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Simplest way to explain the Liar
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Charlie-Boo
Jan 19, 2012, 8:04:57 PM1/19/12
to
Godel said some sentences are neither provable or refutable. The Liar
says some are neither true or false.
To show how/why, we can use simple combinatorial math. But better is
to show formal rules for deriving that conclusion. And how do we
develop a system to do that? Like we do any system: by considering
siblings and generalizing. What are the siblings? Things like:
This is true.
This is false.
This is provable.
This is unprovable.
This is refutable.
This is unrefutable.
This is true or provable.
This is true of "It is false of itself."
Is each true? False? Provable? Refutable? Decidable? Then what is
the general rule? Then the Liar is one simple special case of a
derivation.
C-B
says some are neither true or false.
To show how/why, we can use simple combinatorial math. But better is
to show formal rules for deriving that conclusion. And how do we
develop a system to do that? Like we do any system: by considering
siblings and generalizing. What are the siblings? Things like:
This is true.
This is false.
This is provable.
This is unprovable.
This is refutable.
This is unrefutable.
This is true or provable.
This is true of "It is false of itself."
Is each true? False? Provable? Refutable? Decidable? Then what is
the general rule? Then the Liar is one simple special case of a
derivation.
C-B
LudovicoVan
Jan 19, 2012, 9:50:54 PM1/19/12
to
"Charlie-Boo" <shyma...@gmail.com> wrote in message
news:fe4a3940-744d-4893...@o13g2000vbf.googlegroups.com...
What is the truth-value of "this is false"? Provable/decidable, in? What I
can see looks more like a series of absolute statements, more like axioms...
Anyway, and as far as I recollect (my account should be double-checked), all
that is needed to bootstrap a logical theory is self-contradiction, i.e. the
notion of mutually contradictory (incompatible) assertions/utterances
(within the same context, etc.).
-LV
news:fe4a3940-744d-4893...@o13g2000vbf.googlegroups.com...
can see looks more like a series of absolute statements, more like axioms...
Anyway, and as far as I recollect (my account should be double-checked), all
that is needed to bootstrap a logical theory is self-contradiction, i.e. the
notion of mutually contradictory (incompatible) assertions/utterances
(within the same context, etc.).
-LV
Dan Christensen
Jan 19, 2012, 10:23:34 PM1/19/12
to
"This statement is false."
Such statements have flummoxed philosophers for millennia. What
meaning, if any, can we attach to such a statement? About as much
meaning as we can attach to:
"This statement is true."
When confronted by this statement on its own (not referring to any
other statement), we immediately recognize it as nonsense. Why then
does the former merit any more consideration simply because, in
addition to being nonsense, it is also self-contradictory?
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Also see "The Barber Paradox Video"
LudovicoVan
Jan 20, 2012, 10:49:58 PM1/20/12
to
"Dan Christensen" <Dan_Chr...@sympatico.ca> wrote in message
news:593a6c86-69fa-42b8...@hs8g2000vbb.googlegroups.com...
> On the so-called Liar Paradox...
>
> "This statement is false."
>
> Such statements have flummoxed philosophers for millennia. What
> meaning, if any, can we attach to such a statement? About as much
> meaning as we can attach to:
>
> "This statement is true."
The former is as genuine a logical puzzle as the irrationality of 2 was for
the ancients (and/or every modern student who has to digest it). The latter
would be a perfectly sensible statement for instance in a court. That is,
you should broaden your perspective here, which is not simply "calculistic"
or "the rules of the sell and buy at the market", for instance: i.e., logic
is of interest to areas much broader than basic mathematical "calculus"
only. (And all the more so is philosophy, don't you think??)
> When confronted by this statement on its own (not referring to any
> other statement), we immediately recognize it as nonsense. Why then
> does the former merit any more consideration simply because, in
> addition to being nonsense, it is also self-contradictory?
You are just missing the fundamentals and ultimately approaching the thing
upside down: A statement on its own has no validity to it at all. From a
preliminary notion of self-contradiction (which is: one self-contradicting
himself/herself and how that works in terms of utterances, predicates, and
mutual exclusions), one can rigorously define validity, then logical
entailment, finally proceed to the formal systems.
-LV
news:593a6c86-69fa-42b8...@hs8g2000vbb.googlegroups.com...
> On the so-called Liar Paradox...
>
> "This statement is false."
>
> Such statements have flummoxed philosophers for millennia. What
> meaning, if any, can we attach to such a statement? About as much
> meaning as we can attach to:
>
> "This statement is true."
the ancients (and/or every modern student who has to digest it). The latter
would be a perfectly sensible statement for instance in a court. That is,
you should broaden your perspective here, which is not simply "calculistic"
or "the rules of the sell and buy at the market", for instance: i.e., logic
is of interest to areas much broader than basic mathematical "calculus"
only. (And all the more so is philosophy, don't you think??)
> When confronted by this statement on its own (not referring to any
> other statement), we immediately recognize it as nonsense. Why then
> does the former merit any more consideration simply because, in
> addition to being nonsense, it is also self-contradictory?
upside down: A statement on its own has no validity to it at all. From a
preliminary notion of self-contradiction (which is: one self-contradicting
himself/herself and how that works in terms of utterances, predicates, and
mutual exclusions), one can rigorously define validity, then logical
entailment, finally proceed to the formal systems.
-LV
Dan Christensen
Jan 21, 2012, 1:00:21 AM1/21/12
to
On Jan 20, 10:49 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
only if they were referring to other statements. As I understand this
paradox, however, each statement refers to itself.
> That is,
> you should broaden your perspective here, which is not simply "calculistic"
> or "the rules of the sell and buy at the market", for instance: i.e., logic
> is of interest to areas much broader than basic mathematical "calculus"
> only. (And all the more so is philosophy, don't you think??)
>
I take it you are referring to my recent comments in other threads
here. There, I was arguing only that not all the features of
mainstream logic are required to do mathematical proofs. The logicians
here took exception to that, and off we went. Really quite
exhausting!
> > When confronted by this statement on its own (not referring to any
> > other statement), we immediately recognize it as nonsense. Why then
> > does the former merit any more consideration simply because, in
> > addition to being nonsense, it is also self-contradictory?
>
> You are just missing the fundamentals and ultimately approaching the thing
> upside down: A statement on its own has no validity to it at all. From a
> preliminary notion of self-contradiction (which is: one self-contradicting
> himself/herself and how that works in terms of utterances, predicates, and
> mutual exclusions), one can rigorously define validity, then logical
> entailment, finally proceed to the formal systems.
>
What then do you make of the following statements:
"There are seven words in this sentence." (True)
"There are not eight words in this sentence." (False)
Both statements stand on their own, and refer only to themselves. And
yet we can assign a "truth value" to each. They are not utter nonsense
like the statements:
"This statement is true."
"This statement is false."
> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
>
> news:593a6c86-69fa-42b8...@hs8g2000vbb.googlegroups.com...
>
> > On the so-called Liar Paradox...
>
> > "This statement is false."
>
> > Such statements have flummoxed philosophers for millennia. What
> > meaning, if any, can we attach to such a statement? About as much
> > meaning as we can attach to:
>
> > "This statement is true."
>
> The former is as genuine a logical puzzle as the irrationality of [the square root of] 2 was for
> news:593a6c86-69fa-42b8...@hs8g2000vbb.googlegroups.com...
>
> > On the so-called Liar Paradox...
>
> > "This statement is false."
>
> > Such statements have flummoxed philosophers for millennia. What
> > meaning, if any, can we attach to such a statement? About as much
> > meaning as we can attach to:
>
> > "This statement is true."
>
> the ancients (and/or every modern student who has to digest it). The latter
> would be a perfectly sensible statement for instance in a court.
BOTH statements would make sense, and could even both be true, but
> would be a perfectly sensible statement for instance in a court.
only if they were referring to other statements. As I understand this
paradox, however, each statement refers to itself.
> That is,
> you should broaden your perspective here, which is not simply "calculistic"
> or "the rules of the sell and buy at the market", for instance: i.e., logic
> is of interest to areas much broader than basic mathematical "calculus"
> only. (And all the more so is philosophy, don't you think??)
>
here. There, I was arguing only that not all the features of
mainstream logic are required to do mathematical proofs. The logicians
here took exception to that, and off we went. Really quite
exhausting!
> > When confronted by this statement on its own (not referring to any
> > other statement), we immediately recognize it as nonsense. Why then
> > does the former merit any more consideration simply because, in
> > addition to being nonsense, it is also self-contradictory?
>
> You are just missing the fundamentals and ultimately approaching the thing
> upside down: A statement on its own has no validity to it at all. From a
> preliminary notion of self-contradiction (which is: one self-contradicting
> himself/herself and how that works in terms of utterances, predicates, and
> mutual exclusions), one can rigorously define validity, then logical
> entailment, finally proceed to the formal systems.
>
"There are seven words in this sentence." (True)
"There are not eight words in this sentence." (False)
Both statements stand on their own, and refer only to themselves. And
yet we can assign a "truth value" to each. They are not utter nonsense
like the statements:
"This statement is true."
"This statement is false."
Jack Campin
Jan 21, 2012, 3:09:02 AM1/21/12
to
Dan Christensen <Dan_Chr...@sympatico.ca> wrote:
> I take it you are referring to my recent comments in other threads
> here. There, I was arguing only that not all the features of
> mainstream logic are required to do mathematical proofs. The logicians
> here took exception to that, and off we went. Really quite
> exhausting!
You spent much more time arguing for a system that included *many
> I take it you are referring to my recent comments in other threads
> here. There, I was arguing only that not all the features of
> mainstream logic are required to do mathematical proofs. The logicians
> here took exception to that, and off we went. Really quite
> exhausting!
more* features than were necessary for the job in hand.
-----------------------------------------------------------------------------
e m a i l : j a c k @ c a m p i n . m e . u k
Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland
mobile 07800 739 557 <http://www.campin.me.uk> Twitter: JackCampin
LudovicoVan
Jan 21, 2012, 9:07:09 AM1/21/12
to
"Dan Christensen" <Dan_Chr...@sympatico.ca> wrote in message
news:584aca62-838c-4142...@cf6g2000vbb.googlegroups.com...
> On Jan 20, 10:49 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
>> news:593a6c86-69fa-42b8...@hs8g2000vbb.googlegroups.com...
>>
>> > On the so-called Liar Paradox...
>>
>> > "This statement is false."
>>
>> > Such statements have flummoxed philosophers for millennia. What
>> > meaning, if any, can we attach to such a statement? About as much
>> > meaning as we can attach to:
>>
>> > "This statement is true."
>>
>> The former is as genuine a logical puzzle as the irrationality of [the
>> square root of] 2 was for
>> the ancients (and/or every modern student who has to digest it). The
>> latter
>> would be a perfectly sensible statement for instance in a court.
>
> BOTH statements would make sense, and could even both be true, but
> only if they were referring to other statements. As I understand this
> paradox, however, each statement refers to itself.
Yes, of course. And, as such, the latter has nothing paradoxical to it: if
>> "Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message
>> news:593a6c86-69fa-42b8...@hs8g2000vbb.googlegroups.com...
>>
>> > On the so-called Liar Paradox...
>>
>> > "This statement is false."
>>
>> > Such statements have flummoxed philosophers for millennia. What
>> > meaning, if any, can we attach to such a statement? About as much
>> > meaning as we can attach to:
>>
>> > "This statement is true."
>>
>> The former is as genuine a logical puzzle as the irrationality of [the
>> square root of] 2 was for
>> the ancients (and/or every modern student who has to digest it). The
>> latter
>> would be a perfectly sensible statement for instance in a court.
>
> BOTH statements would make sense, and could even both be true, but
> only if they were referring to other statements. As I understand this
> paradox, however, each statement refers to itself.
it is true it is true, if it is false it is false. The former is
paradoxical in that if it is true it is false and if it is false it is true.
>> That is,
>> you should broaden your perspective here, which is not simply
>> "calculistic"
>> or "the rules of the sell and buy at the market", for instance: i.e.,
>> logic
>> is of interest to areas much broader than basic mathematical "calculus"
>> only. (And all the more so is philosophy, don't you think??)
>
> I take it you are referring to my recent comments in other threads
> here. There, I was arguing only that not all the features of
> mainstream logic are required to do mathematical proofs. The logicians
> here took exception to that, and off we went. Really quite
> exhausting!
you are 100% correct (and who is anyway?), but because of the pure
negativity of what comes back. That said: no, I wasn't referring to those,
and yes, I do think your perspective might be correct/acceptable/useful re
the construction of a logical system with this or that specific scope, but
it isn't (or, to me, doesn't appear to be so) re an approach to logic in
general.
>> > When confronted by this statement on its own (not referring to any
>> > other statement), we immediately recognize it as nonsense. Why then
>> > does the former merit any more consideration simply because, in
>> > addition to being nonsense, it is also self-contradictory?
>>
>> You are just missing the fundamentals and ultimately approaching the
>> thing
>> upside down: A statement on its own has no validity to it at all. From a
>> preliminary notion of self-contradiction (which is: one
>> self-contradicting
>> himself/herself and how that works in terms of utterances, predicates,
>> and
>> mutual exclusions), one can rigorously define validity, then logical
>> entailment, finally proceed to the formal systems.
>
> What then do you make of the following statements:
>
> "There are seven words in this sentence." (True)
> "There are not eight words in this sentence." (False)
>
> Both statements stand on their own, and refer only to themselves.
shouldn't fool you in thinking that they stand out in the void: as explained
above, even just implicitly, there is a chain of assumptions before you can
even sensibly ask yourself the question of whether any statement is valid or
not.
> And
> yet we can assign a "truth value" to each. They are not utter nonsense
> like the statements:
>
> "This statement is true."
> "This statement is false."
former is a perfectly sensible statement in some circumstances (as
self-referential as it is: self-referential statements are not necessarily
problematic statements), while the latter is a genuine paradoxical
statement: but that still is different from "meaningless" or "utter
nonsense". You know: we need to account for the *fact* that liars do
exist...
-LV
Dan Christensen
Jan 21, 2012, 11:39:01 AM1/21/12
to
On Jan 21, 3:09 am, Jack Campin <bo...@purr.demon.co.uk> wrote:
proofs. And, as I saw it, the "job in hand" was the construction of
the real numbers. My "many more features" were, of course, the axioms
of set theory, which I think we all agreed were necessary for the "job
in hand."
Dan
Download my DC Proof 2.0 software at http://wwww.dcproof.com
> Dan Christensen <Dan_Christen...@sympatico.ca> wrote:
> > I take it you are referring to my recent comments in other threads
> > here. There, I was arguing only that not all the features of
> > mainstream logic are required to do mathematical proofs. The logicians
> > here took exception to that, and off we went. Really quite
> > exhausting!
>
> You spent much more time arguing for a system that included *many
> more* features than were necessary for the job in hand.
I argued for using a kind of subset of FOL in writing mathematical
> > I take it you are referring to my recent comments in other threads
> > here. There, I was arguing only that not all the features of
> > mainstream logic are required to do mathematical proofs. The logicians
> > here took exception to that, and off we went. Really quite
> > exhausting!
>
> You spent much more time arguing for a system that included *many
> more* features than were necessary for the job in hand.
proofs. And, as I saw it, the "job in hand" was the construction of
the real numbers. My "many more features" were, of course, the axioms
of set theory, which I think we all agreed were necessary for the "job
in hand."
Dan
Download my DC Proof 2.0 software at http://wwww.dcproof.com
Frederick Williams
Jan 22, 2012, 9:54:14 AM1/22/12
to
Dan Christensen wrote:
>
> On the so-called Liar Paradox...
>
> "This statement is false."
>
> Such statements have flummoxed philosophers for millennia. What
> meaning, if any, can we attach to such a statement? About as much
> meaning as we can attach to:
>
> "This statement is true."
>
> When confronted by this statement on its own (not referring to any
> other statement), we immediately recognize it as nonsense.
Who are they, these "we" who immediately recognize "This statement is
>
> On the so-called Liar Paradox...
>
> "This statement is false."
>
> Such statements have flummoxed philosophers for millennia. What
> meaning, if any, can we attach to such a statement? About as much
> meaning as we can attach to:
>
> "This statement is true."
>
> When confronted by this statement on its own (not referring to any
> other statement), we immediately recognize it as nonsense.
true." as nonsense?
--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
LudovicoVan
Jan 22, 2012, 10:18:09 AM1/22/12
to
"Frederick Williams" <freddyw...@btinternet.com> wrote in message
news:4F1C2316...@btinternet.com...
> Who are they, these "we" who immediately recognize "This statement is
> true." as nonsense?
Wasn't my reply enough? Why do you keep provoking controversy?
-LV
news:4F1C2316...@btinternet.com...
> Who are they, these "we" who immediately recognize "This statement is
> true." as nonsense?
-LV
Charlie-Boo
Jan 25, 2012, 6:59:36 AM1/25/12
to
On Jan 19, 10:23 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
> Why does the "This statement is false." merit any more consideration simply because, in
popular belief that all statements are true or false, is wrong. More
recently, it is of additional interest because it resembles the
statement "This is unprovable." and central to Godel's Theorem is a
formal wff that expresses "This is unprovale."
C-B
> Dan
> Download my DC Proof 2.0 software athttp://www.dcproof.com
wrote:
> Why does the "This statement is false." merit any more consideration simply because, in
> addition to being nonsense, it is also self-contradictory?
Originally it was of interest because it seems to show that the
popular belief that all statements are true or false, is wrong. More
recently, it is of additional interest because it resembles the
statement "This is unprovable." and central to Godel's Theorem is a
formal wff that expresses "This is unprovale."
C-B
> Dan
> Download my DC Proof 2.0 software athttp://www.dcproof.com
Ross A. Finlayson
Jan 26, 2012, 4:03:54 PM1/26/12
to
in logic, it is the explanation, answers to question words, how when
why what where, simply reveals collision of exclusions in excluded
middle, with cause/effect.
The barber from where everyone who _doesn't_ shave themselves is
shaved by that barber, and everyone shaves, has that the negation
applies _both_ to the barber being a person who shaves, and not
himself being a person shaved by the barber. So that would mean he
expands the class of being shaved by the barber to include himself.
It doesn't really matter whether he does or doesn't, except to the
truth value of a) everyone there shaves, b) the barber shaves, c) the
barber shaves himself.
So it doesn't make sense to assume there could be this impossible
class of non-self-shaving barbers who shave everybody who is not who
is there, just to not assume either way and take them as contingent
facts. It makes sense to not care in that sense, or as engineers
might say: "don't care".
So the Liar, "this statement is false", means that statement for
whatever other reasons is false, and if there is thus a circularity in
cancellation, that is just a perfect systolic system, or a reason to
break implication into cases.
If the barber only ever at there is shaved, the barber or anybody
shaves.
This statement is true.
I think I'll just leave it at that.
Close to grammar.
LudovicoVan
Jan 27, 2012, 9:33:06 PM1/27/12
to
"Ross A. Finlayson" <ross.fi...@gmail.com> wrote in message
news:d27f9d84-1197-448e...@v6g2000pba.googlegroups.com...
> So it doesn't make sense to assume there could be this impossible
> class of non-self-shaving barbers who shave everybody who is not who
> is there, just to not assume either way and take them as contingent
> facts. It makes sense to not care in that sense, or as engineers
> might say: "don't care".
What do you make of reductio ad absurdum?
> So the Liar, "this statement is false", means that statement for
> whatever other reasons is false, and if there is thus a circularity in
> cancellation, that is just a perfect systolic system, or a reason to
> break implication into cases.
>
> If the barber only ever at there is shaved, the barber or anybody
> shaves.
>
> This statement is true.
>
> I think I'll just leave it at that.
>
> Close to grammar.
Please state a falsity!
-LV
news:d27f9d84-1197-448e...@v6g2000pba.googlegroups.com...
> So it doesn't make sense to assume there could be this impossible
> class of non-self-shaving barbers who shave everybody who is not who
> is there, just to not assume either way and take them as contingent
> facts. It makes sense to not care in that sense, or as engineers
> might say: "don't care".
> So the Liar, "this statement is false", means that statement for
> whatever other reasons is false, and if there is thus a circularity in
> cancellation, that is just a perfect systolic system, or a reason to
> break implication into cases.
>
> If the barber only ever at there is shaved, the barber or anybody
> shaves.
>
> This statement is true.
>
> I think I'll just leave it at that.
>
> Close to grammar.
-LV
Ross Finlayson
Mar 19, 2023, 3:11:33 PM3/19/23
to
olcott
Mar 19, 2023, 5:08:14 PM3/19/23
to
then whether the barber shaves himself or not derives a contradiction
that proves that this barber is a liar. ZFC set theory figured this out
long ago.
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X in
L. *Axiom(P) ⊨□ X*
False(L,X) means that a semantic connection exists between (a) and ~X in
L. *Axiom(P) ⊨□ ~X*
Provable(L,P,X) means that a semantic connection exists between premises
P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
The above system screens out the Liar Paradox, and Gödel's G as
semantically incorrect expressions of language.
Furthermore it anchors the basis for a consistently correct True(L,X)
predicate that Tarski "proved" cannot possibly exist.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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