The Revenge Paradox in DC Proof

[Mild Shock]

Here we have a stipulation of a
Trivalent solution to the Liar Paradox:

https://dcproof.com/LiarParadox2.htm

My stipulation is that there will be always
the posibility to formulate some Paradox.
For example Trivalently this is a new paradox:

"this sentence does not express a true proposition"

Its based on this truth table for "not express a true proposition":

A A =\= T
T F
U T
F T

[Dan Christensen]

On Monday, July 24, 2023 at 3:21:58 PM UTC-4, Mild Shock wrote:
> Here we have a stipulation of a
> Trivalent solution to the Liar Paradox:
>
> https://dcproof.com/LiarParadox2.htm
>

Sorry, Mr. Collapse, no "trivalent" logic here. Just, good, old-fashioned, true-or-false logic with a smattering of set theory. https://dcproof.com/LiarParadox2.htm

Some sentences (like questions or imperatives, or simply meaningless or ambiguous sentences) are neither true nor false. We can say that their truth values are INDETERMINATE (like the "numerical value" of 1/0). Unlike logical propositions in classical logic, if you have a set of sentences, you can classify their truth values as either (1) true, (2) false, or (3) indeterminate. Now, "This sentence is false" (The Liar) is true if and only if it false. What could be more ambiguous? Logically, it cannot be classed as either true or false. Logically, as shown in my proof (link above), it has to be classified as INDERTIMINATE.

I hope this helps.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Mild Shock]

Trivalent = Trichotomy holds

Its in your proof:

ALL(a):[a e s => [a e t | a e f | a e m] (Trichotomy holds)
& ~[a e t & a e f] & ~[a e t & a e m] & ~[a e f & a e m]]

And it gives a new paradox:

"this sentence does not express a true proposition"

See for yourself for an explanation:
https://plato.stanford.edu/entries/liar-paradox/#InstReve

[olcott]

On 7/25/2023 12:04 AM, Dan Christensen wrote:
> On Monday, July 24, 2023 at 3:21:58 PM UTC-4, Mild Shock wrote:
>> Here we have a stipulation of a
>> Trivalent solution to the Liar Paradox:
>>
>> https://dcproof.com/LiarParadox2.htm
>>
>
> Sorry, Mr. Collapse, no "trivalent" logic here. Just, good, old-fashioned, true-or-false logic with a smattering of set theory. https://dcproof.com/LiarParadox2.htm
>
> Some sentences (like questions or imperatives, or simply meaningless or ambiguous sentences) are neither true nor false.

The key issue that I have focused on since 2004 is pathological self-
reference such that an expression of language contradicts both of its
truth values. This is misnamed as undecidable.

> We can say that their truth values are INDETERMINATE (like the "numerical value" of 1/0).

It is not that no one can determine the truth value, it is that the
truth value is non-existent.

> Unlike logical propositions in classical logic, if you have a set of sentences, you can classify their truth values as either (1) true, (2) false, or (3) indeterminate. Now, "This sentence is false" (The Liar) is true if and only if it false. What could be more ambiguous?

That is not ambiguity that is self-contradiction.

>Logically, it cannot be classed as either true or false. Logically, as shown in my proof (link above), it has to be classified as INDERTIMINATE.
>

It has a non-existent truth value.

> I hope this helps.
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

[Mild Shock]

Calling names "non-existent" etc.. doesn't help. The domain of discourse
was extended by Dan Christensen from "propositions" which are bi-valently
thought in classical logic, to "expressions" which are tri-valently thought

in his proof, i.e. the set s is partitioned into t, f and m:

> ALL(a):[a e s => [a e t | a e f | a e m] (Trichotomy holds)
> & ~[a e t & a e f] & ~[a e t & a e m] & ~[a e f & a e m]]

> https://dcproof.com/LiarParadox2.htm

Question is, whether going from "propositions" to "expressions"
and going from "bi-valent" to "tri-valent" excludes the paradoxes.
The answer is "no". Its just that new paradoxes pop up:

"this sentence does not express a true proposition"

I proposed a tri-valent logic indeed, which would render
the above expression paradoxical, similar like the Liar Paradox
is a paradoxical proposition, namely I proposed:

/* LOGIC OF HERE AND THERE */
A not(T)

T F
U T
F T

Which is the "LOGIC OF HERE AND THERE"
https://en.wikipedia.org/wiki/Three-valued_logic#HT_logic

But you could also try this tri-valent logic, the
"KLEENE LOGIC", which would also yield a paradoxical
situation, if U rippels through bi-conditional as well the

sentence is not true, usually U is not accepted as true:

/* KLEENE LOGIC */
A not(T)
T F
U U
F T

But either way with the change of the domain of discourse,
from "propositions" to "expressions" and from "bi-valent" to
"tri-valent", the problem didn't go away, there

are still expressions and logics (many different ones) where a
Liar Paradox like situation appears. This phaenomenon is called the
Revenge Paradox. You find some mention of it on the internet.

BTW: There is also the "Ultimate Revenge" Paradox.
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

[Mild Shock]

Between Revenge and Ultimate Revenge is Super Revenge:

Revenge Paradox: 3 partition (like in Dan Christensen)
Super Revenge: 4 partition
Ultimate Revenge: n partition

The Ultimate Revenge is usually used in
- Alan Turing Undecidability Theorem,
- Kurt Gödels Incompletness Theorem
- Alonzo Churchs Unsovability Theorem:

An Unsolvable Problem of Elementary Number Theory - Alonzo Church
American Journal of Mathematics, Vol. 58, No. 2. (Apr., 1936), pp. 345-363
https://www.ics.uci.edu/~lopes/teaching/inf212W12/readings/church.pdf

The Ultimate Revenge is there, with natural numbers
starting at one, i.e. 1, 2, 3, ..., in Alonzo Church (page 361 bottom ff):

"this sentence expresses the successor of itself or
one if this sentences is non-normalizing"

[olcott]

On 7/25/2023 9:28 AM, Mild Shock wrote:
> Calling names "non-existent" etc.. doesn't help. The domain of discourse
> was extended by Dan Christensen from "propositions" which are bi-valently
> thought in classical logic, to "expressions" which are tri-valently thought
>
> in his proof, i.e. the set s is partitioned into t, f and m:
>
>> ALL(a):[a e s => [a e t | a e f | a e m] (Trichotomy holds)
>> & ~[a e t & a e f] & ~[a e t & a e m] & ~[a e f & a e m]]
>> https://dcproof.com/LiarParadox2.htm
>
> Question is, whether going from "propositions" to "expressions"
> and going from "bi-valent" to "tri-valent" excludes the paradoxes.
> The answer is "no". Its just that new paradoxes pop up:
>
> "this sentence does not express a true proposition"
>
> I proposed a tri-valent logic indeed, which would render
> the above expression paradoxical, similar like the Liar Paradox
> is a paradoxical proposition, namely I proposed:
>

I will ask you three questions:
(1) Are you less than 50 years old?
(2) Do you weigh more than 100 pounds?
(3) sbejdk ijsu kslndf asdbjkads?

It the problem with the last question that you cannot make up your mind,
or something else?

[Dan Christensen]

On Tuesday, July 25, 2023 at 3:19:10 AM UTC-4, Mild Shock wrote:

> Dan Christensen schrieb am Dienstag, 25. Juli 2023 um 07:04:07 UTC+2:
> > On Monday, July 24, 2023 at 3:21:58 PM UTC-4, Mild Shock wrote:
> > > Here we have a stipulation of a
> > > Trivalent solution to the Liar Paradox:
> > >
> > > https://dcproof.com/LiarParadox2.htm
> > >
> > Sorry, Mr. Collapse, no "trivalent" logic here. Just, good, old-fashioned, true-or-false logic with a smattering of set theory. https://dcproof.com/LiarParadox2.htm
> >
> > Some sentences (like questions or imperatives, or simply meaningless or ambiguous sentences) are neither true nor false. We can say that their truth values are INDETERMINATE (like the "numerical value" of 1/0). Unlike logical propositions in classical logic, if you have a set of sentences, you can classify their truth values as either (1) true, (2) false, or (3) indeterminate. Now, "This sentence is false" (The Liar) is true if and only if it false. What could be more ambiguous? Logically, it cannot be classed as either true or false. Logically, as shown in my proof (link above), it has to be classified as INDERTIMINATE.
> >

> Trivalent = Trichotomy holds
>

Wrong.

> Its in your proof:
>
> ALL(a):[a e s => [a e t | a e f | a e m] (Trichotomy holds)
> & ~[a e t & a e f] & ~[a e t & a e m] & ~[a e f & a e m]]
>

For any set s, there will exist sets t, f and m that satisfies this condition. Hint: Use t=s, f=m= { }

> And it gives a new paradox:
> "this sentence does not express a true proposition"

It also does not express a false proposition.

Other examples : "What time is it?" "Do your homework." Neither sentence is true or false. Their truth values are indeterminate.

> See for yourself for an explanation:
> https://plato.stanford.edu/entries/liar-paradox/#InstReve

Not very helpful.

[Mild Shock]

I 100% agree with Olcott, he is the better Logician:

olcott schrieb am Dienstag, 25. Juli 2023 um 15:18:48 UTC+2:
> > On 7/25/2023 12:04 AM, Dan Christensen wrote:

> > We can say that their truth values are INDETERMINATE (like the "numerical value" of 1/0).

> It is not that no one can determine the truth value, it is that the truth value is non-existent.

https://groups.google.com/g/sci.logic/c/KfDliBm1Hb8/m/S4FUx1w_CAAJ

Dan Christensens usage of the word "indeterminate" is completely wrong.
According to these slides the Liar Paradox doesn't have an "indeterminate"
truth, because "indeterminate" is defined as:

A sentence Φ (or set of sentences Σ) is indeterminate if and only if
there is more than one way to coherently assign it a truth value (or to
assign the sentences contained in it truth values)
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

The correct terminology is "paradoxical", defined as:

A sentence Φ (or set of sentences Σ) is paradoxical if and only if
there is no way to coherently assign it a truth value (or to assign the
sentences contained in it truth values).
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

And its relativey easy to prove that the sentence is paradoxical,
just stay in classical logic and stay with proposition you then get:

/* Law of Non-Contradiction */
∀x(~(Tx & Fx)) &
/* Law of Excluded Middle */
∀x(Tx v Fx) =>
/* Paradoxical */
~∃x(Tx <-> Fx)

Its pretty easy. Even Wolfgang Schwartz tree tool can do it:

(∀x¬(Tx ∧ Fx) ∧ ∀x(Tx ∨ Fx)) → ¬∃x(Tx ↔ Fx) is valid.
https://www.umsu.de/trees/#~6x(~3(Tx~1Fx))~1~6x(Tx~2Fx)~5~3~7x(Tx~4Fx)

Now we have Olcotts "non-existent". But we need to go
back to "propositions" and "classical", to have Olcotts
"non-existent". But Olcott was 100% right.

[Mild Shock]

In set theory one can even prove the ultimate revenge
paradox, for arbitrary domains J = {0,...}. There is the
following set theory theorem, relatively trivial:

Theorem Constant Function:
Assume an arbitrary domain J = {0,...} which is non-empty
and has zero 0, i.e. 0 e J. Then looking at the function space,
i.e. functions for an arity n:

f : J^n -> J

We find that there is always a constant function:

ALL(x1):[x1 e J => .... ALL(xn):[xn e J => f(x1,..,xn) = 0] ...]

Proof:
Just use the Subset Axiom, and construct this function:
f = { (x1,...,xn,0) | x1 e J & ... & xn e J }
Q.E.D:

Corrolary Ultimate Revenge:
In a language that is at least as expressive as set theory,
there is always a Liar Paradox like form, even if we allow
to range expressions over more than B = {0,1}.

Proof:
We had J arbitrary in the previous theorem, so it can be also
different or larger than B. Even excelling the super revenge
paradox which transcends only K = {0,1,u}.
Q.E.D.

[Mild Shock]

I guess the wish to "resolve" the Liar Paradox, is rooted in
Alchemy. I find, the Ouroboros as a symbol of self reference?

Ouroboros (representation of a serpent eating its own tail)
with the words ἕν τὸ πᾶν, hen to pān ("the all is one") from
the Chrysopoeia of Cleopatra the Alchemist in the 3rd
century or 4th century A.D. (Christian era)
https://en.wikipedia.org/wiki/Chrysopoeia

So will Dan O Matik turn dirt to gold. Go Go Dan O Matik
resolve the Liar Paradox, we are all currious how an

Antinomy is not an Antinomy.

[Mild Shock]

The easiest resolution of the Liar Paradox is found in:

"One is the Serpent which has its poison according to two
compositions, and One is All and through it is All, and by
it is All, and if you have not All, All is Nothing."
https://en.wikipedia.org/wiki/Cleopatra_the_Alchemist

Kind of Archimedes Plutonium totality, but we can
use it to resolve the Liar Paradox. Just go Oneotomy
instead of Trichtomoy, i.e. only a single truth value, and

you are done. Just change this into a single truth value:

https://dcproof.com/LiarParadox2.htm

LoL

[Mild Shock]

Here is a new challenge, can you find a revenge paradox to Dan
Christensens trichotomy, that doesn't use the word "indeterminate"
in its text form, respectively that doesn't use Uy in its formula?

Kind of works with a new hidden state, without mentioning it explicitly?

[Mild Shock]

Ok, here is a revenge paradox that doesn't mention Uy:

R'' : This sentences is true iff (if it is true then it is false).
(Ty ↔ (Ty → Fy))

This works:

∀x(Fx ∨ (Tx ∨ Ux)), ∀x¬(Fx ∧ Tx),
∀x¬(Fx ∧ Ux), ∀x¬(Tx ∧ Ux) entails ¬(Ty ↔ (Ty → Fy)).
https://www.umsu.de/trees/#~6x(Fx~2Tx~2Ux),~6x(~3(Fx~1Tx)),~6x(~3(Fx~1Ux)),~6x(~3(Tx~1Ux))|=~3(Ty~4(Ty~5Fy))

I got some help from Prolog. And found it with this query:

?- formula(3, X, Y), antinomy(X, (forall(X=t,Y),forall(Y,X=t))).
Y = (X = f; X = u);
Y = (X = u; X = f);
Y = forall(X = t, X = f);
Y = forall(X = t, X = u);
fail.

Prolog source code here:

formula(1, X, X = C) :- !,
value(_, C).
formula(N, X, Y) :- !,
M is N-1, formula(_, M, X, Y).

formula(0, N, X, (\+ Y)) :- formula(N, X, Y).
formula(1, N, X, (Y ; Z)) :- M is N-1, between(1, M, K), J is N-K,
formula(K, X, Y), formula(J, X, Z).
formula(2, N, X, (Y , Z)) :- M is N-1, between(1, M, K), J is N-K,
formula(K, X, Y), formula(J, X, Z).
formula(2, N, X, forall(Y , Z)) :- M is N-1, between(1, M, K), J is N-K,
formula(K, X, Y), formula(J, X, Z).

antinomy(X, Y) :- \+ (value(_, X), Y).

value(0, f).
value(1, u).
value(2, t).

[Mild Shock]

Since it doesn't mention Uy, we can try whether it also
works in bi-valence. And amazingly its also an Antinomy
in bi-valence. I get this result:

L' : This sentences is true iff (if it is true then it is false).
¬(p ↔ (p→¬p)) is valid.
https://www.umsu.de/trees/#~3(p~4(p~5~3p))

[Mild Shock]

One key ingredient of Dan Christensens approach is to use
classical logic. He models the Liar sentences as (a e T <=> a e F),
so he uses classical logic. For classical propositional logic

there exists the notion of a propositional variable assignment,
and if L is a formula, set(L) would denote all possible variable
assignment. There is this rather trivial theorem:

Theorem: If sat(L)=\={} and sat(L) ⊆ Y \ X, then:
1) L is an Antinomy in X
2) L has a Resolution in Y

So we can use this for another type of Revenge Paradox, namely
that Dan Christensen has only shown a sufficient example for
Resolution, but not a necessary example for Resolution:

Take X = {{0},{1}} classical logic, then:

1) Y= { {}, {0}, {1}}: 3-valued Logic with bottom, the solution proposed
by Dan Christensen, is a Resolution
2) Y= { {0}, {1}, {0,1}: 3-valued Logic with top, sometimes called
Logic of Paradox, would be also a Resolution
3) Y= { {}, {0}, {1}, {0,1}: 4-valued Logic with bottom and top, called
Belnap FOUR, would be also a Resolution

Since there are more Resolutions than only one, he only showed
a sufficient example, its not some necessary example.

Dan Christensen schrieb am Dienstag, 25. Juli 2023 um 07:04:07 UTC+2:

[Julio Di Egidio]

On Saturday, 16 September 2023 at 12:57:22 UTC+2, Mild Shock wrote:

> One key ingredient of Dan Christensens approach is to use
> classical logic. He models the Liar sentences as (a e T <=> a e F),

Which isn't the liar paradox, you spamming demented pieces of shit.

*Agents of the enemy Alert*

Julio

[Mild Shock]

Its the Liar Paradox as modelled by Dan Christensen:

[b e t <=> b e f]
https://dcproof.com/LiarParadox2.htm

Whats your point italian stinky?