Some doubts about interpretation of an atomic formula in predicate calculus

Question

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

Answer 1

I'm going to start from scratch because your post is very long.

Let us start with a language $$ L=\{R_i \,:\, i\in I\}\cup\{f_j \,:\, i\in J\}\cup \{c_k\, :\, k\in K\} $$ The $R_i$ are relation symbols, the $f_j$ function symbols and the $c_k$ constant symbols with $I,J,K$ indexing sets. Now an interpretation/structure/model for this language is a tuple $$ \mathfrak{A}=\langle A,\{\textbf{R}_i \,:\, i\in I\},\{\textbf{f}_j \,:\, i\in J\}, \{\textbf{c}_k\, :\, k\in K\}\rangle$$ Where $A$ is a (depending on your conventions, possibly required to be non-empty) set, if the symbol $R_i$ is $n$-ary then the corresponding $\textbf{R}_i\subseteq A^n$. If the symbol $f_j$ is $n$-ary then the corresponding $\textbf{f}_j$ is a function $A^n\rightarrow A$ and $\textbf{c}_j\in A$.

The idea of this is that you just interpret a relation symbol as a relation on the domain, a function symbol as a function and a constant symbol as an element. Now we can use this to define a satisfaction relation on the formulae in the language $L$. But we have to be careful as we may have free-variables in the formula.

So inductively, let $i:\{\text{variables}\}\rightarrow A$. Then inductively we define an interpretation of terms, and use this to do formulae. $$ c_i^\mathfrak{A}=\textbf{c}_i $$ $$ x_i^\mathfrak{A}=i(x_i) $$ $$ f_i(t_1,t_2,\ldots,t_n)^\mathfrak{A}=\textbf{f}_i(t_1^\mathfrak{A},t_2^\mathfrak{A},\ldots,t_n^\mathfrak{A})$$ So the idea of this definition is that we find out the value of a term by evaluating any constants or variables according to $i$ and the structure $\mathfrak{A}$ and then apply functions as defined in the structure. Now we are ready to define satisfaction of an atomic.

Atomic formulas look like $t_1=t_2$ where $t_i$ are terms or like $R(t_1,\ldots,t_n)$ where $R$ is an $n$-ary relation symbol and $t_i$ are terms. We define $$ (\mathfrak{A},i)\models t_1=t_2 \Leftrightarrow t_1^\mathfrak{A}=t_2^\mathfrak{A} $$ and $$ (\mathfrak{A},i)\models R(t_1,\ldots,t_n) \Leftrightarrow \langle t_1,\ldots,t_n\rangle\in \textbf{R} $$

So we say that a claim that $t_i=t_j$ is true if they are interpreted as the same object in the domain, and a claim that a relation $R$ holds of some $n$-tuple of terms if we have stipulated that the interpretation of the terms are in the domain of $\textbf{R}$.

Hope this clears a few things up.