But the idea that someone would think that a non-bivalent approach was
fuzzy makes me think.   I’ve just said – assume a language where the
liar means what it says and change whatever else has to be changed to
make that happen – and see what else happens.   But I haven’t thought
about what it means.  Obviously I should.  But I’ll have to see what
happens first.

But I am very interested in bivalent approaches also, and will read
yours carefully and follow links.   Some of my postings are at:
http://sandyhodges.artshost.com/crete/scilogic.html

- Hide quoted text — Show quoted text -

peter_corsel…@my-deja.com wrote:

> If the expression:

>      expression which yields a true statement
>      when properly substituted for `X’ in ` "X" is not an X.’

> is quotable, then we may derive a contradiction. This version of
> Grelling’s paradox expression is analogous to Quine’s:

>      expression which yields a falsehood
>      when appended to its own quotation

> Appending this to its own quotation apparently yields a sentence
> which asserts its own falsehood, as does properly substituting
> the first expression for `X’ in ` "X" is not an X’.

> You may be thinking of some non-bivalent approach. Up to now I
> have largely restricted my search for a solution to the paradigm
> of bivalent logic. I will happily concede that reality is
> fundamentally fuzzy, but if this is what you have in mind, you
> will have to suggest some account of non-bivalency if the
> discussion is to proceed. (You seem to have anticipated this
> request in your manifesto, which I have just seen for the first
> time. This will take a couple of days to digest!)

> Alternatively you may have in mind the fact that an expression in
> a given language, may be true of all expressions in that language
> which may be quoted and which are not true of themselves, if that
> expression may not itself be quoted in the language. Such an
> expression would be analogous to a proper class. I agree this is
> a useful concept.

> Am I getting any closer?

> In article <3962B702.A7ED4…@iname.com>,
>   Sandy Hodges <sandy-hod…@iname.com> wrote:
> > Well, I may not have understood the point of your original posting.
> If
> > I understand it now I’m not sure I would call it a solution to the
> > paradox.   I don’t think there is a `solution’ to the paradox, but it
> is
> > at least conceivable that really close attention to the way quotation
> > works would show that a particular apparent paradox was based on a
> > mistake.   If so that would certainly be a `solution.’   But that’s
> not
> > what you’re saying, I think.

> > If I’m taking the role of the defender of the paradox, and you are
> > `solving’ it away, then in effect I’ve made some concessions:

> > My admission that

> > `X’ is heterological, if and only if `X’ is not an X.

> > is a schema, is a point granted, and it may represent an avenue of
> > attack:  If `heterological’ does need a schema to define it, it can’t
> be
> > easily expressed in a formula such as:

> > A heterological expression is an expression which yields a true
> > statement when substituted for `X’ in ` "X" is not an X.’

> > And the fact that it can’t be shows there’s something fishy about
> > `heterological.’

> > The fact that

> >   expression which yields a true statement when substituted for `X’
> in `
> > "X" is not an X.’

> > isn’t the definition of heterological, is another admission.    But if
> > this formula isn’t the definition of heterological, I don’t see what
> > good comes of beating it to death.   The fact that this formula, at a
> > given quotation depth, does not quote itself at the same depth, does
> not
> > matter if it is not what heterological means.

> > Since I have, after seeing your posting, made concessions, I think
> that
> > you must shift your attack.  I am not convinced that it is impossible.

> > Heterological must mean something like

> >   expression which yields a true statement when properly substituted
> for
> > `X’ in ` "X" is not an X.’

> > where proper substitution for a variable X in a text …X… means to
> > add n primes to any quotes surrounding X, for the minimum n such that
> > quotes surrounding X have higher level than any in the inserted
> text.
> > The mere fact that quotation is so complicated is something.

> > peter_corsel…@my-deja.com wrote:

> > > Sandy Hodges wrote:

> > > > The normal operation of the paradox defines (I’m assuming) a
> `Grelling
> > > > expression’ by a formula:

> > [clip]
> > sentence when substituted for “X” in “ `X’ is not an X,” is
> > > not
> > > > the definition of “Grelling expression,”  since it fails to
> capture
> > > > the varying quote levels of the formula A schema.

> > > I did not claim to have achieved the impossible! No expression may
> > > completely capture the varying quote levels of your schema, as such
> an
> > > expression would then be true of all and only those expressions
> which
> > > are not true of themselves. What I claim to have found is a natural
> > > notation for constructing expressions, in which any attempt to
> > > construct Grelling’s paradoxical expression is automatically
> blocked,
> > > but which still allows the representation of self-reference.

> > > Perhaps I should clarify what I mean by a "solution" to Grelling’s
> > > paradox and the paradoxes of set theory. Russell once criticised
> Quine
> > > for developing solutions which were "created ad hoc and not … such
> > > as even the cleverest logician would have thought of if he had not
> > > known of the contradictions" (My Philosophical Development, p.80). I
> > > accept that the standard iterative conception of set is reasonably
> > > natural in this sense. However it involves a complete ban on self-
> > > reference, and I believe a full solution to the paradoxes must
> involve
> > > some natural theory of logic which allows self-reference to be
> > > represented consistently.

> > I don’t follow this paragraph.   It may well be that “a full solution
> > to the paradoxes must involve some natural theory of logic which
> allows
> > self-reference to be represented consistently.”    How is that
> related
> > to quotation levels?

> > > In my previous posting I summarised my solution to Grelling’s
> paradox
> > > by translating expressions from the notation I have developed into
> > > natural language. If I receive some positive feedback on this, I
> shall
> > > then post a longer account, giving formation rules and so forth.

> > > (Apologies if I have missed the point of your comment. Please write
> > > and explain if I have!)

> > > Sent via Deja.com http://www.deja.com/
> > > Before you buy.

> > —
> > — —– —— — —— – - —– – —
> > When my love swears that she is made of truth,
> > I do believe her, though I know she lies.
> > – Sandy Hodges

> Sent via Deja.com http://www.deja.com/
> Before you buy.

–
— —– —— — —— – - —– – —
When my love swears that she is made of truth,
I do believe her, though I know she lies.
– Sandy Hodges

You say all GRAS sentences are bivalent – so do I.
You do not say doohickeys are bivalent – neither to I.
For each doohickey A, I say that `A implies A’ is true and logic should
let you prove it. – You say `A implies A’ is gibberish.

So how am I less bivalent than you are?   We apply bivalence to the same
things.  I’m just more `A implies A’-ish than you are.

Still don’t know what it means. though. – S

- Hide quoted text — Show quoted text -

peter_corsel…@my-deja.com wrote:

> If the expression:

>      expression which yields a true statement
>      when properly substituted for `X’ in ` "X" is not an X.’

> is quotable, then we may derive a contradiction. This version of
> Grelling’s paradox expression is analogous to Quine’s:

>      expression which yields a falsehood
>      when appended to its own quotation

> Appending this to its own quotation apparently yields a sentence
> which asserts its own falsehood, as does properly substituting
> the first expression for `X’ in ` "X" is not an X’.

> You may be thinking of some non-bivalent approach. Up to now I
> have largely restricted my search for a solution to the paradigm
> of bivalent logic. I will happily concede that reality is
> fundamentally fuzzy, but if this is what you have in mind, you
> will have to suggest some account of non-bivalency if the
> discussion is to proceed. (You seem to have anticipated this
> request in your manifesto, which I have just seen for the first
>[clip]

–
— —– —— — —— – - —– – —
When my love swears that she is made of truth,
I do believe her, though I know she lies.
– Sandy Hodges

First I must explain why I believe a solution to Grelling’s paradox
must involve a hierarchy of languages. As Quine wrote (From a Logical
Point of View, pp.134-6):

  Strictly, the notions of the theory of reference, and likewise
  those of the theory of meaning … are relative always to a
  language; the language figures, albeit tacitly, as a parameter.
  …

One reason that I am interested in the paradox, is that I believe it
illustrates in a beautiful way the fact that any system for
communication is limited (also, I dig all that Eastern philosophy
stuff, but I try not to impose it on others!).

This I why commented that (in the context of bivalent logic at
least) it is impossible to view the word `heterological’ as
corresponding to a single expression. For a given language L, the
term `heterological in L’ may only be defined in a separate
language. Language is necessarily dynamic.

The problem I am seeking to address is the fact that, as far as I
am aware, the only intuitive hierarchies of formal languages which
have been developed, involve the restriction that terms in one
language may only be true of terms in some previous language, and so
involve a complete rejection of self-reference. I shall refer to
this as the `standard approach’.

The relevance of quotation levels to a full solution to the paradox
is that, when one makes the use of quotation marks in the process of
definition explicit, a hierarchy of languages magically appears in
which terms in a given language may be true of terms from that
language, and any previous language.

Moreover, as the idea of a hierarchy of quotation marks is implicit
in the use of single and double quotes in standard written English, I
suggest that a logical theory based on it may claim to be more
natural than the standard approach.

In a given language the derivation of the contradictions is avoided
in a manner analogous to that involved in the solution to the
paradoxes of set theory presented by Quine in his New Foundations for
Mathematical Logic. Quine avoided Russell’s contradiction by
introducing an ad hoc restriction on which open sentences may be
abstracted from to form a set. In the languages in my hierarchy a
restriction automatically emerges on which open sentences may be
abstracted from to form a term.

In order to explain the precise nature of this restriction it is
useful to present the simple formal language I mentioned earlier.
However I can summarise this point in terms of the distinction
between first-order and second-order logic if you would prefer.

Interested?

Sent via Deja.com http://www.deja.com/
Before you buy.

I must admit I am not particularly satisfied with it, which is
why I have made no attempt to publicise its existence. It is not
very readable, as I go straight into Russell’s paradox, rather
than starting with Grelling’s paradox as I am doing in this thread.
My intension was to get some feedback from this newsgroup, and then
develop a better account. However if you (or anyone else) has taken
the trouble to read some of it, I’d be very interested to know what
you think!

Best wishes, Peter

In article <03d4a4ec.9292e…@usw-ex0104-026.remarq.com>,
  Stephen <cyberdictionNOcyS…@hotmail.com.invalid> wrote:

- Hide quoted text — Show quoted text -

> I just happened to come across Grelling’s Paradox and had heard
> about it, but had very little understanding of the paradox. So
> this post for others not too familar with Grelling’s Paradox.
> Also I am not sure "heterological" is in a standard dictionary.

> http://www.ti62.dial.pipex.com/#section1 (Peter Corsellis)

> 1.1 – Removing Grelling’s contradiction (begin quote)

> The derivation of Grelling’s contradiction may be stated in
> English as follows:

> The word `word´ is a word, but the word `human´ is not a human.
> Thus the former is true of itself, while the latter is not.
> Say the word `heterological´ is true of all words which are not
> true of themselves.

> Is `heterological´ heterological? From each answer the opposite
> follows. In this discussion I present no analysis of what is
> involved in a term being true of things (I present my analysis
> of this in §5). A term is thus defined as either an expression
> which is defined as true of some things, or an expression which
> refers to just one thing, i.e. a term is a predicate or a
> singular term.

> Another way of representing the derivation of Grelling’s
> contradiction, is to use a singular variable and the phrase
> `is true of´. The contradiction may then be derived from the
> following schema:

> If x is a term, `heterological´ is true of x if, and only if,
> x is not true of x

> by considering the case in which x is the term `heterological´.
> However the approach I propose rests on the fact that in
> sentences
> such as:
> ______`polysyllabic´ is polysyllabic
> the same term appears both inside and outside quotation marks,
> i.e. it is both used and mentioned.(end of quote)

> Regards,
> Stephen

> ———————————————————–

> Got questions?  Get answers over the phone at Keen.com.
> Up to 100 minutes free!
> http://www.keen.com

(My analysis of what is involved in a term being true of things,
is that `is true of’ means the same as `refers to’. Thus a
predicate true of just one thing is a name of that thing, and a
predicate true of nothing is simply an expression defined as not
referring to anything. On my website I relate this idea to
mereology (lit. `the theory of parts’), which I characterise as
the study of how we may talk about many things (or no things) as
though they were a single thing, without assuming the existence
of some separate entity such as a set. I then extend my approach
to Grelling’s paradox to apply to Russell’s paradox, by analysing
a set of things as the symbol formed by fusing all symbols which
refer to just those things.)

Sent via Deja.com http://www.deja.com/
Before you buy.

     expression which yields a true statement
     when properly substituted for `X’ in ` "X" is not an X.’

is quotable, then we may derive a contradiction. This version of
Grelling’s paradox expression is analogous to Quine’s:

     expression which yields a falsehood
     when appended to its own quotation

Appending this to its own quotation apparently yields a sentence
which asserts its own falsehood, as does properly substituting
the first expression for `X’ in ` "X" is not an X’.

You may be thinking of some non-bivalent approach. Up to now I
have largely restricted my search for a solution to the paradigm
of bivalent logic. I will happily concede that reality is
fundamentally fuzzy, but if this is what you have in mind, you
will have to suggest some account of non-bivalency if the
discussion is to proceed. (You seem to have anticipated this
request in your manifesto, which I have just seen for the first
time. This will take a couple of days to digest!)

Alternatively you may have in mind the fact that an expression in
a given language, may be true of all expressions in that language
which may be quoted and which are not true of themselves, if that
expression may not itself be quoted in the language. Such an
expression would be analogous to a proper class. I agree this is
a useful concept.

Am I getting any closer?

In article <3962B702.A7ED4…@iname.com>,
  Sandy Hodges <sandy-hod…@iname.com> wrote:

- Hide quoted text — Show quoted text -

> Well, I may not have understood the point of your original posting.
If
> I understand it now I’m not sure I would call it a solution to the
> paradox.   I don’t think there is a `solution’ to the paradox, but it
is
> at least conceivable that really close attention to the way quotation
> works would show that a particular apparent paradox was based on a
> mistake.   If so that would certainly be a `solution.’   But that’s
not
> what you’re saying, I think.

> If I’m taking the role of the defender of the paradox, and you are
> `solving’ it away, then in effect I’ve made some concessions:

> My admission that

> `X’ is heterological, if and only if `X’ is not an X.

> is a schema, is a point granted, and it may represent an avenue of
> attack:  If `heterological’ does need a schema to define it, it can’t
be
> easily expressed in a formula such as:

> A heterological expression is an expression which yields a true
> statement when substituted for `X’ in ` "X" is not an X.’

> And the fact that it can’t be shows there’s something fishy about
> `heterological.’

> The fact that

>   expression which yields a true statement when substituted for `X’
in `
> "X" is not an X.’

> isn’t the definition of heterological, is another admission.    But if
> this formula isn’t the definition of heterological, I don’t see what
> good comes of beating it to death.   The fact that this formula, at a
> given quotation depth, does not quote itself at the same depth, does
not
> matter if it is not what heterological means.

> Since I have, after seeing your posting, made concessions, I think
that
> you must shift your attack.  I am not convinced that it is impossible.

> Heterological must mean something like

>   expression which yields a true statement when properly substituted
for
> `X’ in ` "X" is not an X.’

> where proper substitution for a variable X in a text …X… means to
> add n primes to any quotes surrounding X, for the minimum n such that
> quotes surrounding X have higher level than any in the inserted
text.
> The mere fact that quotation is so complicated is something.

> peter_corsel…@my-deja.com wrote:

> > Sandy Hodges wrote:

> > > The normal operation of the paradox defines (I’m assuming) a
`Grelling
> > > expression’ by a formula:

> [clip]
> sentence when substituted for “X” in “ `X’ is not an X,” is
> > not
> > > the definition of “Grelling expression,”  since it fails to
capture
> > > the varying quote levels of the formula A schema.

> > I did not claim to have achieved the impossible! No expression may
> > completely capture the varying quote levels of your schema, as such
an
> > expression would then be true of all and only those expressions
which
> > are not true of themselves. What I claim to have found is a natural
> > notation for constructing expressions, in which any attempt to
> > construct Grelling’s paradoxical expression is automatically
blocked,
> > but which still allows the representation of self-reference.

> > Perhaps I should clarify what I mean by a "solution" to Grelling’s
> > paradox and the paradoxes of set theory. Russell once criticised
Quine
> > for developing solutions which were "created ad hoc and not … such
> > as even the cleverest logician would have thought of if he had not
> > known of the contradictions" (My Philosophical Development, p.80). I
> > accept that the standard iterative conception of set is reasonably
> > natural in this sense. However it involves a complete ban on self-
> > reference, and I believe a full solution to the paradoxes must
involve
> > some natural theory of logic which allows self-reference to be
> > represented consistently.

> I don’t follow this paragraph.   It may well be that “a full solution
> to the paradoxes must involve some natural theory of logic which
allows
> self-reference to be represented consistently.”    How is that
related
> to quotation levels?

> > In my previous posting I summarised my solution to Grelling’s
paradox
> > by translating expressions from the notation I have developed into
> > natural language. If I receive some positive feedback on this, I
shall
> > then post a longer account, giving formation rules and so forth.

> > (Apologies if I have missed the point of your comment. Please write
> > and explain if I have!)

> > Sent via Deja.com http://www.deja.com/
> > Before you buy.

> —
> — —– —— — —— – - —– – —
> When my love swears that she is made of truth,
> I do believe her, though I know she lies.
> – Sandy Hodges

Sent via Deja.com http://www.deja.com/
Before you buy.

- Hide quoted text — Show quoted text -

Sandy Hodges wrote:
> The normal operation of the paradox defines (I’m assuming) a `Grelling
> expression’ by a formula:

> `X’ is a Grelling expression, if and only if `X’ is not an X.

> call this formula A.   Put in `Grelling expression’ for X and you have
> your paradox.   For this to work for X such as `an expression
containing
> the word “expression” ‘, formula A will have to have different sorts
> of quotation depending on what X is.    We can think of formula A as a
> schema, which for each X uses a quote level one higher than the
highest
> found in X.

> Now, if we want to replace `Grelling expression’ with its definition,
to
> see if the paradox goes through, then “an expression which yields a
> true sentence when substituted for “X” in “ `X’ is not an X,” is
not
> the definition of “Grelling expression,”  since it fails to capture
> the varying quote levels of the formula A schema.

I did not claim to have achieved the impossible! No expression may
completely capture the varying quote levels of your schema, as such an
expression would then be true of all and only those expressions which
are not true of themselves. What I claim to have found is a natural
notation for constructing expressions, in which any attempt to
construct Grelling’s paradoxical expression is automatically blocked,
but which still allows the representation of self-reference.

Perhaps I should clarify what I mean by a "solution" to Grelling’s
paradox and the paradoxes of set theory. Russell once criticised Quine
for developing solutions which were "created ad hoc and not … such
as even the cleverest logician would have thought of if he had not
known of the contradictions" (My Philosophical Development, p.80). I
accept that the standard iterative conception of set is reasonably
natural in this sense. However it involves a complete ban on self-
reference, and I believe a full solution to the paradoxes must involve
some natural theory of logic which allows self-reference to be
represented consistently.

In my previous posting I summarised my solution to Grelling’s paradox
by translating expressions from the notation I have developed into
natural language. If I receive some positive feedback on this, I shall
then post a longer account, giving formation rules and so forth.

(Apologies if I have missed the point of your comment. Please write
and explain if I have!)

Sent via Deja.com http://www.deja.com/
Before you buy.

If I’m taking the role of the defender of the paradox, and you are
`solving’ it away, then in effect I’ve made some concessions:

My admission that

`X’ is heterological, if and only if `X’ is not an X.

is a schema, is a point granted, and it may represent an avenue of
attack:  If `heterological’ does need a schema to define it, it can’t be
easily expressed in a formula such as:

A heterological expression is an expression which yields a true
statement when substituted for `X’ in ` "X" is not an X.’

And the fact that it can’t be shows there’s something fishy about
`heterological.’

The fact that  

  expression which yields a true statement when substituted for `X’ in `
"X" is not an X.’

isn’t the definition of heterological, is another admission.    But if
this formula isn’t the definition of heterological, I don’t see what
good comes of beating it to death.   The fact that this formula, at a
given quotation depth, does not quote itself at the same depth, does not
matter if it is not what heterological means.  

Since I have, after seeing your posting, made concessions, I think that
you must shift your attack.  I am not convinced that it is impossible.

Heterological must mean something like

  expression which yields a true statement when properly substituted for
`X’ in ` "X" is not an X.’

where proper substitution for a variable X in a text …X… means to
add n primes to any quotes surrounding X, for the minimum n such that
quotes surrounding X have higher level than any in the inserted text.  
The mere fact that quotation is so complicated is something.

peter_corsel…@my-deja.com wrote:

> Sandy Hodges wrote:

> > The normal operation of the paradox defines (I’m assuming) a `Grelling
> > expression’ by a formula:

[clip]
sentence when substituted for “X” in “ `X’ is not an X,” is

- Hide quoted text — Show quoted text -

> not
> > the definition of “Grelling expression,”  since it fails to capture
> > the varying quote levels of the formula A schema.

> I did not claim to have achieved the impossible! No expression may
> completely capture the varying quote levels of your schema, as such an
> expression would then be true of all and only those expressions which
> are not true of themselves. What I claim to have found is a natural
> notation for constructing expressions, in which any attempt to
> construct Grelling’s paradoxical expression is automatically blocked,
> but which still allows the representation of self-reference.

> Perhaps I should clarify what I mean by a "solution" to Grelling’s
> paradox and the paradoxes of set theory. Russell once criticised Quine
> for developing solutions which were "created ad hoc and not … such
> as even the cleverest logician would have thought of if he had not
> known of the contradictions" (My Philosophical Development, p.80). I
> accept that the standard iterative conception of set is reasonably
> natural in this sense. However it involves a complete ban on self-
> reference, and I believe a full solution to the paradoxes must involve
> some natural theory of logic which allows self-reference to be
> represented consistently.

I don’t follow this paragraph.   It may well be that “a full solution
to the paradoxes must involve some natural theory of logic which allows
self-reference to be represented consistently.”    How is that related
to quotation levels?

> In my previous posting I summarised my solution to Grelling’s paradox
> by translating expressions from the notation I have developed into
> natural language. If I receive some positive feedback on this, I shall
> then post a longer account, giving formation rules and so forth.

> (Apologies if I have missed the point of your comment. Please write
> and explain if I have!)

> Sent via Deja.com http://www.deja.com/
> Before you buy.

–
— —– —— — —— – - —– – —
When my love swears that she is made of truth,
I do believe her, though I know she lies.
– Sandy Hodges

`X’ is a Grelling expression, if and only if `X’ is not an X.

call this formula A.   Put in `Grelling expression’ for X and you have
your paradox.   For this to work for X such as `an expression containing
the word “expression” ‘, formula A will have to have different sorts
of quotation depending on what X is.    We can think of formula A as a
schema, which for each X uses a quote level one higher than the highest
found in X.

Now, if we want to replace `Grelling expression’ with its definition, to
see if the paradox goes through, then “an expression which yields a
true sentence when substituted for “X” in “ `X’ is not an X,” is not
the definition of “Grelling expression,”  since it fails to capture
the varying quote levels of the formula A schema.

- Hide quoted text — Show quoted text -

peter_corsel…@my-deja.com wrote:

> How’s this for a solution to Grelling’s paradox?

> (The paradox concerns the distinction between expressions
> which are true of themselves, and expressions which are not
> true of themselves. There can be no expression which is true
> of all and only the latter kind of expressions. Say there was
> such an expression, is this expression true of itself? From
> each answer the opposite follows. This is paradoxical
> because it apparently contradicts the natural assumption that
> for any plurality of things, it is possible to define an
> expression as being true of all, and only, those things.)

> Consider the following expression:

>      expression which yields a true sentence
>      when substituted for "X" in "`X´ is not an X"

[clip]
—
— —– —— — —— – - —– – —
When my love swears that she is made of truth,
I do believe her, though I know she lies.
– Sandy Hodges

http://www.ti62.dial.pipex.com/#section1 (Peter Corsellis)

1.1 – Removing Grelling’s contradiction (begin quote)

The derivation of Grelling’s contradiction may be stated in
English as follows:

The word `word´ is a word, but the word `human´ is not a human.
Thus the former is true of itself, while the latter is not.
Say the word `heterological´ is true of all words which are not
true of themselves.

Is `heterological´ heterological? From each answer the opposite
follows. In this discussion I present no analysis of what is
involved in a term being true of things (I present my analysis
of this in §5). A term is thus defined as either an expression
which is defined as true of some things, or an expression which
refers to just one thing, i.e. a term is a predicate or a
singular term.

Another way of representing the derivation of Grelling’s
contradiction, is to use a singular variable and the phrase
`is true of´. The contradiction may then be derived from the
following schema:

If x is a term, `heterological´ is true of x if, and only if,
x is not true of x

by considering the case in which x is the term `heterological´.
However the approach I propose rests on the fact that in
sentences
such as:
______`polysyllabic´ is polysyllabic
the same term appears both inside and outside quotation marks,
i.e. it is both used and mentioned.(end of quote)

Regards,
Stephen

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