first order structure and its usage

Question

I am trying to wrap my head around "first order structure", What I've come up with so far is the following:

First order structure is a non-empty set equiped with relations, functions and elements (constants)

Relations: $R_1...R_n$

Functions: $f_1...f_m$

Constants: $a_1...a_k$

So am I correct assuming I could build a first order structure like this:

$A = \{A, Scarier, Ancestor \}$

Where $A$ are all animals, living and dead, $Scarier$ is a two-way relation $Scarier(a,b)$, where $a$ would be more scary than $b$ in this case, and $Ancestor(a)$ would mean that $a$ is an ancestor assuming $a \in A$.

Why are why using this? Are we using first order structures simply to constrict and simplify working with "stuff"?

Another question is what the $constants$ are used for in our first order structures, lets say we have:

$R_1 = \{R,+,-,0,1 \} $

$R$ is the real numbers, does this mean we can only use plus and minus with 0 and 1? or why are the constants here?

Thanks beforehand!

Answer 1

Why are why using this? Are we using first order structures simply to constrict and simplify working with "stuff"?

Yes we are constricting what we can say, for good reason. Natural language is ambiguous. This language of first-order logic is precise. After we have stated which symbols stand for functions, predicates and constants, and specified the number of inputs for each function or predicate, then we have a well-defined language in the sense that, if you give me any string of symbols, I can tell you whether or not it is a valid first-order formula over the language you chose.

First-order languages are actually less restricted than natural languages, because the precision (using brackets) allows us to freely express complex statements with arbitrarily many nested quantifiers, which we cannot do in natural language easily without it becoming impossible to unambiguously interpret.

Another question is what the constants are used for in our first order structures, lets say we have:

$R_1=\{R,+,−,0,1\}$

$R$ is the real numbers, does this mean we can only use plus and minus with $0$ and $1$? or why are the constants here?

You can use $+,-$ with any two terms, and the result is also a term. Constants and variables are the base terms, from which you can build up all other terms by applying functions. Note that once you apply a predicate the result is no longer a term but a truth value, and you can only use logical connectives ($\land,\lor,\neg,\rightarrow,\leftrightarrow$) to combine truth values.