# What is the I in this quote "the symbol f is associated with the function I(f)"

I was reading through the Wikipedia page for first-order logic and all made sense to me, though I got confused at this quote "In other words, the symbol $$f$$ is associated with the function $$I(f)$$ which, in this interpretation, is addition." and am unsure of what the I in the last sentence means. Is it a function that maps the function symbol to a function?
Is there a name for it?

Thank you friends

• They define $I$ to be an interpretation (a model) which takes anything from $D^n$ to $D$ which is the domain of discourse over an n-arty input. Takes anything from constants, predicates, and functions, and puts them into the model with their own assignments in the universe (or domain) of the model. I personally don't like the way that they're teaching it, but that's the answer for your question. In this case, "I" is supposed to be understood meta-semantics as addition. Sep 3, 2021 at 3:03

It is defining an interpretation, and I personally don't like that way. Allow me to explain it.
Definition:
An interpretation is a pair ($$\mathbf{M}$$, $$j$$), with $$j$$ being the assignment and $$\mathbf{M}$$ being the model. j takes elements like functions, relations, and constants from a language $$\mathcal{L}$$ and "maps" them into elements in the universe of the interpretation (or domain of), and they're usually denoted as: $$c^{\mathbf{M}}$$, $$F^{\mathbf{M}}$$, $$R^{\mathbf{M}}$$ (unimaginatively). I explained in the comment what they exactly mean by turning it into a function that takes the tuples and maps them into elements in the domain of the model. An interpretation may also be called a structure.
Examples:
($$i$$): The language of group theory has $$e$$ as a constant which will be assigned to $$e^{\mathbf{G}}$$, and $$\circ$$ which is some arbitrary binary operation (predicate constant), which will be assigned as $$\circ^{\mathbf{G}}$$, and at the end we have the universe (domain) $$\mathcal{G}$$, and our interpretation looks like this: $$\mathbf{G}=(\mathcal{G},\, e^{\mathbf{G}},\,\circ^{\mathbf{G}})$$. Now, how is this useful? Fill in the gaps! ($$\mathbb{R}$$, $$\cdot$$, $$1$$) for example. Now you've multiplicative group.
($$ii$$): Take the language of graphs which has $$R$$ as its predicate, and so will will have $$R^{\mathbf{Gr}}$$. Define a domain $$\mathcal{R}$$, and you're done. The interpretation $$\mathbf{Gr}$$ = ($$\mathcal{R}$$, $$R^{\mathbf{Gr}}$$). You can replace $$R^{\mathbf{Gr}}$$ with the Cartesian relation for example, and leave the reals as the domain: ($$\mathbb{R}$$, $$\times$$), and you've a Cartesian graph.

The comment:

They define $$I$$ to be an interpretation (a model) which takes anything from $$D^n$$ to $$D$$ which is the domain of discourse over an $$n$$-arty input. Takes anything from constants, predicates, and functions, and puts them into the model with their own assignments in the universe (or domain) of the model. I personally don't like the way that they're teaching it, but that's the answer for your question. In this case, "$$I$$" is supposed to be understood, in a meta-semantics sense, as addition.

Note that loosely speaking.. this is not exactly correct, and plays around with some words in a misleading way. That's why I suggest seeing the explanation that I wrote.
Lastly an interpretation is of a language, and a model is of a theory. Switching them around is "fine" because they're the "same structure" but it is not recommended, and should only be left for purposes of intuition.

• It might be an idea to give a concrete example of a basic interpretation to let the OP see what's going on? The wiki explanation has manged to turn a really basic idea into something confusing. For instance, $\begin{array}{lll}\Im:&&\\& \text{Domain:} & \{0, 1, 4\}\\& \text{F:} & \{1, 4 \}\\& \text{G:}& \{0, 1\}\\ & \text{H:} & \emptyset \\ & \text{a:} & 0\\ & \text{b:} & 1 \\& \text{c:} & 4\\\end{array}$ Sep 3, 2021 at 4:03
• @TenO'Four Good idea. Will edit. Sep 3, 2021 at 4:10
• The examples really ground it now 👍 Sep 3, 2021 at 5:33