APPENDIX A

Logical Foundations

This appendix discusses 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.

## 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.3

A binary relation *R* ⊆ *S* × *S* is *reflexive* if and only if, for every *a* ∈ *S*, 〈*a, a*〉 ∈ *R*.

##### DEFINITION A.4

A binary relation *R* ⊆ *S* × *S* is *irreflexive* if and only if, for every ...