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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 Sn.

##### DEFINITION A.2

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

##### DEFINITION A.3

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

##### DEFINITION A.4

A binary relation RS × S is irreflexive if and only if, for every ...

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