Is there a sentence that proves itself is not provable?

[Scott Mayers]

Then the Tarski Undefinability Theorem and Gödel's 1931 Incompleteness Theorem die along with it.

If you are asserting these are not sound theorems with respect to the logic they are using, you are wrong. If you are asserting they are not 'sound' with respect to all systems, you are right. But the soundness with respect to reality is what the theorems are speaking about BEGINNING with the assumption of the minimal consistent and complete system they are using. And their inputs to test the validity are from the outside world of the system itself. (like that some theorem of any system is true or false with respect to using self-referential inputs of the system itself.

Because the theorem uses the possible values of 'true' or 'false' of ANY system from the perspective of reality apart from the system they use, it creates a "feedback" loop using the particular systems they are using. This is because within the system, the system's axioms are defaulted to be 'true' ONLY or the system faults. Because you cannot presume the value 'false' of the very axioms of the systems that are consistent and complete, then the system can only test values of reality apart from their own axioms. So the 'soundness' of the theorems these incompleteness theorems speak of are only 'sound' with the limit of assuming consistency and completeness is universally true about reality as a whole.

Thus, what they are basically saying, and implying by extension, is that no mechanism (like a machine or a logical system) can solve all problems universally apart from the axioms being assumed (gambled as 'true')

The only alternative that IS universal is a system that is itself 'complete' and 'consistent' is some real machine that accepts ALL possibilities, which has to include systems that allow for contradiction. This is already understood though.

In essence, these theorems are transferred to philosophy of science that says we can only 'confirm' things about reality as a whole by useful logical means but cannot disprove all things without limiting the domain of inputs of any 'calculator' being used to measure reality.

Another interpretation is that you cannot have some calculating machine (virtual or real) that can encapsulate ALL facts about reality without requiring some machine that is itself bigger than what it is measuring. For instance, you cannot have a REAL machine that can predict the whether precisely at all points in space without some machine that is itself larger than the domain of the space, which then limits the possibility of having some 'ideal' Universal machine that can PREDICT all problems in reality with such a machine. It CAN predict specific parts withing it but requires testing that machine's function in the instances it is using.

A logic that is NOT consistent nor complete CAN be a "Universal Machine" though. But this is considered, "trivial" (meaning literally to have at LEAST three valued logics....that include, say contradiction as a functioning part of it.

[Scott Mayers]

Example of a kind of reasoning (intuitively) is:


It is raining
It is not raining
Therefore, it is raining and not raining.


If we restrict the machine to just DENY this contradiction, we might inappropriately draw the conclusion,
"It is not raining" [something that 'false']

If we ADD a function to deal with this, it could be one that increases the domain, such as adding distinct times where both can occur within the larger reality but separate:

So, not((It is raining) and (It is not raining)) may be possibly true IF we extend the domain in time,
We might do this by first denying the contradiction, such as,


It is raining
It is not raining
It is raining and not raining
X and not-X is true in Dimension Y
X or not X in Dimension Y

Let Dimension Y hold X and Dimension Y-plus-one not-X
Then there is some greater universe that contains Dimensions Y and Y-plus-one, say Z, such that
(X in Y true) and (not-X in not Y true)

One of an infinite possibilities:
That is raining in New York but not raining in Los Angeles.

Note that I'm assuming 'an infinite possibilities' unless we specifically define what the dimensions are because the above possibility may still be wrong. The point of this is to show that reality includes contradiction and precisely a mechanism that segregates the realities. Thus no UNIVERSAL logic can exist that is not inclusive of all possible systems with exclusive domains within the whole.

[PeteOlcott]

If you are asserting these are not sound theorems with respect to the logic they are using, you are wrong.

You can't begin to understand anything that I said you you until you first understand my most recent reply to you.
If forms the foundation for everything else.

[Logik]

The Liar Paradox in English: "This sentence is not true."
The Liar Paradox in Symbolic Logic: LP ↔ ~⊢LP

If a logician hypothesizes that the symbolic logic is a precise translation of
the English sentence, then it is very easy for them to see the error of the Liar Paradox:
LP ↔ ~⊢LP // It can only be true if it can be proven that it is not provable

The system in which you write/present your proofs (piece of paper) IS incomplete and is constraining your thinking.

It lacks one critical aspect: time/change.

In a Turing-complete system the proof is trivial

https://repl.it/repls/BeneficialTiredTitles

Code: Select all

PROPOSITION = 'this sentence is false'

def A(p)
  return true if  p == PROPOSITION
end

def B(p)
  return false if p == PROPOSITION
end

A(PROPOSITION) == (not B(PROPOSITION) )
=> true 

In English:

You gave system A() which somehow evaluates "this sentence is false" as true.
You have system B() which somehow evaluates "this sentence is false" as false.

Is system A or system B the correct one?

I don't know

And the simpler version. Proof that A = ~ A
https://repl.it/repls/GiddyRoughMosaic

Code: Select all

$toggle = false

def a
   $toggle = ( not $toggle )
end

a == ( not a )

I have no other advice for you except: Question everything and break all the "rules".

[Logik]

Example of a kind of reasoning (intuitively) is:


It is raining
It is not raining
Therefore, it is raining and not raining.


If we restrict the machine to just DENY this contradiction, we might inappropriately draw the conclusion,

It's not a contradiction.

07:44:05AM It's raining.
07:44:08AM It's not raining.

It suddenly stopped raining!

Why do you keep suspending time to do "logic" ?

[Scott Mayers]

If you are asserting these are not sound theorems with respect to the logic they are using, you are wrong.

You can't begin to understand anything that I said you you until you first understand my most recent reply to you.
If forms the foundation for everything else.

I hear you saying that I have not understood you. But that is why I asked you to specify a thesis statement first with clarity.

I need to ask what your own axioms are set upon? (consistent or non-consistent rules; do you allow for contradiction or are you beginning with reasoning from a non-contradicting system; Is it strictly binary truth values or does it excludes a possible third or more value?)

That if you begin with a consistent logic, when you input something that is possibly true of something outside of it as inputs, while the inputs can be true and the theorems derived be valid as well as true (ie sound), the conclusions which present themselves as contradictory are themselves 'sound theorems' that are 'true contradictions' of the system but suggest the system is incomplete with respect to the domain of reality that permits the result. So you have to either reduce the domain allowed as inputs or extend the system to include the desired domain.

Since the incompleteness type theorems begin complete and consistent to prove them, the theorems speak only upon their domain. But if you presume an unlimited domain about logic itself, you can't restrict it to speak about the very system you are using to prove it. So exhausting all possible systems of logic based on its own type, a proof that leads to inconsistency/incompleteness is a proof about its own limitations as well when they are required to allow for ALL possible 'sound' inputs, which must test their own system's axioms.

You even showed this with your C program example which means that there is a limit to its capacities. The nature of making something illegal as a test when compiling says that there is at least some real input that cannot be put into that language's axioms. Thus, the C language is unable to COMPLETELY exhaust all problems in reality as part of a larger domain. The 'error' could be just a proof that C cannot be compiled with your input and so you may question the compiler's logic system or something even more beneath it like the architecture's logic.

So you must first show that the C language is complete on its own domain first before you can use it to infer something beyond it.

If it is 'wrong' as you imply, they either they made a mistake in assuming their own system that proves a theorem about incompleteness validly as sound, or your own logic system is different than theirs in some way. Compare: if a given program you input for C can't compile, you need to question whether C can actually take in any information as a program that can compile.

I'm guessing you recognize that a compiler CAN still compile this but that the program, if not checked for this would not halt if ran. This would prove to act in accordance to your expectation of the domain of the C language's correct logic. Thus if you used this logic trusting the computer worked, you'd have a "proof" that the system should hang. So if it were to test all possible programs, by the time it reached this line of programming WHERE the program is itself well-formed, then the nature of not being able to have the machine stop to say there is no end to the hanging would prevent you from passing on to test further possible programs without some outside interference to stop the machine looping.

The incompleteness theorems only assure us that we cannot solve all problems with a single logic system without having at least some program that hangs at some point and make it impossible to determine ALL such infinite possible inputs beyond that hanging one on its own. As soon as some external logic system is required to stop the computer hanging, or let it pass to test the next possible program, you have to then include that outside mechanism's logic system to be a part of the 'completion' of all problems. All that you need to know from outside is that you can have two such hanging programs. Then you know that when it reaches the first, it cannot reach the second hanging one beyond that one to test it, and thus that system would be 'incomplete' on its own.

From my last post above, I showed how reality CAN be contradictory but resolve itself by extending itself to a new dimension. This is like us being able to add parallel processors or other logic systems to test all possible programs. We can keep adding new hardware to enable it to overcome the next hang but we want one to go past all hangs and so have to keep adding more hardware to do this.

But that because reality permits contradiction AND can still have something 'true' result from it, says that only non-complete systems have this power to BE "complete" because it permits dynamic growth of the system. So only if we restrict ourselves to a fixed complete logic system do we run into the problem of it not being able to solve all problems, even where we allow for an infinite set of possible input data.

[Scott Mayers]

Example of a kind of reasoning (intuitively) is:


It is raining
It is not raining
Therefore, it is raining and not raining.


If we restrict the machine to just DENY this contradiction, we might inappropriately draw the conclusion,

It's not a contradiction.

07:44:05AM It's raining.
07:44:08AM It's not raining.

It suddenly stopped raining!

Why do you keep suspending time to do "logic" ?

That is the point though. The resolution of the static (finite) result requires extending to another dimension such as different 'times' (infinite). It only BECOMES non-contradictory once you extend the logic to include a new dimension of time. But then it also doesn't SPECIFY any particular time but has an infinity of possible solutions.

[Logik]

That is the point though. The resolution of the static (finite) result requires extending to another dimension such as different 'times' (infinite).

Que? We are certainly not traversing any dimensions to have this conversation?

It only BECOMES non-contradictory once you extend the logic to include a new dimension of time. But then it also doesn't SPECIFY any particular time but has an infinity of possible solutions.

No. There's no extra-dimensional hoohaa required. This is the implicit vs explicit distinction.

Your statement was made in a context-sensitive grammar.
https://en.wikipedia.org/wiki/Context-sensitive_grammar

And so the truth-value of the claim is relative to the observer. You.

YOU failed to explicitly state the time-window and location of your claim. You left that information out.
In that gap I found ambiguity (room to maneuver).

Demonstration:

07:15:01AM It's raining
07:15:01AM it's not raining

No contradiction. How?!?!?!

07:15:01AM It's raining (In Stockholm)
07:15:01AM it's not raining (In Barcelona)

More?

07:15:01AM It's raining in Stockholm.
07:15:01AM It's not raining in Stockholm.

No contradiction. How?!?!?!?!

27th March 07:15:01AM It's raining in Stockholm.
28th March 07:15:01AM It's not raining in Stockholm.

More????

27th March 07:15:01AM It's raining in Stockholm.
27th March 07:15:01AM It's not raining in Stockholm.

No contradiction. How ?!?!?!?

27th March 2018 07:15:01AM It's raining in Stockholm.
27th March 2019 07:15:01AM It's not raining in Stockholm.

More????? OK!

27th March 2019 07:15:01AM It's raining in Stockholm.
27th March 2019 07:15:01AM It's not raining in Stockholm.

No contradiction. How ?!?!?!?!?!?!?!

27th March 2019 07:15:01AM It's raining in Stockholm, Iowa, USA
27th March 2019 07:15:01AM It's not raining in Stockholm, Sweden.


Context, context, context, context.

The LNC is stated as follows: Contradictory propositions cannot both be true at the same time and in the same sense.

Define "same time"...

If contradictions don't exist in reality, then speaking OF contradictions is self-contradictory.

[Scott Mayers]

That is the point though. The resolution of the static (finite) result requires extending to another dimension such as different 'times' (infinite).

Que? We are certainly not traversing any dimensions to have this conversation?

It only BECOMES non-contradictory once you extend the logic to include a new dimension of time. But then it also doesn't SPECIFY any particular time but has an infinity of possible solutions.

No. There's no extra-dimensional hoohaa required. This is the implicit vs explicit distinction.

Your statement was made in a context-sensitive grammar.
https://en.wikipedia.org/wiki/Context-sensitive_grammar

And so the truth-value of the claim is relative to the observer. You.

YOU failed to explicitly state the time-window and location of your claim. You left that information out.
In that gap I found ambiguity (room to maneuver).

Demonstration:

07:15:01AM It's raining
07:15:01AM it's not raining

No contradiction. How?!?!?!

07:15:01AM It's raining (In Stockholm)
07:15:01AM it's not raining (In Barcelona)

You are just being a contrarian. Your assumption above is the extension of the logic before it. The example I gave begins with a Propositional type calculus that only takes the whole of a position or statement with binary values:

If P = "It is raining", in propositional calculus, the first order of a complete and consistent language,
then not-P = it is not the case that "It is raining".

In propositional calculus, if you assume P "true", then not-P is "false" and vice versa. So if

(P & not-P) is given as true in this system, it concludes not-P where P is posited or P where not-P is posited.

For both to be true, another factor HAS to exist to un-contradict this state. [contradiction -> with-three-states --> P, not-P, or (P & not-P)]

The resolution is to increase the range to include both by having a new axiom that acts to separate the two in distinct places under the same domain. Basically this turns it into a (P or not-P) state by extending the universe to allow for the two to exist in it simultaneously.

In propositional terms, this would be to alter the meaning internally of each by some additional dimension:

Now, let P = "It is raining in/at X" and not-P becomes redefined distinctly as Q = "It is raining in/at some-non-X".

In the style of Noam Chompski, we re-generate a new definition that conserves the reality in a kind of 'updated' system.

Predicate Calculus is an example of an extended logic that depends upon the prior Propositional logic but adds feature that penetrate into those propositions.

[Scott Mayers]

@ logik,

Time is only one possible dimension and is distinct from place. The added dimensions that define one system must add another dimension for every REAL paradox of a valid argument of that system that is still 'sound'.

I don't know why you keep jumping on something I say as though I missed something. (?) All you add is precisely some rephrasing of something I just said but appearing to disagree when you actually don't.

[Logik]

You are just being a contrarian.

Of course you would say that when I rip your "logical" argument apart.

Your assumption above is the extension of the logic before it. The example I gave begins with a Propositional type calculus that only takes the whole of a position or statement with binary values:

Given the propositional calculi that are available to you how and why did you CHOOSE that particular propositional calculus?

If P = "It is raining", in propositional calculus, the first order of a complete and consistent language,
then not-P = it is not the case that "It is raining".

If that is a "complete and consistent language" how come it has so many contingencies?

Surely all those contingencies are evidence AGAINST the "completeness" of your system ?

In propositional calculus, if you assume P "true", then not-P is "false" and vice versa. So if

(P & not-P) is given as true in this system, it concludes not-P where P is posited or P where not-P is posited.

For both to be true, another factor HAS to exist to un-contradict this state. [contradiction -> with-three-states --> P, not-P, or (P & not-P)]

I literally just showed you how your ASSUMPTION fails in practice!

Your "true claims" are RELATIVE. To YOU.

[Logik]

Time is only one possible dimension and is distinct from place. The added dimensions that define one system must add another dimension for every REAL paradox of a valid argument of that system that is still 'sound'.

So now you are drawing distinctions between "real" and "non-real" paradoxes?

And making claims that a contradictory system can still be sound?

I don't know why you keep jumping on something I say as though I missed something. (?) All you add is precisely some rephrasing of something I just said but appearing to disagree when you actually don't.

Because you keep leaving out the most important element of truth. The observer.

The truth-value of the statement "It's raining" is RELATIVE to the observer! Not to the person interpreting the sentence.

Because when you combine the psychological illusion of transparency + shortcommings of propositional logic. That's how miscommunication between humans happens and eternal arguments about "truth" ensue.

https://wiki.lesswrong.com/wiki/Illusio ... ansparency

[PeteOlcott]

If you are asserting these are not sound theorems with respect to the logic they are using, you are wrong.

You can't begin to understand anything that I said you you until you first understand my most recent reply to you.
If forms the foundation for everything else.

I hear you saying that I have not understood you. But that is why I asked you to specify a thesis statement first with clarity.

I need to ask what your own axioms are set upon? (consistent or non-consistent rules; do you allow for contradiction or are you beginning with reasoning from a non-contradicting system; Is it strictly binary truth values or does it excludes a possible third or more value?)

As I have stated repeatedly axioms are expressions of language that have been defined to have the semantic value of true.

A theory T is a conceptual class consisting of certain of these elementary statements. The elementary statements which belong to T are called the elementary theorems of T and said to be true. In this way, a theory is a way of designating a subset of E which consists entirely of true statements. (Haskell Curry 2010).

Think of axioms as a set of Prolog Facts added to a database one at a time, and rejected if they are ever found contradict any facts already in the system.

That if you begin with a consistent logic, when you input something that is possibly true of something outside of it as inputs, while the inputs can be true and the theorems derived be valid as well as true (ie sound), the conclusions which present themselves as contradictory are themselves 'sound theorems' that are 'true contradictions' of the system but suggest the system is incomplete with respect to the domain of reality that permits the result. So you have to either reduce the domain allowed as inputs or extend the system to include the desired domain.

If we query a Prolog datebase that does not have the current weather conditions for the current location, yet has the current location and the current time: raining_outside(current_location, current_date_time) it responds with "no" because it does not have the current weather. We then provide this query not(raining_outside(current_location, current_date_time)) and it replies no. Thus we can verify that the answer is unknown.

Since the incompleteness type theorems begin complete and consistent to prove them, the theorems speak only upon their domain. But if you presume an unlimited domain about logic itself, you can't restrict it to speak about the very system you are using to prove it. So exhausting all possible systems of logic based on its own type, a proof that leads to inconsistency/incompleteness is a proof about its own limitations as well when they are required to allow for ALL possible 'sound' inputs, which must test their own system's axioms.

When we define the prolog rule comparable to this symbolic logic: True(X) ↔ ⊢X and plug the Liar Paradox into this rule
True(LP ↔ ~⊢LP) the answer that I would expect is No, the Liar Paradox is not true.

It may also be rejected by unify_with_occurs_check/2 as cyclic thus infinitely recursive:
unify_with_occurs_check(Materially_Equivalent(~Provable(LP), LP)), in this case it would not be evaluated.

The same thing would occur when evaluating this simplified essence of the conclusion of the 1931 Incompleteness Theorem:
∀F (F ∈ Formal_Systems & Q ⊆ F) → ∃G ∈ L(F) (G ↔ ~(F ⊢ G)) (Panu Raatikainen 2018)

The problem is that when a formal system was asked to evaluate expressions that are erroneous because they are self-contradictory and they failed to evaluate these expressions correctly the blame was placed on the formal system rather than the self-contradictory expression.

The incompleteness theorems only assure us that we cannot solve all problems with a single logic system without having at least some program that hangs at some point and make it impossible to determine ALL such infinite possible inputs beyond that hanging one on its own.

For the Liar Paradox and the simplified essence of the conclusion 1931 Incompleteness Theorem the formal system only needs to detect and reject self-contradictory expressions of language on the basis of this universal Truth predicate: True(X) ↔ ⊢X.

This portion of the simplified essence of the 1931 Incompleteness Theorem: G ↔ ~(F ⊢ G) is unsatisfiable, thus proving that there are no expressions of language that are true and not provable because True is only correctly construed as the conclusion of sound deductive inference and formal proofs are merely a way to symbolically represent deductive inference:

The chain of symbolic manipulations in the calculus corresponds to and represents the chain of deductions in the deductive system. (Braithwaite 1962: 2)

So when we interpret this: True(X) ↔ ⊢X (and all of the corresponding [chains of symbolic manipulations] as the [the chain of deductions in the deductive system] to a sound conclusion, then simplified essence of the conclusion of the 1931 Incompleteness Theorem, the Tarski Undefinability Theorem and the Liar Paradox are all rejected as erroneous.

[Scott Mayers]

You are just being a contrarian.

Of course you would say that when I rip your "logical" argument apart.

and

Your assumption above is the extension of the logic before it. The example I gave begins with a Propositional type calculus that only takes the whole of a position or statement with binary values:

Given the propositional calculi that are available to you how and why did you CHOOSE that particular propositional calculus?

Propositional Calculus is basic and any system you hold onto so dearly relies on it even though you are ignorant not to know this. [all computers rely on Boolean algebra, which is a limited domain of Propositional calculus, for instance.]

You are disrespecting my own argument's examples by attempting to claim it as your own. I, not you, pointed out how nature absolves contradiction by utilizing extensions of dimension, like time or place. I used examples you just repeated in form. If you actually don't see this, I'm not even going to waste any more time with you at all. It's verging on trolling.

[Scott Mayers]

You can't begin to understand anything that I said you you until you first understand my most recent reply to you.
If forms the foundation for everything else.

I hear you saying that I have not understood you. But that is why I asked you to specify a thesis statement first with clarity.

I need to ask what your own axioms are set upon? (consistent or non-consistent rules; do you allow for contradiction or are you beginning with reasoning from a non-contradicting system; Is it strictly binary truth values or does it excludes a possible third or more value?)

As I have stated repeatedly axioms are expressions of language that have been defined to have the semantic value of true.

A theory T is a conceptual class consisting of certain of these elementary statements. The elementary statements which belong to T are called the elementary theorems of T and said to be true. In this way, a theory is a way of designating a subset of E which consists entirely of true statements. (Haskell Curry 2010).

Think of axioms as a set of Prolog Facts added to a database one at a time, and rejected if they are ever found contradict any facts already in the system.

I see you understand some of the facts of logic. But even your responses here are only stating facts about what logic is or what others have already developed or discovered. I don't understand what you are UNIQUELY adding other than some implicit doubt about the Liar paradox.

I haven't yet found a need for investing in learning Prolog, though I still might in the future. Unless you feel you need to rely on programming to prove something about other programs, can you use some other examples where necessary?

I still need to first know what your own answer to, "Is there a sentence that proves itself is not provable?" Is your paper for a "yes" a "no", "both" or "neither"?