Consider a symbol vocabulary that contains
$c$ constant symbols, $p_k$ predicate symbols of each arity $k$, and
$f_k$ function symbols of each arity $k$, where $1\leq k\leq A$. Let the
domain size be fixed at $D$. For any given model, each predicate or
function symbol is mapped onto a relation or function, respectively, of
the same arity. You may assume that the functions in the model allow
some input tuples to have no value for the function (i.e., the value is
the invisible object). Derive a formula for the number of possible
models for a domain with $D$ elements. Don’t worry about eliminating
redundant combinations.
Consider a symbol vocabulary that contains $c$ constant symbols, $p_k$ predicate symbols of each arity $k$, and $f_k$ function symbols of each arity $k$, where $1\leq k\leq A$. Let the domain size be fixed at $D$. For any given model, each predicate or function symbol is mapped onto a relation or function, respectively, of the same arity. You may assume that the functions in the model allow some input tuples to have no value for the function (i.e., the value is the invisible object). Derive a formula for the number of possible models for a domain with $D$ elements. Don’t worry about eliminating redundant combinations.