Is Boundary Logic interesting?

Discussion in 'Physics & Math' started by arfa brane, Aug 21, 2018.

  1. arfa brane call me arf Valued Senior Member

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    It's surprising how a simple enough representation of a boundary which is semantic, i.e. gives it meaning in the exact same sense as Boolean AND, OR, or NOT have meaning, then seems to generate a whole family of formal logical systems, only some of which are Boolean rings.

    One thing I noticed about nested boundaries is that any nesting 'collapses' to either a single or no boundary (i.e. the "non-symbolic" void), depending on whether there is an odd or even number of ( ) symbols.

    Since ((( ))) = ( ), then (( )) = . All the odd nestings are equivalent to ( ), the even ones are void. So you can of course use 1 = ( ), 0 =

     
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  3. origin In a democracy you deserve the leaders you elect. Valued Senior Member

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    No. (just my opinion)
     
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  5. Yazata Valued Senior Member

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    I know next to nothing about it, apart from the fact it exists. I've never studied it.

    But I'd say yes, it's interesting.

    It was apparently originated (like so many other cool ideas) by one of my favorite philosophers C. S. Peirce. Frege invented his own version.

    These remarks (from Bricken below) sum up its motivation:

    "What is the simplest mathematics? In finding a simpler foundation for mathematics we also find a variety of simpler and stronger computational tools, algorithms and architectures."

    There are only a handful of BL rules (and none of the traditional logical operators like and, or or not) and applying the rules typically makes expressions simpler, not more complex. But apparently despite all that, it's equivalent in logical power to (but not isomorphic with) Boolean algebra.

    Here's a relatively simple and straightforward explanation of what it is:

    http://iconicmath.com/mypdfs/bl-commonsense.040717.pdf

    A more elaborate but still rather easy to understand 32 page introduction here:

    http://www.wbricken.com/pdfs/01bm/01math/01bl-about/01bl-from-begin-sbin.pdf

    Many papers from short and simple to more advanced, including the introduction immediately above can be found here:

    http://iconicmath.com/logic/boundary/

    He says that while boundary logic seems alien and incomprehensible at first, it becomes easy and second-nature with practice. Part of what makes it seem unfamiliar at first is that it doesn't correspond to how our languages are structured, the way conventional propositional and predicate logics, and their extensions are. But it's structured in ways that's easier for computers to implement, holding out the potential of better algorithms and more efficient computation.
     
    Last edited: Aug 21, 2018
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  7. iceaura Valued Senior Member

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    This guy played an often underestimated role - his influence extends well beyond those who, as Bricken does, cite him specifically and adopt his vocabulary etc.
    https://en.wikipedia.org/wiki/G._Spencer-Brown
    http://www.enolagaia.com/GSB.html
     
    Last edited: Aug 22, 2018
  8. arfa brane call me arf Valued Senior Member

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    5,711
    There are only two semantic rules in BL:

    1. ( ) ( ) = ( )
    2. (( )) =

    But these are easily combined into a single equality: ( ) ( ) = ((( ))) = ( ). I have to say I find the names given to 1. and 2., namely "calling" and "crossing" a bit semantically perplexing, but of course it doesn't really matter.

    Notice how introducing additional symbols such that ( ) = 1 immediately gives (1) = 0.
    And you have 1 = (0) = ((1)) = (((0))) = . . .

    Rule 1. looks like Boolean OR, but rule 2. doesn't seem to have a direct Boolean counterpart until you define 1 = (0), then you have ((0)) = (1) = 0, so ( ) now looks like Boolean NOT (and of course with OR and NOT relations you have the AND relation: ((A) (B)) ~ A AND B). So it certainly looks like BL is more fundamental (foundational?) than Boolean logic. Note how ( ) denotes the presence of structure--a boundary--and its complement is the absence of structure.

    Ed: mistake corrected in BL expression for AND.
     
    Last edited: Aug 22, 2018
  9. iceaura Valued Senior Member

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    The names are from Laws Of Form, G. Spencer-Brown, which features a slightly different and imho visually easier notation than those links, as well as a different and I think more intuitively compatible introductory approach. They fit naturally, there. Bricken has some clear exposition of some of the advantages of the logic, and has carried on into greater mathematical sophistication, but as an introduction, getting an initial handle on things, I'd prefer LoF.
     
  10. Confused2 Registered Senior Member

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    499
    I am interested but I can't see anything in what I see. The links Iceaura gives (thank you) seem as much out of Alice in Wonderland as the starting point. This could be a notation I could use, perhaps need, but I can't even get started on it. Any help would be appreciated.
     
  11. arfa brane call me arf Valued Senior Member

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    5,711
    Do you know what a Boolean ring is?

    One thing about BL and its notation is that it doesn't seem to have a 'natural' map to any other kind of mathematical object-with-a-symmetry, such as a group or a semigroup. That's according to Bricken in this pdf: http://iconicmath.com/mypdfs/bl-the-difference.080830.pdf

    So perhaps one way to think about the logic in boundary logic is that it isn't symmetrical, although you can give it some by using more symbols. Like, it's a broken kind of language and giving it some more structure fixes it. With 0 and 1 defined in a BL context, you should be able to do arithmetic.
     
    Last edited: Aug 23, 2018
  12. arfa brane call me arf Valued Senior Member

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    5,711
    Some geometric wanderings:

    In three dimensions, a solid object has a two-dimensional boundary; because of gravity, fluids with mass like water or mercury, 'fill' a container and form a flat boundary--the surface of the liquid.

    In two dimensions, areas have one-dimensional boundaries, and in one dimension intervals have zero-dimensional boundaries.

    So 'in' zero dimensions (if that even makes any sense), boundaries are -1 dimensional . . . Here, the logic seems to fail. Physical objects have positive boundaries, distances are positive, time "advances" positively, etc.

    In BL, the notion of (a) void is like a place with no structure--a zero-dimensional point.
     
    Last edited: Aug 24, 2018
  13. Confused2 Registered Senior Member

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    499
    No, I don'y know what a boolean ring is - wiki doesn't seem to help. I don't think I know enough to get to the starting blocks on this but thank you for trying to help - much appreciated.
     
  14. arfa brane call me arf Valued Senior Member

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    5,711
    Well, you surely know what Boolean algebra is.

    This is precisely the algebra used in logic circuits, with OR, AND, NOR, NAND, NOT, XOR types of gates. It's called binary logic because all the gates have two input variables, and every variable is either 1 or 0, 'technically'.
     
  15. arfa brane call me arf Valued Senior Member

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    5,711
    Ok how's this.

    I define a space with nothing in it and call it the void, or just void.
    I introduce a notional boundary which means I also introduce the notion of "inside" along with its complement "outside". The single boundary "condition" means I have defined what the semantics of difference are, in this space.

    Then I show that, by arbitrarily labelling a boundary, or some composition of boundary symbols with the likes of A, B, 1, 0, ... I introduce some algebraic structure in the otherwise blank landscape of arbitrary forms all composed entirely with the symbol "( )".

    So if I choose to define 1 = ( ) (a let's say, pattern-naming, or maybe a rewriting rule), then it follows that (1) = (( )). By the laws of form, the right hand side is the symbol for void (which is, no symbol), but I'm ok with if 1 = ( ), then 0 = (( )).

    That choice of a pair of symbols, 0,1 as a representation of the difference between inside and outside a boundary, along with the rules that, a boundary inside another boundary is void, and a boundary outside another boundary is one boundary, I get Boolean logic (algebra) and its logical structure, a Boolean ring.
     
    Last edited: Aug 25, 2018
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  16. iceaura Valued Senior Member

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    A major advantage of the notation and algebra is that you don't have to do that. It's a hassle, like Roman Numerals, and the cleaner, simpler mark/no mark notation what you want to use.
    Supposedly.
     
  17. Confused2 Registered Senior Member

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    Yay, I think I'm starting to get it - thanks arfa. Quiet contemplation will follow while I try to work out if it's any use in my SQL world of
    A and B and not ((C or D) or (E and F)). Very interesting though - nice one.
     
  18. arfa brane call me arf Valued Senior Member

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    5,711
    Well, notice that I can do this, after just identifying 1 = ( ).

    1 1 = 1, by rule 1.
    (1) = (( )) by rule 2.

    But I've written something on the second line that seems to say 1 is inside 1. Whether I can fix this by now saying 0 = (( )), so the difference is now marked by that between 1 and 0, is probably philosophical.

    You also easily derive that (1 1) = (1), so that (( )1) = (1). This is called occlusion in BL. I say that's just another name for the difference between inside and outside.

    If you start with the difference between no and one boundary, then that introduces another difference, between inside and outside the boundary, then the next difference is between the spatial arrangement of a pair of boundaries. One of these goes back to where you started . .
     
    Last edited: Aug 25, 2018
  19. arfa brane call me arf Valued Senior Member

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    5,711
    Ah, so I now prove that what BL calls occlusion is a valid consequence of rules 1. and 2.:

    Since any expression (called a form) in BL which is valid (in the same sense implication is valid) is equal to either ( ), or is void, then if I show that (0 0) = (0) is equivalent to (( )0) = (0), then given what's in my last post, it must hold for (( )A) if A is a valid form (or parenthetical expression).

    Using the rules you can derive two more kinds of equivalence, involution and what's called pervasion. The first is already here, since ((A)) = A, but the second is (A (A B)) = (A (B)).

    Rewriting, we have (1(1 B)) = (1 (B)) and (0(0 B)) = (0(B)).
     
  20. arfa brane call me arf Valued Senior Member

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    5,711
    I got this.

    You take a valid Boolean expression and rewrite it in BL. So you get the BL equivalences because you can show they hold for say, implication.

    If A then B is equivalent to not A or B, in BL: (A) B.

    So then:

    If A implies B then A or B implies B = If A or B implies B then A implies B, is the Boolean form of BL's (A B) B = (A) B.
     
  21. arfa brane call me arf Valued Senior Member

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    5,711
    Then since (A B) B = (A) B, it follows that (B) B = ( ) B. (not B or B = 1 or B).
    Which is just rules 1. and 2. with two and with one variables, resp.
     
  22. iceaura Valued Senior Member

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    Yes, you can do this.

    You don't want to continue, though. Now that you have confirmed for yourself that the new notation can handle all the same stuff Predicate Logic and Boolean algebra and so forth handles (both Bricken and Brown prove this, formally), you no longer necessarily need them.

    As in learning as new language, you don't want to be always translating. You just want to speak, in the new language.
    Otherwise, you are giving up the power of the notation, and the insight of its algebra.
     
  23. parmalee peripatetic artisan Valued Senior Member

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    This all seems awfully Batesonian--I know Bateson was influenced by Peirce, but I guess I just never realized how much. (I'm thinking particularly here of his analysis of play, but really, it's all over the place.) . In GS Brown, where do "calling" and crossing" come from--surely that was "borrowed" from or inspired by something?
     

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