Identity implies logic

It seems to me that if the law of identity does not hold in a context, then the Law of the Excluded Middle (LEM) and the Law of Non Contradiction (LNC) are irrelevant and cannot be used in that context.   For example:  If we have two different computers hooked up to the semantic web and in one's database an apple is only known as 'apple' and the other it is only known as 'orange',  then logical consequences that are inferred from the combination of those two databases in that regard might  be erroneous.   In KIF I think this would be expressed something like:

  (=>  
      (holdsIn contextX IDENTITY)
      (and 
          (holdsIn contextX LNC)
          (holdsIn contextX LEM)
  )

I have expressed this in the following mentograph:
http://robustai.net/mentography/lawsOfLogic.gif

Don't you think that a model theory for RDF should take this basic assumption about when to apply strong logic into consideration?

Logic is great, survival is better!
Seth Russell