You might have noticed something funny: we said **safe ( a)** depended on the board, but that prime (18) was false. Why are some some relations different than others? To
add to the puzzling, there was a caveat in some fne-print from the previous section: "

**prime (18)**is false

**under the standard interpretation of prime**". Why these weasel-words? Everybody knows what prime is, don't they? Well, if our domain is matrices of integers (instead of just integers), we might suddenly want a different idea "prime".

Consider the formula *E* (*x*, *x*) true for all *x* in a domain? Well, it depends not only on the domain, but also on the specifc binary relation *E*
actually stands for:

- for the domain of integers where
is interpreted as "both are even numbers",**E**is false for some*E*(*x*,*x*)*x*. - for the domain
**{2, 4, 6, 8}**whereis interpreted as "sum to an even number",**E**is true for every*E*(*x*,*x*)*x*. - for the domain of integers where
is interpreted as "greater than",**E****E****(**is false for some*x*,*x*)*x*(indeed, it's false for**every**).*x* - for the domain of people where
is interpreted as "is at least as tall as",**E**is true for every*E*(*x*,*x*)*x*.

Thus a formula's truth depends on the interpretation of the (syntactic, meaning-free) relation symbols in the formula.

**Definition 3.1: Interpretation**

The interpretation of a formula is a domain, together with a mapping from the formula's relation symbols to specific relations on the domain.

One analogy is “Programs are to data, as formulas are to interpretations ". (In particular, the formula is a like a boolean function: it takes its input (interpretation), and returns true or false.)

- 1116 reads