When it is said that G is true and unprovable it never means EXACTLY what it says.
— 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. — ssu

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.

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. — ssu

We must start with the common lack of sufficient precision of your first statement. Most everyone makes tis same mistake.

In mathematics, do you think there can be true, but unprovable statements? — 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.

But we do! We can give an indirect proof. — ssu

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.

In mathematics, do you think there can be true, but unprovable statements? — ssu

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.

It is exactly your fault, Olcott. By my count through at least 450 posts in good will and good faith made in an attempt to gain any clarity about what you are talking about, you have dodged, evaded, and avoided every attempt, content to make and repeat nonsense claims, and when pressed to change the subject.
Not a good look for you, and to my way of thinking making it impossible to have any respect for you. Sympathy? Maybe. Respect - which also implies trust - no. — tim wood

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:

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.

This isn't a particularly productive discussion. — fdrake

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.

I am precisely focused on the central point from which this discussion is pursuant: — TonesInDeepFreeze

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.

Whatever else the poster thinks he is doing, he claimed that classical logic is not truth preserving. I explained why that is false. The poster still refuses to understand the matter. — TonesInDeepFreeze

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.

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. — TonesInDeepFreeze

I am establishing a brand new foundation for analytical truth and simply ignoring that I
am doing this is no actual rebuttal at all.

The poster has to be bot. — TonesInDeepFreeze

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.

Truth preservation is: If the premises are true then the conclusion is true. And that is PROVABLY upheld by classical logic. — TonesInDeepFreeze

That part is correct yet simply ignores the actual point

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. — PL Olcott

Therefore:
True(PA, G) == false
True(PA, ~G) == false.

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! — TonesInDeepFreeze

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.

The poster said that G is untrue. Now he says he did not say it is false. — TonesInDeepFreeze

To understand what I am saying requires knowledge of truth-maker maximalism that you seem to lack.

The Godel-sentence G is proven true in a meta-theory that is ordinary arithmetic. It is not at all controversial that in plain arithmetic the Godel-sentence is true.

It is completely a confused notion that G is false. — TonesInDeepFreeze

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.

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. — fdrake

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.

The poster asks a question anew. He should read the post to which he is replying. — TonesInDeepFreeze

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.

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: — TonesInDeepFreeze

What is there about the semantic meaning of {cats are not cats} that shows that {the Moon is made of green cheese} ???

Let CNC stand for "The cat is not a cat," intended here as a false proposition.
Let MGC stand for, "The moon is made of green cheese," also a false proposition.
Let K stand for the implication, (CNC => MGC).
According to the rules, K is true. Period. — tim wood

Yet when we make sure to NOT IGNORE the semantics underlying the sentential logic that

That is a basic result in sentential logic, known to anyone who has studied the subject. — 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.

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. — TonesInDeepFreeze

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.

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. — TonesInDeepFreeze

Maybe you are overwhelmed by too many details.
The Principle of Explosion claims this: (A & ~A) proves B

Getting back to the poster slipping from the context of contradiction to falsehood: Yes, all contradictions are falsehoods. But not all falsehoods are contradictions. The point here is that mere falsehood is not what's involved in the principle of explosion. — TonesInDeepFreeze

(A & ~A) proves B is the POE
(A & ~A) proves FALSE is the actual correct inference
Two aspects of the same case.

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. — TonesInDeepFreeze

Yet when D is a contradiction we know that D is not true.

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. — TonesInDeepFreeze

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.

The principle of entailment goes very far back in the history of logic. It is in model theory that the principle is given mathematical exactness. The model theoretic version adheres to the general principle: A set of premises entails a conclusion if and only if there are no circumstances in which the premises are all true but the conclusion is false. — TonesInDeepFreeze

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.

It was claimed in this thread that most philosophers believe it is not the case that there are sentences that are true on the basis of their meaning.

What is the basis for that claim? — TonesInDeepFreeze

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.

A contradiction doesn't make a false statement true. No one disagrees with that. And it is not the principle of explosion. — TonesInDeepFreeze

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.

Given C, a contradiction, the expression C => P is true. That is because C is false, and whenever the antecedent is false, the implication is true - them's the rules. But it is an elementary and serious error to suppose this shows that P is true. For P to be true, C must first be affirmed. That is, C ^ (C => P) => P, C being true, affirms P. And this is exactly - or should be - what Tones said. — tim wood

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.

* 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. — TonesInDeepFreeze

Disagrees with this:

(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. — PL Olcott

↪PL Olcott You're incoherent, here. And it looks like you do not understand the distinction between valid and true. — tim wood

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.

↪PL Olcott Do you understand that in terms of these discussions and your replies to me you're talking crazy - most of your replies being either or both nonsense and non-sequiturs? — tim wood

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.

Rather than merely bandying Richard Montague, the poster would do well to start at the beginning with symbolic logic as presented in his textbook: — TonesInDeepFreeze

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.

You ether cannot or will not answer it. You describe what you call a set and make certain claims about it. You have not shown that it exists or can exist, or how it's built, and you certainly have not shown how it can satisfy the claims you make for it. — tim wood

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.

* 'entailment' and 'consequence' are usually taken as specifying the same relation. — TonesInDeepFreeze

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.

* 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. — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Principle_of_explosion Disagrees.

* 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). — 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}.

the set of expressions of specified semantic meanings
— PL Olcott
Please define this. If it is a constructed set, please show how it is constructed. — tim wood

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.

The meanings of sentences are given by the method of models — TonesInDeepFreeze

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.

The principle of explosion adheres to the principle of truth preservation. — TonesInDeepFreeze

Nothing can be semantically derived from the expression that "cats are not cats"

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. — 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.

On may reasonably propose an alternative formalized logic, but a formalized logic requires that we have a purely mechanical method by which to determine whether a given finite sequence of sentences is or is not a proof, which requires a mechanical method by which to determine whether a given sequence of symbols is or is not a sentence. — TonesInDeepFreeze

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.

And I've granted R all day long. But you're not talking just about R, but generalizing your claims beyond R, and as you persist beyond reason, so with reason I call you out and warn against engaging with you. And not to be forgot, you have been asked about R itself and given no answer. That is, R does not exist and I suspect cannot exist, either way, how is R an "ultimate foundation" of anything? By contrast, Godel et al were exactly rigorously clear about what their system(s) are. — tim wood

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.

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. — TonesInDeepFreeze

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.

Simplest is this: if P is undecidable, then neither P nor ~P are provable in R. I don't know what R is, but let's assume it stands for the kinds of systems that are actually relevant to this discussion, and which include arithmetic as described by Godel. As such, for clarity let R = G, and let us refer to P as unprovable in G. — tim wood

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.

Simplest is this: if P is undecidable, then neither P nor ~P are provable in R. I don't know what R is — tim wood

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.