
Self-reference and Logic (2005) [pdf] - idiliv
http://www.imm.dtu.dk/~tobo/essay.pdf
======
hs86
This is great for understanding why simpler logics (Boolean) are computable
but too expressive ones (FO) are not. As soon as the logic is able to make
statements about itself, things get 'weird'. A not formal(!) but imho still
intuitive example:

"This sentence has five words."

This is a true statement. And we know that the conjunction of two true
statements should also be true.

"This sentence has five words and this sentence has five words."

With eleven words each part of this statement is obviously false and here we
can't rely on the "and", which we are used to from the simpler logics,
anymore.

~~~
gerdesj
_" This sentence has five words and this sentence has five words."_ In English
"this" need not always be self referential, so I'll retort with "This sentence
is imprecise".

The two "this"s could refer to two different things. However, the second
_this_ would normally become _that_ to indicate comparison but grammatically
the original is still sound [with notes]:

This sentence [indicates a sentence with five words in it] has five words and
this sentence [indicates another sentence, also with five words in it] has
five words.

~~~
jjnoakes
Even if you try to separate what each 'this' refers to, there is still only
one sentence, so the sentence becomes meaningless if you do that.

~~~
white-flame
No, there are 3 sentences implied in the context, but only 1 sentence is
visible in this specific example.

Natural language normally isn't a fully self-contained representation.

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CliffStoll
An excellent paper, especially reviewing the connections of Peano, Tarski, and
Godel.

I'm surprised that he did not mention Jim Propp's self=referential aptitude
test. It starts with:

> 1\. The first question whose answer is B is question > (A) 1 > (B) 2 > (C) 3
> > (D) 4 > (E) 5

And, of course, Bolander's paper ends with an utterly delightful final
sentence!

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firethief
Footnote 17 mentions a proposed approach of "extending logic to include
_contexts_ ", but doesn't name any names. Anyone have a reference I can
follow?

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yters
If this sentence is true, then Santa Claus exists. Therefore, Santa Claus
exists.

~~~
skh
If P then Q is a true statement when P is false. (Assuming standard two valued
logic.). The statement

If P then Q

being true does not mean Q has to be true. I don’t think you can conclude
Santa Claus exists based on your argument.

~~~
yters
P = P -> Q

P = !P v Q

Q |= _|_

P = !P v _|_

P = !P

~~~
skh
Your reasoning only works for valid statements. That is, statements that have
a truth value. As you’ve demonstrated P does not have one. For no admissible
statement is P equal to not P.

~~~
yters
1\. Premise: P -> Q

2\. Premise: ((P -> Q) -> P) -> P

3\. (P -> Q) -> P, from 1

4\. P, from 2 and 3

5\. Q, from 1 and 4

~~~
skh
The P, in your original post, is not a valid formula since it does not have a
truth value. You can construct all the proofs and whatnot you want but if P is
not a logical statement (one that is either true or false but not both) then
the arguments are not valid.

Your Q was generic and could have been any statement. In particular one that
has been provably shown to be false. If your arguments are correct then you
have the ability to prove any false statement is true.

Do you really think you have found a way to demonstrate that the standard two-
valued logic used in mathematics leads to a contradiction?

~~~
yters
Just repeating Curry's paradox from Wikipedia.

~~~
skh
I see. The difference in our views comes from me being a mathematician.
Curry’s Paradox is not a paradox in mathematical logic due to my objections.
In mathematical logic we don’t allow statements that don’t have a well defined
truth value. The premise in the start of Curry’s Paradox is a statement
without a well defined truth value.

Curry’s Paradox shows the limits if naive set theory. Thus mathematically we
have to be OK with the idea that not all naive set constructions produce valid
sets.

In informal setttings we can argue that

If this sentence is true then Santa Claus exists

can’t be false. But then one usually assumes it must then be true. I would
argue this isn’t the case. There are sentences that have no truth value. For
instance,

This sentence is false.

~~~
yters
Thanks, this clarifies things for me.

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keithnz
Runtime Error: Stack Overflow in recursive function.

