  We give a formal treatment of simple type theories, such as the simply-typed
$\lambda$-calculus, using the framework of abstract clones. Abstract clones
traditionally describe first-order structures, but by equipping them with
additional algebraic structure, one can further axiomatize second-order,
variable-binding operators. This provides a syntax-independent representation
of simple type theories. We describe multisorted second-order presentations,
such as the presentation of the simply-typed $\lambda$-calculus, and their
clone-theoretic algebras; free algebras on clones abstractly describe the
syntax of simple type theories quotiented by equations such as $\beta$- and
$\eta$-equality. We give a construction of free algebras and derive a
corresponding induction principle, which facilitates syntax-independent proofs
of properties such as adequacy and normalization for simple type theories.
Working only with clones avoids some of the complexities inherent in
presheaf-based frameworks for abstract syntax.
