There seem to be at least three syntactic conventions in common use for function terms and predicate formulas in first-order languages. Specifically (where $\mathfrak{F}$ is an arbitrary function or predicate symbol):
- $x \mathfrak{F} y$
- $\mathfrak{F} x y$
- $\mathfrak{F}(x, y)$
Sometimes a single convention for, say, binary predicates isn't followed consistently even within the same book.
This brings up a few related questions:
Besides the three listed above, are there any other syntactic variants to be aware of that are in common use in first-order languages of mathematical logic? (reverse Polish notation might hypothetically be one, but I've never seen it used in first-order logic)
Is any one of these syntactic conventions more "officially correct" than the others? (my guess would be that $\mathfrak{F} x y$ is kind of the "canonical" one, but I'm not at all sure).
In the case of the $\mathfrak{F}(x, y)$ syntax, the comma symbol
,seems to pretty clearly have special meaning as an argument separator. Yet in the books where I've seen this syntax used, the definition of the "alphabet" of first-order languages never mentions,as a special symbol (alongside the other special symbols like parentheses and $\neg$ and $\wedge$). I guess this is just an oversight? Or am I misunderstanding something?Bonus question: I would have thought that books covering mathematical logic would make clear the syntactic issues like the preceding 1-3, but (surprisingly) I have yet to find one that does. Does anyone know of such a book (or other resource like lecture notes)?
- Books I've looked at are: Logic books by Ebbinghaus and Enderton, and Discovering Modern Set Theory by Just & Weese.