Witt

This is a 4 valued system of decision for modal and classical propositional logic that includes the possible (<>) and necessary ([]) operators, as well as the usual bivalent operators: not (~), or (v), and (&), implies (->), and equivalence (<->).

The four truth values are: T = logical truth, t = factual truth, f = factual falsity, F = logical falsity.

Set:T=1, t=2, f=3, F=0.

Any fomula that has all T's is a theorem.
I.e. any formula that has all 1's is a theorem.

Table 1.

p

1
2
3
0


~p

0
3
2
1

not (~):
~1=0, ~2=3, ~3=2, ~0=1.

[]p

1
0
0
0

necessary ([]):
[]1=1, []2=0, []3=0, []0=0.

<>p

1
1
1
0

possible (<>):
<>1=1, <>2=1, <>3=1, <>0=0.

Table 2.

p v q

1 1 1
2 1 1
3 1 1
0 1 1
1 1 2
2 2 2
3 1 2
0 2 2
1 1 3
2 1 3
3 3 3
0 3 3
1 1 0
2 2 0
3 3 0
0 0 0

or (v):
1v1=1, 2v1=1, 3v1=1, 0v1=1,
1v2=1, 2v2=2, 3v2=1, 0v2=2,
1v3=1, 2v3=1, 3v3=3, 0v3=3,
1v0=1, 2v0=2, 3v0=0, 0v0=0.

p & q

1 1 1
2 2 1
3 3 1
0 0 1
1 2 2
2 2 2
3 0 2
0 0 2
1 3 3
2 0 3
3 3 3
0 0 3
1 0 0
2 0 0
3 0 0
0 0 0

and (&):
1&1=1, 2&1=2, 3&1=3, 0&1=0,
1&2=2, 2&2=2, 3&2=0, 0&2=0,
1&3=3, 2&3=0, 3&3=3, 0&3=0,
1&0=0, 2&0=0, 3&0=0, 0&0=0.

p -> q

1 1 1
2 1 1
3 1 1
0 1 1
1 2 2
2 1 2
3 2 2
0 1 2
1 3 3
2 3 3
3 1 3
0 1 3
1 0 0
2 3 0
3 2 0
0 1 0

implies (->):
1->1=1, 2->1=1, 3->1=1, 0->1=1,
1->2=2, 2->2=1, 3->2=2, 0->2=1,
1->3=3, 2->3=3, 3->3=1, 0->3=1,
1->0=0, 2->0=3, 3->0=2, 0->0=1.

p <-> q

1 1 1
2 2 1
3 3 1
0 0 1
1 2 2
2 1 2
3 0 2
0 3 2
1 3 3
2 0 3
3 1 3
0 2 3
1 0 0
2 3 0
3 2 0
0 1 0

equivalence (<->):

1<->1=1, 2<->1=2, 3<->1=3, 0<->1=0,
1<->2=2, 2<->2=1, 3<->2=0, 0<->2=3,
1<->3=3, 2<->3=0, 3<->3=1, 0<->3=2,
1<->0=0, 2<->0=3, 3<->0=2, 0<->0=1.

Examples:

1. p v ~p

1 1 0
2 1 3
3 1 2
0 1 1

2. ~(p & ~p)

1(1 0 0)
1(2 0 3)
1(3 0 2)
1(0 0 1)

3. []p -> p

1 1 1 1
0 2 1 2
0 3 1 3
0 0 1 0

4. p -> <>p

1 1 1 1
2 1 1 2
3 1 1 3
0 1 0 0

5. p -> []<>p

1 1 1 1 1
2 1 1 1 2
3 1 1 1 3
0 1 0 0 0

All of the axioms and theorems of classical propositional logic and modal propositional logic are tautologies.
Many more logical operators are available in this 4-valued system compared to the 2-valued system.

There are an endless number of different multi-valued systems that have these qualities, i.e. any system with 2^n different values, where n is a natural number >0.

Any opinions?

Witt

mnkbdky

If a cherry was a pickle, would a banana be a cucumber?

Like anyone is going to take the time to read this?

Good Almighty Friday!

Witt

mnkbdky:

If a cherry was a pickle, would a banana be a cucumber?

Like anyone is going to take the time to read this?

Good Almighty Friday!
----------------------------

Your idiotic remarks make me happy that you don't like my posts!

Thanks for the warning,

Witt

mnkbdky

I didn't say I didn't like it. I was merely stating I don't think anyone is going to do the work it takes to understand your argument. Maybe some will. But not this puppy. I would rather you spelled it out in Salt Lake street prose. But that is just me.

Thanks,

--mnkbdky

SlateGreySky

Witt,

Interesting logic . . . I'm a little confused as to the difference between a factual falsity and a logical falsity (or a logical truth versus a factual truth, for that matter). Could you please give an example of a proposition that would be factually true but logically false, or logically true but factually false?

Also, I don't understand why "logically true" seems to be equal to []p. It seems like one could replace "F" with "[]~p" and "T" with "[]p." Could you please explain how this is not the case?

Thanks.

Witt

SlateGreySky:
Interesting logic . . . I'm a little confused as to the difference between a factual falsity and a logical falsity (or a logical truth versus a factual truth, for that matter).

Consider the statement, 'My car is blue' and let's assume that it is in fact true.

My car is blue, is factually true.
My car is red, is factually false.
My car is blue or My car is not blue, is logically true.
My car is blue and My car is not blue, is logically false.

SlateGreySky: Could you please give an example of a proposition that would be factually true but logically false, or logically true but factually false?

My car is blue, is true but it is not necessarily true.
That is, Necessarily(my car is blue) is false.
There are no necessary facts at all.
Facts are not theorems.

My car is blue or My car is not blue, is logically true but not factually true, i.e. there cannot be a situation that confirms 'My car is blue or My car is not blue'.
There are no factual necessities at all.
Theorems are not facts.

Situations can confirm what they show but they cannot confirm what they do not display. What is not the case is inferred from what is shown.

We cannot 'see' that, My car is not red.

SlateGreySky: Also, I don't understand why "logically true" seems to be equal to []p. It seems like one could replace "F" with "[]~p" and "T" with "[]p." Could you please explain how this is not the case?

If p is factually true or factually false, []p is contradictory. I.e. material truths are not theorems, rather, they happen to be the case.

If p is: tautologous, analytically true, deducibly true, apriori, then []p is also tautologous etc., and []~p is contradictory.

Witt

mnkbdky

Guess I was wrong! Someone will take the time.

SecularFuture

[deleted]

John Page

How about:

A statement is deemed to TRUE if it is considered to accurately convey a state of affairs.

A statement is deemed to FALSE if it is considered to completely contrary to a state of affairs.

Rather than the Boolean 1 and 0, this could be represented by 1 and -1 with 0 as a neutral value (maybe caused by irrelevance or uncertainty). This is not to say that truth has to be binary!

A & B = 2
~A & ~B = -2

So the overall truthfulness can be evaluated according to the accuracy of the component propositions, if most of them are false then the overall proposition is likely to be false (after reversing the negations!!).

Just thoughts.

Cheers, John

SlateGreySky

But what you point to here as a "logical truth" (i.e. that the car is either blue or not blue) just seems to be the law of non-contradiction, which would be true regardless of any factual truth or falsehood.

My car is blue or My car is not blue, is logically true but not factually true, i.e. there cannot be a situation that confirms 'My car is blue or My car is not blue'.

I disagree. I would say (especially if you agree to phrase the above as "it is either the case that my car is blue or it is not the case that my car is blue") that every logically possible state of affairs confirms "it is either the case that my car is blue or it is not the case that my car is blue." That is just to say that the law of non-contradiction applies universally, right?

Of course we can! If it's blue, we see that it's not red.

What about 1=1? That seems to me to be a factual necessity, in that it is factually true in all possible worlds that 1=1.

Is this just the assertion that there are no such things as necessary truths? If so, I disagree; if not, could you please unpack the first sentence further?

Sorry to try and open so many threads here . . . I just find your "multivalent logic" very worthy of discussion. Hopefully somebody has the time to talk about at least one of these issues . . .