Philosophy Now Forum

I have completely reformulated this making it much more clear:
Tarski Undefinability Theorem Reexamined
https://www.researchgate.net/publicatio ... Reexamined

The above supersedes and replaces the following, except for the way the link provided
below explains how sound deductive inference from true conclusions is represented
by formal proofs to theorem consequences. I also rewrote portions of this paper:

Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf

To put this in laymen's terms all of the truth that can be expressed using
words or math symbols is anchored in sentences that are defined to be true:
“A cat is an animal”. Other true sentences are derived from this basic set:
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.

The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.

Here is how the mathematical notation works:
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
In English means:
{There are sentences of English that prove that "Cats breath"}

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

When we assume this formalization of the notion of Truth:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
Then the above sentence says:

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

Any sentence that proves itself does not exist will be a contradiction. A violation of TLC? Too bad. This violation is itself a contradiction, and a violation of the law of Identity as well.

For all sentences that exist, there is no sentence that exists.

If this sentence exists, it proves that it does not exist.

Ah, but to prove the truth of these statements—there’s the rub. However, with a one sentence limitation, the above claims will half to do for themselves for now.

I phrased the original question incorrectly. That is fixed now.
I have rewritten the linked paper many dozens of times since yesterday.

PeteOlcott wrote: ↑Tue Mar 26, 2019 4:39 pm I phrased the original question incorrectly. That is fixed now.
I have rewritten the linked paper many dozens of times since yesterday.

This sentence proves it is unprovable.

The link is up again.

PeteOlcott wrote: ↑Tue Mar 26, 2019 4:39 pm I phrased the original question incorrectly. That is fixed now.
I have rewritten the linked paper many dozens of times since yesterday.

I tried to read that paper as you've printed above as a start. But it is not 'layman' friendly, if that was your intent. It assumes Predicate (or Quanitifier) logic. But even one who follow the first order languages, I still can't follow your own expressions.

To begin with,

“A cat is an animal”. Other true sentences are derived from this basic set:
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.

If you're going to be appropriately precise in your reasoning, you need to quantify (2) with the "All" you apparently assumed for the conclusion or it may be interpreted as "some" without (because, "Some Animals breathe." is as true of an English sentence.

The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.

Bad comparisons. You need to differentiate validity from soundness and truth.
Axioms (or postulates) are 'begged' rules of a system that cannot themselves be proven or are arbitrary of some set of possible other axioms that could be presented. Calling them "truths" is not sincere unless they are true universally 'true' in all systems. A theorem is the 'form' or general statement from conclusions within the system that are universally true WHEN we conditionally accept the axioms of the system.

To be more relative to all readers without the background, why not just use a direct example, like commonsense used. The liars paradox, is often used. One such form is:

"This sentence is not a real sentence."
If the meaning of the sentence is 'true' then its meaning is false; if it's meaning is false, though, it is assumed by many that this assures that meaning is, "The sentence, 'This sentence is not a real sentence.' is non-existent". I think this one is a bad example in that there is a confusion between whether the 'falseness' of such a statement refers to the existence of the expression or to the truth of its interpretation.


Bertrand Russell uses,

"The barber shaves only those who do not shave themselves."
The 'paradox' assumes that the act of the barber shaving under the condition that the ones who are only those who do not shave themselves, has to include himself. If the barber then doesn't shave himself, then he would be one of those he shaves under the specified condition which is contradictory.

or "There exists a set that contains all sets that do not include themselves."
If it contains ALL sets, then it must include the set itself that defines this but then instantly would include itself.

I treat these kinds of paradoxes as 'true' if they exist in their own isolated universes where they cannot be breached. They are like the concept of a perpetual motion machine. While in one sense they CAN potentially exist, to know that means we had to have some information from the system that tells use it has motion. This is some form of energy release that would imply it can give off more energy than it is given. The reality is that they can exist but for what information that comes out of them requires an equivalent amount going in. [In quantum mechanics, this relates also to the fact that observation imposes upon the system being observed some input information from SOMEWHERE else at least for the system to be conserved.]

This is also the kind of error used in St. Anselm's clever argument about the existence of God. His assumed that since we can conceive something, the very conception itself is a real thing. And since we can imagine the concept, "something such that no greater thing exists", then this fact has to be true also. Thus, then he places the term, "God", as that which he means as that concept. Besides the fault of assuming you can call something by the same label in distinct definitions, the assumption is that the thoughts in your head have a real-world SHARED reality. Thus this logic, though 'clever' is a trick the author confused about the meaning of 'truth'. That is, you can't define the meaning of universal truths SHARED by others objectively as including the particular subjective perception of its members.

So the paradoxical statements can exist if dynamic circularity is permitted, for instance, but we cannot literally interpret them as having meaning where they speak of themselves with a prerequisite that meaning of truth applied to something must remain consistent. You have to separate the meaning's reality from the expression's reality. Both are 'true' in a greater universe but have exclusive domains or have to be speaking about the mechanism of the paradox to cause dynamic changes in value inclusively.

PeteOlcott wrote: ↑Mon Mar 25, 2019 4:51 pm Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf

To put this in laymen's terms all of the truth that can be expressed using
words or math symbols is anchored in sentences that are defined to be true:
“A cat is an animal”. Other true sentences are derived from this basic set:
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.

The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.

Here is how the mathematical notation works:
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
In English means:
{There are sentences of English that prove that "Cats breath"}

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

When we assume this formalization of the notion of Truth:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
Then the above sentence says:

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

I agree with the general point that the paradox of the Liar, and other similar paradoxes, are effectively false.
However, I'm not sure you can prove that conclusively by formal proof. Any formal system specifies its own proof process and formal proofs are therefore all relative to the specific formal system in which they are made. So, formal truth is relative, too.
Your proposed formalisation of the notion of truth is dependent on the definition of the proof sign "⊢". So, now, you can't define it using the notion of truth. How do you define proof and deduction without using the notion of truth?
EB

Scott Mayers wrote: ↑Tue Mar 26, 2019 7:36 pm Bad comparisons. You need to differentiate validity from soundness and truth.

The difference between valid reasoning and sound reasoning within conventional deduction:
A deductive argument is said to be valid if and only if it takes a form that makes it
impossible for the premises to be true and the conclusion nevertheless to be false.
Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises
are actually true. Otherwise, a deductive argument is unsound.
Source: Validity and Soundness https://www.iep.utm.edu/val-snd/

When we define symbolic logic according to (Curry/Braithwaite) then valid deduction to
conclusions is exactly the same as formal proofs to consequences and sound deduction
to conclusions is exactly the same as formal proofs to theorem consequences.

This last part is a very big deal because it provides the entire foundational basis for
refuting Tarski Undefinability and Gödel's 1931 Incompleteness Theorem.

Scott Mayers wrote: ↑Tue Mar 26, 2019 7:36 pm
Axioms (or postulates) are 'begged' rules of a system that cannot themselves be proven or are arbitrary of some set of possible other axioms that could be presented. Calling them "truths" is not sincere unless they are true universally 'true' in all systems. A theorem is the 'form' or general statement from conclusions within the system that are universally true WHEN we conditionally accept the axioms of the system.

An axiom in a formal system (according to Curry 2010) is an expression of the language a formal
system that is defined to be true. There are things just like this in natural language, we can
call them basic facts. Example: { A cat is an animal }.

Scott Mayers wrote: ↑Tue Mar 26, 2019 7:36 pm
To be more relative to all readers without the background, why not just use a direct example, like commonsense used. The liars paradox, is often used.

"This sentence is not true." Is the common form that I began my full-time work on these things in August 2016.
https://philpapers.org/archive/OLCFST.pdf
I have worked on these things quite diligently since 1997, yet no more than 20 hours per week until 2016.

Here is the Liar Paradox in C++:
bool LP = !(LP == true); // The C++ compiler warning indicates the nature of the error.

The Liar Paradox and the simplified essence of Gödel's 1931 Incompleteness Theorem
have identical structure.

The Liar Paradox can only be true when it is proven to be false, thus making it not true.
The Gödel sentence of the Incompleteness Theorem can only be satisfied it is provably unprovable.

Both of the above examples are merely self-contradictory sentences and thus must be
treated the same way as deduction with contradictory premises, unsound.

Speakpigeon wrote: ↑Tue Mar 26, 2019 9:42 pm I agree with the general point that the paradox of the Liar, and other similar paradoxes, are effectively false.
However, I'm not sure you can prove that conclusively by formal proof. Any formal system specifies its own proof process and formal proofs are therefore all relative to the specific formal system in which they are made. So, formal truth is relative, too.
Your proposed formalisation of the notion of truth is dependent on the definition of the proof sign "⊢". So, now, you can't define it using the notion of truth. How do you define proof and deduction without using the notion of truth?
EB

I spent over 3000 hours of full-time work on just the Liar Paradox.
In the process I had to acquire a much better understanding of symbolic logic, because
English simply had too much wiggle room that allowed meaning to slip and slide back and
forth making it impossible to be unequivocally understood.

The Liar Paradox is definitely not false, and it is not true either. Instead of true or false
it is self-contradictory and thus wrong.

Any C++ compiler will tell you the exact error of the Liar Paradox when
you try to compile this expression:

int main()
{
bool Liar_Paradox = !(Liar_Paradox == true);
}

lp.cpp(3) : warning C4700: uninitialized local variable 'Liar_Paradox' used

Formal Truth is relative in the same sort of way that Chinese Truth is relative to Chinese.
It is not relative so much as it depends upon a specific context.
{Five} > {Three} no matter what the language is. It is immutable true, thus not relative.

I get around this philosophical limitation and specify the formal system as the
currently existing entire body of general knowledge as formalized in the language
of Minimal Type Theory (providing the universal lingua franca).

As Haskell Curry posits (and it actually the case in reality) All formal language and natural
language truth is ultimately anchored in expressions of language defined to be true.
We can also derive other truth on the basis of this through sound deduction.
which is represented in symbolic logic as formal proofs to theorem consequences.

PeteOlcott wrote: ↑Tue Mar 26, 2019 10:18 pm The Liar Paradox is definitely not false, and it is not true either. Instead of true or false
it is self-contradictory and thus wrong.

Yes it's designed to be self-contradictory. I would like to call it an "invalid paradox", but validity already has a specific meaning in logic.

Atla wrote: ↑Tue Mar 26, 2019 10:49 pm

PeteOlcott wrote: ↑Tue Mar 26, 2019 10:18 pm The Liar Paradox is definitely not false, and it is not true either. Instead of true or false
it is self-contradictory and thus wrong.

Yes it's designed to be self-contradictory. I would like to call it an "invalid paradox", but validity already has a specific meaning in logic.

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

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

PeteOlcott wrote: ↑Mon Mar 25, 2019 4:51 pm Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf

To put this in laymen's terms all of the truth that can be expressed using
words or math symbols is anchored in sentences that are defined to be true:
“A cat is an animal”. Other true sentences are derived from this basic set:
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.

The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.

Here is how the mathematical notation works:
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
In English means:
{There are sentences of English that prove that "Cats breath"}

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

When we assume this formalization of the notion of Truth:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
Then the above sentence says:

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

this cat is dead

-Imp

Impenitent wrote: ↑Wed Mar 27, 2019 10:39 pm

PeteOlcott wrote: ↑Mon Mar 25, 2019 4:51 pm When we assume this formalization of the notion of Truth:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
Then the above sentence says:

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

this cat is dead

-Imp

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

From the Tarski Undefinability Theorem:
We shall now show that the sentence x is actually undecidable and at the same time true.
...
(3) x ∉ Pr ↔ x ∈ Tr // ~Provable(x) ↔ True(x)

PeteOlcott wrote: ↑Tue Mar 26, 2019 9:49 pm

Scott Mayers wrote: ↑Tue Mar 26, 2019 7:36 pm Bad comparisons. You need to differentiate validity from soundness and truth.

The difference between valid reasoning and sound reasoning within conventional deduction:
A deductive argument is said to be valid if and only if it takes a form that makes it
impossible for the premises to be true and the conclusion nevertheless to be false.
Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises
are actually true. Otherwise, a deductive argument is unsound.
Source: Validity and Soundness https://www.iep.utm.edu/val-snd/

When we define symbolic logic according to (Curry/Braithwaite) then valid deduction to
conclusions is exactly the same as formal proofs to consequences and sound deduction
to conclusions is exactly the same as formal proofs to theorem consequences.

This last part is a very big deal because it provides the entire foundational basis for
refuting Tarski Undefinability and Gödel's 1931 Incompleteness Theorem.

I take issue with presuming that given a theorem is 'sound' WITHOUT respecting the system of logic as 'conditional' only. That is, while a theorem can be 'sound' in reality, that reality is subject to the condition THAT the system of logic is 'true' in reality, NOT that is IS 'true' of reality with exception to that condition.

The 'soundness of theorems' are thus conditional (and thus contingent) to the subset of reality that includes that system ONLY.

PeteOlcott wrote:

Scott Mayers wrote: ↑Tue Mar 26, 2019 7:36 pm
Axioms (or postulates) are 'begged' rules of a system that cannot themselves be proven or are arbitrary of some set of possible other axioms that could be presented. Calling them "truths" is not sincere unless they are true universally 'true' in all systems. A theorem is the 'form' or general statement from conclusions within the system that are universally true WHEN we conditionally accept the axioms of the system.

An axiom in a formal system (according to Curry 2010) is an expression of the language a formal
system that is defined to be true. There are things just like this in natural language, we can
call them basic facts. Example: { A cat is an animal }.

The first sentence can be true but the second you state doesn't follow without defining what you mean by 'facts". "A cat is an animal" can be a well-formed-(instance) of the English language. So by 'fact' are you merely referring to this or are you extending this to the meaning that the sentence stands for external to the language? The 'fact' that cats or animals exist are distinctly about the real world by the semantic interpretation. But this is distinct and can't compare to the coinciding value, "fact" in both instances. The fact of the language being some 'fact' is not dependent upon the fact that the semantic meaning is true and vice versa. ....thus they are not "comparable".

PeteOlcott wrote:

Scott Mayers wrote: ↑Tue Mar 26, 2019 7:36 pm
To be more relative to all readers without the background, why not just use a direct example, like commonsense used. The liars paradox, is often used.

"This sentence is not true." Is the common form that I began my full-time work on these things in August 2016.
https://philpapers.org/archive/OLCFST.pdf
I have worked on these things quite diligently since 1997, yet no more than 20 hours per week until 2016.

Here is the Liar Paradox in C++:
bool LP = !(LP == true); // The C++ compiler warning indicates the nature of the error.

The Liar Paradox and the simplified essence of Gödel's 1931 Incompleteness Theorem
have identical structure.

The Liar Paradox can only be true when it is proven to be false, thus making it not true.
The Gödel sentence of the Incompleteness Theorem can only be satisfied it is provably unprovable.

Both of the above examples are merely self-contradictory sentences and thus must be
treated the same way as deduction with contradictory premises, unsound.

Okay. I understand the paradox. But I'm not sure what you are saying about it. ? You seem to be implying that the theorem itself is wrong with respect to reality as a whole (?) or ? Can you clarify if this is what you are stating?

What is your thesis statement? What is it that you are claiming?

Scott Mayers wrote: ↑Thu Mar 28, 2019 5:46 am
I take issue with presuming that given a theorem is 'sound' WITHOUT respecting the system of logic as 'conditional' only. That is, while a theorem can be 'sound' in reality, that reality is subject to the condition THAT the system of logic is 'true' in reality, NOT that is IS 'true' of reality with exception to that condition.

The 'soundness of theorems' are thus conditional (and thus contingent) to the subset of reality that includes that system ONLY.

If we construe formal proofs in formal languages as merely the way to write down
valid and sound deductive inference, as Braithwaite indicates:
The chain of symbolic manipulations in the calculus corresponds to and
represents the chain of deductions in the deductive system. (Braithwaite 1962: 2)

Then it is impossible for formal proofs to diverge from valid or sound deduction.

The tricky part here is putting all the knowledge into the formal system that
people usually use for deductive inference, so I initially assume that only
one notion of human knowledge is added to the formal system, that is exactly
how the English word: "True" is converted into symbolic logic: ∀x True(x) ↔ ⊢x

In simple English this says that an expression of language is true if there is sound
deductive inference that has this expression as its conclusion.