Truth tables for Modal Logic.

[Witt]

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

Classical binary logic is included in this method.

The four truth values are: 1= logical truth, 2 = factual truth, 3 = factual falsity, 0 = logical falsity.

Any fomula that has all 1's is a theorem.

1. (~): ~1=0, ~2=3, ~3=2, ~0=1.
2. (<>): <>1=1, <>2=1, <>3=1, <>0=0.
3. ([]): []1=1, []2=0, []3=0, []0=0.

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


As in classical logic:

5. p -> q defined ~p v q.
6. p & q defined ~(~p v ~q)
7. p <-> q defined (p -> q) & (q -> p).

Table 1.

p ~p []p <>p

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


Examples:

T1. []p -> p

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

T2. p -> <>p

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

T3. []p -> <>p

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

T4. []p <-> ~<>(~p)

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

T5. <>p <-> ~[](~p)

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

T6. <>p <-> <>(<>p)

T7. <>p <-> [](<>p)

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

T8. []p <-> []([]p)

T9. []p <-> <>([]p)

T10. ~<>(p & ~p)
1 0 1 0 0
1 0 2 0 3
1 0 3 0 2
1 0 0 0 1

T11. [](p v ~p)
1 1 1 0
1 2 1 3
1 3 1 2
1 0 1 1

etc., etc.

All of the axioms and theorems of propositional logic and
modal propositional logic are tautologies, in Wittgenstein's sense.

The truth value analysis of modal logic with two propositional variables requires 16 values, I will show the relevant tables if there is some interest.

Any opinions?

Witt

[Clutch]

Haven't worked through your post, but at a quick glance: 2v3=1?

Either the Cowboys won the Superbowl in 1995 or Henry VIII died of a nosebleed.

That's a logical truth?

Or did you mean to have only a single proposition, as in: p2vp3=1?

Which itself is contentious, inasmuch as it's not constructive.

[Witt]

Clutch:
Haven't worked through your post, but at a quick glance: 2v3=1?

Either the Cowboys won the Superbowl in 1995 or Henry VIII died of a nosebleed.

That's a logical truth?

Or did you mean to have only a single proposition, as in: p2vp3=1?

Which itself is contentious, inasmuch as it's not constructive.
---------------------------------------------------------------------------

These tables apply only to one statement, truth tables for two or more propositions are more complex.

Either, the Cowboys won the Superbowl in 1995, or, the Cowboys did not win the Superbowl in 1995.

This has the form 'factual truth or factual falsity' ie. (2 v 3).
And, it certainly is (1) logically true.

Witt

[John Page]

Originally posted by Witt
All of the axioms and theorems of propositional logic and
modal propositional logic are tautologies, in Wittgenstein's sense.

..but aren't all axioms tautologies w.r.t. the system they define?

Also, in your #3, why is it []2=0?

IMO, nothing is absolutely [] therefore all (1v2) has to have a system that it is true for.

Finally, in modal logic, it seems popular to conceive of results as relating to possible worlds. I think it more accurate to conceive of possible result w.r.t. the system being considered.

Cheers, John

[Witt]

Re: Truth tables for Modal Logic.

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
All of the axioms and theorems of propositional logic and
modal propositional logic are tautologies, in Wittgenstein's sense.

--------------------------------------------------------------------------------

John:
..but aren't all axioms tautologies w.r.t. the system they define?

No. The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.

John: Also, in your #3, why is it []2=0?

While it is factually true that my cat is black, ie. it has the value (2), that it is necessarily true is contradictory.

John: IMO, nothing is absolutely [] therefore all (1v2) has to have a system that it is true for.

(1v2) has the value (1)

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

John:
Finally, in modal logic, it seems popular to conceive of results as relating to possible worlds. I think it more accurate to conceive of possible result w.r.t. the system being considered.

Yes, Saul Kripke has provided a semantics that is based on the Leibnizian principle of possible worlds.

My method provides an alternate solution, ie a different semantics which is an extension of classical truth value analysis.

Witt

[Clutch]

Originally posted by Witt
This has the form 'factual truth or factual falsity' ie. (2 v 3).
And, it certainly is (1) logically true. It is a classical logical truth, but not (eg) intuitionistically valid.

The difference is important. Otherwise facts about vague predicates, for example, will be legislated away by your logical presuppositions.

In any case, what is the payoff supposed to be? You're multiplying senses of "true", to what effect?. Isn't "factual truth" just expressed by:

p & <>~p

with

~p & <>p

expressing "factual falsity"?

[spacer1]

Witt,

I may be way out of my depth here, but if [] means "it is necessary that", then I find the following somewhat strange, to say the least:
The four truth values are: 1= logical truth, 2 = factual truth, 3 = factual falsity, 0 = logical falsity.

Any fomula that has all 1's is a theorem.

1. (~): ~1=0, ~2=3, ~3=2, ~0=1.
2. (<> ): <>1=1, <>2=1, <>3=1, <>0=0.
3. ([]): []1=1, []2=0, []3=0, []0=0.

My problem, therefore, is with your #3. If my interpretation of your [] is correct, then 2=0 (factual truth = logical truth), which makes sense. However, 3=0 (factual falsity = logical truth) as well, which would also mean that 2=3 (factual falsity = factual truth). This seems a bit odd to me, but I'll wait and see if my interpretation of your [] does, in fact, mean "it is necessary that".

[John Page]

Originally posted by Witt
John:
..but aren't all axioms tautologies w.r.t. the system they define?

Witt: No. The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.

I'm still quibbling - an axiom will appear self-evident and therefore be a tautology w.r.t. the system concerned.
Originally posted by Witt
John: Also, in your #3, why is it []2=0?

While it is factually true that my cat is black, ie. it has the value (2), that it is necessarily true is contradictory.

Sorry but I don't get it (yet!). If (your cat is black) then it is necessarily true that (it is black). If we are not talking about any specific cat of yours then, of course, it could be a different color.
Originally posted by Witt
John:
Finally, in modal logic, it seems popular to conceive of results as relating to possible worlds. I think it more accurate to conceive of possible result w.r.t. the system being considered.

Witt: Yes, Saul Kripke has provided a semantics that is based on the Leibnizian principle of possible worlds.

My method provides an alternate solution, ie a different semantics which is an extension of classical truth value analysis.

:) Yes, the world contains the semantics, not the other way round.

Cheers, John

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
This has the form 'factual truth or factual falsity' ie. (2 v 3).
And, it certainly is (1) logically true.
--------------------------------------------------------------------------------

Clutch:
It is a classical logical truth, but not (eg) intuitionistically valid.
The difference is important. Otherwise facts about vague predicates, for example, will be legislated away by your logical presuppositions.


What facts about what vage predicates?

"It is provable that a wff is valid in the Heytin calculus (intuitionist logic) iff its translation is valid in S4, (ie. Modal Logic), McKinsey and Tarski (1948); Fitting (1969)" Susan Haack, Philosophy of Logics, (1978), page 219.


Clutch:
In any case, what is the payoff supposed to be? You're multiplying senses of "true", to what effect?. Isn't "factual truth" just expressed by:

p & <>~p
------------


No. when p has the value 1:
1 & <>(~1), is not factual at all. It has the value 0.

when p has the value 1, p & <>~p has the value 0
when p has the value 2, p & <>~p has the value 2
when p has the value 3, p & <>~p has the value 3
when p has the value 0, p & <>~p has the value 0

<>p & <>~p, says p is factual, that is, it is true iff p has the value 2 or 3.

And ~(<>p & <>~p), ie. <>p -> []p, says that p is analytic.
ie. <>p -> []p, is true iff p has the value 1 or 0.
eg. God exists. is analytic.


Clutch:
with

~p & <>p

expressing "factual falsity"?
-------------------------------------

Again, if p has the value 1: ~1 & <>1, ie. 0 & 1, which resolves to 0.

When p has the value 1, ~p & <>p is 0.
When p has the value 2, ~p & <>p is 3.
When p has the value 3, ~p & <>p is 2.
When p has the value 0, ~p & <>p is 0.

Both of the expressions you have used, 1. p & <>~p and 2. ~p & <>p, are important functions within modal logic.

This truth table method allows us to calculate the truth values of these complicated combinations of modal propositional logic.

Witt

[Witt]

Witt,

spacer1:
I may be way out of my depth here, but if [] means "it is necessary that", then I find the following somewhat strange, to say the least:


quote:
--------------------------------------------------------------------------------
The four truth values are: 1= logical truth, 2 = factual truth, 3 = factual falsity, 0 = logical falsity.

Any fomula that has all 1's is a theorem.

1. (~): ~1=0, ~2=3, ~3=2, ~0=1.
2. (<> ): <>1=1, <>2=1, <>3=1, <>0=0.
3. ([]): []1=1, []2=0, []3=0, []0=0.
--------------------------------------------------------------------------------


spacer1:
My problem, therefore, is with your #3. If my interpretation of your [] is correct, then 2=0 (factual truth = logical truth), which makes sense. However, 3=0 (factual falsity = logical truth) as well, which would also mean that 2=3 (factual falsity = factual truth). This seems a bit odd to me, but I'll wait and see if my interpretation of your [] does, in fact, mean "it is necessary that".


I do mean that, []p defined (it is necessary that p is true).

Your interpretation is not what I have intended, thanks for pointing it out. What do you think about this revised version.

1. (~): (~1)=0, (~2)=3, (~3)=2, (~0)=1.
2. (<>): (<>1)=1, (<>2)=1, (<>3)=1, (<>0)=0.
3. ([]): ([]1)=1, ([]2)=0, ([]3)=0, ([]0)=0.

Clearly, ~(1=2) and ~(1=3) and ~(1=0) and ~(2=3) and ~(2=0).

Witt

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
John:
..but aren't all axioms tautologies w.r.t. the system they define?

Witt: No. The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.
--------------------------------------------------------------------------------

John:
I'm still quibbling - an axiom will appear self-evident and therefore be a tautology w.r.t. the system concerned.

Not so. Self-evidence is as elusive as absolut truth.
Apparent truth belongs to religion not to logic.

Axioms are chosen on the basis of achieving expected results, not on the idea of self evidence.

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
John: Also, in your #3, why is it []2=0?

While it is factually true that my cat is black, ie. it has the value (2), that it is necessarily true is contradictory.

--------------------------------------------------------------------------------

John:
Sorry but I don't get it (yet!). If (your cat is black) then it is necessarily true that (it is black). If we are not talking about any specific cat of yours then, of course, it could be a different color.

Are you saying that factual truth is the same as necessary truth?
There are no logical truths that are factually true, and, there are no factual truths that are logically true.

My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
John:
Finally, in modal logic, it seems popular to conceive of results as relating to possible worlds. I think it more accurate to conceive of possible result w.r.t. the system being considered.

Witt: Yes, Saul Kripke has provided a semantics that is based on the Leibnizian principle of possible worlds.

My method provides an alternate solution, ie a different semantics which is an extension of classical truth value analysis.
--------------------------------------------------------------------------------

John: Yes, the world contains the semantics, not the other way round.

Agreed, possible worlds ..seem to me a vague expression. There is only one universe, even though we admit different universes of discourse, within that one universe.

Witt

[spacer1]

Witt,
What do you think about this revised version.

1. (~): (~1)=0, (~2)=3, (~3)=2, (~0)=1.
2. (<> ): (<>1)=1, (<>2)=1, (<>3)=1, (<>0)=0.
3. ([]): ([]1)=1, ([]2)=0, ([]3)=0, ([]0)=0
I don't see how this changes the meaning, but then I think my lack of knowledge regarding logic may be adding more confusion than helping you in any way, so I might just bow out of the discussion at this point.

[Witt]

Witt,

quote:
--------------------------------------------------------------------------------
What do you think about this revised version.

1. (~): (~1)=0, (~2)=3, (~3)=2, (~0)=1.
2. (<> ): (<>1)=1, (<>2)=1, (<>3)=1, (<>0)=0.
3. ([]): ([]1)=1, ([]2)=0, ([]3)=0, ([]0)=0
--------------------------------------------------------------------------------

spacer1
I don't see how this changes the meaning, but then I think my lack of knowledge regarding logic may be adding more confusion than helping you in any way, so I might just bow out of the discussion at this point.


That would be sad.
Logical calculations are considerably simpler than arithmetic calculations.
I am sure that a little more effort will be benificial.

I can try to express thes points in a different way, if that will help.

Witt

[spacer1]

Witt,
I can try to express thes points in a different way, if that will help.
Ok then. I have no problem understanding #1 and #2, nor with "Clearly, ~(1=2) and ~(1=3) and ~(1=0) and ~(2=3) and ~(2=0)", but I still see the same problem with #3 which I outlined earlier. Perhaps you could write out each term of #3 in natural language.

[Witt]

Witt,
quote:
--------------------------------------------------------------------------------
I can try to express thes points in a different way, if that will help.
--------------------------------------------------------------------------------

spacer1:
Ok then. I have no problem understanding #1 and #2, nor with "Clearly, ~(1=2) and ~(1=3) and ~(1=0) and ~(2=3) and ~(2=0)", but I still see the same problem with #3 which I outlined earlier. Perhaps you could write out each term of #3 in natural language.

original:
1. (~): ~1=0, ~2=3, ~3=2, ~0=1.
2. (<> ): <>1=1, <>2=1, <>3=1, <>0=0.
3. ([]): []1=1, []2=0, []3=0, []0=0.

revised:
1. (~): (~1)=0, (~2)=3, (~3)=2, (~0)=1.
2. (<> ): (<>1)=1, (<>2)=1, (<>3)=1, (<>0)=0.
3. ([]): ([]1)=1, ([]2)=0, ([]3)=0, ([]0)=0

If p is logically true (1), necessary p, is true. ie. ([]1)=1.
If p is factually true(2), necessary p, is false. ie. ([]2)=0.
If p is factually false(3), necessary p, is false. ie. ([]3)=0.
If p is logically false (0), necessary p, is false. ie. ([]0)=0.

Necessarily(its raining or its not raining) is true.
Necessarily(its raining) is false.
Necessarily(its not raining) is false.
Necessarily(its raining and its not raining), is false.

Hope that helps. These concepts are not common within ordinary language, so I can understand your position.

Witt

[John Page]

Originally posted by Witt
Not so. Self-evidence is as elusive as absolut truth.
Apparent truth belongs to religion not to logic.

Axioms are chosen on the basis of achieving expected results, not on the idea of self evidence.

:) To me it is axiomatic that axioms are self-evident.

I'm not sure what evidence you have to claim that axioms are "chosen" on the basis of acheiving expected results! I think that claim would require a lot more research and understanding of the mind. I think "axioms are selected from a trial and error process as to their empirical accuracy" is closer to the truth. ;)
Originally posted by Witt
Are you saying that factual truth is the same as necessary truth?
There are no logical truths that are factually true, and, there are no factual truths that are logically true.

My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?

It is necessary that a truth is in accordance with the facts. If the "necessary truth" is different than an apparent factual truth, we're necessarily talking about a different set of facts!

If your cat is factually black and you can logically prove that the (very same) cat is white, I venture that it is the system of logic employed is not efficacious.

Cheers, John

[Clutch]

Originally posted by Witt
quote:
Clutch:
It is a classical logical truth, but not (eg) intuitionistically valid.
The difference is important. Otherwise facts about vague predicates, for example, will be legislated away by your logical presuppositions.

What facts about what vage predicates?You're making it, by fiat, a fact about logic that there are sharp demarcations between, for instance, something's being red and something's not being red. This seems a poor candidate for decision by pledging allegiance to a particular logic a priori.
Clutch:
In any case, what is the payoff supposed to be? You're multiplying senses of "true", to what effect?. Isn't "factual truth" just expressed by:

p & <>~p
------------

No. when p has the value 1:
1 & <>(~1), is not factual at all. It has the value 0.Utterly irrelevant.

Of course if p is a logical truth, then p & <>~p is false. (Its rightmost conjunct being false.) That's the point of my saying that p & <>~p expresses factual truth.

The point is quite simple. With a single undifferentiated notion of truth, for something to be merely factually true, and not logically true, is just for it to satisfy the schema:

P, but it's a logical possibility that not-P

plausibly formalized as p & <>~p.

So what advantage of explanation or clarity do you see arising from your introduction of an ambiguity to the term 'true'? What do you think is sayable in your terms that is not sayable without the notions of merely factual truth and merely factual falsity?

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
Are you saying that factual truth is the same as necessary truth?
There are no logical truths that are factually true, and, there are no factual truths that are logically true.

My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?

--------------------------------------------------------------------------------

John:
It is necessary that a truth is in accordance with the facts. If the "necessary truth" is different than an apparent factual truth, we're necessarily talking about a different set of facts!

Your empiricism is showing. You are assuming that all truths are facts.
How then, do you show the physical facts that prove p v ~p is true?

John:
If your cat is factually black and you can logically prove that the (very same) cat is white, I venture that it is the system of logic employed is not efficacious.

No one can make such a claim.

Witt

[John Page]

Originally posted by Witt
John:
If your cat is factually black and you can logically prove that the (very same) cat is white, I venture that it is the system of logic employed is not efficacious.

Witt: No one can make such a claim.
:confused: Please can you clarify which claim no one can make?

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
John:
If your cat is factually black and you can logically prove that the (very same) cat is white, I venture that it is the system of logic employed is not efficacious.

Witt: No one can make such a claim.
--------------------------------------------------------------------------------


John: Please can you clarify which claim no one can make?

No one can make the claim that their cat is not white and their cat is white. That is a contradiction.

"If your cat is factually black and you can logically prove that the (very same) cat is white," cannot be true.

If it is black then it is not white, is a presumption we make.
Therefore, It is not white.

Surely, it cannot be white and not be white, can it?


Witt

[ex-xian]

I've only casually looked through the OP; I'll examine it more carefully when time permits.

Do you have a way to express it in terms of reduction trees? I've always found them more conveinent than truth tables.

[ex-xian]

Does including "factual truth" and "factual falsity" amount to trying to incorporate the "real" world into a formal system?

Also, isn't a probabilistic-based logic more useful? It would seem to incorporate a wider universe of possiblities regarding the truth value of statements.

Like spacer1, I kind of feel out of my league. I've only had a couple of formal classes in logic, although I'm taking more this semester and always doing indepedent reading. :)

[Witt]

I've only casually looked through the OP; I'll examine it more carefully when time permits.

Do you have a way to express it in terms of reduction trees? I've always found them more conveinent than truth tables.
------------------

What are reduction trees?

I think there are many equivalent ways of showing the logical truth of propositions, from deduction to truth value analysis.

Witt

[ex-xian]

Originally posted by Witt
What are reduction trees?
See A Practical Introduction to Formal Logic (http://www.uvawise.edu/philosophy/Logic%20Text/Contents.htm) by Dr. David Rouse. It's a text for one year undergraduate logic. The later chapters deal with reduction trees and how to use them with propositions and quantifiers.

I think there are many equivalent ways of showing the logical truth of propositions, from deduction to truth value analysis.

Witt
Agreed, but I'd much rather use a reduction tree than a truth table to analyze something of this sort:
(A & B)-> (C & D) <-> (~C or ~D) -> (~A or ~B)

(of coures this is a tautology since it's the contrapostive, but I wanted to use a complex looking example)

[Witt]

ex-xian:
Does including "factual truth" and "factual falsity" amount to trying to incorporate the "real" world into a formal system?

Yes, insofar as facts are part of a logical truth.

ex-xian:
Also, isn't a probabilistic-based logic more useful? It would seem to incorporate a wider universe of possiblities regarding the truth value of statements.

Probabilistic-based logic is the method of deciding factual truths.
There is no probability that p v ~p is true, ..it is necessarily true, far any p. There are no 'degrees' of necessity.

Witt

[John Page]

Originally posted by Witt
--------------------------------------------------------------------------------
Originally posted by Witt
Are you saying that factual truth is the same as necessary truth?
There are no logical truths that are factually true, and, there are no factual truths that are logically true.

My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?
--------------------------------------------------------------------------------
John: It is necessary that a truth is in accordance with the facts. If the "necessary truth" is different than an apparent factual truth, we're necessarily talking about a different set of facts!

Witt: Your empiricism is showing. You are assuming that all truths are facts.
How then, do you show the physical facts that prove p v ~p is true?

John: If your cat is factually black and you can logically prove that the (very same) cat is white, I venture that it is the system of logic employed is not efficacious.

Witt: No one can make such a claim.

John: Please can you clarify which claim no one can make?

Witt: No one can make the claim that their cat is not white and their cat is white. That is a contradiction.

"If your cat is factually black and you can logically prove that the (very same) cat is white," cannot be true.

If it is black then it is not white, is a presumption we make.
Therefore, It is not white.

Surely, it cannot be white and not be white, can it?

Witt:

You seem to have misunderstood my remark. I first responded to your statement "My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?"

with my statement: "It is necessary that a truth is in accordance with the facts."

Under what circumstances can it be factually true that a cat is black but not logically necessary that the same cat is black?

Cheers, John

[ex-xian]

Originally posted by Witt
ex-xian:
Does including "factual truth" and "factual falsity" amount to trying to incorporate the "real" world into a formal system?

Yes, insofar as facts are part of a logical truth.

ex-xian:
Also, isn't a probabilistic-based logic more useful? It would seem to incorporate a wider universe of possiblities regarding the truth value of statements.

Probabilistic-based logic is the method of deciding factual truths.
There is no probability that p v ~p is true, ..it is necessarily true, far any p. There are no 'degrees' of necessity.

Witt
I become wary when someone tried to map the actual world onto a formal system. As for P v ~p, this would have a probability of 0. But other statements can be modeled that are outside of bivalent logic, just as yours does.

[Witt]

John:
You seem to have misunderstood my remark. I first responded to your statement "My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?"

with my statement: "It is necessary that a truth is in accordance with the facts."

Under what circumstances can it be factually true that a cat is black but not logically necessary that the same cat is black?
-----------------------------------------------------------------------------

IMHO,

There is no circumstances that proves that my cat is black is logically true. It is not logically true at all.
My cat is black, is neither deduced nor calculated, its truth is seen true by perception.

The empirical fact that my cat is black is shown true by the correspondence of: the empirical object -my cat, the empirical property of -blackness, and that my empirical cat has this empirical property. The world displays primary positive facts.
Its blackness is shown by the presentation of the situation percieved.

Even though it is a fact that my cat is black, it is logically true that, my cat is white or it is not white.

Logical truths are not shown true by perceptions of any kind.

They are deduced or calculated.
Their truth is concieved and their truth is not percieved.
The procedures for showing logical truths and the procedures of showing factual truths, are mutually exclusive.

They have no common true statement.
No statement can be both fact and tautology.

Negative facts, eg. my cat is not white, are inferred and not percieved. We do not percieve the empirical property of non-whiteness.
Its truth is deduced by the assumed realisation that if x is percieved to be color Y then it cannot be some other color Z.
We conclude that since, my cat is black, it cannot be another color at the same time and place.
A proposition is factually false if its negation requires the factual truth of some component of the factually false statement.

Facual falsities are not contradictions.
My cat is not white, means, it is not the case that, my cat is white is a true (percieved) fact. That is, it is not the case that the world presented a situation in which my empirical cat has the empirical quality of whiteness.

Logical fasities are contradictions, there are no possible situations that could occur that are contradictory.

When our reasoning concludes with a contradicory situation, then, we have made errors in our assumptions.

The antinomy, paradox, cannot occur.

Witt

[spacer1]

Witt,
Necessarily(its raining or its not raining) is true.
Necessarily(its raining) is false.
Necessarily(its not raining) is false.
Necessarily(its raining and its not raining), is false.
Thanks for taking the time to clarify. I see my error now was to assume the "=" was a mathematical operation, such that 2=3. However, I was merely making a logical fallacy along the lines of:

All cats are black (let's assume)
All blackboards are black
Therefore, all cats are blackboards.

I'll have to go back and reread your OP again with this new perspective and see if I can make sense of the arguments the other posters are making against you. Thanks again for your efforts to explain it.

[John Page]

Originally posted by Witt
John:
You seem to have misunderstood my remark. I first responded to your statement "My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?"

with my statement: "It is necessary that a truth is in accordance with the facts."

Under what circumstances can it be factually true that a cat is black but not logically necessary that the same cat is black?
-----------------------------------------------------------------------------

IMHO,

There is no circumstances that proves that my cat is black is logically true. It is not logically true at all.
My cat is black, is neither deduced nor calculated, its truth is seen true by perception.

The empirical fact that my cat is black is shown true by the correspondence of: the empirical object -my cat, the empirical property of -blackness, and that my empirical cat has this empirical property. The world displays primary positive facts.
Its blackness is shown by the presentation of the situation percieved.

Even though it is a fact that my cat is black, it is logically true that, my cat is white or it is not white.

Logical truths are not shown true by perceptions of any kind.

Witt:

Logical truths are shown by comparing a statement with the rules of a system of logic to determine the statement's truth functionality. Our minds perceive the result.

The color of a cat is shown by comparing the sense data from the relevant cat with a system of color determination. Our minds perceive the result.

Your cat is black iff your cat is black. This statement is true iff this statement is true.

Why do you think logic is above or does not need empirical evidence?

Cheers, John

[Witt]

John:
Logical truths are shown by comparing a statement with the rules of a system of logic to determine the statement's truth functionality. Our minds perceive the result.

I dont agree that any truth is percieved. I would rather say it is concieved.

John:
The color of a cat is shown by comparing the sense data from the relevant cat with a system of color determination. Our minds perceive the result.

That our senses recieve data from a situation, is perception.

John:
Your cat is black iff your cat is black. This statement is true iff this statement is true.

Why do you think those remarks are informative.
What physical fact do you have that shows the truth of, this statement is true iff this statement is true?

p <-> p, says: this statement is true, iff, this statement is true ..for any statement.

What physical facts are required to show its logical truth?

This statementis true, is as arbitrary as is the free variable p.

John:
Why do you think logic is above or does not need empirical evidence?

Because, logical truths are shown true without considering empirical concerns. Of course, mind is required ..because all truth requires mind. And, mind requires physical brain.
But, the abstract entities that logic deals with are not empirical concepts.
For eample what physical thing represents the logical word not.

The world does not show truth, we interpret what is shows as factual truth.

We seem to saying different things with the same words.

Witt

[John Page]

Originally posted by Witt
John:
Logical truths are shown by comparing a statement with the rules of a system of logic to determine the statement's truth functionality. Our minds perceive the result.

I dont agree that any truth is percieved. I would rather say it is concieved.

...and the essential difference between perception and conception is what?
Originally posted by Witt
John:
Your cat is black iff your cat is black. This statement is true iff this statement is true.

Why do you think those remarks are informative.
What physical fact do you have that shows the truth of, this statement is true iff this statement is true?

They are informative because they are facts about the kind of logical truth we are discussing.

The mental operation of truth determination, that we seem both able to perform, is a physical operation. A truth for you true iff you think it.
Originally posted by Witt
This statementis true, is as arbitrary as is the free variable p.

Truth is arbitrary? In what sense do you think p is "free"? Do you think it arbitrary because it is free? (Note: An independent variable may appear free w.r.t. the proposition it is independent of.)
Originally posted by Witt
John:
Why do you think logic is above or does not need empirical evidence?

Because, logical truths are shown true without considering empirical concerns. Of course, mind is required ..because all truth requires mind. And, mind requires physical brain.
But, the abstract entities that logic deals with are not empirical concepts.
Empirical:
a) Relying on or derived from observation or experiment: empirical results that supported the hypothesis.
b) Verifiable or provable by means of observation or experiment: empirical laws.
I hope it is easy to see that the concepts of logic are developed and proved empirically by such techniques as truth tables and specialized language used to represent factual situations, albeit generalized.
Originally posted by Witt
For eample what physical thing represents the logical word not.

The physical word "not"!
Originally posted by Witt
We seem to saying different things with the same words.
Perhaps, but it seems to me that you are proposing a theory of truth where truth exists in some higher epistamic realm than the real, factual, world.

Cheers, John

[spacer1]

Witt,
Sorry, I'm still having a bit of trouble understanding this:
If p is factually true(2), necessary p, is false. ie. ([]2)=0.
[e.g.] Necessarily(its raining) is false.
If p is factually true, then p is necessarily logically false? Is that right? If so, why?

[Edited to add:] Or do you mean:
If p is factually true, then necessary p is logically false, since all factual truths are contingent?

[Witt]

Witt,
Sorry, I'm still having a bit of trouble understanding this:


quote:
--------------------------------------------------------------------------------
If p is factually true(2), necessary p, is false. ie. ([]2)=0.
[e.g.] Necessarily(its raining) is false.
--------------------------------------------------------------------------------


spacer1:

If p is factually true, then p is necessarily logically false? Is that right? If so, why?

[Edited to add:] Or do you mean:
If p is factually true, then necessary p is logically false, since all factual truths are contingent?
--------------------------------------

You are correct. By convention or axiom, we assuume that all factual truths are indeed contingent. Thereby, that they are necessary is contradictory.

We can define different notions of 'necessity' but the one used here seems to represent what we mean by logically true.

With this multivalued logic many more logical operators are available, than are used in classical logic. For example there are at least two different 'or' operators. This aspect allows for this logic to be applied in many more ways than classical logic.

For example, we can define physical necessity as that which is equal to the value 2. ie. ({}1)=0, ({}2)=1, ({}3)=0, ({}0)=0.

I am currently looking into the implications, via this 4-valued logic, to see if sense applies to physical necessity, ({}), defined in this way. I think it might work well.

What do you think about extending the logical operators in this way?

Witt

[John Page]

Originally posted by Witt
You are correct. By convention or axiom, we assuume that all factual truths are indeed contingent. Thereby, that they are necessary is contradictory.

We can define different notions of 'necessity' but the one used here seems to represent what we mean by logically true.

Witt:

I don't see how factual truths are any more or less contingent than logical truths. They are both dependent upon the system that determines/defines facts/logical truths.

Truth is not "necessary", it is a condition that can be defined a number of different ways. See link here to debate on dialetheism (http://www.iidb.org/vbb/showthread.php?s=&threadid=46424&perpage=25&highlight=dialetheism&pagenumber=2) - on the second page I offer some axioms for this contradictory logic.

Please consider that for one to know a fact or a truth, one must first know and understand falsity. If we know falsity, that condition resides in our minds as part of our deductive process.

Cheers, John

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
You are correct. By convention or axiom, we assuume that all factual truths are indeed contingent. Thereby, that they are necessary is contradictory.

We can define different notions of 'necessity' but the one used here seems to represent what we mean by logically true.

--------------------------------------------------------------------------------


John:
I don't see how factual truths are any more or less contingent than logical truths. They are both dependent upon the system that determines/defines facts/logical truths.

It is true that all truths are contingent upon some system of decision, but, empirical truths have a unique dependency (contingency) on physical perception.

That p is dependent (contingent) on some physical sense perception, entails that p is logically true .. is false!

Witt

[John Page]

Originally posted by Witt
It is true that all truths are contingent upon some system of decision, but, empirical truths have a unique dependency (contingency) on physical perception.

That p is dependent (contingent) on some physical sense perception, entails that p is logically true .. is false!
Witt:

1. Empirical truths have no unique dependency on physical perception as you allege. Logical truths are physically perceived.

2. It is irrelevant whether you consider it false that (p being dependent upon physical sense perception entails that p is logically true). Physical sense perception and predicate logic are different systems of truth determination - it seems to me what you are assuming is "All systems of physical sense perception" operate according to the principles of predicate logic. Now that can be proven to be false**.

Cheers, John

** In fact ;) , I prove this every time I'm "wrong".

[John Page]

Originally posted by Witt
By convention or axiom, we assuume that all factual truths are indeed contingent. Thereby, that they are necessary is contradictory.
...but not universally. Facts are truths w.r.t. the system for which they are a fact and thus not contingent within that system.

Nevertheless, facts can be disproven through examining other evidence, this process of examination altering the facts within the system. But this changes the system!!

Repeating this thinking, we can say the definition of a system makes certain facts necessary and given.

Cheers, John

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
It is true that all truths are contingent upon some system of decision, but, empirical truths have a unique dependency (contingency) on physical perception.

That p is dependent (contingent) on some physical sense perception, entails that p is logically true .. is false!
--------------------------------------------------------------------------------


John:
1. Empirical truths have no unique dependency on physical perception as you allege.

Can you provide an example of an empirical truth that is not a physically perceptable truth?

If it is raining, is true ..how do you know it without some sense of empirical perception?

John: Logical truths are physically perceived.

How do you percieve that, 1=1 or i=i? [Silly Rabbit]!

John:
2. It is irrelevant whether you consider it false that (p being dependent upon physical sense perception entails that p is logically true). Physical sense perception and predicate logic are different systems of truth determination - it seems to me what you are assuming is "All systems of physical sense perception" operate according to the principles of predicate logic. Now that can be proven to be false**.


"All systems of physical sense perception" operate according to the principles of predicate logic.

Of course they do.
Are you aware of some sense perception that cannot be expressed in predicate logic?
Please show me how! I don't believe you.

Witt

[John Page]

Originally posted by Witt
John:
1. Empirical truths have no unique dependency on physical perception as you allege.

Can you provide an example of an empirical truth that is not a physically perceptable truth?

Witt - you are reading the sentence illogically. It says empirical truths have no unique dependency on physical dependence, not that they have no dependency.
Originally posted by Witt
John: Logical truths are physically perceived.

How do you percieve that, 1=1 or i=i? [Silly Rabbit]!

By comparing the forms that appear in my mind/brain in a manner consistent with the rules of logic. Outside the rules of logic and its representation I am free to deny the equality.
Originally posted by Witt
Are you aware of some sense perception that cannot be expressed in predicate logic?
Aside from the example I already gave and which you seem to have overlooked - "Blueness".

Cheers, john

[Wondering]

The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.

But what about this? In his Introduction to Mathematical Logic (4th ed (http://www.amazon.com/exec/obidos/tg/detail/-/0412808307?v=glance)), Mendelson says (p.34): "Most often, one can effectively decide whether a given wf [well-formed formula] is an axiom; in such a case, L [a formal theory] is called an axiomatic theory."

Accordingly, the first axiom of Mendelson's system, P -> (Q -> P), can be proven (decided) in that system as such:


Prove: |- P -> (Q -> P)

1. P hypothesis
2. Q hyp
3. P 1, repeat (or Lemma 1.8 & MP)
4. P, Q |- P 1-3
5. P |- Q -> P 4, deduction theorem
6. |- P -> (Q -> P) 5, deduction theorem


Mendelson does not give that proof, but it follows from the rules of his system. I'm not trying to be 'Mr Contrary' here. I know Witt's knowledge of logic far exceeds my own! But as a student of logic I just want to understand why the above statements (ie, Witt's and Mendelson's) seem to conflict. Such apparent conflicts are usaully an indicator that there's something to learn. ~Wondering

[kennethamy]

Witt:

You seem to have misunderstood my remark. I first responded to your statement "My cat is black is factually true, but, it cannot be logically necessary that it is black, can it?"

with my statement: "It is necessary that a truth is in accordance with the facts."

Under what circumstances can it be factually true that a cat is black but not logically necessary that the same cat is black?

Cheers, John


Although is is certainly logically necessary that if this cat is black, then this cat is black; it is not logically necesssary that this cat is black, for it is not self-contradictory (although it is false) that this cat is white.

[Preno]

Welcome to IIDB, Wondering, and enjoy your stay. But one thing about the forum etiquette - don't resurrect 3 year old threads. Start a new one with the same topic if you want. It confuses a lot of people (like me, in this case) if you reply to posts that are more than a year old.

That said, it's impossible to do modal logic with truth tables. Consider the two statements:
(1) "G. Bush is the president of the USA and G. Bush is not the president of the USA"
(2) "G. Bush is the president of the USA and Queen Elizabeth rules Norway"

Both are a conjunction of a (in terms of the OP) 'factually true' and a 'factually false' statement. Yet (1) is 'logically false' and (2) is only 'factually false'.

[Breath]

These tables apply only to one statement, truth tables for two or more propositions are more complex.

Either, the Cowboys won the Superbowl in 1995, or, the Cowboys did not win the Superbowl in 1995.

This has the form 'factual truth or factual falsity' ie. (2 v 3).
And, it certainly is (1) logically true.

Witt

Where in your scheme of things would the following belong?:

Tuesday is black.

In my scheme of things it is neither true nor false, it is meaningless. Because it is different to be meaningless than it is to be false or true, a system that allows only for true or false is always going to have limited applicability in the real world, IMO.

[TheMathGuy]

I'm not sure what evidence you have to claim that axioms are "chosen" on the basis of acheiving expected results! I think that claim would require a lot more research and understanding of the mind. I think "axioms are selected from a trial and error process as to their empirical accuracy" is closer to the truth. ;)

A lot of how axioms are chosen depends on who is doing the choosing. If someone is trying to do something practical, such as an engineer or applied mathematician, then yes, naturally you would try to choose your axioms in such a way as to have a system that was empirically accurate. And the fact that often axioms are chosen for their apparent "self-evident-ness" cannot be denied. But if you're a theoretical mathematician, you mess around with the axioms just for the sheer fun of it! And as sometimes happens, later on somebody actually does find a practical use for a mathematical system that once seemed counterintuitive. However as a theoretical mathematician practicality is not your primary motivation, it's intellectual curiosity. It's like a quote I heard the other day (not sure who first said it): "Theoretical mathematics is a lot like sex. Sure it has a practical application, but that's not why we do it!"

I leave you with this thought: Consider the statement "1 + 1 = 10". It doesn't seem very correct, unless you know something about computer science and the binary number system.

[comiezapr]

I guess i arrived late here, and i dont have alot of time to post so ill keep my comments to a minimum.

In a formal system the first thing you do is define the set of symbols you use. You have done this, sort of, except that you dont make it explicit what sybols there are in the language you want to create. The second thing you do is charachterize the well formed formulas with a (usually recursive) definition. You have not done this; there is no charachterization of the well formed formulas in the system.

The next thing to do is decide whether you are going to use model theoretic methods to analyze the algebra of the system or a proof theoretic system to charachterize the deductive structure of the system. (Some model theoretic acounts that you should be familiar with are truth tables for the standard propositional logic.) After making this decision you follow through with the plan.

You are, apparently, doing model theory. You have, vaguely, said how to evaulate the truth of strings of symbols. How do i evaulate <>p<><><>pvr<>pvvvvvr? You have given an algebraic system alright, but thats hardly the start of a "logic." (Actually you have not given an algebraic system of the sort you think you do. So far the system deals with numbers and functions defined on those numbers, you have not given an acount of how to assign truth values to the various symbols; in other words you have not given an acount of how to forge the link between strings of symbols and evaluation of truth tables.) Edit: You fail to give a model theoretic acount of the other symbols, which is required if you plan to actually make a cogent system. As it stands the "definitions" of symbols in terms of other symbols stand as introduction and elimination rules for those symbols within a proof theory or statements of statements of equivalence within model theory. This cant be done, in your system, because you need to either specify what it means, proof theoretically, for a string of symbols to be "defined" in terms of other strings of symbols or give a model theoretic acount of what it is for two string of symbols to be equivalent. You have done niether of these tasks and so cant "define" new symbols in terms of old.

This is all irrelevant since it you havent even specified what symbols are part of the system in question! Can i use symbols like X, or 2, or @, or $, or \?

There are to many preliminary problems of this sort for me to comment on the actual system you create. The whole thing is just to confused and inexplicit. Logic isnt evaluating truth values.

I can comment on the motivation for the system though. What do you plan to acomplish by adding some truth values and symbols to propositional logic? The point of modal logic is to give an adequate algebraic treatment of the various modal terms, quantificational terms, signular terms ... etc that occur within a (highly idealized) natural language. The big problem is that modal contexts are intensional; if you replace two words that have the same referant within a modal operator you get different truth values. Your system doesnt even attempt to analyze the intentionality of the modalities. It isnt clear to me that your system does anything philosophically or mathematically interesting at all.

Edit 2:

Any fomula that has all 1's is a theorem.

What could this possibly mean? Whats a formula to begin with? What does it mean for a formula to have all 1s? What point is there in defining something as a theorem (it doesnt occur to be significant in the later theory)?

Please repost what you want to say keeping my criticisms in mind. I will comment if the system is regimented to the point required for it to be a formal system. I do not need examples of truth tables, just defining the structure of the system is good enough for me!

[comiezapr]

No. The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.

What does it mean for an axiom to be decidable? (I know the standard definition, but i have no idea what youre talking about.) Usually it is the case that axioms are the paragon of decidability in deductive systems! You take the axioms as all true. Then when you ask "Is it true that axiom 1 is true?" (this is the question of decidability) you get the unequivocal: YES!

Axioms, within a deductive system, are the best example of something that is decidable!

[Wondering]

Welcome to IIDB, Wondering, and enjoy your stay. But one thing about the forum etiquette - don't resurrect 3 year old threads. Start a new one with the same topic if you want. It confuses a lot of people (like me, in this case) if you reply to posts that are more than a year old.
Thanks Preno! Sorry for the resurrection. I was perusing the board for examples of symbols that can be used, and during a search came upon that thread. Seems the "code" method I use above may have to do.


That said, it's impossible to do modal logic with truth tables. Consider the two statements:
(1) "G. Bush is the president of the USA and G. Bush is not the president of the USA"
(2) "G. Bush is the president of the USA and Queen Elizabeth rules Norway"

Both are a conjunction of a (in terms of the OP) 'factually true' and a 'factually false' statement. Yet (1) is 'logically false' and (2) is only 'factually false'.

You may be right about it being impossible to apply truth tables in modal logic, that's Witt's argument, and maybe he's right. But I don't see how your example suggests he's wrong. The 'logical falsity' of (1) means it syntactically entails falsum (ie, contradiction), and the 'factual falsity' of (2) means it is semantically false. Truth tables traditionally involve the semantics of propositional logic. Statement (1) is also semantically false (it's false that Bush is both the president and not the president), and so both (1) and (2) would have truth values of F in such a truth table. But I don't see how that causes a problem for propositional logic (where truth tables are unquestionably used), and so I don't see how it would cause problems for Witt's proposed modal logic application. Note too that he uses multivalent truth values. ~Wondering

[TheMathGuy]

Correct me if I'm wrong, but isn't this system essentially equivalent to a two-universe system (let's call the universes 'U' and 'V') where 1 represents the statements that are true in both universes, 2 represents the statements that are true in U but not in V, 3 represents the statements that are false in U but true in V, and 0 represents the statements that are false in both universes?

Having 4 truth values seems to be an unnecessary complication. I see no reason we couldn't simplify it by letting:

p1 = p(U) & p(V)
p2 = p(U) & ~p(V)
p3 = ~p(U) & p(V)
p0 = ~p(U) & ~p(V)

We could then define:
<>p = p(U) v p(V)
[]p = p(U) & p(V)

and let ~, v, &, ->, and <-> have their usual meanings. I'm fairly convinced that this simpler system is equivalent to the 4-value one.

[Wondering]

What does it mean for an axiom to be decidable?
As an example, in my questioning above (http://www.iidb.org/vbb/showpost.php?p=3722586&postcount=41) of the same comments I give an example of an effective procedure (ie, proof) that expresses decidability in Mendelson's axiomatization of propositional logic. ~Wondering

[Preno]

You may be right about it being impossible to apply truth tables in modal logic, that's Witt's argument, and maybe he's right. But I don't see how your example suggests he's wrong. The 'logical falsity' of (1) means it syntactically entails falsum (ie, contradiction), and the 'factual falsity' of (2) means it is semantically false. Truth tables traditionally involve the semantics of propositional logic. Statement (1) is also semantically false (it's false that Bush is both the president and not the president), and so both (1) and (2) would have truth values of F in such a truth table. But I don't see how that causes a problem for propositional logic (where truth tables are unquestionably used), and so I don't see how it would cause problems for Witt's proposed modal logic application. Note too that he uses multivalent truth values. ~WonderingWell, I know what 'logical' and 'factual' falsity means. Both statements are of the form (given the OP's convention that "the four truth values are: 1= logical truth, 2 = factual truth, 3 = factual falsity, 0 = logical falsity.") 2 AND 3, yet one of them is 3 and the other 0. Which implies that one cannot assign a value to the conjunction of 2 and 3 based simply on knowing that one statement is factually true and the other factually false.

[Witt]

Quote:Preno,

That said, it's impossible to do modal logic with truth tables. Consider the two statements:
(1) "G. Bush is the president of the USA and G. Bush is not the president of the USA"
(2) "G. Bush is the president of the USA and Queen Elizabeth rules Norway"

Both are a conjunction of a (in terms of the OP) 'factually true' and a 'factually false' statement. Yet (1) is 'logically false' and (2) is only 'factually false'.


Thanks Preno!

You may be right about it being impossible to apply truth tables in modal logic, that's Witt's argument, and maybe he's right. But I don't see how your example suggests he's wrong. The 'logical falsity' of (1) means it syntactically entails falsum (ie, contradiction), and the 'factual falsity' of (2) means it is semantically false. Truth tables traditionally involve the semantics of propositional logic. Statement (1) is also semantically false (it's false that Bush is both the president and not the president), and so both (1) and (2) would have truth values of F in such a truth table. But I don't see how that causes a problem for propositional logic (where truth tables are unquestionably used), and so I don't see how it would cause problems for Witt's proposed modal logic application. Note too that he uses multivalent truth values. ~Wondering

In mutivalent systems, we can have different interpretations of the logical operators.

In particular the 'conjunction function' has at least two interpretations:

1a. factual truth & factual falsity = logical falsity.
1b. factual truth ^ factual falsity = factual falsity.

If we apply the ^ function to Preno's statements, then he is correct.

This example, by Preno does not show that, " it's impossible to do modal logic with truth tables" ..in any way!

I am happy to challenge: you, Preno, or anyone else, ..to provide an instance of failure to my system!!

[Preno]

I mean, don't you see that the same argument works for both "interpretations"? The point is that the conjunction of a factual truth and a factual falsity is sometimes false and sometimes true, depending on what particular factual truth or falsity it is. (2 AND 3 cannot be either 3 or 0. You see, if you define 2 AND 3 as 3 (or 0), you define it for all propositions p and q which are 2 and 3 respectively.) This certainly does prove that modal logic cannot be done with truth tables. I don't intend to sound hostile, but if you don't want to accept it, that's your problem.

If you don't consider be a sufficient "authority", let me quote:
One should note that the truth or falsity of []p is not decided by p's being true. Thus [] is not a truth-functional operator (unlike the usual logical connectives, for instance) and so there is no direct way of using truth-tables to analyze propositions containing modal operators.
(source (http://www-formal.stanford.edu/jmc/mcchay69/node22.html))

[Witt]

Quote: Witt,
No. The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.

What does it mean for an axiom to be decidable? (I know the standard definition, but i have no idea what youre talking about.) Usually it is the case that axioms are the paragon of decidability in deductive systems! You take the axioms as all true. Then when you ask "Is it true that axiom 1 is true?" (this is the question of decidability) you get the unequivocal: YES!

Axioms, within a deductive system, are the best example of something that is decidable!

Nonsense!
All axioms are the primitive (assumed truths) of that system which uses them, by definition.

Please demonstrate a proof of any axiom of any logic, within that same system.

For example, how can you prove: (p v p) -> p, within Russell's Principia Mathematica?? Lots Of Luck.

[Witt]

I mean, don't you see that the same argument works for both "interpretations"? The point is that the conjunction of a factual truth and a factual falsity is sometimes false and sometimes true, depending on what particular factual truth or falsity it is. (2 AND 3 cannot be either 3 or 0. You see, if you define 2 AND 3 as 3 (or 0), you define it for all propositions p and q which are 2 and 3 respectively.) This certainly does prove that modal logic cannot be done with truth tables. I don't intend to sound hostile, but if you don't want to accept it, that's your problem.

If you don't consider be a sufficient "authority", let me quote:

Preno:"This certainly does prove that modal logic cannot be done with truth tables. I don't intend to sound hostile, but if you don't want to accept it, that's your problem."


It most certainly does not.
Your understanding of proof will not do!

Where is your proof??
Indeed, where is his proof?

Please provide an argument that shows your so-called proof.

To quote so-called authorities who know less about modal logic than you do does not help your claims.

Please prove your claim that "modal logic cannot be done with truth tables", with or without your hostilities!

Poop or get the hell off the pot!

[Preno]

Chill. :huh:

The point is that you cannot determine the value of 2 AND 3. I gave an example to illustrate this. If you claim you can, what is the value of 2 AND 3?

[Witt]

Chill. :huh:

The point is that you cannot determine the value of 2 AND 3. I gave an example to illustrate this. If you claim you can, what is the value of 2 AND 3?


In particular the 'conjunction function' has at least two interpretations:

1a. factual truth & factual falsity = logical falsity.
1b. factual truth ^ factual falsity = factual falsity.

If we apply the ^ function to Preno's statements, then he is correct.


Why don't you read my posts??

[Preno]

Well, I do read your posts but they don't make much sense. Let me do it step by step:

Let a be "G. Bush is the president of the USA", b be "G. Bush is not the president of the USA" and c be "Queen Elizabeth rules Norway". Then:

a=2
b=3
c=3

But we have:

a AND b = 0

and at the same time:

a AND c = 3

Thus, the value of 2 AND 3 would need to be in some cases 0 and in some cases 3. In does not have two alternative interpretations - it would need to have two interpretations at the same time! Which is a contradiction.

[Wondering]

All axioms are the primitive (assumed truths) of that system which uses them, by definition.

Please demonstrate a proof of any axiom of any logic, within that same system.

Witt, please see my query to you (http://www.iidb.org/vbb/showpost.php?p=3722586&postcount=41). Mendelson (as well as others) says axiomatic theories are precisely those formal theories within which the axioms can be proved. According to Mendelson, that's the case at least for propositional logic. Though the whole point of my inquiry was that I know you know more about these matters than I, so you're probably saying something accurate and once I see what you're saying I'll learn something. It is said that axioms are assumed truths, as you say, but then what's Mendelson talking about? I suspect it's that in some systems the axioms can be proved, but not necessarily in all systems. ~Wondering

[Wondering]

This example, by Preno does not show that, " it's impossible to do modal logic with truth tables" ..in any way!

That's what I was also saying. ~Wondering

[Preno]

That's what I was also saying.It's quite simple, actually. You can to choose which value you will assign to AND(2,3). And whichever value you choose, there will be some statements that will be assigned incorrect truth value (one of my two example statements will be assigned an incorrect truth value whether you choose 3 or 0 as the functional value of AND(2,3)). Which part of this do you disagree with, precisely?

[Wondering]

It's quite simple, actually. You can to choose which value you will assign to AND(2,3). And whichever value you choose, there will be some statements that will be assigned incorrect truth value (one of my two example statements will be assigned an incorrect truth value whether you choose 3 or 0 as the functional value of AND(2,3)). Which part of this do you disagree with, precisely?

Are you saying in your initial comments (http://www.iidb.org/vbb/showpost.php?p=3723122&postcount=43) that what you say means no method of using truth tables in modal logic could ever be devised, or that Witt's method will not work? I assumed the former, yet your critique now more clearly seems to revolve around his method of defining truth values. ~Wondering

[TheMathGuy]

Quote: Witt,
No. The axioms, of deductive systems, are undecidable. That is, they cannot be decided by the system that uses them.
They are unprovable there.

I think we're getting stuck on a technicality here. In any deductive system, the proof of an axiom is the axiom itself. So in that regard, yes, every axiom is provable. However, a desirable property of any deductive system is that that the axioms not be redundant. Besides wanting this for purely aesthetic reasons, it is also a preventative measure, since if an axiom 'a' is provable using some set of axioms 'S', then adding the axiom '~a' to S would result in an inconsistent system of logic. So we try to construct logical systems in such a way that no axiom is provable in it's logic system when that axiom is removed from the system.

[Wondering]

Preno, you may well be right. Based on better understanding your point, what strikes me is that Witt's definitions of truth values mixes syntax ("logical truth") and semantics (factual truth), whereas truth tables in propositional logic only reflect semantic truth. So given a case where you have some statement f(x,y) (where f is some binary operator and x and y are the truth values of its operands) that is both factually and semantically T or F, it seems our valuation function V(f(x,y)) mapping statements to truth values is not a function since it maps to two truth values. But maybe Witt's got an answer.

It should be noted that 'truth' in logic involves semantics (ie, making references from the language to an external world). Tarski's definition of truth makes that clear. So "logical truth" is somewhat of a misnomer. Logic without semantics (the syntax of logic) is concerned with logical consequence, ie, what formulas can be derived from other formulas. And there are no 'true formulas,' only valid or invalid inferences from some set of formulas. Only when formulas become interpreted in some domain and thereby become statements does 'truth' emerge as an issue. So Preno, you may well be right. ~Wondering

[comiezapr]

As an example, in my questioning above (http://www.iidb.org/vbb/showpost.php?p=3722586&postcount=41) of the same comments I give an example of an effective procedure (ie, proof) that expresses decidability in Mendelson's axiomatization of propositional logic. ~Wondering

The notion of decidability that the particular author you mention is standard usage. The decidability question is "can this formula be proved in so-and-so deductive system." Witt, i think, is not using the term standardly; he wants to say that the axioms arent decidable in respect to being provable within a universal system of rationality. This is a philosophical, and not a mathematical issue.

[Wondering]

The notion of decidability that the particular author you mention is standard usage. The decidability question is "can this formula be proved in so-and-so deductive system." Witt, i think, is not using the term standardly; he wants to say that the axioms arent decidable in respect to being provable within a universal system of rationality. This is a philosophical, and not a mathematical issue.

No, he specified decidability within a given system, not in some meta "universal system of rationality." ~Wondering

[comiezapr]

So I think I'm beginning to see how Witt's right, and as I noted, I figured he'd not be inaccurate. ~Wondering

No, hes wrong. Lets charachterize a proof of a formula first. A proof is a sequence of formulas (ill assumed these are already defined), where each formula is either an axiom of the system in question or follows from an inferance rule of the system and the forumlas preceding it. The last formula in the proof of particular formula is the formula to be proved.

Now lets ask the decidability question: Is there a proof of the axiom X? Well, we would need to construct a sequence of forumlas that has as the last forumla the axiom x to be proved and where each formula before that is either an axiom or follows by an inferance rule from the formulas preceding it. How do we construct such a proof? We state axiom x!

Simple. Witt is wrong.

Edit: What Witt wants to say, and what you are implying, is that an axiom cant be proven from other axioms and inferance rules of the deductive system. (At least each axiom should be such that it cant be proven from other axioms and inferance rules. Most axiomatic systems that are actually discussed have this property so ill assume that it is the case.) This is exactly what MathGuy said. This is uninteresting, uninteresting to the point that it is dialectically neutral in a philosophical argument. Considering that it was said AS an argument to something, Witt is probably a little off on his original comment (at least when decidability istaken in the normal sense of the word.)

[comiezapr]

I think we're getting stuck on a technicality here. In any deductive system, the proof of an axiom is the axiom itself. So in that regard, yes, every axiom is provable. However, a desirable property of any deductive system is that that the axioms not be redundant. Besides wanting this for purely aesthetic reasons, it is also a preventative measure, since if an axiom 'a' is provable using some set of axioms 'S', then adding the axiom '~a' to S would result in an inconsistent system of logic. So we try to construct logical systems in such a way that no axiom is provable in it's logic system when that axiom is removed from the system.

It is absolutly great to see someone with a command of formal systems here! Welcome abord MathGuy! You dont happen to be an avid reader of philosophical issues in the modalities too do you?! :)

[TheMathGuy]

So I've managed to convince myself that my simplification of the modal logic system posted by Witt is in fact an equivalent system. I suppose in the end it's just a matter of aesthetics as to which one you prefer. Given my background in computer science, I tend to prefer a binary system myself. Here's how the logic might look in what I believe to be the equivalent two universe system:

A "truth" value would consist of two binary digits, one to indicate if it is true in "this" universe, and another to indicate if it is true in the "alternative" universe.

11 = necessary truth
10 = factual truth
01 = factual falsity
00 = necessary falsity

Then we would have the following:
Truth table for the unary operators ~, <>, and []:
p | ~p | <>p | []p
___|______|_______|_______
00 | 11 | 00 | 00
01 | 10 | 11 | 00
10 | 01 | 11 | 00
11 | 00 | 11 | 11


Truth table for the binary operator v (showing the value of p v q)
q| 00 | 01 | 10 | 11
p__|______|______|______|______
00 | 00 | 01 | 10 | 11
___|______|______|______|______
01 | 01 | 01 | 11 | 11
___|______|______|______|______
10 | 10 | 11 | 10 | 11
___|______|______|______|______
11 | 11 | 11 | 11 | 11

To ensure there is no ambiguity, one could define a syntax that makes heavy use of parentheses:

S
S --> ~(S)
S --> <>(S)
S --> [](S)
S --> (S)v(S)
S --> (S)->(S)
S --> (S)&(S)
S --> (S)<->(S)
S --> p
S --> q

So one example of a valid statement in the system would be ((p->q)v([](~(p))))&((<>(p))<->((p)&(p)))

Yes, it's clunky I know, but if you wanted something better you would need to define a cleverer syntax that included things such as order of operation and proper associativity. Let's just assume for now that as humans we can tell when a statement is ambiguous and where it needs parentheses and where it doesn't (though this is not always so obvious!). Since the variables hold two truth values each, I think it would seem fair to call this a bi-modal system.

[TheMathGuy]

It is absolutly great to see someone with a command of formal systems here! Welcome abord MathGuy! You dont happen to be an avid reader of philosophical issues in the modalities too do you?! :)

I must admit my knowledge is more on the level of formal mathematical training, so I can't say I'm all that well read on the philosophy of logic. I'm well aware that no logical system is "perfect" in it's applicability to the "real world". And even worse, no sufficiently complex logical system is even "perfect" in and of itself ("perfect" in the sense of satisfying certain desirable properties, such as being consistent (the system cannot prove true=false), and complete (every statement expressible by the system is either provably true or provably false)). It's a well known result in mathematics known as Gödel's Incompleteness Theorem, and no doubt had a strong influence on the development of post-modernism.

[Wondering]

Good analyses comiezapr and TheMathGuy. I think we agree that an axiom is provable in a proof that is a sequence of wfs a1, ..., an such that n = 1.

[Wondering]

an axiom is provable in a proof that is a sequence of wfs a1, ..., an such that n = 1.

In fact, from that couldn't derive the following definition of 'axiom' ? :

Formula b is an axiom of a formal theory L iff there is a proof a1, ..., an in L of formula an where b = a1 = an and n = 1.

[Wondering]

A "truth" value would consist of two binary digits, one to indicate if it is true in "this" universe, and another to indicate if it is true in the "alternative" universe.

11 = necessary truth
10 = factual truth
01 = factual falsity
00 = necessary falsity

Does it follow in your system that a truth value is a string x1...xn (where xi is a '0' or '1') such that the number of worlds = n? For example, suppose the set of worlds in a modal frame, or structure, contains three members, would we have this for "this" world:

111 = necessary truth
110 = factual truth a
101 = factual truth b
100 = factual truth c
011 = factual falsity a
010 = factual falsity b
001 = factual falsity c
000 = necessary falsity

?

Then it seems the number of truth values = 2n, where n = number of worlds.

[Preno]

Are you saying in your initial comments that what you say means no method of using truth tables in modal logic could ever be devised, or that Witt's method will not work? I assumed the former, yet your critique now more clearly seems to revolve around his method of defining truth values. ~WonderingWell, I'm saying both. A truth-table based semantic must assign some specific value to AND(2,3), which will then lead to incorrect assignment for some statements.
Preno, you may well be right. Based on better understanding your point, what strikes me is that Witt's definitions of truth values mixes syntax ("logical truth") and semantics (factual truth), whereas truth tables in propositional logic only reflect semantic truth. So given a case where you have some statement f(x,y) (where f is some binary operator and x and y are the truth values of its operands) that is both factually and semantically T or F, it seems our valuation function V(f(x,y)) mapping statements to truth values is not a function since it maps to two truth values.Precisely.
It should be noted that 'truth' in logic involves semantics (ie, making references from the language to an external world). Tarski's definition of truth makes that clear. So "logical truth" is somewhat of a misnomer. Logic without semantics (the syntax of logic) is concerned with logical consequence, ie, what formulas can be derived from other formulas. And there are no 'true formulas,' only valid or invalid inferences from some set of formulas. Only when formulas become interpreted in some domain and thereby become statements does 'truth' emerge as an issue.Good point (although as a short-hand, I think it's possible to speak of true formulas).

[Witt]

Where in your scheme of things would the following belong?:

Tuesday is black.

In my scheme of things it is neither true nor false, it is meaningless. Because it is different to be meaningless than it is to be false or true, a system that allows only for true or false is always going to have limited applicability in the real world, IMO.

Yes, of course.
Propositional Logic only deals with propositions, i.e. those statements which do have truth values.


Tuesday is black, has no truth value.

[Witt]

Well, I do read your posts but they don't make much sense. Let me do it step by step:

Let a be "G. Bush is the president of the USA", b be "G. Bush is not the president of the USA" and c be "Queen Elizabeth rules Norway". Then:

a=2
b=3
c=3

But we have:

a AND b = 0

and at the same time:

a AND c = 3

Thus, the value of 2 AND 3 would need to be in some cases 0 and in some cases 3. In does not have two alternative interpretations - it would need to have two interpretations at the same time! Which is a contradiction.


Wrong.
We cannot apply the 4 valued system to more than one propositional variable.

Again, you did not read my post.
From the original post I state: "The truth value analysis of modal logic with two propositional variables requires 16 values, I will show the relevant tables if there is some interest."

Your example is an instance of 'two propositional variables'.

Once again ..The truth value analysis of modal logic with two propositional variables requires 16 values, I will show the relevant tables if there is some interest.

Are you interested in truth tables for modal logic with more than one variable??

[Witt]

So I've managed to convince myself that my simplification of the modal logic system posted by Witt is in fact an equivalent system. I suppose in the end it's just a matter of aesthetics as to which one you prefer. Given my background in computer science, I tend to prefer a binary system myself. Here's how the logic might look in what I believe to be the equivalent two universe system:

A "truth" value would consist of two binary digits, one to indicate if it is true in "this" universe, and another to indicate if it is true in the "alternative" universe.

11 = necessary truth
10 = factual truth
01 = factual falsity
00 = necessary falsity

Then we would have the following:
Truth table for the unary operators ~, <>, and []:
p | ~p | <>p | []p
___|______|_______|_______
00 | 11 | 00 | 00
01 | 10 | 11 | 00
10 | 01 | 11 | 00
11 | 00 | 11 | 11


Yes.

Truth table for the binary operator v (showing the value of p v q)
q| 00 | 01 | 10 | 11
p__|______|______|______|______
00 | 00 | 01 | 10 | 11
___|______|______|______|______
01 | 01 | 01 | 11 | 11
___|______|______|______|______
10 | 10 | 11 | 10 | 11
___|______|______|______|______
11 | 11 | 11 | 11 | 11

Definitely not!

To ensure there is no ambiguity, one could define a syntax that makes heavy use of parentheses:

S
S --> ~(S)
S --> <>(S)
S --> [](S)
S --> (S)v(S)
S --> (S)->(S)
S --> (S)&(S)
S --> (S)<->(S)
S --> p
S --> q

So one example of a valid statement in the system would be ((p->q)v([](~(p))))&((<>(p))<->((p)&(p)))

Yes, it's clunky I know, but if you wanted something better you would need to define a cleverer syntax that included things such as order of operation and proper associativity. Let's just assume for now that as humans we can tell when a statement is ambiguous and where it needs parentheses and where it doesn't (though this is not always so obvious!). Since the variables hold two truth values each, I think it would seem fair to call this a bi-modal system.

Wrong again!

Modal logic with two propositional variables cannot be expressed by a 4-valued logic. ...however those symbols are expressed. Your concern about binary symbolism is not significant.
Why do you think that (11) is more relevant than (T)??

[Witt]

Good analyses comiezapr and TheMathGuy. I think we agree that an axiom is provable in a proof that is a sequence of wfs a1, ..., an such that n = 1.

What!
Axioms are those expressions such that if we 'assume' they are true, we can "deduce" theorems. Can we say that all theorems are axioms, or, that all axioms are theorems, ..what kind of nonsense is this??

Indeed.
Can we, by your understanding of axiom, say that 22+22=44 is an axiom??

[TheMathGuy]

Does it follow in your system that a truth value is a string x1...xn (where xi is a '0' or '1') such that the number of worlds = n?

Yes, that's how I meant to contruct it.

For example, suppose the set of worlds in a modal frame, or structure, contains three members, would we have this for "this" world:

111 = necessary truth
110 = factual truth a
101 = factual truth b
100 = factual truth c
011 = factual falsity a
010 = factual falsity b
001 = factual falsity c
000 = necessary falsity

?

Then it seems the number of truth values = 2n, where n = number of worlds.

Precisely. So we can dream up tri-modal systems or quad-modal systems, etc... depending on how many universes we would like. As long as you have only finitely many universes one could write a computer program that given a syntatically valid propositional statement and the truth values of it's variables would compute the truth value of the proposition. (In fact this sort of problem would be ideally suited to a computer program since the truth values are already just strings of binary digits.) A more interesting question is that of knowing which propositions are tautologies (meaning they are always true by logical necessity regardless of the truth values of their variables). A good example would be "p -> (q -> p)". If there is a proof of a tautology within your logical system, then it is a theorem.

[Witt]

Yes, that's how I meant to contruct it.



Precisely. So we can dream up tri-modal systems or quad-modal systems, etc... depending on how many universes we would like. As long as you have only finitely many universes one could write a computer program that given a syntatically valid propositional statement and the truth values of it's variables would compute the truth value of the proposition. (In fact this sort of problem would be ideally suited to a computer program since the truth values are already just strings of binary digits.) A more interesting question is that of knowing which propositions are tautologies (meaning they are always true by logical necessity regardless of the truth values of their variables). A good example would be "p -> (q -> p)". If there is a proof of a tautology within your logical system, then it is a theorem.

No, you cannot.
These multivalued systems do not include modal theorems.
Perhaps they have an interest of their own, but it is not a concern of modal logic.


These systems fail

[Wondering]

Check out: Modal logic is "non-tabular" (http://groups.google.com/group/sci.logic/browse_frm/thread/cc3eff7173e8b85a/d551858a912a3b13) by David C. Ullrich. By "non-tabular" he means cannot have truth tables.

[Witt]

Check out: Modal logic is "non-tabular" (http://groups.google.com/group/sci.logic/browse_frm/thread/cc3eff7173e8b85a/d551858a912a3b13) by David C. Ullrich. By "non-tabular" he means cannot have truth tables.


I am Owen. David Ullrich is clearly wrong.

[Wondering]

Can we, by your understanding of axiom, say that 22+22=44 is an axiom??

Hmm. I'd say that if "22+22=44" is said to be a proof of itself, it's said to be an axiom. That said, I think TheMathGuy and comiezapr are correct that an axiom is a trivial proof of itself. I can't find a definition of 'proof' that forbids a one-formula proof.

Can an irreducible atom like P be said to be a proof of itself? If so, that would falsify my proposed definition, unless perhaps we take atoms as axioms or define the formula b as some composition of atoms. ??

[Witt]

Hmm. I'd say that if "22+22=44" is said to be a proof of itself, it's said to be an axiom. That said, I think TheMathGuy and comiezapr are correct that an axiom is a trivial proof of itself. I can't find a definition of 'proof' that forbids a one-formula proof.

Can an irreducible atom like P be said to be a proof of itself? If so, that would falsify my proposed definition, unless perhaps we take atoms as axioms or define the formula b as some composition of atoms. ??

How is it that P proves itself??

Can we say, because of this assumption, that all theorems are axioms,
Give me a break!

How can it be, within any logic, that axioms are theorems???

Clearly, TheMathGuy and comiezapr, are wrong.

How do you assume things into existence,
how do you assume that 1+1=2, is true????

[TheMathGuy]

Truth table for the binary operator v (showing the value of p v q)
q| 00 | 01 | 10 | 11
p__|______|______|______|______
00 | 00 | 01 | 10 | 11
___|______|______|______|______
01 | 01 | 01 | 11 | 11
___|______|______|______|______
10 | 10 | 11 | 10 | 11
___|______|______|______|______
11 | 11 | 11 | 11 | 11

Would you mind pointing out to me exactly where my table of truth values for the binary 'v' operator is wrong? If I am not mistaken, with the substitutions:
1 = 11
2 = 10
3 = 01
0 = 00,
it is in perfect agreement with your version:

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

Modal logic with two propositional variables cannot be expressed by a 4-valued logic. ...however those symbols are expressed.

I'm not entirely sure what it is you are asserting here. If by this you are saying that you would need more than 4 values to express all possible combinations of states of the two variables, I would agree. You would need 16. But the variables themselves would still each only be in one of 4 possible states, no?

Why do you think that (11) is more relevant than (T)??

It is mainly for aesthetic reasons. When I see 'T' I don't know whether I should interpret that as logical truth or merely factual truth. I also think it simplifies the system in that all of the equalities in (4) can now be defined using:

p v q = (p0 v q0, p1 v q1)
where 0v0=0, 0v1=1, 1v0=1, 1v1=1 (i.e. the usual logical meaning of "or")

So while the two systems are equivalent, I think my binary representation of it is more conducive to making the actual computation of the truth values easier (it's the computer scientist in me--I'm thinking about how I might actually code this in something like C or C++).

So if I understand correctly, let me try modifying one of your examples to include two propositional variables and showing what I think is the truth table in both my system and your system:

Example: <>p <-> ~[](~q)

Original system:
<->
<> ~
p []
~
q
______|______|______|______|______|______|______
1 | 1 | 1 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
1 | 1 | 1 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
1 | 1 | 1 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
1 | 1 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______
1 | 2 | 1 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
1 | 2 | 1 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
1 | 2 | 1 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
1 | 2 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______
1 | 3 | 1 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
1 | 3 | 1 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
1 | 3 | 1 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
1 | 3 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______
0 | 0 | 0 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
0 | 0 | 0 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
0 | 0 | 0 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
0 | 0 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______

Binary code system:
<->
<> ~
p []
~
q
______|______|______|______|______|______|______
00 | 00 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
00 | 00 | 00 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
00 | 00 | 00 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
00 | 00 | 00 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______
11 | 01 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
11 | 01 | 11 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
11 | 01 | 11 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
11 | 01 | 11 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______
11 | 10 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
11 | 10 | 11 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
11 | 10 | 11 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
11 | 10 | 11 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______
11 | 11 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
11 | 11 | 11 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
11 | 11 | 11 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
11 | 11 | 11 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______

[Witt]

Would you mind pointing out to me exactly where my table of truth values for the binary 'v' operator is wrong? If I am not mistaken, with the substitutions:
1 = 11
2 = 10
3 = 01
0 = 00,
it is in perfect agreement with your version:

Yes, within a singe monadic function.

[quote]I'm not entirely sure what it is you are asserting here. If by this you are saying that you would need more than 4 values to express all possible combinations of states of the two variables, I would agree. You would need 16. But the variables themselves would still each only be in one of 4 possible states, no?

Certainly not.

[quote]It is mainly for aesthetic reasons. When I see 'T' I don't know whether I should interpret that as logical truth or merely factual truth. I also think it simplifies the system in that all of the equalities in (4) can now be defined using:

p v q = (p0 v q0, p1 v q1)
where 0v0=0, 0v1=1, 1v0=1, 1v1=1 (i.e. the usual logical meaning of "or")

So while the two systems are equivalent, I think my binary representation of it is more conducive to making the actual computation of the truth values easier (it's the computer scientist in me--I'm thinking about how I might actually code this in something like C or C++).

So if I understand correctly, let me try modifying one of your examples to include two propositional variables and showing what I think is the truth table in both my system and your system:

Example: <>p <-> ~[](~q)


<>p <-> ~[](~q), cannot be expressed by a 4-valued logic at all!

Your computer expertise is of no help!!





Original system:
<->
<> ~
p []
~
q
______|______|______|______|______|______|______
1 | 1 | 1 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
1 | 1 | 1 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
1 | 1 | 1 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
1 | 1 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______
1 | 2 | 1 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
1 | 2 | 1 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
1 | 2 | 1 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
1 | 2 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______
1 | 3 | 1 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
1 | 3 | 1 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
1 | 3 | 1 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
1 | 3 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______
0 | 0 | 0 | 1 | 0 | 0 | 1
______|______|______|______|______|______|______
0 | 0 | 0 | 1 | 0 | 3 | 2
______|______|______|______|______|______|______
0 | 0 | 0 | 1 | 0 | 2 | 3
______|______|______|______|______|______|______
0 | 0 | 0 | 0 | 1 | 1 | 0
______|______|______|______|______|______|______

Binary code system:
<->
<> ~
p []
~
q
______|______|______|______|______|______|______
00 | 00 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
00 | 00 | 00 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
00 | 00 | 00 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
00 | 00 | 00 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______
11 | 01 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
11 | 01 | 11 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
11 | 01 | 11 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
11 | 01 | 11 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______
11 | 10 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
11 | 10 | 11 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
11 | 10 | 11 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
11 | 10 | 11 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______
11 | 11 | 00 | 00 | 11 | 11 | 00
______|______|______|______|______|______|______
11 | 11 | 11 | 11 | 00 | 10 | 01
______|______|______|______|______|______|______
11 | 11 | 11 | 11 | 00 | 01 | 10
______|______|______|______|______|______|______
11 | 11 | 11 | 11 | 00 | 00 | 11
______|______|______|______|______|______|______

[TheMathGuy]

Now you've really got me confused. :huh:

I'm afraid you're going to have to explain your last message in further detail if I am to understand. If I am grasping your logic correctly, a variable represents some proposition that can take on one of 4 possible truth values: necessary truth, factual truth, factual falsity, necessary falsity. So in the example of <>p <-> ~[](~q), if p represented a statement that was factually false (as in "The sky is red") and q represented a statement that was factually true (as in "The earth is the third planet from the sun."), then <>p would be necessarily true, ~q would be factually false, [](~q) would be necessarily false, ~[](~q) would be necessarily true, and hence <>p <-> ~[](~q) would be necessarily true. Why should it be necessary to add more truth values? It would seem from this example that we still only need four:
- necessary truth
- factual truth
- factual falsity
- necessary falsity

What am I missing?

[Witt]

Now you've really got me confused. :huh:

I'm afraid you're going to have to explain your last message in further detail if I am to understand. If I am grasping your logic correctly, a variable represents some proposition that can take on one of 4 possible truth values: necessary truth, factual truth, factual falsity, necessary falsity. So in the example of <>p <-> ~[](~q), if p represented a statement that was factually false (as in "The sky is red") and q represented a statement that was factually true (as in "The earth is the third planet from the sun."), then <>p would be necessarily true, ~q would be factually false, [](~q) would be necessarily false, ~[](~q) would be necessarily true, and hence <>p <-> ~[](~q) would be necessarily true. Why should it be necessary to add more truth values? It would seem from this example that we still only need four:
- necessary truth
- factual truth
- factual falsity
- necessary falsity

What am I missing?

You are missing the point that: <>p <-> ~[]~p is valid in a 4-valued logic, but, <->p <-> ~[]~q, is not valid, there.

If you want to talk about two variables then: (11), (10), (01), (00), will not suffice!


A four valued logic does have its interest, but, it is not sufficient to express all functions.
Especially we cannot express modal functions of more than one variable with the four values:(11),(10), (01), (00).

[TheMathGuy]

You are missing the point that: <>p <-> ~[]~p is valid in a 4-valued logic, but, <->p <-> ~[]~q, is not valid, there.

If you want to talk about two variables then: (11), (10), (01), (00), will not suffice!

OK. I think I'm begining to see your point. I believe I misinterpreted 16 values to mean 16 "truth values", whereas now I believe I should think of it more as 16 "states". So on the computer we could represent it with (1111), (1110), (1101), (1100), (1011), (1010), (1001), (1000), (0111), (0110), (0101), (0100), (0011), (0010), (0001), (0000)

As I matter of fact I am interested in the analysis of two-variable logic. If you can post it, I'd be interested in seeing it.

[Witt]

OK. I think I'm begining to see your point. I believe I misinterpreted 16 values to mean 16 "truth values", whereas now I believe I should think of it more as 16 "states". So on the computer we could represent it with (1111), (1110), (1101), (1100), (1011), (1010), (1001), (1000), (0111), (0110), (0101), (0100), (0011), (0010), (0001), (0000)

As I matter of fact I am interested in the analysis of two-variable logic. If you can post it, I'd be interested in seeing it.

I am happy to hear that. Deliberate rejection is not welcome.

In the case of one proposition: logical truth, empirical truth, empirical fasity, logical falsity, are all that is needed, especially for Lewis's S5.


I we want to deal with modal logic that have two propositional variables, then 16-values are required. ..As I said in my original post.

If we allow more than 4 propositions (in a 4-valued logic) then many contradictions arise.

[Witt]

OK. I think I'm begining to see your point. I believe I misinterpreted 16 values to mean 16 "truth values", whereas now I believe I should think of it more as 16 "states". So on the computer we could represent it with (1111), (1110), (1101), (1100), (1011), (1010), (1001), (1000), (0111), (0110), (0101), (0100), (0011), (0010), (0001), (0000)

Yes, clearly!

'States', 'representative computer word',

Computers allow us to persue many things, that would otherwise be impossible.

As I matter of fact I am interested in the analysis of two-variable logic. If you can post it, I'd be interested in seeing it.

I will.