VagueReferringTermsHandout < Discussion

A Precise Formalization of Vagueness.

The simple picture is that a context for a language of vagueness, which will serve as a possible world for a modal semantics, will be a Tarskian structure or a set of contexts for the language of vagueness. Thus, a context may be a structure, or a set of structures, or a set of structures and sets of structures, or ... . The actual picture will be a variant.

A few disclaimers:

The model I am presenting here is intended to show the coherence of certain intuitive considerations, not to replace them. The model is, however, sufficiently complex that I cannot both motivate and present it in the space of a couple of hours.
The model is based on a supervaluationist approach. It does not provide a reductive account of vagueness, only help in fixing the inferential relations between concepts allied to vagueness and between them and the rest of our conceptual economy.
The formal models employed will be selected in terms of the unreduced notion of states of semantic affairs that are possible or indeterminate, a notion taken to be antecedently understood. I do not see the lack of a reduction as a defect. We should not expect a reductive explanation of such a fundamental feature of language as vagueness. (Compare remarks of Field, Indeterminacy, degree of belief, and excluded middle.)
Not every aspect of the mathematical formalism to be presented is driven by philosophical considerations. There may well be alternative models that serve equally well. The model is useful nonetheless, both because it serves to fix ideas and because it shows that my intuitive constraints, since they are all realized here, are in fact compatible.
The formalism is built on top of Tarskian semantics for precise language. If one takes Tarski's theory to be an inflationary correspondence theory, the formalism here will inherit that characteristic. If one reads Tarski's theory in a thin way, so that it is compatible with deflationism, the same will be true of the formalism here.
If one has a semantics for precise language different from that of Tarski, it is very likely that it will be possible to build a semantics for vagueness on top of it in a manner that is precisely analogous to the one adopted here on top of Tarski's theory. $^{\text{footnote}}$
I am rather inclined to prefer a thin reading of the semantic theory adopted here. The theory is useful for making clear what I take to be the inferential roles of various aspects of vagueness, but I take the underlying preformal intuitions to be primary, not the semantics suggested by the formalism.

Some preliminary definitions concerning trees.

Let ${\Bbb T}=\langle T,<\rangle$ be a strict partial order (that is, such that

is a transitive, irreflexive relation on

) with greatest element such that for every

$\{s:t<s\}$ is linearly ordered by

(that is, such that any two distinct elements of the set are comparable).

The structure ${\Bbb T}$ is a well-founded tree if every subset of that is linearly ordered by is finite.

Since we shall only be interested in well-founded trees, I shall drop 'well founded,' referring to well-founded trees simply as trees.

The elements of are called nodes.
The leaves of ${\Bbb T}$ are the -minimal nodes.
An $\subseteq$ -maximal linearly ordered subset of is a branch of ${\Bbb T}$ .
An $\subseteq$ -maximal subset of with pairwise incomparable elements (that is, a set such that for all and in it, if $a\le b$ , then ) is an antichain of ${\Bbb T}$ .

If a subset of includes an antichain, then it is not hard to see that the set of -maximal elements of is an antichain, and that no member of any antichain included in is above any member of that antichain.

That antichain is the high antichain in .
When is a node of ${\Bbb T}$ , then ${\Bbb T}_{n}$ is the structure $\langle\{m:m\le n\},<\cap \{m:m\le n\}^{2}\rangle$ .

Note that ${\Bbb T}_{n}$ is a tree—the subtree of ${\Bbb T}$ below

When is a node of ${\Bbb T}$ , then $N_{n}$ , the set of nodes immediately below , is $\{m:m<n\land (\forall l)(m<l\rightarrow n\le l)\}$ .
A leaf in a tree has level 0. The level of any other node is the supremum of $\{{\text{level}}(m) +1:m<n\}$ .
The height of a tree is the level of its top node.
Two trees are isomorphic if there is a bijection f from the set of nodes of the first to the set of nodes of the second that preserves , that is such that the first tree satisfies if and only if the second tree satisfies .
The function is an isomorphism from the first tree to the second.
A tree ${\Bbb T}$ is homogeneous if ${\Bbb T}_{m}$ is isomorphic to ${\Bbb T}_{n}$ whenever and are nodes of ${\Bbb T}$ of the same level.
An ordered pair is an explicitly homogeneous tree if is a map taking every pair of nodes and of of the same level to an isomorphism from to such that
1. if nodes , , , and are such that , and are of the same level, and $f_{mn}(k)=l$ (and thus and , and are of the same level), then $f_{kl}=f_{mn}\restriction T_{k}$ ,
2. for every the isomorphism $f_{nn}$ is the identity, and
3. for all nodes , , and of the same level $f_{mn} f_{lm}=f_{ln}$ .
A tree is full at level $\alpha$ if there is a node of level $\alpha$ on every branch of the tree.

Note that a tree is always full at the top level and level 0, the levels at the two ends of every branch, a fact that will prove to be of considerable technical use below.

The models

The notion of a context will be something like either a well-founded set of possible worlds or a conventional Tarskian structure (that is, a context is something like a set of Tarskian structures, or a set of Tarskian structures and sets of Tarskian structures, or a set of Tarskian structures, sets of Tarskian structures, and sets of Tarskian structures and sets of Tarskian structures, and so forth). Taking that to be the official definition would not permit the same possible world to be a member of a possible world twice and so we make the following definition instead:

A model of vagueness, ${\Bbb V}$ is an ordered triple $\langle{\Bbb T},f,{\mathfrak A}\rangle$ such that $\langle{\Bbb T}, f\rangle$ is an explicitly homogeneous tree and ${\mathfrak A}$ is a map from the leaves of ${\Bbb T}$ to conventional Tarskian structures all of the same language, ${\mathcal L}$ , the language of the model of vagueness and all with the same domain, , the domain of the model of vagueness.

Note that there will be a proper class of nonisomorphic models of vagueness even of a language with only one constant symbol over a domain of at least two elements.

If is a node in , let ${\Bbb V}_{n}$ be
$\langle{\Bbb T}_{n}, f\restriction \{\langle l,m\rangle:l,m\le n{\text{ and }}l,m{\text{ are of the same level}}\},{{\mathfrak A}\restriction\{l:l\le n{\text{ and }}l{\text{ is a leaf}}\}\rangle.}$

Note that if is the top node of ${\Bbb T}$ , then ${\Bbb V}_{t}={\Bbb V}$ .

The nodes are our contexts (possible worlds of our modal semantics). Leaves play the role that the structures associated with them did in the simplified picture of sets of sets of worlds and every other node plays the role that that the set $N_{n}$ , the "set of its members" did in the simplified picture. It turns out to be simpler for mathematical reasons to take the possible worlds to be the models ${\Bbb V}_{n}$ instead of the nodes themselves, but that is just a matter of technical convenience.

Since our only concern here is with vagueness, the case of interest is the one in which all the structures in the range of ${\mathfrak A}$ are structures for the same language with the same domain: Allowing different domains for the structures would only make sense if it were indeterminate whether certain base-level, determinate objects existed.

By requiring that each referring term refer at every leaf (by building it in to the language of a Tarskian structure at the leaf), I have blocked the possibility of formalizing vague terms such that it is indeterminate whether they refer at all. Allowing such terms would necessitate allowing nonreferring terms in the structures at the leaves, that is, defining vagueness over some free logic. That would be an appropriate thing to do, but there is no generally agreed-upon free logic, and so, since this is, with respect to the concerns of this work, a side issue, I have simply not handled the possibility.

The complete precision of the structures at the leaves is a regulative ideal that need not be obtainable in any permanent sense. Increasingly fine-grained models of vagueness can be obtained by replacing the leaves of a tree by further trees.

Vague Referring Terms

An object for the model of vagueness ${\Bbb V}$ is a map ${\mathcal O}$ from the leaves of ${\Bbb T}$ to members of the domain of the model.
If is in , then ${\mathcal O}_{n}$ is the restriction of ${\mathcal O}$ to the leaves of ${\Bbb T}_{n}$ .

It is tempting to refer to objects as just defined as vague objects, and I shall succumb to that temptation. Note nonetheless that "vague objects" in fact codify indeterminate reference to determinate objects.

To give a simple example, let ${\Bbb T}$ be a tree with three nodes, one top node and two leaves immediately below it, and let ${\mathfrak A}$ associate both leaves with the same structure, one that has Ararat and Ararat in its domain. There is an object for the model of vagueness ${\mathcal O}$ that is Ararat at one of the leaves and Ararat at the other. That object, like Ararat, is indeterminate with respect to whether it is Ararat or Ararat.

A tree is required here: a set of structures won't do since the same structure is on both leaves of the tree, while a structure cannot be a member of a set twice. Once again, to avoid a tempting misunderstanding, the vague object Ararat here is nothing more than an indeterminate reference to Ararat or Ararat, not something new.

An object ${\mathcal O}$ is homogeneous at level $\alpha$ if ${\Bbb T}$ is full at level $\alpha$ and for all nodes and of level $\alpha$ , the equation ${\mathcal O}_{m}={\mathcal O}_{n}f_{mn}$ holds.

Note that every object is homogeneous at level $\alpha$ when $\alpha$ is the height of the tree. $^{\text{footnote}}$

An object ${\mathcal O}$ is an object of level $\alpha$ if $\alpha$ is the least ordinal $\beta$ such that ${\mathcal O}$ is homogeneous at level $\beta$ .

For example, an object ${\mathcal O}$ is of level 0 if and only if it is a constant function, that is, if and only if it picks out one and the same member of the domain at every leaf. When an object is of level $\alpha$ , it is in effect of fixed semantic value at the levels above $\alpha$ .

If $\Bbb T$ is full at level $\alpha$ , is a node at level $\alpha$ and ${\mathcal O}$ is an object, then the homogeneous object associated with the object ${\mathcal O}$ at the node , $H({\mathcal O},n)$ , is the object that maps each leaf to ${\mathcal O}f_{mn}(l)$ , where is the node over that is of the same level as .

The object $H({\mathcal O},n)$ is homogeneous at level $\alpha$ . If the object ${\mathcal O}$ is homogeneous at level $\beta$ , then so is $H({\mathcal O},n)$ . The object $H({\mathcal O},n)$ is the object with a fixed semantic value of ${\mathcal O}_{n}$ .

For example, if is of level 0, then ${\mathcal O}_{n}$ is in effect just a member of the domain of the model of vagueness, and in that case $H({\mathcal O},n)$ is the object with a constant value of that member of the domain.

We shall have exactly one special relation in our formalism, which we shall symbolize by $\succ$ , the binary relation that obtains between the objects ${\mathcal O}$ and ${\mathcal O}'$ when there is a node strictly below the level of ${\mathcal O}$ such that ( $\Bbb T$ is full at the level of and) ${\mathcal O}'=H( {\mathcal O}_{n},n)$ , read " ${\mathcal O}$ is above ${\mathcal O}'$ ."
An assignment for the model of vagueness ${\Bbb V}$ is a function from the set of variables of the language to objects for ${\Bbb V}$ .
When is a node and an assignment, then $s_{n}$ is the assignment for ${\Bbb V}_{n}$ such that for every variable the object $s_{n}(x)$ is ${\mathcal O}_{n}$ for ${\mathcal O}=s(x)$ .
When is a leaf and is an assignment, I simply write $s_{l}$ for the Tarskian assignment that takes each variable to $s_{l}(x)(l)$ , the object in the domain of ${\mathfrak A}_{l}$ designated by the vague object , when no confusion can arise.

Note that a constant symbol, since it may denote different objects in different structures associated with different leaves, determines a function from leaves to objects in the domain of those leaves. Thus, a constant symbol is a vague referring term. I shall say that it denotes a vague object.

Definition of satisfaction

It is now possible to define what it is for an assignment

to satisfy a formula for vagueness $\phi$ in a model of vagueness ${\Bbb V}$ , notation, ${\Bbb V}\models\phi [s]$ , where the language for vagueness for ${\Bbb V}$ is the language of ${\Bbb V}$ augmented by the symbols $\lozenge$ and $\succ$ .

Each node in functions as a possible world: an assignment satisfies a formula $\phi$ at node if the assignment $s_{n}$ satisfies the formula $\phi$ in the model of vagueness ${\Bbb V}_{n}$ .

If $\phi$ is an atomic formula in which $\succ$ does not appear, then satisfies $\phi$ (reference to the model of vagueness, which is fixed by context, is suppressed, here and at similar points below) if ${\mathfrak A}_{l}\models\phi [s_{l}]$ for every leaf , where I have used $s_{l}$ as a shorthand for the Tarskian assignment determined by at and $\models$ stands for ordinary Tarskian satisfaction.
If $\phi$ is $\tau _{1}\succ\tau _{2}$ , and $\tau _{1}$ is a constant symbol denoting ${\mathcal O}$ or a variable assigned ${\mathcal O}$ by and, similarly, $\tau _{2}$ refers to ${\mathcal O}'$ , then satisfies $\phi$ if the object ${\mathcal O}$ is above the object ${\mathcal O}'$ .
If $\phi$ is $(\psi \lor\theta )$ , then satisfies $\phi$ if is the set of all nodes such that ${\Bbb V}_{n}\models\psi [s_{n}]$ or ${\Bbb V}_{n}\models\theta [s_{n}]$ or both ${\Bbb V}_{n}\models\lnot \psi [s_{n}]$ and ${\Bbb V}_{n}\models\lnot\theta [s_{n}]$ , there is a high antichain in (in fact, always exists because all leaves are in , and the leaves form an antichain) and ${\Bbb V}_{n}\models\psi [s_{n}]$ or ${\Bbb V}_{n}\models\theta[s_{n}]$ for every node in .
If $\phi$ is $(\exists x)\psi$ , then satisfies $\phi$ if there is an object ${\mathcal O}$ such that ${\Bbb V}\models\psi\left[s{\left[\!{{x}\atop{\mathcal O}}\!\right]}\right]$ , where $\left[s{\left[\!{{x}\atop{\mathcal O}}\!\right]}\right]$ is the assignment that assigns $\mathcal O$ to and assigns what does to every other variable.
If $\phi$ is $\lozenge\psi$ , then satisfies $\phi$ if ${\Bbb V}_{n}\models\psi [s_{n}]$ for some node .
Negation:
- If $\phi$ is $\lnot\psi$ and $\psi$ is atomic and $\succ$ does not appear in it, then satisfies $\phi$ if ${\mathfrak A}_{l}\models\phi [s_{l}]$ for every leaf . (Note that ${\mathfrak A}_{l}\models\phi [s_{l}]$ is already defined at leaves, since at leaves $\models$ is ordinary Tarskian satisfaction.)
- If $\phi$ is $\lnot\tau _{1}\succ\tau _{2}$ , and $\tau _{1}$ is a constant symbol denoting ${\mathcal O}$ or a variable assigned ${\mathcal O}$ by and, similarly, $\tau _{2}$ refers to ${\mathcal O}'$ , then satisfies $\phi$ if the object ${\mathcal O}$ is not above the object ${\mathcal O}'$ .
- If $\phi$ is $\lnot\psi$ and $\psi$ is $( \theta \lor\zeta )$ , then satisfies $\phi$ if is the set of all nodes such that ${\Bbb V}_{n}\models\theta [s_{n}]$ or ${\Bbb V}_{n}\models\zeta [s_{n}]$ or both ${\Bbb V}_{n}\models\lnot \theta [s_{n}]$ and ${\Bbb V}_{n}\models\lnot\zeta [s_{n}]$ , there is a high antichain in (in fact, always exists because all leaves are in , and the leaves form an antichain) and ${\Bbb V}_{n}\models\lnot\theta [s_{n}]$ and ${\Bbb V}_{n}\models\lnot\zeta [s_{n}]$ forevery node in .
- If $\phi$ is $\lnot\psi$ and $\psi$ is $(\exists x)\theta$ , then satisfies $\phi$ if for every object ${\mathcal O}$ ${\Bbb V}\models \lnot\theta \left[s{\left[\!{{x}\atop{\mathcal O}}\!\right]}\right]$ .
- If $\phi$ is $\lnot\psi$ and $\psi$ is $\lozenge\theta$ , then satisfies $\phi$ if ${\Bbb V}_{n}\models\lnot\theta [s_{n}]$ for every node .
- Finally, If $\phi$ is $\lnot\psi$ and $\psi$ is $\lnot\theta$ , then satisfies $\phi$ if ${\Bbb V}\models\theta [s]$ .

We take ' $\land$ ,' ' $\rightarrow$ ,' ' $\forall$ ,' and so forth to be defined in the usual ways. We leave it to the reader to check that those definitions yield the desired results.

That concludes the definition of satisfaction.

A formula $\phi$ is indeterminate under an assignment , $\text{Ind} \phi$ , if ${\mathbb V}\models(\lozenge\phi\land \lozenge \lnot\phi )[s]$ .

Let $\phi$ be a formula, a variable, and an assignment. An object ${\mathcal O}$ is a fixed $\phi (x)$ under the assignment if $s{\left[{{x}\atop{\mathcal O}}\right]}$ satisfies
$\phi (x)\land(\forall y)\left(x\succ y\rightarrow\lnot\phi\left[\!\!\left[\!{{x}\atop{y}}\!\right]\!\!\right]\lor\text{Ind}\phi\left[\!\!\left[\!{{x}\atop{y}}\!\right]\!\!\right]\right),$
where $\phi \left[\!\!\left[\!{{x}\atop{y}}\!\right]\!\!\right]$ denotes the formula that results from $\phi$ when every occurrence of in $\phi$ is replaced by , systematically renaming bound variables as necessary to prevent collisions.

Thus, an object ${\mathcal O}$ is a fixed $\phi(x)$ under the assignment

if ${\mathcal O}$ is a $\phi(x)$ and every object such that ${\mathcal O}$ is above it is at best indeterminate with respect to whether it is a $\phi(x)$ .

It is straightforward to prove by induction on the sum of the height of ${\mathbb T}$ and the rank of $\phi$ that if $\phi$ is an ordinary formula, without modal operators, then for any assignment , satisfies $\phi$ in ${\mathbb V}$ if and only if for every leaf , $s_{l}$ satisfies $\phi$ in ${\mathfrak A}_{l}$ . Thus, all nonmodal logical truths are true in every model of vagueness, and all nonmodal truths are determinately (that is, not possibly not) true.

Discussion

The formalism will result in sharp transitions in sorites sequences: For example, if one starts with a heap of sand and removes grains one by one, the resulting piles, at least if they are construed, as is arguably natural, as objects that are homogeneous of level 0, will either be heaps at every leaf, hence heaps; heaps at some leaves but not others, in which case it is indeterminate whether they are heaps; or heaps at no leaf, and hence not heaps. There will therefore be sharp transitions at a particular number of grains of sand between heap and indeterminate, and again between indeterminate and not a heap. Recall, however, that it will be indeterminate which model of vagueness is "the" model: there will instead be a family of such models of indeterminate membership. Thus, there will be no particular transition points that are held to be part of the ordinary usage of the term 'heap.' In addition, actual heaps are not homogeneous at level 0, since there will be loose grains of sand near the base, and the like. If we take that phenomenon seriously, then even within a formal model of vagueness there will not be transition points at a fixed number of grains of sand.

To illustrate the use of the formalism for vague objects just introduced, for suitable ${\mathbb V}$ , let the constant symbol ' $\bf A$ ' denote the type, the letter 'A.' That is, the symbol ' $\bf A$ ' denotes an object that is indeterminate with respect to whether it is each object that is a fixed inscription of the letter 'A.' It is indeterminate whether a fixed inscription is $\bf A$ if and only if it is indeterminate whether that object is an inscription of the letter 'A.' No fixed inscription of the letter 'A' is the type $\bf A$ , since the fixed inscriptions are of lower level than the type. In formal detail, to a first approximation, if ${\mathcal A}$ is the object denoted by $\bf A$ in ${\mathbb V}$ , then on every branch there is a least node such that $H({\mathcal A}_{n},n)$ is (defined and) a fixed inscription; $H({\mathcal A}_{n},n)={\mathcal A}$ is indeterminate; for every node above except the top node, ${\mathcal A}_{m}= H({\mathcal A}_{n},n)_{m}$ is true of ${\mathbb V}_{m}$ ; and every fixed inscription of 'A' is ${\mathcal A}_{n}$ for some node . In other words, the object ${\mathcal A}$ at the nodes just below the top is indeterminately equal to all the various fixed inscriptions of 'A.' This construction allows ${\mathcal A}$ to be all the fixed inscriptions of 'A,' and not any intermediate objects. For example, ${\mathcal A}$ does not have the type of all the fixed italic inscriptions of the letter below it.

Unfortunately, the description of ${\mathcal A}$ just given is too simple: it presumes that it is determinate which objects are fixed inscriptions and which objects are fixed inscriptions of the letter 'A.' But such determinacy cannot be assumed, and so there will be different "versions" of ${\mathcal A}$ that represent different specifications of which objects are fixed inscriptions and which objects are fixed inscriptions of the letter 'A.' Accordingly, ${\mathcal A}$ will in fact have to be such that on each branch there is a least node such that ${\mathcal A}_{n}$ is what I have just described as a version of ${\mathcal A}$ in ${\mathbb V}_{n}$ , and for every node above except the top node, ${\mathcal A}_{m}= H({\mathcal A}_{n},n)_{m}$ is true of ${\mathbb V}_{m}$ . It is tedious but routine to check that if there are predicates for inscription and inscription of 'A,' then the description just given of ${\mathcal A}$ can be completely formalized in the language for vagueness.

I am prepared to simply bite the bullet on the odd result that the letter 'A' is an inscription even though it is at best indeterminate whether it is a fixed inscription. First of all, it does not seem to me implausible to say that the letter is an inscription as a way of expressing that there can be no fixed instance that is not an inscription. Indeed, some have held that it is a distinguishing characteristic of types that they, unlike, for example, universals, have the characteristics of their tokens. $^{\text{footnote}}$ The ordinary sentiment that a letter is not an inscription—that is, not a particular inscription—is expressed by the fact that for any fixed inscription it is at best indeterminate whether the letter is that inscription. In some cases, we do, as the theory suggests, have some reluctance to declare outright that the letter is not the inscription: Imagine pointing at an inscription of the letter 'A' and declaring "This is not the letter 'A.'"

Not all abstract objects will have such properties as being inscriptions. (As we shall see below, mathematical objects are more fully abstract and do not have such properties.) Indeed, it is plausible to hold that only the abstract objects Parsons has called quasiconcrete (Mathematics in Philosophy, "Introduction," and "On some difficulties concerning intuition and intuitive knowledge") have the properties of their instances. There is a strong connection of the view that abstractness is vagueness with traditional doctrines that abstract objects arise by a process of abstraction—typically, that they have exactly the properties shared by all of their instances. Such views typically have the same outcome as the one just outlined with respect to a letter being an inscription. Such views have largely been abandoned in the face of scathing criticism of them by the likes of Frege (Grundlagen), Russell (Principles of Mathematics), and even Dummett ("Frege"), but the present proposal answers that criticism since it proposes, not a peculiar sort of object, but a notion of an indeterminate reference. For example, one traditional explanation of the (cardinal) number two on the basis of abstraction associates or identifies it with a set of two distinct objects that have no characteristics besides that distinctness. It is at the very least difficult to see how there could be such objects. On the counterpart of such a theory reformulated in terms of vagueness, one has not two distinct but otherwise identical objects, but two names, each of which can be stipulated to refer to any object, but with the proviso that they may not refer to the same one. In each application, the two names refer to distinct objects, and so our present conventions commit us to the claim that they already refer to two distinct objects, even though there is no way to say in what way the objects differ. Distinct but otherwise identical objects can occur in less peculiarly philosophical contexts as well: Consider a story in which there are two characters who are never differentiated. Say, for example, they are always referred to as "the twins." The characters are then distinct but indiscernible.

-- ShaughanLavine - 14 Mar 2005