Multi-Value Logic And A Topological Problem

[SaxonViolence]

I have several books on Symbolic Logic, and in contrast to my General ineptness at Math, I understand the Logic books fairly well when they're fresh in my mind. {Someday I'll have to study Boolean Algebra.....}

 

They only mentioned "Multi-Valued" Logic Systems tantalizingly. When I learned to Google a few years ago, and happened to search for Multi-Valued Logics--I was quite disappointed to find that by "Multi-Value" they were simply "Fuzzy Logic" systems trying to make room for Values like, "Maybe"; "Probably" or "Probably Not".

 

What I had envisioned was more like--Well, "Rock; Paper; Scissors"--now that's a Multi-Value Logic System--though it is Trivial. "Paper" is not an Intermediate Position, but it is a perfectly valid state all on it's own.

 

I could go on to exhaustively state the same for the other two Values.

 

You could even create a Truth Table.

 

R+P= P

 

P+S=S

 

S+R=R...

 

If you try to combine large statements, you'll quickly find that the Associative Property does not hold true--and the whole exercise seems rather pointless.

 

Let us use three more abstract properties.

 

Just for fun, lets call them "Red; Blue and Yellow" after the Artist's Palate.

 

Envision a Circle, with the three primary colors distributed 120 degrees apart from each other.

 

R+B= V(iolet)

 

B+Y= G(reen)

 

And,

 

Y+R= O(range)

 

Well, if you try to mix them much further, you soon get a muddy gray.

 

Let us stipulate that each Secondary Color is a "Gate" to a whole new Circle. Each Circle has three exits and three entrances. The entrances and exits are such that you can never exit directly back to the Circle that you just left.

 

We choose Violet. "Violet" is one entrance into Circle₂V. Circle₂V has two other entrances.

 

Lets call our entrances V'; V'' and V'''--later we may think of catchy names. {V' is the one that came from Circle₁.

 

V'+V''= "X"

 

V''+V'''= "Y"

 

V'''+V'= "Z"

 

Alright, the number of circles increases for two or three steps--three new Circles are invariably created.

 

But after the third or fourth movement, the Circles start to decrease in number--more than one Circle₃ or Circle₄ Feeds into the same Circle₅.

 

It should take exactly as many "Condensing Steps" to get Down to a Single Circle as it did to expand to the maximum number of Circles.

 

Now the Good Part: Circle_{indeterminate} That reunites the last three Circles, is identical to Circle₁.

 

The three Colors--Red, Blue and Yellow are the Entrances for the last three "Condensing Circles".

 

And no Circle should be preferential to any other Circle. Select any Circle, and it should be possible--using optimum strategy--to Circle back to the starting Circle in the same Number of moves.

 

Questions:

 

Is such a Series of Interconnected Circles Topologically possible without contradiction?

 

What is the Minimum Number of "Expansions" that the Circles must go through before they Start Condensing?

 

Is there a Maximum Number of "Expansions" that can be used, and still meet the Criteria?

 

Can one continue moving through the Network indefinitely, while avoiding a given Circle--or will one eventually be forced to go to the "Prohibited Circle"?

 

Assign each and every Entrance and Exit a Unique Identifier--How would one construct a "Truth Table" for the Network as a Whole?

 

Is this the least bit interesting to anyone but me?

 

Saxon Violence

Edited April 17, 2012 by SaxonViolence

[maddog]

I have several books on Symbolic Logic, and in contrast to my General ineptness at Math, I understand the Logic books fairly well when they're fresh in my mind. {Someday I'll have to study Boolean Algebra.....}

Yeah, Boolean Algebra is quite fun and the basis of all Computer technology today.

 

They only mentioned "Multi-Valued" Logic Systems tantalizingly. When I learned to Google a few years ago, and happened to search for Multi-Valued Logics--I was quite disappointed to find that by "Multi-Value" they were simply "Fuzzy Logic" systems trying to make room for Values like, "Maybe"; "Probably" or "Probably Not".

What I had envisioned was more like--Well, "Rock; Paper; Scissors"--now that's a Multi-Value Logic System--though it is Trivial. "Paper" is not an Intermediate Position, but it is a perfectly valid state all on it's own. I could go on to exhaustively state the same for the other two Values.

You could even create a Truth Table. If you try to combine large statements, you'll quickly find that the Associative Property does not hold true--and the whole exercise seems rather pointless.

In my early years in college, I had professor in a machine language course. His interest was "three-valued logic". He even chaired a conference a

few times on this. It was clear that Fuzzy Logic is a spin-off from this line of thinking (three states: True, False, Unknown = 1, 0, ~).

Actual Hardware today is often using this concept where the '?' state is unstable to know what it is. This is used in all digital forms of communication

where arbitration and deadlocking are issues considered.

 

Let us use three more abstract properties.

Just for fun, lets call them "Red; Blue and Yellow" after the Artist's Palate.

Envision a Circle, with the three primary colors distributed 120 degrees apart from each other.

R+B= V(iolet)

B+Y= G(reen)

Y+R= O(range)

Well, if you try to mix them much further, you soon get a muddy gray.

Questions: Is such a Series of Interconnected Circles Topologically possible without contradiction?

Is this the least bit interesting to anyone but me?

Caution here: Boolean Algebra is discrete. There no notion of linear combination of to get to a resultant.

Logically it is or it isn't. Or in a 3-Algebra it would one of the three yet not a little from here or there.

Because of this what you are describing though not invalid is not discrete in any way. So this could not be

considered useful for a state machine or any kind of multiple valued logic per se.

 

What it appears you are describing is like continuum where mixing can occur. This occurs once you start

using addition to generate other colors not what you started with. Thus either the set of values is not "closed

under the operation (+)" [appears not] or you have not indicated all of the values. In this way, I do not think of

this topologically unless there was more definition on what '+' means here. Addition is a binary operator as

such, it need two things to operate on. You must define from what set these elements are member of.

 

This is a good start. Keep exploring.

 

maddog