The Liar Paradox Revisited... Again!

Following is my attempt, using predicate calculus, to resolve the so-called Liar Paradox. Informally:

A man says, "Everything I say is a lie."

If the sentence, "Everything I say is a lie," is true, then the man does say at least one thing that is true, namely that sentence itself. This a contradiction, therefore the sentence must be false.

The formal argument:

We have 3 predicates:

S(x) means x is sentence
T(x) means x is true
M(x) means man says x

Using indirect proof...

1. S(x) /\ M(x) & Ay(S(y) -> (M(y) -> ~T(y))) Premise

2. S(x) Splitting premise, 1

3. M(x)

4. Ay(S(y) -> (M(y) -> ~T(y)))

5. T(x) Premise

6. S(x) -> (M(x) -> ~T(x)) Universal Specification, 4

7. M(x) -> ~T(x) Detachment, 2, 6

8. ~T(x) Detachment, 3, 7

9. T(x) & ~T(x) Contradiction

10. ~T(x) Conclusion, 5

11. ~Ax((S(x) /\ M(x) & Ay(S(y) -> (M(y) -> ~T(y))) -> ~T(x)) Conclusion, 1


Comments? Does this resolve the paradox?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com

(Correcting typos)
1. S(x) /\ M(x) /\ Ay(S(y) -> (M(y) -> ~T(y))) Premise
11. Ax((S(x) /\ M(x) /\ Ay(S(y) -> (M(y) -> ~T(y))) -> ~T(x)) Conclusion, 1

That isn't the liar paradox because you can resolve it by saying that
the man lies on that occasion.

The liar paradox is

This sentence is false.

which isn't so readily dealt with. Hence "paradox".
Is it really plausible that a stupid cunt like you can resolve the
problem so easily?

--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.

My resolution, if indeed it is one, goes beyond that. It says, in effect, that EVERY such sentence that a man says must be false.
Historically, that is only one form. And I'm not sure that it captures of essence of the original from antiquity, which talks about a potentially infinite number of sentences. The paradox in the original can also be removed by restating the liar's claim: "But for this sentence which is true, everything I say is a lie." I don't think that any such patch can repair "This sentence is false."
Geez, Willy, it seems you have given up already and are starting right in with the personal insults! Get your butt kicked one too many times?

Using the same predicates, "This sentence is false," we obtain:

S(x) /\ (T(x) <-> ~T(x))

where x = "This sentence is false."

This nothing more than a simple contradiction. It's like saying the sky is blue and the sky is not blue. Not very interesting. Have I missed something?

(Important corrections)
5. S(x) -> (M(x) -> ~T(x)) Universal Specification, 4

6. M(x) -> ~T(x) Detachment, 2, 5

7. ~T(x) Detachment, 3, 6

8. Ax((S(x) /\ M(x) /\ Ay(S(y) -> (M(y) -> ~T(y))) -> ~T(x)) Conclusion, 1


There is nothing intrinsically paradoxical about someone always lying. As far as I can tell, a contradiction arises only from the fact that, as in my scenario, our constant liar admits to being one.

5, 9 and 10 aren't needed to conclude 8.
How did you conclude that? Clearly

S(x) & M(x) & Ay(S(y) -> (M(y) -> ~T(y))) -> ~T(x)
Ax(S(x) & M(x) & Ay(S(y) -> (M(y) -> ~T(y))) -> ~T(x))

in contradiction to "conclusion" 1.
No. That's not the pardox which requires 2nd order logic.

The liar's paradox
S(p) & M(p) & (p <-> Ay(S(y) & M(y) -> ~T(y))

Theorem. p -> ~T(p).

Assume ~p. Then Ey(S(y) & M(y) & T(y))
Denote by y such a proposition. Thus T(y) & ~T(y).
Consequently p. Thusly ~T(p).
Is ~T(p) -> ~p a theorem? If so, p & ~p.

Resolution of this paradox is that p -> T(p) is not a theorem.

On Aug 21, 7:42 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
What happens if all the other statements the man makes throughout his
life are statements we know to be false?

"Dan Christensen" <Dan_Chr...@sympatico.ca> wrote in message
news:962c3acf-aa2d-4dfc...@googlegroups.com...
Paraphrasing something from Smullyan's GIT:

Imagine a land where people are either Athenians, or Cretan, or neither.

You are given that every inhabitant of the land either always lies or never
does so, and that Athenians always tell the truth, while Cretans always lye.

What would be a statement from an inhabitant of the land that would convince
you that he (always) tells the truth?

What would be a statement from an inhabitant of the land that would convince
you that he (always) lies?

-LV

It is clear that those predicates alone cannot be used to formalize the
liar because it is about a sentence (or a supposed sentence, if you
prefer) saying something about itself (or supposedly saying something
about itself, if you prefer). The self-referential nature of the liar
is an important aspect of it, this you ignore completely. It was not
long ago that you became an instant expert in group theory, and thus
managed to spout bollocks about it. Have you now become an instant
expert in the liar with a similar outcome?

A glutton for punishment, Willy?

In hindsight, my "proof" did not contribute much to the discussion (just "brainstorming"), but what does self-reference have to do with it? The classical paradox arises from two contradictory assumptions:

1. Everything the liar says is false. (Not always made explicit by writers, including me.)

2. The claim by the liar that everything he says is false.

The negation of the liar's claim is, of course, that something he says is true.

Without either of these two assumptions, there is no paradoxical contradiction. If the liar simply refrained from making such a claim, there would be no contradiction. And self-reference doesn't seem to be the culprit. If the liar had said, "Everything I say is true" (a self-reference), there would also be no contradiction.

Nor did I say it was. What I did say was "The self-referential nature

On Wednesday, August 22, 2012 1:37:15 PM UTC-4, Frederick Williams wrote:

[snip]
How is self-reference important? As I see, it is just a matter of having made two contradictory assumptions (see above). Remove one of them and the paradoxical aspect of this scenario vanishes.

I know that self-reference is often cited in the literature as part of the problem -- as if it was something to be avoided -- but I just don't see it. As I pointed out above, you can still have self-reference without any contradictions arising, with only a slightly modified scenario (where the liar says everything he says is TRUE).

The claim that self-reference is a problem in some particular case
should not be read as a claim that it is always a problem. Only a
witless cunt like you would think otherwise.

[snipping abuse]

So, self-reference is NOT important in this case after all?

It is relevant to the liar.

Ummmm.... OK.

On Aug 23, 8:08 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote
S: IF HALT(S) GOTO S

This is informally,

"If this program itself always stops, then keep running forever and
never stop".

THEREFORE: OMEGA ~e COMPUTABLE-REALS
THEREFORE: |R| > |N|

Correct?

Herc

Suppose that is the only statement that the man has ever made.

What does this have to do with the topic at hand?

(Correction)
I think my analysis still holds up in this case. By assumption, his statement is false. The negation of his statement, of course, is that he must say something that is true -- a contradiction.

Again, the paradox arises from making two contradictory assumptions:
2. The liar himself claims that everything he says is false.

Eliminate either one of these assumptions and the paradoxical nature of this scenario vanishes.

Why are you assuming that? Look, if the man has only ever said
"Everything I say is a lie" then that is equivalent to (him saying)

This statement is false. (*)

Which _is_ the liar (unlike your garbled version). Now, _if_ that is
false then it is true; and _if_ it is true then it is false. That, in
the light of two assumptions, is paradoxical. The two assumptions are:
1: (*) really is a statement, and 2: statements are always either true
or false and never both.
That is not an assumption. If you think in terms of a man actually
making an utterance you might call it an empirical fact.
No, forget 2 because it is a fact over which you have no control.

(Corrections)
I neglected to mention the assumption that everything the liar says is indeed a lie. Without that assumption, there is no contradiction and no paradox. As one reader pointed out, his statement could then be a false and other of his statements could be true.
Mine more closely resembles the original Liar Paradox of antiquity. Much more interesting.
We are talking about a hypothetical situation here. As such, your distinction is moot.
We cannot forget 2. The "fact" is, if the liar never made this famous claim, there would be no contradiction and no paradox. If he had even said, "Everything I say is TRUE," there would be no contradiction and no paradox.

No we're not. At about 19:44 BST I uttered these words: "This statement
is false." That's a fact. In bivalent propositional calculus these are
laws:

(P -> ~P) -> ~P (1)
(~P -> P) -> P (2)

for any statement P. Furthermore, the rule,

from P and P -> Q, Q follows (3)

for any statements P and Q applies in bivalent propositional calculus.
If my utterance was true, then it would be false. Letting P in (1) be
my utterance, and applying (3) we find that

not-(this statement is false).

On the other hand, if my utterance was false, then it would be true.
Letting P in (2) be my utterance, and applying (3) we find that

this statement is false.

There is a contradiction. What have we assumed? (a) that my utterance
is a statement, and (b) that certain laws and rules of bivalent
propositional calculus apply.
There is no need for the liar to do anything. I acted in the necessary
way at about 19:44 BST.
An irrelevance. Clearly you're struggling to understand what the liar
is about; introducing irrelevancies won't help you.

Okay, suppose that 2. is true. A man has just said "Everything
I say is a lie". Suppose further that this is the only thing he
has ever said.

So the question is whether 1 is true. If it is true, then that leads
to a contradiction. But if it is false, that *also* leads to a
contradiction.

Assuming that he only ever said that one thing leads to a contradiction. So, he could not have said only one thing. Good point!!!

If 1 is false, that is, if the man says at least one thing that is true, then the following scenario would not lead to a contradiction: He (1) falsely claims that everything he says is a lie, and (2)truthfully identifies, say, the day of the week. Sometimes he tells the truth. Sometimes he lies.

If 1 was true, then 2 could easily be false. Vacuously so if he never said anything. Or he could always lie without ever actually saying, "Everything I say is a lie."

On Aug 24, 12:47 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
It is a well known method to equate Godel Statements, Russell Sets,
Hang-If-Halt programs et al

with Liar Statements, although the argument is usually trivially
dismissed



Herc

news:2ba299c5-a9ba-4d83...@googlegroups.com...
No, "the liar" is the name we give to the form of a sentence: it does not
matter who and when pronounces it, the statement "this statement is false"
is paradoxical by itself. Then, about resolutions: in a standard formal
treatment, that statement comes out false but not refutable.

-LV

On Aug 24, 6:24 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
As I said at the beginning, I am talking here about the original Liar
Paradox of antiquity -- a version in which someone (the liar, as I
call him) is purported to say only lies, and who even admits it. I
don't know how to formalize the argument, but it seems that the
paradox arises out of two contradictory assumptions:

1. Everything this man says is a lie.

2. He himself says, "Everything I say is a lie."

If this confession, like everything else he says, is a lie, then it
would follow that he must say something that is true. This contradicts
1.

"Dan Christensen" <d...@dcproof.com> wrote in message
news:d78b9423-6049-4ea9...@b10g2000yqc.googlegroups.com...
You are over-assuming: in 1, that "this man" is lying is what you have to
show, all you have is that a man that is lying is always lying, and vice
versa; in 2, all "this man" needs to utter is "This utterance of mine is a
lie". Then you have the formalizations and resolutions provided up-thread.

-LV

Just _what_ are you referring to? I mean, can you name names? Look at
this: http://en.wikipedia.org/wiki/The_liar_paradox#History, there are a
few ancients mentioned there. Are you referring to any of them?

On Aug 24, 12:11 pm, Frederick Williams
Yes, the first reference there is to a version by Greek philosopher,
Epimenides:

"The Epimenides paradox (circa 600 BC) has been suggested as an
example of the liar paradox, but they are not logically equivalent.
The semi-mythical seer Epimenides, a Cretan, reportedly stated that
"The Cretans are always liars."

What is liar in this context? Someone who speaks nothing but lies? Or
someone who only sometimes tells lies. Nothing very interesting about
the latter. If he only tells lies sometimes, there is no paradox. So,
I assume the former was the intent.

(Resending -- transmission difficulties?)
[snip]

Isn't that just a straightforward self-contradiction? No more
information is conveyed than if he had said, "This utterance of mine
is TRUE." We know nothing more about the world as a result. Much more
information is conveyed if you call someone a liar. A nice little
scandal at the very least! ;^)

If I say "The moon is full and the moon is not full." Then I
straightforwardly contradict myself; and there is a straightforward
response: "The moon is full and the moon is not full." is false. But if
I say "This utterance is false" then (in the light of certain
assumptions spelt out elsewhere) the claim that what I said is false
does not resolve the matter. For, if it is false then it is true, and
vice versa.
God, you're thick. It has got nothing to do with calling someone a
liar. The point is, the utterance says _of itself_ that it is false.

On Aug 25, 4:09 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
"I am a Liar" is interesting.

It's the only statement in natural language this is always true.

(it depends on no world historical facts)

Herc

On Aug 24, 2:09 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
As your link to the Wiki article shows, yours is just one version of
the Liar Paradox. I don't know why you feel you have to repeatedly
deny this.

news:72a421a2-ef86-48c1...@m18g2000yqp.googlegroups.com...
<snip>
Under the assumption that Cretans always lie, I have already showed you what
that boils down to, and you have also been shown a formalization. I'd
suggest you re-read what has been said so far, and do play with the way
Smullyan puts it, as I have posted up-thread.
Again, "the liar" is *the form of a sentence*: this is a *logical* paradox.

-LV

news:23cb81c1-85e9-4beb...@f17g2000vbz.googlegroups.com...
> If you eliminate (negate) either of these assumptions, there is no
> contradiction and no paradox.

No, your version just is not "the liar": you are given a sentence, you are
*not* told in advance if the pronouncer is a liar or not. Under the sole
assumption that either a man always lies or is always truthful, the sentence
is paradoxical *in itself*.

-LV

On Aug 24, 9:43 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
I have reviewed what you wrote here and saw no attempt at
formalization.
The paradox I present here is based on the original from Epimenides in
600 BC. I think it can legitimately be called the Liar Paradox. It is
a puzzle to me why "This statement is false" is also called the "Liar
Paradox" -- where is the liar? -- but it does seem to be a common
usage in some circles of academe.

The original paradox from antiquity is also every bit as much a
logical paradox as, say, Russell's famous paradox, which also arises
from faulty assumption(s). (See my analysis here.)

news:d7c0f3ed-1cd8-4465...@u19g2000yqo.googlegroups.com...
The formalization was provided by Frederick Williams: you cannot even read.
You apparently do not even know what logic is.
Ah, I see, it is not that you are totally clueless and close-minded, it is
that you have your own theory... LOL.

Bye,

-LV

Enough to know a contradiction when I see one.
You claim that the original Liar Paradox is not really The Liar
Paradox. You claim that ONLY a sentence which makes no mention of
liars or lying can be called that. That is closed-mined.

news:3d63e473-91f5-4f17...@a14g2000yqc.googlegroups.com...
You are utterly confused: the one who claims that those are two different
problems is you.

-LV

On Aug 25, 10:28 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
Now you are contradicting yourself. You said that my version is NOT
the real Liar Paradox. Now you seem to be saying it is equivalent to
yours. Please clarify.

It seems unlikely that the two are logically equivalent. (That Wiki
article said they are not.) The resolution of the original paradox,
which I outline above, is based on the notions of lies and liars,
while your "This statement is false" has nothing to do lies or liars.

Admittedly, I haven't given your version much thought. It really seems
like pure nonsense to me -- a simple self-contradiction. Whatever the
context, "This statement is TRUE" (referring to itself) would convey
no real information. We could remove it from any argument without
affecting the conclusion. Likewise, I think, for "This statement is
FALSE."

Dan
Download my DC Proof 2.0 software at http://www.dcprof.com

On Aug 25, 12:13 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
Some preliminary thoughts.... The statement "This statement is false"
makes no references to any functions, variables, predicates or
propositions. It has no references to any other statements other than
itself. Could there be a more useless statement in a proof or
argument?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com

On Aug 26, 7:20 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
>
Is it true or false?

Herc

Does it matter? It's a completely useless statement in a proof. When
would you ever use it?

On Aug 26, 8:10 am, Dan Christensen <Dan_Christen...@sympatico.ca>
what about:

Statement(X) is false

where X refers to that statement.

Herc

http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem


--
I have made a thing that superficially resembles music:

http://soundcloud.com/eroneity/we-berated-our-own-crapiness

The Liar Paradox has nothing to do arithmetic (numbers, etc.).

On Aug 26, 1:54 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
> Never mind the X. I was thinking about a special premise with nothing
> more than the the key-word "FALSE". It would not be parsed as a
> variable, constant or logical proposition. You would not be able to do
> anything with it, but you could subsequently introduce other premises
> with variables, etc.. It would perfectly model "This statement is
> false!"
>
> Dan


I just finished my new Help.html file

with new enhanced PROLOG GRAMMAR

microPROLOG V2.0 HELP

WELCOME TO microPROLOG!

SYSTEM COMMANDS

LIST.
DELETE.

microPROLOG Syntax

Rule 1 LINE. --> FACT.
Rule 2 LINE. --> FACT IF TAIL.
Rule 3 LINE? --> TAIL?
Rule 4 TAIL --> FACT FACT ... FACT
Rule 5 FACT --> [term ARGS]
Rule 6 ARGS --> ARG ARG ... ARG
Rule 7 ARG --> term | VAR | FACT


In microPROLOG only Predicate Functions are allowed as theorems.

LINE.
FACT.
[term ARGS].
[term ARG].
[term term].

is the smallest possible statement.

This is a restriction not in usual PROLOG.

PROLOG
thisIsAfact.

since that won't parse into RELATIONAL ALGEBRA (SQL).

Perhaps in L.V.s fuzzy logic

FALSE

and

THIS IS FALSE

would have different outcomes.

FALSE would cause an INCONSISTENCY

since a THEORY is equivalent to the conjunction of it's THEOREMS.

THEORY <-> THEOREM1 ^ THEOREM2 ^ FALSE ^ THEOREM3 ..

THEORY <-> FALSE

whereas:

THEORY <-> TH1 ^ TH2 ^ (TH3<->FALSE) ^ TH4 ^ ...

THEORY could be TRUE (consistent)

Herc
--
http://tinyurl.com/BLUEPRINTS-THEOREM

(Correction)
To do that, all you have to do is introduce the premise ~X. But I
don't think that would capture the essence of "This statement is
false."

I was thinking about a special premise or definition with nothing more
than the key-word "FALSE". It would not be parsed as a statement. It
would be treated like a meaningless comment. It would be essentially
ignored by other statements in the proof. It would perfectly model
"This statement is false" since it would have no connection to any
other statement -- it would not have references to any variables,
functions, predicates or propositions.

On Aug 26, 2:38 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
This branch in the topic is not worth continuing

because all your definitions are already twisted to support this self
consistent argument of yours.

Let's use my 10 symbol theorem syntax.

0 1 a ( , ) ^ v ! =

where a is the simplest variable

!a
82

we can precede this statement !a with

a=82
a=1010010
291010010

So now using 2 statements you have

291010010
82

the second line is

NOT(a)
NOT(82)
NOT(THIS STATEMENT)

or

THIS STATEMENT IS FALSE

Herc

That's not completely true. The Liar Paradox can be used to prove
that there is no arithmetically definable truth predicate for arithmetic.

Fix a coding for arithmetic, that is, a way to associate a unique
natural number with each statement of arithmetic. In terms of this
coding, a truth predicate Tr(x) is a formula with the following
property: For any statement S in the language of arithmetic,

Tr(#S) <-> S

holds (where #S means the natural number coding the sentence S).

If Tr(x) is a formula of arithmetic, then using techniques developed
by Godel, we can construct a sentence L such that

L <-> ~Tr(#L)

But by the definition of a truth predicate, we also have

L <-> Tr(#L)

Together, these two statements are a contradiction. So the assumption
that Tr(x) is a formula of arithmetic leads to a contradiction. So it
must be false. Therefore, there is no truth predicate for arithmetic
definable in the language of arithmetic.

If there were a truth predicate for arithmetic, then L would act like
the Liar. L would say of itself that it is not true.

Yes it does, as demonstrated by the fact that it may be used as part of
a proof of Tarski's undefinability theorem.

[snip]

It does seem silly, doesn't it? But I am exaggerating to make a point:
A statement in a formal proof with no references to any variables,
constants, functions, predicates or propositions is entirely inert --
it is its own little world, completely isolated from any other
considerations. It can be removed without affecting the conclusion in
any way. And "This statement is false" is just such a statement. What
am I missing here?

I shouldn't have to wade through Tarski's proof to get at what meaning
if any can be attached to "This statement is false" (referring only to
itself). Just tell me how can any statement with no reference to any
variable, constant, function, predicate, proposition or any other
statement can possibly affect the outcome of any formal proof in, say,
number theory.

And indeed you don't. You could simply look at the Wikipedia page or
Daryl's reply.
The liar statement /does/ have a reference to a predicate, namely a
truth predicate (or its negation, a falsity predicate). Tarksi showed
that no such predicate can be defined in the language of arithmetic by
assuming to the contrary and using that assumption to construct a
sentence that says of itself that it is not true. The existence of such
a sentence of arithmetic would lead to a contradiction.

Dan Christensen wrote:
>
> [...] What
> am I missing here?

What you're missing is the work that's been done on the problem. Whole
books have been written about The Liar and its relatives. You think
that you can become an instant expert without reading anything. Your
arrogance and contempt for others is staggering. At the very least read
Daryl McCullough's contribution to this thread:
news:16c98d1c-f6fa-42ea...@googlegroups.com.

What you're saying is, by analogy, that if someone on the face of the
planet lied everything that everybody else said would be unreliable.

T |- t1 ^ t2 ^ t3 ^ t4 ^ ....

t3 <-> ~true(#t3)
t3 <-> not(t3)
IS CONSTRUCTABLE

ergo:

T |- t1 ^ t2 ^ t3 ^ (t3<->not(t3)) ^ t4 ^ ...

T |- t1 ^ t2 ^ t3 ^ not(t3) ^ t4 ^ ...

T |- contradiction

T |- any formula

ex contradictione sequitur quodlibet

from a contradiction, anything follows

{t3, !t3} |- W

So by allowing *any formula* you can construct to be a theorem of the
Theory, Tarski proved that ^any formula^ can be a theorem of the
Theory.

So why boycott false(..) and true(..) ?

ZFC uses AXIOMS to remove contradictions.

Dumbest logic proof there ever was!

Herc

On Aug 26, 11:56 pm, Frederick Williams
> Daryl McCullough's contribution to this thread.

Yes it is arrogant, to support every post with your own formal proof
in your own formal proof language verifiable only by the professional
software he wrote without consulting you or anyone first.

Tsk Tsk!

Herc
--
www.microPROLOG.com

[scary SPIDER] if [big SPIDER] [hairy SPIDER].
RULE ADDED

That is interesting. Thanks.

Just to confirm my understanding: The statement "This statement is
false" cannot be formalized in number theory (the language of
arithmetic), which is the basis for pretty much all of mathematics. Is
that correct?

>>> [...]
Correct, but this implies the much more general fact that the concepts
of "false" and "true" as they apply to arithmetic cannot be defined
within arithmetic (in a precise sense given by the statement of Tarski's
theorem).
Wrong.

So, within the context of formal proofs in number theory, algebra,
analysis and geometry (all based on the language of arithmetic plus a
few other axioms as required), the statement "This statement is false"
is not an issue?
I guess we will have to agree to disagree on that. (I don't want to
rehash the definition of mathematics!)

[...]
How do you do, say, group theory in the language of arithmetic?

On Aug 26, 1:32 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
You add a few axioms -- the group axioms -- to the language of
arithmetic et voila! You can't do much group theory with only the
group axioms. Very soon, you need the natural numbers.

You cannot express the axioms of group theory in the language of
arithmetic.

On Aug 26, 3:51 pm, Frederick Williams <freddywilli...@btinternet.com>
The language of arithmetic includes the language of logic and sets,
does it not? So you can postulate the existence of a set G with binary
operator * on it with the required properties (associativity of *,
inverses, etc.).

The first order language of arithmetic does not contain (names for)
sets.
A (name for the) group operation * is not to be found in the language of
arithmetic.

On Aug 26, 4:40 pm, Frederick Williams <freddywilli...@btinternet.com>
How about the set of even numbers? Is that your "language of
arithmetic?"

What does it mean for a statement to be "an issue" within the context of
formal proofs in number theory, algebra, analysis and geometry?

I mean the statement cannot be formalized since in each of these
formal systems, the language of arithmetic is embedded in it.

Try to answer your own question. If a sentence that meant "this
statement is false" could be formalised in a formal system, what would
that tell you about said system?

"Dan Christensen" <Dan_Chr...@sympatico.ca> wrote in message
news:af2f21b0-b27b-4de7...@s15g2000yqo.googlegroups.com...
> On Aug 25, 10:28 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
>> news:3d63e473-91f5-4f17...@a14g2000yqc.googlegroups.com...
>>
>> > You claim that the original Liar Paradox is not really The Liar
>> > Paradox.
>>
>> You are utterly confused: the one who claims that those are two different
>> problems is you.
>
> Now you are contradicting yourself. You said that my version is NOT
> the real Liar Paradox. Now you seem to be saying it is equivalent to
> yours. Please clarify.

The two paradoxes that are equivalent are the Epimenides paradox and its
formal counterpart: these are equivalent problems. Your ruminations are
nothing at all: I just don't know if in this thread you really deserved to
be attacked so badly since the very start, but it is a fact that your
position in this discussion is just less and less sensible.

> Admittedly, I haven't given your version much thought. It really seems

It is not "my" version: you know, I too read books sometimes, I don't just
make stuff up all along...

-LV

news:2cb90726-e69c-42d8...@k17g2000yqp.googlegroups.com...
Are you saying that the incompleteness theorems, recursion theory, proof
theory, and the like are all useless?

-LV

news:f61d4de4-6cc8-4b0e...@b8g2000yqh.googlegroups.com...
Of course not! That statement can indeed be formalized, which is what makes
it interesting... On the other hand, don't be surprised that arithmetic
etc. are always involved: mathematical logic just is not logic, as it should
be clear by now.

-LV

On Aug 27, 6:40 am, Frederick Williams <freddywilli...@btinternet.com>
This branch of the topic seems redundant since Dan stated

ADDING AXIOMS to Arithmetic.

i.e. making_up axioms for whatever and including arithmetic axioms.

not embedding sub-axioms within only arithmetic

using, not only using.

Herc

Just change the topic line to *A* Liar Paradox

Herc

Not as a theory of arithmetic, which is what is generally meant by
formalized IN number theory.
A theorem and the negation of that theorem (even if it's one and the
same) forms a contradiction and an inconsistent theory.

Mind you TARSKI's ERROR is obvious

the same error as Godel's Proof.

A Godel Number implies a SYNTAX (Left Right GRAMMAR)
exists to construct the formula.

e.g. you could number all PREDICATES

A(x) P(x,y,z)

E(x) not( x = x )

These are both FORMULA's of Predicate Calculus.
But they are not both THEOREMS of Predicate Calculus.

There is nothing wrong with having a GODEL NUMBER FUNCTION within a
theory.

TRUE(#phi)

Banishing true(x) and false(x) predicates is just pruning the leaves
of your theory.


-------
Without a THEORY GRAMMAR (axioms)

You start with this

T |- W

a TRUE() predicate just gives you this!

{t3, !t3} |- W
-------




You guys don't know the difference between a SYNTAX GRAMMAR and THEORY
GRAMMAR.

http://www.microprolog.com/help.html


microPROLOG Syntax

Rule 1 LINE. --> FACT.
Rule 2 LINE. --> FACT IF TAIL.
Rule 3 LINE? --> TAIL?
Rule 4 TAIL --> FACT FACT ... FACT
Rule 5 FACT --> [term ARGS]
Rule 6 ARGS --> ARG ARG ... ARG
Rule 7 ARG --> term | VAR | FACT



e.g LINE.
--> FACT. ----------------------------- (Rule 1)
--> [term ARGS].
--> [term ARG].
--> [term term].
--> [lady gaga].

Following SYNTAX does not guarantee valid code!


Herc
--
http://tinyurl.com/TARSKI-PROOF
http://tinyurl.com/BLUEPRINTS-TARSKI

"|-| E R C" <herc.o...@gmail.com> wrote in message
news:5e2e4e18-5cda-4f9f...@ou2g2000pbc.googlegroups.com...
It does not: A&~A with the usual meaning of the logical connectives is
simply false over all assignments. A contradiction is |- A and |- ~A,
meaning that a theorem and its negation can be derived. Don't confuse the
language and the meta-language...
Indeed please disproof Goedel, as the rest is mostly a consequence. In
fact, no: please disprove the diagonal argument...
It depends on what you mean by "valid".

-LV

who the fck would snip 4 paragraphs on GRAMMARS

to contest that final remark??

you should worry more about CONTEXT before you take cheap shots like a
big know it all who can't read.

Herc

On Aug 26, 9:13 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
On what might be called the modern version of the Liar Paradox, it
seems my preliminary thoughts were largely correct. (For my informal
analysis of the classical version, see above.) IIUC, Rotwang has
confirmed that the statement, "This statement is false," cannot be
formalized in the language of arithmetic. He wrote, "The existence of
The language of arithmetic, as I understand the term, is embedded into
most if not all of mathematics. While "This statement is false" may be
useful in some rarefied philosophical discussions, in number theory,
algebra, analysis and geometry with the language of arithmetic being
embedded in each, "useless" is probably not too strong a word.

On Aug 27, 12:26 pm, "LudovicoVan" <ju...@diegidio.name> wrote:

VVVV LOOK BELOW VVVVV

T |- t3
T |- not(t3)
THIS THEOREM IS FALSE

will generate the fact

false(THIS THEOREM IS FALSE)
When you get off your training wheels kid
I just defined it!

I don't think ANYBODY here has read a SINGLE POST THEY REPLY TO.

You're all bots employed by Harper and Collins to refine the next
edition of CANTORS PARADISE.

Herc

Thanks for clearing that up, but even the Wiki article cited here says
they are NOT equivalent. I suppose it doesn't really matter.
Perhaps. Some time ago here, I dismissed "This statement is false" as
nonsense. I reasoned that "This statement is TRUE" made no sense. It
could add nothing to any argument in "real world" mathematics.
Likewise, I reasoned that "This statement is FALSE" made even less
sense, it being a self-contradiction on top of being nonsensical.

Recently I revisited the classical version, and tried to come to grips
with it. I think I made some progress with my informal analysis.

A bit reluctantly, I then revisited the modern version with mixed
results. Rotwang's comments were instructive, but I haven't changed my
mind as far as "real world" mathematical theory goes (i.e. number
theory, algebra, analysis, statistics and geometry). But I will
concede that it may be useful in some other contexts -- just not sure
which.

In article <k1else$39m$1...@dont-email.me>,
An how is your statement different from Herc's?

Doesn't your "|- A" represent the statement that A is a theorem, i.e.,
is provable is the system in questions?
--

On Aug 27, 3:24 pm, Virgil <vir...@ligriv.com> wrote:
> In article <k1else$39...@dont-email.me>,
>
>
>  "LudovicoVan" <ju...@diegidio.name> wrote:
> > "|-| E R C" <herc.of.z...@gmail.com> wrote in message
LVs correction.
A is not a Theorem -> A&~A is a Theorem.

LVs definition
|- A and |- ~A --> contradiction


Not sure what the context is, I was explaining that freely adding a
formula to a theory because_it_is_constructable, as Daryl said, is
inconsistent to start with so Taski's truth predicate conjecture is
wrong.


[DARYL]
Fix a coding for arithmetic, that is, a way to associate a unique
natural number with each statement of arithmetic. In terms of this
coding, a truth predicate Tr(x) is a formula with the following
property: For any statement S in the language of arithmetic,

Tr(#S) <-> S

holds (where #S means the natural number coding the sentence S).
If Tr(x) is a formula of arithmetic, then using techniques developed
by Godel, we can construct a sentence L...

err no you can't!

That's as bad as ex contradictione sequitur quodlibet!

T |- any formula

Godel numbering was not axiomatic, it was lexicographic and included
all strings.

Predicate Calculus is just a SYNTAX GRAMMAR, even it's not axiomatic,
but slightly better than Godel's lexicographic numbering.

GODEL NUMBERED FORMULA f(((,,,ggh))01,fg+0((

PREDICATE CALCULUS SYNTAX E(X) ~(X=X)

AXIOMATIC A(X) (X=X)


Herc