ON QUANTIFICATION AND EXTENSIONALITY

Abstract

We investigate whether ordinary quantification over objects is an extensional phenomenon, or rather creates non-extensional contexts; each claim having been propounded by prominent philosophers. It turns out that the question only makes sense relative to a background theory of syntax and semantics (here called a grammar) that goes well beyond the inductive definition of formulas and the recursive definition of satisfaction. Two schemas for building quantificational grammars are developed, one that invariably constructs extensional grammars (in which quantification, in particular, thus behaves extensionally) and another that only generates non-extensional grammars (and in which quantification is responsible for the failure of extensionality). We then ask whether there are reasons to favor one of these grammar schemas over the other, and examine an argument according to which the proper formalization of deictic utterances requires adoption of non-extensional grammars.

1. Introduction

Is quantification over objects a non-extensional operation? In other words (speaking somewhat loosely for now), does the presence of ordinary quantification in an interpreted formal language ipso facto prevent it from having the property that, whenever in any complex well-formed expression one replaces some proper subexpression by another, coextensive one, the resulting expression is coextensive with the original one?

There are no doubt many readers to whom this question seems silly. Some will want to point out that Quine ¹ gave a categorical affirmative answer long ago:

It is well known, and easily seen, that the conspicuously limited means which we have lately allowed ourselves for compounding sentences—viz., ‘and’, ‘not’, and quantifiers—are capable of generating only extensional contexts. Quine [14, p. 12]

Others, coming from a different tradition, will likewise dismiss the question, but rather because they take it to have been answered in the negative. Nathan Salmon ² may be cited as an example:

The nonextensionality of a quantifier phrase is a surprising but trivial consequence of the way the quantifier works with a variable. (Salmon [17, p. 416])

The very fact that these eminent authors provide different answers to our question suggests that it might be worth taking a closer look. This is what we propose to do in this essay. ³

That contradictory answers should have been given is the more surprising in light of the fact that the two parties appear to agree on what we might call the generating syntax (i.e., the inductive definition of the well-formed expressions, particularly of quantified formulas), the ingredients for the semantic evaluation of well-formed expressions (to wit, model-theoretic structures and variable assignments), and the recursive (Tarskian [23]) steps to be taken in evaluating expressions relative to structures and variable assignments, as well as, presumably, on what it takes for well-formed expressions to be coextensive: ⁴

• • Two sentences are coextensive if and only if they are both true or both false.

• • Two singular terms are coextensive if and only if they designate the same object.

• • Two predicative expressions (“predicators,” as we will call them for brevity) are coextensive if and only if they are true of the same arrangements of arguments (such as n-tuples or sequences of objects) and false of the same arrangements of arguments.

We will see that what allows for reasonable disagreement as to the extensionality of quantification is the fact that the generating syntax and the recursive procedure underlying Tarskian satisfaction do not by themselves suffice to constrain coextensiveness sufficiently in order to answer the question. To do that, we need a grammar. ⁵ As we shall use the term here, a grammar $\mathsf {G}$ comprises all of the following:

1. (G1) A lexicon $\Lambda $ and a definition of the set $\mathsf {WF}^{\mathsf {G}}$ of $\mathsf {G}$ -well-formed expressions ( $\mathsf {G}$ -expressions for short) over $\Lambda $ , together with a relation of $\mathsf {G}$ -constituency on $\mathsf {WF}^{\mathsf {G}}.$

2. (G2) A specification of the logical space relevant to $\mathsf {G}$ , i.e., the class $\mathsf {Mod}(\mathsf {G})$ of $\mathsf {G}$ -models in which $\mathsf {G}$ -expressions are to be evaluated.

3. (G3) An ontology, i.e., a function $\mathsf {Obj}^{\mathsf {G}}$ that assigns to each $\mathsf {G}$ -model $\mathbf {M}$ the set $\mathsf {Obj}^{\mathsf {G}}_{\mathbf {M}}$ of $\mathsf {G}$ -objects $_{\mathbf {M}}.$

4. (G4) For each $\mathsf {G}$ -model $\mathbf {M}$ , a partition of the $\mathsf {G}$ -expressions into three $\mathsf {G}$ -categories $_{\mathbf {M}}$ : $\mathsf {G}$ -sentences $_{\mathbf {M}}$ , $\mathsf {G}$ -terms $_{\mathbf {M}}$ , and $\mathsf {G}$ -predicators $_{\mathbf {M}}.$

5. (G5) A $\mathsf {G}$ -truth operator $\mathsf {true}^{\mathsf {G}}$ that maps each $\mathsf {G}$ -model $\mathbf {M}$ to a function $\mathsf {true}^{\mathsf {G}}_{\mathbf {M}}$ from the set of $\mathsf {G}$ -sentences $_{\mathbf {M}}$ into $\{0,1\}$ , where a $\mathsf {G}$ -sentence $_{\mathbf {M}}$ $\phi $ is said to be $\mathsf {G}$ -true in $\mathbf {M}$ just in case $\mathsf {true}^{\mathsf {G}}_{\mathbf {M}}(\phi )=1$ , and $\mathsf {G}$ -false in $\mathbf {M}$ if $\mathsf {true}^{\mathsf {G}}_{\mathbf {M}}(\phi )=0.$

6. (G6) A $\mathsf {G}$ -designation operator $\mathsf {des^{\mathsf {G}}}$ that maps each $\mathsf {G}$ -model $\mathbf {M}$ to a function $\mathsf {des}^{\mathsf {G}}_{\mathbf {M}}$ all of whose admissible arguments are $\mathsf {G}$ -terms $_{\mathbf {M}}$ and whose values are $\mathsf {G}$ -objects $_{\mathbf {M}}$ (where, if $\mathsf {des}^{\mathsf {G}}_{\mathbf {M}}(t)=m$ , we say that t $\mathsf {G}$ -designates m in $\mathbf {M}$ ).

7. (G7) A $\mathsf {G}$ -predication operator $\mathsf {true{\mbox {-}}of}^{\mathsf {G}}$ that assigns to each $\mathsf {G}$ -model $\mathbf {M}$ a function $\mathsf {true{\mbox {-}}of}^{\mathsf {G}}_{\mathbf {M}}$ which maps each $\mathsf {G}$ -predicator $_{\mathbf {M}}$ $\Pi $ to a function $\mathsf {true{\mbox {-}}of}^{\mathsf {G}}_{\mathbf {M}}(\Pi )$ whose arguments are arrangements of members of the type hierarchy over $\mathsf {Obj}^{\mathsf {G}}_{\mathbf {M}}$ and whose values are in $\{0,1\}$ . If $\mathsf {true{\mbox {-}}of}^{\mathsf {G}}_{\mathbf {M}}(\Pi )(\rho )=1$ (respectively, $=0$ ), we say that $\Pi $ is $\mathsf {G}$ -true of $\rho $ in $\mathbf {M}$ (respectively, $\mathsf {G}$ -false of $\rho $ in $\mathbf {M}$ ).

Each of our two camps assumes, more or less tacitly, a background grammar; and what makes it possible for them to arrive at different answers to the extensionality question is that their grammars differ. In our reconstruction, the camps make the same choices, we shall see, with respect to (G1) and, but for a technicality, (G3). Their model spaces differ in a crucial way that we will explain in due course; but despite this difference in model spaces, the recursive semantics they pick to drive their respective items (G4) through (G7) are essentially the same; it’s just that they let the semantics drive these items in different ways, resulting in different outcomes with respect to (G4) through (G7) and hence in (closely related but nevertheless) different grammars.

Now suppose $\mathsf {G}$ is a grammar. For each $\mathsf {G}$ -model $\mathbf {M}$ , we define the $\mathsf {G}$ -coextensiveness relation in $\mathbf {M}$ , $\mathsf {coext}^{\mathsf {G}}_{\mathbf {M}}$ , as follows:

1. (DC1) $\mathsf {G}$ -sentences $_{\mathbf {M}}$ $\phi $ and $\psi $ are $\mathsf {G}$ -coextensive in $\mathbf {M}$ if and only if $\mathsf {true}^{\mathsf {G}}_{\mathbf {M}}(\phi )=\mathsf {true}^{\mathsf {G}}_{\mathbf {M}}(\psi )$ ; i.e., just in case they are either both $\mathsf {G}$ -true in $\mathbf {M}$ or both $\mathsf {G}$ -false in $\mathbf {M}$ .

2. (DC2) $\mathsf {G}$ -terms $_{\mathbf {M}}$ $t_0,t_1$ are $\mathsf {G}$ -coextensive in $\mathbf {M}$ if and only if they $\mathsf {G}$ -designate the same $\mathsf {G}$ -object $_{\mathbf {M}}$ in $\mathbf {M}$ , i.e., if and only if $t_0$ and $t_1$ are both in the domain of $\mathsf {des}^{\mathsf {G}}_{\mathbf {M}}$ and $\mathsf {des}^{\mathsf {G}}_{\mathbf {M}}(t_0)=\mathsf {des}^{\mathsf {G}}_{\mathbf {M}}(t_1)$ .

3. (DC3) $\mathsf {G}$ -predicators $_{\mathbf {M}}$ $\Pi _0,\Pi _1$ are $\mathsf {G}$ -coextensive in $\mathbf {M}$ if and only if they are $\mathsf {G}$ -true in $\mathbf {M}$ and $\mathsf {G}$ -false in $\mathbf {M}$ , respectively, of the same arrangements of arguments; in other words, just in case $\mathsf {true{\mbox {-}}of}^{\mathsf {G}}_{\mathbf {M}}(\Pi _0)$ and $\mathsf {true{\mbox {-}}of}^{\mathsf {G}}_{\mathbf {M}}(\Pi _1)$ are the same function.

4. (DC4) The $\mathsf {G}$ -coextensiveness relation in $\mathbf {M}$ is the union of the three category-specific $\mathsf {G}$ -coextensiveness relations in $\mathfrak {M}$ defined in (DC1) through (DC3).

Let $\mathsf {G}$ be a grammar and suppose that $\alpha _0$ , $\alpha _1$ , $\beta $ are well-formed $\mathsf {G}$ -expressions. Suppose further that o is an occurrence, as a constituent, of $\alpha _0$ in $\beta $ . Then we say that the replacement of $\alpha _0$ by $\alpha _1$ at the occurrence o in $\beta $ is $\mathsf {G}$ -legitimate if the result of this replacement is also a well-formed $\mathsf {G}$ -expression.

Now let $\mathbf {M}$ be a $\mathsf {G}$ -model. The $\mathsf {G}$ -expressions $\alpha _0$ and $\alpha _1$ are $\mathsf {G}$ -equicategorial in $\mathbf {M}$ if they are both $\mathsf {G}$ -sentences $_{\mathbf {M}}$ , both $\mathsf {G}$ -terms $_{\mathbf {M}}$ , or both $\mathsf {G}$ -predicators $_{\mathbf {M}}$ . A complex $\mathsf {G}$ -expression $\beta $ is said to be $\mathsf {G}$ -extensional in $\mathbf {M}$ if, whenever (i) $\alpha _0$ and $\alpha _1$ are well-formed $\mathsf {G}$ -expressions that are $\mathsf {G}$ -equicategorial and $\mathsf {G}$ -coextensive in $\mathbf {M}$ , (ii) o is an occurrence, as a constituent, of $\alpha _0$ in $\beta $ , and (iii) the replacement of the occurrence o of $\alpha _0$ with $\alpha _1$ in $\beta $ is $\mathsf {G}$ -legitimate, then the result of this replacement is $\mathsf {G}$ -coextensive in $\mathbf {M}$ with $\beta $ . The grammar $\mathsf {G}$ is extensional in $\mathbf {M}$ just in case every complex $\mathsf {G}$ -expression is $\mathsf {G}$ -extensional in $\mathbf {M}$ . Finally, $\mathsf {G}$ is extensional if $\mathsf {G}$ is extensional in every $\mathsf {G}$ -model $\mathbf {M}$ .

Here’s what we will do in the rest of this essay: The grammar-component (G1) common to both parties, as well as the recursive semantics upon which they agree, are laid out in Section 2. In Section 3 we define a “Quinean” grammar schema $\mathsf {Ext}$ that constructs, from any given quantificational lexicon $\Lambda $ , a grammar $\mathsf {Ext}_{\Lambda }$ , shown in Section 4 to be extensional. Against the background of $\mathsf {Ext}$ , the answer to the question whether quantification is extensional is affirmative. In Section 5 we define a “Salmonian” grammar schema $\mathsf {NExt}$ , whose grammars $\mathsf {NExt}_{\Lambda }$ can be construed as generalizations of their counterpart grammars $\mathsf {Ext}_{\Lambda }$ . It is shown that no $\mathsf {NExt}_{\Lambda }$ is extensional, and that it is indeed quantification that is responsible for the failure of extensionality in $\mathsf {NExt}$ -grammars. The question whether quantification is extensional thus has no absolute, grammar-independent answer. A related but different question, to wit, whether we should embrace grammars that make quantification extensional or rather those that make it non-extensional, however, might be answered by showing that one of $\mathsf {Ext}$ and $\mathsf {NExt}$ is preferable to the other. Considerations of simplicity would seem to favor $\mathsf {Ext}$ ; on the other hand, the non-extensional grammars $\mathsf {NExt}_{\Lambda }$ are more general than their extensional counterparts $\mathsf {Ext}_{\Lambda }$ . But generality is not per se a theoretical virtue: Unless the added generality serves a previously unfulfilled purpose, there is no reason to embrace it. This makes us shift attention, in Section 6, from formal and conceptual considerations to applications. We consider an argument for the usefulness of $\mathsf {NExt}$ ’s additional generality that turns on the application of grammars to the analysis of natural-language phenomena, specifically deixis. This argument is shown to be without force in Section 7, where we explain that the expressive advantage of $\mathsf {NExt}_{\Lambda }$ disappears when that grammar is compared to an extensional grammar $\mathsf {Ext}_{\Lambda ^+}$ over a suitably expanded lexicon $\Lambda ^+\supseteq \Lambda $ . The concluding Section 8 pulls together the results and observations obtained.

2. The common core

2.1. Lexicon and well-formed expressions

By a lexicon we shall mean a triple $\Lambda = (\mathcal {C}_\iota , \mathcal {C}_{\mu },\mathcal {P})$ , where:

• • $\mathcal {C}_\iota $ is a set of constant symbols of type e, or individual constants for short,

• • $\mathcal {C}_{\mu }$ is a set of constant symbols of type $(et)t$ , or Montagovian constants for short, and

• • $\mathcal {P}$ is a set of predicate symbols, each $P\in \mathcal {P}$ having a finite arity $\#P\geq 1$ .

$\mathcal {P}$ must contain at least one element. We let $\mathcal {P}_n$ be the set of n-ary members of $\mathcal {P}$ . Either or both of the sets $\mathcal {C}_\iota $ and $\mathcal {C}_{\mu }$ may be empty. Individual constants are the constant symbols familiar from elementary logic; Montagovian constants are, as it were, individual constants lifted to quantifier type. Their availability will only become relevant in Section 7.

All grammars we shall consider use for the construction of their well-formed expressions, in addition to the specific vocabulary provided by its lexicon, the following symbols:

• • the members of a fixed countably infinite set $\mathcal {V}$ of (individual) variables ${\mathsf {v}_0}, {\mathsf {v}_1}, \mathsf {v}_2,\ldots $ ;

• • the Boolean connectives $\neg $ and $\land $ ; and

• • the two quantifier symbols $\forall $ (universal) and $\exists $ (existential).

We make the usual disjointness assumptions, so that no connective may occur in a lexicon, no quantifier symbol may also be a variable, etc. ⁶

The $\Lambda $ -formulas are defined inductively as follows:

1. 1. Whenever $P\in \mathcal {P}_n$ and $t_1,\ldots ,t_n\in \mathcal {V}\cup \mathcal {C}_\iota $ , the result $Pt_1\ldots t_n$ of prefixing the string $t_1\ldots t_n$ with P is a $\Lambda $ -formula.

2. 2. Whenever $\phi $ and $\psi $ are $\Lambda $ -formulas, so are $\neg \phi $ and $(\phi \land \psi )$ .

3. 3. Whenever $\phi $ is a $\Lambda $ -formula, $x\in \mathcal {V}$ , and $C\in \mathcal {C}_{\mu }$ , the result $C x\phi $ of prefixing $\phi $ with $Cx$ is a $\Lambda $ -formula. ⁷

4. 4. Whenever $\phi $ is a $\Lambda $ -formula, $x\in \mathcal {V}$ , and $Q\in \{\forall ,\exists \}$ , the result $Qx\phi $ of prefixing $\phi $ with $Qx$ is a $\Lambda $ -formula.

The set of all $\Lambda $ -formulas is $\mathsf {Fml}_{\Lambda }$ . By a well-formed $\Lambda $ -expression we shall mean a member of $\mathcal {V}\cup \mathcal {C}_\iota \cup \mathcal {C}_{\mu }\cup \mathcal {P}\cup \{\forall ,\exists \}\cup \mathsf {Fml}_{\Lambda } =: \mathsf {WF}(\Lambda )$ .

Where $\alpha $ and $\beta $ are well-formed $\Lambda $ -expressions, we define the notion of an immediate constituent occurrence (ico) of $\alpha $ in $\beta $ as follows. (1) If $\beta $ is not a $\Lambda $ -formula, there are no ico’s of $\alpha $ in $\beta $ . (2) If $\beta $ is $Pt_1\ldots t_n$ , then o is an ico of $\alpha $ in $\beta $ if and only if either (a) $\alpha $ is P and o is $\ast t_1\ldots t_n$ , or (b) for some $i\in \{1,\ldots ,n\}$ , $\alpha $ is $t_i$ and o is $Pt_1\ldots t_{i-1}\ast t_{i+1}\ldots t_n$ . (3) If $\beta $ is $\neg \phi $ , then o is an ico of $\alpha $ in $\beta $ if and only if $\alpha $ is $\phi $ and o is $\neg \ast $ . (4) If $\beta $ is $(\phi \land \psi )$ , then o is an ico of $\alpha $ in $\beta $ if and only if either (a) $\alpha $ is $\phi $ and o is $(\ast \land \psi )$ , or (b) $\alpha $ is $\psi $ and o is $(\phi \land \ast )$ . (5) If $\beta $ is $\Sigma {\mathsf {v}_i}\phi $ with $\Sigma \in \mathcal {C}_\mu \cup \{\forall ,\exists \}$ , then o is an ico of $\alpha $ in $\beta $ if and only if either (a) $\alpha $ is $\Sigma $ and o is $\ast {\mathsf {v}_i}\phi $ , or (b) $\alpha $ is $\phi $ and o is $\Sigma {\mathsf {v}_i}\ast $ . We say that o is a constituent occurrence of $\alpha $ in $\beta $ if and only if o is $\pi [\rho ]$ , where (a) $\rho $ is an ico of $\alpha $ in a $\Lambda $ -formula $\gamma $ , and (b) $\beta $ is $\pi [\gamma ]$ . ⁸ Where $\alpha _0$ , $\alpha _1$ , and $\beta $ are $\Lambda $ -expressions and o is a constituent occurrence of $\alpha _0$ in $\beta $ (so that $\beta $ is $o[\alpha _0]$ ), $\alpha _1$ is legitimately substitutable for $\alpha _0$ at o in $\beta $ if $o[\alpha _1]$ is a well-formed $\Lambda $ -expression.

The set of variables having a free occurrence in a well-formed $\Lambda $ -expression is defined as follows. For $x\in \mathcal {V}$ , $\mathsf {FV}(x)=\{ x\}$ , and for $\alpha \in \mathcal {C}_\iota \cup \mathcal {C}_{\mu }\cup \mathcal {P}\cup \{\forall ,\exists \}$ , $\mathsf {FV}(\alpha )=\varnothing $ . For $\Lambda $ -formulas $\phi $ , the set $\mathsf {FV}(\phi )$ is defined recursively thus:

1. (1) $\mathsf {FV}(Pt_1\ldots t_n) = \bigcup \{ \mathsf {FV}(t_i) : 1\leq i\leq n \}.$

2. (2) $\mathsf {FV}(\neg \phi ) = \mathsf {FV}(\phi ).$

3. (3) $\mathsf {FV}(\phi \land \psi ) = \mathsf {FV}(\phi )\cup \mathsf {FV}(\psi ).$

4. (4) $\mathsf {FV}(Cx\phi ) = \mathsf {FV}(Qx\phi ) = \mathsf {FV}(\phi ) \setminus \{x\}$ for $C\in \mathcal {C}_{\mu }$ , $Q\in \{\forall ,\exists \}$ , and $x\in \mathcal {V.}$

2.2. Recursive semantics

A $\Lambda $ -structure is a pair $\mathfrak {M} = (M,\mathfrak {I})$ , where M, the universe of $\mathfrak {M}$ , is a non-empty set and $\mathfrak {I}$ is a $\Lambda $ -interpretation in M that maps each $\alpha \in \mathcal {C}_\iota \cup \mathcal {C}_{\mu }$ to an element $\mathfrak {I}(\alpha )\in M$ and each $P\in \mathcal {P}_n$ to a function $\mathfrak {I}(P): M^n\to \{0,1\}$ (i.e., to the characteristic function of a subset of $M^n$ ). The class of all $\Lambda $ -structures is $\mathsf {Str}(\Lambda )$ . ⁹

A variable assignment (or just assignment) in a non-empty set M is a function from a finite subset of $\omega $ into M. ¹⁰ The empty function $\varnothing $ is thus an assignment in any M. The set of all assignments in M is $\mathcal {G}(M)$ or just $\mathcal {G}$ when no confusion is likely; where $\mathfrak {M} $ is a structure, $\mathcal {G}(\mathfrak {M})$ is $\mathcal {G}(M)$ . When $g\in \mathcal {G}(M)$ and $i\in \mathsf {dom}(g)$ , we sometimes write $g_i$ instead of $g(i)$ . For $g,h\in \mathcal {G}(M)$ and $i\in \omega $ , we say that h is an i-permissive extension of g, in symbols $g\sqsubset _i h$ , just in case (a) $\mathsf {dom}(h)=\mathsf {dom}(g) \cup \{ i\}$ and (b) for all $j\in \mathsf {dom}(g)\setminus \{ i\}$ : $g(j) = h(j)$ . For $g\in \mathcal {G}(M)$ , $i\in \omega $ , and $m\in M$ , we write $g_i^m$ for the i-permissive extension of g that maps i to m.

Where $g\in \mathcal {G}$ , we say that a variable ${\mathsf {v}_i}\in \mathcal {V}$ is covered by g, or g-covered, if $i\in \mathsf {dom}(g)$ . The set $\mathcal {V}_g$ of g-covered variables is $\{{\mathsf {v}_i} : i\in \mathsf {dom}(g)\}$ ; hence $\mathcal {V}_\varnothing = \varnothing $ . A well-formed $\Lambda $ -expression $\alpha $ is g-closed if $\mathsf {FV}(\alpha )\subseteq \mathcal {V}_g$ , that is, if every variable that has a free occurrence in $\alpha $ is g-covered. Thus the members of $\mathcal {C}_\iota $ , of $\mathcal {C}_{\mu }$ , of $\mathcal {P}$ and of $\{\forall ,\exists \}$ are g-closed for any g; in particular, they are $\varnothing $ -closed or, as we will also say, closed tout court. A $\Lambda $ -expression that is not g-closed is g-open; so the $\varnothing $ -open expressions (i.e., the open expressions tout court) are just those expressions $\alpha $ with $\mathsf {FV}(\alpha )\neq \varnothing $ .

Definition 1 (Tarskian satisfaction values)

For each $\alpha \in \mathsf {WF}(\Lambda )$ , $\alpha $ ’s Tarskian satisfaction value (or just Tarski-value) is a function on $\mathsf {Str}(\Lambda )$ , where for $\mathfrak {M}\in \mathsf {Str}(\Lambda )$ , is the Tarski-value of $\alpha $ in $\mathfrak {M}$ . The definition proceeds by recursion on $\alpha $ . For each $\alpha $ and $\mathfrak {M}$ , is to be a function whose domain is a (not necessarily proper) subset of $\mathcal {G}(\mathfrak {M})$ .

1. 1. For variables ${\mathsf {v}_i}\in \mathcal {V}$ :

   1. (a) ,

   2. (b) and where , .

2. 2. For individual constants $c\in \mathcal {C}_\iota $ :

   1. (a) ,

   2. (b) and where , .

3. 3. For Montagovian constants $C\in \mathcal {C}_{\mu }$ :

   1. (a) ,

   2. (b) and where , with iff $\mathfrak {I}(C)\in X$ .

4. 4. For predicate symbols $P\in \mathcal {P}_n$ :

   1. (a) ,

   2. (b) and where , .

5. 5. For quantifier symbols $Q\in \{\forall ,\exists \}$ :

   1. (a) ,

   2. (b) and where $g\in \mathcal {G}$ , with iff $X=M$ and iff $X\neq \varnothing .$

6. 6. For predications $Pt_1\ldots t_n$ :

   1. (a) ,

   2. (b) and for :

7. 7. For negations $\neg \phi $ :

   1. (a) ,

   2. (b) and where :

8. 8. For conjunctions $(\phi \land \psi )$ :

   1. (a) ,

   2. (b) and where :

9. 9. For formulas $C{\mathsf {v}_i}\phi $ with $C\in \mathcal {C}_{\mu }$ :

   1. (a) ,

   2. (b) and where :, which equals

10. 10. For formulas $\forall {\mathsf {v}_i}\phi $ :

    1. (a) ,

    2. (b) and where :

11. 11. For formulas $\exists {\mathsf {v}_i}\phi $ :

    1. (a) ,

    2. (b) and where :

Note that, if $c\in \mathcal {C}_\iota $ , $C\in \mathcal {C}_{\mu }$ , and $\mathfrak {I}(c)=\mathfrak {I}(C)$ , the formulas $Cx\phi $ and $\exists x (x=c \land \phi )$ have the same Tarski-value in $\mathfrak {M}$ (and so does $\phi _x[c]$ , the result of replacing all free occurrences of x in $\phi $ by c). Montagovian constants are therefore quite innocuous with respect to first-order languages, since they are definable in terms of individual constants, identity, and the existential quantifier (conversely, of course, individual constants are reducible to Montagovian constants as well). As the identity of , , and suggests, the members of $\mathcal {C}_\mu $ are hybrids between individual constants and quantifiers: Their syntactic behavior is that of quantifiers, and they bind variables just like quantifiers do; on the other hand, their interpretations are single objects, just like those of individual constants, and like individual constants, they are essentially scopeless (cf. [27]).

One readily shows:

Lemma 1. Let $\mathfrak {M}$ be a $\Lambda $ -structure and $\alpha $ a $\Lambda $ -expression. Then:

1. 1. For all $g\in \mathcal {G}$ , if and only if $\alpha $ is g-closed.

2. 2. If agree on the indices of all variables in $\mathsf {FV}(\alpha )$ , .

3. 3. If and $h\in \mathcal {G}$ extends g, then and .

For $g\in \mathcal {G}$ , we let $\mathcal {G}_g$ be the set $\{h\in \mathcal {G}\, |\, \mathsf {dom}(g)\cap \mathsf {dom}(h)=\varnothing \}$ . Obviously $\mathcal {G}_\varnothing =\mathcal {G}$ . If F is a function whose domain is a subset of $\mathcal {G}$ , we let $F[g]$ be the function with domain $\{h\in \mathcal {G}_g: (g\cup h)\in \mathsf {dom}(F)\}$ that maps any assignment h in its domain to $F(g\cup h)$ , i.e., $F[g](h) = F(g\cup h)$ . Note that $F[\varnothing ] = F$ . Moreover, $\varnothing \in \mathsf {dom}(F[g])$ if and only if $g\in \mathsf {dom}(F)$ . For any set X, let $\boldsymbol {1}_X$ be the total constant function on X with value $1$ and let $\boldsymbol {0}_X$ be the total constant function on X with value $0$ .

We can then show, using Lemma 1:

Lemma 2. Let $\mathfrak {M}$ be a $\Lambda $ -structure, $g\in \mathcal {G}$ , and $\phi $ a $\Lambda $ -formula. The following are equivalent:

1. (a) $\phi $ is g-closed,

2. (b)

3. (c)

4. (d)

As an immediate consequence of Lemma 2, we obtain:

Corollary 1. Let $\mathfrak {M}$ be a $\Lambda $ -structure and $\phi $ a $\Lambda $ -formula. The following are equivalent:

1. (a) $\phi $ is closed,

2. (b)

3. (c)

4. (d)

3. The grammar schema $\mathsf {Ext}$

In this section we define, for each lexicon $\Lambda $ , a grammar $\mathsf {Ext}_{\Lambda }$ , by providing the necessary ingredients (G1) through (G7); in Section 4 we’ll see that each $\mathsf {Ext}_{\Lambda }$ is extensional. The function $\Lambda \mapsto \mathsf {Ext}_{\Lambda }$ will be called the grammar schema $\mathsf {Ext}$ .

Let $\Lambda $ be an arbitrary lexicon.

(G1) requires the specification of a lexicon; this is our chosen $\Lambda $ . The set of well-formed expressions of $\mathsf {Ext}_{\Lambda }$ is $\mathsf {WF}(\Lambda )$ as defined in Section 2.

(G2) requires the specification of $\mathsf {Ext}_{\Lambda }$ ’s logical space or model class $\mathsf {Mod}(\mathsf {Ext}_{\Lambda })$ . We let the $\mathsf {Ext}_{\Lambda }$ -models be the $\Lambda $ -structures as defined in Section 2; so $\mathsf {Mod}(\mathsf {Ext}_{\Lambda })$ is $\mathsf {Str}(\Lambda )$ .

(G3) requires the specification of a function $\mathsf {Obj}^{\mathsf {Ext}_{\Lambda }}$ that maps each $\Lambda $ -structure $\mathfrak {M}$ to a set $\mathsf {Obj}^{\mathsf {Ext}_{\Lambda }}_{\mathfrak {M}}$ of $\mathsf {Ext}_{\Lambda }$ -objects $_{\mathfrak {M}}$ . For each $\Lambda $ -structure $\mathfrak {M}=(M,\mathfrak {I})$ , let $\mathsf {Obj}^{\mathsf {Ext}_{\Lambda }}_{\mathfrak {M}}$ be M.

(G4) requires, for each $\Lambda $ -structure $\mathfrak {M}$ , a partition of $\mathsf {Ext}_{\Lambda }$ ’s well-formed expressions into $\mathsf {Ext}_{\Lambda }$ -sentences $_{\mathfrak {M}}$ , $\mathsf {Ext}_{\Lambda }$ -terms $_{\mathfrak {M}}$ , and $\mathsf {Ext}_{\Lambda }$ -predicators $_{\mathfrak {M}}$ . In the present case, this partition will in fact be independent of the $\mathsf {Ext}_{\Lambda }$ -model $\mathfrak {M}$ : The $\mathsf {Ext}_{\Lambda }$ -sentences $_{\mathfrak {M}}$ (in any $\mathfrak {M}$ ) are the closed $\Lambda $ -formulas, the $\mathsf {Ext}_{\Lambda }$ -terms $_{\mathfrak {M}}$ (in any $\mathfrak {M}$ ) are the members of $\mathcal {V}\cup \mathcal {C}_\iota $ , and the $\mathsf {Ext}_{\Lambda }$ -predicators $_{\mathfrak {M}}$ (in any $\mathfrak {M}$ ) are the members of $\mathcal {C}_{\mu }$ , the members of $\mathcal {P}$ , the quantifier symbols $\forall $ and $\exists $ , and the open $\Lambda $ -formulas. ¹¹ (We will in future simply speak of $\mathsf {Ext}_{\Lambda }$ -sentences, $\mathsf {Ext}_{\Lambda }$ -terms, and $\mathsf {Ext}_{\Lambda }$ -predicators.)

(G5) requires the definition of a truth operator that maps each $\Lambda $ -structure $\mathfrak {M}$ to a $\{0,1\}$ -valued function $\mathsf {true}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ on the set of $\mathsf {Ext}_{\Lambda }$ -sentences (i.e., closed $\Lambda $ -formulas). Let $\mathsf {true}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ be the function that maps each closed $\Lambda $ -formula $\phi $ to . Equivalently, a closed $\Lambda $ -formula $\phi $ is $\mathsf {Ext}_{\Lambda }$ -true in $\mathfrak {M}$ if and only if its Tarskian value in $\mathfrak {M}$ is $\boldsymbol {1}_{\mathcal {G}}$ .

(G6) requires the definition of a designation operator that maps each $\Lambda $ -structure $\mathfrak {M}=(M,\mathfrak {I})$ to a designation function $\mathsf {des}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ . Let $\mathsf {des}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ be the set that contains, for each individual constant $c\in \mathcal {C}_\iota $ , the pair $\langle c,\mathfrak {I}(c)\rangle $ , and nothing else (so that, in particular, no variable designates anything in any $\mathsf {Ext}_{\Lambda }$ -model).

(G7), finally, requires the definition of a predication operator that maps each $\Lambda $ -structure $\mathfrak {M}$ to a function $\mathsf {true{\mbox {-}}of}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ on the set of $\mathsf {Ext}_{\Lambda }$ -predicators whose value for a given predicator $\Pi $ is a $\{0,1\}$ -valued function on some set of arrangements of members of the type hierarchy over M. We let $\mathsf {true{\mbox {-}}of}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ be the function mapping:

1. (i) each Montagovian constant $C\in \mathcal {C}_{\mu }$ to the function , which is the characteristic function of the unary relation $\{ X\subseteq M\, |\, \mathfrak {I}(C)\in X \}$ on $\mathsf {Pow}(M).$

2. (ii) each $P\in \mathcal {P}_n$ to , i.e., to $\mathfrak {I}(P)$ , which by definition is the characteristic function of an n-ary relation over $M.$

3. (iii) each $Q\in \{\forall ,\exists \}$ to the function , which in either case is the characteristic function of a unary relation on $\mathsf {Pow}(M)$ , viz. of $\{ M\}$ or of $\{ X\subseteq M\, |\, X\neq \varnothing \}$ , respectively.

4. (iv) each open $\Lambda $ -formula $\phi $ to the function , which is the characteristic function of the unary relation on .

With respect to (iv), we take members of $\mathcal {G}$ to be bona fide arrangements of objects, just as members of $M^n$ are. In fact it would not be stretching terminology to call both types of arrangement sequences: A member $\{\langle 0,k \rangle , \langle 1,l \rangle , \langle 2,m \rangle \}$ of $M^3$ , say, is a sequence $\langle k, l, m, \_, \_,\ldots \rangle $ all of whose gaps occur after an initial gapless segment, and is at the same time an assignment in M covering ${\mathsf {v}_0}, {\mathsf {v}_1}, \mathsf {v}_2$ ; whereas $g = \{ \langle 1, k\rangle , \langle 3, l \rangle , \langle 6, m \rangle \}$ is a sequence $\langle \_, k, \_, l, \_, \_, m, \_, \_,\ldots \rangle $ with gaps between entries, and is at the same time an assignment in M, covering ${\mathsf {v}_1}, \mathsf {v}_3$ , and $\mathsf {v}_6$ . We thus take both predicate symbols and open formulas to be true and false of arrangements of objects (which, incidentally, we’ll need to do within the grammar schema $\mathsf {NExt}$ , too).

4. Extensionality of $\mathsf {Ext}_{\Lambda }$

Given $\mathsf {Ext}_{\Lambda }$ , conditions (DC1) through (DC4) determine the $\mathsf {Ext}_{\Lambda }$ -coextensiveness relation in each $\Lambda $ -structure $\mathfrak {M}$ . Applied to $\mathsf {Ext}_{\Lambda }$ , these conditions yield:

1. (DC1 _{Ext Λ} ) Closed $\Lambda $ -formulas $\phi $ and $\psi $ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case they are either both $\mathsf {Ext}_{\Lambda }$ -true in $\mathfrak {M}$ or both $\mathsf {Ext}_{\Lambda }$ -false in $\mathfrak {M}$ , i.e., just in case and , or and . It follows that closed $\Lambda $ -formulas $\phi $ and $\psi $ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case .

2. (DC2 _{Ext Λ} ) Members $t_1$ and $t_2$ of $\mathcal {V}\cup \mathcal {C}_\iota $ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case they are both members of $\mathcal {C}_\iota $ and $\mathfrak {I}(t_1)=\mathfrak {I}(t_2)$ . Equivalently, distinct members $t_1,t_2$ of $\mathcal {V}\cup \mathcal {C}_\iota $ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case $t_1$ and $t_2$ are both members of $\mathcal {C}_\iota $ and . It follows that distinct members $t_1,t_2$ of $\mathcal {V}\cup \mathcal {C}_\iota $ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case . ¹²

3. (DC3 _{Ext Λ} ) Where each of $\alpha _0$ and $\alpha _1$ is either a member of $\mathcal {C}_{\mu }$ , a member of $\mathcal {P}$ , a member of $\{\forall ,\exists \}$ , or an open $\Lambda $ -formula, $\alpha _0$ and $\alpha _1$ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case they are $\mathsf {Ext}_{\Lambda }$ -true and $\mathsf {Ext}_{\Lambda }$ -false in $\mathfrak {M}$ of the same arrangements of members of the type hierarchy over M. It follows that:

   1. (a) If $n\geq 1$ and $\alpha _0$ and $\alpha _1$ are both members of $\mathcal {P}_n$ , they are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case , i.e., just in case (since each is a total constant function on $\mathcal {G}$ ).

   2. (b) If $\alpha _0$ and $\alpha _1$ are both members of $\mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ , they are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case , i.e., just in case (again since each is a total constant function on $\mathcal {G}$ ).

   3. (c) If $\alpha _0$ and $\alpha _1$ are both open $\Lambda $ -formulas, they are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case .

4. (DC4 _{Ext Λ} ) Well-formed $\Lambda $ -expressions belonging to distinct $\mathsf {Ext}_{\Lambda }$ -categories are not $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ .

What will matter for the extensionality of $\mathsf {Ext}_{\Lambda }$ in $\mathfrak {M}$ is to what extent distinct $\mathsf {Ext}_{\Lambda }$ -equicategorial $\Lambda $ -expressions $\alpha _0$ and $\alpha _1$ that are legitimately substitutable for each other are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ . Among $\mathsf {Ext}_{\Lambda }$ -equicategorial expressions, there are five cases of legitimate substitutability:

1. (GS1) A member of $\mathcal {V}\cup \mathcal {C}_\iota $ may be legitimately replaced by another member of $\mathcal {V}\cup \mathcal {C}_\iota $ .

2. (GS2) A member of any $\mathcal {P}_n$ may be legitimately replaced by another member of $\mathcal {P}_n$ .

3. (GS3) A member of $\mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ may be legitimately replaced by a member of $\mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ .

4. (GS4) An open $\Lambda $ -formula may be legitimately replaced by another open $\Lambda $ -formula.

5. (GS5) A closed $\Lambda $ -formula may be legitimately replaced by another closed $\Lambda $ -formula.

In case (GS1), we already know that distinct members $\alpha _0$ and $\alpha _1$ of $\mathcal {V}\cup \mathcal {C}_\iota $ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case . In case (GS2), we likewise already know that distinct members $\alpha _0$ and $\alpha _1$ of $\mathcal {P}_n$ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case , and again in case (GS3), we’ve already seen that distinct members $\alpha _0$ and $\alpha _1$ of $\mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case . In cases (GS4) and (GS5), we already know that two open (respectively, two closed) formulas $\alpha _0$ and $\alpha _1$ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ just in case . Thus:

Lemma 3. $\Lambda $ -expressions $\alpha _0$ and $\alpha _1$ that are $\mathsf {Ext}_{\Lambda }$ -equicategorial and legitimately intersubstitutable are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ iff .

A Quinean digression. We note that (DC1 $_{\mathsf {Ext}_{\Lambda }}$ ) through (DC4 $_{\mathsf {Ext}_{\Lambda }}$ ) accord with Quine’s notion of coextensiveness (if we take into account that, unlike us, Quine treats quantifier expressions syncategorematically, and assume that he leaves (DC4 $_{\mathsf {Ext}_{\Lambda }}$ ) implicit). Of particular relevance here is (DC3 $_{\mathsf {Ext}_{\Lambda }}$ )(c), the treatment of open formulas, which is the critical difference between our two grammar schemas. This comes out most clearly in the following passage from “Confessions of a Confirmed Extensionalist” [15], where we’ve indicated in boldface which of (DC1 $_{\mathsf {Ext}_{\Lambda }}$ ) through (DC3 $_{\mathsf {Ext}_{\Lambda }}$ ) we take Quine to be referring to. ¹³

(DC1 $_{\mathsf {Ext}_{\Lambda }}$ ) I shall call two closed [formulas] coextensive if they are both true or both false. (DC3 $_{\mathsf {Ext}_{\Lambda }}$ ) Two predicates or general terms or open [formulas] are coextensive, of course, if they are true of just the same objects or sequences of objects. (DC2 $_{\mathsf {Ext}_{\Lambda }}$ ) Two singular terms are coextensive if they designate the same object. […] In defining coextensiveness, I lumped predicates, general terms, and open [formulas] together. They are what can be predicated of objects or sequences of objects, and in that capacity they all three come to the same thing. […] (DC3 $_{\mathsf {Ext}_{\Lambda }}$ )(c) Two open [formulas] are coextensive if they have the same free variables and agree with each other in truth-value for all values of those variables. [15, p. 215]

Note that two open formulas are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ (i.e., have the same Tarski-value in $\mathfrak {M}$ ) if and only if they have the same free variables and have the same truth value in $\mathfrak {M}$ for all assignments covering those variables, just as Quine requires in the final sentence of the quoted passage. End of Quinean digression.

We are now in a position to show that $\mathsf {Ext}_{\Lambda }$ is extensional in every $\Lambda $ -structure $\mathfrak {M}$ . This requires showing that, for any $\mathfrak {M}$ and any $\Lambda $ -formula $\phi $ , if we construct a new $\Lambda $ -formula $\phi '$ from $\phi $ by legitimately replacing some constituent $\alpha _0$ of $\phi $ by some $\alpha _1$ that is $\mathsf {Ext}_{\Lambda }$ -equicategorial and $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ with $\alpha _0$ , the formula $\phi '$ is $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ with $\phi $ .

As a first step, we observe that this holds if we replace “constituent” with “immediate constituent”. Call this property immediate-constituent substitutivity. ¹⁴

This follows by Lemma 3 from inspection of clauses 6 through 11 in Definition 1. For suppose $\alpha _0$ and $\alpha _1$ are legitimately intersubstitutable, $\mathsf {Ext}_{\Lambda }$ -equicategorial, and $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ . Then by Lemma 3. Now if $\phi $ is $Pt_1\ldots t_n$ , as long as and for each $1\leq i\leq n$ , we have by clause 6 of Definition 1, and so . Similarly, by clause 7, if $\phi $ is $\neg \alpha _0$ , as long as , we have , and thus . The case of conjunction works analogously. Finally, by clauses 9–11, for any $\alpha _0,\alpha _1\in \mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ and $\Lambda $ -formula $\psi $ , if $\phi $ is $\alpha _0{\mathsf {v}_i}\psi $ , as long as , we will have and thus ; and for any $Q\in \mathcal {C}_\mu \cup \{\forall ,\exists \}$ and $\Lambda $ -formulas $\alpha _0,\alpha _1$ , if $\phi $ is $Q{\mathsf {v}_i}\alpha _0$ , as long as , and hence . Thus in all cases . Therefore $\phi $ and $\phi '$ are either both closed or both open $\Lambda $ -formulas (otherwise ), hence $\mathsf {Ext}_{\Lambda }$ -equicategorial and $\mathsf {Ext}_{\Lambda }$ -legitimately substitutable in $\mathfrak {M}$ and so, by Lemma 3, $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ .

This observation generalizes easily to arbitrary-constituent substitutivity. ¹⁵

Observation 1 (General substitutivity)

Let $\mathfrak {M}$ be a $\Lambda $ -structure and let $\alpha _0$ be be legitimately replaceable by $\alpha _1$ at a constituent occurrence o in $\phi $ . Let $\alpha _0$ and $\alpha _1$ be $\mathsf {Ext}_{\Lambda }$ -equicategorial and $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ . Let $\phi '$ be the result of replacing $\alpha _0$ at o by $\alpha _1$ . Then $\phi $ and $\phi '$ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ .

Proof. Since $\alpha _0$ and $\alpha _1$ are $\mathsf {Ext}_{\Lambda }$ -equicategorial and legitimately substitutable, they are either both in $\mathcal {C}_\iota \cup \mathcal {V}$ , both in $\mathcal {P}_n$ for the same n, both in $\mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ , both open $\Lambda $ -formulas, or both closed $\Lambda $ -formulas. Since $\alpha _0$ and $\alpha _1$ are $\mathsf {Ext}_{\Lambda }$ -coextensive in $\mathfrak {M}$ , . We show by induction on $\phi $ that (from which $\mathsf {Ext}_{\Lambda }$ -coextensiveness of $\alpha _0$ and $\alpha _1$ follows as above). For the induction basis, note that any constituent occurrence of $\alpha _0$ in an atomic $\Lambda $ -formula $\phi $ is an immediate constituent occurrence, and hence the desired result follows from immediate-constituent substitutivity.

If $\phi $ is $\neg \chi $ and $\alpha _0$ occurs once in $\phi $ , $\alpha _0$ is either $\chi $ and thus an immediate constituent (so we’re done by immediate-constituent substitutivity), or $\alpha _0$ ’s occurrence is within $\chi $ . In the latter case, by induction hypothesis and so by immediate-constituent substitutivity. The case of conjunctions is quite analogous.

If finally $\phi $ is $Q{\mathsf {v}_i}\chi $ with $Q\in \mathcal {C}_{\mu }\cup \{\forall ,\exists \}$ and $\alpha _0$ has a single occurrence in $\phi $ , $\alpha _0$ occurs either as (i) one of Q, ${\mathsf {v}_i}$ , and $\chi $ and thus as an immediate constituent (so the desired result follows from immediate-constituent substitutivity), or else (ii) within $\chi $ . In the latter case, by induction hypothesis and again, by immediate-constituent substitutivity, , as required.

Corollary 2. For any $\Lambda $ -structure $\mathfrak {M}$ , $\mathsf {Ext}_{\Lambda }$ is extensional in $\mathfrak {M}$ .

Corollary 3. For any lexicon $\Lambda $ , $\mathsf {Ext}_{\Lambda }$ is extensional.

5. The grammar schema $\mathsf {NExt}$

We now define a grammar schema $\mathsf {NExt}$ , i.e., a function $\Lambda \mapsto \mathsf {NExt}_{\Lambda }$ mapping lexicons $\Lambda $ to grammars $\mathsf {NExt}_{\Lambda }$ , that will be able to underwrite Salmon’s claim regarding the non-extensionality of quantification. As with $\mathsf {Ext}_{\Lambda }$ , we need to provide, for each lexicon $\Lambda $ , the ingredients (G1) through (G7) required for the specification of a grammar. We begin by picking an arbitrary lexicon $\Lambda =(\mathcal {C}_\iota ,\mathcal {C}_\mu ,\mathcal {P})$ .

(G1) requires the specification of a lexicon; this is our chosen $\Lambda $ . The set of well-formed expressions of $\mathsf {NExt}_{\Lambda }$ is $\mathsf {WF}(\Lambda )$ . Thus $\mathsf {Ext}_{\Lambda }$ and $\mathsf {NExt}_{\Lambda }$ agree on (G1).

(G2) requires the specification of $\mathsf {NExt}_{\Lambda }$ ’s model class. We let the $\mathsf {NExt}_{\Lambda }$ -models be pairs $\mathfrak {M}^g := (\mathfrak {M},g)$ consisting of a $\Lambda $ -structure $\mathfrak {M}$ and an $\mathfrak {M}$ -assignment g. We will call such pairs $\Lambda $ -context systems, g being the context and $\mathfrak {M}$ the underlying structure of the context system $\mathfrak {M}^g$ . (By a minimal context system we will mean a context system $\mathfrak {M}^\varnothing $ .) The logical space $\mathsf {Mod}(\mathsf {NExt}_{\Lambda })$ of $\mathsf {NExt}_{\Lambda }$ is thus the class of all $\Lambda $ -context systems. Thus every $\mathsf {NExt}_{\Lambda }$ -model $\mathfrak {M}^g$ contains an $\mathsf {Ext}_{\Lambda }$ -model $\mathfrak {M}$ as its underlying structure.

(G3) requires the specification of a function $\mathsf {Obj}^{\mathsf {NExt}_{\Lambda }}$ that maps each $\Lambda $ -context system $\mathfrak {M}^g$ to a set $\mathsf {Obj}^{\mathsf {NExt}_{\Lambda }}_{\mathfrak {M}^g}$ of $\mathsf {NExt}_{\Lambda }$ -objects $_{\mathfrak {M}^g}$ . Where the structure $\mathfrak {M}$ underlying the context system $\mathfrak {M}^g$ is $(M,\mathfrak {I})$ , we let $\mathsf {Obj}^{\mathsf {NExt}_{\Lambda }}_{\mathfrak {M}^g}$ be M. Thus $\mathsf {Ext}_{\Lambda }$ and $\mathsf {NExt}_{\Lambda }$ essentially agree, modulo the difference in their notions of model, on (G3).

(G4) requires, for each $\Lambda $ -context system $\mathfrak {M}^g$ , a partition of $\mathsf {NExt}_{\Lambda }$ ’s well-formed expressions into $\mathsf {NExt}_{\Lambda }$ -sentences $_{\mathfrak {M}^g}$ , $\mathsf {NExt}_{\Lambda }$ -terms $_{\mathfrak {M}^g}$ , and $\mathsf {NExt}_{\Lambda }$ -predicators $_{\mathfrak {M}^g}$ . The $\mathsf {NExt}_{\Lambda }$ -sentences $_{\mathfrak {M}^g}$ are the g-closed $\Lambda $ -formulas, the $\mathsf {NExt}_{\Lambda }$ -terms $_{\mathfrak {M}^g}$ are the members of $\mathcal {V}\cup \mathcal {C}$ , and the $\mathsf {NExt}_{\Lambda }$ -predicators $_{\mathfrak {M}^g}$ are the members of $\mathcal {C}_{\mu }$ , the members of $\mathcal {P}$ , the members of $\{\forall ,\exists \}$ , and the g-open $\Lambda $ -formulas. Thus the $\mathsf {NExt}_{\Lambda }$ -categories are model-dependent, unlike the $\mathsf {Ext}_{\Lambda }$ -categories. Note that relative to minimal context systems, the $\mathsf {NExt}_{\Lambda }$ -notions of sentence, term, and predicator are just those of $\mathsf {Ext}_{\Lambda }$ .

(G5) requires the definition of a truth operator that maps each $\Lambda $ -context system $\mathfrak {M}^g$ to the set $\mathsf {true}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ of $\mathsf {NExt}_{\Lambda }$ -sentences $_{\mathfrak {M}^g}$ (i.e., g-closed $\Lambda $ -formulas) that are $\mathsf {NExt}_{\Lambda }$ -true in $\mathfrak {M}^g$ . Let $\mathsf {true}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ be the set of all g-closed $\Lambda $ -formulas $\phi $ for which (equivalently, for which ). Note that by Lemma 2, a g-closed formula $\phi $ that is not in $\mathsf {true}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ is such that , or equivalently, . Also, since , $\mathsf {true}_{\mathfrak {M}^\varnothing }^{\mathsf {NExt}_{\Lambda }}=\mathsf {true}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ .

(G6) requires the definition of a designation operator that maps each $\Lambda $ -context system $\mathfrak {M}^g$ to a function $\mathsf {des}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ whose domain consists of $\mathsf {NExt}_{\Lambda }$ -terms $_{\mathfrak {M}^g}$ (i.e., variables and constant symbols) and whose values are $\mathsf {NExt}_{\Lambda }$ -objects $_{\mathfrak {M}^g}$ (i.e., members of M). Let $\mathsf {des}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ be the set that contains, for each individual constant $c\in \mathcal {C}_\iota $ , the pair $\langle c,\mathfrak {I}(c)\rangle $ , and for each g-covered variable ${\mathsf {v}_i}$ , the pair $\langle {\mathsf {v}_i},g(i)\rangle $ ; and nothing else (so that variables not covered by g do not designate anything). Thus $\mathsf {des}_{\mathfrak {M}^\varnothing }^{\mathsf {NExt}_{\Lambda }}=\mathsf {des}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ .

(G7), finally, requires the definition of a predication operator that maps each $\Lambda $ -context system $\mathfrak {M}^g$ to a function $\mathsf {true{\mbox {-}}of}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ mapping each $\mathsf {NExt}_{\Lambda }$ -predicator $_{\mathfrak {M}^g}$ (i.e., member of $\mathcal {C}_{\mu }$ , member of $\mathcal {P}$ , member of $\{\forall ,\exists \}$ , or g-open $\Lambda $ -formula) to a $\{0,1\}$ -valued function on a set of arrangements of members of the type hierarchy over M. We let $\mathsf {true{\mbox {-}}of}_{\mathfrak {M}^g}^{\mathsf {NExt}_{\Lambda }}$ be the function mapping:

1. (i) each Montagovian constant $C\in \mathcal {C}_{\mu }$ to the function , which is the characteristic function of the subset $\{ X\subseteq M\, |\, \mathfrak {I}(C)\in X \}$ of $\mathsf {Pow}(M)$ .

2. (ii) each $P\in \mathcal {P}_n$ to , i.e., to $\mathfrak {I}(P)$ , which by definition is the characteristic function of a subset of $M^n$ .

3. (iii) each $Q\in \{\forall ,\exists \}$ to the function , which in either case is the characteristic function of a subset of $\mathsf {Pow}(M)$ , viz. of $\{ M\}$ or of $\{ X\subseteq M\, |\, x\neq \varnothing \}$ , respectively.

4. (iv) each g-open $\Lambda $ -formula $\phi $ to the function , which is obviously the characteristic function of some subset of the domain of .

Note that $\mathsf {true{\mbox {-}}of}_{\mathfrak {M}^\varnothing }^{\mathsf {NExt}_{\Lambda }}=\mathsf {true{\mbox {-}}of}_{\mathfrak {M}}^{\mathsf {Ext}_{\Lambda }}$ .

Obviously, modulo the identification of minimal $\Lambda $ -context systems $(\mathfrak {M},\varnothing )$ with their underlying $\Lambda $ -structures $\mathfrak {M}$ , the grammar $\mathsf {Ext}_{\Lambda }$ is simply the result of restricting the logical space of $\mathsf {NExt}_{\Lambda }$ to minimal context systems, i.e., context systems with an empty context. In this precise sense, then, $\mathsf {NExt}_{\Lambda }$ is a generalization of $\mathsf {Ext}_{\Lambda }$ . It also follows that $\mathsf {Ext}_{\Lambda }$ -coextensiveness in $\mathfrak {M}$ and $\mathsf {NExt}_{\Lambda }$ -coextensiveness in $\mathfrak {M}^\varnothing $ coincide, so any violation of extensionality in $\mathsf {NExt}_{\Lambda }$ will have to occur in non-minimal context systems.

The coextensiveness condition (DC1) plays out as follows in $\mathsf {NExt}_{\Lambda }$ : ¹⁶

1. (DC1 _{NExt Λ} ) g-closed $\Lambda $ -formulas $\phi $ and $\psi $ are $\mathsf {NExt}_{\Lambda }$ -coextensive in $\mathfrak {M}^g$ just in case they are either both $\mathsf {NExt}_{\Lambda }$ -true in $\mathfrak {M}^g$ or both $\mathsf {NExt}_{\Lambda }$ -false in $\mathfrak {M}^g$ , i.e., just in case and , or and . Since, for g-closed formulas $\chi $ , , we have that g-closed $\Lambda $ -formulas $\phi $ and $\psi $ are $\mathsf {NExt}_{\Lambda }$ -coextensive in $\mathfrak {M}^g$ just in case .

Violations of extensionality occur within $\mathsf {NExt}_{\Lambda }$ regardless of the nature of $\Lambda $ : By definition $\mathcal {P}\neq \varnothing $ . Suppose without loss of generality that $\mathcal {P}$ contains a one-place predicate symbol P. Let $\mathfrak {M}$ be such that $\mathfrak {I}(P)(a)=1$ and $\mathfrak {I}(P)(b)=0$ . Let $g(0)=g(1)=a$ . Then both $P{\mathsf {v}_0}$ and $P{\mathsf {v}_1}$ are $\mathsf {NExt}_{\Lambda }$ -true in $\mathfrak {M}^g$ , yet $\forall {\mathsf {v}_0} P{\mathsf {v}_0}$ will be $\mathsf {NExt}_{\Lambda }$ -false and $\forall {\mathsf {v}_0} P{\mathsf {v}_1}$ , $\mathsf {NExt}_{\Lambda }$ -true in $\mathfrak {M}^g$ . (Clearly a variant of this strategy can be used for a lexicon containing a predicate symbol of an arity $n>1$ .) ^{17 , 18} And since atomic formulas can be shown to be $\mathsf {NExt}_{\Lambda }$ -extensional in any $\Lambda $ -context system, it is indeed quantification that is responsible for this failure of extensionality. ¹⁹ Thus:

Observation 2. $\mathsf {NExt}_{\Lambda }$ is never extensional.

6. Context systems and deixis

In $\mathsf {Ext}$ and $\mathsf {NExt}$ , we have two grammar schemas that build on the same generating syntax and the same recursive semantics but that nevertheless differ with respect to the extensionality of their generated grammars. Since grammars of either kind are, in an intuitive but clear sense, grammars of quantification, we are led to conclude that the question whether quantification is extensional is, unless further qualified, ill-posed, because it is answerable only relative to an antecedently chosen grammar.

It is nevertheless conceivable that one of our two grammar schemas should, all things considered, be preferable to the other, and if this were the case, it might indirectly support the answer to the extensionality question given by the camp endorsing the superior grammar schema. Are there arguments, then, that might be advanced in favor of $\mathsf {Ext}$ over $\mathsf {NExt}$ or vice versa?

For a start, as we’ve seen in the preceding section, there is a precise sense in which $\mathsf {NExt}$ generalizes $\mathsf {Ext}$ (if we identify a $\Lambda $ -structure $\mathfrak {M}$ with the minimal $\Lambda $ -context system $\mathfrak {M}^\varnothing $ that it underlies). It is thus tempting to argue that it is the more general schema, i.e., $\mathsf {NExt}$ , that should win out; adherents of $\mathsf {Ext}$ , according to this argument, take too narrow a view of logical space by considering only minimal context systems, and are misled into adopting a grammar schema that, because of its myopia, makes quantification look extensional.

One might counter on behalf of the extensionalist that not every generalization is an appropriate generalization. Perhaps the generalization $\mathsf {NExt}$ of $\mathsf {Ext}$ is a spurious one, one that accidentally destroys extensionality but that otherwise has no point. If that is right, it should be $\mathsf {Ext}$ that rules; after all, its grammars are not only more parsimonious ²⁰ than $\mathsf {NExt}$ ’s, but also extensional; and as Lewis [9, p. 256] notes, “extensionality itself is generally thought to be an important dimension of simplicity.” Thus, if there’s no point to adopting $\mathsf {NExt}$ , we should embrace $\mathsf {Ext}$ on grounds of theoretical simplicity.

It seems, however, that this argument of the extensionalist’s won’t fly. Granted, for the purpose of formalizing mathematics there is perhaps little point in replacing simple structures with arbitrary context systems. But within philosophy of language and linguistic semantics, context systems are routinely appealed to in the analysis of deictic utterances involving third-person singular pronouns, and thus certainly appear to have a point.

The idea that references for deictic pronouns are to be furnished by a contextual variable assignment goes back at least to Montague [10], is prominent in Kaplan [8], and has gained wide currency through the influential textbook by Heim and Kratzer [6]. ²¹ In brief, the proposal is that the context g is the result of various demonstrative acts on the part of the speaker, and that the objects demonstrated can be designated via variables co-indexed with the objects’ places in g.

To take a simple example, suppose we wanted to formalize the following utterance:

1. (A) Pointing at Venkatraman Ramakrishnan: He is a genius.

According to the view under discussion, the speaker’s pointing gesture places Ramakrishnan into a suitable position in the context g, and it is assumed that the deictic pronoun (tacitly, as it were) carries the index that corresponds to that position. The utterance is therefore to be represented more precisely as

1. (A _i ) Pointing at Venkatraman Ramakrishnan: $\text {He}_i$ is a genius,

where the context g generated by the pointing gesture is such that $g_i = \text {Ramakrishnan}$ .

Now (A $_i$ ) can be formalized, in any grammar $\mathsf {NExt}_{\Lambda }$ whose lexicon contains the one-place predicate symbol $\texttt {Genius}$ , as

1. (B) $\texttt {Genius}({\mathsf {v}_i})$ .

In any context system $\mathfrak {M}^h$ whose context h covers ${\mathsf {v}_i}$ , formula (B) is $\mathsf {NExt}_{\Lambda }$ -true just in case $h_i$ belongs to $\mathfrak {I}(\texttt {Genius})$ . So in the intended context system where $\mathfrak {I}(\texttt {Genius})$ is the set of geniuses and g is determined by the speaker’s pointing gesture as assumed above, (B) will be true just in case $g_i$ , i.e., Venki Ramakrishnan, is a genius, which is precisely what is required of an adequate formalization of (A). ²²

An analysis of deixis of this nature is unavailable to adherents of $\mathsf {Ext}$ . No grammar $\mathsf {Ext}_{\Lambda }$ counts the open formula (B) as a sentence and thus neither its truth nor its falsity can even be entertained—after all, no $\mathsf {Ext}_{\Lambda }$ -model, i.e., no $\Lambda $ -structure $\mathfrak {M}$ (equivalently, no minimal $\Lambda $ -context system $\mathfrak {M}^\varnothing $ ) supplies a value for the variable ${\mathsf {v}_i}$ . The grammars $\mathsf {Ext}_{\Lambda }$ therefore seem severely limited in their applicability to natural-language analysis in that they cannot handle deixis—surely a serious weakness. ²³

It thus looks like we have a substantive argument, based on the logical analysis of a natural-language phenomenon, in favor of the more general grammars $\mathsf {NExt}_{\Lambda }$ , and thus, indirectly, in favor of the non-extensionality of quantification.

7. An extensionalist approach to deixis

What might an adherent of $\mathsf {Ext}$ do in the face of utterances like (A)? More precisely, how might they treat the pronoun he as it occurs in (A)? Given the available lexical resources, and given that they cannot model he as a free variable, there would seem to be two options: The pronoun might correspond to an individual constant, or it might correspond to a bound rather than a free variable. Let us consider these in turn.

If we want to render deictic pronouns as individual constants, we need to think of the lexicon as being expanded, at the time of a deictic utterance like (A), by a new individual constant d corresponding to the deictic pronoun he, and to think of a demonstrative gesture not as assigning an object m to a certain hitherto valueless variable ${\mathsf {v}_i}$ that thereby becomes covered by the context (as a $\mathsf {NExt}$ -theorist would), but as introducing, and fixing m as the interpretation of, the new individual constant d.

Put somewhat differently, where the received view regards a deictic utterance like (A), or rather (A $_i$ ), as transforming an initial context system $(\mathfrak {M},g)$ via the gesture pointing at Venki Ramakrishnan, into a richer context system $(\mathfrak {M},g_i^{\text {Venki}})$ , Venki thus becoming the value of the variable ${\mathsf {v}_i}$ , $\mathsf {Ext}$ -grammarians rather think of (A) as transforming an initial $\Lambda $ -structure $(M,\mathfrak {I})$ via the pointing gesture into a richer $\Lambda _{d}$ -structure $(M,\mathfrak {I}_{d}^{\text {Venki}})$ , where $\Lambda _{d}$ is the expansion $(\mathcal {C}_\iota \cup \{ d \},\mathcal {C}_{\mu },\mathcal {P})$ of $\Lambda $ by the previously uninterpreted individual constant d, and $\mathfrak {I}_{d}^{\text {Venki}}$ is $\mathfrak {I} \cup \{ \langle d, \text {Venki}\rangle \}$ .

It should be clear that, if the extensionalist follows this route, they have no expressiveness problems in dealing with deixis. After all, in order to say what is expressed in $\mathsf {NExt}_{\Lambda }$ by $\phi (\mathsf {v}_{i_0},\ldots ,\mathsf {v}_{i_n})$ , with the free variables $\mathsf {v}_{i_0},\ldots ,\mathsf {v}_{i_n}$ having been assigned values through pointing, our extensionalist can use $\phi (d_{0},\ldots ,d_{n})$ , where each $d_{j}$ is interpreted by their $\mathsf {Ext}_{\Lambda _{d_0,\ldots ,d_n}}$ -model as the object that the $\mathsf {NExt}$ -theorist would assign to the variable $\mathsf {v}_{i_j}$ as its contextual value.

So it looks as if the pendulum has swung back towards $\mathsf {Ext}$ : the logical analysis of deixis does not, after all, favor $\mathsf {NExt}$ over $\mathsf {Ext}$ , if we’re willing to expand the lexicon by individual constants for deictic pronouns. But then simplicity considerations would seem to push us back towards $\mathsf {Ext}$ . ²⁴

Friends of $\mathsf {NExt}_{\Lambda }$ may not be willing to admit defeat, though. While they can hardly object to the extensionalist’s claim of being able to formalize deictic utterances in a truth-conditionally adequate manner, they can point to an implausible feature of the proposed representation. The issue comes out quite clearly if we compare:

1. (A) Pointing at Venkatraman Ramakrishnan: He is a genius

with

1. (C) Every living US President is such that he is a genius.

According to the current extensionalist proposal, the formalization of (A) would be

1. (B′) $\texttt {Genius}(d)$

whereas the uncontentious ²⁵ formalization of (C) is

1. (D) $\forall \mathsf {v}_j\, \big (\texttt {Pres}(\mathsf {v}_j) \to \texttt {Genius}(\mathsf {v}_j)\big )$ .

Both (A) and (C) contain the pronoun he, but the formalization (B $'$ ) of (A) renders it as d while the formalization (D) of (C) renders the very same pronoun as $\mathsf {v}_j$ . Granted, in (A) he is used deictically while in (C) it is used like a bound variable, but the pronoun itself is morphologically the same in both cases; indeed this seems to be the situation in a wide range of human languages. Formalizing the two occurrences of the same pronoun by distinct symbols, i.e., in effect positing an ambiguity, for the sole purpose of safeguarding extensionality, therefore seems difficult to justify. ²⁶

Now a similar charge of ambiguity may perhaps be leveled against $\mathsf {NExt}$ . Recall that the $\mathsf {NExt}_{\Lambda }$ -formalization of (A), or rather of (A $_i$ ), is

1. (B) $\texttt {Genius}(\mathsf {v}_i)$ ,

in which he corresponds to a variable (here ${\mathsf {v}_i}$ ), just as it does in

1. (D) $\forall \mathsf {v}_j\, \big (\texttt {Pres}(\mathsf {v}_j) \to \texttt {Genius}(\mathsf {v}_j)\big )$ ,

where it corresponds to $\mathsf {v}_j$ . So it is certainly true that the pronoun is represented as a variable in both cases, which is not the case in the $\mathsf {Ext}$ -proposal we just examined. They are nevertheless distinct variables—unless one happens to choose $j=i$ for the formalization (D) of (C), for which there is no particular reason (after all, the choice of index for bound variables is largely irrelevant). What is disconcerting is that the $\mathsf {NExt}$ -strategy requires the choice of a particular index i for the formalization of the deictic pronoun, i.e., a decision that (A) is really (A $_i$ ) rather than, say, (A $_k$ ), without an obvious story as to how this might work; a requirement of choice that has no counterpart in our $\mathsf {Ext}$ -strategy. Stojnić [18, chap. 3] argues that this commits the received analysis of deixis, i.e., the one appealed to by the $\mathsf {NExt}$ -theorist, to a substantive ambiguity problem. We are sympathetic to this objection but will set it aside here. ²⁷

All this said, the extensionalist should, we believe, concede that her strategy of expanding the lexicon by new individual constants, while able to establish expressive parity with $\mathsf {NExt}_{\Lambda }$ , is nevertheless problematic as an analysis of deixis.

We’ve left one potential $\mathsf {Ext}$ -strategy for the formalization of deixis unexplored, however, namely the option of rendering the deictic pronoun as a bound rather than a free variable. Suppose that, instead of introducing the new individual constant d into the lexicon, with the understanding that its interpretation is to be the demonstrated object m (which, for the $\mathsf {NExt}$ -theorist, would be the new value of ${\mathsf {v}_i}$ ), we introduce a new Montagovian constant D into the lexicon, with the understanding that the interpretation of D is the object m. We then have that and thus in particular that, for any $h\in \mathcal {G}$ , if and only if , if and only if $m\in \mathfrak {I}(\texttt {Genius})$ . Accordingly, under this proposal, the $\mathsf {Ext}_{\Lambda }$ -theorist may formalize (A) as

1. (B'') $D\mathsf {v}_j\, \texttt {Genius}(\mathsf {v}_j)$

provided that Venkatraman Ramakrishnan is the interpretation of D.

Thus, where the previous extensionalist proposal was to think of (A) as transforming an initial $\Lambda $ -structure $(M,\mathfrak {I})$ via the pointing gesture into the richer $\Lambda _{d}$ -structure $(M,\mathfrak {I}_{d}^{\text {Venki}})$ , the new proposal is to think of (A) as transforming $(M,\mathfrak {I})$ into the richer $\Lambda _{D}$ -structure $(M,\mathfrak {I}_{D}^{\text {Venki}})$ . It should be clear that this, too, solves the extensionalist’s expressiveness problem, for they can now do with a closed formula $D_{1}x_1\ldots D_{n}x_n\,\phi (x_1,\ldots ,x_n)$ whatever the non-extensionalist can do with the open formula $\phi (\mathsf {v}_{i_1},\ldots ,\mathsf {v}_{i_n})$ . ²⁸ Consequently, $\mathsf {Ext}_{\Lambda _{D_1,\ldots ,D_n}}$ can formalize any deictic utterance formalizable in $\mathsf {NExt}_{\Lambda }$ .

But do these new Montagovian constants solve the representational problem of unwanted deictic–bound ambiguity in pronouns? Well, yes. For where the extensionalist’s first attempt at sprucing up the expressive power of $\mathsf {Ext}_{\Lambda }$ was unsatisfactory in that it formalized:

1. (A) Pointing at Venkatraman Ramakrishnan: He is a genius

as

1. (B′) $\texttt {Genius}(d)$

with an individual constant d rather than a variable, as in $\mathsf {NExt}_{\Lambda }$ ’s formalization

1. (B) $\texttt {Genius}(\mathsf {v}_i)$ ,

she can now, in $\mathsf {Ext}_{\Lambda _{D}}$ , render (A) as

1. (B'') $D\mathsf {v}_j\, \texttt {Genius}(\mathsf {v}_j)$ ,

with a variable $\mathsf {v}_j$ in the argument place of G, as desired. ²⁹ Taking that variable to be the formal equivalent of the deictic pronoun he in (A), she thus avoids the spurious ambiguity of pronouns that was objectionable about her initial idea: bound and deictic pronouns alike now correspond to variables. ³⁰

But wait—if the deictic pronoun in (A) corresponds to the bound variable occurrence in the argument place of $\texttt {Genius}$ in (B $"$ ), what in (A) corresponds to the Montagovian constant D in (B $"$ )? In other words, what is the justification for including $D\mathsf {v}_j$ in the logical form of (A)?

A comparison of (A) to its proposed $\mathsf {Ext}_{\Lambda _{D}}$ -formalization (B $"$ ) suggests an answer:

1. (A) $\boxed{Pointing\ at\ Venkatraman\ Ramakrishnan{:}}$ He is a genius.

2. (B'') $\boxed{D\textsf{v}_j}$ Genius $(\mathsf {v}_j)$

What the extensionalist can say is that the predicate symbol $\texttt {Genius}$ represents the verb phrase is a genius, the variable ${\mathsf {v}_j}$ occurring in its argument place represents the pronoun he, and the Montagovian constant ${D}$ represents the pointing gesture, aimed at Ramakrishnan, that accompanies the speaker’s production of the pronoun. Thus the pointing gesture itself, the extensionalist will maintain, functions as a Montagovian constant and is represented as such at the level of logical form.

While we don’t have the space to enter into much detail here, we want to point out that the recent philosophical and linguistic literature has seen a sophisticated, comprehensive proposal [18–21] for the semantic analysis of deixis which centrally includes the requirement that demonstrative gestures be represented at the level of logical form, and indeed in the form of variable binders, much as we have just suggested. ³¹

The formal analysis proposed by Stojnić [18] is, setting aside the facts that her overall framework is a dynamic semantics rather than the “static” one with which we are concerned in this paper, and that she aims to provide a mechanism for pronoun resolution as well, very similar to the one at which we have arrived on behalf of the extensionalist: Stojnić’s demonstration operator $\langle \pi kd\rangle $ is essentially our $D\mathsf {v}_k$ , if we follow the convention that D is the Montagovian constant corresponding to the individual constant d; in other words, a formula $\langle \pi kd\rangle \, \phi $ in Stojnić’s formalism is essentially equivalent to the gloss $\exists \mathsf {v}_k (\mathsf {v}_k = d\land \phi )$ we gave for $D\mathsf {v}_k\phi $ under the assumption that $\mathfrak {I}(d)=\mathfrak {I}(D)$ . ³²

Deictic pronoun use does of course also occur without an accompanying demonstrative gesture. Does this fact complicate the position of the $\mathsf {Ext}$ -theorist? ³³ We are inclined to think not. Pointing, the extensionalist will acknowledge, is just one way of introducing a (Montagovian) constant into the language, a way that makes the mechanism responsible for the identification of a particular individual as its interpretation particularly transparent. But there are other rule-based mechanisms that can serve the same purpose, as Stojnić et al. [19] have shown in some detail, using the theory of coherence relations. And such cases of deixis without ostension, too, Stojnić’s proposed logical-form correlate of introducing into the discourse an object made salient by coherence relations is essentially a Montagovian constant that binds the variable representing the deictic pronoun.

8. Conclusion

In order better to understand the disagreement between Quine and Salmon over the extensionality of quantification that we saw in Section 1, we developed (or perhaps better: made explicit) two schemas for constructing quantificational grammars. Both schemas base their syntax and semantics on the Tarskian techniques reviewed in Section 2, but the first, $\mathsf {Ext}$ , builds an extensional grammar $\mathsf {Ext}_{\Lambda }$ from any lexicon $\Lambda $ (Sections 3 and 4), while the second, $\mathsf {NExt}$ , builds a non-extensional grammar $\mathsf {NExt}_{\Lambda }$ (in which quantification is responsible for the failure of extensionality) from any $\Lambda $ (Section 5). There is thus a sense in which both Quine and Salmon are right: depending on the ambient grammar, quantification can turn out extensional, as Quine claims, or non-extensional, as Salmon has it. At the same time, there is a sense in which they are both wrong—quantification per se is neither extensional nor non-extensional; it is only relative to an entire grammar environment that it turns out one way or another.

In Section 6, we imagined our two camps not being content with this annoyingly philosophical outcome, and wondered what arguments there might be to favor the grammar schema $\mathsf {Ext}$ over $\mathsf {NExt}$ or vice versa, arguments that would then indirectly support either the extensionality or the non-extensionality of quantification. In terms of purely conceptual arguments, we ended up in something of a stalemate: On the one hand, $\mathsf {Ext}$ ’s grammars are simpler than $\mathsf {NExt}$ ’s in that they are extensional and their internal machinery is more parsimonious; on the other hand, $\mathsf {NExt}$ ’s grammars seem to generalize $\mathsf {Ext}$ ’s. At this point in the dialectic, we had the extensionalist challenge their opponent to demonstrate that the ostensibly greater generality of $\mathsf {NExt}$ ’s grammars wasn’t just a pointless complication. The non-extensionalist came back with a well-established application of $\mathsf {NExt}$ -grammars in the logical analysis of natural language, namely to deixis. According to a widely held view, $\mathsf {Ext}$ -type grammars are inadequate for the modeling of deixis since deictic pronouns are to be rendered as free variables, and consequently, deictic utterances as open formulas; within the grammars $\mathsf {Ext}_{\Lambda }$ , however, open formulas are never truth-eligible, and thus the formalization of deixis appears to fall outside the range of applicability of the extensionalist’s grammar schema.

The final dialectical move we considered was the extensionalist’s attempt to refute their opponent’s contention that $\mathsf {Ext}$ is unable to handle natural-language deixis. To this end, we examined two strategies available within $\mathsf {Ext}$ , namely, on the one hand, rendering deictic pronouns as individual constants, and on the other, rendering them as variables bound by a Montagovian constant. While the first strategy was found to be problematic in that it postulates a morphologically unattested ambiguity in natural-language pronouns between deictic occurrences (which would correspond to individual constants) and bound occurrences (which would correspond to variables), the second strategy was seen to avoid this problem, but is committed to loading additional material into the logical forms of deictic utterances, to wit, Montagovian constants. We suggested on behalf of the $\mathsf {Ext}$ -theorist that there is a natural interpretation of this additional material, namely that it is the logical-form counterpart of the demonstrative gesture. Indeed, as it happens, recent proposals for the linguistic treatment of deixis by Stojnić and by Ebert (and respective collaborators) require just such material in the relevant logical forms, so the presence of Montagovian constants in the $\mathsf {Ext}$ -formalizations of deictic utterances appears to be an independently motivated feature of the grammar of deixis.

Where does this leave us? It is still the case that, technically speaking, depending on the kind of grammar that controls, quantification may present as extensional or as non-extensional. But there now seems to be little reason to adopt a grammar schema that makes quantification non-extensional, given that extensional grammars are not only simpler but, by all appearances, also no less expressive than their non-extensional generalizations. In the field of quantification theory, extensionalism seems to be the most attractive option.