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A first-order structure and an assignment determine the value of every term recursively: a variable has its assigned value, a constant has its interpreted value, and a compound function term is evaluated by applying the interpreted function to the recursively evaluated arguments. This operation supplies the input values for the satisfaction clauses of atomic formulas.

The first-order theory of a bijection $f$ with named inverse $g$ and no finite cycles has first-order models consisting of disjoint integer orbits. Atomic logical formulas reduce to $x_j=f^k(x_i)$ for integer $k$. An existential logical variable is either fixed by such a logical equality or must avoid finitely many points. This proves quantifier elimination and completeness. There is one complete one-type; the complete two-types are the integer offsets and the nonprincipal different-orbit partial type. There are countably many finite-arity partial types but countable first-order models with different numbers of orbits are not isomorphic.

In the first-order language of logical equality, assert that there are at least $n$ elements for every positive integer $n$. Eliminating one existential logical variable either substitutes a logical equality or avoids finitely many points, so this first-order theory has quantifier elimination and is complete. It is categorical in every infinite cardinal. Its complete $n$-types are the logical equality partitions of $n$ coordinates and are counted by the Bell numbers.

The unique readability theorem asserts uniqueness of the outer formation rule and its immediate components in a logical formula. For the fully parenthesized language with negation and implication, every formula is uniquely a variable, a negation of one formula, or an implication between an ordered pair of formulas. It gives a well-defined syntax tree and licenses recursive definitions and structural induction on formulas.