TITLE: In logic, is 'P = Q,' where P and Q are propositions, ill-conceived?
QUESTION [7 upvotes]: In formal/mathematical logic, what are the rules governing "identity" (equals sign) use? Why is it that a conditional (or in certain instances a biconditional) sign is used instead to show a relation between propositions.
I'm a high school student with a developing interest in logic, so any insight into the philosophy guiding the semantics of logical languages would be appreciated. 
Thank you for your time. 
Edit:
Thank you for your responses. I have two fallacious arguments written propositionally: If P, then Q If P, then R Therefore: If Q, then R And If P, then Q If R, then Q Therefore: If P, then R However, if these particular propositions were interpreted as being connected not by a conditional sign but by an "=" (identity) sign, wouldn't we have examples of the transitive property (i.e. P=Q, P=R, so Q=R using the rule of identity elimination in Fitch calculus)? However, neither conditional/biconditional introduction/elimination would be able to prove this in Fitch. Is this why it's a bad idea to use "=" when translating to the more primitive language of propositional logic?

REPLY [3 votes]: First-order logic already uses = for identity on the domain under discussion (sets, integers or whatever). So even though we could conceivably use = for identity on truth values, it's clearer to have a different symbol $\leftrightarrow$.
Plus the notation $P \leftrightarrow Q$ has the advantage of suggesting $P \rightarrow Q$ and $Q \rightarrow P$.