I have some doubt related to the interpretation of atomics formulas in predicate calculus.
In predicate calculus a formula will be interpreted on a specific domain that is where I take the allowed values for the formula constants.
So I have a formula A such that:
$\{p_1,.....,p_n\}$: is the set of all the predicates into the A formula.
$\{t_1,.....,t_k\}$: is the set of all the constants that appear in the n predicates
So an interpretation of my A formula is a triad build in this way:
$I = \{D, \{R_1,....,R_n\}, \{d_1,....,d_k\}\}$ where:
1) $D$: is a not empty domain (so the constants are in $D$)
2) $R_i$ represents an assignment of a relation $n$-ary that goes in D (in a valid constant) for each p_i predicate
3) $d_i$: represents an assignment of a $d_j$ element, that belong to $D$, for each constant $a_j$ (this thing say also that a constant symbol of my language must be a value that is in the domain $D$)
So for example if I have this formula:
A valid interpretation for the previous formula could be:
In which I say that:
1) The domain D is all the Natural Number set
2) The relation associated to the p predicate is the minor or equal relation
3) The value of the constant a must be into the domain D and is the natural number 0
And this is true because this say that in the natural number set exist a special element that is 0 that is the minimum element of the set
Is my reasoning correct until now? i hope yes...
Now I have a big doubt about the meaning of what on my book is called as INTERPRETATION OF ATOMICS FORMULA.
It say that give an atomic formula its interpretation is:
if it is TRUE that:
and FALSE otherwise
I have some problem to unserstand what means this assertion.
I think that $P_i$ is a predicate that is into the set of the predicates of my language: **$\{p_1,.....,p_n\}$
but what is $a_1,...,a_n$ ? I think that are the constants used in this predicate.
I am not understanding what it means.
Someone can help me?
Tnx
Andrea