In The Logical Syntax of Language, Rudolf Carnap uses "=" to represent if and only if, noting in passing that his usage is non-standard. I've been reading that recently, and it occurred to me to wonder why that was non-standard. A logician commented that we feel like the "2+3" on the one side and the "5" on the other side of "2+3=5" name the same thing, while the reference of the sentence letter names is more controversial, so we're less comfortable saying that the named things on both sides of an if and only if are the same merely because their truth values are.
The named things? When did we all become mathematical Platonists? I don't remember getting any memos about it. Last I checked, more than a few of us were still skeptical of any notion of there being "things" corresponding to either "2+3" or "5", much less the same thing corresponding to both. But we don't quibble about "=" in mathematics. Why do we not use it for the same thing in logic?
