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 modules, classes, objects, methods, fields, and locals 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~\cite{vanemden1976-predicate-logic-semantics, reynolds1972-definitional-interpreters, aczel1977-inductive-definitions}, denotational, and computable by tabling~\cite{tamaki1986-tabled-resolution}, even for cyclic inheritance hierarchies. These three properties extend to the $λ$-calculus, since Böhm tree equivalence~\cite{barendregt1984-lambda-calculus} is fully abstract for the first-iteration approximation of a sublanguage of inheritance-calculus. As a corollary, this establishes a convergence hierarchy $\text{eager} \subsetneq \text{lazy} \subsetneq \text{fixpoint}$ among $λ$-calculi sharing the same $λ$-syntax. Inheritance-calculus is distilled from MIXINv2, a practical implementation in which the same code acts as different function colors~\cite{nystrom2015-function-color}; ordinary arithmetic yields the relational semantics of logic programming~\cite{vanemden1976-predicate-logic-semantics}; $\mathtt{this}$ resolves to multiple targets; and programs are immune to nonextensibility in the sense of the Expression Problem~\cite{wadler1998-expression-problem}. This makes inheritance-calculus strictly more expressive than the $λ$-calculus in both common sense and Felleisen's sense~\cite{felleisen1991-expressive-power}.

翻译：正如 $λ$-演算使用三个基本要素（抽象、应用、变量）作为函数式编程的基础一样，继承演算使用三个基本要素（记录、定义、继承）作为声明式编程的基础。该演算通过将模块、类、对象、方法、字段和局部变量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，组合操作本质上是可交换、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~\cite{vanemden1976-predicate-logic-semantics, reynolds1972-definitional-interpreters, aczel1977-inductive-definitions}、指称的，并且可通过表格化~\cite{tamaki1986-tabled-resolution}计算，即使对于循环继承层次结构也是如此。这三个性质可推广至 $λ$-演算，因为对于继承演算某个子语言的第一迭代近似而言，Böhm 树等价~\cite{barendregt1984-lambda-calculus} 是完全抽象的。作为推论，这在共享相同 $λ$-语法的 $λ$-演算之间建立了一个收敛层次结构 $\text{eager} \subsetneq \text{lazy} \subsetneq \text{fixpoint}$。继承演算提炼自 MIXINv2，这是一个实用的实现，其中相同的代码充当不同的函数颜色~\cite{nystrom2015-function-color}；普通算术产生逻辑编程的关系语义~\cite{vanemden1976-predicate-logic-semantics}；$\mathtt{this}$ 解析为多个目标；并且程序在表达式问题~\cite{wadler1998-expression-problem} 的意义上对不可扩展性免疫。这使得继承演算无论是在常识意义上还是在 Felleisen 的意义上~\cite{felleisen1991-expressive-power}，都严格比 $λ$-演算更具表达力。