The concept of equivalence class is basic math, its for
example used in rational numbers construction since 2/3 ~ 4/6.

https://en.wikipedia.org/wiki/Equivalence_class#Properties

Equivalence classes are usually written as [a]. So under
the ~ equivalence relation, instead of dealing with 3 subsets:

+---------------+----------------+----------------+
| f | t | m |
+---------------+----------------+----------------+

You can take 3 values, that represent the 3 subsets,
with F = [a] where a e f, T = [a] where a e t, U = [a] where a e m.

F, T and U

But in DC Proof, I cannot write uppercase for a value,
so I had to use lower case:

49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48

Thats a technical problem of DC Poop.

Mild Shock schrieb am Donnerstag, 21. September 2023 um 10:20:57 UTC+2:
> You can use this equivalence relation:
>
> a ~ b :<=> [[a e t & b e t] v [a e f & b e f] v [a e m & b e m]]
>
> To get equivalence classes:
>
> https://en.wikipedia.org/wiki/Equivalence_class
>
> The equivalence classes are the truth values.
> Thats missing in your take. If you then further
> abstract, you get this nifty result:
> 49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
> Rem DNeg, 48
> Dan Christensen schrieb am Mittwoch, 20. September 2023 um 23:03:13 UTC+2:.
> > What ARE you talking about?

Ultimately I am using slightly different equivalence classes.
On the left hand side is the set condition, on the right hand
side the equivalence class:

Model 3-
~a e t & ~a e f ~~> x = u
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f

Model 3+
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u

Model 4
~a e t & ~a e f ~~> x = u
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u

Quite impressive Zoo of resolutions, already.

Mild Shock schrieb am Donnerstag, 21. September 2023 um 10:27:59 UTC+2:
> The concept of equivalence class is basic math, its for
> example used in rational numbers construction since 2/3 ~ 4/6.
>
> https://en.wikipedia.org/wiki/Equivalence_class#Properties" rel="nofollow" target="_blank">https://en.wikipedia.org/wiki/Equivalence_class#Properties
>
> Equivalence classes are usually written as [a]. So under
> the ~ equivalence relation, instead of dealing with 3 subsets:
>
> +---------------+----------------+----------------+
> | f | t | m |
> +---------------+----------------+----------------+
>
> You can take 3 values, that represent the 3 subsets,
> with F = [a] where a e f, T = [a] where a e t, U = [a] where a e m.
>
> F, T and U
>
> But in DC Proof, I cannot write uppercase for a value,
> so I had to use lower case:
> 49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
> Rem DNeg, 48
> Thats a technical problem of DC Poop.
> Mild Shock schrieb am Donnerstag, 21. September 2023 um 10:20:57 UTC+2:
> > You can use this equivalence relation:
> >
> > a ~ b :<=> [[a e t & b e t] v [a e f & b e f] v [a e m & b e m]]
> >
> > To get equivalence classes:
> >
> > https://en.wikipedia.org/wiki/Equivalence_class
> >
> > The equivalence classes are the truth values.
> > Thats missing in your take. If you then further
> > abstract, you get this nifty result:
> > 49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
> > Rem DNeg, 48
> > Dan Christensen schrieb am Mittwoch, 20. September 2023 um 23:03:13 UTC+2:.
> > > What ARE you talking about?

On Thursday, September 21, 2023 at 4:20:57 AM UTC-4, Mild Shock wrote:
> You can use this equivalence relation:
>
> a ~ b :<=> [[a e t & b e t] v [a e f & b e f] v [a e m & b e m]]
>

You are making this needlessly complicated. Why would you say, for example, that the sentences "What time is it?" and "Wash your hands" are somehow equivalent? They are simply sentences of indeterminate truth value.

There is no need for an equivalence relation here. The only structure you need here is the simple trichotomy: Every sentence is either (1) a true sentence, (2) a false sentence, or (3) one of indeterminate truth value. (See line 6 here: https://dcproof.com/LiarParadoxV2.htm )

Dan

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

There is no constant for the truth value "T" in DC Proof.
In classical logic you could defined the truth values
"T" and "F" as follows:

a ~ "T" :<=> (a <=> p & ~p)

a ~ "F" :<=> (a <=> p v ~p)

Now concerning your non-classical logic that resolves the
Liar Paradox, how do you define the involved truth values?
Assume we have already defined a ~ "T" and a ~ "F" in

your system. I guess you want to defined "U" as follows, right?

a ~ "U" :<=> ~a ~ "T" & ~a ~ "F"

But then there is a bug in DC Proof, I cannot use
capital letter for constant values. So I cannot
use the STANDARD constant values "F", "T" and "U" from here:

https://en.wikipedia.org/wiki/Three-valued_logic

Because DC Proof doesn't accept:

1 ALL(x):[Value(x) <=> x=F | x=T | x=U]
Axiom

It only accepts:

1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
Axiom

:-( Is this a bug or feature?

Dan Christensen halucinated complete bullshit:
> Every sentence is either (1) a true sentence,
(2) a false sentence, or (3) one of indeterminate truth value.

Take these two propositional formulas:
a) p => ~p
b) ~p => p

They are not equivalent to these two propositional formulas:
1) p v ~p /* true sentence */
2) p & ~p /* false sentence */

Do you claim they are equivalent to the Liar Paradox?

Mild Shock schrieb am Donnerstag, 21. September 2023 um 19:53:15 UTC+2:
> There is no constant for the truth value "T" in DC Proof.
> In classical logic you could defined the truth values
> "T" and "F" as follows:
>
> a ~ "T" :<=> (a <=> p & ~p)
>
> a ~ "F" :<=> (a <=> p v ~p)
>
> Now concerning your non-classical logic that resolves the
> Liar Paradox, how do you define the involved truth values?
> Assume we have already defined a ~ "T" and a ~ "F" in
>
> your system. I guess you want to defined "U" as follows, right?
>
> a ~ "U" :<=> ~a ~ "T" & ~a ~ "F"
>
> But then there is a bug in DC Proof, I cannot use
> capital letter for constant values. So I cannot
> use the STANDARD constant values "F", "T" and "U" from here:
>
> https://en.wikipedia.org/wiki/Three-valued_logic
>
> Because DC Proof doesn't accept:
>
> 1 ALL(x):[Value(x) <=> x=F | x=T | x=U]
> Axiom
>
> It only accepts:
> 1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
> Axiom
> :-( Is this a bug or feature?

On Thursday, September 21, 2023 at 1:53:15 PM UTC-4, Mild Shock wrote:
> There is no constant for the truth value "T" in DC Proof.

Neither is there in most textbook math proofs. As I have shown here, it isn't necessary for a resolution of The Liar Paradox. Basic set theory and ordinary logic are enough.

> In classical logic you could defined the truth values
> "T" and "F" as follows:
>
> a ~ "T" :<=> (a <=> p & ~p)
>
> a ~ "F" :<=> (a <=> p v ~p)
>

Why bother?

> Now concerning your non-classical logic ...

There is nothing "non-classical" about it. It's the ordinary logic of most textbook math proofs.

> ... that resolves the
> Liar Paradox, how do you define the involved truth values?

I prove a theorem of basic set theory:

ALL(s):ALL(t):ALL(f):ALL(m):[Set(s) & Set(t) & Set(f) & Set(m) <--------- 4 Arbitrary sets

=> [ALL(a):[a in t => a in s] <------------- 3 Subsets of s
& ALL(a):[a in f => a in s]
& ALL(a):[a in m => a in s]

& ALL(a):[a in s => [a in t | a in f | a in m] <---------- Trichotomy Rules

& ~[a in t & a in f]
& ~[a in t & a in m]
& ~[a in f & a in m]]

=> ALL(b):[b in s => [[b in t <=> b in f] => b in m]]]]

Since "This sentence is false" is a true sentence if and only if it is false sentence, we can resolve LP by interpreting:

s = a set of sentences
t = the subset of true sentences in s
f = the subset of false sentence in s
m = the set sentences "of indeterminate truth value") in s

Thus, "This sentence is false" can be proven to be a sentence of indeterminate truth value using only basic set theory and ordinary logic with no need to postulate any other structures.

> But then there is a bug in DC Proof, I cannot use
> capital letter for constant values.

Not a bug. The first character of the name of a variable/constant cannot be an uppercase letter. The first character of the name of a logical proposition or predicate must be an uppercase letter. Deal with it.

> So I cannot
> use the STANDARD constant values "F", "T" and "U" from here:
>
> https://en.wikipedia.org/wiki/Three-valued_logic
>
> Because DC Proof doesn't accept:
>
> 1 ALL(x):[Value(x) <=> x=F | x=T | x=U]
> Axiom
>
> It only accepts:
> 1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
> Axiom
> :-( Is this a bug or feature?

OMG! More excuses for not doing your homework, Mr. Collapse??? Why not use lower case letters?

Dan

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