What does paradox reveal about the nature of truth?

Don't sweat it igm.
For the purposes of this discussion you are allowed to speak in certainties to some extent. Within the realm of mathematics you are allowed to speak of such things as true and proof.

I agree Frank.
Concepts of belief, consciousness, self-ness, and (I suspect) reality itself are intimately linked to self-referencing within and between systems.

Off topic:
Frank yesterday I was thinking about starting a thread regarding the topics in that video I posted. You know the "Frank Ipisa mantra".
I know you have already thought and written about a lot of them, would you be interested in starting such a thread?

I'd prefer not to start it myself...but if you were to start it, I certainly would participate.

I have long been very much against videos being used in these discussions...so it would be inappropriate for me to "start" such a thread. But as I mentioned, something caused me to start watching the piece...and I was hooked immediately and watched the entire 19 minutes.

It is an excellent video...the guy is a terrific speaker...and with the one minor exception I mentioned, he and I are in agreement.

If you build it...they will come.

Hello,

looking for a way around the liar and other paradoxes
I focused on logic as the culprit:
By introducing a new logical dimension, most problems can be solved.

Statements are not absolutely true or false anymore
but true or false related to a viewing angel
or kind of logical layer or meta-level.

Most contradictions are not contradicional anymore,
as the truth values belong to different layers.

The good news (in my theory):
The liar´s paradox, Cantor´s diagonal argument, Russell´s set and
(not yet fully proved) Goedel´s incompleteness theorem are valid no more.

The bad news: There is no more absolute truth
and we have to get used to a new mathematics
where numbers might have multiple prime factorisations.

Over all, infinity and paradoxes will be much easier to handle in layer theory,
finite sets and natural numbers more complicated, but possible.

The theory was in the beginning just a ´Gedankenexperiment´,
and my formal description and axioms may still be incorrect an incomplete.
Perhaps someone will help me?

Here my axioms of layer logic:

Axiom 0: There is a inductive set T of layers: t=0,1,2,3,…
(We can think of the classical natural numbers, but we need no multiplication)

Axiom 1: Statements A are entities independent of layers,
but get a truth value only in connection with a layer t, referred to as W(A,t).

Axiom 2: All statements are undefined (=u) in layer 0.
VA: W(A,0)=u
(We need u to have a symmetric start.)

Axiom 3: All statements in positive layers have either the truth value ´w´ (true)
or ´-w´ (false) or ‘u’ (undefined).
Vt>0:VA: W(A,t)= either w or –w or u.

Axiom 4: Two statements A an B are equal in layer logic,
if they have the same truth values in all layers t=0,1,2,3,...
VA:VB: ( A=B := Vt: W(A,t)=W(B,t) )

Axiom 5: (Meta-)statements M about a layer t are constant = w or = -w for all layers d >= 1.
For example M := ´W(-w,3)= -w´, then w=W(M,1)=W(M,2)=W(M,3)=...
(Meta statements are similar to classic statements)

Axiom 6: (Meta-)statements about ´W(A,t)=...´ are constant = w or = -w for all layers d >= 1.

Axiom 7: A statement A can be defined by defining a truth value for every layer t.
This may also be done recursively in defining W(A,t+1) with W(A,t).
It is also possible to use already defined values W(B,d) and values of meta statements (if t>=1).
For example: W(H,t+1) := W( W(H,t)=-w v W(H,t)=w,1)

A0-A7 are meta statements, i.e. W(An,1)=w.

Although inspired by Russell´s theory of types, layer theory is different.
For example there are more valid statements (and sets) than in classical logic
and set theory (or ZFC), not less.
And (as we will see in layer set theory) we will have the set of all sets as a valid set.
And self referring statements and sets are allowed in layer logic and layer set theory.
Last not least a look onto the liar in layer theory:

Classic: LC:= This statement LC is not true (LC is paradox)

Layer logic: We look at: ´The truth value of statement L in layer t is not true´
And define L by (1): Vt: W(L, t+1) :=w iff W(L,t) -= w and W(L, t+1:=-w else.

Axiom 2 gives us: W(L,0)=u
(1) with t=0 gives us: W(L,0) = u-=w -> W(L,1) = w
(2) with t=1 : W(L,1) = w -> W (L,2) = -w
(3) with t=2 : W(L,2) = -w-=w -> W (L,3)= w ; W(L,4)= -w; etc.

L is a statement with different truth values in different layers,
but L is not paradox.

That should be enough for a start

Now what do you think:
Is this ´changing the rules´ with the help of a new dimension worth to be tried,
or too wild a shot?

Yours
Trestone

Your thinking seems to reflect that of of Hegel's "dialectic" (thesis /antithesis /synthesis), and you might note that formalisms for nested layers have been investigated in "Systems Theory". Note too that layer shifting implies a time dimension along which the process of conceptualization operates. It is this time dimension which is lacking in traditional (static) logic in which axiomatization goes hand in hand with naive realism, in which "states of affairs" are (wrongly IMO) taken to be independent of the perception of observers.

Hello fresco,

to Hegel I have little to say, as I did not understand him 30 years ago and never tried since …
Connections to “Systems Theory” I did not find, but I had only a short glimpse and do not know this branch.
The conection of shifting layers with time is an interesting thought.
I still have no good practical interpretation or model for the layers in layer logic.
My best model are cause and effect, the effect always belonging to a higher layer than the cause.
In a way this is also a time order.
But thince the theory of realtivity time seems to be more complicated than cause and effect,
so there might be only some aspects (or dimensions?) of time in my logic layers.

I think we still have to try to understand layer theory better
(e.g. where do we need layers and where is classical logic enough?),
for up to now nobody uses my logic, and even I am only experimenting with it.

On the other hand a reasonable new logic could give us a lot of new inpulses in many fields,
and nearly all theories would have to be checked.

I can imagine, that there are not many who like this thought …

Yours,
Trestone

You're close to the fallacy of arguing from the realm of metaphysics into the realm of real/physical things, i.e. there is nothing in reality which corresponds to the (this sentence is false) statement.

In the case of simple numbers, the fact of their being metaphysical is obvious, i.e. there is no store which you can walk into and buy a three or tree from which you could pick an eight. Numbers are only as real as there are things in the real world to which they correspond. Rational numbers correspond to the invention of knives (cut a pie into three pieces and all of a sudden, you have to be able to think in terms of thirds etc.). Negative numbers correspond to the reverse gear in your car and your credit card etc.

The decimal for Pi, 3.14159..., seems like a paradox, a number with no end, but it exists in nature in the form as circles.

This is a delightful thread
Truth compared to validity
Simple, yet profound

I haven't read through it all

yet

the liar paradox is false and not in any way true thus not a paradox

Nothing here follows

What am I missing here ...nothing that I can read .

There is nothing logical about the quoted statement

So long there isn't true infinity compute all the relative motions and time dilations and you get into a final resting order all the same. You can look at motion as we can look at a frozen piece of film...if there is time dilation from A relative to B in the film it doesn't change its absolute state in the piece of film at all. What it does is making us think on the secondary nature of motion itself.
This is what Fresco keeps missing all the time, he throws away the baby with the bath water and seems to be totally unaware of it.

Both observer and observed can be framed in the same way. No matter how entangled they are, they boil down to a final resting place of relations that is factual. Itself an object. There is no construction, no one is constructing anything, the relation is timeless as an object frozen into a frame of film.

Back to your layer Logic, cause and effect are just a timely explanation regarding the correlations of order within the general frozen film of reality. They show the order by which things are related to each other or entangled with each other.

Finally you may be wondering how do I dismiss infinity so easily once we can think of it...to put it simply I think of infinity as a loop whole that goes through all possible arrangements within a given X amount of space, while it doesn't stop anywhere the number of options is finite hence repetition follows. This space is reduced in its final form to one single dimension. You can think of it somewhat as a metaphysical sort of space in the sense it cannot be accessed by us. Its timeless.