We have to revise the semantics in Gödel’s incompleteness theorem, Plato´s theorem and E. Gettie’s example. This conclusion stunnes even me, because Gödel, Plato and Gettie were no hacks!
Gödel’s incompleteness theorem:
In any logical system for mathematics, there are statements of speech that are true, but that cannot be proved.
This statement cannot be true
Must be either true or false.
If the claim is false, it can be proved. Then it must be true. Which is a contradiction, therefore, the claim is true.
This is therefore a mathematical claim that is true, but cannot be proven.
What if the Riemann hypothesis would prove to be true, but is impossible to prove?
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It seems to me, Gödel’s incompleteness theorem gets entangled on its semantics! It is certainly a logical argument based on the theorem, but you cannot use the order in which the words follow, mathematically. What do I mean? Well, the sentence: ”This statement cannot be true” must indeed be either false or true, but if it is false then you should – if it is possible to translate it into a mathematical formula that says something about something other than semantics – replace the words ”cannot be true” with the words ”is not true”, which also makes it true, without the inconsistencies. Y can stand for ”is not true”, and X can represent ”must be true.” A can stand for the opening words ”this statement”.
——————————— X ”must be true” (+)
”This statement” A X or Y, but it cannot be both!
——————————— Y ”is not true” (-) (cannot be true)
The theorem as it appears above the line, perhaps proves that something semantically can be either false or true, though it cannot be proven. But does it prove anything, with mathematics, of the nature of world beyond it? No, it rather seems to disprove the theorem itself! The theorem doesn´t help us understand the world. Perhaps one cannot conclude a solution from the first (”This statement”) with both (”must be true”) and (”cannot be true”) for it to be correct mathematics? Either “This statement” is true or it´s not true, so Y should read ”is not true” if it should be adjacent with ”This statement” A! Furthermore, “Can” is a statement that says that something either is, or is not, but not both at the same time. When you put “not” after “can” (cannot), you are either saying +(can; as in must)–(not) = (-) = (“is not true”) which translates into a mathematical language + (-)= - or with other words it is negative. Or you are saying –(can´t as in not)-(not) = (+) = (“must be true”) which translates into a mathematical language – (-)= + or with other words it is positive and henceforth must be true. Conclusion; “is not true” is the correct wording, and not “cannot be true”! I also have other philosophical examples of how semantics can mess it up, when trying to convey it into logical theorems (read below). The presentation of the criteria (for how we could be considered to have knowledge of anything) is construed by Plato, and problematized by the renowned philosopher E. Gettie. It has been considered an unsolvable problem for many years. The problem is related to Gödel’s incompleteness theorem, because of their semantic nature. I consider myself to have solved the enigma of Gettie’s problematisation of Plato’s theorem:
AN EPISTEMELOGICAL AND RATIONAL CONCLUSION FROM PLATO’S THEOREM AND E. GETTIES EXAMPLE WITH THE WOLF. (Version nine)
1:st example: There runs a train on the tracks past a meadow. On the meadow there is a wolf. The passengers can see the wolf from the train.
According to Plato we require three criteria for enabling us to have knowledge of it.
(1) It should be a conviction.
(2) It must be consistent with reality.
(3) We must have rational reasons to accept it.
All three criteria are met.
2:nd example: Now suppose that, as in E. Getties example, the wolf is actually a dog dressed up as a wolf. A little further behind the dog on the meadow there stands a real wolf. The three criteria are still met, and this is E. Getties problematisation of Plato´s theorem, for the wolf we see is not a wolf at all, and hence the theorem is faulty even if it is true, according to Gettie.
Can we have knowledge that there is a wolf on the meadow (that the theorem is satisfied) by observing the dog, and applying the three criteria? The answer is that we cannot! The theorems correctness is completely independent of our observation of the dog (we do not know that ”wolf” is our costumed dog or that there is a real wolf just behind the dog on the meadow).
Or should we perhaps say that the theorem, on the contrary, are totally dependent on our observations, because our observations results in our belief (1), and our rational reasons to accept it (3). But thereby follows that our observation leads to a faulty conclusion, for the visible wolf is false. The theorem is still true, but Plato’s theorem requires an alternation applied to the unique situation.
(1) It should be a conviction.
(2) It must be consistent with reality.
An omniscient archangel must be the judge of whether the theorem is consistent with the real situation. Or in other words:
(3) ONE must have rational reasons to accept it.
Thus premise (3)’s rationality (as above) is not based on observations from our side. By changing premise (3) to; one must have rational reasons to accept the belief, we move the decision for what is rational from our self, to an omniscient archangel. One objection you might come up with is that one can say that premise (3) is not needed then, because to claim premise (3), is the same as to claim premise (2).
The ideal type theorem itself is not critical to getting an epistemological answer to an investigation of the rational conclusion of the theorem. The key is to know when a rational answer emanates from the premises, not when a premise is rational. ”A rational answer” is comprehensive of the whole situation with the wolf and takes into account both the wolf and the dog as distinctive entities (even in mathematics!). The premises (1),(2) and (3) have not led to a rational answer to Plato’s theorems inconsistencies in this unique situation from Gettie’s example with the wolf and the dog simultaneously located on the meadow but where we only see the dog, because that is what the whole point with Gettie´s fictional example is, that Platos theorem is inconsistent. Here the archangel in my modified third premise, come into the picture.
In the following example from Plato’s original theorem, all of Plato’s original premises are not fulfilled: Say that on one occasion there is a dog dressed up as a wolf on the meadow (premises 1 and 3 are satisfied), while there is not a wolf behind the dog (premise 2 is not fulfilled), then we cannot draw a correct conclusion that the ”wolf” is actually a dressed up dog, as long as we are on the train. If we had been able to draw a correct conclusion, it would not have been our belief that there is a wolf on the meadow.
In our second example from above (read 2:nd example in bold red and black letters above), from Plato’s original theorem, a wolf is standing behind the dog, and all three premises are met. On this occasion, we cannot draw a correct conclusion based on our position on the train (a correct conclusion would be as seen from a correct supervision of an omniscient archangels judgment about what constitutes a proper conclusion), that there is a wolf in the meadow, because we do not see it, we only see the dressed up dog. We believe however that the conditions are in order, (which they actually are, but not as we think, because we believe that the dog is the wolf on the meadow), and from it derives a conclusion that happens to be true, based on our false beliefs and Plato’s original premise, (from which I say that we have achieved an “Accidental Conclusion”, which we may call it). It also requires that the dressed up dog really looks like a wolf for us to be able to draw a true (but not overall correct) conclusion. If there had been a water fountain or a Dachshund dressed up in front of the wolf rather than a German shepherd dog dressed up, we had never come to the conclusion that there is a wolf on the meadow, by looking at the fountain or Dachshund. The conclusion is true in this our other example, where all three premises complied with the conclusion, but it is not a correct one. For this to be a correct conclusion requires that the premises implicitly take into account all the underlying facts. (Read and compare with Gödel’s incompleteness theorem!) Again; the theorem itself is not of crucial importance. The key is to determine when the premises involve a rational response. And here is where the archangel and my modified third premise comes to use, for here it is the archangel’s insight that is the standard, and not my insight, and from that follows a rational answer to the theorem. The fact that the original theorem is true in this unique situation where we see the dog but not the wolf, is a pure coincidence (read blot on Plato´s behalf) and not relevant to how we should set up the premises properly. To draw a true conclusion based on faulty underlying facts is something that has happened before in history. For example, there was an ancient Greek who said that the earth was round long before anyone else had thought of it, and he founded this conclusion from that the shadow the earth cast on the moon could not be a likeness of the Earth’s shape, if the earth was flat. He believed that the earth cast its shadow on the moon, when in fact the moon (usually) is shaded by itself and its position relative to the sun as seen from our perspective. In light of this, Plato’s original theorem and Gettie’s situation with the wolf appears quite absurd. The theorem ”proves” more than it can prove, just as the moon’s shadow can do for those who have certain beliefs. The belief ((1) we believe there is a wolf in the meadow) and the rational reasons ((3) we have rational reasons to accept that there is a wolf in the meadow) with ((2) There is a wolf in the meadow) may seem to be waterproof as a logical theorem, but the premise (2) should be read/understood and set up like this: (the wolf is false, but there is another wolf in the meadow that we do not see), if the belief is to conform to reality. This is how we must adopt the adapting of the situation with the wolf and the dressed up dog in premise (2) I think. Had we just said (2), there is a wolf in the meadow, yes, it would have been correct, but should we allow the reality of our second premise to be so simplified as to say “It must be consistent with reality”? If so, the premise would not be completely true, or at least not entirely complete. It is a semantic debate in itself! Look at the example with the costumed dog which had a wolf behind it. We have rational reasons to accept the belief that there is a wolf on the meadow when we drive by in the train, according to the original theorem. We have the illusion of the dog as a wolf. But coincidently there was a wolf on the meadow. Leaving aside premise (2), here in the form: ”It must be consistent with reality,” is premise (1) and premise (3) merely cosmetic? They are at least “ideal types” construed from our own shortsighted perspective, but still inconclusively construed since they in Plato’s original theorem are not based on any actual situation in an all in all complete situation with at least as in this case, the dog and the wolf in E. Gettie’s example. Premise (1) and premise (3) are merely convictions, which by chance happens to mess it up in at least one of the cases above, where the wolf and the dog coexisted on the meadow simultaneously, in Gettie’s example – hence ”Accidental Conclusions”. In conclusion, we have to revise both Plato´s theorem and E. Gettie’s example. This conclusion stunnes even me, because Plato and Gettie were no hacks!
Conclusion 1: One has to have rational grounds for accepting the belief.
Conclusion 2: Convictions leads to ”Accidental Conclusions.”
Conclusion 3: The costumed dog must look like a wolf, and not a Dachshund or a water fountain, for the theorem to work.
Conclusion 4: The theorem proves more than it can prove, by the principle “the earth casts its distinctive shadow on the moon, and therefore the Earth is round”, which is false for some part.
In a textbook used at Lund University in the B course, called ”Philosophy of Language a contemporary introduction” by William G. Lycan from University of North Carolina, chapter 13 on ”Implicative relations”, page 198 it says to read in the first lines: ”Sentences entail other sentences, and in that strong sense imply them. But there are several ways in which sentences or utterances also linguistically imply things they do not strictly entail.”
It describes the chapter’s content very briefly. Anyway, in this chapter you can read an interesting thing that you can directly connect to and make of use to Getties problem without that Lycan, or rather Grice, seems to have had any intentions in that direction.
There you can read: ”-Here as in many cases, a good way to investigate the nature of these different kinds of implications, is to ask about the penalty or sanction that ensues when an implicatum is false. When S:1 entails S:2 and S:2 is false, the penalty is that S:1 is false. When S:1 semantically presupposes S:2 and S:2 is false, then S:1 is sent ignominiously to zip. When someone utters S:1, thereby conversationally implicating S:2, and the conveyed meaning or invited inference S:2 is false, then the penalty is that, even if S:1 is true, the speaker´s utterance is misleading. If S:1 conventionally implicates S:2 and S:2 is false, then S:1 is misworded even if not false.”
One can imply and translate this into Gettie’s example with the wolf directly like this:
”-Here as in many cases, a good way to investigate the nature of these different kinds of implications, is to ask about the penalty or sanction that ensues when an implicatum is false. When a ”wolf” on the meadow (S:1) entails a belief (S:2) and the belief (S:2) is false, the penalty is that the wolf (S:1) is false. When the wolf (S:1) (semantically) (I here chose to put this word within parentheses) presupposes a belief (S:2) and the belief (S:2) is false, then the wolf (S:1) is sent ignominiously to zip. When someone utters wolf (S:1), thereby conversationally implicating a belief (S:2), and the conveyed meaning or invited inference of the belief (S:2) is false, then the penalty is that, even if the wolf (S:1) is true, the speaker´s utterance is misleading. If the wolf (S:1) conventionally implicates a belief (S:2) and the belief (S:2) is false, then the wolf (S:1) is misworded even if not false.”
To translate this, one must resort to some drastic interpretations. Among other things, one must interpret the following sentence – ”When someone utters Wolf (S: 1), thereby conversationally implicating a belief (S: 2), and the conveyed meaning or invited inference of the belief (S: 2) is false, then the penalty is that, even if the wolf (S: 1) is true, the speakers utterance is misleading.” – as utterances always are misleading regardless of whether they are true! But thereafter an interesting thing is mentioned, namely: - ”If the wolf (S: 1) conventionally implicates a belief (S: 2) and the belief (S: 2) is false, then the wolf (S: 1) is misworded even if not false.”
Also the philosopher Bertrand Russell addressed the self-contradictory logical problems one can construe with semantics and set up in equally contradictory theorem, in Russel’s paradox or ”Performative Contradiction”. The paradox is as follows: When people say; ”all truths are relative” they make an absolute claim, and thus it becomes a contradiction in terms. I can answer with saying that; if all truths are relative, they are not truths, they are but a misch-masch or a composite of separate truths and non-truths that need to be figured out separately, just like with this statement in red by Bertrand Russel, or as I did with Gödel´s incompleteness theorem above. Either “the truth” (or in other words – the claim) is true, or it is false, but it cannot be half true in between!
Roger Klang, Lund Scaania Sweden, updated version again (version 9), first translated into English 9/3/2011.
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