=Paper= {{Paper |id=Vol-3249/paper3-FMKD |storemode=property |title=Reference, Predication and Quantification in the Presence of Vagueness and Polysemy |pdfUrl=https://ceur-ws.org/Vol-3249/paper3-FMKD.pdf |volume=Vol-3249 |authors=Brandon Bennett |dblpUrl=https://dblp.org/rec/conf/jowo/Bennett22 }} ==Reference, Predication and Quantification in the Presence of Vagueness and Polysemy== https://ceur-ws.org/Vol-3249/paper3-FMKD.pdf 
Reference, Predication and Quantification in the
Presence of Vagueness and Polysemy
Brandon Bennett1
1
    University of Leeds, Leeds, UK


                                         Abstract
                                         Classical semantics assumes that one can model reference, predication and quantification with respect to
                                         a fixed domain of possible referent objects. Non-logical terms and quantification are then interpreted in
                                         relation to this domain: constant names denote unique elements of the domain, predicates are associated
                                         with subsets of the domain and quantifiers ranging over all elements of the domain. The current paper
                                         explores the wide variety of different ways in which this classical picture of precisely referring terms can
                                         be generalised to account for variability of meaning due to factors such as vagueness, context and diversity
                                         of definitions or opinions. Both predicative expressions and names can be given either multiple semantic
                                         referents or be associated with semantic referents that have some structure associated with variability. A
                                         semantic framework Variable Reference Semantics (VRS) will be presented that can accommodate several
                                         different modes of variability that may occur either separately or in combination. Following this general
                                         analysis of semantic variability, the phenomenon of co-predication will be considered. It will be found
                                         that this phenomenon is still problematic, even within the very flexible VRS framework.

                                         Keywords
                                         Vagueness, Polysemy, Predication, Quantification, Copredication




1. Introduction
The notion of reference and the operations of predication and quantification are fundamental
to classical first-order logic. The standard semantics for this logic assumes a fixed domain of
possible referent objects, with naming constants referring to unique elements of the domain,
predicates being associated to subsets of the domain and quantifiers ranging over all the elements
of the domain. Thus, if π’Ÿ is the domain of objects that can be referred to, then a constant name,
say c, will donate an object 𝛿(c), such that 𝛿(𝑐) ∈ π’Ÿ, and a predicate 𝑃 will be taken to denote a
set of objects 𝛿(𝑃), with 𝛿(𝑃) βŠ† π’Ÿ. Then one can straightforwardly interpret the predicating
expression 𝑃(c) as a proposition that is true if and only if 𝛿(c) ∈ 𝛿(𝑃).
   But what if we are dealing with a language that is subject to variability in the interpretation of
its symbols: a name may not always refer to a unique, precisely demarcated entity; a predicate
need not always correspond to a specific set of entities. Accounting for such semantic variability
requires some generalisation or other modification of the classical denotational semantics.
   The aim of the current paper is not to propose a single theory but rather to explore some

The Eighth Joint Ontology Workshops (JOWO’22), August 15-19, 2022, JΓΆnkΓΆping University, Sweden
Envelope-Open B.Bennett@leeds.ac.uk (B. Bennett)
GLOBE https://bb-ai.net (B. Bennett)
Orcid 0000-0001-5020-6478 (B. Bennett)
                                       Β© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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representational possibilities. In the first part of this paper I consider different ways in which
semantics can be given to predication in the presence of semantic variability. I shall first
consider some general ideas regarding different views of vagueness, in particular the de re
and de dicto accounts of this phenomenon. I shall suggest that these need not be mutually
exclusive, but could be describing different aspects of semantic variability. I explore possible
models of denotation, predication and quantification in the presence of such variability, first
informally, with the aid of some diagrams and then in terms of a formal framework, based on
standpoint semantics [1, 2, 3], within which variable references can be modelled. In the final
section of the paper I shall consider the problem of co-predication. This has been the subject of
considerable debate in recent years and is a phenomenon that throws up many examples that
pose difficulties for different approaches and has been used to try to support/reject models of
semantic variability.

1.1. Types of Vagueness
The literature on vagueness has generally assumed that the phenomena of vagueness could
arise from three potential sources (see e.g. Barnes [4]):

   1. indeterminacy of representation (linguistic a.k.a de dicto vagueness),
   2. indeterminacy of things in the world (ontic a.k.a. de re vagueness),
   3. limitations of knowledge (epistemic vagueness).

   The epistemic view of vagueness has some strong advocates [5, 6, 7] and many other take
the view that the logic of multiple possible interpretations takes a similar form to logics of
knowledge and belief (e.g. Lawry [8], Lawry and Tang [9]). Indeed, the standpoint semantics,
which will be used in our following analysis can be regarded as being of this form. However,
the question of whether this is a deep or superficial similarity is not relevant to our current
concerns; and the distinction between de dicto and de re aspects of multiple reference, would
also arise within an epistemic account. So in the current paper we shall not further consider
the epistemic view.

1.2. De Dicto Vagueness
A widely held view is that all vagueness is essentially de dicto, and that any kind of vagueness
that seems to come from another source can actually be explained in de dicto terms [10, 11].
A fairly typical version of such an attitude is that of Varzi who (focusing on the domain of
geography, within which vagueness is pervasive) takes a strong position against ontic vagueness.
Varzi’s view of the relationship between vague terms and their referents is summarised in the
following quotation:

      β€œ[To] say that the referent of a geographic term is not sharply demarcated is to say
      that the term vaguely designates an object, not that it designates a vague object.”
      [11]
    Advocates of exclusively de dicto vagueness typically favour some variety of supervaluationist
account of linguistic vagueness, within which the meanings of vague terms are explained in
terms of a collection (a set or some more structured ensemble) of possible precise interpretations
(often called precisifications). An early proposal that vagueness can be analysed in terms
of multiple precise senses was made by Mehlberg [12], and a formal semantics based on a
multiplicity of classical interpretations was used by van Fraassen [13] to explain β€˜the logic
of presupposition’. This kind of formal model was subsequently applied to the analysis of
vagueness by Fine [14], and thereafter has been one of the more popular approaches to the
semantics of vagueness adopted by philosophers and logicians, and a somewhat similar approach
was proposed by Kamp [15], which has been highly influential among linguists.
    What one might call absolutist versions of supervaluationism are those that, following Fine,
hold to a doctrine of super-truth.1 This is the tenet that the truth of an assertion containing vague
terminology should be equated with it being true according to all admissible precisifications
[14, 17, 18]. In such theories the set of admissible precisifications is generally taken as primitive,
just as is the set of possible worlds in Kripke semantics for modal logics. Other, supervaluation
inspired, theories propose a similar semantics based on the idea that the truth of a vague assertion
can only be evaluated relative to a locally determined precisification (or set of precisifications),
but reject the idea that the notion of super-truth is useful. One might call such theories relativist
supervaluation theories. Such theories include that of Shapiro [19] and my own standpoint
semantics [1, 2].2
    Apart from providing a general framework for specifying a de dicto semantics of vagueness,
the supervaluationist idea is also attractive in that it can account for penumbral connection
[14], which many believe to be an essential ingredient of an adequate theory of vagueness.
This is the phenomenon whereby logical laws (such as the principle of non-contradiction) and
semantic constraints (such as mutual exclusiveness of two properties β€” e.g. β€˜... is red’ and
β€˜.. is orange’) are maintained even for statements involving vague concepts. The solution, in
a nutshell, being that, even though words may have multiple different interpretations, each
admissible precisification of a language makes precise all vocabulary in a way that ensures
mutual coherence of the interpretation of distinct but semantically related terms.

1.3. De Re Vagueness
In natural language, objects are often described as vague. We commonly encounter sentences
such as: β€˜The smoke formed a vague patch on the horizon’; β€˜The vaccine injection left a vague
mark on my arm’; β€˜He saw the vague outline of a building through the fog’. In this kind of usage,
β€˜vague’ means something like β€˜indefinite in shape, form or extension’. Of course, β€˜indefinite’ is
almost synonymous with β€˜vague’ so this definition is far from fully explanatory. However, if we
take sentences like the aforegoing examples at face value, they seem to indicate that vagueness
can be associated with an object in virtue of some characteristic of its spatial extension. One
may then argue that spatial extension is an intrinsic property of an object, and that, if an object

1
    What I call β€˜absolutist’ supervaluationism is what Williamson [16] refers to as β€˜traditional’ supervaluationism.
2
    According to standpoint semantics, vague statements are true or false relative to a standpoint. My notion of
    standpoint is formally identified with a set of precisifications determined by a set of (consistent) judgements that
    are accepted as true in a given context. This idea is similar to that presented by Lewis [20], and elaborated by [21].
has a vague intrinsic property, this indicates vagueness β€˜of the thing’. Such vagueness may be
called de re or ontic.
  The idea that vagueness of objects is primarily associated with vagueness of spatial extension
has been endorsed and examined by Tye [22], who gives the following criterion for identifying
vague objects: β€œA concrete object π‘œ is vague if and only if: π‘œ has borderline spatio-temporal parts;
and there is no determinate fact of the matter about whether there are objects that are neither
parts, borderline parts, nor non-parts of π‘œ.” (The second, rather complex condition concerns
the intuition that we cannot definitely identify borderline parts of a vague object. The current
paper will not consider this second-order aspect of vagueness.)

1.4. Combining De Re and De Dicto and the Idea of De Sensu
Rejecting the possibility of de re vagueness requires one to argue that the forms of language by
which vagueness is apparently ascribed to objects are in fact misleading idioms, whose correct
interpretation does not involve genuine ontological commitment to vague objects. However,
contrary to what many proponents of one or other of the two explanations of vagueness often
maintain, accepting that vagueness may be de re does not require one to deny that vagueness is
often de dicto. Indeed, Tye [22] suggests that vagueness can be present both in predicates and
also in objects. He argues that the vagueness of objects cannot simply be explained by saying
that they are instances of vague predicates.
   But the idea of vague objects in the physical world may be hard to accept. Although at the
quantum level of atomic particles we may conceive of the position of an electron might be
vague in a physical sense, vagueness of macroscopic physical objects seems a very odd idea.
For example, in consider whether a particular twig is a part of some pile of twigs, it seems
unintuitive to consider the twig pile as a vague physical object, which may or may not include
certain twigs. As a palliative to this worry, I propose the existence of de sensu vagueness.
   De sensu vagueness is a kind of indeterminism, in the form of a multi-faceted structure, that
is located within the sense, that is in the semantic denotation term. In the case of names and
nominal variables, this means that they can refer to semantic objects that are indeterminate
with regard to any exact physical entity. Both the model of vagueness proposed in fuzzy logic
[23, 24, 25] and also the idea of dot cateogories and dot objects proposed in some accounts of
co-predication (e.g. [26]) can be regarded as based on a de sensu conception of vagueness.


2. Illustrating the Semantics of Predication
Predication involves a predicate and a nominal expression. Both the predicate and the nominal
can be given a semantics that allows either de dicto vagueness or de sensu vagueness or both
of these. De dicto vagueness is modelled within that part of the semantics that maps symbolic
expressions (predicates and nominals) to their referents. If there is no de sensu vagueness then
the referents will be precise entities, and these will normally be taken as being actual entities in
the world. Thus, the reference of a nominal will be a particular precisely demarcated material
entity and the reference of a property (or relation) predicate will be a determinate set (or set
of pairs) of material entities. If there is de sensu vagueness then the references of symbols are
semantic objects which can be vague in so far as their correspondence with precisely demarcated
material entities is not fully determinate.
   Before specifying a formal semantic framework, it may be useful to consider possible models
of predication by means of diagrams. The image in Figure 1(a) depicts an aerial view of a hilly
region with rocky crags. As we see, the terrain is irregular and there is no obvious unique way
of dividing it into separate β€˜crag’ objects. The name β€˜Arg Crag’ has been given to one of the
rocky outcrops. However, there may be different opinions regarding exactly which of outcrop
is Arg Crag. Indeed, some people might use the name to refer to the whole of this rocky area,
whereas others would consider that it refers to a more specific rock structure.
   In the standard classical semantics each conceptual term denotes a fixed set of entities and
each name must refer to a single precise entity. Thus, event before specifying denotations we
will need to divide the craggy region into specific individual objects, to make a set of possible
referents:




Figure 1: (a) Arg Crag and surrounding area.     (b) Classical denotational semantics.


   The best-known approach to modelling vague concepts that has been taken in computer
science and AI is that of fuzzy sets. A fuzzy set is one such at rather than having a definite
true or false membership condition, there are degrees of membership. Figure 2(a) depicts the
standard fuzzy logic model of vagueness: each singular term denotes a single precise referent
entity and each property predicate denotes a fuzzy set of precise entities. Figure 2(b) depicts
the form of semantics arising for a de dicto theory of vagueness, where both properties and
objects are precise but the predicate symbols and names of the language that are referentially
indeterminate.
   One might assume that the blurry concept boundary of the fuzzy logic model provides a
model of predicate denotation that is radically different from that given by the supervaluationist
multi-denotation model. But it can be argued that this difference is not greatly significant
because we can consider the fuzzy set as essentially equivalent to a densely nested structure of
classical set denotations (one for each degree of membership value). However, the semantics
for evaluating a predicating expression, 𝐢(a) is significantly different, since in fuzzy logic the
truth value of 𝐢(a) would be the degree of membership of object denoted by a with respect to
the fuzzy set denoted by 𝐢; whereas, in the multiple denotation model we simply get different
truth values for different choices and then (as we shall see in detail below) can determine truth
in relation to a β€˜standpoint’ regarding what choices we consider admissible.
   A limitation of both the standard fuzzy model and the classical multiple reference model is that
                      C(a)




                     (a)                                                (b)
Figure 2: (a) Fuzzy model of a vague predicate. (b) Multiple denotation model, such as supervaluation
semantics.


only variability in the terms is modelled, not variability in the object referred to. However, both
approaches can be modified to incorporate vague referents, such as objects with indeterminate
physical boundaries. For instance, objects with fuzzy extensions can be modelled as fuzzy sets
of points. And within non-fuzzy multiple-reference semantics, one could also associate a set of
different extensions for different precise versions of an object (or perhaps maximal and minimal
extensions, as in the β€˜egg-yolk’ representation of Cohn and Gotts [27]).




Figure 3: Possible multiplicity of reference when predicates, names and objects are all indeterminate.


  Figure 3 depicts an extended form of multi-reference semantics (The Full Multi). Here we
see not only that the predicates and names can have variable reference, but also there can be
multiple possible precise versions of each reference object. This variability in the objects could
correspond to de re vagueness, but I prefer to think of it as a form of de sensu vagueness, with
the objects being vague semantic objects (visualised as the small black discs) that correspond to
multiple precise physical objects.
  It is worth noting that Figure 3 still represents a considerable simplification of the potential
semantic variability that might arise. In particular, I have assumed that the global set of vague
objects, together with their associations to precise entities, remains fixed, even though the
subset associated with predicate C may vary. In other words C only varies in how it classifies
objects, not in how these objects are individuated. The VRS semantics presented below is more
general. It allows different senses of sortal predicates to be associated with different ways of
individuating objects (for instance under some interpretations of β€˜Crag’, all three of the roundish
craggy objects within the innermost circle of the diagram, might be considered as parts of a
single large crag.


3. Semantic Analysis of Variable Reference
We now consider what kind of semantics can account for the general form of variable denotation
illustrated in Figure 3. Standpoint Semantics provides a quite general framework within which
polysemy can be modelled in terms of the symbols of a formal language having multiple possible
denotations. We shall start by specifying a simple propositional standpoint logic and then
elaborate this to a first-order formalism, Variable Reference Semantics, within which we can
model predication and quantification. But first we explain the basic idea of standpoint semantics.
   Standpoint Semantics is based on a formal structure that models semantic variability in terms
of the following two closely connected aspects:

        β€’ A precisification is a precise version of a vague language. It is used as an index to assign
          precise denotations to vague terms. The model is based on having a set of precisification,
          each corresponding to a consistent precise interpretation of the language.

        β€’ A standpoint is modelled as a set of precisifications. Each standpoint corresponds to a
          range of possible precise interpretations that are compatible with a particular context
          of language understanding. It could capture explicit specifications of terminology given
          by some organisation or implied constraints on meanings that arise whenever some
          vague statement is made in conversation (e.g. β€œThat rock formation is a crag, but this is a
          boulder”).

3.1. Propositional Standpoint Logic
3.1.1. Syntax
The language of propositional standpoint logic π•Š0 is based on a non-empty finite set of proposi-
tional variables 𝒫 = {𝑃1 , … , 𝑃𝑛 }. It extends the usual syntax of propositional logic by adding
a set of standpoint operators Ξ£ = {░𝑠1 , … , ░𝑠𝑛 , β–‘βˆ— }, where βˆ— is used to designate the universal
standpoint.3 So the language of π•Š0 is the smallest set 𝑆 of formulae such that 𝒫 βŠ† 𝑆 and all
formulae of the forms {Β¬πœ™, (πœ™ ∧ πœ“ ), β–‘πœŽ πœ™} βŠ† 𝑆 for each 𝜎 ∈ Ξ£ and every πœ™, πœ“ ∈ 𝑆. One can
easily extend the language by defining additional connectives (e.g. ∨, β†’) and the dual operators
(♦𝑠1 … , ♦𝑠𝑛 , β™¦βˆ— ) in the usual way.



3
    The language is often augmented by also specifying a (partial) order relation over the standpoint operators, which
    indicates that one standpoint may be more specific or more general than another. For the purposes of the current
    paper, we do not consider this elaboration the basic system.
3.1.2. Semantics
In order to characterise the semantics of standpoint logic π•Š0 , we specify a class π”π•Š0 of structures
of the form ⟨Π, 𝑆, π›ΏβŸ© where:

        β€’ Ξ  = {… , πœ‹π‘– , …} is a set of precisifications (which are analogous to possible worlds),

        β€’ 𝑆 = is a set {𝑠1 , ..., 𝑠𝑛 , βˆ—} of subsets of Ξ ,

        β€’ 𝛿 ∢ 𝒫 β†’ β„˜(Ξ ) is a function mapping each propositional variable 𝑝 ∈ 𝒫 to the set 𝛿(𝑝) βŠ† Ξ 
          of precisifications at which it is true. (Here, β„˜(Ξ ) denotes the powerset of Ξ .)

   The most distinctive elements of the model are the 𝑠𝑖 , which model the notion of standpoint
via a set of precisifications that are admissible for that standpoint. Thus, if πœ‹ ∈ 𝑠𝑖 then all
propositions that are unequivocally true according to standpoint 𝑠𝑖 are true at precisification πœ‹.
   For a model structure β„³ ∈ π”π•Š0 , we write (β„³, πœ‹) ⊧ πœ™ to mean that formula πœ™ is true at a
precisification πœ‹ ∈ Ξ  in β„³. For a model β„³ = ⟨Π, 𝑆, π›ΏβŸ©, this relationship is defined by:

        β€’ (β„³, πœ‹) ⊧ 𝑃 if and only if πœ‹ ∈ 𝛿(𝑃),

        β€’ (β„³, πœ‹) ⊧ ¬𝛼 if and only if (β„³, πœ‹) ⊭ 𝛼,

        β€’ (β„³, πœ‹) ⊧ 𝛼 ∧ 𝛽 if and only if (β„³, πœ‹) ⊧ 𝛼 and (β„³, πœ‹) ⊧ 𝛽,

        β€’ (β„³, πœ‹) ⊧ ░𝑠𝑖 𝛼 if and only if (β„³, πœ‹ β€² ) ⊧ 𝛼 for all πœ‹ β€² ∈ 𝑠𝑖 .

        β€’ (β„³, πœ‹) ⊧ β–‘βˆ— 𝛼 if and only if (β„³, πœ‹ β€² ) ⊧ 𝛼 for all πœ‹ β€² ∈ Ξ .

  A formula πœ™ is valid if it is true at every precisification of every model in π”π•Š0 . In this case
we write βŠ§π•Š0 πœ™.
   The logic π•Š0 enables one to formalise the content of statements such as. β€œArg is definitely
either a crag or a boulder. This map labels it a crag, but I would say it could be called either,
although a boulder is different from a crag.” With Ba and Ca meaning respectively β€˜Arg is a
boulder’ and β€˜Arg is a crag’, one could write:

                       β–‘βˆ— (Ba ∨ Ca) ∧ β–‘map Ca ∧ ♦me Ba ∧ ♦me Ca ∧ β–‘me Β¬(Ba ∧ Ca)

3.2. Variable Reference Logic
We now generalise the standpoint semantics framework to define a first-order variable reference
logic 𝕍1 that can represent predication and quantification.4



4
    In this presentation, we omit specifying the semantics for propositional connectives and standpoint operators and
    just give the semantics for predication and quantification in terms of truth conditions at a particular precisification
    in a given model. A more comprehensive semantics could be given by incorporating, with slight modification, the
    relevant specifications from the propositional semantics.
3.2.1. Syntax
The language of 𝕍1 is built from a vocabulary 𝒱 = βŸ¨π’¦ , 𝒫 , 𝒬, 𝒩 , 𝒳 ⟩, comprising the following
symbols:
    β€’ 𝒦 = {… , K𝑖 , …} is a set of count-noun symbols (sortals),
    β€’ 𝒫 = {… , P𝑖 , …} is a set of individual property predicates,
    β€’ 𝒬 = {… , Q𝑖 , …} is a set of precise entity property predicates, (e.g. exact spatial properties)
    β€’ 𝒩 = {… , n𝑖 , …} is a set of proper name symbols.
    β€’ 𝒳 = {… , π‘₯𝑖 , …} is a set of nominal variable symbols.
   The symbols of 𝒦, 𝒫 and 𝒬 can all be applied as predicates, with the sortal symbols of 𝒦
also being used to specify a range of quantification. Symbols of both 𝒩 and 𝒳 can both occur as
arguments of predicates, although the variable symbols of 𝒳 are only meaningful in the context
of quantification.
   The set 𝑆 of formulae of 𝕍1 is the smallest set such that:

    β€’ {𝛼(𝜏 ) | 𝛼 ∈ (𝒦βˆͺ𝒫βˆͺ𝒬), 𝜏 ∈ (𝒩βˆͺ𝒳 )} βŠ† 𝑆             (contains all atomic predication formulae)
    β€’ {Β¬πœ™, (πœ™ ∧ πœ“ ), ░𝑠𝑖 πœ™, β–‘βˆ— πœ™} βŠ† 𝑆 for every πœ™, πœ“ ∈ 𝑆                (closed under connectives)
    β€’ (βˆ€K ∢ π‘₯)[πœ™] ∈ 𝑆 for every K ∈ 𝒦 every π‘₯ ∈ 𝒳 and every πœ™ ∈ 𝑆               (includes quantified
      formulae)

3.2.2. Semantics
The semantics for variable reference logic 𝕍1 will be based on structures ⟨𝐸, Ξ , 𝒱 , π›ΏβŸ© where:
    β€’ 𝐸 is the set of precise entities.
    β€’ Ξ  is the set of precisifications.
    β€’ 𝒱 = βŸ¨π’¦ , 𝒫 , 𝒬, 𝒩 , 𝒳 ⟩ is the non-logical vocabulary, as specified above,
    β€’ 𝛿 = βŸ¨π›Ώπ’¦ , 𝛿𝒫 , 𝛿𝒬 , 𝛿𝒩 , 𝛿𝒳 ⟩ is a denotation function that can be considered as divided into
      components specifying the denotations for each type of non-logical symbol (see below).
   To simplify the explanation of the semantics, I first define the semantic representation of
a indefinite individual. (Here, and in the following, 𝐡𝐴 denotes the set of all functions from
domain set 𝐴 into the range set 𝐡):
    β€’ 𝐼 = 𝐸 Ξ  is the set of (indefinite) individuals, each of which is a mapping from the set of
      precisificaiton indices Ξ  to the set of precise entities.
For each individual 𝑖 ∈ 𝐼 and each precisification πœ‹ ∈ Ξ , the value of 𝑖(πœ‹) will be a precise version
of individual 𝑖 according to precisification πœ‹.
The denotation functions for all the non-logical vocabulary of the language as follows:
    β€’ 𝛿𝒦 ∢ 𝒦 β†’ β„˜(𝐼 )Ξ  is a function mapping each sortal concept (count noun) in 𝒦 to a
      function from precisifications to sets of indefinite individuals.
On the basis of 𝛿𝒦 we define πΌπœ‹ = ⋃{𝛿𝒦 (K)(πœ‹) | K ∈ 𝒦 }, the set of all individuals of any sort
according to precisification πœ‹. We can now define:

    β€’ 𝛿𝒫 ∢ 𝒫 β†’ β„˜(𝐼 )Ξ  , such that, for all P ∈ 𝒫 we must have 𝛿𝒫 (P)(πœ‹) βŠ† πΌπœ‹ .

    β€’ 𝛿𝒬 ∢ 𝒬 β†’ β„˜(𝐸)      (each precise predicate is associated with a set of precise entities)

    β€’ 𝛿𝒩 ∢ 𝒩 β†’ β„˜(𝐼 )Ξ  , subject to the condition that, for all n ∈ 𝒩 we must have 𝛿𝒩 (n)(πœ‹) ∈ πΌπœ‹ .

    β€’ 𝛿𝒳 ∢ 𝒳 β†’ β„˜(𝐼 ), (but the semantically relevant denotations of variables are determined
      by sortals occurring in quantifiers)

3.2.3. Interpretation of Reference and Predication
For a model β„³ = ⟨𝐸, Ξ , 𝒱 , π›ΏβŸ© and precisification πœ‹ ∈ Ξ , the truth conditions for atomic
predication formulae are as follows:

    β€’ (β„³, πœ‹) ⊧ K(n) if and only if (𝛿𝒩 (n))(πœ‹) ∈ (𝛿𝒦 (K))(πœ‹)

    β€’ (β„³, πœ‹) ⊧ P(n) if and only if (𝛿𝒩 (n))(πœ‹) ∈ (𝛿𝒫 (P))(πœ‹)

    β€’ (β„³, πœ‹) ⊧ Q(n) if and only if ((𝛿𝒩 (n))(πœ‹))(πœ‹) ∈ 𝛿𝒬 (Q)

    β€’ (β„³, πœ‹) ⊧ K(π‘₯) if and only if 𝛿𝒳 (π‘₯) ∈ (𝛿𝒦 (K))(πœ‹)

    β€’ (β„³, πœ‹) ⊧ P(π‘₯) if and only if 𝛿𝒳 (π‘₯) ∈ (𝛿𝒫 (P))(πœ‹)

    β€’ (β„³, πœ‹) ⊧ Q(π‘₯) if and only if (𝛿𝒳 (π‘₯))(πœ‹) ∈ 𝛿𝒬 (Q)

   To make sense these specifications you need to be aware that evaluation of a symbol may
require zero, one, or two levels of de-referencing in relation to the precisification index πœ‹. You
should first note that the K and P predications are semantically equivalent. When applied
to a name constant, n, both the constant and the predicate get evaluated with respect to a
precisification, so that the name denotes a particular individual and the predicate denotes a set
of such individuals. When the argument is a variable rather than a name constant, the variable
directly denotes an individual without any need for evaluation relative to a precisification.
In the case of (exact) Q predications, individuals need to be further evaluated relative to the
precisification in order to obtain a precise entity, which can be tested for membership of the
precise set denoted by property Q. So although Q predicates are not themselves subject to
variation in relation to πœ‹ the impose an extra level of variability in the interpretation their
argument symbol.
   It may seem curious that the same precisification index πœ‹ is used both for mapping names
(and predicates) to individuals (and sets of individuals), and also for mapping from individuals
to precise entities. Thus the individual denoted by a name n in precisification πœ‹ is (𝛿𝒩 (n))(πœ‹)
and according to πœ‹ it also refers to the precise entity ((𝛿𝒩 (n))(πœ‹))(πœ‹). This slightly simplifies
the specification and does not appear to place any constraint on the semantics.
3.2.4. Interpretation of Quantification
To facilitate specification of the semantics for quantification, I define a meta-level operation
β„³ (π‘₯𝑖 β‡’πœ‰ ) on interpretation structures that enables us to replace the value of a variable with
a new value. For β„³ = ⟨𝐸, Ξ , 𝒱 , π›ΏβŸ©, with 𝒱 = βŸ¨π’¦ , 𝒫 , 𝒬, 𝒩 , 𝒳 ⟩, 𝛿 = βŸ¨π›Ώπ’¦ , 𝛿𝒫 , 𝛿𝒬 , 𝛿𝒩 , 𝛿𝒳 ⟩, and
                                                                                                  β€² ⟩ and
πœ‰ ∈ 𝐸 Ξ  let β„³ (π‘₯𝑖 β‡’πœ‰ ) be the structure β„³ β€² = ⟨𝐸, Ξ , 𝒱 , 𝛿 β€² ⟩, where 𝛿 β€² = βŸ¨π›Ώπ’¦ , 𝛿𝒫 , 𝛿𝒬 , 𝛿𝒩 , 𝛿𝒳
 β€²                                         β€²
𝛿𝒳 (π‘₯𝑗 ) = 𝛿𝒳 (π‘₯𝑗 ) for every π‘₯𝑗 β‰  π‘₯𝑖 and 𝛿𝒳 (π‘₯𝑖 ) = πœ‰.
   Finally we can specify the interpretation of a quantified formula:

    β€’ (β„³, πœ‹) ⊧ (βˆ€ K ∢ π‘₯)[πœ™(π‘₯)]       if and only if (β„³ (π‘₯𝑖 β‡’πœ‰ ) , πœ‹) ⊧ πœ™(π‘₯) for all πœ‰ ∈ 𝛿𝒦 (K)(πœ‹)

    β€’ (β„³, πœ‹) ⊧ (βˆƒ K ∢ π‘₯)[πœ™(π‘₯)]      if and only if (β„³ (π‘₯𝑖 β‡’πœ‰ ) , πœ‹) ⊧ πœ™(π‘₯) for some πœ‰ ∈ 𝛿𝒦 (K)(πœ‹)

  This is much the same as how one would specify quantification in a classical sorted logic.
The universally quantified formula is true just in case its immediate sub-formula is true for all
possible values of the variable, taken from the range of the sortal predicate.

3.2.5. What Can 𝕍1 Express?
So was the result worth all the work of defining that complicated semantics? Does it help
us understand and represent semantic variability and its effect on reference, predication and
quantification. Yes, I think so. Although the final definitions of the quantifiers seem simple,
one needs to consider that significant work has been done by the denotation functions for the
different kinds of symbol and the semantics given above for the different cases of predication.
Their effect is to allow quantification to operate at an intermediate level in the interpretation
of semantic variability of reference. The individuation of potential referents occurs prior to
quantification by establishing individuals in relation to a given interpretation of sortal predicates.
But these individuals can still be indeterminate in that they may correspond to many exact
entities.
   Consider the statement β€œThere is definitely a mountain in Equador, that some say is on the
equator and others say it is not”. This might be represented as:

        β–‘βˆ— (βˆƒ Mountain ∢ π‘₯)[ InEquador(π‘₯) ∧ β™¦βˆ— [OnEquator(π‘₯) ∧ β™¦βˆ— [Β¬OnEquator(π‘₯)] ]

    I believe that this kind of statement cannot be represented without using a semantics that
can account for both de dicto and de re forms of vagueness. Its truth conditions require that
there is something that is definitely a mountain and definitely in Equador but whose extent is
ill defined such that it could reasonably be said either that it does or does not lie on the equator.


4. Co-Predication and Deep Polysemy of Sortal Concepts
I now turn to another issue involving polysemy and reference, that I admit not having properly
considered during the whole time I was constructing the VRS logic described above: the issue
of co-predication [28, 26, 29, 30, 31]. When I first became aware that people were studying this
phenomenon as an issue in its own right, rather than just being a particular case of polysemy, I
was somewhat surprised. And even after preliminary consideration, I assumed that it could
be handled without special difficulty within the double level model of semantic indeterminacy
provided by VRS. However, once I had got more deeply acquainted with the problem I came to
realise that it does require special attention. This section will be a largely informal discussion
of the topic.

4.1. The Problem of Co-Predication
A sortal concept is one that carries with it criteria for individuating and counting entities. Hence
one might expect that sortals should at least be unambiguous when it comes to the fundamental
criteria for being an instance or at least the general ontological category of the objects that can
instantiate them. However this is not the case. Consider the sentence β€œThere is a book by Olaf
Stapledon on my bookshelf.” With the simplification of treating being on my bookshelf as a
simple predicate we would get the following β€˜naive’ classical representation:

                         βˆƒπ‘₯[Book(π‘₯) ∧ By(π‘₯, OS) ∧ OnMyBookShelf(π‘₯)]

A problem with this representation is that the predicate Book appears to be polysemous between
the sense of being an informational artifact and being a physical object. And, moreover, whereas
the authorship predicate, By, applies to the informational sense, the predicate OnMyBookShelf
applies to the physical sense. It is this application to a single referent of two predicates that
seem to describe very different kinds of object that is called co-predication. The issue is not just
that Book has two senses but that the predications seem to imply the existence of an object that
is an instance of both senses of Book.
   One might expect that this problem could be avoided by adopting the view that books are a
type of object that has both an information and a physical aspect and that different types of
predicate that can be applied to books must be interpreted with respect to the aspect appropriate
to the type of predicate (authorship applying to the informational aspect and physical properties
to the physical aspect of book). But, unfortunately, the idea of multi-aspect objects can only
account for very simple examples. Consider, the sentence: β€œThere are two books by Olaf
Stapledon on my bookshelf.”


                    βˆƒπ‘₯βˆƒπ‘¦[π‘₯ β‰  𝑦 ∧ Book(π‘₯) ∧ Book(𝑦)
                                ∧ By(π‘₯, OS) ∧ By(𝑦, OS)
                                ∧ OnMyBookShelf(π‘₯) ∧ OnMyBookShelf(𝑦)]

   The problem is that determining how many books by Olaf Stapledon are on my self depends
upon whether I count in terms of informational artifacts or physical volumes. I might have
two copies of the same book, or two book titles contained within the same volume (or some
more complex combination of volumes and contents). Thus, a claim regarding the number of
books can only have definite meaning once I choose what kind of book object I wish to count.
But accepting this leads to a recapitulation of the original problem. Once I choose between
informational and physical books, I am no longer dealing with entities that can support both
the informational property of the books content originating from a particular author and also
the physical property of being a physical object located on a particular shelf.
   Before considering some proposals made in the literature regarding how one might account for
co-predication, let us first note that the VRS theory of predicates and vague objects given above
(Section 3.2) fails to provide a straightforward explanation of typical co-predication examples.
That semantics both allows vagueness of sortal predicates that categorise and individuate objects
over which we may quantify and also allows these objects to have properties that are not fully
determinate, such as a mountain, whose boundary is unclear. So maybe the complicated VRS
semantics could make sense of the following formulation:

                          (βˆƒ Book ∢ π‘₯)[ByOS(π‘₯) ∧ OnMyBookShelf(π‘₯)]

But for this to work, we would need to assume that the polysemy of book is within the semantic
object denoted by π‘₯ so that the two predicates can be evaluated with respect to different aspects
of some multi-faceted book object that has both physical and informational aspects. However,
the problem of individuation and counting still remains. Although VRS allows that Book can
have different senses both in terms of individuation and in terms of exact entities, it is based on
the assumption that any ambiguity in individuation criteria is resolved prior to determining
the set of individuals over which the quantifier (βˆƒ Book ∢ π‘₯) will range. Hence, although we
can have both indeterminate counting criteria and indeterminate individuals, we cannot have
indeterminate individuals whose precise correlates satisfy different counting criteria. In the case
of vague of geographic features, for example, one might say that β€˜individuation must precede
demarcation’.

4.2. Towards a Solution
Many approaches, some very clever and most rather complex, have been proposed for accounting
for co-predication. I will not consider these in the current paper except to mention that some
have proposed that co-predication involves mechanism for coercion or inheritance between
different categories of entity that can be referred to by polysemous nouns that exhibit co-
predication phenomena [28, 30], some have proposed special kinds of compound ontological
categories (e.g. [26]) and some have proposed complex semantics of individuation (e.g. [29]).
None of the proposals I have encountered so far seems completely satisfactory to me.
   My view is that different individuation conditions must be associated with different precisifi-
cations of the count noun. I don’t think we can individuate objects unless we have restricted the
interpretation of the term to the extent that it enables individuation. The VRS approach does
allow different modes of individuation to be associated with different senses of a sortal concept,
which could account for the radically different ontological types associated with certain count
nouns such as β€˜book’. But if this is the case, we still need to account for co-predication examples
where it seems that individual objects exhibit multiple ontologically diverse aspects. Such pos-
sibilities would need to be specified in terms of type conversions that can occur due to context,
and are often unambiguous due to corresponding entities of a different type being uniquely
determined in many cases (e.g. in many particular situations a physical book instance uniquely
determines and informational book and a reference to an informational book determines a
unique physical book).
5. Conclusions and Further Work
The paper has explored the issue of multiple possible references of linguistic terms, that may
arise due to vagueness of terms or differences in opinions on how they should be interpreted,
and how such variability effects the semantics of reference, predication and quantification, as
conceived within a denotational approach to semantics. I have proposed the framework of
variable reference semantics to interpret a logical language in a way that can account for both de
dicto vagueness in predicates and also de re, or, as I would prefer to call it de sensu variability in
the β€˜objects’ that are referred to.
   The system presented is intended more as a proof of concept than a workable formal language.
The semantics is rather complex but when formulating representations within the object
language of the logic 𝕍1 much of this complexity is hidden under the bonnet. However, the
utility of the system remains to be demonstrated beyond a few relatively simple examples. The
main direction for further work would be to consider a much wider range of examples and
evaluate the strengths and limitations of the proposed formalism. As for formal logic in general,
I do not really envisage the system being used in its full blown form, but rather some aspects
of the language and semantics might be used in a restricted form to support various kinds of
application, such as, for example, querying of information systems.
   The other direction in which I would like to extend the work is with further consideration of
issues such as co-predication, which would test the limitations of the system in dealing with
semantic phenomena related to semantic variability.


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