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Re: layer logic: a new dimension to logic?
Posted:
Oct 15, 2011 5:41 PM
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Hello,
perhaps I should tell a little bit more about the motivation for my axioms:
?Axiom 1: Statements A are entities independent of layers,
but get a truth value only in connection with a layer t,
refered to as W(A,t).?
My introduction of layers to logic was a little bit similar to Max Planck introducing quants in physics:
Not very plausible and I do not really like it, but it seems to work ?
With the layers, a liar statement was easier to handle:
If we define statement L by the following:
For all layers t: W(L,t+1) := W (W(L,t) = -w ,1)
?The value of thus statement L is true (in layer t+1) iff the value of L is not true (in layer t?
With the (universal) start W(L,0) = u we get: W(L,1) = -w, W(L,2) = w, W(L,3) = w, W(L,4) = w, ?
So L , which is similar to the liar statement, is a statement with alternating truth values and this is allowed and no problem in layer theory thanks to the layers.
The next very special axioms are the axioms about meta statements:
?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.?
For the motivation of these axioms, we have a look at axiom 1:
There we find formulations like ?for all layers t?, so axiom 1 is a statement about all layers.
As there is a hierarchy of layers and we are only allowed to use smaller layers for defining a truth value
of a statement in a certain layer t0
(I have not formalized this completely yet) the axiom 1 can not belong to a certain layer t1.
On the other side I did not want to have infinite ordinal numbers in my layers,
so I made statements about one ore more layers independent of layers by defining axiom 5.
(I am not sure, if axiom 6 is needed at all, as layers are always in connection with truth values up to now.)
Now we can also handle another liar statement LA:
For all layers t: W(LA,t+1) := W (For all d>0: W(LA,d) = -w ,1)
LA is a meta statement, therefore we can write:
For all layers t: W(LA,t+1) := W (For all d>0: W(LA,1) = -w ,1)
With t=0: W(LA,1) := W (W(LA,1) = -w ,1)
Case 1: W(LA,1)=w then W(LA,1)= W (w = -w ,1) = -w , what is not allowed.
Case 2: W(LA,1)=-w then W(LA,1)= W (-w = -w ,1) = w , what is not allowed.
So LA is not a well defined (meta) statement in layer theory ? and therefore no problem.
Similar with liar statement LE:
For all layers t: W(LE,t+1) := W (It exists a layer d0>=0: W(LE,d0+1) = -w ,1)
LE is a meta statement, therefore we can write:
For all layers t: W(LE,t+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1)
With t=d0: W(LE,d0+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1)
Case 1: d0 exists: W(LE,d0+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1) = w, what is not allowed.
Case 2: d0 does not exist: W(LE,d0+1) := W (It exists a layer d0>0: W(LE,d0+1) = -w ,1) = -w, what is not allowed.
So LE is not a well defined (meta) statement in layer theory ? and therefore no problem.
I do not know if we have to define (and with what more details), that all statements A need a definition of the form:
?For all layers t: W(A, t+1) = ??
and on the right sides only statements of layers smaller than t+1 or meta statements are allowed.
Yours,
Trestone
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