Wednesday, June 01, 2005

Three Valued Logic

In logic there is the usual bi-valued "logic" where everything is supposed to be either true or false, and there are three valued logics, where I did a quick search this morning and saw stuff like true, false, or unknown, among other possibilities, like using numbers where 1 is true, 1/2 is in-between, or unknown and 0 is false.

I'll make the case that three valued logic is a necessity.

I suggest a three valued logic, where a statement has a logical value of 1, 0, or -1, where 1 is true, -1 is false, and negatably true, while 0 is unresolvable.

For instance

1+1 equals 2

is a logical statement with a truth value of 1.

1+1 equals 3

is an illogical statement with a truth value of -1.

Notice it is negatable, to

1+1 does not equal 3

with a truth value of 1.

So the negative of a false statement i.e. a statement with a truth value of -1, is a true statement with a truth value of 1.

So what about 0?

Consider the statement:

If a=b and c=d then gooses lay eggs.

That statement cannot be said to be false and negatable, as the negative is, maybe,

If a does not equal b and c does not equal d then gooses do not lay eggs.

It remains nonsensical, despite negations where there are different ways you can negate.

So it has a truth value of 0.

Notice that in bi-valued systems the nonsensical statement can give you problems. Possibly some would just say it's false, but then again, it just doesn't make sense.

As "doesn't make sense" is not very rigorous, I like the term "malformed".

The statement is illogical as it is malformed.

The correct form in abstract is,

If a=b and b=c then a = c.

That has a truth value of 1.

Now let's consider a "paradox".

"This sentence is false."

Is an example of what I've seen called the "Liar's Paradox".

It has a truth value in the system I've outlined of -1.

So why?

"This sentence is not false."

is the negation, with a truth value of 1.

The sentence is negatable, so it has a truth value of -1.

No paradox.

That may have happened so fast that you may not realize how quickly this system just processed a "paradox" that has challenged logicians for quite some time, so here's a link:

http://www.iep.utm.edu/p/par-liar.htm

Now I emphasize three values in a system that can handle the "Liar Paradox" with only two, but that's because of malformed statements, which force you to have a formedness value.

So I see logic as about formedness, where a logical statement connects two truths, and it has a truth value of 1.

A false statement has a truth value of -1, and is negatable to a well-formed statement.

A statement that is not true nor can it be negated to a true statement is malformed with a truth value of 0.

If you disagree and believe that a bi-valued system can work, then you need to handle bizarre statements like:

"If the world were made of chocolate, and you were my friend, then robots would bleep like sheep."

Now then, in a two valued system, make sense of that statement!

James Harris