Hello Paul Pistea,

Formal systems rely on a set of assumptions, which are propositions assumed to be true even without proofs. They are called axioms.However they should be consistent with every other formal systems or observations: That is, they differ from dogma.

They are sufficiently fundamental  to be perceived, understood and judged by common sense.

It is this barrier called axioms which limits the process of down-going definition of concepts of a scientific or more precisely mathematical theories

To go even further, it comes out that all our observations rely on perception: That is the point of Kant about Phenomena and Noumena. We cannot go farther than our perception of the world. The world is therefore what we commonly perceive.

So it is on top of these foundations, that we talk about FORMAL.

Finally, I would like to take an example with the question you asked. This question was expressed in a natural language namely English, which is not a FORMAL language, so by definition it is not even precise. But I am almost sure that you think everyone will get you

dear Franklin, thank you! this is well known to me. one more example of an Axiom is: Speed of light is c. needn`t be demonstrated :))

I`m talking about Definitions; not about axioms.

Dear Paul,

I thank you for your quick reply. I address the Problem of Definitions too or may be I was not so clear.

The thing is:

- Formal systems are made up of a set of  Definitions. 

- Those Definitions are expressed in a formal language

- For a formal language. there is a formal Grammar( a formal mechanism to generate syntactically correct sentences), a well-defined vocabulary(a formal mechanism to generate lexically correct words ), an analyzer( a formal mechanism to check syntactic correctness of sentences and words) and an Interpreter( a mechanism to extract the meaning/semantic of sentences only based on the syntax FIRST)

- Since those Formal Theories are built on top of the others, trying to formally semantically, lexically or syntactically analyze a formal definition  would bring you to the first axioms: Propositional logic which is also lexically and syntactically formal. However, we need a more lower formal system to fully get it semantically. There is no such lower formal system except the COMMON SENSE.

 Example:

1-A is Human        :H(A)

2-B is Human       :H(B)

3-C is Human       :H(C)

4-if A can beat B, and B can beat C, then A can beat C:

    Beat(A,B) ^ Beat(B,C) => Beat(A,C)

To understand this 0-order logic text, we repeatedly analyze syntactically:

Sentence 4: It is simply states that if the first two propositions are true then the last is too. We cannot prove it but it is commonly integrated like that(Hypothetical syllogism but compacted).

But what is A can beat B => what does A mean. A is a Human. Common sense must be used to understand "A is a Human " if Human is not further defined.

...

That is why, I said:

"So it is on top of these foundations, that we talk about FORMAL."

If this Answer is not convincing enough, I would like you to clearly state what Goedel said in his theorem that you think it is wrong.

dear Franklin, You are right. I just think that words implicitly make notions static, while they are not. therefore it is impossible to exactly define something while everything changes. that`s why I would introduce an epi-logic Level, versus a logic Level. epi-logic (logic about the logic) is boolean, and logic itself is fuzzy.

Ηι Paul Pistea,

Regardless of any definition there are some principles out of which the logical communication is impossible. The logical communication includes the classical logic the sufficient reason principle and a hidden axion which states that there is anterior-posterior (one letter is put after another, a word is put after another etc.). If we denote by Λ a logic consiting of the classical logic and the sufficient reason principlke it can be proven that:

"A system including logic  Λ   and a statement which is not theorem of  Λ  leads to contradiction."

The anterior-posterior axiom is not a theorem of  Λ (e.g. at the begining of a text the existence of a symbol cannot derive logically from its non existence); therefore our logical communication system is contradictory.

This imposes silence which can be broken by a claim for minimum contradictions since the contradictions never can be vanished. This leads to a minimum contadictions way of thinking which is a kind of fuzzy logic.

This way of thinking is indepened of definitions beyond axioms verified by the experience and can lead to conclusions applicable to reality.Thus, it can lead to a space time quantum mechanics including quantum gravity.

For more details you can search in Google  for "Theorem proving the existence of contradictions...Bowen Publishing."

Logic Λ  is not so simpe; it includes the sufficient reason princiole. Please see the proof of this statement by searching  in Google  for "Theorem proving the existence of contradictions...Bowen Publishing."

I am sending this answer because after sending the first one I noticed the existence of  an additional text.

What precisely is this Λ ?

Logic Λ consists of  the classical logic and the sufficient reason principle.  Please see the proof of the following statement by searching  in Google  for "Theorem proving the existence of contradictions...Bowen Publishing."

"A system including logic  Λ   and a statement which is not theorem of  Λ  leads to contradiction."

dear Athanassios, This is not 100% true, because NonA includes Zero (nothing, indeterminacy) too. (see. neutrosophic, Smarandache, for Details)

dear Franklin, it`s a Long time since I the Notion of POINT: a Point is smaller than itself :))

like I said: I do separate the Logic about the Logic (LaL) and the logic itself. LaL must be boolean. Logic can be neutrosophic, of fuzzy. conclusions from LaL to Logic are allowed. But conclusions Logic->LaL are not allowed

In logic and phisics all terms should be define by others for consist of theory. Definition is precisely if adeqacy describing state. Theory could be closed if is complete, complete is if new definition is hypothesis ad hoc. Logic about logic is meatalogic or philosophy of logic, for science we got philosophy of science resp. methodology. Logic->LaL not allowed becaouse object couldnt desrcribe language who describe object. LaL is describe of Logic. Is Tarski diferenciation of language and metalaguage.

Dear Paul,

Beyond any definition, according to classical Aristotelian logic:

It is non valid  that      non A   proves   A                                                                                  where  A: " there is an object- symbol "  

I feel that the purpose of logic sciences should be the solution of the problems of the reality. If through definitions and excellent inferences we create a  new world the question is raised of which is this relation of this world with the reality itself. Of course I am not against this effort that has done incredible achievments but please leave some space to Aristotelian logic which is in immediate relation with reality. I would like to mention that through this logic and the claim for minimum contradictions one can reach Schroedinger relativistic equation and quantum gravity

dear Athanassios, You are right, but aristotelian logic should be used only as Epi-Logic, i.e. Logic about the (fuzzy) logic, in other words aristotelian logic is indispensable for the construct of levery logic theory (fuzzy, neutrosophic, multivalent, boolean, aso.). Aristotelian Logic is the Beginning, the Basic, The Epi-Logic!

Dear Paul, I agree with you but I would like to draw your attention to the fact that classical logic leads to the fact that our logical communication system is contadictory.

Through the claim for minimum contradictions we obtain  a metalogical fuzzy system whithot the need of extra definitions and assumptions which can explain  the uncertainty principle in physics.

As for Goedels theorem Hillary Putnam has shown that it is based on an arbitrary hypothesis;therefore it can apply only under certain restrictions. This is a problem that Sir Roger Penrose  met in his theory for quanum gravity and consciousness.

Hallo Paul,

Why do you say that  "the point is smaller than itself" ?  It is like assigning a volume to a point and this is only possible when you start connecting mathematical objects to concrete world object. I think Mathematics is about the abstract world even if there might be inspirations from the real world(Formalization) and vice-versa(Modeling).

Now about logic and metalogic or language and metalanguage or data and metadata or algorithm and metaalgorithm. To be Meta- is not an intrinsic or absolute property of something. It is only extrinsic: computer can be seen as a metaprogram, that is a program which takes as parameter another program. This way there will metametaprogram, metametametaprogram,...

The thing with Meta- is that you always want to express the properties of a system, whose the system itself cannot. Therefore the use of a Meta-: That is pointed by the Tarski's undefinability theorem for semantics. If you force, you get cyclic definitions and it is an illusion to know everything.

Meta- only provides more expressiveness of properties but does not bound.

We cannot fully do without perception, since it allows us to interact with each other. However, it is only under the assumption that we can commonly perceive basics that we can start formalizing from those basics.

You said that you separate Logic and Logic about Logic.  There is not only a logic  of propositions or of predicates or of descriptions or  which is fuzzy or boolean or probabilistic... You can create your own logic as formal system having a language for expressiveness, some axioms and rules of inferences. So if LAL meets those requirements it is a logic too and should not be separated from the concept of Logic.I think what you wanted to say something different.

  For example, either for Logic as you mentioned or for LaL, propositions would be string of characters and would be fully interpreted under some circumstances by assigning  values to them. Then apply inference rules to get more.

Type of definitione is few. I recomend talk on philpapers, I talk with few philosopers about this trouble. https://philpapers.org/bbs/thread.pl?tId=1063#p20178

Dear Athanassios, You are right. e.g. to a (fuzzy) logic Proposition a true value must be attached. so one can say in which degree the Proposition is true. for instance: means that, compared to other rainy days, it rains pretty hard. the Logic about this Logic has to confirm if that 75%-true-Proposition is TRUE or FALSE. either true or false, noting else. this becomes more interesting when one has to decide if a Proposition is true to 50%! Yes, or no!

@Franklin that`s why the logics LaL and (fuzzy)-Logic must be separated. In LaL the assigned values are merely 100%, or 0%.

dear Piotr, LaL not allowed becaouse object couldnt desrcribe language who describe object. LaL is describe of Logic. Is Tarski diferenciation of language and metalaguage.> it is exactly what I mean! I only Name it Epi-Logic (not meta-)

hi Franklin, that a Point is smaller than itself, is ment to be an aphorism! cause as a Primary Notion a Point doesn`t Need to be defined :))

Hi Paul,

"@Franklin that`s why the logics LaL and (fuzzy)-Logic must be separated. In LaL the assigned values are merely 100%, or 0%."

It seems,you have not got my point.

LaL as you define it now is just Boolean logic on Fuzzy assertions.

Example: it rains[75%] is a fuzzy assertion. your LaL may assign a True or False value to this fuzzy assertion, not because you defined LaL merely as a LOGIC ABOUT LOGIC but because you defined it as a BOOLEAN LOGIC.

However, you can also define LaL as a Fuzzy logic on fuzzy assertions.

Example: it rains[75%] is a fuzzy assertion. Your LaL as fuzzy logic may assign a value between 0%-100% to this assertion as to any other assertion.

Shortly, There is not only THE CERTAINTY OF UNCERTAINTY,

BUT ALSO THE UNCERTAINTY OF THE UNCERTAINTY,

THE UNCERTAINTY OF THE CERTAINTY,

THE CERTAINTY OF THE CERTAINTY,

THE UNCERTAINTY OF (THE CERTAINTY OF THE UNCERTAINTY),...

IT FOLLOWS SIMPLY LOGIC, LOGIC ABOUT LOGIC, LOGIC ABOUT(LOGIC ABOUT LOGIC)

To go further, we can try to show that those logics are equivalent. The meta-definitions just account for a clean and easy interpretation of assertions or complex assertions.

That is for instance, a Boolean Logic about Fuzzy logic is a Fuzzy logic  .

Example: It is true that it rains[75%] = it rains[75%](notice that the true of the LaL is implicit as usual)

                   It is false that it rains[75%] = it does not rain at 75%[100%]

The system behaves as the concatenation of  NOT-operation,

NOT NOT NOT= NOT

Conclusion: I think , the problem with you is that  you absolutely want a system(LaL) to formally represent the semantic or interpretation of the end axioms: There is no such system as I said in my previous contributions

Example: in Fuzzy logic, it rains with value [75%], axiomatically already means it is true it rains at 75%[we know it implicitly]. But the problem is that you still want to represent this interpretation in a formal system called LaL: in this case LaL augments nothing, that is why I talked about Equivalence above.

Now, it rains with value [75%], axiomatically already means it is true at 75%[we know it implicitly] . Now if LaL indicates FALSE, either you misinterpret the axioms of  Fuzzy Logic or it is no more about Fuzzy logic.

Dear Gentlemen,

I would like to speak about probabilities in Fuzzy systems; I consider as most credible the systems which are related to reality itself.

According to what I mentioned in this discussion, our logical communication is contradictory; this leads to silence. We can brake the silence in a "logical" way through   a claim of minimum contradictions which implies that we decide not to  make mistakes on purpose; however this does not mean that this kind of communication is a pure classikal logic since the contradictions (according to the statement  "A system including logic  Λ   and a statement which is not theorem of  Λ  leads to contradiction"  cannot be vanished. This communication constitutses a kind  of fuzzy logic and more precisely it is a minimum contadictions fuzzy logic.

On this basis new principles (e.g. GRT, QM etc.)  are sources of futher contradictions; therefore we have minimum contradictions when the axioms we use are restricted to the ones of our logical communication.

Taking into account the above mentioned we can reach to the conclution that reality (everything) under the claim for minimum contradictions is described by a stochastic quantum space time  and obeys to the Schroedinger relativistic equation (Gordon Klein); because of this the probability density function P(r,t) has an integral (without any manipulation e.g. renormalization) equal to 1.

 On this basis we use the probabilities of a minimum contradictions Fuzzy system in an absolutely logical way; however P(r,t) of the Schroedinger relativistic equation can be negative which is not understandable; in this point we should remember that contradicions are never vanished. However these negative values can correspond to negative values of mass (quantum space time formation) or to a negative charge (imaginary quantum space time formation). 

Thus,  classical logic might  lead us to entities which seem to be ubnormal but they constitute parts of the whole reality.

describe something  by having as basis the principles of 

hallo Franklin, I would use boolean LaL to avoid uncertainty about Logic-Propositions. why should we ever start building a theory, when the biginng itself says: I am not sure that I am (not) sure :))

dear Franklin, when then make it true by measuring 

NOT NOT is not true, it can be NOTHING too.

Dear Paul,

", when < It is false that it rains[75%] > then make it true by measuring "

After you measure the truth value of that fuzzy assertion, I will measure the truth value of your measurement. And it will never end.

dear Franklin, you are right: it could never end... therefore I would use LaL: if we both consider that the fuzzy valued proposition is true, then we can make the next step. if not: we have to make more measurements until they are acceptable.

when I said boolean, I ment nothing but the evaluation of the truth: is it true? yes or no!? not true does not implicate false. not true could be: not existent, false, not to determine, uncertain, neutrosophy, aso. for instance: is the propositon true? yes or no. if it is not true, then find out to ?% it is true. the rest could be false and not-to-determine toghether.

Dear Paul,

I broadened the area of discussion. I am noticing that you go beyond formal sciences. I even visited your Profile to get your areas of research. 

We know that a subjective/personal assertion differs from an objective assertion, but they can meet.

In empirical sciences(informal logic) as Physics, objective assertion is the meeting of several subjective assertions and anyone measuring correctly should get the same result. whereas in formal sciences (formal logic) objective assertion is correctly deduced from a formal system and anyone reasoning correctly should reach the same result: that is the meeting. That objective assertion either in empirical or in formal sciences   is what LaL delivers. When personal assertion does not meet objective assertion, then LaL returns FALSE otherwise returns TRUE.

AGAIN LAL IS REDUNDANT. IT RECLAIMS WHAT SCIENCE CLAIMS ALREADY: ANY VALID SCIENTIFIC CLAIM IS TRUE

LAL is Science.

dear Franklin, instead of false, I would say: not true. if not true, then make it true. I would reduce the answers to True? yes or No!

As it came to aberrations in physics, my motto is NOT: read your… manuals; remake them! That is the name of the game :))

@Franklin, for re-considering entire physics I wrote a decalogue (it is published here on RG).

Dear Paul,

I am back again. I will read your Publication on RG.

"According to you, Fuzzy assertions should be assigned a last boolean interpretation namely True or not True. If it is not true then, then make it true. And consequently we reduce the entire problem to what I will call STRONG DECIDABILITY OF FUZZY ASSERTIONS: given that a fuzzy assertion F is TRUE, LaL should answer yes or no.  "

1. I assume that we are talking about scientific(either formal or informal) TRUE:  because Absolute TRUE is not reachable

2. I understand how much Importance you attach to the difference between NOT TRUE and FALSE. It might be because you explicitly define both concepts TRUE  and FALSE at the same time. If I strictly have to do with only two(boolean) complementary concepts, I formally define one and the other is simply everything the first is not. In this way,  you and I refer to the same when talking about NOT TRUE and FALSE, but it is important like you did to make the difference when talking about neutrosophic logic where we have False True and Undetermined.

3.Now my question is: In which way is LaL different from Science(All existing scientific theories, except LaL if it would be one)? In oder words, is there a claim that Science can fully validate  as TRUE that LaL cannot or is there a claim that LaL can validate as TRUE that Science cannot?

dear Franklin, about Point 3.

to axiomatise physics proceed as follows: first list a number of axioms needed (s. decalogue below). then play ping-pong between hypotheses and conclusions: case one conclusion is false, the axioms` system is to revise and get a new theory. suspicious conclusions are anew revised by changing corresponding axioms and so on. in this way axioms are deduced to be necessary and sufficient.

 in this way a theory becomes better. but I haven`t demonstrated if there exists a last one (theory).

Dear Paul,

Now, we are at the end of our discussion. I really enjoyed the discussion and will give feedbacks after reading the Decalogue. I thank you for the Topic

1.The way you described the research method above is normally how Science works. Permanent Critics and Reevaluations of scientific claims constitute a great part  in the essence of Science and fundamentally makes difference between Science and Dogma. Example: Evolution of Newton Theory of Gravitation to Einstein's one, Evolution of  2D-Euclidean Geometry, Evolution of Georg Cantor 's set theory to Fraenkel-Zermelo's set theory...

2. About whether there exists a last theory or not:

Scientific TRUE is what we have still now and it evolves with the time. Absolute or Last TRUE is not reachable.

thank You, Franklin!

ps. did You know that Zermelo`s real Name is: Walzermelodie? (waltz melody)

I am waiting for constructive criticism

Dear Paul and Franklin,

I would like to speak about 

" Permanent Critics and Reevaluations of scientific claims"

According to what I mentioned in this discussion, our logical communication is contradictory; this leads to silence. We can brake the silence in a "logical" way through   a claim of minimum contradictions which implies that we decide not to  make mistakes on purpose; however this does not mean that this kind of communication is a pure classikal logic since the contradictions according to the statement  "A system including logic  Λ   and a statement which is not theorem of  Λ  leads to contradiction"  cannot be vanished. 

On this basis new principles (e.g. GRT, QM etc.)  are sources of futher contradictions; therefore we have minimum contradictions when the axioms we use are restricted to the ones of our logical communication.

Therefore we do not need any new axioms and therefore new critics if we insist on logiic despite the fact  that  our logical communication system appears to be contradictory (claim for minimum contradictions).

Note that taking into account the above mentioned we can reach to the conclusion that reality (everything) under the claim for minimum contradictions can be  described by a  space time quantum mechanics which can describe   quantum gravity and which,  under certain simplifications,  is compatible to the existing physics. 

All these can be found extensivelly in my book entitled:

"Minimum Contradictions Everything" reviewed by M.C. Duffy and  edited by C.K. Whitney. ISBN 1-57485-051-X.  Publ.Hadronic Press (2008). Amazon.  I could add this in my researchgate contributions if I got the permission by the publisher.

I would be happy if a discussion was opened on the basis of the question:

Does reality need further axioms beyond the ones of logical comunication even it appears this communication  to be contradictory? (without this communication we cannot communicate in a logical way).

Dear Athanassios ,

I thank you for your reply. Before I make a point, it is really important that I first read your article.

Soon.

dear Athanassios, let us beging with the 1st contradiction: please name it., and we`ll discuss it step by step  the briefer the text, the overseeable.

Dear Paul,

I fully agree with you. I wanted to say that, but I found maybe important to first have a look on "Minimum Contradictions Everything" to see if we are talking about the same things and avoid Non sequiturs.

Dear Paul and Franklin,

As for the first contradiction it can be found in my book mentioned. I will ask Prof. Santilli to give me the permition to add it in me researchgate contributions,

For the time being you can search in Google for:

bowen publishing sjmr Vol.3, No 6, 2013.

( Theorem proving...pdf)

This version, in my opinion  is the most complete I have written

Dear Athanassios,

we got you. The thing is, could you clearly as a thesis state, what you contradict to our(Paul and I) conclusions. The point is, your book can clearly present valid contradictions maybe related to the topics but unrelated to our conclusions. In this case, we can open a new discussion for a new topic: That is why I talked about Non sequiturs. But if  your contradictions reject our conclusions, please briefly state the contradiction to help us know how to start the discussion.

Dear Paul and Franklin,

In order to avoid any misunderstanding the only I can prove is:

1. The theorem:

 "A system including logic  Λ   and a statement which is not theorem of  Λ  leads to contradiction"  (first contradiction)

2. The physics which can derive from this theorem under the claim for minimum contradictions (quantum gravity, existing physics-under certain simplifications).

3. The fact that we do not need any new axioms and therefore new critics if we insist on logiic despite the fact  that  our logical communication system appears to be contradictory (claim for minimum contradictions) 

Dear Paul and Franklin,

In my point of view the discussion we had was very important. For this reason I am doing the best to be continued.

You say

But if  your contradictions reject our conclusions, please briefly state the contradiction to help us know how to start the discussion.

The basic  point is whether you agree that the statement: 

  "A system including logic  Λ   and a statement which is not theorem of  Λ  leads to contradiction"  

is valid or not. This could be a first point for discussion.

Dear Athanassios,

Thank you again. As reacting to your contradiction of  our conclusions, I cannot answer your question because :

                        1:  I don't know if the question is related to your contradiction of our conclusions since you have neither  till yet clearly stated your contradiction nor our conclusions.

                        2: Starting the discussion as you suggest would be beginning anyhow a new discussion instead of continuing the discussion.

Maybe I should sketch what I am asking for: Briefly state

"Franklin and Paul, you conclude that: ......."

" However I maintain the thesis that: ......"{I don't need prove, but just thesis}