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76 LIEFKE
of a streamlined variant of Montague’s type theory from (Montague 1970a; 1973), that is due
to Gallin (1975). The (at most) double application of CT to Gallin’s set of basic types yields the
set of single-type candidates from Table 1.
Sections 3.1 to 3.4 successively eliminate single-type candidates on the basis of their failure to
satisfy Properties 1, 2, and 4. The decorations in Table 1 summarize the reasons for the
persistence or drop-out of each candidate. In the table, Montague types that violate the
requirement of Familiarity (Property 0) are marked in grey.
3.1. Eliminating Non-Boolean Types
The lack of algebraic structure (Property 1) constitutes one of the most effective criteria for
the exclusion of single-type candidates from the set in Table 1. The latter relates to the
difficulty of interpreting linguistic connectives in non-algebraic domains, and of analyzing
linguistic entailment in the absence of a domination relation. In Sections 2.1 and 2.4, we have
already suggested that the domain of the type t has an algebraic structure, and that all
domains of some type inherit this structure through the lifting of
algebraic operations on the set of truth-values. As a result, all candidates from the right-side
partitions of Table 1 are suitable single basic types from the point of view of Property 1.
On the basis of Property 1, candidates from the left-side partition of Table 1 disqualify as
suitable single-type candidates, such that they cannot serve as the single semantic basis for
natural language. This is a result of the absence of an algebraic structure on the domains of
entities and indices, and the attendant ‘primitiveness’ of all domains of some type or . Since the latter constitute two thirds of the members of
the set from Table 1, the algebraicity requirement from Property 1 already enables us to
exclude most of the available candidates as strong single-type candidates.
Notably, our elimination of entities from the set of types in Table 1 also excludes a common
type for entities and propositions (or for entities and truth-values) as a suitable single
semantic basis for natural language. The latter has been proposed by some semanticists 3 as
an obvious single-type candidate, and has been motivated with reference to Frege’s
characterization of truth-values as Gegenstände (cf. Frege 1891). On this assumption, Frege’s
linguistic ontology can be construed as a semantics for natural language that obtains
representations of all Montagovian objects from the single basic type e. However, given the
identification of this type as a non-Boolean type, a common type e for entities and
propositions is ruled out as a suitable single-type candidate. We will see in Section 3.3 that a
semantics based on the type satisfies all requirements.
This completes our elimination of non-Boolean types from the set of single-type candidates.
We next investigate the exclusion of non-representational types from the remaining set.
3.2. Eliminating Non-Representational Types
The ability of representing different Montagovian objects is a more elusive criterion for the
exclusion of single-type candidates than their algebraicity. This is due to the impossibility of
inferring a type’s satisfaction of Property 2 from its outer type structure. As a result, we need
to check the representability of the remaining single-type candidates from the top right-side
partition in Table 1 one-by one.
To this aim, we will first consider single-type candidates (i.e. the types t, , ,
, and ), which fail to provide index-relative (local) representations of
Montagovian entities and propositions. We will then consider candidates (i.e. the types and ) which succeed in providing local, but which fail at giving suitable
3
Proponents include Chierchia and Turner (1988) and Zoltán Gendler Szabó (p.c.).