Appendix A

# Logical Foundations

Abstract We discuss the logical foundations of the event calculus: relations, inductive definitions, first-order logic, many-sorted first-order logic, second-order logic, real numbers, lists, and circumscription.

Keywords Relations, Inductive definitions, First-order logic, Many-sorted first-order logic, Second-order logic, Circumscription

## A.1 Relations

Relations are used in the definitions of the semantics of first-order logic and many-sorted first-order logic, and in the section on state constraints (Section 3.4).

Definition A.1

An n-ary relation R on a set S is a subset of S^{n}.

Definition A.2

A binary relation R on a set S is a subset of S × S.

Definition A.4

A binary relation R ⊆ S × S is reflexive if and ...