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.
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.