Just as the $λ$-calculus uses three primitives (abstraction, application, variable) as the foundation of functional programming, inheritance-calculus uses three primitives (record, definition, inheritance) as the foundation of declarative programming. By unifying classes, methods, objects, and properties under a single record abstraction, the calculus models inheritance simply as set union. Consequently, composition is inherently commutative, idempotent, and associative, structurally eliminating the multiple-inheritance linearization problem. Its semantics is first-order, denotational, and fixpoint-computable. These three properties extend to the lazy $λ$-calculus, since Böhm tree equivalence~\cite{barendregt1984lambda} is fully abstract for a sublanguage of inheritance-calculus, except that recursive $λ$-programs that diverge under $β$-reduction may converge under $\mathrm{lfp}(T_P)$. Inheritance-calculus is distilled from MIXINv2, a practical implementation in which the same code acts as different function colors~\cite{nystrom2015color}; ordinary arithmetic yields the relational semantics of logic programming~\cite{vanemden1976semantics}; self-reference resolves to multiple targets; and programs are immune to the Expression Problem~\cite{wadler1998expression}. This makes inheritance-calculus strictly more expressive than the $λ$-calculus in both common sense and Felleisen's sense~\cite{felleisen1991expressive}.

翻译：正如 $λ$-演算使用三个基本原语（抽象、应用、变量）作为函数式编程的基础，继承演算使用三个基本原语（记录、定义、继承）作为声明式编程的基础。该演算通过将类、方法、对象和属性统一在单一的记录抽象之下，将继承简单地建模为集合的并集。因此，组合运算本质上是可交换、幂等且可结合的，从而在结构上消除了多重继承的线性化问题。其语义是一阶的、指称的且可不动点计算的。这三个性质可推广至惰性 $λ$-演算，因为对于继承演算的一个子语言而言，Böhm 树等价性~\cite{barendregt1984lambda} 是完全抽象的，唯一的例外是：在 $β$-归约下会发散的递归 $λ$-程序可能在 $\mathrm{lfp}(T_P)$ 下收敛。继承演算提炼自 MIXINv2——一个实用的实现，在该实现中，同一段代码可作为不同的函数颜色~\cite{nystrom2015color}；普通算术运算可产生逻辑编程的关系语义~\cite{vanemden1976semantics}；自引用可解析到多个目标；且程序对表达式问题~\cite{wadler1998expression} 免疫。这使得继承演算无论在常识意义上还是在 Felleisen 的意义上~\cite{felleisen1991expressive}，其表达能力都严格强于 $λ$-演算。