
Re: layer logic: a new dimension to logic?
Posted:
Oct 15, 2011 5:41 PM


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

