What does notation like $A^1_0$ and $f^2_1$ mean? My logic book uses these symbols to represent predicates and constants, respectively (with varying numbers in the bottom/top), but I don't really know what they mean.
Any help?
From the book:
"Officially we think of ourselves as working for each $k > 0$ with a fixed denumerable stock of $k$-place predicates:
$A^1_0$   $A^1_1$  $A^1_2$...
$A^2_0$   $A^2_1$  $A^2_2$ ...
$A^3_0$   $A^3_1$  $A^3_2$ ....
...
and with a fixed denumerable stock of constant:
$f^0_0$   $f^0_1$  $f^0_2$ ..."
 A: From the additional information supplied, it seems clear that the superscripts are the arities of the predicate symbols and function symbols. The subscripts distinguish between symbols of the same arity. 
A constant symbol can be thought of as a function symbol that denotes a function of $0$ variables. Hence the superscripts $0$ for the constant symbols. 
An example of a predicate symbol of arity $1$ is a symbol intended to be interpreted as a subset of our domain $M$. A predicate symbol of arity $2$ is a symbol intended to be interpreted as a binary relation on $M$, that is, as a subset of $M\times M$.
A function symbol of arity $1$ is intended to be interpreted as a function from $M$ to $M$. A function symbol of arity $2$ is intended to be interpreted as a function from $M\times M$ to $M$, that is, as a function of $2$ variables.
But at the syntactic level, the arities determine only what is meant by a term, or an atomic formula. For example, if $P_1^2$ is a predicate symbol of arity $2$, and $s$ and $t$ are terms, then $P_1^2(s,t)$ is an atomic formula, but $P_1^2(s)$ is not a well-formed formula at all. 
