Ɐx (x=x) is a statment in predicate logic, which extends propositional logic .
A proposition is a truth valued variable. The symbol
A in the law
A v ~A represents a proposition.
Where as the symbol
a in the law
Ɐa.(a=a), represents a variable (sometimes called an object, or entity), where as the equality symbol
= is a relational operator, and the predicate
a=b is a function that is valued as true if "
A predicate is a function which maps its variable(s) to a truth value. These variables are not themselves truth-valued; they reference objects in the domain of discorse -- whatever it is that is being discussed, which might often be, for example, integers, rational numbers, real numbers, people, quokas, et cetera.
We usually write predicates as capital letters followed by a bracketed list of lower letters (the variables), such as
P(x), Q(y), R(x,y) or such, but some bivariate predicates (relations) are written with the predicate operator (often a capital letter or special symbol) between two variables:
a R b, a = b, a > b or such.
The law doesn't state that "each thing is identical only to itself".
Indeed, it does not. The law is not that restrictive, by itself.
Ɐa.(a=a) merely states that "anything is equal to itself".
I mean even if a thing is equal to itself (a=a) then it maybe true that a=b.
It may indeed, and we have other laws that allow us to make inferences when that happens.
Why we need to state that?
It is a seemingly self-evident statement and in many proofs it is surprisingly useful to be able to state that something is equal to itself. We may as well take it as an axiom.
[In some systems we do this by a rule of inference, 'equality introduction'. Effectively the same thing.]