or a category of objects have to do with anything? Why do certain people bother to summon up the interest you think?

My best hope? Since it's already applicable to QM, albeit not as a classical logic, that someday arrived a while ago. But no matter, since it wasn't you who applied it.

 

Your question was "Is Boundary Logic interesting?" I replied.

So you can't explain yourself.

That doesn't explain how it would be useful.
Still, never mind, we're going nowhere fast or somewhere but much too slowly for me. You obviously don't like to provide informative answers, not to me at the very least, so there is no point going on about this.
EB

 

And that reply indicates that you think BL doesn't "look" interesting.
I have to ask, is anything interesting to you? You seem to want to discuss logic, but you also seem to show little interest here at sciforms.

What does explaining myself have to do with anything? Perhaps I could start at the beginning: I was born, I grew up and went to school . . .

Quantum Mechanics doesn't do that either. "Useful" is a subjective thing.

 

I'm interested in logic as per our sense of logic, "sense" to be taken like in "sense of hearing", for example. I take our sense of logic to be one of the most effective of our capabilities as human beings and therefore I would assume that logic is useful. Better logic would mean more useful. That's what is interesting to me.

Explaining yourself is the only way you have to show you're rational and the only way you have to convince rational people of whatever you may want to claim.

Sure, everything we know is subjective. Interesting is subjective, no? Usefulness is just as subjective as logic is, or as the whole material world is. Me, I take usefulness to be a very effective guide to understand things.
EB