In Freindly introduction to mathematical logic, there is a procedure to convert the first order logic formulas into propositional formulas, this is a natural way to convert FOL's formulas into propositional formulas, here is the procedure:
My question is, If I have the first order logic formula $(=x_1x_2) \rightarrow (=x_2x_1)$ and I want to convert it into a propositional formula using the above procedure, Should I get $A\rightarrow A$ or $A\rightarrow B$? ($A$ and $B$ are two propositional variables)
In other words, Are the two formulas $(=x_1x_2)$ and $(=x_2x_1)$ distinct?
I don't know whether I should deal with them syntactically and so they will be distinct or semantically and so they are the same (not different).
So, What is the right thing?
Yes, they are distinct.
Thus, if $\beta$ is $(=x_1,x_2) → (=x_2,x_1)$, then $\beta_P$ must be : $A \rightarrow B$ (with $A,B$ sentential letters).
The transormation is a "syntactical" one; thus you are considering a formula of the "logical form" :
and you do not know that $R$ is symmetric.
The issue is that every instance of a (propositional) tautology is a valid formula, but not vice versa.
From the tautology $A \lor \lnot A$ you can derive the valid instance $(x=y) \lor \lnot (x=y)$ but the "propositional transformation" of the valid $x=x$ is simply $A$, which of course is not a tautology.