A Totally Abstract Multi-Valued Logic (?)

[SaxonViolence]

I have several basic books on Symbolic Logic.

 

Scattered and Arcane references to "Multi-Valued Logics" are.

 

Traditional Logic only has two states "T" and "F".

 

In Boolean Algebra that becomes "1" and "0" as in Binary.

 

Basically "1" trumps "0".

 

0+1=1

 

On the other hand,

 

0 X 1= 0

 

La-La-Lah!

 

I was really disappointed when I finally got to Examine a Multi-Valued Logic and found that it was nothing but Fuzzy Logic in Drag.

 

I played with Idealized Colors for a good long while, searching for an Abstract Multi-Valued System—and I haven't given up...

 

But just the other day I realized that the Children's game of "Rock, Paper and Scissors" is an Abstract Multi-Valued Logic.

 

R+P=P

 

P+S=S

 

S+R=R

 

What is R+P+S=?

 

In our System, that is a nonsensical statement.

 

One cannot directly Compare Multiple Terms and neither The Associative or Distributive Properties apply.

 

Get away from "rock, paper and scissors"...

 

Think of "R" "P" & "S" as purely abstract states of a 3-Value Logic...

 

Just like "T" and "F" can be thought of as purely abstract operators in 2-Value Logic.

 

Now 2 Value Logic seems to have a great deal of correspondence to the Real World.

 

{Yes! I Acknowledge and Respect the Significant Minority who don't believe in a "Real World"...}

 

Whereas my 3-Value System seems to represent precisely nothing except abstract states—or perhaps those Sub-Atomic Particles that have 1/3 Spin...

 

But can we develop the 3-Value System any Farther? Discover any Interesting Theorems?

 

Now you could, in principle, arrange any number of Elements like this into a Ring...

 

But lets not. So far as I can see, it would only be Repetitious.

 

But can anyone come up with any other True Multi-Valued Logic Systems?

 

 

Saxon Violence

Edited August 15, 2013 by SaxonViolence

[phillip1882]

But can we develop the 3-Value System any Farther? Discover any Interesting Theorems?

i suppose that depends on what your axioms are.

for example; let's make some.

the identity laws:

Rock & Rock <==> Rock

Paper & Paper <==> Paper

Scissor & Scissor <==> Scissor

Rock or Rock <==> Rock

Paper or Paper <==> Paper

Scissor or Scissor <==> Scissor

 

the commutative laws:

Paper & Rock <==> Rock & Paper

Scissor & Paper <==> Paper & Scissor

Rock & Scissor <==> Scissor & Rock

Paper or Scissor <==> Scissor or Paper

Rock or Paper <==> Paper or Rock

Rock or Scissor <==> Scissor or Rock

 

the negation laws:

~Rock <==> Paper or Scissor

~Scissor <==> Rock or Paper

~Paper <==> Rock or Scissor

~Rock & ~Paper <==> Scissor

~Paper & ~Scissor <==> Rock

~Scissor & ~Rock <==> Paper

 

the double negation laws:

~~Rock <==> Rock

~~Paper <==> Paper

~~Scissor <==> Scissor

 

the cycle laws:

Rock & Paper => Paper

Paper & Scissor => Scissor

Scissor & Rock => Rock

 

 

 

now we can derive some observations.

since ~Rock <==> (Paper or Scissor)

and ~~Rock <==> Rock

we can substitute and find:

~(Paper or Scissor) <==> Rock

and of course:

~(Rock or Paper) <==> Scissor

~(Rock or Scissor) <==> Paper

then we also have:

~Rock & ~Paper <==> ~(Rock or Paper)

~Paper & ~Scissor <==> ~(Paper or Scissor)

~Scissor & ~Rock <==> ~(Rock or Scissor)

further more:

(Paper or Scissor) & (Rock or Scissor) <==> Scissor

(Rock or Scissor) & (Rock or Paper) <==> Rock

(Rock or Paper) & (Scissor or Paper) <==> Paper

as well as:

(~Paper & ~Rock) or (~Paper & ~Scissor) <==> ~Paper

(~Paper & ~Rock) or (~Rock & ~Scissor) <==> ~Rock

(~Rock & ~Scissor) or (~Paper & ~Scissor) <==> ~Scissor

can you find any more interesting ones?

[CraigD]

If you consider any system with values and operators to me a multi-valued logic system, you might find some examples from computer programming languages illuminating.

 

In the now-obscure MIIS programming language, for example, the “logical” & (and) and ! (or) binary operators are actually

“first collating string” and “last collating sting” operators, and the usual comparison operators, such as = (equal), > (greater than), < (less than), return either the lowest collating empty string or a high-collating ASCII character 127. So the expression (A=1)&(B=2)!(C=3) (where A, B and C are variables) returns either the empty string or character 127, expressions like “ALICE”!”BOB”&”CAROL” return “BOB”, which is considered “true” (that is, the command

IF “ALICE”!”BOB”&”CAROL” DO SOMETHING

invokes the subroutine SOMETHING) but not “truth” (that is not ASCII character 127). This allows very much programs that have far fewer source code characters than most procedural languages – for example, the one-line/command program

SET A=B=1&”JOE”!(B=2&”KAREN”)!”LARY”

perform the same as

IF B=1 SET A=”JOE”

ELSE IF B=2 SET A=”KAREN”

ELSE SET A=”LARY”

 

As MIIS was used on multi-user computers with little available memory common in the 1960s-80s, such terseness was valuable.

 

Another, slightly (or perhaps not) less obscure programming language that stirs interesting multi-value logic system thoughts is Sinclair BASIC. Although it included the traditional true/false boolean-valued operators and commands such as AND, OR, and IF...THEN, it also allowed flow control without them, by allowing any numeric value expression to be a line reference, and treating undefined line references as referring to the next highest numbered defined line. For example, the program

1000 GOTO 2000+A-B

1999 PRINT “A is less than B” : GOTO 3000

2000 PRINT “A is equal to B” : GOTO 3000

2999 PRINT “A is greater than B”

3000 END

behaves, for values of A and B less than 1000, the same as

1000 IF A<B THEN PRINT “A is less than B” : GOTO 3000

1010 IF A>B THEN PRINT “A is greater than B” : GOTO 3000

1020 PRINT “A is equal to B”

3000 END

 

Programming languages like these suggest that a zero-valued logic system, or a practically infinite-valued one, are practically useful.

 

But just the other day I realized that the Children's game of "Rock, Paper and Scissors" is an Abstract Multi-Valued Logic.

R+P=P

P+S=S

S+R=R

 

What is R+P+S=?

In our System, that is a nonsensical statement.

If you define the + operator, as you have, as “the play that wins”, and further, define an implicit order of same-operator-precedence order of execution of left-to-right,

“Rock” + “Paper” + “Scissors”

which with explicit execution ordering is

(“Rock” + “Paper” ) + “Scissors”

evaluates sensically to

“Scissors”.

 

Change the order to right-to-left, and the explicit expression is

“Rock” + ( “Paper” + “Scissors” )

which evaluates to

“Rock”.

 

You can also treat the “the play that wins” operator as having more than 2 arguments (that is, rather than writing the algebra-like

“Rock” + “Paper” + “Scissors”

, write the LISP-like

(+, “Rock”, “Paper”, “Scissors”)

). One way to define this operator is as an unordered “plays not beaten” operator returning a variable length list of values, which gives a results table of

(+, “Rock”, “Rock”, “Rock”) = (“Rock”, “Rock”, “Rock”)

(+, “Rock”, “Rock”, “Paper”) = (“Paper”)

(+, “Rock”, “Rock”, “Scissors”) = (“Rock”, “Rock”)

(+, “Rock”, “Paper”, “Paper”) = (“Paper”, “Paper”)

(+, “Rock”, “Paper”, “Scissors”) = ( )

(+, “Rock”, “Scissors”, “Scissors”) = (“Rock”)

(+, “Paper”, “Paper”, “Paper”) = (“Paper”, “Paper”, “Paper”)

(+, “Paper”, “Paper”, “Scissors”) = (“Scissors”)

(+, “Paper”, “Scissors”, “Scissors”) = (“Scissors”, “Scissors”)

(+, “Scissors”, “Scissors”, “Scissors”) = (“Scissors” , “Scissors” , “Scissors”)

 

In philosophical logic, such systems, and non-2-value logic systems in general, can be considered to be ones in which the law of excluded middle is not in force.

 

One can make a philosophical arguments that the law of excluded middle is an artifact of reason, rather than of insentient nature – though such arguments turn paradoxical very quickly!

[C1ay]

But can we develop the 3-Value System any Farther? Discover any Interesting Theorems?

I believe so based on my own experiments with a 3 value system...

 

In the '70s I was very intrigued with the birth of electronic computing. I was building my own logic gates from collections of various transistors and studying how it all worked. Processors of the time were coming out and most were built entirely using NMOS transistors or from PMOS transistors. NMOS circuits usually assigned Ve+ to the value 1 and 0V as 0. This was typically referred to at the time as positive logic. PMOS circuits on the other hand usually assigned Ve- to the value 1 and 0V as 0 and this was referred to as negative logic.

 

As I played with these on my breadboard I wondered why you couldn't mix them and use a power supply that provided Ve+, 0 and Ve-. The system would still be digital, as opposed to analog, and the capacity of a byte would be exponentially increased with the application of a ternary system made possible by combining technologies. I was excited by the possibilities.

 

At the time I used a cassette tape recorder to save and load programs for my computer, a Sinclair ZX80. I did some math and found that a binary to ternary converter could pack about 25 times as much information on tape. I built a small proof of concept circuit to produce some ternary values and successfully recorded them on tape. I thought this would have a big impact on storage and transmissions. I envisioned future processors that used ternary logic for increased computing power and efficiency over their binary counterparts.

 

I went to speak to a patent attorney for some advice on how to proceed. He advised that electronic patents were some of the most expensive because of the requirements to look for prior works with an in depth patent search. The patent search he explained would be the most costly part of seeking any kind of related patent and quoted me an initial price of $500 just to look for any existing binary to ternary converters. It took me a few years actually to save up for this and when I finally got it done he found several patents dating back years. That let the wind out of my sails on my pursuit of what I called trinary logic at the time.

 

To this day I see no usage of any such technology. I still think it could be used in some areas like digital transmissions over copper. I think it could be useful in certain processors since it would allow ternary algorithms and operations that might be more efficient in some cases over their binary counterpart. It puzzles me sometimes why I see nothing that utilizes it.

[SaxonViolence]

I remember studying Fortran at Purdue in 1976—never mastered it.

 

But I remember the "GOTO" Statements and the danger of creating infinite loops if you weren't careful.

 

{Wasn't no biggie—after a few loops, the computer would spit it out with an "Error" statement...}

 

Had to study Visual Basic many years later and they told me that all the "GOTO" statements had been eliminated from Programs and there was something called "Structured Programming"...

 

Sigh...

 

This all needs beaucoup study.

 

Sometimes my mind is swift and sometimes I can't really follow complex lines of reasoning.

 

Thanks for the input.

 

 

Saxon Violence