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?
Best Answer
There is nothing ill-conceived about thinking of what is usually written as $P \Leftrightarrow Q$ or $P \leftrightarrow Q$ as an equation between the propositions denoted by $P$ and $Q$. However, it is conventional in mathematical logic not to use the equals sign for this relation.
This convention avoids various kinds of ambiguity and fits in well with the concepts of first-order logic (where we have a specific universe of discourse in mind, like natural number arithmetic, in which equality denotes equality of natural numbers, while bi-implication denotes equivalence of propositions about the natural numbers).
There are higher-order logics in which the world of propositions forms part of the universe of discourse and then bi-implication just becomes a special case of equality.
There is a separate question of notation for syntactic identity between linguistic constructs. I.e., the notation to use when we are thinking of $1 + 2$ or $A\land B$ as sequences of symbols not expressions that denote numbers or truth values. The symbol $=$ is often used for syntactic identity, but this usage can give rise to confusion: using $\equiv$ or $:=$ is a common way of avoiding this.