🚀 go-pugleaf

On May 4, 10:47 am, Newberry <newberr...@gmail.com> wrote:
> We will utilize the following three definitions.
>
> Definition 1: A sentence is meaningful if and only if it is a picture
> of a possible state of affairs.
>
> Definition 2: A sentence is true if and only if it corresponds to an
> actual state of affairs.
>
> Definition 3: A sentence is false if its negation corresponds to an
> actual state of affairs.

The problem isn't mapping  SENTENCE => {true,false}.  The problem with
the Liar is when you assume a given sentence is true or false, you can
map that to a contradiction.  The above definitions can be used to
decide if a simple sentence e.g. "Snow is white." is true or not.  But
the Liar has nothing to do with that.  It has to do with formal
implication producing falsehood.

1. Any solution to the Liar must be formal.
2. Any formalism can be used to generate variations of anything that
it is used to formalize.
3. Thus any solution to the Liar must be able to generate variations
of the Liar.

And when you do this, you see what the real problem is.

The Liar is not that hard - it really isn't.  It is simply 4
properties of systems that are (a) desirable, (b) easy to have in a
system by themselves, and (c) contradictory to have all 4 of them.

Formalized (1), we have (2) variations of "This is not true." and (3)
a theorem that applies to other systems - e.g. Turing Machines, Set
Theory, Axiomatic Theorem-Proving.

"This is not true." uses 4 words, and each word provides a capability
of English.  These 4 capabilities are common in formal systems, and a
simple (though not trivial) theorem shows they are not compatible.
Applied to other systems, we get theorems about Turing Machines, sets,
etc.

BTW You are right that "This is not true." has no truth value.  That
is the property that you have to give up in English: "Every sentence
is true or false." does not hold.

Here are some variations (I posted 1,000 generated ones once):

1. It is true.
2. It is not true.
3. It is false.
4. It is not false.
5. "It is true" is true of "This is false."
6. "It is not true of itself." is true of "It is not true of itself."

# 6 is the direct application of the contradiction that exists in
every system that has the 4 properties, in the case of English.  It
reduces to "This is not true."

C-B

> To further characterize possible we can say that it means imaginable.
> Ludwig Wittgenstein says something very similar. He alludes to
> imaginability a few times: If I can imagine objects combined in states
> of affairs, I cannot imagine them excluded from the possibility of
> such combinations. [TLP 2.0121], We picture facts to ourselves. [TLP
> 2.1] A picture is a model of reality. [TLP 2.12] Finally he says: A
> picture represents a possible situation in a logical space [TLP 2.202]
> But what does "possible" mean? What is thinkable is possible too. [TLP
> 3.02] So we have: A picture represents a thinkable situation in a
> logical space. What is thinkable? 'A state of affairs is thinkable':
> what this means is that we can picture it to ourselves. [TLP 3.001] So
> finally we have: A situation in a logical space is a state of affairs
> we that can picture to ourselves.  ...
>
> http://xnewberry.tripod.com/SentencesLackTruthValues.html