THOMAS SCHWARTZ

DE RE LANGUAGE, DE RE ELIMINABILITY, AND THEESSENTIAL LIMITS OF BOTH?

ABSTRACT. De re modality is eliminable if there is an effective translation of all wffsinto non-de re equivalents. We cannot have logical equivalence unless ‘logic’ has oddtheses, but we can have material equivalence by banning all essences, something the non-de re facts let us do, or by giving everything such humdrum essences as self-identity andbanning the more interesting ones. Eliminability cannot be got from weaker assumptions,nor independent ones of even modest generality. The net philosophical import is that,quite apart from the merits of essentialism, de re language has scant utility.

KEY WORDS: logic, language, modal, de re, elimination, eliminability, Quine, essen-tialism.

INTRODUCTION

Quine’s modal devil is de re modal language. There open sentencessport � and the like, making it hard to tell what satisfies them or whatit would take to tell (does 9, the number of planets, satisfy ‘�x > 7’?)but easy to return to those thrilling days of essentialism, when propertieswere had by necessity and otherwise as well.1 Exorcism would comefrom a proof that ‘de re modality is eliminable’: for some eliminationfunction e, an effective translator of all formulas A into non-de re ones,each A is equivalent to e(A), so definable that way if de re.2 Equivalenthow? Not logically unless ‘logic’ has odd theses. But the only obstacleto material equivalence is essentialism. The non-de re facts let us banall essences, and that ensures the general truth of A↔ e(A) for one e.Or we can give everything such essences as self-identify and ban themore interesting ones; that does the same for a more interesting e. Elim-inability cannot be got from weaker assumptions, nor independent onesof even modest generality. So I explain (§1), then show (§§2–11), thenapply to essentialism (§12). I conclude with Quine, but without knockingessentialism, that de re language has scant utility (§13).

? Thanks to W. V. Quine for helpful comments on an ancestral squib. Research sup-ported by NSF grants SES 8612120 and SES 8896228 and UCLA Senate grants.

Journal of Philosophical Logic 26: 521–544, 1997.c© 1997 Kluwer Academic Publishers. Printed in the Netherlands.

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522 THOMAS SCHWARTZ

1. OUTLINE AND EXPLANATION OF FINDINGS

To make all this more digestible, let me cut it in nine pieces.

PIECE 1, SEMANTICS. A modal logic is a category of modal structures,each comprising possible worlds, one real, their domains of denizens,none empty, and the relation borne a world by alternative worlds (theirfacts are its possibilities). I insist that only denizens of a world haveproperties in it (of what is not we can say naught), but not that it sharesdenizens with other worlds, even alternative ones, nor that it has anyalternatives, even itself. To a modal structure M, add extensions of pred-icates at worlds, and you have an interpretation. Truth therein is real-world truth, and truth at world w (for any values of free variables, allin w) is understood au Tarski with this necessary wrinkle: �A is trueat w iff A is true at every alternative to w. Validity in M is truth in allits interpretations, and validity in a logic is validity in all its Ms.

PIECE 2, LOGICAL ELIMINABILITY LIMITED. In odd enough Ms,A↔ e(A) is valid for some e and all A. To make a small story short,ignore As with nested �. If ¬∃x�x = x is valid in M then so is everyopen ¬�B, so A ↔ e(A) is too if e rewrites each open �B in Aas ∀xx 6= x. Or if ∃x�∀yy = x and every �B→ B are valid in M thenso is every �B↔ � ∀B (universal closure ∀B), so A↔ e(A) is too ife rewrites each open �B in A as �∀B.

Of course, the no-essence ¬∃x�x = x and Parmenidean ∃x�∀yy =x are not valid conventionally. But if neither is valid in M, then forevery e, some A ↔ e(A) is not. Better: for every non-de re B andnonlogical predicate Pn,∃x1 · · ·xn�Pnx1 · · ·xn ↔ B is not.

PIECE 3, ELIMINABILITY ABSENT ESSENCES. Conventional valid-ity aside, ¬∃x�x = x might still be true, indeed true at every world.Let interpretation K be thus essenceless. Write ε(A) by rewriting eachoutermost open �B in A as ∀xx 6= x. Then ε applies to K: A↔ ε(A)is true for every A at every K-world.

PIECE 4, ESSENCELESSNESS CONSISTENT WITH NON-DE RETRUTH. Essencelessness is consistent with the non-de re facts of any Kwhere each world has some alternative besides itself. For we can turn Kinto an essenceless ΣK that preserves non-de re truth: change domainsso no two worlds have overlapping ones. ΣK also preserves logic (itsmodal structure belongs to any containing K′s) unless logic constrainsdomains to block the change – as it often does by mandating ∀x�x = x.

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DE RE LANGUAGE AND DE RE ELIMINABILITY 523

PIECE 5, ELIMINABILITY ABSENT PD ESSENCES. Essencelessnessmakes �x = x and �(Fx → Fx) false though x = x and Fx→ Fx arenecessarily true on most accounts. For eliminability, however, we cangive everything such humdrum essences as self-identity and ban poten-tially distinguishing ones, PD essences. If Bill Clinton is essentially ratio-nal then rationality is a PD essence of Bill unless necessarily everythingis rational. Or if Bill essentially admires Newt Gingrich then admirationis a PD essence of that pair of worthies unless necessarily everythingadmires everything else. For this it is irrelevant that Mother Theresaneed not admire herself, but that makes self-admiration a PD essence ofany essential self-admirer. By thus respecting ‘self’ and ‘other’ we canhave more humdrum essences: maybe difference is a humdrum essence ofBill and Newt though not Bill and himself, equality-in-height of MotherTheresa and herself though not her and Dustin Hoffman.

We need no direct ban on PD essences expressed by de re opensentences: they vanish by implication. Maybe ‘x admires y’ express-es a PD essence of some ‘x’ values for a given ‘y’ value. But to bansuch parametric essences is to ban even self-identity by not letting any‘y’ value satisfy �x = y as ‘x’ value. Think of Ax1 · · ·xn as any opensentence whose free variables are exactly x1, . . . ,xn. Then K is PD-essenceless if every non-de re Ax1 · · · xn meets this condition: when�Ax1 · · ·xn is satisfied at a K-world by a1, . . . , an, Ax1 · · ·xn is sat-isfied at every alternative world by denizens b1, . . . , bn thereof so longas bi = bj iff ai = aj . To capture humdrum essences, call K cumulativeif ∀x�x = x is true at each K-world w: w′s domain is included in thoseof its alternatives.

An elimination function δ applies to all interpretations fulfilling cumu-lativity and PD-essencelessness, or C + PDE. This fact complements fourmore. First, C + PDE is necessary for δ-applicability – as essenceless-ness is for ε, by the way. Second, δ captures a natural reading of de re �:� attributes necessary truth to open sentences. For �-free Ax, that letsus read �Ax as �∀xAx, which is δ(�Ax)(δ(�Ax1 · · · xn) is morecomplicated when n ≥ 2). So three things are equivalent: C + PDE,the general truth of A ↔ δ(A), and the general use of � to attributenecessary truth. Two more facts are two more pieces.

PIECE 6, PD-ESSENCELESSNESS CONSISTENT WITH NON-DE RETRUTH AND MORE. The non-de re facts can block cumulativity: if∃x∃yx 6= y∧¬�∃x∃yx 6= y is true in K then K′s real world has a denizenabsent from an alternative. But PD-essencelessness is always consistentwith those facts, also with cumulativity and all but the most outre logics.

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524 THOMAS SCHWARTZ

For we can turn any K into a PD-essenceless ∆K that preserves non-de retruth, cumulativity, and most any logic: replace each w with all (w, p)whose p is a permutation of w′s domain, and make predicates applyin (w, p) to their p-permuted applicata in w – so Bill is a Republicanin (w, p) if Newt is in w and p(Newt) = Bill. That is all: (w, p) hasw′s domain, ∆K′s real world is K′s coupled with the identity mappingon its domain (same nonmodal truths), and (v, q) is a ∆K-alternative to(w, p) iff v is a K-alternative to w.

Apart from the inessential representation of worlds as pairs, the onlydifference between the modal structures of K and ∆K is that ∆K′s mayhave more worlds, including more alternatives to some and more pos-sessors of them. For ∆K′s worlds can be mapped onto K′s so that realworlds match, alternatives match alternatives, and matching worlds havethe same domain. But the always-available (w, p)-to-w mapping is many-one, and maybe any such mapping must be. So ∆K preserves logic unlesslogic limits the number of worlds, alternatives, or possessors.

To see what ∆ does to essences, suppose rationality is a PD essenceof Bill Clinton in K-world w: Bill is rational in every alternative world,but in some, say v, a co-occupant, say First Cat Socks Clinton, is not.Let permutation q of v′s domain switch Bill with Socks. Because Socks,not Bill, is rational in (v, q), Bill is not essentially rational in any (w, p).Still he is essentially rational-or-nonrational and essentially different fromSocks in (w, p): permutations cannot deplete all-inclusive categories orundo differences. So ∆ subtracts essences, though only certain ones, byadding possibilities, but only certain ones: ∆K has no new denizens orroles for them to fill, only new ways, all possible new ways, for olddenizens to fill old roles. Think of a repertory company that lets eachactor play any role. No more is needed to free every actor from all con-straints of typecasting, all potentially distinguishing theatrical essences.

PIECE 7, LIMITS OF ELIMINABILITY. C + PDE may be too strongor needlessly strong even for some anti-essentialist tastes. One relishesessenceless worlds, barred by cumulativity, but it should be gratified byessenceless interpretations.3 Others stomach some Aristotelian interpre-tations: they may have essenceless worlds, but each flouts C + PDEotherwise. Quine (1986: p. 292) tolerates essences not necessarily hadby everything so long as none is also an accident. Von Wright (1951:p. 27) tolerates less: no essence or its negate is an accident. He thoughtthis Principle of Predication (PP) solved all de re problems. Prior (1955:p. 211) took him to mean that eliminability can be got from PP, fortifiedas need be with conventional logic.

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DE RE LANGUAGE AND DE RE ELIMINABILITY 525

We can refute all such conjectures in one fell swoop. Stop requir-ing elimination functions to be effective. Still, eliminability cannot begot from PP or other Aristotelian relaxations of C + PDE, nor fromany assumptions, however strong, that tolerate even one Aristotelian Kand this much PD-essencelessness: when K satisfies them, so does thePD-essenceless but otherwise similar ∆K. For no elimination functionapplies to both K and ∆K if K is Aristotelian.

An Aristotelian K can still enjoy an elimination function of sorts.Assume a bounty of PD-essences, but suppose the essential possessors(humans, apes, and porpoises) of any property or relation (languagecapacity) always fit some non-de re description (‘primate or cetacean’)that describes no accidental possessors (computers), though it may somenonpossessors (monkeys, whales). Then for every Ax1 · · · xn, some non-de re Bx1 · · ·xn makes ∀(�Ax1 · · ·xn ↔ (Ax1 · · ·xn ∧ Bx1 · · ·xn))true, and we can let e(�Ax1 · · ·xn) = e(Ax1 · · ·xn) ∧Bx1 · · ·xn. Butsuch an e is essence specific: its definition depends on what has whatessences. Far from having an e that clarifies de re truth conditions andlets us avoid essentialism, we have no e until we have settled thoseconditions by adopting one version of essentialism in fine and full detail.

In general, an elimination schema A↔ e(A) once fitted to the non-de re facts allows no variation in the other facts: we cannot define ewithout specifying all essences and accidents and facts about them. Forif A ↔ e(A) holds in K-world w and K′-world w′ that have the samenon-de re facts, then w and w′ must be alike in all their facts. When Kand K′ satisfy C + PDE, this trivial theorem has no bite: the desiccatedessences allowed by C + PDE are determined anyway by non-de re facts(‘� ∀x x is rational’). But given the non-de re facts, all other essences –all interesting ones – are optional, PD-essencelessness being consistentwith those facts, and they block eliminative flexibility: an A ↔ e(A)that tolerates them can tolerate no variation in them or facts about them.

PIECE 8, MOTHER OF ALL THESE RESULTS. The results of Pieces 3and 5 gain most of their import from the others, which come from another.Let K′ be a projector of K: we can map its worlds onto K′s so that realworlds match, alternatives match alternatives, and matching worlds areisomorphic in their domains and predicate-extensions. Then K has thesame non-de re truths as K′ at matching worlds.

PIECE 9, ORGANIZATION OF LABOR. Following some syntax andsemantics (§§2, 3), I prove this last result (§4), then the one aboutmodal structures in which ¬∃x�x = x and ∃x�∀yy = x are not valid:

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526 THOMAS SCHWARTZ

∃x1 · · · xn�Pnx1 · · · xn ↔ B is not valid therein for any nonlogical Pn

and non-de re B (§5). Next I show that ε applies to all and only essence-less Ks (§6) and that Σ turns practically every K into an essencelessΣK that preserves non-de re truth (§7). After defining δ and explaininghow it reads de re language (§8), I show that δ applies to all and onlycumulative, PD-essenceless Ks (§9) and that ∆ turns every K into a PD-essenceless ∆K that preserves non-de re truth and cumulativity (§10).Now come the limitative theorems: an elimination schema that toleratesinteresting essences must be specific to them, and essenceless worldsaside, no schema that tolerates a modicum of PD-essencelessness canbreak the bounds of C + PDE (§11). I end with lessons about essential-ism (§12) and the utility of de re language (§13).

2. SYNTAX

Our vocabulary has→,∀,�, parentheses, one or more variables, includ-ing x, and one or more predicates, including 6=; A→ ∀xx 6= x plays ¬A.Because no more is assumed about predicates and variables, nothing willdepend on how many distinctions or degrees of relationship are express-ible. Wffs are as usual except there may be any limit on nesting of �:maybe �A is wellformed only when A is nonmodal.

A, B, C, A1, etc. are any wffs; x, y, z, x1, etc., any variables;Pn, any n-place predicate; and Ax1 · · ·xn, Bx1 · · ·xn, etc., any wffswhose n ≥ 1 free variables (defined as usual) are x1, . . . ,xn in orderof first occurrence. A is open if it has a free variable, closed if it hasnone, de re if it contains an open �B. Define ¬A for (A → ∀xx 6=x), x = y for ¬x 6= y, ∀Ax1 · · ·xn for ∀x1 · · · ∀xnAx1 · · ·xn, and∃x1 · · ·xnA, (A↔ B), (A1 ∨A2 ∨ · · · ∨An), (A1 ∧A2 ∧ · · · ∧An)as usual.

3. SEMANTICS

My actualist semantics differs from Kripke’s (1963), agreeing with Hin-tikka’s (1961): only denizens of w satisfy A at w. There is no differenceif, as in the classics of Kripke (1959), Hughes and Cresswell (1968),and Thomason (1970), all worlds share a domain. Having left domainsunconstrained (alternativeness too) except in special cases, I regainedsome lost control with actualism. I see no cost: in any world there isonly what exists, yet it may still be that some things necessarily exist(∃x�x = x) and others do not, that soandsos possibly exist (¬�¬∃xSx)but actually do not, that some things are ‘extant’ (∃xEx) and others not.

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A modal structure is a vector (g,W,R, d) where (real world) g belongsto set W (of worlds), R is a binary relation (alternativeness) on W, andfunction d turns every w in W into a nonempty set d(w) (w′s domain).Let M = (g,W,R, d) be any modal structure, w and v any membersof W, and R[w] ≡ {v | vRw}.

An interpretation function over M turns every Pn and w into an n-ary relation (extension) ϕ(Pn, w) on d(w), nonidentity if Pn is 6=. Aninterpretation is a (g,W,R, d, ϕ) whose ϕ is an interpretation functionover (g,W,R, d). Let K = (g,W,R, d, ϕ) and K′ = (g′,W′,R′, d′, ϕ′)be any interpretations.

An assignment is a function of variables. Let σ and τ be any such.In K, σ is an A-w-assignment iff σ(x) ∈ d(w) for all free x in A, and τan x-w-variant of σ iff τ(x) ∈ d(w) and also τ(y) = σ(y) when y 6= x.

In K, σ satisfies A at w according to this recursion:

σ satisfies Pnx1 · · ·xn at w iff(σ(x1), . . . , σ(xn)

)∈

ϕ(Pn, w),

(B→ C) iff σ is a (B→ C)-w-assignment satisfying C

if B at w,

∀xB iff σ is a ∀xB-w-assignment whose x-w-variants

all satisfy B at w,

and

�B iff σ is a B-w-assignment satisfying B throughout R[w].

A is true at w in K iff every A-w-assignment satisfies A at w in K,true in K iff true at g in K, and valid in M iff true in (g,W,R, d, ϕ) forevery ϕ over M.

4. PROJECTORS AND THEIR PRESERVATION OF NON-DE RE TRUTH

I first prove the theorem on which most of the others are based: every Kshares non-de re truths with its projectors at matching worlds.

K′ is projector of K under functions F and T iff

F maps W′ onto W, F(g′) = g, vR′w iff F(v)RF(w),

T turns every w in W′ into a one-one function Twon d(F(w)) onto d′(w),

andϕ′(Pn, w) ≡ {

(Tw(a1), . . . ,Tw(an)

)|(a1, . . . , an

)∈

ϕ(Pn,F(w)

)}.

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528 THOMAS SCHWARTZ

LEMMA 4.1. Suppose K′ is a projector of K under F and T, w ∈W′, Ais not de re, and σ is an A-F(w)-assignment. Then σ satisfies A at F(w)iff Tw ◦ σ satisfies A at w.

Proof. By induction on the complexity of A. 2

Case 1. A = Pnx1 · · ·xn. Then σ satisfies A at F(w) iff (σ(x1), . . . ,σ(xn)) ∈ ϕ(Pn,F(w)). But that happens iff (Tw(σ(x1)), . . . ,Tw(σ(xn)))∈ ϕ′(Pn, w), i.e., Tw ◦ σ satisfies A at w.

Case 2. A = (B → C). By inductive hypothesis, σ satisfies B atF(w) iff Tw ◦σ satisfies B at w, and likewise C. But Tw ◦σ is an A-w-assignment since Tw maps d(F(w)) into d′(w). So σ satisfies A at F(w)iff Tw ◦ σ satisfies A at w.

Case 3. A = ∀xB. By inductive hypothesis, any x-F(w)-variant τ ofσ satisfies B at F(w) iff Tw ◦ τ does at w. But since Tw maps d(F(w))into d′(w), each such τ makes Tw ◦ τ an x-w-variant of Tw ◦ σ, andsince Tw is one-one and onto, each x-w-variant of Tw ◦ σ is Tw ◦ τ forsome such τ . So every x-F(w)-variant of σ satisfies B at F(w) iff everyx-w-variant of Tw ◦ σ satisfies B at w: σ satisfies A at F(w) iff Tw ◦ σdoes at w.

Case 4. Closed A = �B. Since vR′w iff F(v)RF(w), it suffices toshow, for each v in R′[w], that σ satisfies B at F(v) iff Tw ◦σ does at v.Since B is closed, σ is a B-F(v)-assignment, so by inductive hypothesis,σ satisfies B at F(v) iff Tv ◦ σ does at v. But since B is closed, oneassignment satisfies B at v iff all do, so Tv ◦ σ does iff Tw ◦ σ does.

THEOREM 4.1. Suppose K′ is a projector of K under F and T, w ∈W′,and A is not de re. Then A is true at F(w) in K iff A is true at w in K′.

Proof. We may suppose A is closed, else take ∀A. Then any σ sat-isfies A at F(w) iff all do, and likewise w, whence the theorem followsby the lemma. 2

5. THE LIMIT OF LOGICAL ELIMINABILITY

Theorem 4.1 makes short work of showing that ∃x1 · · ·xn�Pnx1 · · ·xn↔ B is not valid for any nonlogical Pn and non-de re B in any M where¬∃x�x = x and ∃x�∀yy = x are not valid, where g shares a denizenwith all its alternatives and one has another.

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THEOREM 5.1. In M, suppose a ∈ d(g) ∩ d(v) for all v in R[g] anda 6= b ∈ d(v∗) for some v∗ in R[g]. Then for any non-de re B andPn different from 6=, (∃x1 · · · xn�Pnx1 · · · xn ↔ B) is not valid in M.

Proof. Define ϕ and Ψ over M so that, for all predicates Q differentfrom 6=,

ϕ(Q, w) = Ψ(Q, w) = ∅ if Q 6= Pn or a /∈ d(w),

ϕ(Pn, w) = Ψ(Pn, w) = {a}n if a ∈ d(w) but w 6= v∗,

ϕ(Pn, v∗) = {a}n, and Ψ(Pn, v∗) = {b}n.

Then Kϕ = (g,W,R, d, ϕ) is a projector of KΨ = (g,W,R, d,Ψ) underF, T with F(w) ≡ w, Tv∗(a) = b, Tv∗(b) = a, and Tw(c) = c wheneverw 6= v∗ or c /∈ {a, b}.

Obviously ∃x1 · · ·xn�Pnx1 · · ·xn is true in Kϕ but not KΨ. So if(∃x1 · · ·xn�Pnx1 · · ·xn ↔ B) were valid in M, non-de re B would betrue in Kϕ but not KΨ, contrary to Theorem 4.1. 2

6. ELIMINABILITY BY ε WHEN THERE ARE NO ESSENCES

Here I show that elimination function ε applies to all and only essence-less Ks.

K is essenceless iff, for all w and a in d(w), a /∈ d(v) for some vin R[w]. Define:

ε(Pnx1 · · · xn

)= Pnx1 · · ·xn.

ε(A↔ B) =(ε(A)→ ε(B)

).

ε(∀xA) = ∀xε(A).

ε(�A) = �ε(A) if A is closed.

ε(�Ax1 · · · xn

)= ∀xx 6= x.

THEOREM 6.1. If K is essenceless then (A↔ ε(A)) is true at every w.Proof. By induction on the complexity of A. If A = Pnx1 · · ·xn

then ε(A) = A. If A is (B → C) or ∀xB, then (B ↔ ε(B)) and(C ↔ ε(C)) are true at w by inductive hypothesis, so (A ↔ ε(A)) istoo. If closed A = �B, then (B ↔ ε(B)) is true throughout R[w] byinductive hypothesis, so B is iff ε(B) is, so (�B ↔ �ε(B)) = (A ↔ε(A)) is true at w. And if A = �Bx1 · · ·xn, then since K is essenceless,no σ satisfies A at w, and of course none satisfies ∀xx 6= x at w, so(A↔ ∀xx 6= x) = (A↔ ε(A)) is true at w. 2

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530 THOMAS SCHWARTZ

THEOREM 6.2. If (A ↔ ε(A)) is true in K for all A at every w, thenK is essenceless.

Proof. Suppose not: a ∈ d(w)∩d(v) for all v in R[w]. Let σ(x) = a.Then σ satisfies �x = x at w but not ε(�x = x) = ∀xx 6= x, so(�x = x↔ ε(�x = x)) is not true at w. 2

7. HOW Σ CREATES ESSENCELESSNESS AND PRESERVES NON-DE-RETRUTH

The importance of Theorem 6.1 rests on two others: ΣK is almost alwaysessenceless, and it always has the same non-de re truths as K.

To construct ΣK, replace the d and ϕ of K with the d′ and ϕ′ forwhich

d′(w) ≡ {(a,w) | a ∈ d(w)}

and

ϕ′(Pn, w) ≡ {((a1, w), . . . , (an, w)

)|(a1, . . . , an

)∈ ϕ(Pn, w)}.

THEOREM 7.1. If, in K, vRw for all w and some v 6= w, then ΣK isessenceless.

Proof. If instead we had (a,w) ∈ d′(w)∩d′(v) for all v in R[w], thenfor all such v we should have (a,w) = (a, v), whence w = v, contraryto hypothesis. 2

THEOREM 7.2. If, in K, vRw for all w and some v 6= w, then (A ↔ε(A)) is true in ΣK at every w.

Proof. Immediate from Theorems 6.1 and 7.1. 2

THEOREM 7.3. Any non-de re A is true at w in K iff A is true at win ΣK.

Proof. By Theorem 4.1: ΣK is a projector of K under F(w) ≡ w andTw(a,w) ≡ a. 2

8. THE ELIMINATION FUNCTION δ: DEFINITION AND RATIONALE

If Ax is not de re, δ(�Ax) = �∀xAx. The rationale takes a detourthrough the de dicto. Necessity embraces valid statements and often more,

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notably nonvalid but analytic statements, got from valid ones by puttingsynonyms for synonyms, and synthetic truths of mathematics. But allversions embrace open sentences too. ‘Everything is or is not material’is valid, but so is ‘x is or is not material.’ ‘Every brother is male’ isanalytic, but so likewise is ‘x is brother only if male.’ ‘∀xx + 0 = x’is true, but so is ‘x + 0 = x.’ Necessity of every sort is implicationby background laws, and open sentences follow from any. So it seemsnatural to read �Ax as an attribution of necessity to Ax. We do notuse ‘yellow’ to attribute yellow to all objects but bananas and applyit to bananas for another purpose. Why use � to attribute necessity toall sentences but open ones and apply it to open sentences for anotherpurpose? Since Ax follows from background laws iff ∀xAx does, itseems natural to read �Ax as � ∀xAx.

But for non-de re Bxy, we cannot always read �Bxy as� ∀x∀yBxy.Let ∀x�x = x and ∃x∃yx 6= y be true. Then ∃x∃y�x = y is too but�∀x∀yx = y is not. Also ∀x�(Px → Px) is, so ∃x∃y�(Px → Py)is too, but � ∀x∀y(Px → Py) need not be. Again, ∃x∃y�x 6= y isbut � ∀x∀yx 6= y is not. The reason we cannot read ∃x∃y�x = y as�∀x∀yx = y is that the former is true only because �x = y is sat-isfied by shared values of x and y. Similarly for ∃x∃y�(Px → Py).And the reason we cannot read ∃x∃y�x 6= y as � ∀x∀yx 6= y is thatthe former is true only because �x 6= y is satisfied by distinct val-ues.

Though we cannot always read �Bxy as � ∀x∀yBxy, as an unqual-ified attribution of necessity to Bxy, we can still read � so it doesno more than attribute necessity. Distinct variables can share a valueor not, and that, we saw, can affect the truth of �Bxy when �Bxyis not simply asserted but wrapped in a wider quantificational context.To our unqualified reading of �Bxy, add the obvious qualification: incontext, �Bxy might attribute necessity to Bxy only for shared val-ues of x and y or only for distinct values. But Bxy is necessary forshared values iff � ∀x∀y(x = y → Bxy) is true, and distinct ones iff�∀x∀y(x 6= y → Bxy) is. Now �Bxy says ((x = y ∧ � ∀x∀y(x =y→ Bxy)) ∨ (x 6= y ∧� ∀x∀y(x 6= y→ Bxy))), which is δ(�Bxy).Simply to assert �Bxy is to assert ∀x∀y�Bxy, and δ(∀x∀y�Bxy) =∀x∀yδ(�Bxy) amounts to our unqualified �∀x∀yBxy. It is for widercontexts that we have to read �Bxy as attributing necessity to either oftwo sentences, according as it treats therein of one or two things.

For n ≥ 1 wffs A1, . . . ,An, define vector S(A1, . . . ,An) of wffs:

S(A1) = (¬A1,A1).

If S(A2, . . . ,An) = (B1, . . . ,Bk) then

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532 THOMAS SCHWARTZ

S(A1,A2, . . . ,An) =((¬A1 ∧B1),

(¬A1 ∧B2), . . . , (¬A1 ∧Bk),

(A1 ∧B1), (A1 ∧B2), . . . , (A1 ∧Bk)).

It has 2n state descriptions got from A1, . . . ,An. For n ≥ 2 variablesx1, . . . ,xn, define:

U(x1,x2) = (x1 6= x2).

If U(x2, . . . ,xn) = (B1, . . . ,Bk) then

U(x1,x2, . . . ,xn) = (x1 6= x2,x1 6= x3, . . . ,

x1 6= xn,B1, . . . ,Bk).

It has one x 6= y for every two of those variables, or (n2−n)/2 altogeth-er. So S(U(x1, . . . ,xn)) has 2(n2−n)/2 wffs, each a consistent conjunctionof equalities or unequalities or both, complete for the listed variables,each expressing one way the values of x1, . . . ,xn might be identical ordistinct, each way expressed by one of them: if σ(xi) ∈ d(w) for all ithen σ satisfies exactly one such wff at w. Now δ:

δ(Pnx1 · · ·xn

)= Pnx1 · · · xn.

δ(A→ B) = (δ(A) → δ(B)).

δ(∀xA) = ∀xδ(A).

If A is closed then δ(�A) = �δ(A).

δ(�Ax) = �∀xδ(Ax).

If A = Ax1 · · · xn, n ≥ 2, and

S(U(x1, . . . ,xn)

)= (B1, . . . ,Bk), then

δ(�A) =((

B1 ∧� ∀(B1 → δ(A)))∨ · · ·

∨(Bk ∧ � ∀(Bk → δ(A))

)).4

The definition construes de re necessity thus: a thing necessarilyhas a property (expressed, perhaps, with more than one variable, allsharing a value) iff necessarily everything has it, and n things neces-sarily stand in an n-ary relation (expressed, perhaps, with more thann variables, altogether taking n values) iff necessarily any n thingsstand in that relation. So to attribute a supposed essence to things isto say nothing distinctive about them but only this about it: an essenceit is but not PD. Cumulativity follows: ∀x�x = x is true becauseδ(∀x�x = x) = ∀x�∀xx = x is.

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DE RE LANGUAGE AND DE RE ELIMINABILITY 533

9. ELIMINABILITY BY δ WHEN THERE ARE NO PD ESSENCES BUTCUMULATIVITY HOLDS

We can rigorize these last two observations by proving that δ applies toall cumulative, PD-essenceless interpretations and them alone.

Call w locally cumulative in K iff d(w) ⊆ d(v) for all v in R[w].Then K is cumulative iff every w is locally cumulative in K.

Given K, call τ an (x1, . . . ,xn)-w-variant of σ iff τ(xi) ∈ d(w) forall i and τ(xi) = τ(xj) just when σ(xi) = σ(xj). Call A = Ax1 · · ·xna PD expressor at w iff A is non-de re and, for some σ and v in R[w],σ satisfies �A at w but not every (x1, . . . ,xn)-v-variant of σ satisfiesA at v. Then K is PD-essenceless iff no w has a PD expressor.

LEMMA 9.1. Suppose w is locally cumulative in K and A = Ax1 · · · xnis neither de re nor a PD expressor at w. Then (�A ↔ δ(�A)) is trueat w.

Proof. Since A is not de re, δ(A) = A. If n = 1, so δ(�A) =�∀x1A, let C1 be x1 = x1. Otherwise let S(U(x1, . . . ,xn)) =(C1, . . . ,Ck), so δ(�A) = ((C1∧�∀(C1 → A))∨· · ·∨(Ck∧�∀(Ck →A))). Either way, every A-w-assignment σ satisfies one Ci at w, and itsuffices to show that σ satisfies �A at w iff ∀(Ci → A) is true at every vin R[w]. Suppose the latter. But for every such v, σ is an A-v-assignmentby local cumulativity, so σ satisfies Ci at v, and thus σ satisfies A at v.Hence, σ satisfies �A at w. Conversely, suppose σ satisfies �A at wand vRw. By hypothesis all (x1, . . . ,xn)-v-variants of σ satisfy A at v.But they alone satisfy Ci at v. So ∀(Ci → A) is true at v. 2

THEOREM 9.1. If K is cumulative and PD-essenceless, then (A ↔δ(A)) is true for every A at every w.

Proof. By induction on the complexity of A. If A = Pnx1 · · ·xnthen δ(A) = A. If A is (B → C) or ∀xB, then (B ↔ δ(B)) and(C ↔ δ(C)) are true at w by inductive hypothesis, so (A ↔ δ(A)) istoo. Let A = �B. By cumulativity, every A-w-assignment is an A-v-assignment for all v in R[w]. And by inductive hypothesis, B ↔ δ(B)is true at every such v, so B is iff δ(B) is. Therefore (�B↔ �δ(B)) istrue at w, and it suffices to show that (�δ(B)↔ δ(�B)) is too. Say B isopen, else δ(�B) = �δ(B). But δ(B) is not de re, and by hypothesis itis not a PD expressor at w. By the lemma, then, (�δ(B)↔ δ(�δ(B))) =(�δ(B)↔ δ(�B)) is true at w. 2

LEMMA 9.2. Suppose (�A→ δ(�A)) is true at w in K. Then A is nota PD expressor at w, and if A is x = x then w is locally cumulative.

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534 THOMAS SCHWARTZ

Proof. For the first part let A = Ax1 · · ·xn be non-de re, elsethe lemma is trivial. Suppose σ satisfies �A at w, vRw, and τ isan (x1, . . . ,xn)-v-variant of σ; to prove that τ satisfies A at v. Byhypothesis, σ satisfies δ(�A) at w. If n = 1 then δ(�A) = � ∀x1A,so σ satisfies ∀x1A at v, whence τ satisfies A at v. Otherwise letδ(�A) = ((C1 ∧� ∀(C1 → A))∨ · · · ∨ (C1 ∧�∀(C1 → A))). Then σsatisfies one Ci and also � ∀(Ci → A) at w. At v, then, τ satisfies Ci

and (Ci → A) and therewith A.For local cumulativity, suppose a ∈ d(w) and vRw. Let σ(x) = a.

Since δ(�x = x) = � ∀xx = x is true at w, σ satisfies x = x at v,so a ∈ d(v). 2

THEOREM 9.2. If (A↔ δ(A)) is true in K for every A at every w, thenK is cumulative and PD-essenceless.

Proof. Immediate from the lemma. 2

10. HOW ∆ CREATES PD-ESSENCELESSNESS, PRESERVING NON-DE RETRUTH AND CUMULATIVITY

Theorem 9.1 gains import from the fact that ∆ turns every K into PD-essenceless ∆K, like K in non-de re truth and cumulative if K is.

∆K is the K′ for which

W′ = {(w, p) | w ∈W and p is a permutation of d(w)},g′ = (g, t) where t is the identity mapping on d(g),

(v, q)R′(w, p) iff vRw, d′(w, p) ≡ d(w),

and

ϕ′(Pn, (w, p)

)≡ {

(p(a1), . . . , p(an)

)| (a1, . . . , an)

∈ ϕ(Pn, w)}.

LEMMA 10.1. ∆K is a projector of K under F(w, p) ≡ w and T(w,p) ≡ p.Proof. Immediate from the definition of ∆. 2

LEMMA 10.2. Suppose A is not de re, σ is an A-w-assignment, and pa permutation of d(w). Then σ satisfies A at w in K iff p ◦ σ satisfies Aat (w, p) in ∆K.

Proof. Immediate from Lemmata 10.2 and 4.1. 2

THEOREM 10.1. ∆K = K′ is PD-essenceless.

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DE RE LANGUAGE AND DE RE ELIMINABILITY 535

Proof. Suppose not: A = Ax1 · · ·xn is non-de re, σ satisfies �Aat (w, p), (v, q)R′(w, p), and τ is an (x1, . . . ,xn)-(v, q)-variant of σ thatdoes not satisfy A at (v, q). Then by Lemma 10.2, q−1◦τ does not satisfyA at v (in K). Since τ(xi) = τ(xj) iff σ(xi) = σ(xj), t(q

−1(τ(xi))) ≡σ(xi) for some permutation t of d(v) = d′(v, q). By Lemma 10.2 again,t ◦ q−1 ◦ τ does not satisfy A at (v, t), and since σ(xi) ≡ t(q−1(τ(xi))),nor does σ. But (v, q)R′(w, p), so vRw, so (v, t)R′(w, p). Thus, σ cannotsatisfy �A at (w, p) after all. 2

THEOREM 10.2. Any non-de re A is true at w in K iff A is true at(w, p) in ∆K.

Proof. Immediate from Lemmata 10.1 and Theorem 4.1. 2

LEMMA 10.3. (w, p) is locally cumulative in ∆K = K′ iff w is in K.Proof. We have d′(w, p) ≡ d(w), and also (v, q)R′(w, p) iff vRw.

So d′(w, p) ⊆ d′(v, q) for all (v, q) ∈ R′[(w, p)] iff d(w) ⊆ d(v) forall v ∈ R[w].

THEOREM 10.3. ∆K is cumulative iff K is.Proof. Immediate from the lemma. 2

THEOREM 10.4. If K is cumulative then (A↔ δ(A)) is true in ∆K atevery (w, p).

Proof. Immediate from Theorems 10.1, 10.3, and 9.1. 2

11. LIMITS OF ELIMINABILITY

The limitative results are two: If K-world w and K′-world w′ have thesame non-de re truths and (A ↔ e(A)) holds in both, then w and w′

must have all the same truths. Also if e applies to both K and ∆K, thenK cannot be Aristotelian.

An elimination function turns wffs into non-de re wffs. Let e be anysuch.

THEOREM 11.1. If (A ↔ e(A)) is true for all A at w in K and w′

in K′, and if w and w′ have the same non-de re truths, then they haveall the same truths.

Proof. For all A, the non-de re e(A) is true at w iff at w′ and (A↔e(A)) is true at both, so A is true at w iff at w′.

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536 THOMAS SCHWARTZ

K is Aristotelian iff, for some w and a in d(w), a ∈ d(v) for all v inR[w] (w is not essenceless) but w either is not locally cumulative or hasa PD expressor. 2

THEOREM 11.2. Assume for all A that (A ↔ e(A)) is true in K and∆K = K′ throughout W and W′. Then K is not Aristotelian.

Proof. Suppose it is: a ∈ d(w) ∩ d(v) for all v in R[w] but w eitheris not locally cumulative or has a PD expressor. 2

Case 1. w is not locally cumulative: b ∈ d(w) and v∗Rw but b /∈d(v∗). Let σ(x) = a. Then σ satisfies �x = x and therewith e(�x = x)at w. Let p be a permutation of d(w) with p(a) = b, and q one of d(v∗).By Lemma 10.2, since σ satisfies non-de re e(�x = x) at w, p ◦ σsatisfies e(�x = x) and hence �x = x at (w, p). But p(σ(x)) = b and(v∗, q)R′(w, p), so b ∈ d′(v∗, q) = d(v∗), an impossibility.

Case 2. w is locally cumulative, so any (w, p) is too by Lemma 10.3,but B is a PD expressor at w. Then (�B ↔ δ(�B)) is not true at wby Lemma 9.2. But (w, p) has the same non-de re truths as w by The-orem 10.2, so (w, p) has all the same truths as w by hypothesis andTheorem 11.1, whence (�B ↔ δ(�B)) is not true at (w, p), and thusB is a PD expressor at (w, p) by Lemma 9.1 and local cumulativity,contrary to Theorem 10.1.

12. QUINE’S ESSENCES

Quine (1986) is now right: it is not modal logic, as he (1966) had thought,that breeds notable essentialism. Yet he was never far wrong: in theonly interesting sense, the ‘third grade of modal involvement’ assumesessentialism.

Let modal logic require cumulativity and C be contingently true.Then, Quine saw, modal logic requires the copossession of essences andaccidents. For CONC, or (C ∧ ¬�C), implies ∀x(�x = x ∧ CON(x =x ∧C)): each thing has an essential and an accidental property. But theone is essential to all things, and Quine now concedes that an essentialismof naught but universal essences is ‘benign, as essentialism goes’ (1986:p. 292).

More malign for Quine are two stronger types of essentialism. In one,‘essential traits ... are peculiar to single objects, shared by none’ (1986:p. 114). But CONC also implies ∀x∃y(�x = y ∧ CON(x = y ∧ C)):

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for each thing a, something determines a property that is essential andpeculiar to a and another that is accidental and peculiar to a.

In the other type, ‘an essential trait of one object can be an accidentof another’ (1986: p. 292). This is stronger than PD essentialism: anessential trait of one object need not be had by all. Of the latter typeand with it any stronger ones, modal logic is now innocent – by itself,in combination with any non-de re facts, and in all its forms. It takeslittle more than a ban on that type to ensure a benign reading of de relanguage, hence of the examples above.

From any interpretation incorporating PD essentialism, the essentialistappendix is surgically removable at no cost but essences lost: ∆ preserveseverything of interest – factual, logical, ontological – but PD essences(§1, Theorems 10.1–10.3). Assuming cumulativity, A ↔ δ(A) must betrue, post-operatively, for all A at all worlds (Theorem 10.4): de remodality is eliminable, de re language benign.

Eliminability does not require cumulativity, only a suitable ban onoptional essences. If K is not cumulative then neither is ∆K (Theo-rem 10.3) and δ does not apply to either (Theorem 9.2). But if K isessenceless then ε applies to K (Theorem 6.1), and if not then ΣK, likeK in non-de re truth (Theorem 7.3), is almost certainly essenceless (The-orem 7.1). Suppose, however, that K is neither cumulative nor essence-less and we insist on preserving this feature. Then the charted routes toeliminability are blocked (Theorems 6.2, 9.2), but only because we haveinsisted that K incorporate a strong if narrow type of essentialism, theexistential type: K is co-occupied (at the same or different worlds) byfulfillers and flouters of �x = x, necessary and contingent things.

Of the grades of modal involvement, the first has a necessity predicatefor statements but no �; the second, � for statements alone; the third,� for open sentences too. Citing his coposession example, Quine (1966)argued that the third assumes essentialism. In a sense he was wrong:at the third grade we might have A ↔ δ(A), which blocks notableessentialism (Theorem 9.2). In a deeper sense he was right.

A new locution might be nominal or real, definitionally eliminableor not. To the language of mathematical analysis, add y = d

dx(· · · x · · ·).Read as expressing a function of infinitesimals, the new notation is realand assumes new things and with them new facts. Read instead as shortfor y = lim

ε→0

[···x···+(···x+ε···)]ε , it is nominal.

De re locutions are like that. The second-grade facts almost alwaysallow essencelessness (Theorems 7.1, 7.3), and likewise C + PDE (Theo-rems 10.1–10.3). If either anti-essentialist assumption is true, then essen-tialism is false and distinctive third-grade language is nominal or we can

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make it so: A ↔ ε(A) is generally true (Theorem 6.1) or A ↔ δ(A)is (Theorem 9.1), so de re A can be defined as ε(A) or δ(A). If not,then essentialism is true and distinctive third-grade language real: weare stuck with either PD or existential essentialism, and we can have noeliminability result of the most modest generality (Theorems 11.1, 11.2),no nominal reading of distinctive third-grade language general enoughto be of any interest. Graduation to the real third grade occurs just whenessentialism kicks in.

13. THE UTILITY OF DE RE LANGUAGE

Quine’s paramount point (1966) about de re language is that it has nopoint, or no very good one: it is there only because it seemed naturalto make � a general sentence operator when modal logic got quantified.I agree, near enough, but my argument does not turn on the merits ofessentialism.

There are six ways to use de re language or not.

Way 1. Do not. Essentialism is not thereby ineffable: instead of ‘�xis rational’ the essentialist might use ‘rationality is essential to x.’ Is thatnot de re? It is in a way but not to make hay: modes of possession arenot merged notationally with modes of truth as if birds of modal feather.It was Quine, after all, who called attention to essentialism expressedby � de re. He did it to show how easily essentialism enters, unbiddenand unannounced, through the open door of de re syntax. As so often, heillustrated the Preacher’s Paradox: a vivid and engaging sermon promptedthe sin it condemned. But sin for him was the de re use of �, not theexpression of essentialism otherwise. In praying us shut the de re door,Quine himself needed some way to express the deplored doctrine: evento gainsay is to say in a way. Often he borrowed ‘is essentially’ or‘is essential to’ from English, where such phrases are used instead toconserve matter: they mark aspects of a thing that matter, or matter most,while avoiding tedious repetition of what they matter to. (Essentialismso expressed suggests that some aspects matter absolutely: it matters notto what. But that is another matter.)

Way 2. Let primitive syntax include the de re but assume essence-lessness, and with it A ↔ ε(A) (Theorem 6.1). This way has no pointbut to make a point: because the non-de re facts almost always let usban all de re-attributed essences (Theorems 7.1, 7.3), adoption of de relanguage requires no essentialism of even the most austere sort – thoughmodal logic may (§1).

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DE RE LANGUAGE AND DE RE ELIMINABILITY 539

Way 3. Retain Way 1’s idea that the purpose of � is to attributenecessary truth, but now to open sentences. This compels us to acceptA ↔ δ(A) (§8), allowed by the non-de re facts if cumulativity holds(Theorems 10.2, 10.4), and that leaves no room for de re essentialism:A↔ δ(A) is tantamount to the anti-essentialist (if less austere) C + PDE(Theorems 9.1, 9.2). One reason for not adopting so natural an exten-sion of Way 1 is a limited interest in open sentences: their job is doneas parts of statements, free-standing Ax1 · · ·xn serving no purpose notserved by ∀Ax1 · · ·xn. Another is that A↔ δ(A) requires cumulativity(Theorem 9.2), sometimes blocked by non-de re facts (§1). A third isthe want of any desire for the humdrum essences allowed by C + PDE(if we can forswear Stagiran stew for Quinian cuisine, how essential cancrumbs like ∃x�x = x be to our diet?). A fourth is to make way forWay 4, 5, or 6.

Way 4. Adopt an elimination schema A ↔ e(A) incompatible withessencelessness and C + PDE. We cannot reject this way as impossi-ble (§1), but e must suffer from extremely limited applicability (Theo-rems 11.1, 11.2).

Way 5. Start with Way 2, 3, or 4, then simplify semantic theory bydumping de re wffs from primitive syntax and reclaiming them as abbre-viations for their non-de re transforms. Ignoring near worthless Way 4,the definitional schema A↔ ε(A) or A↔ δ(A) still bars de re essen-tialism (Theorems 6.2, 9.2).

Way 6. Forego any abbreviative advantages of de re locutions to makethem available for new purposes. Intolerant of A ↔ ε(A) and A ↔δ(A), those purposes must include tolerance of a type of essentialismmore interesting than Quine’s first. For if ε and δ do not apply to K thenK flouts essencelessness (Theorem 6.1) and C + PDE (Theorem 9.1): itincorporates either PD or existential essentialism (§12).

This way exemplifies the common and invaluable practice of metaphor-ical transfer: locutions used for one purpose (attribution of necessity)in one realm (statements) are adapted to an analogous purpose (essen-tialism) in another realm (open sentences), where they had no previ-ous use or none we care to retain. Because people are not mineralsor vegetables, Jesus could call one of them a rock and my wife cancall another a couch potato. Because phones are not parts of organ-isms, buildings, or the like, engineers could invent cellular ones. Unlike‘rock’ and ‘couch potato,’ ‘cellular phone’ does not ring metaphorical.No matter: ‘metaphorical’ is not to be taken literally. ‘Cellular phone’

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exemplifies metaphorical transfer because it cannot be construed by look-ing up those old words in an old dictionary. Way 6 does too becausemodal logic began with de dicto �, whose de re extension by Way 6 isnot peculiarly natural or automatic, as witness Ways 2–5 and the boun-ty of conceivable essentialisms. The value of metaphorical transfer isexpressive economy: the novel use of old locutions saves on vocabularywhile fusing ideas (pictorial and discursive) in succinctly informativeways.

One man’s informative fusion can be another’s irritating confusion.Because people are not beasts, ‘snake’ is available for my political oppo-nents. But even my allies might protest that the transfer would be unfair,the fusion unwarranted by the facts. Because people are not words, ‘gen-der’ (‘is of — gender’) could be applied to people. But the fusion ofsex and gender encouraged editors to demand the likes of ‘Every con-gressperson serves only his or her constituents,’ good or bad according toone’s point of view: a textual bumper-sticker for sexual equality, it inti-mates adherence to a wide-compass ideology not necessarily the author’s.Metaphorical transfer can be as consequential and controversial as anystatement of fact or value.

So it is with Way 6. An economy is gained by using old locutionsfor new purposes. The fusion of ideas calls attention to �(A ∧ B) ↔(�A ∧ �B) and its ilk, plausible in both realms, and lets us expoundand exploit a common semantic theory. But it can also lead us to see tooclose a fusion, no notable new commitment; so Quine argued. Way 6 ismore consequential than that, I shall argue.

Let our language be severely monadic: it has one variable and nonested ∀ or �. Assume cumulativity and read de dicto � as logical truth,as validity or analyticity. To capture what we need of this idea, it isenough to assume, when �∃xAx is true, that some Bx makes ∃xBxvalid and � ∀(Ax ↔ Bx) true: maybe Ax = Bx, or maybe Ax is gotfrom Bx by synonym substitutions.

For true � ∃xAx find a suitable Bx. Then ∃xBx is valid, monadic,and nonmodal, so ∀xBx is likewise valid and �∀xBx true. Since�∀x(Ax ↔ Bx) is too, so is � ∀xAx. Hence, � ∃xAx → � ∀xAxis always true. But so are �Ax → � ∃xAx and � ∀xAx → �Ax (bycumulativity). Hence so is �Ax ↔ � ∀xAx and with it C ↔ δ(C):again no de re essentialism (Theorem 9.2).5

Now drop monadicity and the logical-truth assumption. Let some trueinstance of PD essentialism be monadic: typical examples are. Assume,whenever monadic �∃xAx does meet our logical-truth condition, thata matching �∃xBx is likewise monadic: ‘logical truth’ can still be read

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in a reasonably broad way if not the very broadest reasonable way. Thennot every such �∃xAx meets that condition: Way 6, de re essential-ism, has blocked the most common reading of de dicto �, the one thatoccasioned modal logic in the first place.6 Other readings abound, butWay 6 often blocks them too: we cannot read de dicto necessity as impli-cation by physical or other laws if the fancied instance of essentialismoccupies any language fragment where those laws have no distinctiveconsequences, none but logical truths.

We are left with three options and maybe a fourth. One is not toexpress essentialism. Another is to express it by non-de re means. Athird is to express it by de re means but deny it, to affirm A ↔ ε(A)or A ↔ δ(A). The fourth is to express it by de re means and tolerateit, but only after proving that the metaphorical transfer is consistent, thatde re essentialism is consistent with the intended de dicto use of � –very possibly an impossibility.7

The utility of de re language does not amount to much. Eliminableuses are dispensable, their definitional elimination a theoretical economy,their value solely abbreviative, their abbreviative value small. Yes, someof those uses are widely available (Theorems 10.2, 10.4) and worthi-ly interpretable (§8; Theorems 9.1, 9.2). But they are interpretable inthe worthiest way, by definition, hence worthless but for brevity. Andyes, �Ax1 · · · xn is much briefer than δ(�Ax1 · · ·xn) if n ≥ 2, just asy = d

dx(· · · x · · ·) is much briefer than its definiens. But the utility ofδ-eliminable �Ax1 · · ·xn unlike that of y = d

dx(· · · x · · ·), was small tobegin with.

Other uses are useless except to tolerate essentialism, and they tooare dispensable. Their value, when they have any, is the expressive econ-omy of metaphorical transfer. But it comes dear: the cost is the fre-quent inconsistency of de re with de dicto discourse. That cost may bebearable, consistency cheaply had, in special contexts. But to expressessentialism one way there and otherwise elsewhere is to forego anoth-er economy while inviting confusion. That cost might have been bear-able more broadly if, like y = d

dx(· · · x · · ·) and closed �A and even‘Quine believes x to be a spy,’ essentialist attributions has served apopular purpose all along. But museum pieces aside, they are mostlyencountered, in de re form, as the curiously fascinating wasteproductsof modal syntax, akin to y = x

0 more than y = ddx(· · · x · · ·). It does

not follow that such attributions, somewhat expressed, are unworthy ofstudy (who can say that the essentialist muse, charmless to some, willnot lead others to hidden delights?). It does follow that their expres-sion by de re means, means that must also serve a more common

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542 THOMAS SCHWARTZ

and commonly conflicting end, is a false economy. Of course, to con-cede this point is to lose a prevalent motive for taking up essential-ism.

NOTES

1 The chief sources are Quine (1961, 1966), but my formulation is consistent with thequalified Quine (1986); see §12 above.

2 The quoted phrase comes (near enough) from Hughes and Cresswell (1968: p. 184),the idea from Prior (1955: p. 211). The most thorough treatment is that of Fine (1978),who anticipates some of my own results (of §§5, 9, and less directly 6 and 10) butwithin the context of S5 with constant domain. Most novel are my limitative resultsand philosophical lessons, but Fine goes further in other ways, chiefly by incorporatingindividual constants and related problems of cross-world identification. A difference offramework makes detailed comparison hard.

3 But suppose K is PD-essenceless and bipolar: each K-world is locally essenceless(¬∃x�x = x is true at it) or locally cumulative (∀x�x = x is) and both typesexist. Suppose some non-de re B makes B ↔ ∀x�x = x true at every K-world.Let η be like δ except that, for open A, η(�A) has conjunct B (so if A is nonmodal,η(�A) = B∧δ(�A)). Then η applies to K. Also η(A) amounts to δ(A) if B = ∀xx = x,and to ε(A) if B = ∀xx 6= x. So a single eliminability result of sorts (η varies with Band K) subsumes those for δ and ε and covers bipolarity too. I say no more because thispaper is long and bipolarity weird.

4 The schema A↔ δ(A) is tantamount to schemata of Parsons (1969) and Fine (1978).It is also the syntactic version of Kaplan’s (1986) fungibility. His semantic versionamounts to C + PDE and Fine’s homogeneity. Besides §8’s independent argument forA ↔ δ(A), what my paper adds are theorems about the equivalence of syntactic andsemantic fungibility (§9) and ∆’s creation of semantic fungibility while preserving non-de re truth and more (§10). It also identifies the type of essentialism, different fromany discussed by Quine or Kaplan, that blocks fungibility (§12). And it shows that nominimally general eliminability result can flout fungibility (§11).

Kaplan (p. 283, fn. 69) is wrong that fungibility distinguishes a narrowly logical read-ing of � from others: for cumulative K,∆K is always fungible and it always preservesany construction of de dicto necessity, logical or other, found in K (§10). Still, if de dicto� expresses logical truth, then any monadic violation of fungibility is blocked (§13).Kaplan’s quest for the distinguishing marks of logical necessity is apt; see note 6.

5 This argument is adapted from Schwartz (1979), where its import for essentialismwas not appreciated.

Objection: The argument is needless. A logical reading of de dicto � automaticallyfalsifies the essentialist ‘� Bill is rational but ¬� Socks is rational.’

Reply: The example is essentialist only if ‘Bill’ and Socks’ are rigid designators, butthen it is de re on any interesting account.

6 Not without problems. One is analyticity. Never mind: let de dicto necessity = validi-ty, making � a Frege–Russell assertion operator and suggesting, as Kaplan (1986: p. 283)recommends, a logic specifically of logical necessity in the best sense. Now we have clar-ity, and some utility to boot: mathematical writing is fraught with statements, otherwisemetatheoretic and complicated to formalize, of consistency, implication, and the like. But

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here the problem – one of utility rather than clarity – is that formalization is impossible,else we should have an effective enumeration of all closed valid wffs of the form ¬�A,hence all nonvalid wffs of first-order predicate logic. Though they had logical necessityin mind, early proponents of modal predicate logic did not see this because they proposedpure systems. There, ¬� ∃xFx is not a thesis, despite the nonvalidity of ∃xFx, becauseF is a predicate variable, open to such validating substitutes as Gx→ Gx. To have anyvalue, however, logic must apply outside pure systems. Kaplan did not see this because,pradoxically, he saw nonvalid ∃xFx as having exactly the same logical form as valid∃xx = x (ibid.). No, if F is parsed as a nonlogical predicate, open thereby (unlike =) tovarying interpretations, then it is essential to the logic form of ∃xFx that F is just that,hence that ∃xFx is not valid, hence that ¬� ∃xFx is valid (true by logical form) in therecommended logic of logical necessity.

7 Objection: A fifth option is two interpreted languages, de dicto � meaning in onewhatever it had meant, and in the other whatever essentialism makes it mean.

Reply: This option really is a special case of the second. If we can always tell,morphologically, which language we are reading, their difference does the same jobin near enough the same way as the morphological difference, arbitrary in its details,between ‘�x is rational’ and ‘rationality is essential to x’ – and otherwise we had betterpreserve that difference. Likewise does the morphological difference between ‘hell’ and‘light’ in English do the same job as the difference, also morphological, between Englishand German contexts of ‘hell.’

REFERENCES

Fine, Kit (1978): Model theory for modal logic – Part II: the elimination of de re modal-ities, Journal of Philosophical Logic 7: 277–306.

Hintikka, Jaako (1961): Modality and quantification, Theoria 27: 119–128.Hughes, G. E. and Cresswell, M. J. (1968): An Introduction to Modal Logic. Methuen,

London.Kaplan, David (1986): Opacity, In Hahn, Lewis, E. and Schilpp, Paul A. (eds.), The

Philosophy of W. V. Quine, pp. 229–288. Open Court, La Salle, IL.Kripke, Saul (1959): A completeness theorem in modal logic, Journal of Symbolic Logic

23: 1–14.Kripke, Saul (1963): Semantical considerations on modal logic, Acta Philosophica Fen-

nica 16: 83–94.Parsons, Terence (1969): Essentialism and quantified modal logic, Philosophical Review

78: 35–52.Prior, Arthur N. (1955): Formal Logic. Oxford University Press, Oxford.Quine, W. V. (1961): Reference and modality. In Quine, W. V. From a Logical Point of

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Lewis, E. and Schilpp, Paul A. (eds), The Philosophy of W. V. Quine, pp. 114–115,290–294. Open Court, La Salle, IL.

Schwartz, Thomas (1979): Necessary truth as analyticity and the eliminability of monadicde re formulas, Notre Dame Journal of Formal Logic 20: 336–340.

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Thomason, Richmond H. (1970): Some completeness results for modal predicate calculi.In Karel Lambert (ed.), Philosophical Problems in Logic, pp. 56–76. Reidel, Boston.

Wright, G. H. von (1951): An Essay in Modal Logic. North-Holland, Amsterdam.

Department of Political Science,UCLA,Los Angeles, CA 90095-1472,U.S.A.

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