Introduction

In this chapter a number of existence proofs and theoretical discussions are presented. These are related to the earlier chapters, but were not presented there in order not to distract too much from the main line of those chapters. Sections 9.2 and 9.3 are related to Chapter 1. Sections 9.4 and 9.5 are related to Chapters 2 and 3, respectively. Finally Sections 9.6 and 9.7 are related to Chapter 5.

Undefinedness revisited

In this section we explain precisely how the truth and falsity of COLD-K assertions with respect to the partial many-sorted algebras is established. In particular the issue of undefinedness deserves a careful treatment. In this section we focus on the terms and assertions as presented in in Chapter 1 (see Tables 1.1 and 1.2).

Recall that a partial many-sorted Σ-algebra M is a system of carrier sets SM (one for each sort name S in Σ), partial functions fM (one for each function name f in Σ), and relations rM (one for each relation name r in Σ). The functions fM must be compatible with their typing in the following sense: if f : S1#…# Sm→ V1 #…# Vn is in £ we have that fM is a partial function from to. Similarly the predicates must be compatible with their typing, i.e. if r : S1×…× Sm is in Σ we have that rM is a relation on.