2. The common core

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.Footnote 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 .Footnote 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:

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” [Reference Quine, Floyd and Shieh15], where we’ve indicated in boldface which of (DC1 $_{\mathsf {Ext}_{\Lambda }}$ ) through (DC3 $_{\mathsf {Ext}_{\Lambda }}$ ) we take Quine to be referring to.Footnote ¹³

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.Footnote 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.Footnote ¹⁵

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 $\mathsf {NExt}_{\Lambda }$ :Footnote 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 $n>1$ .)Footnote ^{17 ,} Footnote 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.Footnote ¹⁹ Thus:

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 parsimoniousFootnote ²⁰ than $\mathsf {NExt}$ ’s, but also extensional; and as Lewis [Reference Lewis, Munitz and Unger9, 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 [Reference Montague10], is prominent in Kaplan [Reference Kaplan, Almog, Perry and Wettstein8], and has gained wide currency through the influential textbook by Heim and Kratzer [Reference Heim and Kratzer6].Footnote 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).Footnote 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.Footnote 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 $\mathsf {Ext}$ .Footnote 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 uncontentiousFootnote 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.Footnote 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 [Reference Stojnić18, 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.Footnote 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 $\phi (\mathsf {v}_{i_1},\ldots ,\mathsf {v}_{i_n})$ .Footnote 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.Footnote 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.Footnote 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 [Reference Stojnić18–Reference Stojnić, Stone and Lepore21] 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.Footnote ³¹

The formal analysis proposed by Stojnić [Reference Stojnić18] 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 $\mathfrak {I}(d)=\mathfrak {I}(D)$ .Footnote 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?Footnote 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. [Reference Stojnić, Stone and Lepore19] 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.