Re: OUTGOEDELING A HUMAN?

On Feb 22, 10:55 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
abo says...

Here define "universally true" to be "true in any domain (in
which it is defined)" and consider

"This sentence is not universally true."

Your response (I imagine) is the same as LauLuna - to forbid the
saying of the sentence.

I'm not forbidding the saying of anything. I can give a perfectly
good (in my own humble opinion) of that sentence.

First of all, let's get rid of the context-dependent phrase
"this sentence" by introducing a name "Abo's sentence" and
defining it to be the sentence

Abo's sentence is not universally true.

Okay, now let's make the definition of "universally true"
more precise:

A sentence S is universally true if
for all interpretations i, if S is given
any interpretation in i, then it is interpreted
as true in i.

which is equivalent to

A sentence S is universally true if there does not
exist an interpretation i such that S is interpreted
as false in i.


So Abo's sentence is equivalent to the following claim:

There is an interpretation i in which Abo's sentence
is interpreted to be false.

So what's an interpretation? An interpretation assigns meanings
to all noun phrases and predicate phrases and provides a domain
for any quantifiers. The above sentence involves quantification
over interpretations. So if interpretation i_1 gives any
interpretation to the above sentence, then i_1 must provide a
domain D_1 which is the set of interpretations quantified over.

So the interpretation of the above sentence in interpretation i_1
is:

There is an interpretation i in D_1 in which Abo's
sentence is interpreted to be false.

If the domain D_1 is empty, then i_1's interpretation of
Abo's sentence is false. If i_2 is a second interpretation
such that D_2 = { i_1 }, then i_2 would interpret Abo's
sentence as:

There is an interpretation i in D_2 in which Abo's
sentence is interpreted to be false.

which, in the case where i_1 is the only element of D_2,
is equivalent to

Abo's sentence is interpreted to be false in i_1.

which is true.

(*) > So Abo's sentence is interpreted to be true in some
interpretations, and false in others.

It's late here and I need to turn in, so I could well be incorrect.
But your conclusion (*) seems to imply that Abo's sentence is not true
in L (for some fixed language L). So Abo's sentence is not
universally true. So Abo's sentence is true...


Basically, all that you've shown is that natural language is not a
language with a mathematically precise semantics.
But we already knew that (although maybe we didn't have a proof of it!).

Yes, we knew that already. We already knew that the Liar sentence
that says

The Liar sentence is not true.

is meaningless (cannot be interpreted as true, and also cannot
be interpreted as false). However, the question is where the
problem lies. I claim that the problem is not self-reference,
but the notion of "truth" independent of interpretation.

I don't think the problem is self-reference ("This sentence is a
sentence" is true), and I don't think grounding "truth" in terms of an
interpretation helps, because that's not the natural-language "truth",
which can talk about all interpretations and all L.


We also would seem to know what "true" means in "All statements are
true or not true" or "Most things Nixon said are not true"?

We only "sort-of" know what those things mean. We understand those
general claims only in the sense that we understand most specific
instances of them. That's good enough for most purposes.

When you speak of "instances of them," you really sound like you're
talking about "tokens."

No. The statement "All statements are true or not true" involves
a quantifier. Quantifiers have instances. For example, one instance
is

"Snow is white" is true or "Snow is white" is not true.

another instance is

"Grass is purple" is true or "Grass is purple" is not true.


OK OK enough already, I just said "sound like", I didn't say "was."
Sorry for not being clearer.

When I say, "All statements are true or not
true," I really do mean all statements, and I really do mean an
unrestricted true.

But you don't know what "unrestricted true" means. In particular,
you don't know what it means for the Liar sentence to be true.

Sure I do. That is, I understand the sentence "This sentence is not
true," and I understand ""This sentence is not true" is not true" and
""This sentence is not true" is true."


I don't agree. Every instance of the non-problematic use of the
word "true" has an implicit domain.

There is nothing problematic about "All statements are true or not
true".

Whether there is or not depends on what you mean by "true".
You haven't specified that. I claim that it doesn't *have*
a meaning, in general. It only has a meaning in restricted
cases.

Nixon: "Every statement Dean said about Watergate is not true."

If Dean said,
Dean: "Every statement Nixon said about Watergate is not true,"
then Nixon's statement becomes problematic. If he didn't (and didn't
say anything like it), Nixon's statement isn't problematic.

But surely Nixon's sentence has meaning regardless of whether Dean
uttered a particular sentence or not.

I claim that it does not. If Nixon said "Every statement Dean said
about Watergate is not true", then the meaning of Nixon's statement
depends on what Dean said. You can't assign any meaning to Nixon's
statement until you know what Dean said.

Well, that's bizarre. Here is a grammatical sentence, and it would
seem prime facie that we know what it means.

Okay, but that's like calling the use of "truth" in the Liar
"paradoxical" and saying the non-paradoxical uses of "truth" fit very
well. It doesn't solve anything.

Look, there's nothing new about the claim that some sentences
are meaningless. If I say "Green ideas sleep furiously", I haven't
said anything true, and I haven't said anything false.

Wait wait wait. Now you're making the change from "not true" to
"false." No good. Not allowed. I only discuss strengthened Liar,
thank you. Note that "It is not true that green ideas sleep
furiously" is a true statement.

I haven't
said anything meaningful at all. You can certainly dismiss the
Liar as such a meaningless statement.

I can't disagree with you completely here (darn!), but I think it's
lack of content rather than meaninglessness. Ok, so this could just
be considered a semantic matter (you mean "meaningless" as I mean
"lack of content"), but I imagine the difference is more profound,
since I have a specific idea of content, and how it fits in with the
solution to the Liar, which does not look equivalent to yours. In any
case simply labelling the Liar by "X", whatever X is, is not a
solution, because it doesn't explain why being X stops the Liar
reasoning. Any solution to the the Liar must explain why X stops the
reasoning (and not just coming up with "X").

However, if you aren't careful,
calling the Liar meaningingless is itself paradoxical:

Let the Liar sentence be the sentence:

The Liar sentence is not true.

We reason as follows:

(1) The Liar sentence is meaningless.
(2) No meaningless sentence is true, therefore we conclude:
(3) The Liar sentence is not true.

So even though we claimed that the Liar sentence is meaningless,
it seems that we can *prove* it. That is, it is the conclusion of
what is usually considered a valid argument form:

A is a B.
No B is a C.
Therefore, A is not a C.

where A is "the Liar sentence", B is "meaningless sentence" and
C is "true sentence". So calling the Liar meaningless *doesn't*
help, unless you revise your notion of what "true sentence" means.

I'd agree with this and myself I assert (1), (2), and (3), again
modulo talking about content rather than meaning. The solution to the
Liar explains which notion of "true sentence" has to be revised. This
is where we disagree.

.

Relevant Pages

  • Re: This sentence is not true
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  • Re: This sentence is not true
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  • Re: Is Truth Mysterious?
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  • Re: OUTGOEDELING A HUMAN?
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  • Re: 2nd-order logic in lower-order language
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