Formal theories dealing with non-commutattive and non-transitive notion of equality This question is inspired by a philosophical discussion which I don't want to bother you with. 
As far as I know when we use (or define) the statement "$x$ is equal to $y$" in logic and ordinary mathematics we mean a "commutative" and "transitive" notion of $=$. i.e. We have

$$\forall x,y~~~(x=y\leftrightarrow y=x)$$
  $$\forall x,y,z~~~(x=y~\wedge~y=z\rightarrow x=z)$$

In some philosophical cases I came across some intuitively contradictory statements which could be consistent if we consider a "non-commutativity" or "non-transitive" binary relation $=$. In fact if we assume a weird situation between two objects $x$, $y$ such that "$x$ is $y$ but $y$ is not $x$" or have a situation that "$x$ is $y$ and $y$ is $z$ but $x$ is not $z$" we may get around of some contradictions. Note that it seems in theses type of using the $=$ a new intuition about the meaning of the word "is" is also needed because our ordinary point of view says that "is" is a commutative transitive verb.  
Question: I would like to ask if anybody is aware of any formal logical system/theory  which deals with a non-commutative or non-transitive equality (identity) notion? If there is such a system I am more interested in its corresponding semantics and model theory. Any philosophical references about any use of non-commutative or non-transitive equality notion are also welcome. 
 A: You'll have to get rid of the replacement property of "=" in order to talk about a non-symmetric equality, so long as you agree that a=a.  In other words, you'll have to get rid of the rule which allows you to replace either side of the "=" by the other side of the "=" wherever those terms appear.
Let all lower case letters qualify as terms.
Demonstration that if we have the replacement rule of inference for terms, that if a=a, then a=b if and only if b=a:
Suppose that a=a, and suppose that a=b.  Then replacing the left or first "a" with "b" in "a=a", we obtain b=a.  Thus, under the hypothesis that a=a, if a=b, then b=a.  Now suppose that b=a.  Replacing the right or second "a" with "b" in "a=a" we obtain "a=b".  Thus, if b=a, then a=b.  Therefore, if a=a, then a=b if and only if b=a.
Or more symbolically $\forall$a $\forall$b {(a=a)$\rightarrow$[(a=b)↔(b=a)]}
This effectively rules out truth tables as possible in a relevant model, since truth tables all rely on having a rule of replacement (even though the formulas in propositional logic are not terms... they behave just like terms in a certain sense).  So, the underlying propositional logic will end up as infinite-valued.  But you can't use say Lukasiewicz infinite-valued logic or any fuzzy logic that I know of either, since that also has a rule of replacement for "=" as permissible.
It's not at all clear how anything could get computed without a rule of replacement also. 
A: I think the best source to start an investigation on the notion of "equality", its history and its rules (also other basic laws of thought) would be here:
http://en.wikipedia.org/wiki/Law_of_thought
