The logic of presuppositions solves the Liar paradox

X.Y. Newberry

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Oct 10, 2015, 4:49:53 PM10/10/15

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P.F. Strawson proposed to translate “No F is G” as ~(∃x)(Fx & Gx) &
(∃x)Fx & (∃x)Gx. (Strawson, 1952, Introduction to Logical Theory,
London, Methuen, pp. 163-179.) Furthermore he proposed to take the term
(∃x)Fx & (∃x)Gx as a PRESUPPOSITION. That is if (∃x)Fx & (∃x)Gx is not
true then ~(∃x)(Fx & Gx) is neither true nor false. So for example the
sentence “All John's children are asleep” is ~(T v F) if John has no
children. How does this help with the Liar paradox? Consider

This sentence is not true (Y)

The argument usually goes as follows.

(1) Assume that Y is true
(2) Then what it says is the case
(3) It says it is not true
(4) Then it is not true
Result: (4) contradicts (1).

Let us now modify the argument and make the assumption (1) part of the
sentence itself:

This true sentence is not true (Y')

If we go through the steps (1)-(4) again we find that the sentence is
not true, and therefore there is no such thing as “this true sentence”.
A presupposition of Y' is that its subject term “this true sentence” has
a referent. But it does not! Hence Y' is ~(T v F).

But then the sentence is not true, and what it says is the case!! Well,
in order to resolve the paradox being ~(T v F) is not sufficient - in
addition the sentence must be meaningless. But we in fact do know that Y
(and Y') is meaningless.

https://groups.google.com/forum/#!topic/sci.logic/FOEZFcy0ZR4

But Strawson developed his theory precisely because he thought that the
sentence “All John's children” are asleep” was perfectly meaningful. (We
agree.) He rejected the trichotomy T, F, meaningless. (We agree.)

What follows is my addition. The meaningless sentences are a proper
subset of the sentences that are ~(T v F). In particular when the
non-existence of the subject is necessary [e.g. ~(Ex)(x # x)] then the
sentence is meaningless, when the non-existence of the subject is
merely contingent [e.g. John has no children] then the sentence is
merely ~(T v F).

Gödel's sentence behaves the same way as Y':

~(Ex)(Ey)(Prf(x,y) & This(y)) (G)

where This() is satisfied only by the Gödel number of G. Let us substitute

~(Ex)(Prf(x,<#G#>) & This(<#G#>)) (J)

We observe that (Ex)(Prf(x,<#G#>) is a presupposition of (J), and it is
not true. But then J is not true [it is ~(T v F).] No sound system
should derive either it or G, although there is no reason why it could
not derive

~(Ex)(Prf(x,<#G#>) (K)

Note that (J) and (K) are not the same sentences. The latter talks about
numbers, the former about unicorns. It says that no unicorn is the Gödel
sentence.

It is possible to specify a semantics such that the latter is true, but
the former is ~(T v F).
http://arxiv.org/abs/1509.06837

--
X.Y. Newberry

If Jack says ‘What I am saying at this very moment is not true’, we can
successfully and truly assert that he did not utter a truth: ‘What Jack
said is not true’. But it is hardly conceivable that Jack’s utterance is
true by virtue of its success in attributing non-truth to itself.

Haim Gaifman

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Charlie-Boo

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Oct 10, 2015, 8:30:31 PM10/10/15

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On Saturday, October 10, 2015 at 4:49:53 PM UTC-4, Newberry wrote:
> P.F. Strawson proposed to translate “No F is G” as ~(∃x)(Fx & Gx) &
> (∃x)Fx & (∃x)Gx. (Strawson, 1952, Introduction to Logical Theory,
> London, Methuen, pp. 163-179.) Furthermore he proposed to take the term
> (∃x)Fx & (∃x)Gx as a PRESUPPOSITION. That is if (∃x)Fx & (∃x)Gx is not
> true then ~(∃x)(Fx & Gx) is neither true nor false. So for example the
> sentence “All John's children are asleep” is ~(T v F) if John has no
> children. How does this help with the Liar paradox? Consider
>
> This sentence is not true (Y)
>
> The argument usually goes as follows.
>
> (1) Assume that Y is true
> (2) Then what it says is the case
> (3) It says it is not true
> (4) Then it is not true
> Result: (4) contradicts (1).

Every sentence is both true and false because you are allowing a syntax with the semantics of expressing truth, which is inconsistent. It's like having a programming language with a function that solves the halting problem.

C-B

Martin Shobe

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Oct 11, 2015, 11:05:44 AM10/11/15

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Just so you know, there are no meaningless sentences in FOL unless the
theory is inconsistent, in which case every sentence is meaningless.

> Gödel's sentence behaves the same way as Y':
>
> ~(Ex)(Ey)(Prf(x,y) & This(y)) (G)
>
> where This() is satisfied only by the Gödel number of G. Let us substitute
>
> ~(Ex)(Prf(x,<#G#>) & This(<#G#>)) (J)
>
> We observe that (Ex)(Prf(x,<#G#>) is a presupposition of (J), and it is
> not true.

According to the Strawson definition, it is not as it follows from (J)
that ~(Ex)(Prf(x,<#G#>)).

> But then J is not true [it is ~(T v F).] No sound system
> should derive either it or G, although there is no reason why it could
> not derive
>
> ~(Ex)(Prf(x,<#G#>) (K)
>
> Note that (J) and (K) are not the same sentences. The latter talks about
> numbers, the former about unicorns. It says that no unicorn is the Gödel
> sentence.

Both (J) and (K) are about numbers.

Martin Shobe

X.Y. Newberry

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Oct 11, 2015, 9:12:13 PM10/11/15

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That is why we need new logic.

> unless the
> theory is inconsistent, in which case every sentence is meaningless.
>
>> Gödel's sentence behaves the same way as Y':
>>
>> ~(Ex)(Ey)(Prf(x,y) & This(y)) (G)
>>
>> where This() is satisfied only by the Gödel number of G. Let us
>> substitute
>>
>> ~(Ex)(Prf(x,<#G#>) & This(<#G#>)) (J)
>>
>> We observe that (Ex)(Prf(x,<#G#>) is a presupposition of (J), and it is
>> not true.
>
> According to the Strawson definition, it is not as it follows from (J)
> that ~(Ex)(Prf(x,<#G#>)).

In classical logic ~(Ex)(Prf(x,<#G#>)) follows from (J). But in
Strawson's logic, if ~(Ex)(Prf(x,<#G#>)) is true then (J) is neither
true nor false.

>

>> But then J is not true [it is ~(T v F).] No sound system
>> should derive either it or G, although there is no reason why it could
>> not derive
>>
>> ~(Ex)(Prf(x,<#G#>) (K)
>>
>> Note that (J) and (K) are not the same sentences. The latter talks about
>> numbers, the former about unicorns. It says that no unicorn is the Gödel
>> sentence.
>
> Both (J) and (K) are about numbers.

Is the following sentence true?

No unicorn is the code of Gödel's sentence.

>
> Martin Shobe

Martin Shobe

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Oct 12, 2015, 10:37:37 AM10/12/15

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Why do we need meaningless sentences?

>> unless the
>> theory is inconsistent, in which case every sentence is meaningless.
>>
>>> Gödel's sentence behaves the same way as Y':
>>>
>>> ~(Ex)(Ey)(Prf(x,y) & This(y)) (G)
>>>
>>> where This() is satisfied only by the Gödel number of G. Let us
>>> substitute
>>>
>>> ~(Ex)(Prf(x,<#G#>) & This(<#G#>)) (J)
>>>
>>> We observe that (Ex)(Prf(x,<#G#>) is a presupposition of (J), and it is
>>> not true.
>>
>> According to the Strawson definition, it is not as it follows from (J)
>> that ~(Ex)(Prf(x,<#G#>)).
>
> In classical logic ~(Ex)(Prf(x,<#G#>)) follows from (J). But in
> Strawson's logic, if ~(Ex)(Prf(x,<#G#>)) is true then (J) is neither
> true nor false.

But does ~(Ex)(Prf(x,<#G#>)) follow from (J)? Could you give us the
rules for handling existential quantifiers in his system?

>>> But then J is not true [it is ~(T v F).] No sound system
>>> should derive either it or G, although there is no reason why it could
>>> not derive
>>>
>>> ~(Ex)(Prf(x,<#G#>) (K)
>>>
>>> Note that (J) and (K) are not the same sentences. The latter talks about
>>> numbers, the former about unicorns. It says that no unicorn is the Gödel
>>> sentence.
>>
>> Both (J) and (K) are about numbers.
>
> Is the following sentence true?
>
> No unicorn is the code of Gödel's sentence.

Yes, but both (J) and (K) are still about numbers.

Martin Shobe

X.Y. Newberry

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Oct 12, 2015, 10:14:14 PM10/12/15

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Because that is what some of them actually are, and "classical" logic
does not model them correctly.

>
>>> unless the
>>> theory is inconsistent, in which case every sentence is meaningless.
>>>
>>>> Gödel's sentence behaves the same way as Y':
>>>>
>>>> ~(Ex)(Ey)(Prf(x,y) & This(y)) (G)
>>>>
>>>> where This() is satisfied only by the Gödel number of G. Let us
>>>> substitute
>>>>
>>>> ~(Ex)(Prf(x,<#G#>) & This(<#G#>)) (J)
>>>>
>>>> We observe that (Ex)(Prf(x,<#G#>) is a presupposition of (J), and it is
>>>> not true.
>>>
>>> According to the Strawson definition, it is not as it follows from (J)
>>> that ~(Ex)(Prf(x,<#G#>)).
>>
>> In classical logic ~(Ex)(Prf(x,<#G#>)) follows from (J). But in
>> Strawson's logic, if ~(Ex)(Prf(x,<#G#>)) is true then (J) is neither
>> true nor false.
>
> But does ~(Ex)(Prf(x,<#G#>)) follow from (J)? Could you give us the
> rules for handling existential quantifiers in his system?

Thus far we only have semantics, and in this semantics (J) is ~(T v F),
so nothing follows from it. Please refer to P. F. Strawson, 1952,
Introduction to Logical Theory, London, Methuen, pp. 173. He showed that
given the interpretation below

A: ~(∃x)(Fx & ~Gx) & (∃x)Fx & (∃x)~Gx
E: ~(∃x)(Fx & Gx) & (∃x)Fx & (∃x)Gx
I: (∃x)(Fx & Gx) v ~(∃x)Fx v ~(∃x)Gx
O: (∃x)(Fx & Gx) v ~(∃x)Fx v ~(∃x~)Gx

all the laws of the traditional syllogism will hold. Furthermore he
proposed to take the terms (∃x)Fx & (∃x)Gx etc. as PRESUPPOSITIONS. But
Strawson considered only simple sentences with two terms. His semantics
has to be generalized to arbitrary monadic and polyadic sentences. I
have done so here http://arxiv.org/ftp/arxiv/papers/1509/1509.06837.pdf
utilizing truth-relevant logic.

>
>>>> But then J is not true [it is ~(T v F).] No sound system
>>>> should derive either it or G, although there is no reason why it could
>>>> not derive
>>>>
>>>> ~(Ex)(Prf(x,<#G#>) (K)
>>>>
>>>> Note that (J) and (K) are not the same sentences. The latter talks
>>>> about
>>>> numbers, the former about unicorns. It says that no unicorn is the
>>>> Gödel
>>>> sentence.
>>>
>>> Both (J) and (K) are about numbers.
>>
>> Is the following sentence true?
>>
>> No unicorn is the code of Gödel's sentence.
>
> Yes, but both (J) and (K) are still about numbers.

But (J) is also about unicorns and (K) is not.

Martin Shobe

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Oct 13, 2015, 11:05:51 AM10/13/15

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"Classical" logic excludes them from the discussion completely, so why
does it need them?

The information on page 173 doesn't answer the question I asked and
neither does the rest of your response. So, once again, what are the

rules for handling existential quantifiers in his system?

(BTW, You still haven't fixed the issues I mentioned last time you
pointed me to this paper. For example, either your first example is
wrong, or you are using a non-standard valuation which you neither warn
the reader of nor explain.)

>>>>> But then J is not true [it is ~(T v F).] No sound system
>>>>> should derive either it or G, although there is no reason why it could
>>>>> not derive
>>>>>
>>>>> ~(Ex)(Prf(x,<#G#>) (K)
>>>>>
>>>>> Note that (J) and (K) are not the same sentences. The latter talks
>>>>> about
>>>>> numbers, the former about unicorns. It says that no unicorn is the
>>>>> Gödel
>>>>> sentence.
>>>>
>>>> Both (J) and (K) are about numbers.
>>>
>>> Is the following sentence true?
>>>
>>> No unicorn is the code of Gödel's sentence.
>>
>> Yes, but both (J) and (K) are still about numbers.
>
> But (J) is also about unicorns and (K) is not.

Unless you've decided to use a many-sorted logic, (J) and (K) have to be
about exactly the same things. If you are using a many sorted logic, you
should say so and indicate which sort is being used by which statement.
In the usual treatment of these things, there is only one sort, and its
domain is N. From this it follows that (J) and (K) are both about
numbers and neither is about unicorns.

Martin Shobe

X.Y. Newberry

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Oct 13, 2015, 9:20:42 PM10/13/15

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I do not know if classical logic needs them. But we need a new logic
because classical logic is not able to model them.

I have said above "Thus far we only have semantics." Having said that,
no sensible derivation system will derive truth from non-truth.

>
> (BTW, You still haven't fixed the issues I mentioned last time you
> pointed me to this paper.

It is entirely possible that the paper is swarming with errors, but you
have not come anywhere close to finding any.

> For example, either your first example is
> wrong, or you are using a non-standard valuation which you neither warn
> the reader of nor explain.)

Do you think I have to warn the reader that I am using non-standard
valuation when the whole purpose of the paper is to explain the
non-standard validation?

>
>>>>>> But then J is not true [it is ~(T v F).] No sound system
>>>>>> should derive either it or G, although there is no reason why it
>>>>>> could
>>>>>> not derive
>>>>>>
>>>>>> ~(Ex)(Prf(x,<#G#>)
>>>>>> (K)
>>>>>>
>>>>>> Note that (J) and (K) are not the same sentences. The latter talks
>>>>>> about
>>>>>> numbers, the former about unicorns. It says that no unicorn is the
>>>>>> Gödel
>>>>>> sentence.
>>>>>
>>>>> Both (J) and (K) are about numbers.
>>>>
>>>> Is the following sentence true?
>>>>
>>>> No unicorn is the code of Gödel's sentence.
>>>
>>> Yes, but both (J) and (K) are still about numbers.
>>
>> But (J) is also about unicorns and (K) is not.
>
> Unless you've decided to use a many-sorted logic, (J) and (K) have to be
> about exactly the same things. If you are using a many sorted logic, you
> should say so and indicate which sort is being used by which statement.

I am not using many-sorted logic, never have, and never shall.

> In the usual treatment of these things, there is only one sort, and its
> domain is N. From this it follows that (J) and (K) are both about
> numbers and neither is about unicorns.

Does not (J) say that the empty set is a subset of the empty set?

Charlie-Boo

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Oct 13, 2015, 9:37:08 PM10/13/15

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1. Go through all unicorns - for each:
a. If it is the code of Godel's sentence, then halt.
2. Go back to 1.

The question is logically equivalent to "Does the above not halt?".

>
> >
> > Martin Shobe
> >
>
>
> --
> X.Y. Newberry
>
> If Jack says ‘What I am saying at this very moment is not true’, we can
> successfully and truly assert that he did not utter a truth: ‘What Jack
> said is not true’. But it is hardly conceivable that Jack’s utterance is
> true by virtue of its success in attributing non-truth to itself.

Why not? Your English (since it has a truth predicate) is an inconsistent system so it is both true and false.

C-B

Charlie-Boo

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Oct 13, 2015, 9:41:06 PM10/13/15

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On Sunday, October 11, 2015 at 9:12:13 PM UTC-4, Newberry wrote:

No, we need new English - without a truth predicate. And why have it? Instead of saying "x is true." just say "x." and instead of saying "x is false." say "x not."

C-B

Martin Shobe

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Oct 14, 2015, 10:56:19 AM10/14/15

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I know they exist in natural languages, but why do we logic to model
them? What's wrong with leaving such statements outside the scope of logic?

Yet you claim Strawson was able to show (derive) all the laws of the
traditional syllogism from it. How did he do that?

>> (BTW, You still haven't fixed the issues I mentioned last time you
>> pointed me to this paper.

> It is entirely possible that the paper is swarming with errors, but you
> have not come anywhere close to finding any.

You wish.

>> For example, either your first example is
>> wrong, or you are using a non-standard valuation which you neither warn
>> the reader of nor explain.)

> Do you think I have to warn the reader that I am using non-standard
> valuation when the whole purpose of the paper is to explain the
> non-standard validation?

Everything else in the paper relies on it and it's simply not present.
(It's not even clear that you are explaining a non-standard valuation.
The only reason I know that you are using a non-standard valuation is
that you told me you were when I pointed out this error previously.)

>>>>>>> But then J is not true [it is ~(T v F).] No sound system
>>>>>>> should derive either it or G, although there is no reason why it
>>>>>>> could
>>>>>>> not derive
>>>>>>>
>>>>>>> ~(Ex)(Prf(x,<#G#>)
>>>>>>> (K)
>>>>>>>
>>>>>>> Note that (J) and (K) are not the same sentences. The latter talks
>>>>>>> about
>>>>>>> numbers, the former about unicorns. It says that no unicorn is the
>>>>>>> Gödel
>>>>>>> sentence.
>>>>>>
>>>>>> Both (J) and (K) are about numbers.
>>>>>
>>>>> Is the following sentence true?
>>>>>
>>>>> No unicorn is the code of Gödel's sentence.
>>>>
>>>> Yes, but both (J) and (K) are still about numbers.
>>>
>>> But (J) is also about unicorns and (K) is not.
>>
>> Unless you've decided to use a many-sorted logic, (J) and (K) have to be
>> about exactly the same things. If you are using a many sorted logic, you
>> should say so and indicate which sort is being used by which statement.
>
> I am not using many-sorted logic, never have, and never shall.

From which it follows, that (J) and (K) both have to be about exactly
the same things.

>> In the usual treatment of these things, there is only one sort, and its
>> domain is N. From this it follows that (J) and (K) are both about
>> numbers and neither is about unicorns.
>
> Does not (J) say that the empty set is a subset of the empty set?

(J) says that there isn't a natural number that both encodes a proof of
(G) and satisfies this(<#G#>). If you want to say that the empty set is
a subset of the empty set, you say something like

∅⊂∅

Martin Shobe

X.Y. Newberry

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Oct 15, 2015, 12:15:44 AM10/15/15

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Wow!! In today's academia this is a big no no!

> but why do we logic to model
> them? What's wrong with leaving such statements outside the scope of logic?

Because vacuous propositions ARE ~(T v F). In fact in mathematics they
are outright meaningless. It is wrong to pretend that they are true.
Secondly, treating vacuous propositions as ~(T v F) gets rid of semantic
incompleteness.

It is strange to say about a paper that describes a non-standard
valuation that it is an "error" because it does not conform to the
standard valuation. It is even more strange to say about a paper whose
only purpose is to explain a non-standard validation that it is not
present. I guess I am not explaining it too well if you did not even
realize that is what I was doing. BTW, what did you think the paper was
about?

Is S = {x| Prf(x,<#G#>} the empty set?

Martin Shobe

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Oct 15, 2015, 8:48:20 AM10/15/15

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No it isn't.

>> but why do we logic to model
>> them? What's wrong with leaving such statements outside the scope of
>> logic?

> Because vacuous propositions ARE ~(T v F). In fact in mathematics they
> are outright meaningless. It is wrong to pretend that they are true.
> Secondly, treating vacuous propositions as ~(T v F) gets rid of semantic
> incompleteness.

No they aren't outright meaningless. Formally, the have a well defined
semantics, so there's no pretending that they are true. They *are* true.

No response?

>>>> (BTW, You still haven't fixed the issues I mentioned last time you
>>>> pointed me to this paper.
>>
>>> It is entirely possible that the paper is swarming with errors, but you
>>> have not come anywhere close to finding any.
>>
>> You wish.
>>
>>>> For example, either your first example is
>>>> wrong, or you are using a non-standard valuation which you neither warn
>>>> the reader of nor explain.)
>>
>>> Do you think I have to warn the reader that I am using non-standard
>>> valuation when the whole purpose of the paper is to explain the
>>> non-standard validation?
>>
>> Everything else in the paper relies on it and it's simply not present.
>> (It's not even clear that you are explaining a non-standard valuation.
>> The only reason I know that you are using a non-standard valuation is
>> that you told me you were when I pointed out this error previously.)
>
> It is strange to say about a paper that describes a non-standard
> valuation that it is an "error" because it does not conform to the
> standard valuation.

What's strange here is that you think I said that. That's not what I said.

> It is even more strange to say about a paper whose
> only purpose is to explain a non-standard validation that it is not
> present.

That's easy to explain. You don't know what you're doing. (This also
explains why you claim that something can't be derived yet you can't
state the derivation rules. See above.)

> I guess I am not explaining it too well if you did not even
> realize that is what I was doing. BTW, what did you think the paper was
> about?

From the contents, you appear to be classifying statements.

If (G) is true, yes. Of course, {x|
Prf(x,<#G#>)} doesn't appear in (J).

Martin Shobe

Rupert

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Oct 15, 2015, 11:04:50 AM10/15/15

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There were some parts of your post that I had a bit of a hard time following. I don't think it is reasonable to take "This sentence is not true" as equivalent with "This true sentence is not true". As far as Gödel's sentence goes the point is it is a sentence in the first-order language of arithmetic and so for that reason surely has a truth-value.

X.Y. Newberry

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Oct 15, 2015, 9:30:40 PM10/15/15

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They don't. For example the sentence

(x)(x = x+1 -> x = x+2)

does not have any interpretation in the domain of the natural numbers.

I do not have the reference on me right now. I will post it this weekend.

>
>>>>> (BTW, You still haven't fixed the issues I mentioned last time you
>>>>> pointed me to this paper.
>>>
>>>> It is entirely possible that the paper is swarming with errors, but you
>>>> have not come anywhere close to finding any.
>>>
>>> You wish.
>>>
>>>>> For example, either your first example is
>>>>> wrong, or you are using a non-standard valuation which you neither
>>>>> warn
>>>>> the reader of nor explain.)
>>>
>>>> Do you think I have to warn the reader that I am using non-standard
>>>> valuation when the whole purpose of the paper is to explain the
>>>> non-standard validation?
>>>
>>> Everything else in the paper relies on it and it's simply not present.
>>> (It's not even clear that you are explaining a non-standard valuation.
>>> The only reason I know that you are using a non-standard valuation is
>>> that you told me you were when I pointed out this error previously.)
>>
>> It is strange to say about a paper that describes a non-standard
>> valuation that it is an "error" because it does not conform to the
>> standard valuation.
>
> What's strange here is that you think I said that. That's not what I said.

What did you say?

>
>> It is even more strange to say about a paper whose
>> only purpose is to explain a non-standard validation that it is not
>> present.
>
> That's easy to explain. You don't know what you're doing. (This also
> explains why you claim that something can't be derived yet you can't
> state the derivation rules. See above.)

That I have not stated the rules yet does not mean that they cannot be
stated. Do you understand that no sound derivation system will derive
truth from non-truth?

>
>> I guess I am not explaining it too well if you did not even
>> realize that is what I was doing. BTW, what did you think the paper was
>> about?
>
> From the contents, you appear to be classifying statements.

Suppose this is true. How is it an error?

Are you suggesting that any logic that does not conform to classical
logic merely classifies statements?

Is S' = {x| x = unicorn}

the same set as

S = {x| Prf(x,<#G#>)}

> Of course, {x|
> Prf(x,<#G#>)} doesn't appear in (J).

But 'Prf(x,<#G#>)' does.

X.Y. Newberry

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Oct 15, 2015, 9:41:48 PM10/15/15

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Rupert wrote:
> There were some parts of your post that I had a bit of a hard time
> following. I don't think it is reasonable to take "This sentence is
> not true" as equivalent with "This true sentence is not true".

The proof of contradiction of the Liar typically starts with the
assumption that the sentence is true. Liar + this assumption make "This
true sentence is not true." So it seems to me that at the very least you
cannot walk through the proof without stepping through "This true

sentence is not true."

> As far
> as Gödel's sentence goes the point is it is a sentence in the
> first-order language of arithmetic and so for that reason surely has
> a truth-value.

"first-order language of arithmetic" usually implies classical logic. In
the present context I am suggesting that there is an alternative
logic/interpretation such that it does NOT have a truth value. Are you
claiming that no such logic/interpretation is possible, or are you just
injecting an orthogonal statement for fun?

Martin Shobe

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Oct 15, 2015, 10:37:43 PM10/15/15

to

On 10/15/2015 8:30 PM, X.Y. Newberry wrote:
> Martin Shobe wrote:
>> On 10/14/2015 11:15 PM, X.Y. Newberry wrote:
>>> Martin Shobe wrote:
>>>> On 10/13/2015 8:20 PM, X.Y. Newberry wrote:
>>>>> Martin Shobe wrote:
>>>>>> On 10/12/2015 9:13 PM, X.Y. Newberry wrote:
>>>>>>> Martin Shobe wrote:
>>>>>>>> On 10/11/2015 8:11 PM, X.Y. Newberry wrote:
>>>>>>>>> Martin Shobe wrote:
>>>>>>>>>> On 10/10/2015 3:49 PM, X.Y. Newberry wrote:
>>>> but why do we logic to model
>>>> them? What's wrong with leaving such statements outside the scope of
>>>> logic?
>>
>>> Because vacuous propositions ARE ~(T v F). In fact in mathematics they
>>> are outright meaningless. It is wrong to pretend that they are true.
>>> Secondly, treating vacuous propositions as ~(T v F) gets rid of semantic
>>> incompleteness.
>>
>> No they aren't outright meaningless. Formally, the have a well defined
>> semantics, so there's no pretending that they are true. They *are* true.
>
> They don't. For example the sentence
>
> (x)(x = x+1 -> x = x+2)
>
> does not have any interpretation in the domain of the natural numbers.

Yes it does. Furthermore, the statement is true.

That's fine.

>>>>>> (BTW, You still haven't fixed the issues I mentioned last time you
>>>>>> pointed me to this paper.
>>>>
>>>>> It is entirely possible that the paper is swarming with errors, but
>>>>> you
>>>>> have not come anywhere close to finding any.
>>>>
>>>> You wish.
>>>>
>>>>>> For example, either your first example is
>>>>>> wrong, or you are using a non-standard valuation which you neither
>>>>>> warn
>>>>>> the reader of nor explain.)
>>>>
>>>>> Do you think I have to warn the reader that I am using non-standard
>>>>> valuation when the whole purpose of the paper is to explain the
>>>>> non-standard validation?
>>>>
>>>> Everything else in the paper relies on it and it's simply not present.
>>>> (It's not even clear that you are explaining a non-standard valuation.
>>>> The only reason I know that you are using a non-standard valuation is
>>>> that you told me you were when I pointed out this error previously.)
>>>
>>> It is strange to say about a paper that describes a non-standard
>>> valuation that it is an "error" because it does not conform to the
>>> standard valuation.
>>
>> What's strange here is that you think I said that. That's not what I
>> said.
>
> What did you say?

It's still there. I can't help you further since I have absolutely no
idea where you go wrong.

>>> It is even more strange to say about a paper whose
>>> only purpose is to explain a non-standard validation that it is not
>>> present.
>>
>> That's easy to explain. You don't know what you're doing. (This also
>> explains why you claim that something can't be derived yet you can't
>> state the derivation rules. See above.)
>
> That I have not stated the rules yet does not mean that they cannot be
> stated. Do you understand that no sound derivation system will derive
> truth from non-truth?

Uh no. I don't understand that as it's false. What a sound system won't
do is derive a non-truth from a truth. (That is, after all, what it
means to be sound.)

>>> I guess I am not explaining it too well if you did not even
>>> realize that is what I was doing. BTW, what did you think the paper was
>>> about?
>>
>> From the contents, you appear to be classifying statements.
>
> Suppose this is true. How is it an error?

I didn't say that was an error?

> Are you suggesting that any logic that does not conform to classical
> logic merely classifies statements?

No. My statement is only intended to be about your paper as it doesn't
describe any semantics and only defines categories of statements.

Not in FOL. "unicorn" is being used as an individual constant, so it
must refer to something. As such, S' is a singleton and not empty.
Therefore, S is not the same set as S'.

>> Of course, {x|
>> Prf(x,<#G#>)} doesn't appear in (J).
>
> But 'Prf(x,<#G#>)' does.

So? That doesn't mean that (J) says that the empty set is a subset of
the empty set. In fact, (J) doesn't say anything about sets at all.
(Unless they are part of the domain, but that would be a bit unusual for
something like this. Usually, the domain is restricted to just natural
numbers.)

Martin Shobe

X.Y. Newberry

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Oct 16, 2015, 12:23:39 AM10/16/15

to

Which natural numbers is it talking about?

> Furthermore, the statement is true.

How does it manage to do that?

That is not important. The important thing is to know that I did go wrong.

>
>>>> It is even more strange to say about a paper whose
>>>> only purpose is to explain a non-standard validation that it is not
>>>> present.
>>>
>>> That's easy to explain. You don't know what you're doing. (This also
>>> explains why you claim that something can't be derived yet you can't
>>> state the derivation rules. See above.)
>>
>> That I have not stated the rules yet does not mean that they cannot be
>> stated. Do you understand that no sound derivation system will derive
>> truth from non-truth?
>
> Uh no. I don't understand that as it's false. What a sound system won't
> do is derive a non-truth from a truth. (That is, after all, what it
> means to be sound.)

That is because the idea that a system would derive truth from non-truth
did not even cross anybody's mind.

>
>>>> I guess I am not explaining it too well if you did not even
>>>> realize that is what I was doing. BTW, what did you think the paper was
>>>> about?
>>>
>>> From the contents, you appear to be classifying statements.
>>
>> Suppose this is true. How is it an error?
>
> I didn't say that was an error?

You said "either your first example is wrong, or you are using a

non-standard valuation which you neither warn the reader of nor

explain." So your position is that that my description is not semantics
but rather statement categorization therefore it does not "explain" the
non-standard valuation?

Is S' = {x| Unicorn(x)}

the same set as

S = {x| Prf(x,<#G#>)} ?

>
>>> Of course, {x|
>>> Prf(x,<#G#>)} doesn't appear in (J).
>>
>> But 'Prf(x,<#G#>)' does.
>
> So? That doesn't mean that (J) says that the empty set is a subset of
> the empty set. In fact, (J) doesn't say anything about sets at all.
> (Unless they are part of the domain, but that would be a bit unusual for
> something like this. Usually, the domain is restricted to just natural
> numbers.)
>
> Martin Shobe
>

Ross A. Finlayson

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Oct 16, 2015, 12:35:39 AM10/16/15

to

On Sunday, October 11, 2015 at 6:12:13 PM UTC-7, Newberry wrote:
>
> That is why we need new logic.
>

We need "new" consistent, complete, and "concrete"
logic as possible, that is still only extra-classical,
extra-standard, and extra-regular, with "all that".

Mathematics could use this, and physics could use that.

Martin Shobe

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Oct 16, 2015, 9:35:27 AM10/16/15

to

Unless otherwise specified (and you didn't) it would be the standard one.

>> Furthermore, the statement is true.

> How does it manage to do that?

There's no x such that x = x + 1. By the semantics of the -> operator,
it follows that x=x+1 -> x=x+2 is true regardless of which value of x is
used. Therefore, (x)(x=x+1 -> x=x+2) is true.

Then I can't help you as I have absolutely no idea where you got your
interpretation from.

>>>>> It is even more strange to say about a paper whose
>>>>> only purpose is to explain a non-standard validation that it is not
>>>>> present.
>>>>
>>>> That's easy to explain. You don't know what you're doing. (This also
>>>> explains why you claim that something can't be derived yet you can't
>>>> state the derivation rules. See above.)
>>>
>>> That I have not stated the rules yet does not mean that they cannot be
>>> stated. Do you understand that no sound derivation system will derive
>>> truth from non-truth?
>>
>> Uh no. I don't understand that as it's false. What a sound system won't
>> do is derive a non-truth from a truth. (That is, after all, what it
>> means to be sound.)
>
> That is because the idea that a system would derive truth from non-truth
> did not even cross anybody's mind.

Not hardly.

>>>>> I guess I am not explaining it too well if you did not even
>>>>> realize that is what I was doing. BTW, what did you think the paper
>>>>> was
>>>>> about?
>>>>
>>>> From the contents, you appear to be classifying statements.
>>>
>>> Suppose this is true. How is it an error?
>>
>> I didn't say that was an error?

> You said "either your first example is wrong, or you are using a
> non-standard valuation which you neither warn the reader of nor
> explain." So your position is that that my description is not semantics
> but rather statement categorization therefore it does not "explain" the
> non-standard valuation?

Your so doesn't follow from the statement (it does follow from other
things I've said.) Now, how are you getting to me saying that
classifying statements is an error?

Maybe. How is Unicorn(x) defined? If it's defined in such a way that
(x)(~Unicorn(x)), then yes S' = S. But, to put it into the larger
picture, if that is sufficient to claim that (J) is about Unicorns, then
it should also be sufficient to claim that (K) is about Unicorns. Once
again, both (J) and (K) have to be about exactly the same things unless
you are using different sorts (and you've said that you aren't).

Martin Shobe

X.Y. Newberry

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Oct 17, 2015, 2:29:42 AM10/17/15

to

Please name the standard integer.

>>> Furthermore, the statement is true.
>
>> How does it manage to do that?
>
> There's no x such that x = x + 1.

So the sentence does not talk about anything.

> By the semantics of the ->
> operator,

So it is all about the semantics of -> and has nothing to do with the
natural numbers per se.

> it follows that x=x+1 -> x=x+2 is true regardless of which
> value of x is used. Therefore, (x)(x=x+1 -> x=x+2) is true.

How is not talking about anything true? You got me on this one.

The error is that I thought you said there was an error. In fact there
is no error.

So the Gödel sentence ~(Ex)(Ey)(Prf(x,y) & This(y)) is equivalent to
~(Ey)(Unicorn(y) & This(y)).

> But, to put it into the larger
> picture, if that is sufficient to claim that (J) is about Unicorns,
> then it should also be sufficient to claim that (K) is about
> Unicorns.

I do not see any unicorns in K. It says that there is no proof of G.

> Once again, both (J) and (K) have to be about exactly the
> same things

No, they don't.

> unless you are using different sorts (and you've said
> that you aren't).

Never have and never shall.

> Martin Shobe
>

I am not sure what exactly you are asking about Strawson pp. 173, 174.

Rupert

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Oct 17, 2015, 2:38:27 AM10/17/15

to

On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
> Rupert wrote:
> > There were some parts of your post that I had a bit of a hard time
> > following. I don't think it is reasonable to take "This sentence is
> > not true" as equivalent with "This true sentence is not true".
>
> The proof of contradiction of the Liar typically starts with the
> assumption that the sentence is true. Liar + this assumption make "This
> true sentence is not true." So it seems to me that at the very least you
> cannot walk through the proof without stepping through "This true
> sentence is not true."

But it is not equivalent to the original sentence.

> > As far
> > as Gödel's sentence goes the point is it is a sentence in the
> > first-order language of arithmetic and so for that reason surely has
> > a truth-value.
>
> "first-order language of arithmetic" usually implies classical logic. In
> the present context I am suggesting that there is an alternative
> logic/interpretation such that it does NOT have a truth value. Are you
> claiming that no such logic/interpretation is possible, or are you just
> injecting an orthogonal statement for fun?

Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.

Nam Nguyen

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Oct 17, 2015, 3:47:53 AM10/17/15

to

On 17/10/2015 12:38 AM, Rupert wrote:
> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
>> Rupert wrote:
>>> There were some parts of your post that I had a bit of a hard time
>>> following. I don't think it is reasonable to take "This sentence is
>>> not true" as equivalent with "This true sentence is not true".
>>
>> The proof of contradiction of the Liar typically starts with the
>> assumption that the sentence is true. Liar + this assumption make "This
>> true sentence is not true." So it seems to me that at the very least you
>> cannot walk through the proof without stepping through "This true
>> sentence is not true."
>
> But it is not equivalent to the original sentence.
>
>>> As far
>>> as Gödel's sentence goes the point is it is a sentence in the
>>> first-order language of arithmetic and so for that reason surely has
>>> a truth-value.

Except surely nobody knows the truth value of G(PA + cGC),
G(PA + ~cGC), ... and infinitely many Gödel's sentences.

>> "first-order language of arithmetic" usually implies classical logic. In
>> the present context I am suggesting that there is an alternative
>> logic/interpretation such that it does NOT have a truth value. Are you
>> claiming that no such logic/interpretation is possible, or are you just
>> injecting an orthogonal statement for fun?
>
> Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.

"Standard semantics" is very much philosophical here.

--

Is there something in your cultural background that
makes it impossible for you to admit to your errors?

Justin Thyme (being racist) -- in sci.logic

Jim Burns

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Oct 17, 2015, 8:55:34 AM10/17/15

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On 10/17/2015 3:47 AM, Nam Nguyen wrote:
> On 17/10/2015 12:38 AM, Rupert wrote:
>> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
>>> Rupert wrote:

>>>> There were some parts of your post that I had a bit of a hard time
>>>> following. I don't think it is reasonable to take "This sentence is
>>>> not true" as equivalent with "This true sentence is not true".
>>>
>>> The proof of contradiction of the Liar typically starts with the
>>> assumption that the sentence is true. Liar + this assumption make "This
>>> true sentence is not true." So it seems to me that at the very least you
>>> cannot walk through the proof without stepping through "This true
>>> sentence is not true."
>>
>> But it is not equivalent to the original sentence.
>>
>>>> As far
>>>> as Gödel's sentence goes the point is it is a sentence in the
>>>> first-order language of arithmetic and so for that reason surely has
>>>> a truth-value.
>
> Except surely nobody knows the truth value of G(PA + cGC),
> G(PA + ~cGC), ... and infinitely many Gödel's sentences.

You seem to think that this is relevant at this point.

And that would seem to indicate you think "nobody knows the
truth value of" opposes "has a truth value" somehow.
Is it true that you think that, Nam?

If you think it does, you should say it loud and clear,
so that everyone knows what you're saying.

>>> "first-order language of arithmetic" usually implies classical logic. In
>>> the present context I am suggesting that there is an alternative
>>> logic/interpretation such that it does NOT have a truth value. Are you
>>> claiming that no such logic/interpretation is possible, or are you just
>>> injecting an orthogonal statement for fun?
>>
>> Of course you can always come up with some nonstandard semantics with
>> respect to which some given sentence does not have a truth-value, but
>> why is that supposed to be interesting? When we are talking about the
>> first-order language of arithmetic then surely we usually have the
>> standard semantics in mind.
>
> "Standard semantics" is very much philosophical here.

Standard semantics is what you don't use when you (NN) interpreted
(Ax)P(x) as "There _exists_ an x such that P(x)".
If the standard semantics is not mathematical/logical, then there is
no mathematics/logic.

Is that your point, Nam? That there is no mathematics/logic?

If you think there isn't, you should say it loud and clear,
so that everyone knows what you're saying.

Martin Shobe

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Oct 17, 2015, 5:58:56 PM10/17/15

to

This should have read, "Unless otherwise specified (and you didn't) it
would be the standard ones.

> Please name the standard integer.

Some of them are called, 0, 1, 2, 3, 4, ....

>>>> Furthermore, the statement is true.
>>
>>> How does it manage to do that?
>>
>> There's no x such that x = x + 1.
>
> So the sentence does not talk about anything.

No. It talks about natural numbers.

>> By the semantics of the ->
>> operator,
>
> So it is all about the semantics of -> and has nothing to do with the
> natural numbers per se.

It has something to do with the natural numbers, every x in the domain
is a natural number.

>> it follows that x=x+1 -> x=x+2 is true regardless of which
>> value of x is used. Therefore, (x)(x=x+1 -> x=x+2) is true.

> How is not talking about anything true? You got me on this one.

Your presupposition is incorrect.

>>>>>>> I guess I am not explaining it too well if you did not
>>>>>>> even realize that is what I was doing. BTW, what did you
>>>>>>> think the paper was about?
>>>>>>
>>>>>> From the contents, you appear to be classifying statements.
>>>>>
>>>>> Suppose this is true. How is it an error?
>>>>
>>>> I didn't say that was an error?
>>
>>> You said "either your first example is wrong, or you are using a
>>> non-standard valuation which you neither warn the reader of nor
>>> explain." So your position is that that my description is not
>>> semantics but rather statement categorization therefore it does not
>>> "explain" the non-standard valuation?
>>
>> Your so doesn't follow from the statement (it does follow from other
>> things I've said.) Now, how are you getting to me saying that
>> classifying statements is an error?
>
> The error is that I thought you said there was an error. In fact there
> is no error.

There is no error is pointing out the error in your paper. The error is
not the fact that you are classifying statements. The error is your
failure to state which non-standard semantics you are using.

(x)(~Unicorn(x)), then yes, ~(Ey)(Unicorn(y)&This(y)) is equivalent to
~(Ex)(Ey)(Prf(x,y) & This(y)). Of course, this is because
~(Ex)(Unicorn(x) is equivalent to ~(Ex)(Prf(x,<#G#>). So once again, we
see that (J) is about unicorns if and only if (k) is about unicorns.

>> But, to put it into the larger
>> picture, if that is sufficient to claim that (J) is about Unicorns,
>> then it should also be sufficient to claim that (K) is about
>> Unicorns.
>
> I do not see any unicorns in K. It says that there is no proof of G.

A more accurate translation would be, "There is no natural number that
encodes a proof of G."

Likewise, I do not see any unicorns in (J). It says that there is no
natural number that both encodes a proof of G and satisfies the
statement This(<#G#>).

>> Once again, both (J) and (K) have to be about exactly the
>> same things
>
> No, they don't.

Yes they do. The semantics of FOL don't allow for any for anything else
(unless you are using multiple sorts).

>> unless you are using different sorts (and you've said
>> that you aren't).
>
> Never have and never shall.
>
>> Martin Shobe
>>
>
> I am not sure what exactly you are asking about Strawson pp. 173, 174.
>

I'm not asking about Stawson pp. 173, 174. I'm asking about the rules
for the existential quantifiers in Strawson's system. They are not
present on those pages.

Martin Shobe

X.Y. Newberry

unread,

Oct 18, 2015, 12:03:34 AM10/18/15

to

Rupert wrote:
> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
>> Rupert wrote:
>>> There were some parts of your post that I had a bit of a hard
>>> time following. I don't think it is reasonable to take "This
>>> sentence is not true" as equivalent with "This true sentence is
>>> not true".
>>
>> The proof of contradiction of the Liar typically starts with the
>> assumption that the sentence is true. Liar + this assumption make
>> "This true sentence is not true." So it seems to me that at the
>> very least you cannot walk through the proof without stepping
>> through "This true sentence is not true."
>
> But it is not equivalent to the original sentence.

This is true [no pun intended], they are not equivalent. "This sentence"
has a referent, "This true sentence" does not. Another interesting
observation is that "This true sentence is not true" is unable to refer
to itself. In Strawson's logic "This true sentence is not true" is ~(T v
F) because the subject term does not have a referent. I think this much
is clear.

I am also suggesting that given
"This sentence is not true" (Y)
then
Y & Y is not true
is equivalent to "This true sentence is not true". Is it an unreasonable
conclusion?

>
>>> As far as Gödel's sentence goes the point is it is a sentence in
>>> the first-order language of arithmetic and so for that reason
>>> surely has a truth-value.
>>
>> "first-order language of arithmetic" usually implies classical
>> logic. In the present context I am suggesting that there is an
>> alternative logic/interpretation such that it does NOT have a truth
>> value. Are you claiming that no such logic/interpretation is
>> possible, or are you just injecting an orthogonal statement for
>> fun?
>
> Of course you can always come up with some nonstandard semantics with
> respect to which some given sentence does not have a truth-value, but
> why is that supposed to be interesting?

It is interesting for a number of reasons

a) The idea that empty sentences are true is implausible - it does not
correspond to the intuitive notion of truth

b) The semantics I am proposing formalizes the notion of
meaninglessness. I.e. sentences that do not have interpretation are
meaningless.

c) Truth and derivability can be equated. It gets rid of the paradoxical
situation that there are true but unprovable sentences.

d) Since a sound derivation system (i.e. system that does not derive
non-truth) will not derive G even if it did derive ~(Ex)Prf(x,<#G#>)
more axiom(s) can be added to Peano Arithmetic without causing a
contradiction.

> When we are talking about the
> first-order language of arithmetic then surely we usually have the
> standard semantics in mind.

Yes, but in this thread we are NOT talking about the standard semantics.
Sounds like you want us back in the box.

X.Y. Newberry

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Oct 18, 2015, 12:55:41 AM10/18/15

to

You really confused me here. So 0, 1, 2, 3, 4 have the property that
1+1=1, 2+1=2, 3+1=3, 4+1=4?

>>>>> Furthermore, the statement is true.
>>>
>>>> How does it manage to do that?
>>>
>>> There's no x such that x = x + 1.
>>
>> So the sentence does not talk about anything.
>
> No. It talks about natural numbers.

Which natural numbers is the sentence talking about?

>>> By the semantics of the -> operator,
>>
>> So it is all about the semantics of -> and has nothing to do with
>> the natural numbers per se.
>
> It has something to do with the natural numbers, every x in the
> domain is a natural number.

I know that A(x) denotes the domain of the natural numbers [which also
includes unicorns]. I am asking what natural numbers is (x)(x=x+1 ->
x=x+2) talking about.

>
>>> it follows that x=x+1 -> x=x+2 is true regardless of which value
>>> of x is used. Therefore, (x)(x=x+1 -> x=x+2) is true.
>
>> How is not talking about anything true? You got me on this one.
>
> Your presupposition is incorrect.

What is the sentence talking about then?

>
>>>>>>>> I guess I am not explaining it too well if you did not
>>>>>>>> even realize that is what I was doing. BTW, what did
>>>>>>>> you think the paper was about?
>>>>>>>
>>>>>>> From the contents, you appear to be classifying
>>>>>>> statements.
>>>>>>
>>>>>> Suppose this is true. How is it an error?
>>>>>
>>>>> I didn't say that was an error?
>>>
>>>> You said "either your first example is wrong, or you are using
>>>> a non-standard valuation which you neither warn the reader of
>>>> nor explain." So your position is that that my description is
>>>> not semantics but rather statement categorization therefore it
>>>> does not "explain" the non-standard valuation?
>>>
>>> Your so doesn't follow from the statement (it does follow from
>>> other things I've said.) Now, how are you getting to me saying
>>> that classifying statements is an error?
>>
>> The error is that I thought you said there was an error. In fact
>> there is no error.
>
> There is no error is pointing out the error in your paper. The error
> is not the fact that you are classifying statements. The error is
> your failure to state which non-standard semantics you are using.

The entire paper is the description of the semantics. Is your position
a) classification of statements does not amount to a definition of semantics
or
b) I failed to state which non standard semantics I am using?

My semantics shows otherwise.

>>> But, to put it into the larger picture, if that is sufficient to
>>> claim that (J) is about Unicorns, then it should also be
>>> sufficient to claim that (K) is about Unicorns.
>>
>> I do not see any unicorns in K. It says that there is no proof of
>> G.
>
> A more accurate translation would be, "There is no natural number
> that encodes a proof of G."
>
> Likewise, I do not see any unicorns in (J). It says that there is no
> natural number that both encodes a proof of G and satisfies the
> statement This(<#G#>).

If there is no natural number that encodes a proof of G then you cannot
attribute anything to it. You cannot attribute something to nothing.

>
>>> Once again, both (J) and (K) have to be about exactly the same
>>> things
>>
>> No, they don't.
>
> Yes they do. The semantics of FOL

But in this thread we are NOT talking about the [standard] semantics of
FOL. Hasn't this been abundantly clear from the beginning? We are
talking about the logic of presuppositions. And (K) is a presupposition
of (J).

> don't allow for any for anything
> else (unless you are using multiple sorts).
>
>>> unless you are using different sorts (and you've said that you
>>> aren't).
>>
>> Never have and never shall.
>>
>>> Martin Shobe
>>>
>>
>> I am not sure what exactly you are asking about Strawson pp. 173,
>> 174.
>>
>
> I'm not asking about Stawson pp. 173, 174. I'm asking about the rules
> for the existential quantifiers in Strawson's system. They are not
> present on those pages.

Of course they are not present. If I am not mistaken Strawson believed
that logic was not formalizable. I am using him only as a source of
inspiration. He nevertheless did show that if Aristotelian logic is
interpreted as below

A: ~(∃x)(Fx & ~Gx) & (∃x)Fx & (∃x)~Gx
E: ~(∃x)(Fx & Gx) & (∃x)Fx & (∃x)Gx
I: (∃x)(Fx & Gx) v ~(∃x)Fx v ~(∃x)Gx
O: (∃x)(Fx & Gx) v ~(∃x)Fx v ~(∃x~)Gx

then all the laws of the traditional syllogism hold together. But he
proposed to take the last two terms, such as '(∃x)Fx & (∃x)Gx', as
PRESUPPOSITIONS. Now in order to formalize this, the concept has to be
extended to arbitrary monadic first order sentences. This is what I have
done in my paper, and it turns out it is just a mirror image of the
truth-relevant logic. Derivation rules are a small technical detail that
will be added in due course.

>
> Martin Shobe

Martin Shobe

unread,

Oct 18, 2015, 11:58:23 AM10/18/15

to

Of course not, but why would you think that they do? The statement under
discussion here certainly doesn't say they do.

>>>>>> Furthermore, the statement is true.
>>>>
>>>>> How does it manage to do that?
>>>>
>>>> There's no x such that x = x + 1.
>>>
>>> So the sentence does not talk about anything.
>>
>> No. It talks about natural numbers.
>
> Which natural numbers is the sentence talking about?

Every last one of them.

>>>> By the semantics of the -> operator,
>>>
>>> So it is all about the semantics of -> and has nothing to do with
>>> the natural numbers per se.
>>
>> It has something to do with the natural numbers, every x in the
>> domain is a natural number.
>
> I know that A(x) denotes the domain of the natural numbers [which also
> includes unicorns]. I am asking what natural numbers is (x)(x=x+1 ->
> x=x+2) talking about.

It's talking about all of them. (And the domain of natural numbers does
not include any unicorns.)

>>>> it follows that x=x+1 -> x=x+2 is true regardless of which value
>>>> of x is used. Therefore, (x)(x=x+1 -> x=x+2) is true.
>>
>>> How is not talking about anything true? You got me on this one.
>>
>> Your presupposition is incorrect.
>
> What is the sentence talking about then?

It says that for every natural number, x, if x is equal to x + 1 then x
is also equal to x + 2.

There are no descriptions of semantics in your paper. None at all.

Describe your semantics please. (And no, your paper doesn't describe any
semantics.)

>>>> But, to put it into the larger picture, if that is sufficient to
>>>> claim that (J) is about Unicorns, then it should also be
>>>> sufficient to claim that (K) is about Unicorns.
>>>
>>> I do not see any unicorns in K. It says that there is no proof of
>>> G.
>>
>> A more accurate translation would be, "There is no natural number
>> that encodes a proof of G."
>>
>> Likewise, I do not see any unicorns in (J). It says that there is no
>> natural number that both encodes a proof of G and satisfies the
>> statement This(<#G#>).
>
> If there is no natural number that encodes a proof of G then you cannot
> attribute anything to it. You cannot attribute something to nothing.

Who is doing such a thing? I certainly haven't in this discussion.

>>>> Once again, both (J) and (K) have to be about exactly the same
>>>> things
>>>
>>> No, they don't.
>>
>> Yes they do. The semantics of FOL
>
> But in this thread we are NOT talking about the [standard] semantics of
> FOL. Hasn't this been abundantly clear from the beginning?

Of course not. I've even explicitly mentioned using FOL. Furthermore,
you have only mentioned using something else (though you never give
details on that something else) when it leads to a result you don't
like. (So much for your claim of not being able to attribute something
to nothing, you've managed to do that repeatedly in this thread.)

> We are
> talking about the logic of presuppositions. And (K) is a presupposition
> of (J).

Not using the usual definition of presupposition. (Or for that matter,
Strawson's).

>> don't allow for any for anything
>> else (unless you are using multiple sorts).
>>
>>>> unless you are using different sorts (and you've said that you
>>>> aren't).
>>>
>>> Never have and never shall.

>>> I am not sure what exactly you are asking about Strawson pp. 173,
>>> 174.
>>>
>>
>> I'm not asking about Stawson pp. 173, 174. I'm asking about the rules
>> for the existential quantifiers in Strawson's system. They are not
>> present on those pages.
>
> Of course they are not present. If I am not mistaken Strawson believed
> that logic was not formalizable.

That is incorrect. What he said was that natural language doesn't have a
precise logic. That's almost, but not quite, totally unlike saying that
logic is not formalizable.

> I am using him only as a source of
> inspiration. He nevertheless did show that if Aristotelian logic is
> interpreted as below
>
> A: ~(∃x)(Fx & ~Gx) & (∃x)Fx & (∃x)~Gx
> E: ~(∃x)(Fx & Gx) & (∃x)Fx & (∃x)Gx
> I: (∃x)(Fx & Gx) v ~(∃x)Fx v ~(∃x)Gx
> O: (∃x)(Fx & Gx) v ~(∃x)Fx v ~(∃x~)Gx
>
> then all the laws of the traditional syllogism hold together. But he
> proposed to take the last two terms, such as '(∃x)Fx & (∃x)Gx', as
> PRESUPPOSITIONS.

Do you even understand the question? Do you know what a rule is in the
context of logic? Repeatedly asserting that Strawson was able to do
something with it indicates that you don't.

> Now in order to formalize this, the concept has to be
> extended to arbitrary monadic first order sentences. This is what I have
> done in my paper, and it turns out it is just a mirror image of the
> truth-relevant logic. Derivation rules are a small technical detail that
> will be added in due course.

You've done nothing of the sort in your paper.

Martin Shobe

X.Y. Newberry

unread,

Oct 18, 2015, 9:17:18 PM10/18/15

to

No, but it talks about those that do.

>>>>>>> Furthermore, the statement is true.
>>>>>
>>>>>> How does it manage to do that?
>>>>>
>>>>> There's no x such that x = x + 1.
>>>>
>>>> So the sentence does not talk about anything.
>>>
>>> No. It talks about natural numbers.
>>
>> Which natural numbers is the sentence talking about?
>
> Every last one of them.
>
>>>>> By the semantics of the -> operator,
>>>>
>>>> So it is all about the semantics of -> and has nothing to do
>>>> with the natural numbers per se.
>>>
>>> It has something to do with the natural numbers, every x in the
>>> domain is a natural number.
>>
>> I know that A(x) denotes the domain of the natural numbers [which
>> also includes unicorns]. I am asking what natural numbers is
>> (x)(x=x+1 -> x=x+2) talking about.
>
> It's talking about all of them. (And the domain of natural numbers
> does not include any unicorns.)

So if you say "All John's children are asleep", are you talking about
John's children or about all the objects in the domain?

If you have a discussion about John's children, another person joins the
conversation, and asks "what are you talking about?" do you answer "we
are talking about all the objects in the domain"?

>>>>> it follows that x=x+1 -> x=x+2 is true regardless of which
>>>>> value of x is used. Therefore, (x)(x=x+1 -> x=x+2) is true.
>>>
>>>> How is not talking about anything true? You got me on this
>>>> one.
>>>
>>> Your presupposition is incorrect.
>>
>> What is the sentence talking about then?
>
> It says that for every natural number, x, if x is equal to x + 1
> then x is also equal to x + 2.

Which numbers have the property x=x+1?

I do not know if you can conceive a semantics that differs from the
"classical" one. Even when you are confronted with one you claim that it
is not semantics but "sentence classification". All you can see is that
it does not conform to the standard one, and you are not able to square
it your head. The thrust of all your arguments in essence is that if it
does not conform to ZFC/classical logic/standard semantics then it must
be wrong.

>>>>> But, to put it into the larger picture, if that is
>>>>> sufficient to claim that (J) is about Unicorns, then it
>>>>> should also be sufficient to claim that (K) is about
>>>>> Unicorns.
>>>>
>>>> I do not see any unicorns in K. It says that there is no proof
>>>> of G.
>>>
>>> A more accurate translation would be, "There is no natural number
>>> that encodes a proof of G."
>>>
>>> Likewise, I do not see any unicorns in (J). It says that there
>>> is no natural number that both encodes a proof of G and
>>> satisfies the statement This(<#G#>).
>>
>> If there is no natural number that encodes a proof of G then you
>> cannot attribute anything to it. You cannot attribute something to
>> nothing.
>
> Who is doing such a thing? I certainly haven't in this discussion.

You have just done it: "(J). It says that there is no natural number

that both encodes a proof of G and satisfies the statement This(<#G#>)"

If John has no children you do not get around it by saying "there is no
such object that it is both a John's child and it is asleep." If there
are no John's children there is nothing to say about them no matter how
you twist or turn.

>>>>> Once again, both (J) and (K) have to be about exactly the
>>>>> same things
>>>>
>>>> No, they don't.
>>>
>>> Yes they do. The semantics of FOL
>>
>> But in this thread we are NOT talking about the [standard]
>> semantics of FOL. Hasn't this been abundantly clear from the
>> beginning?
>

> Of course not. I've even explicitly mentioned using FOL.ic.

> Furthermore, you have only mentioned using something else

OK, so I have "only mentioned" something else. You injected an
orthogonal statement, which trumps it because it was "explicitly
mentioned." This is the best I can make of your logic.

I guess not. No matter how many times I answer it, it is still not what
you want to hear. Obviously I am missing something.

>> Now in order to formalize this, the concept has to be extended to
>> arbitrary monadic first order sentences. This is what I have done
>> in my paper, and it turns out it is just a mirror image of the
>> truth-relevant logic. Derivation rules are a small technical
>> detail that will be added in due course.
>
> You've done nothing of the sort in your paper.

You mean I have not specified semantics, I have merely classified sentences?

Martin Shobe

unread,

Oct 18, 2015, 10:54:40 PM10/18/15

to

Which ones that do? (I wasn't aware that there were any.)

>>>>>> By the semantics of the -> operator,
>>>>>
>>>>> So it is all about the semantics of -> and has nothing to do
>>>>> with the natural numbers per se.
>>>>
>>>> It has something to do with the natural numbers, every x in the
>>>> domain is a natural number.
>>>
>>> I know that A(x) denotes the domain of the natural numbers [which
>>> also includes unicorns]. I am asking what natural numbers is
>>> (x)(x=x+1 -> x=x+2) talking about.
>>
>> It's talking about all of them. (And the domain of natural numbers
>> does not include any unicorns.)
>
> So if you say "All John's children are asleep", are you talking about
> John's children or about all the objects in the domain?

I would say John's children, but that's because we use multiple sorts as
a matter of course in natural languages.

> If you have a discussion about John's children, another person joins the
> conversation, and asks "what are you talking about?" do you answer "we
> are talking about all the objects in the domain"?

No, since it's a natural language. It's issues like this that lead to
Strawson saying that natural languages don't have a precise logic.

>>>>>> it follows that x=x+1 -> x=x+2 is true regardless of which
>>>>>> value of x is used. Therefore, (x)(x=x+1 -> x=x+2) is true.
>>>>
>>>>> How is not talking about anything true? You got me on this
>>>>> one.
>>>>
>>>> Your presupposition is incorrect.
>>>
>>> What is the sentence talking about then?
>>
>> It says that for every natural number, x, if x is equal to x + 1
>> then x is also equal to x + 2.
>
> Which numbers have the property x=x+1?

None, and the statement doesn't say (or presuppose) that there are any.

I can. I've read about several and even used some.

> Even when you are confronted with one you claim that it
> is not semantics but "sentence classification". All you can see is that
> it does not conform to the standard one, and you are not able to square
> it your head. The thrust of all your arguments in essence is that if it
> does not conform to ZFC/classical logic/standard semantics then it must
> be wrong.

This is not the thrust of my argument. It's not even close.

>>>>>> But, to put it into the larger picture, if that is
>>>>>> sufficient to claim that (J) is about Unicorns, then it
>>>>>> should also be sufficient to claim that (K) is about
>>>>>> Unicorns.
>>>>>
>>>>> I do not see any unicorns in K. It says that there is no proof
>>>>> of G.
>>>>
>>>> A more accurate translation would be, "There is no natural number
>>>> that encodes a proof of G."
>>>>
>>>> Likewise, I do not see any unicorns in (J). It says that there
>>>> is no natural number that both encodes a proof of G and
>>>> satisfies the statement This(<#G#>).
>>>
>>> If there is no natural number that encodes a proof of G then you
>>> cannot attribute anything to it. You cannot attribute something to
>>> nothing.
>>
>> Who is doing such a thing? I certainly haven't in this discussion.
>
> You have just done it: "(J). It says that there is no natural number
> that both encodes a proof of G and satisfies the statement This(<#G#>)"

That is not attributing something to natural number that doesn't exist.

> If John has no children you do not get around it by saying "there is no
> such object that it is both a John's child and it is asleep." If there
> are no John's children there is nothing to say about them no matter how
> you twist or turn.

>>>>>> Once again, both (J) and (K) have to be about exactly the
>>>>>> same things
>>>>>
>>>>> No, they don't.
>>>>
>>>> Yes they do. The semantics of FOL
>>>
>>> But in this thread we are NOT talking about the [standard]
>>> semantics of FOL. Hasn't this been abundantly clear from the
>>> beginning?
>>
>> Of course not. I've even explicitly mentioned using FOL.ic.
>> Furthermore, you have only mentioned using something else
>
> OK, so I have "only mentioned" something else. You injected an
> orthogonal statement, which trumps it because it was "explicitly
> mentioned." This is the best I can make of your logic.

Then you need to spend some time to learn the basics of the subject.
It's not even close.

So once again, my advice to you has to be, "go and learn the basics."

>>> Now in order to formalize this, the concept has to be extended to
>>> arbitrary monadic first order sentences. This is what I have done
>>> in my paper, and it turns out it is just a mirror image of the
>>> truth-relevant logic. Derivation rules are a small technical
>>> detail that will be added in due course.
>>
>> You've done nothing of the sort in your paper.
>
> You mean I have not specified semantics, I have merely classified
> sentences?

In that paper, yes.

Martin Shobe

Ross A. Finlayson

unread,

Oct 19, 2015, 12:05:05 AM10/19/15

to

On Sunday, October 18, 2015 at 7:54:40 PM UTC-7, Martin Shobe wrote:
> On 10/18/2015 8:16 PM, X.Y. Newberry wrote:
> >>
> >> It's talking about all of them. (And the domain of natural numbers
> >> does not include any unicorns.)
> >
> > So if you say "All John's children are asleep", are you talking about
> > John's children or about all the objects in the domain?
>
> I would say John's children, but that's because we use multiple sorts as
> a matter of course in natural languages.
>
> > If you have a discussion about John's children, another person joins the
> > conversation, and asks "what are you talking about?" do you answer "we
> > are talking about all the objects in the domain"?
>
> No, since it's a natural language. It's issues like this that lead to
> Strawson saying that natural languages don't have a precise logic.
>

"Natural languages" have as precise a logic
within them as any "symbolic language", it
is just usually that the "elements" include
the imprecise. Symbolic language is contained
within natural language.

If there's no ambiguity in definition there's
no ambiguity in terms.

Some natural words as match quantifiers and
other primitives in the symbolic, can establish
the quantifier separations but only as maintained,
where the quantifiers themselves may reflect
structural properties of the domain(s), whether
it's the impredicative or the collective, for examples.

So, a "precise logic" is yet _within_ natural language,
then for the disambiguation via definition and convention
of the imprecise.

Thank you for reviewing my opinion.

Rupert

unread,

Oct 20, 2015, 4:54:20 AM10/20/15

to

On Saturday, October 17, 2015 at 9:47:53 AM UTC+2, Nam Nguyen wrote:
> On 17/10/2015 12:38 AM, Rupert wrote:
> > On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
> >> Rupert wrote:
> >>> There were some parts of your post that I had a bit of a hard time
> >>> following. I don't think it is reasonable to take "This sentence is
> >>> not true" as equivalent with "This true sentence is not true".
> >>
> >> The proof of contradiction of the Liar typically starts with the
> >> assumption that the sentence is true. Liar + this assumption make "This
> >> true sentence is not true." So it seems to me that at the very least you
> >> cannot walk through the proof without stepping through "This true
> >> sentence is not true."
> >
> > But it is not equivalent to the original sentence.
> >
> >>> As far
> >>> as Gödel's sentence goes the point is it is a sentence in the
> >>> first-order language of arithmetic and so for that reason surely has
> >>> a truth-value.
>
> Except surely nobody knows the truth value of G(PA + cGC),
> G(PA + ~cGC), ... and infinitely many Gödel's sentences.

Those two examples of Gödel sentences you just gave are examples where nobody *currently* knows the truth-value, although you haven't offered any good reason for thinking that there is any deep obstacle to us knowing the truth-value one day.

> >> "first-order language of arithmetic" usually implies classical logic. In
> >> the present context I am suggesting that there is an alternative
> >> logic/interpretation such that it does NOT have a truth value. Are you
> >> claiming that no such logic/interpretation is possible, or are you just
> >> injecting an orthogonal statement for fun?
> >
> > Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.
>
> "Standard semantics" is very much philosophical here.

Well, it has a precise mathematical definition.

Rupert

unread,

Oct 20, 2015, 5:04:04 AM10/20/15

to

The sentence Y lacks a truth-value, right? So how can it be equivalent to anything?

> >>> As far as Gödel's sentence goes the point is it is a sentence in
> >>> the first-order language of arithmetic and so for that reason
> >>> surely has a truth-value.
> >>
> >> "first-order language of arithmetic" usually implies classical
> >> logic. In the present context I am suggesting that there is an
> >> alternative logic/interpretation such that it does NOT have a truth
> >> value. Are you claiming that no such logic/interpretation is
> >> possible, or are you just injecting an orthogonal statement for
> >> fun?
> >
> > Of course you can always come up with some nonstandard semantics with
> > respect to which some given sentence does not have a truth-value, but
> > why is that supposed to be interesting?
>
> It is interesting for a number of reasons
>
> a) The idea that empty sentences are true is implausible - it does not
> correspond to the intuitive notion of truth
>
> b) The semantics I am proposing formalizes the notion of
> meaninglessness. I.e. sentences that do not have interpretation are
> meaningless.
>
> c) Truth and derivability can be equated. It gets rid of the paradoxical
> situation that there are true but unprovable sentences.
>
> d) Since a sound derivation system (i.e. system that does not derive
> non-truth) will not derive G even if it did derive ~(Ex)Prf(x,<#G#>)
> more axiom(s) can be added to Peano Arithmetic without causing a
> contradiction.

Well, that is the case anyway.

The point is, you are introducing a non-standard truth for the first-order language of arithmetic, so you need to say something about why it should be thought better than the standard notion of truth that we have. It needs to be a bit more than that it enables us to do the things you listed IMHO.

> > When we are talking about the
> > first-order language of arithmetic then surely we usually have the
> > standard semantics in mind.
>
> Yes, but in this thread we are NOT talking about the standard semantics.
> Sounds like you want us back in the box.

You would need to spell out in more detail the reasons why I should be interested in your non-standard semantics. You wrote up a treatment of it somewhere, didn't you?

Nam Nguyen

unread,

Oct 20, 2015, 9:53:22 AM10/20/15

to

On 20/10/2015 2:54 AM, Rupert wrote:
> On Saturday, October 17, 2015 at 9:47:53 AM UTC+2, Nam Nguyen wrote:
>> On 17/10/2015 12:38 AM, Rupert wrote:
>>> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
>>>> Rupert wrote:
>>>>> There were some parts of your post that I had a bit of a hard time
>>>>> following. I don't think it is reasonable to take "This sentence is
>>>>> not true" as equivalent with "This true sentence is not true".
>>>>
>>>> The proof of contradiction of the Liar typically starts with the
>>>> assumption that the sentence is true. Liar + this assumption make "This
>>>> true sentence is not true." So it seems to me that at the very least you
>>>> cannot walk through the proof without stepping through "This true
>>>> sentence is not true."
>>>
>>> But it is not equivalent to the original sentence.
>>>
>>>>> As far
>>>>> as Gödel's sentence goes the point is it is a sentence in the
>>>>> first-order language of arithmetic and so for that reason surely has
>>>>> a truth-value.
>>
>> Except surely nobody knows the truth value of G(PA + cGC),
>> G(PA + ~cGC), ... and infinitely many Gödel's sentences.
>
> Those two examples of Gödel sentences you just gave are examples where nobody *currently* knows the truth-value, although you haven't offered any good reason for thinking that there is any deep obstacle to us knowing the truth-value one day.

I did, through undecide(cGC). You're just unable to understand basic
notions used in the offer, such as "meta proof". In fact you weren't
telling sci.logic the truth I had never explained what I'd mean by
"meta proof"!

>>>> "first-order language of arithmetic" usually implies classical logic. In
>>>> the present context I am suggesting that there is an alternative
>>>> logic/interpretation such that it does NOT have a truth value. Are you
>>>> claiming that no such logic/interpretation is possible, or are you just
>>>> injecting an orthogonal statement for fun?
>>>
>>> Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.
>>
>> "Standard semantics" is very much philosophical here.
>
> Well, it has a precise mathematical definition.

Yeah sure, very precisely conveniently so. It's a buzzword-definition
though.

There are those who would precisely defined any number greater than
10^500 to be an infinite number. So what?

unread,

Oct 20, 2015, 10:35:07 AM10/20/15

to

On Tuesday, October 20, 2015 at 3:53:22 PM UTC+2, Nam Nguyen wrote:
> On 20/10/2015 2:54 AM, Rupert wrote:
> > On Saturday, October 17, 2015 at 9:47:53 AM UTC+2, Nam Nguyen wrote:
> >> On 17/10/2015 12:38 AM, Rupert wrote:
> >>> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
> >>>> Rupert wrote:
> >>>>> There were some parts of your post that I had a bit of a hard time
> >>>>> following. I don't think it is reasonable to take "This sentence is
> >>>>> not true" as equivalent with "This true sentence is not true".
> >>>>
> >>>> The proof of contradiction of the Liar typically starts with the
> >>>> assumption that the sentence is true. Liar + this assumption make "This
> >>>> true sentence is not true." So it seems to me that at the very least you
> >>>> cannot walk through the proof without stepping through "This true
> >>>> sentence is not true."
> >>>
> >>> But it is not equivalent to the original sentence.
> >>>
> >>>>> As far
> >>>>> as Gödel's sentence goes the point is it is a sentence in the
> >>>>> first-order language of arithmetic and so for that reason surely has
> >>>>> a truth-value.
> >>
> >> Except surely nobody knows the truth value of G(PA + cGC),
> >> G(PA + ~cGC), ... and infinitely many Gödel's sentences.
> >
> > Those two examples of Gödel sentences you just gave are examples where nobody *currently* knows the truth-value, although you haven't offered any good reason for thinking that there is any deep obstacle to us knowing the truth-value one day.
>
> I did, through undecide(cGC). You're just unable to understand basic
> notions used in the offer, such as "meta proof".

You *think* that's where the problem lies, as opposed to the problem being in the fact that the notions used in your proof are not particularly precise or coherent.

> In fact you weren't
> telling sci.logic the truth I had never explained what I'd mean by
> "meta proof"!

If I recall correctly what I said was that I'd never read any explanation from you that I'd found particular enlightening.

> >>>> "first-order language of arithmetic" usually implies classical logic. In
> >>>> the present context I am suggesting that there is an alternative
> >>>> logic/interpretation such that it does NOT have a truth value. Are you
> >>>> claiming that no such logic/interpretation is possible, or are you just
> >>>> injecting an orthogonal statement for fun?
> >>>
> >>> Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.
> >>
> >> "Standard semantics" is very much philosophical here.
> >
> > Well, it has a precise mathematical definition.
>
> Yeah sure, very precisely conveniently so. It's a buzzword-definition
> though.
>
> There are those who would precisely defined any number greater than
> 10^500 to be an infinite number. So what?

So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.

Nam Nguyen

unread,

Oct 20, 2015, 12:00:45 PM10/20/15

to

I know, not just think.

> as opposed to the problem being in the fact that the notions used in your proof are not particularly precise or coherent.

"Precise or coherent" in precisely what way? Do you even have ability
to muster a disproof or counter example to these basic notions? I mean
the cranks, the trolls, the inquisitors, etc... can always whine,
creating smoke-screens like "the notions used in your proof are not
particularly precise or coherent" against their opponent's argument!

>
>> In fact you weren't
>> telling sci.logic the truth I had never explained what I'd mean by
>> "meta proof"!
>
> If I recall correctly what I said was that I'd never read any explanation from you that I'd found particular enlightening.

I took it back: you did have "which I found enlightening" then.
But again, that you not being able to understand my _simple_
definition of "meta proof" isn't my problem and is a clear indication
you're unable to comprehend my meta proof of undecide(cGC), and
everything else related to such as MR, etc..., whatever you wish to
say as a smokescreen.

>>>>>> "first-order language of arithmetic" usually implies classical logic. In
>>>>>> the present context I am suggesting that there is an alternative
>>>>>> logic/interpretation such that it does NOT have a truth value. Are you
>>>>>> claiming that no such logic/interpretation is possible, or are you just
>>>>>> injecting an orthogonal statement for fun?
>>>>>
>>>>> Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.
>>>>
>>>> "Standard semantics" is very much philosophical here.
>>>
>>> Well, it has a precise mathematical definition.
>>
>> Yeah sure, very precisely conveniently so. It's a buzzword-definition
>> though.
>>
>> There are those who would precisely defined any number greater than
>> 10^500 to be an infinite number. So what?
>
> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.

(In sci.logic, I don't speak to them, any more than you do!)

No logical differences: both (your side and those people) would ignore
standard definitions and permissible rules of reasoning in meta level,
and both _believe_ they are right, while actually being wrong.

The insignificant difference is non-technical: one of them of higher
social or certificate ranking so the ignorance (or even idiocy) is not
supposed to be visible.

Justin Thyme

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Oct 20, 2015, 12:12:55 PM10/20/15

to

Nam Nguyen wrote:

> But again, that you not being able to understand my _simple_
> definition of "meta proof" isn't my problem and is a clear indication
> you're unable to comprehend my meta proof of undecide(cGC), and
> everything else related to such as MR, etc..., whatever you wish to
> say as a smokescreen.

You have never defined "meta proof". At best you say that there are
certain statements such a proof begins with and certain rules for
deriving statements from other statements. But you have never listed
all the initial statements nor all the rules. How, then, can anyone
tell if a purported meta proof is valid or not?

Rupert

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Oct 20, 2015, 12:17:27 PM10/20/15

to

Well, you think you know.

> > as opposed to the problem being in the fact that the notions used in your proof are not particularly precise or coherent.
>
> "Precise or coherent" in precisely what way?

In the way normally expected of well-defined mathematical concepts.

> Do you even have ability
> to muster a disproof or counter example to these basic notions?

You have to have a well-defined mathematical statement before I can give a counter-example.

> I mean
> the cranks, the trolls, the inquisitors, etc... can always whine,
> creating smoke-screens like "the notions used in your proof are not
> particularly precise or coherent" against their opponent's argument!

Yes, and sometimes they can be correct.

> >
> >> In fact you weren't
> >> telling sci.logic the truth I had never explained what I'd mean by
> >> "meta proof"!
> >
> > If I recall correctly what I said was that I'd never read any explanation from you that I'd found particular enlightening.
>
> I took it back: you did have "which I found enlightening" then.
> But again, that you not being able to understand my _simple_
> definition of "meta proof" isn't my problem and is a clear indication
> you're unable to comprehend my meta proof of undecide(cGC), and
> everything else related to such as MR, etc..., whatever you wish to
> say as a smokescreen.

As I say, you *think* the problem lies with some kind of failure on comprehension on my part, as opposed to your inability to formulate and present a cogent mathematical argument.

> >>>>>> "first-order language of arithmetic" usually implies classical logic. In
> >>>>>> the present context I am suggesting that there is an alternative
> >>>>>> logic/interpretation such that it does NOT have a truth value. Are you
> >>>>>> claiming that no such logic/interpretation is possible, or are you just
> >>>>>> injecting an orthogonal statement for fun?
> >>>>>
> >>>>> Of course you can always come up with some nonstandard semantics with respect to which some given sentence does not have a truth-value, but why is that supposed to be interesting? When we are talking about the first-order language of arithmetic then surely we usually have the standard semantics in mind.
> >>>>
> >>>> "Standard semantics" is very much philosophical here.
> >>>
> >>> Well, it has a precise mathematical definition.
> >>
> >> Yeah sure, very precisely conveniently so. It's a buzzword-definition
> >> though.
> >>
> >> There are those who would precisely defined any number greater than
> >> 10^500 to be an infinite number. So what?
> >
> > So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
>
> (In sci.logic, I don't speak to them, any more than you do!)
>
> No logical differences: both (your side and those people) would ignore
> standard definitions and permissible rules of reasoning in meta level,
> and both _believe_ they are right, while actually being wrong.

No, I'm afraid you're mistaken about that.

Nam Nguyen

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Oct 20, 2015, 12:22:07 PM10/20/15

to

That's what you *think*.

>>>> There are those who would precisely defined any number greater than
>>>> 10^500 to be an infinite number. So what?
>>>
>>> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
>>
>> (In sci.logic, I don't speak to them, any more than you do!)
>>
>> No logical differences: both (your side and those people) would ignore
>> standard definitions and permissible rules of reasoning in meta level,
>> and both _believe_ they are right, while actually being wrong.
>
> No, I'm afraid you're mistaken about that.

I'm afraid you're mistaken about that, Rupert.

Nam Nguyen

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Oct 20, 2015, 12:27:36 PM10/20/15

to

On 20/10/2015 10:12 AM, Justin Thyme wrote:
> Nam Nguyen wrote:
>
>> But again, that you not being able to understand my _simple_
>> definition of "meta proof" isn't my problem and is a clear indication
>> you're unable to comprehend my meta proof of undecide(cGC), and
>> everything else related to such as MR, etc..., whatever you wish to
>> say as a smokescreen.
>
> You have never defined "meta proof".

Liar.

> At best you say that there are
> certain statements such a proof begins with and certain rules for
> deriving statements from other statements. But you have never listed
> all the initial statements nor all the rules. How, then, can anyone
> tell if a purported meta proof is valid or not?

Easy: just listing enough finitely many initial statements and all the
rules *required* (and if any one of them is in doubt it can be
addressed individually).

Presenting a meta proof isn't the same as writing an education book
teaching about Meta-mathematical logic!

Ross A. Finlayson

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Oct 20, 2015, 12:31:28 PM10/20/15

to

On Tuesday, October 20, 2015 at 9:00:45 AM UTC-7, Nam Nguyen wrote:
>
> I know, not just think.
>

You're hallucinating!

Rupert

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Oct 20, 2015, 12:33:15 PM10/20/15

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Do you have any thoughts on why I would fail to understand if you were giving me a satisfactory presentation of a cogent mathematical argument? Do you just think I'm mathematically incompetent, do you?

> >>>> There are those who would precisely defined any number greater than
> >>>> 10^500 to be an infinite number. So what?
> >>>
> >>> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
> >>
> >> (In sci.logic, I don't speak to them, any more than you do!)
> >>
> >> No logical differences: both (your side and those people) would ignore
> >> standard definitions and permissible rules of reasoning in meta level,
> >> and both _believe_ they are right, while actually being wrong.
> >
> > No, I'm afraid you're mistaken about that.
>
> I'm afraid you're mistaken about that, Rupert.

So which standard definitions are we ignoring, then?

Nam Nguyen

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Oct 20, 2015, 12:46:56 PM10/20/15

to

Yes I do, and I said that many times already. You're incapable of
understanding that my presentation isn't FOL formal system theorem/proof
work: I've informed you this is meta reasoning about the FOL
reasoning framework _in meta level_ .

How many more time do you need to be reminded of these premisses into
the presentation?

> Do you just think I'm mathematically incompetent, do you?

You're either incompetent or of the mindset of an inquisitor when it
comes to making arguments about foundation/logic of mathematics.

>>>>>> There are those who would precisely defined any number greater than
>>>>>> 10^500 to be an infinite number. So what?
>>>>>
>>>>> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
>>>>
>>>> (In sci.logic, I don't speak to them, any more than you do!)
>>>>
>>>> No logical differences: both (your side and those people) would ignore
>>>> standard definitions and permissible rules of reasoning in meta level,
>>>> and both _believe_ they are right, while actually being wrong.
>>>
>>> No, I'm afraid you're mistaken about that.
>>
>> I'm afraid you're mistaken about that, Rupert.
>
> So which standard definitions are we ignoring, then?

The ones that when used will see, e.g., Completeness as invalid,
undecide(cGC), etc...

For sure you're unable to understand that.

Nam Nguyen

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Oct 20, 2015, 12:47:33 PM10/20/15

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Idiotic ranting of a crank.

Justin Thyme

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Oct 20, 2015, 12:50:04 PM10/20/15

to

Nam Nguyen wrote:
> On 20/10/2015 10:12 AM, Justin Thyme wrote:
>> Nam Nguyen wrote:
>>
>>> But again, that you not being able to understand my _simple_
>>> definition of "meta proof" isn't my problem and is a clear indication
>>> you're unable to comprehend my meta proof of undecide(cGC), and
>>> everything else related to such as MR, etc..., whatever you wish to
>>> say as a smokescreen.
>>
>> You have never defined "meta proof".
>
> Liar.
>
>> At best you say that there are
>> certain statements such a proof begins with and certain rules for
>> deriving statements from other statements. But you have never listed
>> all the initial statements nor all the rules. How, then, can anyone
>> tell if a purported meta proof is valid or not?
>
> Easy: just listing enough finitely many initial statements and all the
> rules *required* (and if any one of them is in doubt it can be
> addressed individually).

Ok, so why not get back to the logic/mathematics by proving nK(cGC) and
then following the proof by a list of the initial statements and
required rules used in that proof? And then if anyone has doubts about
any statement or rule you can address it.

Writing down the proof might be best done off-line and then you could
start a new thread by posting the finished whole. Take a few days over
it, get it nice and clear before you post it. How about that?

> Presenting a meta proof isn't the same as writing an education book
> teaching about Meta-mathematical logic!
>
>

--

Nam Nguyen

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Oct 20, 2015, 12:54:50 PM10/20/15

to

All that for the trolls and the incompetent, who are capable of
comprehending what a meta proof is - *despite clear cut examples*?

Nah.

>
>> Presenting a meta proof isn't the same as writing an education book
>> teaching about Meta-mathematical logic!

--

Justin Thyme

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Oct 20, 2015, 12:58:37 PM10/20/15

to

Nam Nguyen wrote:

> On 20/10/2015 10:33 AM, Rupert wrote:

>> Do you have any thoughts on why I would fail to understand if you were
>> giving me a satisfactory presentation of a cogent mathematical argument?
>
> Yes I do, and I said that many times already. You're incapable of
> understanding that my presentation isn't FOL formal system theorem/proof

That can't be it. Almost all of mathematics isn't proofs in first order
logic.

> work: I've informed you this is meta reasoning about the FOL
> reasoning framework _in meta level_ .

Any text on first order logic and first order theories (like
Shoenfield's /Mathematical logic/, for example) is almost all on the
meta level: such books prove things *about* theories rather more than
the prove things *in* theories.

So you seem to be saying that Rupert hasn't read (or, at least, hasn't
read with understanding) a single maths book or logic book. Is that
what you think?

Justin Thyme

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Oct 20, 2015, 1:00:39 PM10/20/15

to

Surely the best example of all would be the meta proof of nK(cGC)? What
is your journal submission going to consist of, if not the meta proof of
nK(cGC)?

> Nah.
>
>>
>>> Presenting a meta proof isn't the same as writing an education book
>>> teaching about Meta-mathematical logic!
>
>
>

--

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Oct 20, 2015, 1:16:15 PM10/20/15

to

Good. Any thought about when it will be ready?

Nam Nguyen

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Oct 20, 2015, 1:17:52 PM10/20/15

to

On 20/10/2015 11:15 AM, Nam Nguyen wrote:
> On 20/10/2015 11:10 AM, Nam Nguyen wrote:
>> On 20/10/2015 10:58 AM, Justin Thyme wrote:
>>
>>> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
>>> read with understanding) a single maths book or logic book. Is that
>>> what you think?
>>
>> Of course not. I think Rupert is capable of understanding 2+2=4 being
>> true in the natural numbers. It's just Rupert is incapable of
>> understanding it's unknowable - impossible to know - the truth value
>> of the sentence cGC, or of the Goldbach conjecture (if the conjecture
>> happens to be true).
>
> I wonder outloud if Rupert himself has ever done an actual PhD research
> into the subject of something that's (logically) unknowable - impossible
> to know?

If he has, how did he go around proving such a meta statement?

If he hasn't, well ... that would explain a lot of things here.

unread,

Oct 20, 2015, 1:36:15 PM10/20/15

to

On 10/20/2015 12:27 PM, Nam Nguyen wrote:
> On 20/10/2015 10:12 AM, Justin Thyme wrote:

>> At best you say that there are
>> certain statements such a proof begins with and certain rules for
>> deriving statements from other statements. But you have never listed
>> all the initial statements nor all the rules. How, then, can anyone
>> tell if a purported meta proof is valid or not?
>
> Easy: just listing enough finitely many initial statements and all the
> rules *required* (and if any one of them is in doubt it can be
> addressed individually).

Rules? Do you mean the rules about which you said
"Caveat: NN won't list all the valid rules of derivation
in _meta level_ in the thread: the listing is only on
the as-needed or case-by-case basis." [1]

You say your "explanation" of your "meta proof" is FINITE.
All-righty, then: End it.

Say what your intrinsic statements and your rules of derivation
are, and then say "NO MORE".

> Presenting a meta proof isn't the same as writing an education book
> teaching about Meta-mathematical logic!

Presenting what you mean by "meta proof" is the exact equivalent
to writing an education book about meta-mathematical logic.

[1]
Subject: Re: A few difficult-to-impossible mathematical problems
From: Nam Nguyen <namduc...@shaw.ca>
Message-ID: <Z6LLx.794$8d3...@fx08.iad>
Date: Sun, 20 Sep 2015 21:49:10 -0600

Justin Thyme

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Oct 20, 2015, 1:39:14 PM10/20/15

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Nam Nguyen wrote:
> On 20/10/2015 11:20 AM, Justin Thyme wrote:
>> Nam Nguyen wrote:
>>> On 20/10/2015 10:58 AM, Justin Thyme wrote:
>>>
>>>> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
>>>> read with understanding) a single maths book or logic book. Is that
>>>> what you think?
>>>
>>> Of course not. I think Rupert is capable of understanding 2+2=4 being
>>> true in the natural numbers. It's just Rupert is incapable of
>>> understanding it's unknowable - impossible to know - the truth value
>>> of the sentence cGC, or of the Goldbach conjecture (if the conjecture
>>> happens to be true).
>>
>> That's hardly his fault - you have given us no treason to think that the
>> truth values of the sentence cGC, or of the Goldbach conjecture, are
>> impossible to know. It's that word "impossible" that seems to be the
>> problem: the truth value of the Goldbach conjecture is not known, fine,
>> but impossible to know? Why impossible?
>
> I think I'm talking to a walking deaf. Because it's not possible to
> verify or prove in meta level using any available and valid means of
> the reasoning framework!

How do we know that if you don't tell us what the available and valid
means of the reasoning framework are?

Suppose that it is announced that the Goldbach conjecture is true. More
specifically, some mathematician has submitted a proof to a journal and
that journal's reviewers accept that the proof is valid. How will you
respond? Will you say that the proof (or alleged proof if you prefer)
is invalid without even reading it?

> Would this be a very difficult notion for a PhD to comprehend?
>

--

Justin Thyme

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Oct 20, 2015, 1:40:25 PM10/20/15

to

Nam Nguyen wrote:
> On 20/10/2015 11:22 AM, Justin Thyme wrote:
>> Nam Nguyen wrote:
>>
>>> I wonder outloud if Rupert himself has ever done an actual PhD research
>>> into the subject of something that's (logically) unknowable - impossible
>>> to know?
>>
>> Rarely do people do two PhDs.
>
> Talking about a troll who couldn't take "PhD research" to mean
> PhD-level research.

I had supposed that "PhD research" meant research done towards a PhD.
My apologies.

Nam Nguyen

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Oct 20, 2015, 1:43:31 PM10/20/15

to

On 20/10/2015 11:36 AM, Jim Burns wrote:
> On 10/20/2015 12:27 PM, Nam Nguyen wrote:
>> On 20/10/2015 10:12 AM, Justin Thyme wrote:
>
>>> At best you say that there are
>>> certain statements such a proof begins with and certain rules for
>>> deriving statements from other statements. But you have never listed
>>> all the initial statements nor all the rules. How, then, can anyone
>>> tell if a purported meta proof is valid or not?
>>
>> Easy: just listing enough finitely many initial statements and all the
>> rules *required* (and if any one of them is in doubt it can be
>> addressed individually).
>
> Rules? Do you mean the rules about which you said
> "Caveat: NN won't list all the valid rules of derivation
> in _meta level_ in the thread: the listing is only on
> the as-needed or case-by-case basis." [1]
>
> You say your "explanation" of your "meta proof" is FINITE.
> All-righty, then: End it.
>
> Say what your intrinsic statements and your rules of derivation
> are, and then say "NO MORE".
>
>> Presenting a meta proof isn't the same as writing an education book
>> teaching about Meta-mathematical logic!
>
> Presenting what you mean by "meta proof" is the exact equivalent
> to writing an education book about meta-mathematical logic.

Idiotic. When JB wrote down a proof for his daily physics quiz,
did he write an "education book" about physics?

Nam Nguyen

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Oct 20, 2015, 1:53:25 PM10/20/15

to

On 20/10/2015 11:39 AM, Justin Thyme wrote:
> Nam Nguyen wrote:
>> On 20/10/2015 11:20 AM, Justin Thyme wrote:
>>> Nam Nguyen wrote:
>>>> On 20/10/2015 10:58 AM, Justin Thyme wrote:
>>>>
>>>>> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
>>>>> read with understanding) a single maths book or logic book. Is that
>>>>> what you think?
>>>>
>>>> Of course not. I think Rupert is capable of understanding 2+2=4 being
>>>> true in the natural numbers. It's just Rupert is incapable of
>>>> understanding it's unknowable - impossible to know - the truth value
>>>> of the sentence cGC, or of the Goldbach conjecture (if the conjecture
>>>> happens to be true).
>>>
>>> That's hardly his fault - you have given us no treason to think that the
>>> truth values of the sentence cGC, or of the Goldbach conjecture, are
>>> impossible to know. It's that word "impossible" that seems to be the
>>> problem: the truth value of the Goldbach conjecture is not known, fine,
>>> but impossible to know? Why impossible?
>>
>> I think I'm talking to a walking deaf. Because it's not possible to
>> verify or prove in meta level using any available and valid means of
>> the reasoning framework!
>
> How do we know that if you don't tell us what the available and valid
> means of the reasoning framework are?

Does JT have to know all the infinitely many induction axioms to prove,
say, 2+2=4 in PA?

>
> Suppose that it is announced that the Goldbach conjecture is true. More
> specifically, some mathematician has submitted a proof to a journal and
> that journal's reviewers accept that the proof is valid. How will you
> respond?

I'll say the both (the mathematician and the journal) are wrong.

> Will you say that the proof (or alleged proof if you prefer)
> is invalid without even reading it?

Yes of course: because I have a meta proof of undecide(cGC) which
include a counter proof the the alleged above. This where I my argument
style is different from the PhD Rupert: I'd present a counter proof,
instead of just saying something like "You haven't convinced me, because
I say so and I have a PhD but you only a Bachelor one"!

Suppose JT claims 2+2=5 being true in the naturals, do I have to
read his "proof".

Justin Thyme

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Oct 20, 2015, 1:57:13 PM10/20/15

to

No, no induction axioms are needed to prove 2+2=4 in PA.

Jim Burns

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X.Y. Newberry

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Oct 21, 2015, 12:40:03 AM10/21/15

to

Rupert wrote:
> On Sunday, October 18, 2015 at 6:03:34 AM UTC+2, Newberry wrote:
>> Rupert wrote:
>>> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
>>>> Rupert wrote:
>>>>> There were some parts of your post that I had a bit of a
>>>>> hard time following. I don't think it is reasonable to take
>>>>> "This sentence is not true" as equivalent with "This true
>>>>> sentence is not true".
>>>>
>>>> The proof of contradiction of the Liar typically starts with
>>>> the assumption that the sentence is true. Liar + this
>>>> assumption make "This true sentence is not true." So it seems
>>>> to me that at the very least you cannot walk through the proof
>>>> without stepping through "This true sentence is not true."
>>>
>>> But it is not equivalent to the original sentence.
>>
>> This is true [no pun intended], they are not equivalent. "This
>> sentence" has a referent, "This true sentence" does not. Another
>> interesting observation is that "This true sentence is not true" is
>> unable to refer to itself. In Strawson's logic "This true sentence
>> is not true" is ~(T v F) because the subject term does not have a
>> referent. I think this much is clear.
>>
>> I am also suggesting that given "This sentence is not true"
>> (Y) then Y & Y is not true is equivalent to "This true sentence is
>> not true". Is it an unreasonable conclusion?
>
> The sentence Y lacks a truth-value, right? So how can it be
> equivalent to anything?

There are number of ways sentences can be equivalent to other sentences.
For example Y is not merely ~(T v F) it is also meaningless. (That
meaningless sentences are a proper subset of ~(T v F) sentences is my
theory.) A meaningless sentence is indeed not equivalent to anything
except perhaps to all other meaningless sentences. In this sense Y and
"This true sentences is not true" are equivalent.

You can also compare sentences component by component. In this sense the
two sentences above are not equivalent. It is possible for all
components to have referents yet the entire sentence can be meaningless.
About this please see http://www.columbia.edu/~hg17/gaifman6.pdf
(Gaifman of course does not use the term "meaningless".)

Sentences that are ~(T v F) but meaningful can be equivalent by their
meaning. For example "All John's children are asleep" and "All the
individuals whose father is John are asleep."

Hope this is sufficiently confusing.

>>>>> As far as Gödel's sentence goes the point is it is a sentence
>>>>> in the first-order language of arithmetic and so for that
>>>>> reason surely has a truth-value.
>>>>
>>>> "first-order language of arithmetic" usually implies classical
>>>> logic. In the present context I am suggesting that there is an
>>>> alternative logic/interpretation such that it does NOT have a
>>>> truth value. Are you claiming that no such logic/interpretation
>>>> is possible, or are you just injecting an orthogonal statement
>>>> for fun?
>>>
>>> Of course you can always come up with some nonstandard semantics
>>> with respect to which some given sentence does not have a
>>> truth-value, but why is that supposed to be interesting?
>>
>> It is interesting for a number of reasons
>>
>> a) The idea that empty sentences are true is implausible - it does
>> not correspond to the intuitive notion of truth
>>
>> b) The semantics I am proposing formalizes the notion of
>> meaninglessness. I.e. sentences that do not have interpretation
>> are meaningless.
>>
>> c) Truth and derivability can be equated. It gets rid of the
>> paradoxical situation that there are true but unprovable
>> sentences.
>>
>> d) Since a sound derivation system (i.e. system that does not
>> derive non-truth) will not derive G even if it did derive
>> ~(Ex)Prf(x,<#G#>) more axiom(s) can be added to Peano Arithmetic
>> without causing a contradiction.
>
> Well, that is the case anyway.

No, it is not. I meant that ~(Ex)Prf(x,<#G#>) of T will be provable in T.

> The point is, you are introducing a non-standard truth for the
> first-order language of arithmetic, so you need to say something
> about why it should be thought better than the standard notion of
> truth that we have. It needs to be a bit more than that it enables us
> to do the things you listed IMHO.

I thought the things I listed were interesting enough.

>>> When we are talking about the first-order language of arithmetic
>>> then surely we usually have the standard semantics in mind.
>>
>> Yes, but in this thread we are NOT talking about the standard
>> semantics. Sounds like you want us back in the box.
>
> You would need to spell out in more detail the reasons why I should
> be interested in your non-standard semantics. You wrote up a
> treatment of it somewhere, didn't you?

http://arxiv.org/abs/1509.06837

--
X.Y. Newberry

If Jack says ‘What I am saying at this very moment is not true’, we can
successfully and truly assert that he did not utter a truth: ‘What Jack
said is not true’. But it is hardly conceivable that Jack’s utterance is
true by virtue of its success in attributing non-truth to itself.

Haim Gaifman

---
This email has been checked for viruses by Avast antivirus software.
https://www.avast.com/antivirus

Rupert

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Oct 21, 2015, 2:55:50 AM10/21/15

to

On Tuesday, October 20, 2015 at 6:46:56 PM UTC+2, Nam Nguyen wrote:
> On 20/10/2015 10:33 AM, Rupert wrote:
> > On Tuesday, October 20, 2015 at 6:22:07 PM UTC+2, Nam Nguyen wrote:
> >> On 20/10/2015 10:17 AM, Rupert wrote:
> >>> On Tuesday, October 20, 2015 at 6:00:45 PM UTC+2, Nam Nguyen wrote:
> >>>> On 20/10/2015 8:35 AM, Rupert wrote:
> >>>>> On Tuesday, October 20, 2015 at 3:53:22 PM UTC+2, Nam Nguyen wrote:
> >>>>>> On 20/10/2015 2:54 AM, Rupert wrote:
> >>>>>>> On Saturday, October 17, 2015 at 9:47:53 AM UTC+2, Nam Nguyen wrote:

> >>>>>>>> On 17/10/2015 12:38 AM, Rupert wrote:
> >>>>>>>>> On Friday, October 16, 2015 at 3:41:48 AM UTC+2, Newberry wrote:
> >>>>>>>>>> Rupert wrote:
> >>>>>>>>>>> There were some parts of your post that I had a bit of a hard time
> >>>>>>>>>>> following. I don't think it is reasonable to take "This sentence is
> >>>>>>>>>>> not true" as equivalent with "This true sentence is not true".
> >>>>>>>>>>
> >>>>>>>>>> The proof of contradiction of the Liar typically starts with the
> >>>>>>>>>> assumption that the sentence is true. Liar + this assumption make "This
> >>>>>>>>>> true sentence is not true." So it seems to me that at the very least you
> >>>>>>>>>> cannot walk through the proof without stepping through "This true
> >>>>>>>>>> sentence is not true."
> >>>>>>>>>
> >>>>>>>>> But it is not equivalent to the original sentence.
> >>>>>>>>>

> >>>>>>>>>>> As far
> >>>>>>>>>>> as Gödel's sentence goes the point is it is a sentence in the
> >>>>>>>>>>> first-order language of arithmetic and so for that reason surely has
> >>>>>>>>>>> a truth-value.
> >>>>>>>>

> >>>>>>>> Except surely nobody knows the truth value of G(PA + cGC),
> >>>>>>>> G(PA + ~cGC), ... and infinitely many Gödel's sentences.
> >>>>>>>
> >>>>>>> Those two examples of Gödel sentences you just gave are examples where nobody *currently* knows the truth-value, although you haven't offered any good reason for thinking that there is any deep obstacle to us knowing the truth-value one day.
> >>>>>>
> >>>>>> I did, through undecide(cGC). You're just unable to understand basic
> >>>>>> notions used in the offer, such as "meta proof".
> >>>>>
> >>>>> You *think* that's where the problem lies,

> >>>>
> >>>> I know, not just think.
> >>>

> >>> Well, you think you know.
> >>>
> >>>>> as opposed to the problem being in the fact that the notions used in your proof are not particularly precise or coherent.
> >>>>
> >>>> "Precise or coherent" in precisely what way?
> >>>
> >>> In the way normally expected of well-defined mathematical concepts.
> >>>
> >>>> Do you even have ability
> >>>> to muster a disproof or counter example to these basic notions?
> >>>
> >>> You have to have a well-defined mathematical statement before I can give a counter-example.
> >>>
> >>>> I mean
> >>>> the cranks, the trolls, the inquisitors, etc... can always whine,
> >>>> creating smoke-screens like "the notions used in your proof are not
> >>>> particularly precise or coherent" against their opponent's argument!
> >>>
> >>> Yes, and sometimes they can be correct.
> >>>
> >>>>>
> >>>>>> In fact you weren't
> >>>>>> telling sci.logic the truth I had never explained what I'd mean by
> >>>>>> "meta proof"!
> >>>>>
> >>>>> If I recall correctly what I said was that I'd never read any explanation from you that I'd found particular enlightening.
> >>>>
> >>>> I took it back: you did have "which I found enlightening" then.
> >>>> But again, that you not being able to understand my _simple_
> >>>> definition of "meta proof" isn't my problem and is a clear indication
> >>>> you're unable to comprehend my meta proof of undecide(cGC), and
> >>>> everything else related to such as MR, etc..., whatever you wish to
> >>>> say as a smokescreen.
> >>>
> >>> As I say, you *think* the problem lies with some kind of failure on comprehension on my part, as opposed to your inability to formulate and present a cogent mathematical argument.
> >>
> >> That's what you *think*.
> >
> > Do you have any thoughts on why I would fail to understand if you were giving me a satisfactory presentation of a cogent mathematical argument?
>
> Yes I do, and I said that many times already. You're incapable of
> understanding that my presentation isn't FOL formal system theorem/proof
> work: I've informed you this is meta reasoning about the FOL
> reasoning framework _in meta level_ .

Where do you get this idea of yours that a meta-proof cannot be formalised in a FOL formal system? I have read many meta-proofs and have never found an example of one which cannot be formalised in a FOL formal system.

> How many more time do you need to be reminded of these premisses into
> the presentation?

Oh, I've heard it plenty of times. As far as I'm concerned, saying that your work can't be formalised in FOL is pretty much tantamount to admitting that you're not doing mathematics.

> > Do you just think I'm mathematically incompetent, do you?
>
> You're either incompetent or of the mindset of an inquisitor when it
> comes to making arguments about foundation/logic of mathematics.
>
> >>>>>> There are those who would precisely defined any number greater than
> >>>>>> 10^500 to be an infinite number. So what?
> >>>>>
> >>>>> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
> >>>>
> >>>> (In sci.logic, I don't speak to them, any more than you do!)
> >>>>
> >>>> No logical differences: both (your side and those people) would ignore
> >>>> standard definitions and permissible rules of reasoning in meta level,
> >>>> and both _believe_ they are right, while actually being wrong.
> >>>
> >>> No, I'm afraid you're mistaken about that.
> >>
> >> I'm afraid you're mistaken about that, Rupert.
> >
> > So which standard definitions are we ignoring, then?
>
> The ones that when used will see, e.g., Completeness as invalid,
> undecide(cGC), etc...

And which are those?

> For sure you're unable to understand that.

Rupert

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Oct 21, 2015, 2:59:07 AM10/21/15

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On Tuesday, October 20, 2015 at 7:15:46 PM UTC+2, Nam Nguyen wrote:

> On 20/10/2015 11:10 AM, Nam Nguyen wrote:
> > On 20/10/2015 10:58 AM, Justin Thyme wrote:
> >
> >> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
> >> read with understanding) a single maths book or logic book. Is that
> >> what you think?
> >
> > Of course not. I think Rupert is capable of understanding 2+2=4 being
> > true in the natural numbers. It's just Rupert is incapable of
> > understanding it's unknowable - impossible to know - the truth value
> > of the sentence cGC, or of the Goldbach conjecture (if the conjecture
> > happens to be true).
>

> I wonder outloud if Rupert himself has ever done an actual PhD research
> into the subject of something that's (logically) unknowable - impossible
> to know?

My own research has not been in that topic but I was very interested in Peter Koellner's essay "On the Question of Absolute Undecidability", and my work on large cardinals has a bearing on Peter Koellner's claim that the statement "Every real number is constructible" is indeed an example of an absolutely undecidable statement relative to the limited view that Gödel held at the time when he formulated it as an example.

Nam Nguyen

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Oct 21, 2015, 3:23:24 AM10/21/15

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Because you got mixed up, confused between two different edifices or
paradigms of mathematical reasoning: truth and FOL formal system
provability (proof) aren't the same. The meta assertion that there
being infinitely many counter examples of Goldbach Conjecture can be
formalized and provable in the FOL formal system PA + {cGC}, but so
what?

Besides, why don't you formalize undecide( Natural-numbers |= cGC) so
that you can show to sci.logic you'd know what you're talking about
(which you don't)?

> I have read many meta-proofs and have never found an example of one which cannot be formalised in a FOL formal system.
>
>> How many more time do you need to be reminded of these premisses into
>> the presentation?
>
> Oh, I've heard it plenty of times. As far as I'm concerned, saying that your work can't be formalised in FOL is pretty much tantamount to admitting that you're not doing mathematics.

Yeah. I'm doing meta mathematics just as Godel tried to do in 1931.
And you're surprised?

>>> Do you just think I'm mathematically incompetent, do you?
>>
>> You're either incompetent or of the mindset of an inquisitor when it
>> comes to making arguments about foundation/logic of mathematics.
>>
>>>>>>>> There are those who would precisely defined any number greater than
>>>>>>>> 10^500 to be an infinite number. So what?
>>>>>>>
>>>>>>> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
>>>>>>
>>>>>> (In sci.logic, I don't speak to them, any more than you do!)
>>>>>>
>>>>>> No logical differences: both (your side and those people) would ignore
>>>>>> standard definitions and permissible rules of reasoning in meta level,
>>>>>> and both _believe_ they are right, while actually being wrong.
>>>>>
>>>>> No, I'm afraid you're mistaken about that.
>>>>
>>>> I'm afraid you're mistaken about that, Rupert.
>>>
>>> So which standard definitions are we ignoring, then?
>>
>> The ones that when used will see, e.g., Completeness as invalid,
>> undecide(cGC), etc...
>
> And which are those?

The standard definition of formal system consistency, for one.

>> For sure you're unable to understand that.

Nam Nguyen

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Oct 21, 2015, 3:28:23 AM10/21/15

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But that falls short of a _meta proof_ of unknowable - impossible to
know. Correct?

Anyway, you're still confused between a meta proof and a FOL formal
system proof, so the communication is still "sluggish".

Rupert

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Oct 21, 2015, 4:53:26 AM10/21/15

to

On Wednesday, October 21, 2015 at 9:28:23 AM UTC+2, Nam Nguyen wrote:
> On 21/10/2015 12:59 AM, Rupert wrote:
> > On Tuesday, October 20, 2015 at 7:15:46 PM UTC+2, Nam Nguyen wrote:
> >> On 20/10/2015 11:10 AM, Nam Nguyen wrote:
> >>> On 20/10/2015 10:58 AM, Justin Thyme wrote:
> >>>
> >>>> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
> >>>> read with understanding) a single maths book or logic book. Is that
> >>>> what you think?
> >>>
> >>> Of course not. I think Rupert is capable of understanding 2+2=4 being
> >>> true in the natural numbers. It's just Rupert is incapable of
> >>> understanding it's unknowable - impossible to know - the truth value
> >>> of the sentence cGC, or of the Goldbach conjecture (if the conjecture
> >>> happens to be true).
> >>
> >> I wonder outloud if Rupert himself has ever done an actual PhD research
> >> into the subject of something that's (logically) unknowable - impossible
> >> to know?
> >
> > My own research has not been in that topic but I was very interested in Peter Koellner's essay "On the Question of Absolute Undecidability", and my work on large cardinals has a bearing on Peter Koellner's claim that the statement "Every real number is constructible" is indeed an example of an absolutely undecidable statement relative to the limited view that Gödel held at the time when he formulated it as an example.
>
> But that falls short of a _meta proof_ of unknowable - impossible to
> know. Correct?

Well, part of the issue here obviously is that there might be philosophical disagreements about what it is possible for us to know. Philosophers of set theory draw a distinction between "intrinsic" and "extrinsic" justifications of candidates for new axioms. An axiom is said to be intrinsically justified if there are some grounds for thinking that it is in some way implicit in the iterative conception of set. For example Gödel claimed that axioms asserting the existence of Mahlo cardinals were implied by the iterative conception of set. We say that an axiom is extrinsically justified, on the other hand, if there are consequences of the axiom, or theorems about the theory obtained by adjoining the axiom to the existing axioms, which we find desirable in some way. When Peter Koellner gave an example of a statement that is (arguably) absolutely undecidable relative to the view that Gödel held at a certain stage in the development of his thinking, he means that it is absolutely undecidable by axioms which can be intrinsically justified. But many might claim that the statement is settled by extrinsic justifications. So the question of whether or not we can know the statement's truth-value turns out to hinge on a philosophical debate about whether we should regard extrinsic justifications of axioms in set theory as giving us knowledge. I think you'll find that most philosophers of set theory would believe that these matters are not amenable to being settled by mathematical proof. You can put forward a philosophical claim that a particular statement is absolutely undecidable but such a claim does not admit of mathematical proof.

You obviously have a different view of this issue, but one of the problems with your stance is that you want to claim that you have an argument for the truth-value of cGC being unknowable, which ought to be called a mathematical proof, but you also want to say that this argument does not admit of being formalised in any formal system of first-order logic. This, to me, is a puzzling conception of what a mathematical proof is. Surely part of the whole notion of mathematical proof is that there should be precise criteria for what counts as a sound argument. Do you have such precise criteria?

> Anyway, you're still confused between a meta proof and a FOL formal
> system proof, so the communication is still "sluggish".

I've never been able to understand what you think you mean by "meta-proof".

Rupert

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Oct 21, 2015, 4:59:36 AM10/21/15

to

On Tuesday, October 20, 2015 at 7:53:25 PM UTC+2, Nam Nguyen wrote:
> On 20/10/2015 11:39 AM, Justin Thyme wrote:

> > Nam Nguyen wrote:
> >> On 20/10/2015 11:20 AM, Justin Thyme wrote:

> >>> Nam Nguyen wrote:
> >>>> On 20/10/2015 10:58 AM, Justin Thyme wrote:
> >>>>
> >>>>> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
> >>>>> read with understanding) a single maths book or logic book. Is that
> >>>>> what you think?
> >>>>
> >>>> Of course not. I think Rupert is capable of understanding 2+2=4 being
> >>>> true in the natural numbers. It's just Rupert is incapable of
> >>>> understanding it's unknowable - impossible to know - the truth value
> >>>> of the sentence cGC, or of the Goldbach conjecture (if the conjecture
> >>>> happens to be true).
> >>>

> >>> That's hardly his fault - you have given us no treason to think that the
> >>> truth values of the sentence cGC, or of the Goldbach conjecture, are
> >>> impossible to know. It's that word "impossible" that seems to be the
> >>> problem: the truth value of the Goldbach conjecture is not known, fine,
> >>> but impossible to know? Why impossible?
> >>
> >> I think I'm talking to a walking deaf. Because it's not possible to
> >> verify or prove in meta level using any available and valid means of
> >> the reasoning framework!
> >
> > How do we know that if you don't tell us what the available and valid
> > means of the reasoning framework are?
>
> Does JT have to know all the infinitely many induction axioms to prove,
> say, 2+2=4 in PA?
>
> >

> > Suppose that it is announced that the Goldbach conjecture is true. More
> > specifically, some mathematician has submitted a proof to a journal and
> > that journal's reviewers accept that the proof is valid. How will you
> > respond?
>
> I'll say the both (the mathematician and the journal) are wrong.
>
> > Will you say that the proof (or alleged proof if you prefer)
> > is invalid without even reading it?
>
> Yes of course: because I have a meta proof of undecide(cGC) which
> include a counter proof the the alleged above. This where I my argument
> style is different from the PhD Rupert: I'd present a counter proof,
> instead of just saying something like "You haven't convinced me, because
> I say so and I have a PhD but you only a Bachelor one"!

That is not really a fair paraphrase of anything I've ever said. I've never said anything that would in any way imply that my PhD degree somehow proves that I am right.

I have said that I find your arguments impossible to follow and that I think they lack force. You always try to insist that I make specific objections, and I do my best, but it is very hard to do that because they are so incoherent. It is hard to say much more than "This is not a cogent mathematical argument".

Your proposed style of arguing is actually not that much different from what you complain of in mine because you are proposing to "refute" the argument without actually reading it.

> Suppose JT claims 2+2=5 being true in the naturals, do I have to
> read his "proof".

Well, Edward Nelson claimed to have derived 2+2=5 from the axioms of Primitive Recursive Arithmetic and the mathematical community felt an obligation to check the argument.

Rupert

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Oct 21, 2015, 5:14:21 AM10/21/15

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What's that got to do with anything? It is trivial that you can prove anything from some appropriately chosen set of axioms. But we find some axiom systems more convincing than others. With a mathematical argument the usual situation is that we can formulate as a deduction from some set of axioms which we find intuitively acceptable. Are you saying that your "meta-proofs" are in some way different? To my mind you have never given a satisfactory explanation of how this can work.

> Besides, why don't you formalize undecide( Natural-numbers |= cGC) so
> that you can show to sci.logic you'd know what you're talking about
> (which you don't)?

That's not my job. It's your argument, if it admits a formalisation then you should do the formalisation. I can't formalise it if I don't know how the argument is supposed to work.

> > I have read many meta-proofs and have never found an example of one which cannot be formalised in a FOL formal system.
> >
> >> How many more time do you need to be reminded of these premisses into
> >> the presentation?
> >
> > Oh, I've heard it plenty of times. As far as I'm concerned, saying that your work can't be formalised in FOL is pretty much tantamount to admitting that you're not doing mathematics.
>
> Yeah. I'm doing meta mathematics just as Godel tried to do in 1931.
> And you're surprised?

Meta-mathematics is a special case of mathematics.

> >>> Do you just think I'm mathematically incompetent, do you?
> >>
> >> You're either incompetent or of the mindset of an inquisitor when it
> >> comes to making arguments about foundation/logic of mathematics.
> >>
> >>>>>>>> There are those who would precisely defined any number greater than
> >>>>>>>> 10^500 to be an infinite number. So what?
> >>>>>>>
> >>>>>>> So, actually, these people of whom you speak have not shown how to make that a part of a consistent mathematical theory, so the case is completely different.
> >>>>>>
> >>>>>> (In sci.logic, I don't speak to them, any more than you do!)
> >>>>>>
> >>>>>> No logical differences: both (your side and those people) would ignore
> >>>>>> standard definitions and permissible rules of reasoning in meta level,
> >>>>>> and both _believe_ they are right, while actually being wrong.
> >>>>>
> >>>>> No, I'm afraid you're mistaken about that.
> >>>>
> >>>> I'm afraid you're mistaken about that, Rupert.
> >>>
> >>> So which standard definitions are we ignoring, then?
> >>
> >> The ones that when used will see, e.g., Completeness as invalid,
> >> undecide(cGC), etc...
> >
> > And which are those?
>
> The standard definition of formal system consistency, for one.

So, you want to make a claim that, when a mathematician puts forward a definition of the standard semantics for the first-order language of arithmetic, he is in some way ignoring the standard definition of formal system consistency?

Rupert

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Oct 21, 2015, 5:22:40 AM10/21/15

to

On Tuesday, October 20, 2015 at 7:57:58 PM UTC+2, Jim Burns wrote:
> On 10/20/2015 1:15 PM, Nam Nguyen wrote:
>
>

> > I wonder outloud if Rupert himself has ever done an actual
> > PhD research into the subject of something that's (logically)
> > unknowable - impossible to know?
>

> A while back, Rupert linked to a paper of his,
> http://arxiv.org/pdf/1403.8058.pdf
> A PROPOSED CHARACTERISATION OF THE
> INTRINSICALLY JUSTIFIED REFLECTION PRINCIPLES
> RUPERT McCALLUM
>
> In his abstract, he refers to
> An open-ended family of strengthenings of this principle is
> tentatively proposed as exhausting everything that can be
> justified on the basis of the iterative conception of set.
>
> I would think that, for each strengthening of the reflection
> principle, there would be statements undecidable without
> that strengthening, otherwise, what would be the point of
> strengthening?
>
> In other words, Nam: Yes.

Some remarks about this paper.

It has been on arxiv for some time and has not yet been submitted for publication. I have put it on the backburner while working on other things. I am aware of at least one error in one of the definitions which would have to be fixed before submitting it for publication. However, coincidentally I had formulated the plan of doing some work on it today.

When I propose to characterise those set-theoretic axioms which can be intrinsically justified, this should not be confused with a claim that anything left undecided by the intrinsically justified set-theoretic axioms as characterised in my paper has a truth-value which is impossible to know. That would be a more controversial philosophical claim which I wouldn't feel able to defend adequately in my paper. Rather the claim is just that these axioms might be plausibly claimed to exhaust everything which can be justified on the basis of the iterative conception of set.

One example of a statement which is left undecided by all of the intrinsically justified set-theoretic axioms as characterised in my paper is the assertion "Every real number is constructible". Gödel put this forward as a possible example of an absolutely undecidable statement, and if my characterisation of the intrinsically justified set-theoretic axioms is accepted, then it can indeed be proved to be absolutely undecidable *relative to axioms which are intrinsically (as opposed to extrinsically) justified*. This would be an example of a "meta-proof" which Nam wanted.

Justin Thyme

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Oct 21, 2015, 8:33:25 AM10/21/15

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> How many more time do you need to be reminded of these premisses into
> the presentation?
>

>> Do you just think I'm mathematically incompetent, do you?
>
> You're either incompetent or of the mindset of an inquisitor when it
> comes to making arguments about foundation/logic of mathematics.
>
>>>>>>> There are those who would precisely defined any number greater than
>>>>>>> 10^500 to be an infinite number. So what?
>>>>>>
>>>>>> So, actually, these people of whom you speak have not shown how to
>>>>>> make that a part of a consistent mathematical theory, so the case
>>>>>> is completely different.
>>>>>
>>>>> (In sci.logic, I don't speak to them, any more than you do!)
>>>>>
>>>>> No logical differences: both (your side and those people) would ignore
>>>>> standard definitions and permissible rules of reasoning in meta level,
>>>>> and both _believe_ they are right, while actually being wrong.
>>>>
>>>> No, I'm afraid you're mistaken about that.
>>>
>>> I'm afraid you're mistaken about that, Rupert.
>>
>> So which standard definitions are we ignoring, then?
>
> The ones that when used will see, e.g., Completeness as invalid,
> undecide(cGC), etc...

So these definitions allow you to prove things not otherwise provable?
They are more than just short expressions standing in place of longer ones?

>
> For sure you're unable to understand that.
>

--

Justin Thyme

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Oct 21, 2015, 8:38:56 AM10/21/15

to

Nam Nguyen wrote:

> Yeah. I'm doing meta mathematics just as Godel tried to do in 1931.

You are so funny!

Gödel's incompleteness theorem (if that's what you're referring to) is
formalizable in PRA. So your claim that there is a distinction between
meta reasoning and formalized reason seems to be untrue.

> And you're surprised?

Rupert

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Oct 21, 2015, 11:16:53 AM10/21/15

to

On Tuesday, October 20, 2015 at 7:35:59 PM UTC+2, Nam Nguyen wrote:

> On 20/10/2015 11:24 AM, Nam Nguyen wrote:
> > On 20/10/2015 11:20 AM, Justin Thyme wrote:

> >> Nam Nguyen wrote:
> >>> On 20/10/2015 10:58 AM, Justin Thyme wrote:
> >>>
> >>>> So you seem to be saying that Rupert hasn't read (or, at least, hasn't
> >>>> read with understanding) a single maths book or logic book. Is that
> >>>> what you think?
> >>>
> >>> Of course not. I think Rupert is capable of understanding 2+2=4 being
> >>> true in the natural numbers. It's just Rupert is incapable of
> >>> understanding it's unknowable - impossible to know - the truth value
> >>> of the sentence cGC, or of the Goldbach conjecture (if the conjecture
> >>> happens to be true).
> >>
> >> That's hardly his fault - you have given us no treason to think that the
> >> truth values of the sentence cGC, or of the Goldbach conjecture, are
> >> impossible to know. It's that word "impossible" that seems to be the
> >> problem: the truth value of the Goldbach conjecture is not known, fine,
> >> but impossible to know? Why impossible?
> >
> > I think I'm talking to a walking deaf. Because it's not possible to
> > verify or prove in meta level using any available and valid means of
> > the reasoning framework!
> >

> > Would this be a very difficult notion for a PhD to comprehend?
>
> Oh I forgot, when push comes to shove, Rupert would have "belief"
> in his sleeves.

I have no idea what this is supposed to mean. If you want to attribute some opinion to me then I'd prefer that you back it up with a quotation.