Logic - Satisfaction of a statement I was curious whether this statement is satisfied by some model?
And is it satisfied by Herbrand's model?

$∃xR(x)∧¬R(c)$ 

For the first question, I can take a model where domain is the real numbers and R is whether x is a natural number or not.
So this is satisfied by taking x=4 and c=3 for instance.
But regarding the second question - if it is not satisfied by some Herbrand model, isn't it suppose to be not satisfied by any model?
Or maybe it is indeed satisfied by Herbrand model? But then, what if Herbrand's model is just has only the constant c?
Thanks in advance.
 A: Every model of $∃xR(x) ∧ ¬R(c)$ must have at least two elements.
We can satisfy the formula with e.g. the domain $D = \{ 0, 1 \}$ and interpreting $c$ with $0$ and $R$ with the subset $\{ 1 \}$ of $D$, i.e. with the "property": $(x \ne c)$.
If the language has only the constant $c$ and no function symbol, then the Herbrand universe will be $U = \{ c \}$.
Thus, any Herbrand structure built up on the universe $U$ will not satisfy the formula.

See :

*

*Mordechai Ben-Ari, Mathematical Logic for Computer Science, Springer (3rd ed, 2012), page 179:


Theorem 9.24 A set of clauses $S$ has a model iff it has an Herbrand model.
Theorem 9.24 is not true if $S$ is an arbitrary formula.
Example 9.25 Let $S = p(a) ∧ ∃x¬p(x)$. Then

$(\{ 0, 1 \}, \{ \{ 0 \} \}, \{ \}, \{ 0 \} )$

is a model for $S$ since $v(p(0)) = T$ , $v(p(1)) = F$.

See page 179 : "An interpretation $\mathfrak I$ is a 4-tuple :
$I = (D, \{ R_1,\ldots, R_l \}, \{ F_1,\ldots, F_m \}, \{ d_1,\ldots, d_n \})$,
where $D$ is a non-empty set called the domain, $R_i$ is an $n_i$-ary relation on $D$ that is assigned to the $n_i$-ary predicate $p_i$, $F_j$ is an $n_j$-ary function on $D$ that is assigned to the function symbol $f_j$ and $d_i ∈ D$ is assigned to the constant $a_i$."

$S$ has no Herbrand models since there are only two Herbrand interpretations and neither is a model:

$( \{ a \}, \{ \{ a \} \}, \{ \}, \{ a \}), (\{ a \}, \{ \{ \} \}, \{ \}, \{ a \})$.


