Eliminating Decision Problem Undecidability
I forgot to say this: L is the formal language of a formal system
My True(L,x) predicate is defined to return true or false for every
finite string x on the basis of the existence of a sequence of truth
preserving operations that derive x from
A set of finite string semantic meanings that form an accurate
verbal model of the general knowledge of the actual world that
form a finite set of finite strings that are stipulated to have the
semantic value of Boolean true.
False(L,x) is defined as True(L,~x).
Truthbearer(L,x) ? (True(L,x) ? True(L,~x))
Finite string expressions that are not truth-bearers are rejected
as a type mismatch error for every formal system of bivalent logic.
Truthbearer(English, "This sentence is not true") is false.
Truthbearer(English, "This sentence is true") is false.
Truthbearer(English, "a fish") is false.
Truthbearer(English, "some fish are alive") is true.
Truthmaker Maximalism (is what the above ideas are anchored in)
https://plato.stanford.edu/entries/truthmakers/#Max
My True(L,x) predicate is defined to return true or false for every
finite string x on the basis of the existence of a sequence of truth
preserving operations that derive x from
A set of finite string semantic meanings that form an accurate
verbal model of the general knowledge of the actual world that
form a finite set of finite strings that are stipulated to have the
semantic value of Boolean true.
False(L,x) is defined as True(L,~x).
Truthbearer(L,x) ? (True(L,x) ? True(L,~x))
Finite string expressions that are not truth-bearers are rejected
as a type mismatch error for every formal system of bivalent logic.
Truthbearer(English, "This sentence is not true") is false.
Truthbearer(English, "This sentence is true") is false.
Truthbearer(English, "a fish") is false.
Truthbearer(English, "some fish are alive") is true.
Truthmaker Maximalism (is what the above ideas are anchored in)
https://plato.stanford.edu/entries/truthmakers/#Max
Comments (192)
From what?
Quoting PL Olcott
Semantic and meaning mean the same thing. Is the quote above supposed to mean "meaningful strings" as for example "the dog bites the ball" instead of "gorbyr dortug equerxi"?
Quoting PL Olcott
What's "general knowledge" supposed to mean as opposed to just "knowledge"? Also, if the answer to the question in the previous quote is "Yes", "string semantic meanings" and "verbal model" approximately mean the same thing, if they mean anything at all.
Quoting PL Olcott
?
You are writing things from a train of thought but the purpose of writing is communication, you need to include the train of thought, not just its destination.
Quoting PL Olcott
We have an X that forms a Y that forms a Z, but X and Z seem awfully similar, as if they mean the same thing.
Quoting PL Olcott
Do you mean False(L,x) is defined as True(L,¬x)?
Quoting PL Olcott
Yes, something is a truth-bearer if it is true or false.
Quoting PL Olcott
That is the naïve reply to sentences such as "This sentence is a lie". Claiming that it is not a truth-bearer is alike hand-waving, you must give some account as to how it is not a truth bearer.
Another further issue is that by the law of non-contradiction, something is X or it is not-X. Something is true or it is not true. Replying that "a fish" is neither true or false while nevertheless defining "false" as not-true violates the LNC. The "type mismatch" is encompassed in "not true" just like "green" is encompassed in "not salty" when you ask the equally nonsensical question "Is the colour green salty or not salty?" do we really need to come up with a concept of "salty-bearer" or can't we just say things that taste salty are salty and everything else is not salty?
It is far too much to unpack all oat once.
I don't say: "This sentence is a lie",
I refer to the strengthened Liar Paradox: "This sentence is not true."
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
The means LP is rejected as not a truth bearer in Prolog because
it has an infinite cycle in its evaluation graph.
This sentence is not true.
What it is not true about?
It is not true about being not true.
What is it not true about being not true about?
Those two mean the same.
Quoting PL Olcott
No clue what that means.
Quoting PL Olcott
It is the criticism of the liar paradox refering to nothing. It was discussed in the thread I linked.
I have spent two decades on this. It
work because Truthbearer(L,x) ? (True(L,x) ? True(L,~x)) screens out
epistemological antinomies that Tarski get stuck on.
That's rad.
Quoting PL Olcott
@jgill@fishfry
General knowledge can be expressed in a finite set of finite strings.
Specific knowledge of everything is unmanageably large and infinite.
I'm familiar with Pete's work from other forums.
Expressions that are {true on the basis of meaning} are ONLY
(a) A set of finite string semantic meanings that form an accurate
model of the general knowledge of the actual world.
(b) Expressions derived by applying truth preserving operations to (a).
True(English, "a cat is an animal") is true.
It is simply all of the details of every fact of the world. General knowledge is a finite set of axioms.
That is not the point. The set of verified facts of the world is defined to exist, it is merely
not written down all in one place yet. LLM AI models might be able to achieve this
within a few years.
The point is that when we know all of the general knowledge facts of the world then
we can easily screen out every epistemological antinomy (as a type mismatch error
non-truth-bearer) that many of the undecidability proofs depend on. Tarski undefinability
proof depends on this.
https://liarparadox.org/Tarski_247_248.pdf
https://liarparadox.org/Tarski_275_276.pdf
If we merely encoded all of the rules of algorithms, logic, and programming in a single
formal system then when when no sequence of truth preserving operations from these
basic axioms derives x or ~x then x can be rejected as a type mismatch error on the basis
that all formal systems of bivalent logic require every expression to be a truth-bearer.
{All cats are animals}
{All animals are living things}
therefore {All cats are living things}
The principle of explosion is not truth preserving.
{All cats are animals} // axiom
{No cats are animals} // false assumption
therefore FALSE
Expressions that are {true on the basis of meaning} are ONLY
(a) A set of finite string semantic meanings that form an accurate
model of the general knowledge of the actual world.
(b) Expressions derived by applying truth preserving operations to (a).
The above algorithm specifies True(L,x) and False(L,x) defined
as True(L, ~x).
https://en.wikipedia.org/wiki/Ontology_(information_science)
of Rudolf Carnap / Richard Montague {meaning postulates} that stipulate relations
between finite strings as providing the semantic meanings that form an accurate
model of the general knowledge of the actual world.
Rudolf Carnap told Willard Van Orman Quine that the otherwise totally
meaningless finite string of "Bachelor(x)" is defined as the otherwise totally
meaningless finite string "~Married(x)" and Quine just could not get it.
The full definition of "Married(x)" entails (at least) billions of other meaning
postulates defining "Human(x)".
Facts are sentences that are defined as true. Cats
The otherwise totally meaningless sequence of letters of "cats" and "animals" are defined to have the
All of the facts about the world work this same way. You wanted it simple and provided a complex example. In one case "?" is a Greek letter. Even this begs the question: What is Greek? and What is a letter?
https://www.mathsisfun.com/algebra/sigma-notation.html requires a whole mathematical infrastructure.
Yes it is the case that only humans have a complex language, however, apes have learned a symbolic language known as Yerkish. https://en.wikipedia.org/wiki/Yerkish
There is a data structure known as a knowledge ontology that is based on a directed graph.
https://en.wikipedia.org/wiki/Ontology_(information_science)
A knowledge ontology has a unique integer (such as the CYC project's use of the 128-bit GUIDs) for each sense meaning of every word. https://en.wikipedia.org/wiki/Cyc
A word is a string (AKA sequence) of characters such as "dog". The first sense meaning is the most common one: https://www.merriam-webster.com/dictionary/dog# These differing sense meanings have an integer index in dictionaries. A knowledge ontology might require an ISO standard dictionary so that its unique sense meanings expressed as 128-bit integers can correspond to their sequence of characters in this IOS standard dictionary.
A knowledge ontology is an inheritance hierarchy of these sense meanings. This means that the sense meaning of {dog} (the animal) gets most of its meaning from {animal} and only adds details that distinguish a {dog} from other {animals}.
In other words you seem to believe that "a cat is probably an animal" and "a cat is probably not a fifteen story office building". I disagree.
We simply correctly encode all of the true facts of the world. When the discussion
devolves into "facts according to who" I lose interest because the discussion has
devolved away from actual truth.
I am referring to the finite set of finite strings that encode the actual general knowledge true facts of the world. I refer to general knowledge because this is finite. That no one has written them all down in one place does not mean that they do not exist.
The only reason that "cats are animals" is true is that it is stipulated to be true thus allocating semantic meaning to otherwise utterly meaningless finite strings. Language different than English does this same thing with their own set of finite strings.
Assuming then it returns true for all true strings and false for all false ones, right?
So are you going here for the solution for the Entscheidungsproblem? Seems something like that.
Quoting PL Olcott
So with this assumption you think you can state that the Church-Turing thesis has no truth-bearing?
Yes and non truth-bearers are screened out.
Truthbearer(L,x) ? (True(L,x) ? True(L,~x))
False ? True(L,~x))
Quoting ssu
Yes. https://simple.wikipedia.org/wiki/Entscheidungsproblem
...14 Every epistemological antinomy can likewise be used for a similar undecidability
proof... (Gödel 1931:43-44)
Truthbearer(L,x) ? (True(L,x) ? True(L,~x))
Epistemological antinomies no longer for the basis for any undecidability proof they are
simply rejected as a type mismatch error for every formal system of bivalent logic.
Quoting ssu
The only issue that I am correcting is the notion of decidability.
The notion of computation remains the same.
If there are no sequence of truth preserving operations from the verbal semantic meaning of expression x {true on the basis of its meaning} to x or ~x then the expression is rejected as not a truth bearer thus a type mismatch error for every system of bivalent logic. LP := ~True(L, LP) is the best known example of this.
When we have all of the truth facts of the world or even all the true facts about logic, math and computation then it is easy to see that epistemological antinomies and their negation cannot be derived from these true facts. LP := ~True(L, LP) then True(L, LP) is false and True(L, ~LP) is false.
It is much simpler to see what Tarski did, Gödel hid the missing inference steps
behind Gödel numbers and diagonalization.
This is Tarski's formalized Liar Paradox
x ? True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf
This is stated more simply as LP := ~True(L ,LP)
Tarski found out that ~True(L, LP) is true (in his meta theory) and
True(L,LP) is not provable in his theory and this got him confused.
This sentence is not true: "This sentence is not true" is true because
"This sentence is not true" is not true.
https://liarparadox.org/Tarski_275_276.pdf
[b]There has never actually been any need for this enrichment, it has always been
expressible in a single formal system with a single formal language as I elaborate below.[/b]
x ? True if and only if p
where the symbol 'p' represents the whole sentence x
His intention was for formalize the actual Liar Paradox
It would then be possible to reconstruct the antinomy of the liar in the
metalanguage, by forming in the language itself a sentence x such that
the sentence of the metalanguage which is correlated with x asserts that
x is not a true sentence. https://liarparadox.org/Tarski_247_248.pdf
The above is Is better stated as: p ? p ? True
his self-reference is a little clumsy and the above is still not quite
actual self-reference.
It is standard convention to formalize self-reference incorrectly
?(x) there is a sentence ? such that S ? ? ? ????.
*The sentence ? is of course not self-referential in a strict sense*,
but mathematically it behaves like one.
https://plato.stanford.edu/entries/self-reference/#ConSemPar
That whole article was ONLY about self-reference
x := y means x is defined to be another name for y
https://en.wikipedia.org/wiki/List_of_logic_symbols
Thus Tarski's p is better stated as: p := p ? True
even better as: p := ~True(Tarski-theory, p)
p says that it is not true in Tarski's-theory.
[b]It took me 500 hours studying those four pages of Tarski to get to that point.
You probably won't be able to get there by skimming those two pages.[/b]
Even if I'm just an amateur on these issues, I think here's a mistake.
Gödel isn't just coming up with the Liar paradox and "hiding" the missing inference steps behind Gödel numbers and diagonalization. Many people do think that Gödel has fallen into the trap of self reference and is talking about basically the paradox, but he isn't.
He is talking about provability and makes a formal mathematical statement not of "this sentence is not true", but "this sentence is unprovable" or "the sentence (s) in unprovable". It isn't a paradox as the statement simply is unprovable, not illogical. While proving or giving a proof in mathematics is very close and usually the same thing to computation, then it's no wonder that the undecidability results are so close to each other (Gödel, Turing, Tarski).
And they aren't confused. Their findings aren't something that just can be assume away, it simply would be illogical to do that.
For myself the clearest example of this diagonalization is for you to say something that you don't say. Now, does there exist something that you don't ever say? Of course! But simply you cannot say it. It doesn't go away by assumption.
Did you notice that I changed the subject to Tarski?
Tarski does the same thing as Gödel yet shows his work.
We can see that when we formalize the Liar Paradox correctly
LP := ~True(L, LP)
and not the clumsy way that Tarski formalized it :
x ? True if and only if p
where the symbol 'p' represents the whole sentence x
There is no need for any separate theory and meta-theory.
True(L, LP) is false
True(L, ~LP) is false
Truthbearer(L,x) ? (True(L,x) ? True(L,~x))
Truthbearer(L,LP) is false
and we get the same result as the Tarski meta-theory directly in L
~True(L, LP) is true.
Yet these findings aren't a repeat finding of a paradox. As also @tim wood stated, what Gödel shows is that there's a true, but unprovable sentences.
Referring to the https://plato.stanford.edu/entries/truthmakers/#Max about truthmakers, it seems that you make a similar argument like Rodriguez-Pereyra (2006c) to the refutation of Milne (2005). Hence you want to refer to the Liar Paradox here. (According to Rodriguez-Pereyra, because (M) is akin to the Liar sentence theres no reason to suppose that (M) is meaningful either).
Or as you say in the OP:
Quoting PL Olcott
Yet Milne has the gist of this: the problem here is that there indeed are true, but unprovable truths. It really doesn't go away by thinking that forbidding paradoxes, forbidding negative self reference or diagonalization and then assuming that everything can be done and all is well.
Hence if the maximalist view is that "For every truth, then there must be something in the world that makes it true" it still holds, with the simple caveat: yet not all of these truths can be proven to be true.
Hopefully you get what I'm trying to say here.
He doesn't actually show that and if he didn't hide his work we could see that he doesn't really show that. He doesn't even claim that, yet what he does claim is a little incoherent. G is true in PA yet not provable in PA. The way that we know G is true is that G is provable in meta-math.
Quoting ssu
I have studied these actual papers. Milne is actually saying that there are some expressions that we know are true yet have no way what-so-ever to know that they are true. If an expression utterly lacks any criterion measure showing that it is true then it remains untrue.
The thing that all of these writers currently lack is shown below:
TT := True(L, TT)
TM := Has-a-Truthmaker(L, TM)
LP := ~True(L, LP)
True(L, TT) is false
True(L, ~TT) is false
True(L, TM) is false
True(L, ~TM) is false
True(L, LP) is false
True(L, ~LP) is false
Copyright 2024 PL Olcott
HERE IS WHY NONE OF THEM ARE TRUTH-BEARERS
BEGIN:(Clocksin & Mellish 2003:254)
equal(X, X).
?- equal(foo(Y), Y).
that is, they will allow you to match a term against an uninstantiated
subterm of itself. In this example, foo(Y) is matched against Y, which
appears within it. As a result,[b]Y will stand for foo(Y), which is
foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))),
and so on. So Y ends up standing for some kind of infinite structure.[/b]
END:(Clocksin & Mellish 2003:254)
:brow:
Gödel is incoherent?
Quoting PL Olcott
Interesting to talk about the same issue in two threads at the same time, but anyway...
Ok, let's think about some of these expressions. And no, I haven't seen the actual Milne's paper, so I cannot say more when I have just the one link. (Is it free and obtainable by the net?)
Now, can you give an answer you don't give? It would be illogical if you could. Are there in existence these kind of answers? Obviously yes. What defines them? Obviously the answers that you, @PL Olcott gives.
The above shows just what the problem is when you "Cantor's diagonalization" or basically negative self reference.
So what your problem in using diagonalization?
I want to mostly Gödel and focus on how a True(L,x) predicate would actually apply to the properly formalize Liar Paradox.
LP := ~True(L, LP)
We shall show that the sentence x is actually undecidable
and at the same time true ... (page 275)
the proof of
the sentence x given in the metatheory can automatically be
carried over into the theory itself: the sentence x which is
undecidable in the original theory becomes a decidable sentence
in the enriched theory.
https://liarparadox.org/Tarski_275_276.pdf (page 276)
When we stick with theory L we get the same results, thus no need for any metatheory
True(L, LP) is false
True(L, ~LP) is false
Truthbearer(L,x) ? (True(L,x) ? True(L,~x))
So what Tarski says is undecidable in his theory is actually not a truth-bearer in his theory.
What Tarski said is provable in his metatheory making it true in his theory is
~True(L, LP) is true in his theory because LP is not a truth-bearer in L.
Not every truth has a truthmaker II PETER MILNE
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=e5f9578348844874f5d2542dcaff8d481e016483
Truthmaker Maximalism defended GONZALO RODRIGUEZ-PEREYRA
https://philarchive.org/archive/RODTMD
Quoting ssu
When we can directly see the cycle in the directed graph of the evaluation
sequence of an expression (thus not an acyclic directed graph) then we can
see the expression is not provable because there is something wrong with it.
When we hide this behind Gödelization and diagonalization we can still see
that that expression X is unprovable yet lose the fact that X is unprovable
because there is something wrong with it.
Undecidability, Provability, True(L,x) and Tarski Undefinability are all inherently interrelated.
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)
The key missing piece is that no one ever noticed that epistemological antinomies are simply not truth-bearers, thus must be rejected by any formal system of bivalent logic. Once rejected they cannot form any basis for any undecidability proof.
In plain English:
"? is not true."
What is ? not true about?
? is not true about being not true.
What is ? not true about being not true about?
? is not true about being not true about being not true...
Ok so ? NEVER gets to the actual point.
Hence conclusively proving that ? cannot bear the truth value of true or the truth value of false.
That does not make True(L, ?) inconsistent. When True(L, ?) is false and True(L, ~?) is false then ? is rejected are inherently incorrect. No sense moving beyond this point until after you totally get it.
A truth-bearer is an expression of language X that can be possibly evaluated to a Boolean value.
What the logicians call an undecidable expression X the philosophers of logic correctly assess
as not truth-bearer X.
Here is a much more formal way of saying the same thing:
Back in 2019 I created a formal system that detects cycles in the evaluation of an expression:
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
Initially it took any MTT expression and output the directed graph
of the evaluation sequence of this expression. The current system
only outputs the XML of the expression yet the directed graph can
still be derived manually.
The directed graph nodes numbers are on the left (00,01,02) and the nodes
that they transition to (directed graph edges) are on the right (01,02,00)
LP := ~True(L, LP)
00 Not 01
01 True 02, 00 // cycle detected
02 L
It turns out the Prolog can also detect cycles in the directed
graph of the evaluation sequence of an expression.
The SWI-Prolog implementation of unify_with_occurs_check/2 is cycle-safe
and only guards against creating cycles, not against cycles that may
already be present in one of the arguments.
https://www.swi-prolog.org/pldoc/man?predicate=unify_with_occurs_check/2
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
In other words Prolog has detected a cycle in the directed graph of the
evaluation sequence of the structure of the Liar Paradox. Experts seem
to think that Prolog is taking "not" and "true" as meaningless and is
only evaluating the structure of the expression.
Any cycle in the directed graph of the evaluation sequence of any expression conclusively proves that this expression is not a truth-bearer thus must be rejected by any formal system of bivalent logic.
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)
This is also true for any {epistemological antinomy} AKA self-contradictory expression. If X cannot possibly be evaluated to true or false (for whatever reason) then X must be rejected by any formal system of bivalent logic.
I either have to explain it in technical terms that you don't understand or explain
it in plain English where too much important meanings slip through the cracks.
[b]The key most important point that can be summed up using the technical terms
of philosophy is that {epistemological antinomies} are not {truth-bearers}.[/b]
"x is true if and only p" is not, according to Tarski or anyone who has reasonably studied this subject, the liar paradox nor the liar sentence. Moreover, as has been explained several times to the poster, Tarski does no use the liar sentence as a premise in any proof. Rather, Tarski assumes, toward a contradiction, that in the interpreted language there is a truth predicate for that language, and then shows that that assumption would allow the formation of the liar sentence and its contradiction, therefore that the assumption is contradictory and there is no such truth predicate.
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)
My whole point in the post is that all epistemological antinomies such as the liar paradox
must be recognized and rejected thus not allowed to be any part of any undecidability proof.
It seems that you are saying that Tarski did not do that.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
And again, for about the tenth time: Tarski does not use the liar sentence as a premise in his proofs. Rather, for undefinability, he makes a reductio ad absurdum assumption that there is a truth predicate, from which he shows that that assumption provides a liar sentence that is a contradiction, thus refuting the reductio ad absurdum assumption. And in another proof, for incompleteness (undecidability actually), he explicitly says that he uses the predicate of provability not the predicate of truth. And one can verify in the actual proofs that the liar sentence is never a premise.
To put this in most stark form:
Assume P.
Derive Q.
Show that Q is contradictory.
Conclude ~P.
P is "a truth predicate can be formed"
Q is "the liar sentence can be formed"
The point is not that we adopt any liar sentence, but the opposite: that the liar sentence cannot be formed therefore there a truth predicate cannot be formed.
Tarski very much stresses that we do NOT deploy the liar sentence, since the liar sentence cannot even be formed in these kinds of languages.
To say that Tarski deploys the liar sentence in his proofs is to brazenly reverse what he wrote.
One more time: One can look at the actual proof steps for incompleteness and undefinability to see that the liar sentence is not used as a premise.
That does not matter. That quote proved that he did not have the very basic understanding
that epistemological antinomies (AKA self-contradictory expressions) are not truth bearers
thus cannot be used for any undecidability proof. This showed that he generally had a poor
understanding of undecidability proofs.
According to my understanding of your explanation of Tarski, Tarski made an equivalent mistake.
He some how derived the Liar Paradox and did not reject it as not a truth bearer. Non-truth-bearers are a type mismatch error for any formal system of bivalent logic.
Saying that Tarski derived the liar paradox is to yet again ignore the plain hard fact that he did not. Not only did Tarski not claim that the liar sentence is a truth bearer in the relevant formal systems, but even more fundamentally he showed the the liar sentence cannot even be formed in those formal systems. It really helps to actually study the subject of undefinability and incompleteness rather than to brazenly misrepresent it, as a seemingly perpetual strawman, as saying the opposite of what it actually says.
Tarski did not use the liar sentence as a premise, and he did not derive the liar sentence. Rather, he showed that a certain assumption would provide that the liar sentence could be formed in the interpreted language ((1) not even as a theorem, but merely as a sentence having the liar property, especially since there is no system of theorems involved but only an interpreted language, thus this is not a "derivation" of the liar sentence in the sense of a theorem, but only a proof that it could be formed as a sentence; moreover the derivation of the existence of the sentence is not a conclusion in the argument, but only a conditional result based on the reductio ad absurdum assumption) but if the liar sentence could be formed then it would be both true and false in the given interpretation, which contradicts that no sentence can be both true and false in an interpretation, thus we conclude the denial of the original assumption.
I have not said anything like that. Here is what I said:
Quoting TonesInDeepFreeze
And the quote of Godel just mentioned drops the context that Godel explicitly wrote this as a matter of "ANALOGY".
The wings of birds are analogous to the wings of airplanes, but they are not the same. The liar sentence "I am not true" is analogous to the Godel sentence "I am not provable", but they are not the same.
And the Tarski quote regards not the theorem that the poster previously cited (pages 275-276). Jumping around, changing contexts like that, is incoherent. But it goes right along with what I wrote. [b]IF[/b] we had a truth predicate then we would have the liar paradox.
Look is up in the proof yourself. Its one page 40
[i][b]One cannot correctly use epistemological antinomies in undecidability proofs,
they are not truth bearers and must be rejected on this basis.[/b][/i]
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf
"The ANALOGY of this argument [...]" [emphasis added ] which is the context of the footnote quoted.
Looking at the specific argument that Godel mentions as "this argument" we see that the liar sentence is not in that argument. Rather, Godel mentions that the liar sentence has an ANALOGOUS role.
So, again, I point out that the poster is quoting while intentionally omitting the crucial context.
Again, the wings of birds are analogous to the wings of airplanes, but they are not the same. The liar sentence "I am not true" is analogous to the Godel sentence "I am not provable", but they are not the same.
And again, if one actually reads the proof, then one will see that the liar sentence is not used anywhere in the proof.
Moreover, Godel could not use it as a line in any step of the proof, because the liar sentence cannot even be formulated in such systems that are the subject of the proof, which is what Tarski proved.
Gödel is terribly wrong about this, these words are dead false:
His proof is an "undecidability proof" and he just proved that made a big mistake with
his understanding of undecidability proofs.
It's right there in the paper. Footnote 14 pertains to the passages that begin, "The analogy of this argument [...]"
There is no context in which those words of Gödel are not a terrible mistake.
The poster seems to have a problem: Posting the quote from the footnote over and over again, as if the passage that it is a footnote to does not exist, even as the poster included my own quote of that passage.
It seems that you are trying to take the words figuratively. That does not work.
Try and see how the literal meaning can be dismissed.
(2) That argument is then developed in full detail in a full proof. That proof does not use the liar sentence in any premise, line or conclusion. And the argument doesn't need to use the liar sentence in any premise, line or conclusion. Moreover, the liar sentence could not be used, since the liar sentence cannot even be formulated in the system under consideration.
(3) Textbooks that prove incompleteness do so without using the liar sentence as a premise, line or conclusion.
(4) Tarski even proved that the liar sentence cannot be formulated in such systems.
In all that context, it is seen that Godel did not use the liar sentence in the proof and did not say he did. What he said is that such antinomies can be use for such proofs, which in all this context, can only be understood not as used in the proofs formally but as an analogy that is adapted to the proofs. That adaptation is using 'provable' rather than 'true'. And that is also exactly what Tarski mentions explicitly in one of his proofs of incompleteness. Anyone with a sincere interest in understanding this subject sees this clearly, as opposed to someone who is interested only in taking a footnote out of context (to the extent of ignoring the very passage to which it is a footnote) for the purpose of insisting on a claim to be right no matter how ridiculous.
So lets get back to Tarski. He did anchor his proof in the Liar Paradox and he says so.
Try and show all of the details of otherwise.
I do not concede the prior point, but, let's move on.
For the 100th time, Tarski himself said that instead of "true" he used "provable. It's in the exact text of the paper.
Rational discussion is barely possible with a person who doen't have the honesty to recognize the connection between a footnote and what it footnotes.
He did start with this Liar paradox. He said so.
Also this is how he encoded his Liar Paradox
x ? True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf
Actually the paper says he swapped "Tr" for "Pr"
That is not the part I am not conceding. I am saying no matter
what anything says anywhere else THAT WAS A BIG MISTAKE
That is not the liar paradox.
So if one concedes, by actually reading the paper, that the footnote pertains to those passages, then one doesn't have grounds to claim there is a mistake.
This stands:
https://thephilosophyforum.com/discussion/comment/907058
The proof uses the Godel sentence that is analogous to the liar sentence but is very different in crucial ways from the liar sentence. The proof itself does not have any mention of the liar sentence. The proof does not need the liar sentence. It would not even make sense for the proof to use the liar sentence. No one who has studied this subject thinks that the proof uses the liar sentence. The proof of just incompleteness doesn't even have to mention the notion of truth or semantics and can be formulated purely in finitisitc arithmetic and regarding only syntax.
Against all those facts, the only person who would claim that Godel is mistaken in this regard is someone obsessed with insisting that Godel is wrong no matter what, no matter how factually and logically wrong it is required to maintain that insistence.
(1) In the Tarski proof of undecidability lately discussed here, Tarski did not use the liar sentence, but rather he used a different formulation involving provability rather than truth. And Tarski himself not only did not claim that the liar sentence has a truth value, but in the undefinability proofs he showed that in the relevant languages, it cannot even be formulated.
(2) In the Godel in the incompleteness paper, he did not claim that the liar sentence has a truth value, and he did not use the liar sentence in the actual incompleteness proof.
You seem to be consistently denying easily verified facts.
One thing that we do definitely do know about
Tarski's use of the "antinomy of the liar" in his Undefinability proof
is that he did not recognize it and reject it as a type mismatch error
for every formal system of bivalent logic.
Tarski's Undefinability Theorem Proof
https://liarparadox.org/Tarski_275_276.pdf
And to evade that fact, the poster switches to a different theorem and proof elsewhere in the article, even though the context of that proof does not support the poster's false claim.
The poster just reposts over and over and over his same dogmatic and already rebutted out of context claims. At a certain point, replies are futile. The poster is out of reach of rational discussion.
Formalized as:
x ? True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf
adapted to become this
x ? Pr if and only if p // line 1 of the proof
Here is (first three steps of) the Tarski Undefinability Theorem proof
(1) x ? Provable if and only if p // assumption (see above)
(2) x ? True if and only if p // Tarski's convention T
(3) x ? Provable if and only if x ? True. // (1) and (2) combined
When L is Tarski's theory:
For the sentence "x is not provable in L" to be true in L requires a sequence of
inference steps in L that prove that they themselves do not exist.
In that proof, Tarski does refer to the set of true sentences, but he does not use the liar sentence. Before the proof, Tarski specifically explains that he does not use the liar sentence, which pertains to truth, but rather a sentence regarding provability. And in the proof itself, he makes good on that by not using the liar sentence.
Notice also that the set Tr is defined in the meta-theory and not in the object theory.
And the poster repeats for the 10000th time his misconception that the theory would have to prove that the sentence is not provable. It's the opposite. The proof that the sentence is not provable does not occur in the theory but rather in the meta-theory, as the theory would be inconsistent if it proved that the sentence is not provable.
has this as its original source:
Which is the paper that I have been citing.
Let's FULLY address this one single point before we attempt to address any other points.
[b]In other words he does not understand that the derived liar sentence must be rejected as a non-truth-bearer thus a type mismatch error for any formal system of bivalent logic.
You never seem to directly address these exact words and seem to always form a rebuttal by talking around these exact words.[/b]
/
I am not opining about a notion of a "type mismatch error" so I don't have to rebut it. I do not address the notion of "type mismatch error" without a crisp definition of it, and I wouldn't approach it without the poster at least first understanding his misconceptions about the actual proofs, which I have more than fully addressed:
(1) The liar sentence is not in any step of Godel's incompleteness proof and not in any step in Tarski's undecidability proof. Instead, the Godel sentence "I am not provable" is used, which is crucially different from the liar sentence "I am not true". The poster still does not grasp this, but instead he cites, like a disinformation bot, a Godel footnote out of context of the passage it footnotes, as in that passage Godel explicitly says that reference to the liar paradox is by analogy. The poster needs to stop talking around this.
(2) And the liar sentence itself is not a step, and especially not a premise, in Tarski's undefinability proofs, but rather the proofs start with the assumption, toward a contradiction, that the language can defined its own truth predicate, then shows that that would allow the language to form the liar sentence, which would yield a contradiction, so we conclude the denial of the assumption that the language can define its own truth predicate. The poster still refuses to understand this point that is at the very heart of undefinability. The poster needs to stop talking around this.
(3) The incompleteness proof does not at all entail that the theory itself would have to prove that the Godel sentence is not provable. The exact opposite: We prove that the theory itself does not prove that the Godel sentence is not provable. This has been explained to the poster probably at least 20 times in this forum. The poster needs to stop talking around this.
Also:
(4) In another thread, that the definition of 'antinomy' (especially in philosophy and logic) is not merely 'self-contradictory', which can be verified by looking at a number of dictionaries and articles. The poster still has not recognized this fact.
¬(p ? ¬p) Law of non-contradiction
(p ? ¬p) Law of excluded middle
p = p Law of identity
Because of Quine's paper: https://www.ditext.com/quine/quine.html most philosophers have been confused into believing that there is no such thing as expressions of language that are {true on the basis of their meaning}.
The unique contribution I have made to this is that the semantic meaning of these expressions is always specified by other expressions. When we can derive x or ~x by applying truth preserving operations to a set of semantic meanings then this perfectly aligns with Wittgenstein's concise critique of Gödel: https://www.liarparadox.org/Wittgenstein.pdf
Unless P or ~P has been proved in Russell's system P has no truth value and thus cannot be a proposition according to the law of the excluded middle. Sometimes this "proof" requires an infinite sequence of steps.
You simply ignored most of what I said.
I didn't read beyond the point where you proved that you ignored my definition of R.
(expressions of language specifying semantic meanings)
I am simultaneously carrying many other conversations so I must stop reading as
soon as I hit the first big mistake.
I already specified that the R I am referring to is the set of semantic meanings specified as expressions of language. This is the key foundation of my whole point and cannot be ignored. This R is the ultimate foundation of the truth of all expressions of language that are {true on the basis of their meaning}.
Truth preserving operations applied to these expressions that fail to derive P or ~P prove that P is not a proposition because it violates the law of excluded middle.
For a given language, we have different models. A model is an interpretation of the meaning of the symbols of the language. Per a given model, every sentence receives exactly one of the two truth values. That is, per a given model, no sentence is both true and false, and every sentence is either true or it is false. Moreover, no proof has an infinite number of steps, since we cannot mechanically check an infinite number of steps.
Some sentences are true in every model (these are called 'logical truths')
Some sentences are true in some models and false in other models (these are called contingent sentences')
Some sentences are false in all models (these are called ''logical falsehoods').
If P is a sentence and t is a closed term term, then
~(P & ~P) is a theorem in every theory and it is true in every model (non contradiction)
P v ~P is a theorem in every theory and it is true in every model (excluded middle)
t = t is a theorem in every theory and it is true in every model (identity).
Moreover, we have the meta-theorem that a sentence is true in every theory if and only if it is provable in every theory.
/
With formal theories, it is required to have a mechanical method to check whether a given sequence of formulas is a proof, and for that we need a mechanical method to check whether a given sequence of symbols is a formula of a certain kind, and for that we need a mechanical method to check whether a given sequence of symbols is a formula. And we need a mechanical method to check whether a given formula is a sentence.
It would be circular if, to know whether a given formula is a sentence, we needed first to know whether a given formula is such that either it or its negation is provable. To know whether it is a sentence we would need to know whether it is provable, but to know whether it is provable, we would need whether either it or its negation is provable.
Moreover, we have the meta-theorem that there are theories such that there are sentences such that neither the sentence nor its negation are provable in the theory. This does not contradict the law of excluded middle (P v ~P), since the law of excluded middle semantically is that either P is true or ~P is true, and the law of excluded middle syntactically is that P v ~P is provable in all theories, but the law of excluded middle is not that in all theories either P is provable of ~P is provable.
That may make conventional sense. In my system semantic meaning is fully integrated directly into the language. This makes things such as the principle of explosion impossible. (A & ~A) semantically entail FALSE.
When any expression P or ~P has no connection through truth preserving operations to elements of the set of expressions of specified semantic meanings then P is not a proposition. This is almost the same thing that Wittgenstein says. I generalized what Wittgenstein said to apply to every expression that is {true (or false) on the basis of its meaning}. No connection to any meaning: then meaningless.
One expression of formal language or formalized natural P or ~P can either be connected to a set of
semantic meanings specified as formal language or formalized natural language through a set of truth preserving operations or not. Some aspects of classical logic (such as the Principle of Explosion) are not truth preserving. If neither P nor ~P can be connected to elements of the set of semantic meanings then P and ~P are meaningless.
The principle of explosion adheres to the principle of truth preservation.
The principle of truth preservation is: All cases in which the premises are true are cases in which the conclusion is true. Put another way: There are no cases in which the premises are true but the conclusion is false.
Since there are no cases in which a contradiction is true, there are no cases in which both a contradiction is true and the conclusion is false.
I am NOT doing it that way. The meanings of terms are specified in a knowledge ontology type hierarchy. The compositional meaning of expressions is derived through something like Montague grammar.
Quoting TonesInDeepFreeze
Nothing can be semantically derived from the expression that "cats are not cats"
Quoting TonesInDeepFreeze
That is simply not good enough. X is semantically entailed by a set of premises if and only if X is a necessary consequence of all of the premises.
It has taken me twenty years to derive the architectural overview that I just provided. A key aspect of this is defining expressions that are {true on the basis of their meaning} where meaning is expressed using other expressions.
I started with absolute truth and found the most people believe that absolute truth only comes from God and they don't believe in God. The ten years after that I started talking about analytical truth only to find that Quine successfully convinced most people that it does not exist.
Recently I came up with the above expressions that are {true on the basis of their meaning} where meaning is expressed using other expressions.
My whole system is just like expanding the syllogism so that it applies to every expression that is {true on the basis of its meaning}.
It is anchored in a type hierarchy knowledge ontology to specify the semantic meaning of terms of a formal or formalized natural language. https://en.wikipedia.org/wiki/Ontology_(information_science)
The compositional meaning of expressions of this language are derived from something like Montague grammar.
* If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P.
* Montague semantics is based on compositionality as with the method of models (though with extended aspects such as types, modality, intensionality and possible world models).
As with any subject, before purporting to critique mathematical logic and model theory, one should know something about it. The number of decades one has been floundering in ignorance and confusion on the subject is not a positive index of the cogency of one's critique.
Logic: Techniques Of Formal Reasoning, 2nd ed. - Kalish, Montague and Mar
Start there, with the basics of the subject before pretending to speak meaningfully about more advanced topics.
I am talking about semantic entailment that has nothing to do with model theory.
The reason the error of the Principle of Explosion has slipped through the cracks
is that semantics was divorced from logic.
When we try to answer what is it about {A cat is not a cat} that makes the
{Moon is made from green cheese} true and we come up with nothing
then the ruse of the POE is exposed.
Quoting TonesInDeepFreeze
https://en.wikipedia.org/wiki/Principle_of_explosion Disagrees.
Quoting TonesInDeepFreeze
I am talking about how to fully integrate semantics directly in the language and have no need for model theory. If we don't do this then we will not understand that the POE is simply wrong. There is nothing about {cats are not cats} that proves {the Moon is made from green cheese}.
It took me twenty years to unequivocally prove that something just like the analytic side of the analytic synthetic distinction really exists. Most philosophers remain convinced by Quine there is no such thing as {true on the basis of meaning}.
When I anchor this in {the meanings must be specified as expressions} of language and X or ~X must be derived by applying truth preserving operations to these expressions of semantic meaning then this {true on the basis of meaning} is proven to exist. Also undecidable sentences are rejected as failing the Law of excluded middle.
In other words you did not understand that I just provided the essence of the foundation of expressions that are {true on the basis of their meaning} thus establishing that something just like the analytic side on the analytic/synthetic distinction has been proven to exist.
Let just talk about the POE. (A & ~A) prove B no matter what A and B are.
(1) We know that "Not all lemons are yellow", as it has been assumed to be true.
(2) We know that "All lemons are yellow", as it has been assumed to be true.
(3) Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "or" means that if even one part of the statement is true, the statement as a whole must be true as well.
You disagreed with the above.
I do know the distinction between valid and true. https://iep.utm.edu/val-snd/
So see how you can use this distinction to explain how what you said
diverges from what Wikipedia said.
Disagrees with this:
Quoting PL Olcott
/
If a proposal for a logic does not include models than it is fundamentally different from Montague grammar/semantics.
/
"If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P."
(1) That is exactly correct. (2) Wikipedia is not a reliable source on the subject of logic. (3) I highly doubt that Wikipedia disagrees with what I wrote anyway.
/
I have not opined on the analytic-synthetic distinction.
A sentence is true or not per a given model.
A sentence is valid if and only if it is true in every model.
We don't need to bring in the question of knowledge in this context, and one cannot know that a false assumption is true. One can assume statements that are false. People make false assumptions often. People even assume contradictory premises fairly often. But one cannot know a statement to be true when it is false. It is not the case that one knows a statement to be true simply by assuming it is true. In particular, one does not know that both lemons are yellow and lemons are not yellow simply by assuming that lemons are yellow and lemons are not yellow.
And the example is unnecessarily overcomplicated, not only with the gratuitous and incorrect remarks about knowing, but also in its logical form. It would be better written:
The assumption "lemons are yellow and lemons are not yellow" implies the conclusion "unicorns exist".
Or, even better, to avoid questions about pluralization and generalization:
The assumption "the Cartier Sunrise Ruby is red and the Cartier Sunrise Ruby is not red" implies the conclusion "the Empire State Building is a unicorn".
And my point is that we don't thereby conclude, independent of the contradiction, that the Empire State Building is a unicorn.
What is the basis for that claim?
In other words you disagree with the Wikipedia quote.
I disagree with the Principle of Explosion itself.
There are no semantics passed from any contradictory premise to any conclusion.
(A ? ¬A) only proves FALSE. There is nothing about the semantics of {Cat's are not cats}
that makes {the Moon is made form green cheese} true.
The POE says the everything is logically entailed by a contradiction and it is simply wrong about this.
A & ~ A proves FALSE and nothing more.
That most philosophers were convinced by Quine that the analytic/synthetic distinction does not exist, thus it is impossible to divide expressions {true on the basis of their meaning} from expressions that are true on some other basis such as observation.
I asked for the basis for the claim. The poster replies by merely repeating the claim.
/
The principle of explosion is that any sentence is entailed by a contradiction. That does not contradict:
If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P.
If C is a contradiction and P is any sentence, then we have:
(1)
C -> P
and
C |- P
Neither of those give us P, since we would first need to have C, which we don't have.
(2)
((C -> P) & C) -> P
and
(C -> P) & C |- P
Neither of those give us P, since we would first need to have C, which we don't have.
The point of this is to fend against people who don't understand the principle and think that it allows us to make any ridiculous conclusions we wish to make.
I am talking about how the categorical propositions of the syllogism directly encode semantics thousands of years before anyone every heard of model theory. The Tarski Undefinability theorem was before model theory.
And model theory adheres to the ancient notion of entailment: A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true but the conclusion is false.
That seems to not be restrictive enough. From your idea Donald Trump is Christ is entailed by the Moon is made from green cheese because the Moon is made from green cheese is false.
Consider two sentences D and M.
If D is false, then M might be true or false. The falsity of D does not entail the truth of M. But the conditional D -> M is true.
If D is a contradiction, then M might be true or false. From the mere fact that D is logically impossible we do NOT infer that M is true. But the conditional D -> M is not just true, it is logically true. THAT is the principle of explosion and it does NOT imply that M is true.
If the sentence D is merely false, we cannot infer M. But if we assume the sentence D & ~D then we can infer M. But, again, we would be mistaken to have D & ~D as a premise, so if M is false, thus a mistake to assert it, the mistake of asserting M would occur only by the mistake of asserting D & ~D.
Yet when D is a contradiction we know that D is not true.
But if any of the sentences in the set are quantified, then the set might be inconsistent (yielding a contradiction) though we don't know it's inconsistent, and a given system might not provide a mechanical means to verify whether a given set of sentences is inconsistent or not.
Moreover, the principle of explosion regards entailment, no matter what we happen to know about the truth, falsehood or inconsistency of the sentences. The principle of entailment, even in ancient form, concerns the impossibility of states of affairs whatever our knowledge or lack of knowledge about those states of affairs.
When the poster tries to justify the slip by noting that all contradictions are falsehoods, he commits the illogic of getting the matter backwards: It's not a matter of all contradictions being falsehoods (which is true) but rather it's that the converse does not hold. The poster can't reason himself out of the proverbial paper bag.
(A & ~A) proves B is the POE
(A & ~A) proves FALSE is the actual correct inference
Two aspects of the same case.
It's not a matter of the conclusion being false but rather that the poster previously tried to slip the discussion from the inconsistency of the premise to the falsehood of the premise.
But to address the latest post anyway:
Let C be a sentential letter and, for some sentence P, we define: C <-> (P & ~P).
Then for any sentence Q:
C |- Q
But since we don't have C, so we don't have Q.
And of course, in any model C is false, while depending on Q, in a given model, Q could be true or false.
Maybe you are overwhelmed by too many details.
The Principle of Explosion claims this: (A & ~A) proves B
Meanwhile, yes of course as I have agreed at least a dozen times by now and as in the very post he just now responded to (!):
If C is any contradiction and Q is any sentence, then:
C |- Q
That is a basic result in sentential logic, known to anyone who has studied the subject.
But the poster just keeps posting it over and over though no one disagrees with it.
He might as well say, "2+2 = 4. Ha! Take that!"
OK we finally have agreement on one point and I am exhausted that it took this long.
I am not going to bother to extend beyond this point with you because it seems to me that
you may be trying as hard as possible to make sure to avoid any honest dialogue.
Yet when we make sure to NOT IGNORE the semantics underlying the sentential logic that
Quoting TonesInDeepFreeze
said is correct then we understand that there is no semantic connection between
{a cat is not a cat} and {the Moon is made from green cheese} thus {the Moon is made from green cheese} is not entailed by the semantic meaning of {a cat is not a cat}.
We can encode the same thing as a syllogism and see the same thing yet I will not bother with that degree of detail while you two seem to insist on being as disagreeable as possible.
As to honest dialogue, there is no point of dishonesty that the poster can point to in anything I've written, while meanwhile, only a few posts ago, the poster tried to evade a key point by dishonestly conflating contradiction with falsehood, as he continues to dishonestly evade that point.
As to the claim that I am as disagreeable as possible, there is so much confused disinformation written by the poster that there is indeed a great amount to disagree with.
/
The poster cites "semantic connection". That is not a defined term. However, the semantics are clear, as I have mentioned over and over but the poster refuses to recognize:
The notion of semantic entailment is:
A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true and the conclusion is false.
If the premises include a contradiction, then there are no circumstances in which all the premises are true and the conclusion is false. Therefore, if the premises include a contradiction, then those premises entail any conclusion.
By the way, mutatis mutandis it works the other way too: If the conclusion is logically true, then that conclusion is entailed by any set of premises.
Little doubt the poster still does not understand these points. He was too exhausted to understand them decades ago when he began not learning even the basics of this subject.
On the other hand, there is an alternative approach to logic call 'relevance logic' that does formalize the notion of the content of sentences and bases a different notion of entailment in that regard. And the advantage of that work, in contradistinction with the confused, ignorant and intractable handwaving of cranks is that the work is (as far as I know) rigorous and it is grounded in clear understanding of the subject - both classical and alternative. Some people prefer such approaches as relevance logic though they are more complicated than plain classical logic. But the existence of alternative approaches does not refute that my own reports of classical logic have been correct. And the takeaway here is that study of alternative logics requires prior understanding of the most basic logic that the alternatives include in some parts, reject in some parts and extend in certain ways. We need to be correct in what we say about classical logic if we are to properly critique it or propose a supposedly remedial alternative to it.
What is there about the semantic meaning of {cats are not cats} that shows that {the Moon is made of green cheese} ???
But I'll say it again in yet different terms:
'shows' in this context does not mean that there is contentual relationship but merely that there are no circumstances in which all the premises are true and the conclusion is false. For an alternative, one would study relevance logics.
The poster's earlier claim that started this part of the discussion was that classical logic is not truth preserving. But it is truth preserving since there are no permitted inferences from true premises to false conclusions. That is the case no matter that classical logic is not a relevance logic.
/
Still interested to know the basis for the claim that most philosophers reject the analytic-synthetic distinction.
Still interested whether the poster will ever admit that he improperly conflated contradiction with falsehood by overlooking that, while contradiction implies falsehood, falsehood does not imply contradiction.
Still interested whether the poster now understands that a footnote is in context of the passage to which it is a footnote.
And meanwhile, the poster still has not understood that he gets both Godel and Tarski exactly backwards. Most specifically that Godel does not claim that the system proves that the there is a proof in the system of the Godel-sentence, but instead proves that the system does not prove it; and Tarski does not use the liar sentence as a premise in any proof (as "this sentence is not provable" is crucially different from "this sentenced is false") but instead proves that the liar sentence cannot be formed in the language.
How to explain cranks? They take a position that they have a system for mathematics that is correct to the exclusion of mathematicians who are terribly wrong. In order to make an impression with that position, the crank postures that he dismantles the work of the mathematicians. But in order to carry out that dismantling, the crank must get the mathematics quite wrong. Not only is the crank's supposed system vague, impressionistic, illogical (to the point of being incoherent) and uninformed, but the crank's remarks about the work of mathematicians are woefully ignorant, confused and disinformational while self-fortified against being corrected on virtually any point ranging from fundamental to incidental, no matter how utterly clear it is that the corrections are sound. And sometimes this goes on for literally decades of spammed repetitions. The only explanation I can think of is that the crank's need is not to understand mathematics or even well conceived alternative mathematics, but rather the crank has a deep need to be taken to be an exceptional, remarkable person who single-handedly has debunked the work of mathematicians. The crank seems to want to live out a kind of hero fantasy and "triumph over conformity" fantasy no matter the facts or logic about the subject, no matter that thereby he makes an abysmal fool of himself though, of course, not in his own hopelessly blinkered mind.
You have proven to be overwhelmed by the detail of the original thread so I simplified it.
I didn't dumb it down you are very smart. I simplified it so rejecting out-of-hand looks foolish.
Again, the answer begins in the post to which the poster was replying and then elaborated on in my next post. In hopes that it is not ignored yet again:
'shows' in this context does not mean that there is contentual relationship but merely that there are no circumstances in which all the premises are true and the conclusion is false. For an alternative, one would study relevance logics.
The poster's earlier claim that started this part of the discussion was that classical logic is not truth preserving. But it is truth preserving since there are no permitted inferences from true premises to false conclusions. That is the case no matter that classical logic is not a relevance logic.
Also to the best of my ability, it doesn't look as if you understand what Tones is writing. "Epistemological antimony" isn't a technical term in any of the proofs you've criticised.
It would be worth reading SEP's articles on the various diagonalisation results. Godel's and Tarski's and Turing's. Moreover, even something like first order logic isn't decidable.
It looks very much like a case where how you're using the words, PL, is not how the literature is using them. And in that regard your ideas - as criticisms of the literature - are off target.
Thank you, fdrake, for those useful words.
Yes, 'epistemological antinomy' is not mentioned as a formal mathematical rubric in Godel's famous paper. The poster abysmally fails to read - fails even to recognize that it has been pointed out to him - that Godel confines the significance to that of analogy not of formal application.
I went back to my original post and still stand firmly behind it. The terms that I use
are relevent to truth-maker maximalism yet probably establish brand new ideas in
this field that have no preexisting terms.
It is probably very very difficult for people that know the conventional notions of
Decision Problem Undecidability to have any idea how to apply the words
of this original post to the subject of Decision Problem Undecidability.
To anchor my ideas in Gödel's 1931 Incompleteness I would say that:
G and ~G are not linked by any sequence of truth preserving operations from the
axioms of PA thus are untrue in PA.
G is linked by a sequence of truth preserving operations from the axioms of
meta-math thus are untrue in meta-math.
To people very accustomed to Gödel numbers and diagonalization this may see very strange.
It Is however, the same idea that Wittgenstein had in mind and I know this because I derived his exact same idea about a year before I ever heard of him. https://www.liarparadox.org/Wittgenstein.pdf
Most people very familiar with conventional notions mistakenly conflate boiling ideas down to their bare essence as a simplistic view of these same ideas. That is what everyone here has done.
It is completely a confused notion that G is false.
/
If the poser were sincere about discovery in mathematics, he'd use his own terminology and define it, rather than piling on confusion by conflation with standard terminology. Let alone that he'd sit himself down to learn the basics of the subject starting with a textbook in introductory symbolic logic rather than misconstruing and misrepresenting stray bits of poorly written and intellectually disorganized Wikipedia quotes.
And, just as I discussed, there we have the poster saying that he's a source of brand new ideas while all his interlocuters are wrong: the fantasy that he's a brilliant maverick math hero. And The Philosophy Forum is just one of the Internet forums that he uses to act out his fantasy.
But I applaud The Philosophy Forum for its toleration, allowing even the most incorrigible cranks to start thread after redundant thread, spewing disinformation like a crudely written bot.
I didn't even say that G was false.
According to the new foundation of True(L, x) that I provided in my original post
when neither G nor ~G can be proven in PA then G is neither true nor false in PA.
Diagonalization and and Gödelization are not any ordinary arithmetic what-so-ever.
Here is all that there is to PA https://en.wikipedia.org/wiki/Peano_axioms
There is no Diagonalization or Gödelization in PA.
/
Godel numbering and diagonalization use only arithmetic. Godel numbering and diagonalization use arithmetical operations in very complicated and surprising ways, but nothing beyond arithmetic is used. When one actually reads the proofs, one sees that each step is utterly unassailable mathematics, well within arithmetic even if a complicated and ingenious use of arithmetic. Indeed, the proof can be done in finitistic intuitionistically acceptable arithmetic. If there is a context that is epistemologically safer than that, then I'd like to know what it is.
To understand what I am saying requires knowledge of truth-maker maximalism that you seem to lack.
More pertinent is that understanding this subject requires knowing, as the poster does not, at least the basics of symbolic logic with the basics of mathematical logic to follow.
But that is the way of the crank: The requirement that everyone else accept the crank's undefined terminology, impressionistic musings, and illogical arguments while the crank himself has no responsibility to learn even page one of an introductory textbook.
As if that makes any point at all here!
A specification of PA itself does not include mention of all kinds of results in arithmetic, from the fact that there is no greatest prime to the fundamental theorem of arithmetic to the most advanced and complicated theorems about the natural numbers ... and to Godel-numbering and diagonalization. So what? It doesn't entail that those developments are not provided for, or pertain to PA.
Here's a pretty good analogy: The rules of chess make no mention of the incredibly complicated strategies of chess masters, but that does not entail that those strategies are not permitted by the rules nor that analysis of those strategies does not pertain to chess.
And as the poster touts the Wikipedia article, he conveniently omits including that the article itself DOES mention the incompleteness theorem!
What a seriously risible argument the poster makes! Really, the poster is as hopelessly ignorant, confused and irrational as they come. I've seen some that are more dishonest, but the poster ranks fairly high in dishonesty too, as just witnessed that he touts a Wikipedia article that actually shows the OPPOSITE of his own claim!
All you have is ad hominem and cannot point out any actual errors in the essence of my reasoning.
Here is the essence of my reasoning:
Every expression of language X that is {true on the basis of its meaning} can only be verified as true on the basis of a connection to this meaning.
I've given exact detailed technical explanations of how the poster has been wrong in so many ways. Discussing also that the poster is a crank does not erase the technical explanations.
The poster is flat out lying that I have not shown errors in his reasoning. My refutations have been copious.
And notice that the poster yet again evades dealing with the very quote of mine that he posts. To wit:
The poster's reasoning includes giving evidence that Wikipedia's article on the Peano axioms does not mention Godel numbering and diagonalization. But (1) The axioms don't mention a lot of things, not even such things as the fundamental theorem of arithmetic. The axioms don't mention Godel numbering and diagonalization but that doesn't entail that they are not basically applications of arithmetic. Indeed, finitistic arithmetic. (2) The Wikipedia article actually DOES discuss the incompleteness theorem regarding PA.
And that is just the latest in a long chain of evasions and misrepresentations by the poster, not even counting all the other redundant threads he's propagated on these subjects.
And his latest message misses the point also:
The question is not ascertaining what is true. Mathematical logic has a rigorous definition of 'true' in which sentences are evaluated first with their atomic components such that an atomic sentence is true if and only if it corresponds to the given state of affairs.
Rather, the question has been about entailment. And the principle of explosion doesn't say that the conclusion is true, only that a contradiction entails any statement. That is, since there are no states of affairs in which a contradiction is true, there are no states of affairs in which a contradiction and any other sentence are together true. Regardless of how we reckon the truth of atomic statements, there are no circumstances in which a contradiction is true thus no circumstances in which both a contradiction and any conclusion are both true.
/
Getting back to the key point that started this part of the discussion:
The poster claims that classical logic is not truth preserving. But it is, as I have explained (and it is proven). And the fact that the poster prefers his own vague, undefined and confused outlook on logic does not entail that classical logic is not truth preserving.
Truth preservation is: If the premises are true then the conclusion is true. And that is PROVABLY upheld by classical logic.
That part is correct yet simply ignores the actual point
Quoting PL Olcott
Therefore:
True(PA, G) == false
True(PA, ~G) == false.
And True(PA, G) has no apparent meaning.
A sentence is true or not per a model, not per a theory. Though, of course, a sentence may be true in all models of a theory, which reduces to the sentence being a theorem of the theory.
In other words you believe that there are are sequence of truth
preserving operations from the axioms of PA to G or to ~G.
When I prove my point ALL YOU HAVE IS AD HOMINEN.
The poster lies that I believe that PA proves G or it proves ~G. It is the opposite. PA proves neither G nor ~G. That is the very statement of incompleteness.
The poster not only fails to desist from (let alone retract) his previous lie that I have not addressed the subject substantively, but he reposts it in all capital letters.
I am establishing a brand new foundation for analytical truth and simply ignoring that I
am doing this is no actual rebuttal at all.
Can you manage to stay focused on the point at hand?
Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system.
Contrary to the poster's false claim, classical logic is truth preserving.
And these points that the poster lacks focus to understand:
Contrary to the poster's confusion, the principle of explosion accords with truth preservation.
Contrary to the poster's ignorance, classical logic handles the notion of truth in terms of correspondence with states of affairs.
Contrary to the poster's addlement, truth and entailment are different notions. A set of premises entails a conclusion if and only if there are no states of affairs in which the premises are all true and the conclusion is false. Whether or not the conclusion is or is not false is a different question.
Contrary to the poster's ineducation and lack of intellectual curiosity, there is an approach to logic called 'relevance logic' in which the conditional is handled with regard to the contentual features of the antecedent and consequent, as that study is, unlike the poster's pronouncements, informed and rigorous.
Contrary to the poster's intellectual recalcitrance, merely to refute the poster's falsehoods about classical logic, one is not required to indulge the poster in his confused notions about some claimed new foundation he thinks he has created.
That the poster is monomaniacal about some pet idea of his own doesn't entail that anyone else who points out his copious errors about classical logic lacks focus for not nodding in acceptance of his spammy gibberish.
Can you manage to stay focused on the point at hand?
Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system.
It is not my fault that people want to change the subject away from the original post.
I have found that allowing people to do this to play Trollish head games is not
productive. The strawman deception of changing the subject as a form of fake
rebuttal never works with me.
[b]I just told you what I want to talk about so we can skip all of the other posts
I simplified what I want to talk about so that it will be easier for you to focus
your attention on this one single idea the follows:[/b]
Every expression of language x that is {true on the basis of its meaning}
can only be verified as true on the basis of a connection to this meaning.
This does enable a True(L, x) predicate to be defined where L is a formal
language of a formal system.
I agree.
But sometimes a tree looks nice to bang your head into. I would be very thankful if someone gave their time to answer my ideas as @TonesInDeepFreeze and @tim wood have given to @PL Olcott on this thread.
Quoting PL Olcott
Let's debate this from another angle.
In mathematics, do you think there can be true, but unprovable statements?
Do you think that all true statements are also provable?
And finally, can an indirect reductio ad absurdum proof prove something?
Yes on No answers would be appreciated (of course with reasoning too).
When I provide a simple yes/no answer all that I get is ad hominem attacks
without anyone even looking at what I said. So I encode my yes/no answer
in the reasoning used to derive that yes/no answer.
True and unprovable never means EXACTLY what it says:
We know that X is true and have no way what-so-ever to know
that X is true yet we know it is true anyway, as if by majick.
I've not made any ad hominem attacks. Please understand that just repeating the same thing will get tempers to rise. Always true to really think what the other one is trying to say.
Quoting PL Olcott
But we do! We can give an indirect proof.
Do you understand how an reductio ad absurdum proof goes? That's why I asked the third question.
This is quite crucial here in my view.
Then is never really was literally unprovable.
True yet cannot possibly be proved in any way what-so-ever
does not allow indirect proof.
A sequence of inference steps in PA that do not derive G only says G
cannot be proved in PA it does not say that G cannot be proved.
OK, it seems where the problem lies and just why you had this long argument with @TonesInDeepFreeze and @tim wood (both or one, some pages ago). This is very important to understand here. Giving a direct proof and giving an indirect proof aren't the same thing. Also proving something and a mathematical statement being true aren't exactly the same thing.
Because let's assume the following statement S
S = This statement is unprovable
So how can you prove this? Well, you prove it by reductio ad absurdum. So let's assume the opposite is true and hence statement S is provable and then go and prove that this cannot be. . Did you give a direct proof? No. You didn't prove S. You proved that not-S is false.
Hence I'll put again to you the last question:
Can an indirect reductio ad absurdum proof prove something in your view?
(Cantor's diagonalization is the easiest way in my view to understand this: with diagonalization we have shown that not all reals would be in the list, but of course this opens up questions like the Continuum Hypothesis.)
We must start with the common lack of sufficient precision of your first statement. Most everyone makes tis same mistake.
Quoting ssu
There cannot possibly be any expression of language that is true and does not have a truth-maker making it true. When it is said that G is true and unprovable it never means EXACTLY what it says.
If we don't start from the exact same common ground then we never get to mutual agreement, thus we must first agree that true and not provable by any means what-so-ever is contradictory.
Either a mathematical statement S is true or not-S (the negation of S) is true. S and not-S cannot be both true in mathematics. (Either the statement 1+1=3 is true or then 1+1=3 is false is true. That 1+1=3 is false is true.)
A direct proof would be simply to prove S. And indirect proof is to prove not-S is false. Since we assume that mathematics (or logic) is consistent, we do admit the indirect proof and say that S is true if not-S is false. But it's not a direct proof.
Quoting PL Olcott
Truth-maker making it true means that there's a proof that it is true? Sorry, with negative self reference you can easily do that.
Like try to give an answer here that you don't give here. Are there those kinds of answers? Obviously yes. Can you give them? No.
Quoting PL Olcott
What do you mean by this? Again, the ability to give a direct proof and something to be true are two different things.
The logic behind negative self reference is quite easy to understand.
If there is no possible way to know that expression X is true then we can't possibly know
that expression X is true. AKA when X lacks a truth-maker then X is not true.
We must get through this key point first because it is the core foundation of everything
that I am saying. I oversimplified this a little bit so that you can get the gist of what I am saying.
As I said, we know it from an indirect proof. We can prove that not-X is false, hence we assume that mathematics is consistent, hence X is true (as either X or not-X is true).
Cantor's diagonalization argument is one of the easiest examples of this.
Cantor start from the assumption that all reals are listed, and makes from this list a real number that cannot be on the list. Hence the reductio ad absurdum proof.
Quoting tim wood
Lol, well, if math is consistent, you'll get the proof. :razz:
Quoting tim wood
It's typical to underestimate the people on this Forum. But the issue here is that there's still a lot of debate on exactly what the undecidability results mean. What exactly is the realm of "true, but unprovable mathematical statements" or the role of non-computable mathematics. Obviously something that at first mathematicians don't want to find their answers to be in the realm of.
It sounds like an oxymoron and for our banned member it was totally nonsense and he obviously didn't get even after your and @TonesInDeepFreeze's effort.
Quoting tim wood
This clearly shows the person simply doesn't grasp the Undecidability results. And just to repeat again and again that Gödel is referring to the Liar paradox (and hence you can disregard it) is the typical error here.
In my view many people just don't understand that negative self-reference a simply logical limitation, especially when your primary idea has refered to all or everything. Then you just show even one example that it's impossible and that's it.
Quoting tim wood
Especially in logic, mathematics, but also in philosophy I think there's a good guideline. If someone says that you are wrong or misunderstand something, that isn't yet worrysome. But if two or more say that, you really have to consider going over what were saying. Now if people especially after some discussion simply fall silent, then you perhaps can have a point. Either nothing to add, or it goes totally over their head (which is also a possibility).
Hence I do urge the people here to correct mistakes, it truly is something that others can also learn from.
Exactly. And referring to the significance of Gödel's results usually gets a response of someone questioning you exactly how the difficult proof goes and repeating it (and sidelining the part that you were talking about: the significance of the results).
Quoting tim wood
That's an absolutely great question.
Because now everything on this field does hang on the (negative) self-reference. Every undecidability result has it. Please correct me now (anybody) if it isn't so! One possibility I've been thinking about is that the procedure defines a whole realm of mathematics. Of course, getting an undecidability result without negative self reference would be really revolutionary and counter my idea.
Yet even to admit just how important the undecidability results are would be important too. The way that first the paradoxes and then the undecidability results are seen as something against the idea of logicism is the problem. I think that mathematics is certainly logical, yet logicism shouldn't be so controversial. The thinking should be turned around: these undecidability results are a cornerstone for mathematics to be logical. Not everything is simply computable and hence what defines non-computable mathematics is a very important law for all of mathematics. In my view if you get in mathematics a paradox simply means that your premises, or more properly the axioms you think are self evident actually aren't so. Yet it seemed that with the paradoxes, they were thought as of a mistake and with the undecidability results, people simply hope to avoid them and hope them to be not important. Hence the idea was first going with Type theory (like Russell did) or simply make axioms that ban the paradox (as ZF did). This didn't erase the issue, as the famous undecidability results showed. It's like when the Greeks at first thought that all numbers have to be rational, and then when confronted by irrational numbers, they didn't like it. Because math was perfect so everything had to be rational, right? The basic problem then in Russell's time (and even now) was the idea that all the essential building blocks of mathematics and it's foundations are already in place. I think they aren't. The paradoxes and then the undecidability results simply show this.
Which makes then you question even more important. It's still just a working hypothesis that everything is based on negative self reference, in the way of "with negative self reference, you can lose computability/the ability to give a direct proof". Obviously a good counter-question is what you asked.
Loved to hear comments.
And in something as logical and rigorous as mathematics, the last thing is for us to accept that we have feelings about how it should be. Or that they would matter. It's like the person who declares: "Philosophy is meaningless to me, I do just science" has a quite specific philosophical view about science.
Quoting tim wood
In economics one way to model self reference or basic interaction of the variables is to use dynamic models. And there you have to make quite careful models that stay in some equilibrium. The amount of premises just increase.
Quoting tim wood
Hm, I'm not sure what you mean by this.
I just think they are quite interesting. For example, here's a thought experiment to clarify (hopefully) my reasoning why this is such a general finding:
Is there a limitation on what kind of answer you can write on PF?
- Obviously not, you can write anything you want (you might be banned later, but aside of that).
If so, can we deduce then from the above that you can write EVERY answer there exists, if you would have infinite time and the ability write anything, any kind of length answer (the typical Busy-Beaver argumentation etc.)?
- Actually not, because you cannot write an answer that you don't write.
Wait a minute! Wasn't it so that there isn't any limitation on what to write?
- Yes, but assuming that you can write everything isn't equivalent to there not being any limitations on what you can write.
As this is purely theoretical and I have infinite time and ability to write, how can there be any limitation?
- Because of diagonalization. From everything you written taken as whole, you can use this to create something you haven't written.
Is this totally trivial? Obviously a person cannot write what he or she doesn't write.
In mathematics when it comes to computing and proving something, it isn't so trivial. But the mechanism is the same.
I guess this is the confirmation I was not wrong when everytime I saw one of those threads I had the feeling it was a crackpot rambling. In my bouts of self-doubt, I wondered if it wasn't my ignorance of the topic that didn't allow me to follow the conversation.
Tones was doing a good job refuting all the crankery in a highly educational manner. That can make for a good thread. Albeit a frustrating one for whoever is doing the hard teaching work.
It's quite difficult to draw a principled line between someone who misunderstands something repeatedly in a rude manner and an incorrigible, disruptive crank.
Anyway they're banned now, thanks @TonesInDeepFreeze for persistently unraveling the cranking.
Same here. I remember I took part in one of @PL Olcott' s threads when he focused on 'Carol' (yes/no incorrect questions. Here). I did my best arguing with him, but he always twisted the words with the aim of making circular points. I ended up with the conclusion that it was actually my fault for not being capable of following his points. I felt very ignorant about myself.