[extropy-chat] Non-classic logics

[scerir]

From: "Ian Goddard" 
> Using '*' for multiplication, it is the case
> that (a*b*c) = a*(b*c) = (a*b)*c
> because multiplication is associative.

In the '20s P.Jordan turned attention,
within the usual QM, from the non-commutative
product of operators x*y =/= y*x to the
commutative-non-associative product
(x*y + y*x) of variables.

(Indeed one can see non-associativity
as a higher sort of non-commutativity,
i.e. the non-commutativity of "left"
multiplication with "right" multiplications.
And indeed non-associativity might be
related to reflexive processes, processes
that act on themselves, while associativity
might be related to non-reflexive processes.) 

But Jordan's non-associative product does not 
have the meaning of "y after x" or "x after y".
I was interested exactly in that meaning:
"y after x", as a fundamental product representing 
material processes, concatenations, compositions.
In chemistry, i.e., H2(SO4) =/= (H2S)O4.
Even in Set Theory [(a,b),c]; [a,(b,c)],
[a,b,c] are different objects, due to Cantor
set formation rules.

Of course there are non-associative algebras
(i.e. the algebra of octonions) somehow
involved in explaining the internal
degree of freedom of quarks, or the strong
force, etc.

I was asking whether there is something about that
at a more abstract level, the level of logic,
or non-standard logic.

Thanks,
s.

(In another post I've wrote, perhaps,
non-distributivity instead of non-associativity
:-)