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I've never seen the following comparison before. Let me start with a specific example:

Given a finite structure with two symmetric binary relations, i.e. a graph $G$ with one vertex set $V$ and two edge sets $E_1$, $E_2$.

Giving an explicit description of $G$ in a formal language can be seen as a special case of defining a function $f$ from a set $T$ with a betweenness relation $B$ to the set $V\ \cup \big( E_1 \times \lbrace E_1 \rbrace \big) \cup \big( E_2 \times \lbrace E_2 \rbrace \big)$ such that $(v,w) \in E_i$ iff

$(\exists x,y,z) f(x) = v \wedge f(y) = ((v,w),E_i) \wedge f(z) = w \wedge B(x,y,z)$

Such a tuple $(x,y,z)$ represents the "sentence" that $(v,w) \in E_i$. The sentences may overlap and reuse "symbols", but this can be avoided. For example ($T = \mathbb{N}$):

|u|v|vwE|uwE|w| | | |... 

represents the sentences $(u,w) \in E$ and $(v,w) \in E$. But also does:

|u|uwE|w|v|vwE|w| | | |... 

In fact, it's enough to consider functions $f$ from $T$ to $V\ \cup \lbrace E_1 \rbrace \cup \lbrace E_2 \rbrace$ such that $(v,w) \in E_i$ iff

$(\exists x,y,z) f(x) = v \wedge f(y)=E_i \wedge f(z) = w \wedge B(x,y,z)$

This comes closer to the normal usage of formal languages. So

|u|E|w|v|E|w| | | |... 

represents the sentences $(u,w) \in E$ and $(v,w) \in E$, which are oftenly written as $uEw$ and $vEw$.

This is what "symbolic" description essentially is: a "structure preserving" function from a medium to the structure. (Using intermediate symbols from an alphabet isn't essential: the elements of the structure may symbolize themselves.)

In contrast, "analog" description essentially is a "structure preserving" function from the structure to a medium:

Consider a function $f$ from the set $V\ \cup \big( E_1 \times \lbrace E_1 \rbrace \big) \cup \big( E_2 \times \lbrace E_2 \rbrace \big)$ to a set $T$ with a betweenness relation $B$ such that $(v,w) \in E_i$ iff $f((v,w),E_i))$ is between $f(v)$ and $f(w)$, or:

$(\exists x,y,z) f(v) = x \wedge f(((v,w),E_i)) = y \wedge f(w) = z \wedge B(x,y,z)$

For example ($T = \mathbb{N}^2$):

|u|uwE|w| | |vwE| | |v|   | | 

Again, we can ignore the specific edges and write/draw for short:

|u|E|w| | |E| | |v| | | 

We even can ignore the specific vertices and write/draw for short (note, how this looks like a unlabelled graph!):

|V|E|V| | |E| | |V| | | 

Analog description can be generalized for relations of arbitrary arity, but that's a bit tedious. In principle, it works.

Is this comparison in any sense enlightening, or is it just baublery?

As I said before, I've never seen this comparison made explicit. So can any references be given?

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    I wanted to show the similarity between the two formulas.2011-02-25

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