This work introduces the theory of illative combinatory algebras,

which is closely related to systems of illative combinatory logic. We

thus provide a semantic interpretation for a formal framework in which

both logic and computation may be expressed in a unified

manner. Systems of illative combinatory logic consist of combinatory

logic extended with constants and rules of inference intended to

capture logical notions. Our theory does not correspond strictly to

any traditional system, but draws inspiration from many. It differs

from them in that it couples the notion of truth with the notion of

equality between terms, which enables the use of logical formulas in

conditional expressions. We give a consistency proof for first-order

illative combinatory algebras. A complete embedding of classical

predicate logic into our theory is also provided. The translation is

very direct and natural.