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SINGLE-TYPE ONTOLOGY 81
For greater understandability, we again illustrate our representational strategy by means of
an example: Consider the rich -representation of the proposition ‘John loves Mary’
(abbreviated Lmj) at @ in a universe consisting of three indices @, w 1, and w 2, and three
(distinct) entities John (j), Mary (m), and Bill (b). Assume further that, at @, the propositions
‘John runs’ (Rj), ‘Mary does not run’ (¬Rm), and ‘Bill runs’ (Rb) are true, that, at the index
w 1, the propositions ‘John loves Mary’, ‘John runs’, ‘Mary doesn’t run’, and ‘Bill doesn’t run’
are true (such that Lmb, Rj, ¬Rm, and ¬Rb obtain at w 1), and that, at the index w 2, the
propositions ‘John loves Mary’, ‘John runs’, ‘Mary runs’, and ‘Bill doesn’t run’ are true (such
that Lmj, Rj, Rm, and ¬Rb obtain at w 2):
Then, the propositions ‘John runs’ and ‘Mary doesn’t run’ are the only true propositions at @
which carry information about the aboutness subjects of the proposition ‘John loves Mary’. As
a result, we represent the truth-value of ‘John loves Mary’ at @ by the subset of the set {@,
w1, w2} at whose members the propositions ‘John runs’, ‘Mary doesn’t run’, and ‘John loves
Mary’ are true. 6 We identify the latter with the singleton set {w1} (underbraced in Fig. 3).
To ensure a correspondence between propositions and their rich single-type representations
(in analogy with (10)), we represent propositions φ by an index-general variant, (14), of their
representation from (13):
(14) { | w ∈ φ & for all p , if w 1 ∈ p &
for some x, (φ ∧ p) ‘is about’ x, then w ∈ p}
We will sometimes denote objects of the above form by φ † .
This completes our elimination of single-type candidates on the basis of their failure to satisfy
the representability requirement from Property 2.
Our investigation of the Booleanness and Representability requirements from Sections 3.1
and 3.2 has reduced the set of single-type candidates from Table 1 to all but two members: the
types and . Since the requirements of Intensionality and Partiality depend
only on the domanial structure of an algebraic type, they are unable to exclude further singletype
candidates. However, we will see below that they place specific constraints on the
candidates’ associated objects. The latter are specified below. Since the requirement of
Intensionality is trivially satisfied in virtue of the types’ partiality (cf. Sect. 2.4), we only
discuss the requirement of Partiality.
3.4. Obtaining Partial Types
The previous subsection has described representations of entities and propositions in the type
as relations between pairs of indices of the form from (10) and (14).
Correspondingly, at the index w 1 from Figure 1, the entity John and the proposition ‘John
whistles’ are represented by the set of indices from (15):
6
The last requirement compensates for the fact that ‘John loves Mary’ is undefined at @. The latter is
analogous to the existence requirement on the representation of entities from (9).