semantics of first order logic - when defining structure

Question

So say L is a structure, which has constant, function and predicate symbols C, F and P.

An L-structure S = (D, I) gives meaning to symbols in C, F and P. • D is a set — the domain of the structure S. • I interprets the constant function and predicate symbols. So I has three parts I = (Ic, If,Ip).

Up to here I have no problem understanding.

However, I do not understand the following definitions.

 Ic : C -> D
 If(f^n) : D^n -> D
 Ip(r^n) ⊆ D^n

I am confused about the meaning of ":" and "->". how does ic : C entails D, where D is a set? I kept googling but I still could not find explanations for these definitions.

Answer 1

$\mathcal L$ is the language with a set $C$ of constants symbols, a set $F$ of function symbols and a set $P$ of predicate symbols.

The $\mathcal L$-structure $S= \langle D, I \rangle$ is made of a set $D$, the domain and a set $I = \{ I_C, I_F, I_P \}$ of functions, where e.g.:

$I_C : C \to D$

is the function that interpret the constant symbols; it assign to each constant $c$ of the language $\mathcal L$ an object of the domain $D$, usually denoted as $c^S$.

For example, if $\mathcal L$ is the first-order language of arithmetic, with constant $0$, functions $+$ and $\times$ and predicate $<$, and we interpret it with the domain $\mathbb N$ of natural numbers, then the function $I_C$ will assign to the symbol $0$ the number $\text {zero}$.

In the same way, the symbol $+$ is mapped by $I_F$ to a binary operation in $\mathbb N$, i.e.:

$+^{\mathbb N} : \mathbb N^2 \to \mathbb N$

i.e. the $\text {sum}$ operation.

Thus, we have that $I_F(+)=+^{\mathbb N}$, i.e. the "interpreting" function $I_F$ maps the symbol $+$ (a binary function symbol) into the $\text{sum}$ operation on natural numbers.

The same for the binary predicate symbol $<$; the interpretation $I_P$ maps it into the (binary) relation $\text {less than}$: a bianry relation on $\mathbb N$ is a subset of $\mathbb N^2$.