[extropy-chat] Non-classic logics

[Ian Goddard]

scerir wrote:

> In algebra (or algebra of operators) 
> there is something called non-distributivity: 
> (AxBxC) =/= Ax(BxC) =/= (AxB)xC. 
> (Non-distributivity is different from 
> non-commutativity). 


I'm not sure what you mean. If by 'x' you mean the
multiplication operator, then what you're showing is
not true. Using '*' for multiplication, it is the case
that

(a*b*c) = a*(b*c) = (a*b)*c

because multiplication is associative. It's also
commutative

a*b*c = b*c*a = c*a*b

and distributive over addition

a*(b + c + ... + n) = a*b + a*c + ... + a*n 


> Is there a non-standard logic reproducing, 
> somehow, a property like this? 

Some classic-logic operators such as AND (&) and OR
(v) like multiplication are also commutative and
associative such that (where '::' indicates that the
lefthand statement can be replaced without a loss of
meaning by the righthand statement and visa versa):

P & Q & R  ::  Q & R & P

P v Q v R  ::  R v P v Q


P & (Q & R)  ::  (P & Q) & R

P v (Q v R)  ::  (P v Q) v R


There are also distributive properties:

P & (Q v R)  ::  (P & Q) v (P & R)

P v (Q & R)  ::  (P v Q) & (P v R)

and for the IF-THEN operator '->' here:

P -> (Q v R)  ::  (P -> Q) v (P -> R)

P -> (Q & R)  ::  (P -> Q) & (P -> R)


~Ian




		
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