Here is a counterexample to Dan Christensens claim:

1) Every sentence is either (1) a true sentence, (2) a false
sentence, or (3) one of indeterminate truth value.

2) The Liar Sentence (L) is of indeterminate truth value.
https://dcproof.com/LiarParadoxV2.htm

The counter example does neither violate Trichotomy, nor
the conclusion that Dan Christensen proved. The Trichotomy
and the conclusion are respected.

Its more based on the fact that the conlusion ALL(b):
.... [b e t <=> b e f] ..., which does not imply or require
.... [L e t <=> L e f] ..., since this was nowhere stated.

So how is the counter model constructed? Very easily, take
as the set s, the set of propositional formulas, encoded as strings,
i.e. "p", "~p", "q", "p /\ q", etc...

Now define:

t = { "a" e s | a <=> pv~p}
f = { "a" e s | a <=> p&~p}
u = s \ t \ f

Then take L = "p<->~p". We find:

~L e u

Because L e f.

So why does ALL(b): ... [b e t <=> b e f] ..., not imply or require
.... [L e t <=> L e f] ...? Because [b e t <=> b e f] is on the left hand
side of a material implication:

=> ALL(b):[b e s => [[b e t <=> b e f] => b e m]]]]
https://dcproof.com/LiarParadoxV2.htm

For a sentence L it could be equally well the case that
[L e t <=> L e f] is false, and that the material implication is
vacously true. The axioms of Dan nowhere talk about L!

So how comes that L e f. By construction our L = "p<->~p",
and to have L e f, we must show (p<->~p) <=> p&~p, which
is easy to show, for example using Wolfgang Schwarz tree tool:

(p↔¬p) ↔ (p∧¬p) is valid.
https://www.umsu.de/trees/#%28p~4~3p%29~4%28p~1~3p%29

Mild Shock schrieb am Donnerstag, 21. September 2023 um 22:58:15 UTC+2:
> Here is a counterexample to Dan Christensens claim:
>
> 1) Every sentence is either (1) a true sentence, (2) a false
> sentence, or (3) one of indeterminate truth value.
>
> 2) The Liar Sentence (L) is of indeterminate truth value.
> https://dcproof.com/LiarParadoxV2.htm
>
> The counter example does neither violate Trichotomy, nor
> the conclusion that Dan Christensen proved. The Trichotomy
> and the conclusion are respected.
>
> Its more based on the fact that the conlusion ALL(b):
> ... [b e t <=> b e f] ..., which does not imply or require
> ... [L e t <=> L e f] ..., since this was nowhere stated.
>
> So how is the counter model constructed? Very easily, take
> as the set s, the set of propositional formulas, encoded as strings,
> i.e. "p", "~p", "q", "p /\ q", etc...
>
> Now define:
>
> t = { "a" e s | a <=> pv~p}
> f = { "a" e s | a <=> p&~p}
> u = s \ t \ f
>
> Then take L = "p<->~p". We find:
>
> ~L e u
>
> Because L e f.

Mild Shock wrote:

> Here is a counterexample to Dan Christensens claim:
> 1) Every sentence is either (1) a true sentence, (2) a false sentence,
> or (3) one of indeterminate truth value.
> 2) The Liar Sentence (L) is of indeterminate truth value.
> https://dcproof.com/LiarParadoxV2.htm
> The counter example does neither violate Trichotomy, nor the conclusion
> that Dan Christensen proved. The Trichotomy and the conclusion are
> respected. Its more based on the fact that the conlusion ALL(b):
> ... [b e t <=> b e f] ..., which does not imply or require ... [L e t
> <=> L e f] ..., since this was nowhere stated.

nothing. Kiss my ass.

On Thursday, September 21, 2023 at 4:58:15 PM UTC-4, Mild Shock wrote:
> Here is a counterexample to Dan Christensens claim:
>
> 1) Every sentence is either (1) a true sentence, (2) a false
> sentence, or (3) one of indeterminate truth value.
>
> 2) The Liar Sentence (L) is of indeterminate truth value.
> https://dcproof.com/LiarParadoxV2.htm
>
> The counter example does neither violate Trichotomy, nor
> the conclusion that Dan Christensen proved. The Trichotomy
> and the conclusion are respected.
>
> Its more based on the fact that the conlusion ALL(b):
> ... [b e t <=> b e f] ..., which does not imply or require
> ... [L e t <=> L e f] ..., since this was nowhere stated.
>
[snip]

See line 13.

In my proof, x is to be interpreted as the sentence, "This sentence is false." I prove that [x in t <=> x in f] => x in m (lines 13-36)
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