Vagueness in Predicates and Objects

TL;DR

This paper introduces Variable Reference Semantics, a framework that models the variability and vagueness in predicates and objects, extending classical semantics to better handle context, opinions, and definitional diversity.

Contribution

It proposes a novel semantic framework that generalizes classical models to incorporate multiple and variable referents for predicates and objects.

Findings

Framework accommodates multiple semantic referents.

Handles vagueness and contextual variability.

Extends classical semantics to more flexible models.

Abstract

Classical semantics assumes that one can model reference, predication and quantification with respect to a fixed domain of precise referent objects. Non-logical terms and quantification are then interpreted directly in terms of elements and subsets of this domain. We explore ways to generalise this classical picture of precise predicates and objects to account for variability of meaning due to factors such as vagueness, context and diversity of definitions or opinions. Both names and predicative expressions can be given either multiple semantic referents or be associated with semantic referents that incorporate some model of variability. We present a semantic framework, Variable Reference Semantics, that can accommodate several modes of variability in relation to both predicates and objects.

Equations

$$t^{.I,v}=\{\begin{array}{ll}v(x)& \text{if }t=x\in X,\\ a^{.I}& \text{if }t=a\in N.\end{array}$$

$$M,\pi ,v\models \text{mbox}\mathsf{P}(t_1,\dots ,t_k)$$

$$M,\pi ,v\models \neg \varphi$$

$$M,\pi ,v\models \varphi \wedge \psi$$

$$M,\pi ,v\models \forall x\varphi$$

$$M,\pi ,v\models {\square }_s\varphi$$

$$M,\pi \models \varphi$$

$$M\models \varphi$$

$$\exists a{\square }_{\ast }(\text{mbox}\mathsf{AridArea}(a)\wedge \text{mbox}\mathsf{Near}(\text{mbox}\mathsf{house},\text{mbox}\mathsf{a}))\wedge {◊}_{\ast }\text{mbox}\mathsf{Desert}(\text{mbox}\mathsf{a})$$

$${\square }_{\ast }\exists a(\text{mbox}\mathsf{AridArea}(a)\wedge \text{mbox}\mathsf{Near}(\text{mbox}\mathsf{house},\text{mbox}\mathsf{a})\wedge \neg \text{mbox}\mathsf{PartOf}(\text{mbox}\mathsf{house},\text{mbox}\mathsf{a})\wedge \text{mbox}\mathsf{Desert}(\text{mbox}\mathsf{a}))$$

$$\exists a{\square }_{\ast }(\text{mbox}\mathsf{AridArea}(a)\wedge \text{mbox}\mathsf{Desert}(a))\wedge {◊}_{\ast }\text{mbox}\mathsf{PartOf}(\text{mbox}\mathsf{house},\text{mbox}\mathsf{a})\wedge \neg {\square }_{\ast }\text{mbox}\mathsf{PartOf}(\text{mbox}\mathsf{house},\text{mbox}\mathsf{a})$$

$${\tau }^{.\pi ,v}=\{\begin{array}{ll}v(x)& \text{if }\tau =x\in X,\\ \text{mbox}{\mathsf{n}}^{.{\delta }_N(\pi )}& \text{if }\tau =\mathsf{n}\in N.\end{array}$$

$$M,\pi ,v\models \text{mbox}\mathsf{K}(\tau )$$

$$M,\pi ,v\models \text{mbox}\mathsf{A}({\tau }_1,\dots ,{\tau }_k)$$

$$M,\pi ,v\models \text{mbox}\mathsf{Q}({\tau }_1,\dots ,{\tau }_k)$$

$$M,\pi ,v\models \neg \varphi$$

$$M,\pi ,v\models \varphi \wedge \psi$$

$$M,\pi ,v\models \forall x\varphi (x)$$

$$M,\pi ,v\models {\square }_s\varphi$$

$$\exists a({\square }_{\ast }\text{mbox}\mathsf{AridArea}(a)\wedge {◊}_{some}\text{mbox}\mathsf{Desert}(a)$$

$$\wedge {◊}_{others}(\exists b,c(b\ne c)\wedge \text{mbox}\mathsf{PartOf}(b,a)\wedge \text{mbox}\mathsf{PartOf}(c,a)\wedge \text{mbox}\mathsf{Desert}(b)\wedge \text{mbox}\mathsf{Desert}(c)))$$

$$\wedge {◊}_{\ast }\exists x\text{mbox}\mathsf{Desert}(a)\wedge \text{mbox}\mathsf{PartOf}(\text{mbox}\mathsf{house},x)$$

$$\mathrm{trans}(\text{mbox}\mathsf{K}(\tau ))=\text{mbox}\mathsf{K}(\tau )$$

$$\mathrm{trans}(\text{mbox}\mathsf{A}(\vec{\tau }))=\text{mbox}\mathsf{A}(\vec{\tau })$$

$$\mathrm{trans}(\forall x\varphi (x))=\forall x(\text{mbox}\mathsf{ink}(x)\to \mathrm{trans}(\varphi ))$$

$$\displaystyle\mathrm{trans}(\mbox{\sf Q}(\tau_{1},\hbox to8.00003pt{.\hss.\hss.},\tau_{k}))=\exists e_{1},\hbox to8.00003pt{.\hss.\hss.},e_{k}(\mbox{\sf Q}(e_{1},\hbox to8.00003pt{.\hss.\hss.},e_{k})\mathrel{\wedge}\mbox{\sf prec}(\tau_{1},e_{1}))\ \wedge\ \hbox to8.00003pt{.\hss.\hss.}\ \wedge\ \mbox{\sf prec}(\tau_{k},e_{k}))\$$

$$\Phi =$$

$$\bigwedge {}_{\text{scaleobj}1\mathsf{Q}\in Q}\forall \vec{x}(\text{mbox}\mathsf{Q}(\vec{x})\leftrightarrow {\square }_{\ast }\text{mbox}\mathsf{Q}(\vec{x}))$$

$$\forall x\exists y(\text{mbox}\mathsf{ind}(x)\to \text{mbox}\mathsf{prec}(x,y)),$$

$$\forall x_1,..,x_k(\text{mbox}\mathsf{A}(x_1,..,x_k)\to \text{mbox}\mathsf{ink}(x_1)\wedge to8.00003pt.\text{hss}.\text{hss}.\wedge \text{mbox}\mathsf{ink}(x_k)),$$

$$\forall x(\text{mbox}\mathsf{ink}(x)\to \text{mbox}\mathsf{ind}(x))\wedge (\text{mbox}\mathsf{ink}(x)\leftrightarrow \bigvee {}_{\text{scaleobj}0.75\mathsf{K}\in K}\text{mbox}\mathsf{K}(x))\}$$

$$\mathrm{Trans}(\psi ):=\mathrm{trans}(\psi )\wedge \underset{\varphi \in \Phi }{\bigwedge }{\square }_{\ast }\varphi$$

Full text

and

1 Introduction

The notion of reference and the operations of predication and quantification are fundamental to classical first-order logic. Its standard semantics assumes a fixed domain of objects, with name constants referring to unique elements of the domain, predicates associated with subsets of the domain and quantifiers ranging over all the elements of the domain. Thus, if ${D}$ is the domain of objects, then a constant name, say c, will denote an object $\delta(\mbox{\sf c})$, such that $\delta(\mbox{\sf c})\in{D}$, and a predicate $P$ will be taken to denote a set of objects $\delta(P)$, with $\delta(P)\subseteq{D}$. In this way, one can straightforwardly interpret the predicating expression $P(\mbox{\sf c})$ as a proposition that is true if and only if $\delta(\mbox{\sf c})\in\delta(P)$. However, in natural languages there is considerable variability in the interpretation of 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.

Building on [1], this paper explores how the classical semantics can be modified to account for semantic variability. We shall first consider some general ideas regarding the nature of vagueness and its sources. The distinction between linguistic (de dicto) and ontological (de re) vagueness will be considered, and we will suggest that the sources of vagueness within the interaction between language and the world are more diverse than is often assumed. We proceed to consider possible models of denotation, predication and quantification in the presence of indeterminacy, first informally, with the aid of some diagrams and then in terms of a formal framework, based on standpoint semantics [2, 3, 4], within which variable references can be modelled.

2 Theories of Vagueness

The literature on vagueness has generally assumed that the phenomena of vagueness could arise from three potential sources [5]: (a) indeterminacy of representation (linguistic a.k.a de dicto vagueness), (b) indeterminacy of things in the world (known as ontic of de re vagueness) and (c) limitations of knowledge (epistemic vagueness). The epistemic view has some strong advocates (e.g. [6]) and it is apparent that semantics of multiple possible interpretations take a similar form to logics of knowledge and belief [7]. Indeed, the standpoint semantics, which will be used in our following analysis can be regarded as being of this form. But the question of whether vagueness is an epistemic phenomena, or merely shares a similar logic, does not seem to have any direct bearing on our semantic analysis, so in the current paper we shall not further consider the epistemic view.

2.1 Vagueness as exclusively De Dicto

A widely held view is that all vagueness is essentially de dicto. That is, it is entirely a property of language, not of the world: it consists in the variability or lack of precision in the way that words and phrases correspond to the precise entities and properties of the real world that they aim do describe. According to this view, forms of natural language that seem to attribute vagueness directly to things in the world are misleading and can be paraphrased in de dicto terms [8, 9]. 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:

“[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.” [9]

The view that vagueness is a linguistic phenomenon leaves open many possibilities regarding which aspects of language are involved, and how one might model vagueness within a theoretical semantic framework. These will be explored in detail below.

2.2 The possibility of De Re or Ontic Vagueness

Natural language sentences commonly describe objects as ‘vague’: ‘The smoke formed a vague patch on the horizon’; ‘The vaccine injection left a vague mark on my arm’; ‘He saw the vague outline of a building through the fog’. Here ‘vague’ means something like ‘indefinite in shape, form or extension’. If we take such sentences at face value, they seem to indicate that vagueness can be associated with an object in virtue of its spatio-temporal extension, its material constituency or even abstract characteristics. One may then argue that such properties are intrinsic to objects, and that, if an object has a vague intrinsic property, this indicates vagueness ‘of the thing’, often called de re vagueness. And, in so far as things are identified with entities existing in the world, such vagueness is often described as ontic (or ontological) meaning that it pertains to what exists in reality.

Note that the view that vagueness may be de re or ontic does not require one to deny that vagueness is often de dicto. Even if the world consisted entirely of precisely demarcated objects with exact physical properties, there could still be vague terminology (e.g. adjectives such as ‘large’) and hence vague ways to describe the world. Proponents of de re vagueness typically only claim that some kinds of vagueness are ontic [10, 11].

The idea that vagueness of objects is primarily associated with vagueness of spatial extension has been endorsed and examined by Tye [10], who gives the following criterion for identifying vague objects: “A concrete object $o$ is vague if and only if: $o$ 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 $o$.” 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. However, we will be presenting a semantics in which there can be objects whose spatial extension, and hence material constituents, are indeterminate.

The case against ontic vagueness is typically based on two contentions: (a) the idea of an object in the world being in itself indeterminate is mysterious and implausible; (b) statements that appear to imply ontic vagueness are readily paraphrased into forms where the vagueness is evidently de dicto. While it is true that, at the level of atomic particles, we may accept that the position of an electron might be indeterminate (as is strongly indicated by quantum mechanics), this kind of indeterminacy seems very far from what we would need to account for vagueness in the demarcation of macroscopic physical objects. Consider a pile of twigs upon a twig-strewn forest floor. The indeterminacy of whether a particular twig is part of the pile does not arise from any indeterminacy in the physical locations of twigs. Rather, it seems that, for exactly the same physical situation, there can be different judgements about which twigs should be counted as part of the pile. Nevertheless, it may still be reasonable (in accord with Tye) to take the view that the extension of the pile of twigs is indeterminate.

So perhaps it is reasonable to consider that a ‘thing’ such as a pile of twigs could have an indeterminate extension, even though there is no ontic indeterminacy in the reality within which the twigs are manifest. Hence, if a pile of twigs is considered a ‘thing’, the indeterminacy of its extension would be de re, though not ontic. Here, we see that there is an ambiguity in the meaning of ‘thing’ (and hence in the term de re). One can interpret it in the sense of an entity existing in a concrete reality or in the sense of an object of discourse, whose physical manifestation need not be completely determinate.

2.3 Established Accounts of Vagueness: Fuzzy Logic and Supervauationism

Among computer scientists fuzzy logic [12] is the dominant approach to modelling vagueness. This theory modifies the classical denotations of expressions. Vague concepts denote fuzzy sets such that the degree of membership of an object in a fuzzy set is graded (typically represented by a real number in the range [0..1]). This degree of membership also determines fuzzy truth values in that the truth of $\mbox{\sf C}(c)$ is simply the degree of membership of the object denoted by $c$ in the fuzzy set denoted by C and the truth value of complex formulae is then determined by interpreting logical operators in terms of fuzzy truth functions. Although the original presentation of fuzzy logic does not treat objects as vague, many researchers have also modelled vague objects as fuzzy sets of points [13].

Many philosophers favour some variety of supervaluationist account, which models linguistic vagueness in terms of variability in the relation between vocabulary terms and their semantic reference. This is modelled in terms of a set 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 [14], and a formal semantics based on a multiplicity of classical interpretations was used by van Fraassen [15] to explain ‘the logic of presupposition’. This kind of formal model was subsequently applied to the analysis of vagueness by Fine [16] and a similar approach was proposed by [17].

An attraction of supervaluationism is its account penumbral connections [16], which many believe to be an essential characteristic of vagueness. This is the phenomenon whereby logical laws (such as the principle of non-contradiction) and semantic constraints (such as mutual exclusiveness of properties such as ‘… is short’ and ‘… is tall’) 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.

3 Semantic Aspects of Predicates and Objects in Relation to Vagueness

The semantic content of a predicate can be analysed into many aspects. We pick out the following as being of particular relevance to the study of vagueness:

1. Classification: Given a set of objects, a predicate classifies them into members or non-members of the set.
2. Individuation: Given a relevant information about a state of the world (its spatial and material properties) the meaning of a sortal predicate provides the criteria by which individual instances of the predicate are recognised. (In terms of axioms these criteria can often be divided into existential conditions and identity criteria.)
3. Demarcation/Constituency: The meaning of sortal predicates also provides criteria for spatially demarcating and determining the material constituents of its instances.

In classical logic ‘classification’ is the most prominent, perhaps even the only, aspect of predication. This is because in the usual semantics for first-order logic, the ‘universe’ or ‘domain of quantification’ is typically presented as if ontologically prior to predication, in that we ‘interpret’ predicates in relation to a given domain. However, in order to name an object or count instances of a category of object we need to be able to individuate the object or objects to which we refer, and this implies that individuation is necessary for (and one could say ontologically prior to) classification. Furthermore, as illustrated by the ‘desert’ examples below, individuating objects as instances of a certain kind does not necessarily require that they be identified with precisely demarcated portions of the real world. Hence, the objects of discourse referred to by names or by quantification may not be completely definite in all their attributes. Accounting for the independence between criteria for individuation, classification and demarcation in the presence of vagueness is a motivation for the semantics that we shall present below.

3.1 Classifying, Individuating and Demarcating Deserts

To illustrate how the different aspects of predication operate, we consider the concept ‘desert’. Several factors may be relevant to deciding what is a desert, in particular: lack of precipitation (aridity), temperature and to a lesser extent vegetation, land form and land use. For simplicity, we consider a situation where a measure of aridity is taken to be the only relevant factor (we may assume that other factors are either uniform or co-vary with aridity within the region under consideration). Consider the situations illustrated in Fig. 1 depicting several states of a region with ‘my house’ indicated by a black icon:

• (a)

The region was cool and fertile although two of the most arid areas were arguably deserts. My house was near to one arid area that was arguably a desert.

• (b)

The climate warmed and the arid areas became true deserts, but with a fertile valley between them. My house was then near to but unequivocally not part of a desert.

• (c)

Eventually the desert areas expanded and merged into one big desert. Then, my house was at its boundaries, arguably though not unequivocally in the desert.

• (d)

Recently a little moisture has returned along the valley. Some argue that the whole arid area is still a desert. Others refer to two deserts in that area, separated by the valley. Either way, my house is still arguably in a desert.

The narrative sequence is not relevant to present concerns, except that it shows that situations of the same general type may exhibit different aspects of vagueness depending on specific details. What we should particularly note is that: in (a) the classification of the arid area as desert is equivocal; in (b) and (c) we definitely have two deserts but each has an indeterminate boundary; in (c) we definitely have exactly one desert, but it is debatable whether my house is within the desert area; and, in (d) the number of deserts present is indeterminate. The semantics that we shall develop is motivated by considering situations such as those depicted Fig. 1, where there is a separation between the aspects of classification, individuation and demarcation with respect to vagueness.

3.2 Possibilities for Semantic Modelling of Vague Predicates and Objects

Vagueness of both predicates and names can be modelled either in terms of the mapping from vocabulary to some form of ‘semantic denotation’ or in the correspondence between semantic denotations and the real world. Or, indeed, each of these relationships could model a different aspect of vagueness. Fig. 2 illustrates some possibilities. Fig. 2(a) depicts a hilly region with rocky crags, where the terrain is irregular and there is no clear way of dividing it into separate ‘crag’ objects. The name ‘Arg Crag’ has been given to one of the rocky outcrops. However, there may be different opinions regarding exactly which outcrop is Arg Crag. Indeed, some people may use the name to refer to the whole of this rocky area, whereas others consider that it refers to a more specific rock structure.

In classical semantics each conceptual term denotes a fixed set of entities and each name refers to a single precise entity. Thus, even before specifying denotations, we will need to divide the craggy region into specific individual objects, to make a set of possible referents. Figs. 2(c)–(f) depict various ways in which vaguenes of predicates and objects can be modelled. Sub-figures (c) and (d) show fuzzy-logic and supervaluation semantics models. In (c) C denotes a fuzzy set, whereas in (d) both the predicate and object have a variable reference (with each precisification determining a unique reference). Sub-figure (e) depicts a possible combination of the fuzzy and supervaluation approaches, where predicate symbols can have multiple references, each of which is a fuzzy set.

A limitation of models (c)–(e) is that fuzziness/variability is only associated with the predicate and/or the reference relation but not the object referred to. However, both approaches can be modified to incorporate vague referents, that might correspond to objects with indeterminate physical boundaries. In fuzzy logic, objects with indefinite extensions can be modelled as fuzzy sets of points [13]. And within a multiple-reference semantics, one could also associate a set of different extensions for different precise versions of an object (see e.g. the ‘egg-yolk’ representation of [18]).

Fig. 2(f) 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 be called de re vagueness, but we regard it as semantic vagueness, with the objects being vague semantic objects (visualised as the small black discs) that potentially correspond to many precise physical extensions.

It may appear that the two stages of multiple reference are redundant. The relation goes first from the name a to one of the indefinite objects and then to a precise extension; but surely we would get the same potential precise referents for a if we simply assign all the precise referents we could reach as possible precise referents of a, with no need for an intervening indefinite object. However, this would force us to conflate the different types of vagueness illustrated in the desert examples above. So we could not have something that was definitely a desert or a mountain but whose extension was indefinite. Moreover, we believe that quantification operates in relation to vague individuals not their precise possible extensions. This will be made explicit in our formal semantics and underpins our solution to the ‘Problem of the Many’, which we shall discuss in Sec. 6.

Figure 2(f) still considerably simplifies the potential semantic variability that might arise. It presupposes 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 it individuates them. The logic ${\mathbb{V}_{1}}$ 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 ‘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.

4 A Semantic Theory of Variable Reference

We now consider what kind of semantics can account for the general form of variable denotation illustrated in Fig. 2(f). Standpoint Semantics provides a reasonably general framework within which semantic variability can be modelled in terms of the symbols of a formal language having multiple possible denotations. We shall start by introducing first-order standpoint logic and then elaborate this to the formalism Variable Reference Logic, within which we can model predication and quantification of vague objects.

4.1 Standpoint Logic

Standpoint Semantics is based on a formal structure that models semantic variability in terms of the following two closely connected components:

• •

A precisification is a precise and consistent interpretation of a vague language, and it coherently assigns precise denotations to all its vague elements. A model for a vague sentence contains a set of admissible precisifications of that sentence.

• •

A standpoint is modelled as a set of precisifications which are compatible with a point of view or context of language understanding. A standpoint can capture both explicit specifications of terminology and implicit constraints on meanings that arise in conversation (e.g. “That person is tall” constrains the meaning of tallness).

4.1.1 Syntax of First-Order Standpoint Logic

First-order standpoint logic (${\mathbb{S}_{{\mathrm{FO}}}}$) is based on a vocabulary $\langle{P},{N},{S}\rangle$, consisting of predicate symbols ${P}$ (each associated with an arity $n\in\mathbb{N}$), constant symbols ${N}$ and standpoint symbols ${S}$, usually denoted with $s,s^{\prime}$, such that \hbox{*}\in{S}, where $*$ is used to designate the universal standpoint. There is also a set ${X}=\{x,y,\hbox to8.00003pt{.\hss.\hss.}\}$ of variables, and the set ${T}={N}\cup{X}$ of terms contains all constants and variables. The set ${\mathbb{S}_{{\mathrm{FO}}}}$ of first-order standpoint formulae is given by

$\phi,\psi:=\mbox{\sf P}(t_{1},\dots,t_{k})\mid\neg\phi\mid\phi\ \wedge\ \psi\mid\forall x\phi\mid\mathop{\square}_{s}\phi\ ,$

where $\mbox{\sf P}\in{P}$ is a $k$-ary predicate symbol, $t_{1},\dots,t_{k}\in{T}$ are terms, $x\in{X}$, and $s\in{S}$. The definable connectives and operators $\mathbf{t}$, $\mathbf{f}$, $\phi\mathrel{\vee}\psi$, $\phi\rightarrow\psi$, $\exists x\phi$, and $\mathop{\lozenge}_{s}\phi$ are as usual.

4.1.2 Semantics of First-Order Standpoint Logic

Given a vocabulary $\langle{P},{N},{S}\rangle$, a first-order standpoint structure ${M}$ is a tuple \langle\Delta,\hbox{\Pi},\sigma,\delta\rangle, where $\Delta$ is the (non-empty) domain of ${M}$, $\Pi$ is a non-empty set of precisifications, $\sigma$ is a function mapping each standpoint symbol from ${S}$ to a set of precisifications (i.e., a subset of $\Pi$), and $\delta$ is a function mapping each precisification from $\Pi$ to an ordinary first-order structure ${I}$ over the domain $\Delta$, whose interpretation function $\cdot^{{I}}$ maps:

• •

each predicate symbol $\mbox{\sf P}{\,\in\,}{P}$ of arity $k$ to an $k$-ary relation $\mbox{\sf P}^{.{I}}{\,\subseteq\,}\Delta^{k}$,

• •

each constant symbol $a{\,\in\,}{N}$ to a domain element $a^{.{I}}{\,\in\,}\Delta$.

The semantics of [19] required that a^{.\delta(\hbox{\pi}_{1})}=a^{.\delta(\hbox{\pi}_{2})} for any \hbox{\pi}_{1},\hbox{\pi}_{2}\in\hbox{\Pi} and $a\in{N}$. Here we allow non-rigid constants that may denote different objects in different precisifications. But (in contrast to the more expressive ${\mathbb{V}_{1}}$ that will be developed below) all precisifications of a first-order standpoint structure will implicitly share the same interpretation domain $\Delta$; that is, the constant domain assumption is adopted. The most distinctive elements of the model are the $s_{i}$, which model the notion of standpoint as a set of precisifications that are admissible for that standpoint. A proposition is unequivocally true according to standpoint $s_{i}$ iff it is true at every precisification \hbox{\pi}\in\hbox{s_{i}}.

Let {M}=\langle\Delta,\hbox{\Pi},\sigma,\delta\rangle be a first-order standpoint structure for the vocabulary $\langle{P},{N},{S}\rangle$ and ${X}$ be a set of variables. A variable assignment is a function $v:{X}\to\Delta$ mapping variables to domain elements. Given a variable assignment $v$, we denote by $v_{\left\{x\mapsto\epsilon\right\}}$ the function mapping $x$ to $\epsilon\in\Delta$ and any other variable $y$ to $v(y)$.

An interpretation function $\cdot^{{I}}$ and a variable assignment specify how to interpret terms as domain elements:111 Here, and in the following, $\sigma^{.\gamma}$ is a concise notation for $\gamma(\sigma)$. It gives the semantic denotation of symbol $\sigma$ according to a function $\gamma$.

[TABLE]

Then, let \hbox{\pi}\in\hbox{\Pi} and $v:{X}\to\Delta$ be a variable assignment. The satisfaction relation $\models$ is:

[TABLE]

When ${M}\models\phi$ we say that ${M}$ is a model for $\phi$.

The logic ${\mathbb{S}_{{\mathrm{FO}}}}$ enables one to formalise the content of some statements discussed in Section 3.1, namely (1), “My house was near to one arid area that was arguably a desert” and (2), “My house was then near to but unequivocally not part of a desert”. We assume that Desert and AridArea are vague predicates, but (my) house has definite coordinates and PartOf is a definite predicate.

[TABLE]

It should however be apparent that addressing examples involving vague objects with ${\mathbb{S}_{{\mathrm{FO}}}}$ becomes less straightforward and would involve, at least, a reasonable amount of paraphrasing. For instance, one may be inclined to formalise sentence (3) as

[TABLE]

However, if the domain only contains precise individuals (e.g. all possible areas), each specific area $a$ has the same extension in all precisifications, so house cannot be arguably but not definitely part of it. Hence, this formula is unsatsifiable in ${\mathbb{S}_{{\mathrm{FO}}}}$, unless PartOf can be indefinite, which is contrary to its intended (purely geometrical) meaning.

4.2 Variable Reference Logic

We now generalise standpoint semantics to define variable reference logic, ${\mathbb{V}_{1}}$, that can represent predication and quantification over indefinite objects. The most significant complication of ${\mathbb{V}_{1}}$ in relation to ${\mathbb{S}_{{\mathrm{FO}}}}$ is that rather than the domain being comprised only of atomic entities, we also introduce ‘indefinite individuals’ that are modelled as functions from precisifications to precise entities. Another complication is that predicates are divided into: ‘sortals’ (corresponding to count nouns such as ‘desert’ or ‘cat’) that determine subdomains of the domain of indefinite individuals, ‘indefinite predicates’ that classify or relate indefinite individuals, and ‘precise entity predicates’ that represent properties or relations of exact spatio-temporal and/or material extensions in reality.

With respect to sortals, individual properties and quantification, indefinite individuals are treated as basic objects (e.g. a particular mountain or fluffy cat). But in the application of a ‘precise entity predicate’ (e.g. ‘… contains $7.3{\times}10^{26}$ atoms’) the individual is evaluated with respect to a precisification to obtain a precise entity (the individual’s extension) that determines the truth of the precise predicate. Although our semantics allows indefinite individuals to be any function from precisifications to precise entities, instances of sortals will typically by highly constrained by axioms expressing conditions arising from the meaning of the sortal. For example, if a desert is a maximal area of the Earth’s surface satisfying certain conditions (e.g. low precipitation) then, even if exact thresholds on the parameters of these conditions are not given, the set of indefinite individuals that could correspond to a desert is hugely restricted.

4.2.1 Syntax

The language of ${\mathbb{V}_{1}}$ is built from a vocabulary ${V}=\langle{K},{A},{Q},{N},{S}\rangle$, comprising:

• $\bullet$ 

${K}=\{\ldots,\mathsf{K}_{i},\ldots\}$,    sortal predicates (e.g. Desert, Mountain, Cat),

• $\bullet$ 

${A}=\{\ldots,\mathsf{A}_{i},\ldots\}$,    indefinite predicates (e.g. Dry, Fluffy, Climbed),

• $\bullet$ 

${Q}=\{\ldots,\mathsf{Q}_{i},\ldots\}$,    precise entity predicates (e.g. exact spatial properties),

• $\bullet$ 

${N}=\{\ldots,\mathsf{n}_{i},\ldots\}$,    proper name symbols (e.g. everest, tibbles).

• $\bullet$ 

{S}\,=\{\ldots,s_{i},\ldots,\hbox{*}\},    standpoint symbols.

Let ${P}={K}\cup{A}\cup{Q}$ be the set of predicate symbols, containing all sortals, indefinite predicates and precise entity predicates. Again, there is a set of variables ${X}$, and the set ${T}={N}\cup{X}$ contains all proper name symbols and variables. Then, the set ${\mathbb{V}_{1}}$ of variable reference logic formulae is given by

$\phi,\psi\ \ :=\ \ \mbox{\sf P}(\tau)\ \mid\ \neg\phi\ \mid\ \phi\!\ \wedge\ \!\psi\ \mid\ \forall x\,\phi\ \mid\ \mathop{\square}_{s}\phi\ ,$

where $\mbox{\sf P}\in{P}$, $\tau\in{T}$, $x\in{X}$, $\mbox{\sf K}\in{K}$ and $s\in{S}$.

4.2.2 Semantics

Given a vocabulary ${V}=\langle{K},{A},{Q},{N},{S}\rangle$, a variable reference logic structure is a tuple $\langle E,\Pi,\sigma,\delta\rangle$ where:

1. $\bullet$

$E$ is the set of precise entities. 2. $\bullet$

$\Pi$ is the set of precisifications. 3. $\bullet$

$\sigma$ is a function mapping standpoint symbols to precisifications, and 4. $\bullet$

$\delta=\langle\delta_{{K}},\delta_{{A}},\delta_{{Q}},\delta_{{N}}\rangle$ is a denotation function divided into components specifying the denotations for each type of non-logical symbol (see below).

We define the set of indefinite individuals $I_{*}=Maps(\Pi,E)$ as the set of all mappings from precisifications to precise entities. For each indefinite individual $i\in I_{*}$ and $\pi\in\Pi$, $i(\pi)$ will be an element of $E$, which constitutes the precise version of individual $i$ according to precisification $\pi$. At each precisification $\pi\in\Pi$, each name symbol $\mathsf{n}\in{N}$ denotes some element of $I_{*}$ and each sortal $\mathsf{K}\in{K}$ and individual attribute predicate $\mathsf{A}\in{A}$ denotes a subset of $I_{*}$. The set of indefinite individuals at a precisification $\pi$ depends on the interpretation of the sortal predicates at $\pi$. The denotation functions map:

1. $\bullet$

$\delta_{{K}}:$ each precisification $\pi\in\Pi$ into a function from each sortal symbol $\mbox{\sf K}{\,\in\,}{K}$ to a subset $\mbox{\sf K}^{.\delta_{{K}}(\pi)}\subseteq I_{*}$ — i.e. a set of indefinite individuals.

On the basis of $\delta_{{K}}$ we define I_{\hbox{\pi}}=\bigcup\{\mathsf{K}^{.{\delta_{{K}}(\pi)}}\ |\ \mathsf{K}\in{K}\}, the domain of (indefinite) individuals according to precisification $\pi$. I_{\hbox{\pi}} is the set of all individuals of any sort, where the sortals are interpreted according to precisification $\pi$. 2. $\bullet$

$\delta_{{A}}:$ each precisification $\pi\in\Pi$ into a function from each indefinite predicate symbol $\mbox{\sf A}{\,\in\,}{A}$ of arity $k$ to a $k$-ary relation $\mbox{\sf A}^{.\delta_{{A}}}{\,\subseteq\,}I_{\pi}^{k}$, 3. $\bullet$

$\delta_{{Q}}:$ each entity predicate symbol $\mbox{\sf Q}{\,\in\,}{Q}$ of arity $k$ to $k$-ary relations $\mbox{\sf Q}^{.\delta_{{Q}}}{\,\subseteq\,}E^{k}$, 4. $\bullet$

$\delta_{{N}}:$ each precisification $\pi\in\Pi$ into a function from each indefinite name symbol $\mbox{\sf N}{\,\in\,}{N}$ to an indefinite individual $\mathsf{N}^{.{\delta_{{N}}(\pi)}}\in I_{\pi}$.

Let us now deal with the variable assignments in variable reference logic. Let ${M}=\langle E,\Pi,\sigma,\delta\rangle$ be a model for the vocabulary $\langle{K},{A},{Q},{N},{S}\rangle$ and ${X}$ be a set of variables. A variable assignment is a function $v:{X}\to I_{*}$ mapping variables to individuals. Given a variable assignment $v$, we denote by $v_{\left\{x\mapsto i\right\}}$ the function mapping $x$ to $i\in I_{*}$ and any other variable $y$ to $v(y)$.

The interpretation function $\delta_{\tiny{N}}$ and a variable assignment specify how to interpret terms by domain elements in each precisification $\pi\in\Pi$:

[TABLE]

Then, let \hbox{\pi}\in\hbox{\Pi} and $v:{X}\to\Delta$ be a variable assignment. The satisfaction relation $\models$ is:

[TABLE]

To understand the interpretation of atomic predications, one must be aware that the evaluation of a symbol may require one or two levels of de-referencing in relation to the precisification index $\pi$. First, notice that the K and A predications are interpreted in the same way. When applied to a name symbol n, both the constant and the predicate get evaluated with respect to a precisification, so that the names denote particular individuals and the predicate denotes a set of such individuals. When the argument is a variable rather than a name symbol, 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 precise entities, which can be tested for membership of the precise set denoted by the property Q. So, although Q predicates are not themselves subject to variation in relation to $\pi$, they require an extra level of disambiguation in the interpretation of their argument symbol.

4.2.3 Classification, Individuation and Demarcation with ${\mathbb{V}_{1}}$

The elaboration in the denotation functions for the different kinds of symbol and the semantics given above for the different cases of predication allow quantification to operate at an intermediate level in the interpretation 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 different exact entities. The formulas (a) and (b) in Section 4.1.2 are also satisfiable ${\mathbb{V}_{1}}$ formulas, and correctly formalise the scenarios described. Furthermore, the formula (c), that was unsatisfiable in ${\mathbb{S}_{{\mathrm{FO}}}}$, is satisfiable in ${\mathbb{V}_{1}}$, because area is modelled as an indefinite individual that is unequivocally classified as a Desert and at some but not all precisifications is associated to a precise extension containing house. Finally, statement (d), “Some argue that the whole arid area is still a desert. Others refer to two deserts in that area, separated by the valley. Either way, my house is still arguably in a desert”, can be formalised as follows:

[TABLE]

Thus, all statements in our example involving the classification, individuation and demarcation of vague objects can be easily represented in ${\mathbb{V}_{1}}$.

5 From ${\mathbb{V}_{1}}$ back to First Order Standpoint Logic

We next turn to the question of the expressive power of ${\mathbb{V}_{1}}$, specifically in comparison to ${\mathbb{S}_{{\mathrm{FO}}}}$. We will provide a translation from ${\mathbb{V}_{1}}$ into ${\mathbb{S}_{{\mathrm{FO}}}}$, which not only settles the above question but is also interesting in its own right, since it spells out the restrictions that ${\mathbb{V}_{1}}$ imposes on the semantics and facilitates the discussion on some ontological commitments that are arguably implicit when favouring the de dicto translation of a ${\mathbb{V}_{1}}$ formula.

The function $\mathrm{trans}:{\mathbb{V}_{1}}\to{\mathbb{S}_{{\mathrm{FO}}}}$, mapping ${\mathbb{V}_{1}}$ formulae to formulae in first-order standpoint logic, is recursively defined as follows:

$\displaystyle\mathrm{trans}(\mbox{\sf K}(\tau))=\mbox{\sf K}(\tau)$ $\displaystyle\mathrm{trans}(\neg\phi)=\neg\mathrm{trans}(\phi)$

$\displaystyle\mathrm{trans}(\mbox{\sf A}(\vec{\tau}))=\mbox{\sf A}(\vec{\tau})$ $\displaystyle\mathrm{trans}(\phi_{1}\!\ \wedge\ \!\phi_{2})=\mathrm{trans}(\phi_{1})\wedge\mathrm{trans}(\phi_{2})$

$\displaystyle\mathrm{trans}(\forall x\,\phi(x))=\forall x\ (\mbox{\sf ink}(x)\rightarrow\mathrm{trans}(\phi))$ $\displaystyle\mathrm{trans}(\mathop{\square}_{s}\phi)=\mathop{\square}_{s}\mathrm{trans}(\phi)$

\displaystyle\mathrm{trans}(\mbox{\sf Q}(\tau_{1},\hbox to8.00003pt{.\hss.\hss.},\tau_{k}))=\exists e_{1},\hbox to8.00003pt{.\hss.\hss.},e_{k}(\mbox{\sf Q}(e_{1},\hbox to8.00003pt{.\hss.\hss.},e_{k})\mathrel{\wedge}\mbox{\sf prec}(\tau_{1},e_{1}))\ \wedge\ \hbox to8.00003pt{.\hss.\hss.}\ \wedge\ \mbox{\sf prec}(\tau_{k},e_{k}))\

Then, we consider the set $\Phi$ of axioms in ${\mathbb{S}_{{\mathrm{FO}}}}$ such that:

[TABLE]

These axioms state that every domain element is either an individual or an extension but not both, with ind and ext rigid predicates (line 1); that every precise entity property predicate is rigid; that $\mbox{\sf prec}(x,y)$ relates an individual $\mbox{\sf ind}(x)$ to a precise entity $\mbox{\sf ext}(y)$ (line 2), and that every individual $\mbox{\sf ind}(x)$ has (at each precisification) a unique precise extension $y$, $\mbox{\sf prec}(x,y)$ (line 3); that every element of a domain with an indefinite predicate A must be an individual in that precisification (ink) (line 4), and finally that an individual in that precisification is an individual of some sort $\mathsf{k}\in\mathsf{K}$ (line 5).

Finally, for a ${\mathbb{V}_{1}}$ formula $\psi$ , we specify:

[TABLE]

A ${\mathbb{V}_{1}}$ formula $\psi$ and its translation $\mathrm{Trans}(\psi)$ are equisatisatisfiable, that is, $\psi$ is ${\mathbb{V}_{1}}$-satisfiable if and only if $\mathrm{Trans}(\psi)$ is ${\mathbb{S}_{{\mathrm{FO}}}}$-satisfiable. Correctness of the translation can be shown by establishing a correspondence between the two kinds of model in each direction. The proof is by induction and is straightforward, yet lengthy, thus it is omitted in the current manuscript.

It is worth remarking that the translation is linear in the size of the formula and preserves the monodic fragment of standpoint logic, where modalities occur only in front of formulas with at most one free variable. The monodic fragment of first-order modal logic is known to be decidable [20], and tight complexity bounds have been shown for monodic fragments of standpoint logic [21]. This highlights the usefulness of the translation, since reasoning support for ${\mathbb{V}_{1}}$ can be provided through its implementation.

6 Some Philosophical Views Considered in relation to the ${\mathbb{V}_{1}}$ Semantics

From the translation, we may observe that we move from having a domain with only precise entities to also having a potentially large collection of indefinite entities that are abstract in nature and only relate to their extension via the predicate prec. This issue regarding whether multiple precise versions of vague things can exist simultaneously (especially fuzzy boundaried objects like clouds) is known to philosophers as ‘The Problem of the Many’ [22], and has been the subject of considerable debate [23].

One view is that the issue can be resolved by modifying the identity relation. According to Geach [24], the different precise versions of a cat are identical relative to the high-level count noun ‘cat’, whereas Lewis [25] suggests that physical objects that are almost the same with respect to spatial extension and material composition are ordinarily regarded as identical. Our semantics has some affinity with Geach’s approach in that we consider that it is count nouns that determine individuation and hence the sets of possible precise versions of individuals. However, in our theory there is no need to modify identity, since quantification is over individuals not their possible precise extensions. We believe that Lewis’s identity condition cannot account for many examples of objects with vague boundaries, where there can be large difference between what we may take to be their spatial extension. For instance, small variations in our interpretation of ‘desert’ may in certain circumstances result in very different spatial extensions being ascribed.

Lewis, along with Varzi [9], also suggests a supervaluationist account where the different candidate extensions of a vague object correspond to their extension in different precisifications. So, at a given precisification there is always a single physical extension of each object. But what is distinctive about our semantics is that we model variability in the extension of an individual as an additional kind of vagueness, not subsumed by the indeterminacy of a name (or sortal) in referring to an individual (or class of individuals).

Lowe [26] posits an ontological difference between objects of a high-level category such as ‘cat’ and physical portions of feline tissue present in the world, which he calls ‘cat-constituters’. This is very much in line with our semantics, especially as presented in the flat (translated) form that contains both definite and indefinite basic elements. Critics of Lowe (e.g. Lewis) have argued that the existence of both cats and cat-constituters is ontologically profligate (or incoherent). We consider instances of the high-level categories to be semantic objects modelling the way language operates rather than carrying ontological commitment. This is clearest in the semantics of ${\mathbb{V}_{1}}$, where modelling indefinite objects as functions from precisifications to extensions is a way of packaging indeterminacy within a semantic object. In the flat version of our semantics, indefinite objects are present (along with precise extensions) as basic elements of the domain; however we still consider them semantic rather than ontic elements.

7 Conclusions and Further Work

We have analysed the phenomenon of vagueness in both predicates and objects, and considered examples suggesting that the aspects of individuation, classification and demarcation associated with predication need to be considered separately. We have proposed the framework of variable reference logic, which takes the general form of a supervaluation semantics but allows variability both in the set of individuals falling under a predicate and also in the spatial extension and physical constitution of ‘objects’ denoted by names or quantified variables. This separation not only supports flexible expressive capabilities of our formal language, but enables the representation to capture definite aspects of information despite other aspects being affected by vagueness.

There remains much interesting work to be done investigating what can be expressed and inferred by means of ${\mathbb{V}_{1}}$. We envisage reasoning support for ${\mathbb{V}_{1}}$ being of practical interest for specific kinds of application, such as querying of geographical information systems, and plan to develop a case study in this area. The linear translation to flat standpoint logic and ongoing progress on implementations for that logic are a strong indicator that query answering functionality may be feasible.

Appendix A Proofs

Translation Correctness Proof.

We now show that for a formula $\psi$ in ${\mathbb{V}_{1}}$, ${\mathbb{V}_{1}}$-satisfiability of $\psi$ coincides with ${\mathbb{S}_{{\mathrm{FO}}}}$-satisfiability of $\mathrm{Trans}(\psi)$.

($\Rightarrow$) Assuming the satisfiability of $\psi$ with ${\mathbb{V}_{1}}$-vocabulary ${V}^{\scaleobj{0.75}{\mathbb{V}}}=\langle{K},{A},{Q},{N},{S}\rangle$, consider some ${\mathbb{V}_{1}}$ model ${M}^{\scaleobj{0.75}{\mathbb{V}}}=\langle E^{\scaleobj{0.75}{\mathbb{V}}},\Pi^{\scaleobj{0.75}{\mathbb{V}}},\sigma^{\scaleobj{0.75}{\mathbb{V}}},\delta^{\scaleobj{0.75}{\mathbb{V}}}\rangle$ of it.

Let now ${M}=\langle\Delta,\Pi,\sigma,\delta\rangle$ denote the ${\mathbb{S}_{{\mathrm{FO}}}}$-interpretation for the vocabulary ${V}=\langle{K}\cup{A}\cup{Q},{N},{S}\rangle$ such that:

1. (I.1)

$\Delta=E^{\scaleobj{0.75}{\mathbb{V}}}\cup I_{*}$, i.e. the domain is the union of the precise entities and the indefinite individuals, 2. (I.2)

$\Pi=\Pi^{v}$ and $\sigma=\sigma^{v}$, i.e. the set of precisifications and standpoint structure are as in ${M}^{\scaleobj{0.75}{\mathbb{V}}}$, 3. (I.3)

the interpretation function $\delta$ is such that

1. (I.3.1)

$\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{K}}\cup\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{A}}\cup\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{N}}\subseteq\delta$, 2. (I.3.2)

$\mbox{\sf Q}^{.\delta(\pi)}=\mbox{\sf Q}^{.\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{Q}}}$ for each $\pi\in\Pi$ and $\mbox{\sf Q}\in{Q}$, 3. (I.3.3)

for each $\pi\in\Pi$,

1. (I.3.3.1)

$\mbox{\sf ext}^{.\delta(\pi)}=E^{\scaleobj{0.75}{\mathbb{V}}}$, 2. (I.3.3.2)

$\mbox{\sf ind}^{.\delta(\pi)}=I_{*}$, 3. (I.3.3.3)

$\mbox{\sf ink}^{.\delta(\pi)}=I_{\pi}$, and 4. (I.3.3.4)

$\mbox{\sf prec}^{.\delta(\pi)}=\{(i,e)\mid i\in I_{*},i(\pi)=e\}$.

First, we show that ${M}\models\bigwedge_{\phi\in\Phi}\square_{{*}}\phi$ by construction, with $\Phi$ the set of axioms defined in the translation.

1. (1.a):

Notice that $\Delta=E^{\scaleobj{0.75}{\mathbb{V}}}\cup I_{*}$ and clearly $E^{\scaleobj{0.75}{\mathbb{V}}}\cap I_{*}=\emptyset$, hence by construction (I.3.3.1 and I.3.3.2) we have that for $d\in\Delta$ then $d\in\mbox{\sf ind}^{\delta(\pi)}\cup\mbox{\sf ext}^{\delta(\pi)}$ and $\mbox{\sf ind}^{\delta(\pi)}\cap\mbox{\sf ext}^{\delta(\pi)}=\emptyset$ for all $\pi\in\Pi$, thus ${M},\pi\models\forall x(\mbox{\sf ind}(x)\leftrightarrow\neg\mbox{\sf ext}(x))$ and finally ${M}\models\square_{{*}}\forall x(\mbox{\sf ind}(x)\leftrightarrow\neg\mbox{\sf ext}(x))$ as desired. 2. (1.b):

From (I.3.3.2) if $d\in\mbox{\sf ind}^{.\delta(\pi)}$ then for all $\pi^{\prime}\in\Pi$ also $d\in\mbox{\sf ind}^{.\delta(\pi^{\prime})}$ and from (I.3.3.1) $d\in\mbox{\sf ext}^{.\delta(\pi)}$ then for all $\pi^{\prime}\in\Pi$ also $d\in\mbox{\sf ext}^{.\delta(\pi^{\prime})}$, hence by the semantics ${M},\pi\models\forall x(\mbox{\sf ind}(x)\!\leftrightarrow\!\mathop{\square}_{*}\mbox{\sf ind}(x))\!\mathrel{\wedge}\!(\mbox{\sf ext}(x)\!\leftrightarrow\!\mathop{\square}_{*}\mbox{\sf ext}(x))$ and thus ${M}\models\square_{{*}}\forall x(\mbox{\sf ind}(x)\!\leftrightarrow\!\mathop{\square}_{*}\mbox{\sf ind}(x))\!\mathrel{\wedge}\!(\mbox{\sf ext}(x)\!\leftrightarrow\!\mathop{\square}_{*}\mbox{\sf ext}(x))$ as desired; 3. (2.a):

It follows from (I.3.2) that if $\vec{d}\in\mbox{\sf Q}^{.\delta(\pi)}$ then for all $\pi^{\prime}\in\Pi$ also $\vec{d}\in\mbox{\sf Q}^{.\delta(\pi^{\prime})}$, hence by the semantics ${M},\pi\models\bigwedge{}_{\scaleobj{1}{\mathsf{Q}\in{Q}}}\forall\vec{x}(\mbox{\sf Q}(\vec{x})\leftrightarrow\mathop{\square}_{*}\mbox{\sf Q}(\vec{x}))$, and thus ${M}\models\square_{{*}}\bigwedge{}_{\scaleobj{1}{\mathsf{Q}\in{Q}}}\forall\vec{x}(\mbox{\sf Q}(\vec{x})\leftrightarrow\mathop{\square}_{*}\mbox{\sf Q}(\vec{x}))$ as desired; 4. (2.b):

by construction from (I.3.3.4) we have that $\mbox{\sf prec}^{.\delta(\pi)}=\{(i,e)\mid i\in I_{*},i(\pi)=e\}$, and by the semantics $i(\pi)\in E^{\scaleobj{0.75}{\mathbb{V}}}$. Together with (I.3.3.1) and (I.3.3.2), we clearly have ${M},\pi\models\forall x,y(\mbox{\sf prec}(x,y)\rightarrow(\mbox{\sf ind}(x)\mathrel{\wedge}\mbox{\sf ext}(y)))$, and thus ${M}\models\square_{{*}}\forall x,y(\mbox{\sf prec}(x,y)\rightarrow(\mbox{\sf ind}(x)\mathrel{\wedge}\mbox{\sf ext}(y)))$; 5. (3.a), (3.b):

by construction from (I.3.3.2) we have that if $i\in\mbox{\sf ind}^{.\delta(\pi)}$ for some $\pi\in\Pi$ then $i\in I_{*}$, and from (I.3.3.4), we then have that $(i,i(\pi))\in\mbox{\sf prec}^{.\delta(\pi)}$, hence clearly ${M}\models\forall x\exists y(\mbox{\sf ind}(x)\!\rightarrow\!\mbox{\sf prec}(x,y))$. And since for each $\pi\in\Pi$ and $i\in I_{*}$ only $(i,i(\pi))\in\mbox{\sf prec}^{.\delta(\pi)}$, then also ${M},\pi\models\forall x,y(\mbox{\sf prec}(x,y)\ \wedge\ \mbox{\sf prec}(x,z)\rightarrow y=z)$ and this ${M}\models\square_{{*}}(\forall x,y(\mbox{\sf prec}(x,y)\ \wedge\ \mbox{\sf prec}(x,z)\rightarrow y=z))$ as desired; 6. (4):

for A of arity $k$, if $\vec{d}\in\mbox{\sf A}^{.\delta(\pi)}$ then by construction (I.3.1) and by the semantics we have $\vec{d}\in I_{\pi}^{k}$. From (I.3.3.3) we have that for all $d\in I_{\pi}$ also $d\in\mbox{\sf ink}^{.\delta(\pi)}$ and hence ${M}\models\square_{{*}}(\forall x_{1},..,x_{k}(\mbox{\sf A}(x_{1},..,x_{k})\!\rightarrow\!\mbox{\sf ink}(x_{1})\!\ \wedge\ \!\hbox to8.00003pt{.\hss.\hss.}\!\ \wedge\ \!\mbox{\sf ink}(x_{k})))$ as desired; 7. (5):

Finally, by construction we have that if $d\in\mbox{\sf ink}^{.\delta(\pi)}$ then $d\in I_{\pi}$, and thus by the semantics both $d\in I_{*}$ and $d\in\bigcup\{\mathsf{K}^{.{\delta_{{K}}(\pi)}}\ |\ \mathsf{K}\in{K}\}$. Then from (I.3.1), (I.3.3.2) and (I.3.3.3) we obtain ${M}\models\square_{{*}}(\forall x\ (\mbox{\sf ink}(x)\!\rightarrow\!\mbox{\sf ind}(x))\wedge({\mbox{\sf ink}}(x)\!\leftrightarrow\!\bigvee{}_{\scaleobj{0.75}{\mathsf{K}\in{K}}}\mbox{\sf K}(x)))$ as desired;

We now show by structural induction on $\psi$ that, for every \pi\in\hbox{\Pi}, if ${M}^{\scaleobj{0.75}{\mathbb{V}}},\pi,v^{\scaleobj{0.75}{\mathbb{V}}}\models\psi$ then ${M},\pi,v\models\mathrm{trans}(\psi)$ for $v$ an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$.

$\psi=\mbox{\sf K}(\tau)$:

Assume ${M}^{\scaleobj{0.75}{\mathbb{V}}},\pi,v^{\scaleobj{0.75}{\mathbb{V}}}\models\mbox{\sf K}(\tau)$. Then we show that ${M},\pi,v\models\mathrm{trans}(\mbox{\sf K}(\tau))$ for $v$ an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$. Notice that $\mathrm{trans}(\mbox{\sf K}(\tau))=\mbox{\sf ind}(\tau)\mathrel{\wedge}\mbox{\sf K}(\tau)$. By the semantics of ${\mathbb{V}_{1}}$ we have $\mathsf{K}^{\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{K}}(\pi)}\subseteq I_{*}$ and $\tau^{\pi,v^{\scaleobj{0.75}{\mathbb{V}}}}\in\mathsf{K}^{\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{K}}(\pi)}$. Then, since $\mbox{\sf ind}^{\delta(\pi)}=I_{*}$ for all $\pi\in\Pi$, and $v$ is an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$, then $\tau^{\pi,v}\in\mbox{\sf ind}^{\delta(\pi)}$ for all $\pi\in\Pi$. Similarly, since $\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{K}}\subseteq\delta$ and $v$ is an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$, then $t^{\pi,v}\in\mbox{\sf K}^{\delta(\pi)}$ and hence ${M},\pi,v\models\mbox{\sf ind}(\tau)\mathrel{\wedge}\mbox{\sf K}(\tau)$.

$\psi=\mbox{\sf A}(\tau)$:

Proceeds similarly to the previous case.

$\psi=\mbox{\sf Q}(\tau)$:

Assume ${M}^{\scaleobj{0.75}{\mathbb{V}}},\pi,v^{\scaleobj{0.75}{\mathbb{V}}}\models\mbox{\sf Q}(\tau)$. Then we show that ${M},\pi,v\models\mathrm{trans}(\mbox{\sf Q}(\tau))$ for any $v$ that is an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$. Notice that $\mathrm{trans}(\mbox{\sf Q}(\tau))=\mbox{\sf ind}(\tau)\mathrel{\wedge}\exists x(\mbox{\sf Q}(x)\mathrel{\wedge}\mbox{\sf prec}(\tau,x))$. Let $\tau^{\pi,v^{\scaleobj{0.75}{\mathbb{V}}}}=i$. By the semantics of ${\mathbb{V}_{1}}$ we have $i\in I_{*}$ and $i(\pi)\in\mathsf{Q}^{\delta^{\scaleobj{0.75}{\mathbb{V}}}_{{Q}}}$, with $i(\pi)\in E^{\scaleobj{0.75}{\mathbb{V}}}$. Then, since $\mbox{\sf ind}^{\delta(\pi)}=I_{*}$ for all $\pi\in\Pi$, and $v$ is an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$, then $t^{\pi,v}\in\mbox{\sf ind}^{\delta(\pi)}$ for all $\pi\in\Pi$. Now, from the translation we have ${M},\pi,v\models\exists x(\mbox{\sf Q}(x)\mathrel{\wedge}\mbox{\sf prec}(\tau,x))$. By construction, for $i(\pi)=e$ we have that $e\in\mbox{\sf Q}^{\delta(\pi)}$ from (I.3.2) and $(i,e)\in\mbox{\sf prec}^{\delta(\pi)}$ from (I.3.3.4). And since $e\in E^{\scaleobj{0.75}{\mathbb{V}}}$, then it is not mapped by $v^{\scaleobj{0.75}{\mathbb{V}}}$, so there is a variable assignment $v_{[x\rightarrow e]}$ extending $v^{\scaleobj{0.75}{\mathbb{V}}}$ and with $v(x)=e$. Consequently, ${M},\pi,v\models\mbox{\sf ind}(\tau)\mathrel{\wedge}\exists x(\mbox{\sf Q}(x)\mathrel{\wedge}\mbox{\sf prec}(\tau,x))$ as desired.

$\psi=\neg\psi^{\prime}$:

Trivial.

$\psi=\psi^{\prime}\mathrel{\wedge}\psi^{\prime\prime}$:

Trivial.

$\psi=\forall x\,\psi^{\prime}(x)$:

Assume ${M}^{\scaleobj{0.75}{\mathbb{V}}},\pi,v^{\scaleobj{0.75}{\mathbb{V}}}\models\forall x\,\psi^{\prime}(x)$. Then we show that ${M},\pi,v\models\mathrm{trans}(\forall x\,\psi^{\prime}(x))$ for any $v$ that is an extension of $v^{\scaleobj{0.75}{\mathbb{V}}}$. Notice that $\mathrm{trans}(\forall x\,\psi^{\prime}(x))=\forall x\,(\mbox{\sf ink}(x)\ \rightarrow\ \mathrm{trans}(\psi^{\prime}))$. Hence we must show that for all $d\in\Delta$, if $d\in\mbox{\sf ink}^{\delta(\pi)}$ then ${M},\pi,v_{x\rightarrow d}\models\mathrm{trans}(\psi^{\prime})(x)$. By construction (I.3.3.3) we have $\mbox{\sf ink}^{\delta(\pi)}=I_{\pi}$, and by the assumption and the semantics of ${\mathbb{V}_{1}}$ we have that ${M}^{\scaleobj{0.75}{\mathbb{V}}},\pi,v^{\scaleobj{0.75}{\mathbb{V}}}_{\left\{x\mapsto i\right\}}\models\forall x\,\psi^{\prime}(x)$ for all $i\in I_{\pi}$. With this and by the inductive hypothesis it is clear that ${M},\pi,v\models\mathrm{trans}(\forall x\,\psi^{\prime}(x))$

$\psi=\mathop{\square}_{s}\psi^{\prime}$:

Trivial.

It is then a direct consequence that ${M}$ is a model of both conjuncts of $\mathrm{Trans}(\phi)$. Hence ${M}$ is a model of $\mathrm{Trans}(\phi)$, witnessing its satisfiability.

($\Leftarrow$) Consider now a ${\mathbb{S}_{{\mathrm{FO}}}}$-model ${M}^{\scaleobj{0.75}{\mathbb{S}}}$ of $\mathrm{Trans}(\psi)$ with vocabulary ${V}^{\scaleobj{0.75}{\mathbb{S}}}=\langle{P},{N},{S}\rangle$. Define the vocabulary ${V}=\langle{K},{A},{Q},{N},{S}\rangle$, by letting

• •

${K}=\{\mbox{\sf P}\in{P}\backslash\{\mbox{\sf ind}\}\mid\mbox{\sf P}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\cap\mbox{\sf ind}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\neq\emptyset\}$

• •

${A}=\emptyset$

• •

${Q}=\{\mbox{\sf P}\in{P}\backslash\{\mbox{\sf ext}\}\mid\mbox{\sf P}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\cap\mbox{\sf ext}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\neq\emptyset\}$

We now define the ${\mathbb{V}_{1}}$ model ${M}=\langle E,\Pi,\sigma,\delta\rangle$ with vocabulary ${V}$, by letting

1. (M1)

$\Pi$ and $\sigma$ be as in ${M}^{\scaleobj{0.75}{\mathbb{S}}}$ 2. (M2)

$E=\mbox{\sf ext}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ for some $\pi\in\Pi$ 3. (M3)

$I_{*}=Maps(\Pi,E)$ is the set of all mappings from precisifications to precise entities. 4. (M4)

$I_{\pi}=\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, $\mbox{\sf K}^{\delta_{K}(\pi)}=\mbox{\sf K}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ and $\mbox{\sf N}^{\delta_{N}(\pi)}=\mbox{\sf N}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ for all $\pi\in\Pi$, $\mbox{\sf K}\in{K}$ and $\mbox{\sf N}\in{N}$, 5. (M5)

$\mbox{\sf Q}^{\delta_{Q}}=\mbox{\sf Q}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ for any $\pi\in\Pi$ and all $\mbox{\sf Q}\in{Q}$.

First, we show that the structure thus defined is indeed a ${\mathbb{V}_{1}}$, recalling that ${M}^{\scaleobj{0.75}{\mathbb{S}}}\models\bigwedge_{\phi\in\Phi}\square_{{*}}\phi$ with $\Phi$ the set of axioms defined in the translation. By axiom (1.b) we have that for all $\pi,\pi^{\prime}$, $\mbox{\sf ind}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}=\mbox{\sf ind}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi^{\prime})}$ and $\mbox{\sf ext}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}=\mbox{\sf ext}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi^{\prime})}$. For the sake of abbreviation we will refer to these sets as $\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ and $\mbox{\sf ext}^{\scaleobj{0.75}{\mathbb{S}}}$. Following, by (1.a) we have that $E^{\scaleobj{0.75}{\mathbb{S}}}=\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}\cup\mbox{\sf ext}^{\scaleobj{0.75}{\mathbb{S}}}$ and $\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}\cap\mbox{\sf ext}^{\scaleobj{0.75}{\mathbb{S}}}=\emptyset$. Hence, the set of indefinite individuals and the set of precise entities are disjoint in ${M}^{\scaleobj{0.75}{\mathbb{S}}}$ and independent from $\Pi$. Moreover, by axiom (3.a) every individual $i\in\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ has a precise extension at a given precisification $\pi$, i.e. there is some $e$ such that $(i,e)\in\mbox{\sf precs}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ and by (2.b) $e\in\mbox{\sf ext}^{\scaleobj{0.75}{\mathbb{S}}}$. And, by (3.b) there is no other $e^{\prime}$ such that $(i,e^{\prime})\in\mbox{\sf precs}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ and hence we can set $i(\pi)=e$ as desired, thus obtaining from each $i\in\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ an indefinite individual $i\in I_{*}$. Finally, $I_{\pi}$, the set of indefinite individuals in $\pi$, corresponds to $\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ because by axiom (4.b) we have $\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\subseteq\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ and $\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}=\bigcup{}_{\scaleobj{0.75}{\mathsf{K}\in{K}}}\mbox{\sf K}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, which given that $\mbox{\sf K}^{\delta_{K}(\pi)}=\mbox{\sf K}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, clearly encodes the definition of $I_{\pi}$.

From this we have that ${M}$ is a ${\mathbb{V}_{1}}$ structure and

1. A0

every $i\in\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ corresponds to an indefinite individual $i\in I_{*}$ such that for each \pi\in\hbox{\Pi} we have $i(\pi)=e$ if $(i,e)\in\mbox{\sf precs}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, and 2. A1

the set $I_{\pi}$ of indefinite individuals in $\pi$ corresponds to $\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$.

To show that ${M}$ is a model of $\psi$, we show by structural induction over $\psi$ that, for every \pi\in\hbox{\Pi} and variable assignment $v$, if ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mathrm{trans}(\psi)$ then ${M},\pi,v\models\psi$, with $v$ the restriction of $v^{\scaleobj{0.75}{\mathbb{S}}}$ to the range of $I_{*}$. Thus, the established modelhood of ${M}$ ensures satisfiability of $\phi$.

$\mathrm{trans}(\mbox{\sf K}(\tau))$

Assume that ${M}^{\scaleobj{0.75}{\mathbb{V}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mathrm{trans}({\mbox{\sf K}(\tau)})$. Then ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mbox{\sf K}(\tau)$. Consequently, there is an element $i\in E^{\scaleobj{0.75}{\mathbb{S}}}$ such that $i=\tau^{\pi,v^{\scaleobj{0.75}{\mathbb{S}}}}$, $i\in\mbox{\sf K}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, and by axiom (4.b) $i\in\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$. By [A0], $i$ translates to an indefinite individual in $I_{*}$. By the axiom (4.b) we have that for all $\pi$, $\mbox{\sf K}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\subseteq\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ and thus again by [A0], $\mbox{\sf K}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ translates to a set of indefinite individuals $\mbox{\sf K}^{\delta_{K}(\pi)}\subseteq I_{*}$ such that $i\in\mbox{\sf K}^{\delta_{K}(\pi)}$ as desired.

$\mathrm{trans}(\mbox{\sf A}(\tau))$

Assume that ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mathrm{trans}({\mbox{\sf A}(\tau)})$. Then ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mbox{\sf A}(\tau)$. Consequently, there is an element $i\in E^{\scaleobj{0.75}{\mathbb{S}}}$ such that $i=\tau^{\pi,v^{\scaleobj{0.75}{\mathbb{S}}}}$, $i\in\mbox{\sf A}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, and by axioms (4.a, 4.b) $i\in\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$. By [A0], $i$ translates to an indefinite individual in $I_{*}$. By axioms (4.a,4.b) it is clear that for all $\pi$ we have $\mbox{\sf A}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\subseteq\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ and $\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}\subseteq\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$. Thus, again by [A0], $\mbox{\sf A}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ translates to a set of indefinite individuals $\mbox{\sf A}^{\delta_{A}(\pi)}\subseteq I_{\pi}$ such that $i\in\mbox{\sf A}^{\delta_{A}(\pi)}$ as desired.

$\mathrm{trans}(\mbox{\sf Q}(\tau))$

Assume that ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mathrm{trans}({\mbox{\sf Q}(\tau)})$. Then ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\exists x(\mbox{\sf Q}(x)\mathrel{\wedge}\mbox{\sf prec}(\tau,x))$. Consequently, there are $i,e\in E^{\scaleobj{0.75}{\mathbb{S}}}$ such that $i=\tau^{\pi,v^{\scaleobj{0.75}{\mathbb{S}}}}$, $e=v^{\scaleobj{0.75}{\mathbb{S}}}(x)$, $(i,e)\in\mbox{\sf prec}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, $e\in\mbox{\sf Q}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, and by axiom (2.b) $i\in\mbox{\sf ind}^{\scaleobj{0.75}{\mathbb{S}}}$ and $e\in\mbox{\sf ext}^{\scaleobj{0.75}{\mathbb{S}}}$. By [A0], $i$ translates to an indefinite individual in $I_{*}$ such that $i(\pi)=e$. Also, from axiom (2.a) we have that for all $\pi,\pi^{\prime}$, $\mbox{\sf Q}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}=\mbox{\sf Q}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi^{\prime})}$, and we recall that $\mbox{\sf Q}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi^{\prime})}=\mbox{\sf Q}^{\delta_{{Q}}}$ for some $\pi^{\prime}\in\Pi$. Finally, since $v$ is a restriction on $v^{\scaleobj{0.75}{\mathbb{S}}}$ to the range of $I_{*}$, then $\tau^{.\pi,v}=\tau^{.\pi,v^{\scaleobj{0.75}{\mathbb{S}}}}$ and consequently $\tau^{.\pi,v}(\pi)\in\mbox{\sf Q}^{\delta_{{Q}}}$. Thus ${M},\!\pi,\!v\models\mbox{\sf Q}(\tau)$ as desired.

$\mathrm{trans}({\neg\psi})$

Trivial.

$\mathrm{trans}(\psi\wedge\psi^{\prime})$

Trivial.

$\mathrm{trans}(\mathop{\square}_{s}\psi)$

Trivial.

$\mathrm{trans}(\forall x\psi)$

Assume that ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mathrm{trans}({\forall x\psi})$. Then ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\forall x(\mbox{\sf ink}(x)\rightarrow\mathrm{trans}(\psi))$. Then, by the semantics we have ${M},\!\pi,\!v\models(\mbox{\sf ink}(x)\rightarrow\mathrm{trans}(\psi))\text{ for all }v:{X}\to\Delta$. This means that for all $i\in\Delta$ if $i\in\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$ then ${M}^{\scaleobj{0.75}{\mathbb{S}}},\pi,v^{\scaleobj{0.75}{\mathbb{S}}}\models\mathrm{trans}(\psi)$ with $v(x)=i$. By the inductive hypothesis we have that ${M},\!\pi,\!v\models\psi(x)$ for $v$ the restriction on $v^{\scaleobj{0.75}{\mathbb{S}}}$ over the range of $I_{*}$. Then since by [A1] we have that $I_{\pi}=\mbox{\sf ink}^{\delta^{\scaleobj{0.75}{\mathbb{S}}}(\pi)}$, we obtain that ${M},\!\pi,\!v_{\left\{x\mapsto i\right\}}\models\ \psi(x)\hbox{~{}~{}for all}\ i\in I_{\pi}$, and thus ${M},\!\pi,\!v_{\left\{x\mapsto i\right\}}\models\ \forall x\psi$ as desired.

∎