DANIEL BONEVAC
PARADOXES
OF FULFILLMENT
A1 Martinich (1983) has observed that the liar paradox has analogues
in illocutionary acts other than assertion. Each of the following, for
example, exhibits something like the paradoxical character of the liar:
(1)
I bet that I lose this bet.
(2)
Don't obey this command!
(3)
I promise not to keep this promise.
(4)
I order you not to follow this order.
(5)
I advise you not to take this piece of advice.
(6)
I propose that this proposal be rejected.
(7)
I suggest that this suggestion not be taken.
(8)
I intend not to act on this intention.
(9)
I want this desire to go unfulfilled.
(10)
I need for this need not to be met.
(11)
This obligation should not be fulfilled.
(1) is paradoxical because the bet is won if and only if it is lost. One
obeys (2), similarly, just in case one does not obey it. The promise in
(3) is kept if and only if it is not. Similarly for the other cases.l For
reasons that will become clear, I shall call these paradoxes offulfillment.
Throughout this paper I shall treat promising, commanding, needing, wanting, obligation, and so on, as well as truth, as predicates of
sentences. Call predicates such as
(12)
I promise
(13)
you command
(14)
is obligatory
Journal of Phdosophical Logic 19: 229-252, 1990.
c~ 1990 Kluwer Academic Publishers. Printed m the Netherlands.
230
DANIEL BONEVAC
commitment predicates. Languages containing commitment predicates
and the resources for diagonalization techniques do not always allow
for the construction of paradoxes of fulfillment. Without additional
features, they give rise to paradoxes only with the assumption of
dubious principles concerning the commitment predi_cates at issue.
Furthermore, the paradoxes that then result are "tricky," like the
paradox of the knower, rather than immediate and straightforward
like the liar paradox.
The paradoxical character of (1)-(11) depends essentially on the
notion of fulfillment: the concept of a promise being kept, fulfilled, or
carried out; of an obligation being fulfilled, discharged, met or satisfied; of a command being obeyed, executed, carried out or followed;
and so on. Michael Dummett (1978) has observed that obedience is to
orders, and winning is to bets, as truth is to assertions. Much the
same relation appears to hold between fulfillment and promising,
fulfillment and obligation, acceptance and proposal, satisfaction and
desire, and action and intention. In this paper I shall develop a logic
for fulfillment predicates to show how the paradoxes involved in
(1)-(11) arise and may be resolved.
1. C O M M I T M E N T
PREDICATES
Suppose that L is a language of first-order logic containing enough
resources to name its own sentences. Suppose further that L does not
contain its own truth predicate but does contain a commitment predicate C. Are paradoxes derivable in L? That depends, of course, on the
logic of the commitment predicate. The following are principles that
might be adopted in such a logic:
(FOL)
C[q~], if ~o is an axiom of first-order logic
(RA)
C[cp], if q~ is an axiom of Robinson arithmetic
(DIS)
C[~ -~ 44 ~ ( C H ~ C[4,])
(CON)
-7c[•
(TR)
c[~] ~ c[c[~]]
(SR)
C[C[~] ~ ~]
PARADOXES OF FULFILLMENT
(AB)
C[-7 C[~o]] -~ --1 C[r
(CA)
C[r
(RC)
If ~-q~ then ~-C[g0]
231
if r is an instance of (FOL), (RA), (DIS) or (AB)
(Here C[go] abbreviates c(rrpT), where rrp7 is the name of sentence rp.)
Various combinations of these principles are inconsistent. Thomason
(1980) has shown that (FOL), (DIS), (TR) and (SR) are jointly inconsistent. Similarly, Koons (1987) has shown that {(FOE), (RA), (DIS),
(AB), (CA)} and {(DIS), (CON), (TR), (RC)} are both inconsistent.
These results might suggest that the paradoxes implicit in (1)-(11) are
essentially paradoxes of commitment.
But some principles needed to derive the Thomason-Koons paradoxes do not hold of commitment predicates in general. (DIS), to
take just one case, entails (AG):
(AG)
(C[r
& C[qJ]) ~ C[r & qJ]
which, in the case of obligation, Williams (1973) has called the
agglomeration principle and Marcus (1982) has called the factoring
principle. Arguments against this principle for obligation are wellknownfl They moreover extend readily to other commitment predicates. Consider, for example, these arguments involving promising
and betting:
(15)
John promises to marry Susan.
John promises to marry Wanda.
.. John promises to marry Susan and Wanda.
(16)
Mary bet that Chicago would win the Super Bowl.
Mary bet that Miami would win the Super Bowl.
.. Mary bet that Chicago and Miami would win the
Super Bowl.
Together with (CON), (DIS) implies that commitments never conflict. That is, (DIS) and (CON)jointly entail
(NC)
--1 (C[r
& C(~ cp]).
(NC) is notoriously problematic for obligation, and outright silly for
promising, commanding, and the like. People can make conflicting
232
DANIEL BONEVAC
promises or suggestions, give conflicting orders or advice, and have
conflicting needs, desires and intentions.
If (DIS) ought to be rejected, then none of the Thomason-Koons
paradoxes afflict commitment predicates. 3 In any case, the paradoxes
resulting from the inconsistent combinations of dubious principles are
not those with which this paper began. The paradoxical character of
(1)-(1 l) seems evident almost immediately. Yet the Thomason-Koons
paradoxes are not so obvious; it takes substantial cleverness to see
that they can be derived at all. Consider the argument to a contradiction from the set {(FOL), (RA), (DIS), (AB), (CA)} (Koons, 1987,
1988b):
(17)
1.
~ ~
--1
C[0t]
diagonalization
2. C[~ C[ct]] -~ --1 CM
(AB)
3. C[~ ~ --1 C[c~]]
(RA), (DIS), 1
4. C[ct] ~ C[--q C[~]]
(DIS), 3
5. C[ct] ~ "-1 C[~]
2, 4
6. ~ C[~]
5
7. C[-q C[cq]
(FOL), (CA), (DIS), 6
8. C[e]
4, 7
This is a sophisticated line of reasoning. Arguing to the ThomasonKoons paradoxes is tricky; such arguments bear little similarity to the
almost automatic, intuitive argument to the paradoxes of fulfillment.
Even accepting all the required principles, therefore, we cannot construct the paradoxes sprouting quickl5 from (1)-(11) in L.
2. FULFILLMENT PREDICATES: A FIRST ATTEMPT
The paradoxical character of (1)-(11) depends on the notion of fulfillment. Promises may be kept, fulfilled, or carried out. Obligations,
desires, wishes and needs may be fulfilled, discharged, met or satisfied.
Orders and commands may be obeyed, executed, carried out or followed.
Bets may be won. Advice, proposals, suggestions and intentions may
be fulfilled, carried out, acted on or followed.
PARADOXES OF FULFILLMENT
233
In this section I shall present a first try at developing a logic of
fulfillment predicates. I shall represent fulfillment, like commitment,
as a sentence predicate. Suppose that language &a contains first-order
logic, enough resources to name its own sentences, and a commitment
predicate C. Say that a predicate f of ~ is a fulfillment predicate for
C in ~o just in case it satisfies, for all sentences q9 and ~b of 5r
(I)
f[C[~0]] ~ ~0, and
(E)
If ~-(o ~ ~b, then ~-f[q~] .--, f[~9].
(I) requires that a fulfillment predicate serve as a kind of inverse to a
commitment predicate. (E) requires that fulfillment predicates not be
hyperintensional; they cannot distinguish between logically equivalent
expressions.
Conditions (I) and (E) are all we need to obtain the paradoxes of
fulfillment. We need to assume nothing about commitment predicates
except that they are predicates of sentences. (1), for example, says of
itself that it should not be fulfilled: we may represent the content of
(1), that is, as 0[-7 f[(1)]]. The arguments from (1)-(11) to contradictions all have the same simple form:
(18)
1. O ~ C[--7 f[0]]
diagonalization
2. f[O] ~ f[C[-7 f[0]]]
(E), 1
3. f[O] .-~ -7 f[O]
(I), 2
This explains why the paradoxical character of (1)-(11) is so evident.
Two additional sentences lead to paradox if we grant additional,
rather plausible assumptions.
(19)
1. ( ~ C[f[-7 (]]
diagonalization
2. f[~] ,--~f[C[f[-7 ~]]]
(E), 1
3. f[~] +--+f [ 7 ~]
(I), 2
This is a contradiction if we grant that f never applies to both a sentence and its negation. If we grant
(NEG)
f [ ~ ~01 ,-o -7 f[~p]
234
DANIEL BONEVAC
we obtain another contradiction from 'It is not the case that this
obligation should be fulfilled':
(20)
1. r *---,---nC[f[~]]
diagonalization
2. f[r
(E), 1
~ f[-n C[f[Q]]
3. f[~] ,--, -7 f[C[f[~]]]
(NEG), 2
4. f [ ~ ] ~ --1 f [ ~ ]
(I), 3
We have seen that commitment predicates, without a corresponding
fulfillment predicate, do not themselves lead to paradox. The known
paths to paradox all involve implausible assumptions. Do fulfillment
predicates, by themselves, get into trouble? Not always. We might try
to construct a liar analogue by choosing the diagonalization instance
(21)
fl ~ --1 f[fl].
But this leads nowhere; (I) and (E) do not imply that fulfillment is
redundant, so we cannot obtain fl .--, f[fl]. Indeed, where commitment
predicates are involved, we would not want fl and f[fl] to be equivalent. More generally, without a commitment predicate in the language, (I) has no force, so a predicate counts as a fulfillment predicate just in case it satisfies (E). But any extensional or even weakly
intensional predicate does this. The predicate 'is a sentence', for
example, which holds of every sentence, satisfies (E), but does not
lead to paradox.
All languages containing commitment and fulfillment predicates
contain, implicitly, their own truth predicate. In language ~ , we may
define truth-in-~L,r T, as follows:
(22)
T[cp] *--~f[C[rp]].
That is, r is true if and only if any commitment to r is fulfilled. So,
it is not surprising that such languages admit paradoxes. But argum e n t (18), though closely related to the liar, is not just that paradox
expressed in terms of C and f rather than T. Using this strategy, we
could obtain the paradox
(23)
1. ~ ~ f[C[--n ?]]
diagonalization
2. 7 *--~--'3];
(I), 1.
PARADOXES OF FULFILLMENT
235
But it is hard to see how to express the content of the first premise in
English. Using the definability of truth in 2~o, we can also construct a
version of the strong liar:
(24)
1. 6 ~ ~ f[C[6]]
diagonalization
2. 6 *-~ --7 6
(I), 1
The first premise here, too, is hard to express; perhaps 'A commitment to this would be unfulfilled' is close. In any event, the paradoxes
derivable from (1)-(11) have the form, not of (23) and (24), but of
(18). What I have called paradoxes of fulfillment are not "strictly
analogous" (Martinich 1983, 63, 65) to the liar paradox; they are
related, but do not have precisely the same form.
Nevertheless, a particular paradox of fulfillment is almost the liar
paradox. Any first-order language containing sentence predicates C
and f satisfying (I) and (E) admits paradox. A language containing its
own truth predicate fits this description. If we put T in for both C
and f, (I) and (E) hold. It follows that a first-order language contains
its own truth predicate if and only if it is susceptible to paradoxes of
fulfillment. We can obtain a variant of (18) for truth:
(25)
1. 0 ~ T [ ~ T[0]]
diagonalization
2. T[O] ,--, T[T[--qT[0]]]
(E), 1
3. T[0] ~ -7 T[~gl
(I), 2
What is strictly analogous to the paradoxes of fulfillment, therefore, is
not the liar or strong liar, derived from 'This sentence is false' or 'This
sentence is not true', but a paradox derived from 'It is true that this
sentence is not true'. This paradox differs from the liar and strong liar
in an important way; it relies on (I) and (E) rather than Convention
T. That is, it arises not from T[q~] ~ q~ but the weaker T[T[q~]] ~ q~.
3. FULFILLMENT PREDICATES: AN ALTERNATIVE ACCOUNT
In the last section, I took fulfillment predicates as inverses to commitment predicates. Principle (I) expressed the central idea: f[C[~0]] *--, ~o.
On reflection, however, this seems unfaithful to the English commitment and fulfillment predicates I have discussed. 'John's promise to
236
DANIEL BONEVAC
come has been fulfilled', for example, seems equivalent not to 'John
has come' but to 'John promised to come, and he has'. This would
lead to an alternative principle:
(I')
f [ C H ] ,--, ( C H & q~).
According to (I), any sentence q~ entails f[C[q~]], a sentence asserting
the fulfillment of a commitment to r But most sentences have no
logical implications about the fulfillment of promises, satisfaction of
desires, and so on. From 'Jack lost weight' we can infer that Jack's
wish to lose weight has been fulfilled only given the additional premise
that Jack wanted to lose weight. This is precisely what (I') predicts: ~o
entails f[C[q~]] only given the additional assumption that C[q)].
We might defend the logic of the previous section by reading
f[C[~0]] as asserting, not that a commitment to ~0 is fulfilled, but that
any commitment to q~ would be fulfilled. On this reading, f[C[q~]] does
not entail C[q~]. That any commitment Jack might have made to lose
weight would have been fulfilled does not imply that Jack made such
a commitment. Indeed, any such commitment would be fulfilled if
and only if Jack lost weight, in keeping with (I).
However this may be, for the remainder of this section I shall
adopt the more natural reading off[C[~0]] as 'a (or the) commitment
to ~0 is fulfilled', taking (I'), rather than (I), to characterize the logic
of fulfillment predicates. So, I shall now count as a fulfillment predicate any sentence predicate f satisfying both (I') and (E).
The central paradox afflicting fulfillment predicates as previously
understood does not follow from (I'). Nevertheless, (I') has some
peculiar consequences. The reasoning in (18), adapted to (I'), leads to
the following:
(26)
1. 0 ~-~ C [ ~ f[O]]
diagonalization
2. f[O] ~ f[C[-n f[0]]]
(E), 1
3. f[O] ,--, (C[-n f[0]] & -1 f[O])
0% 2
4. --7 f[O]
3
5. --7 C[-n f[0]l
3
6.-70
5,1
PARADOXES
OF FULFILLMENT
237
This is not a contradiction. But it means that some commitments cannot be made, even though the sentence expressing the object of the
commitment is true. 4
(26) reveals an important feature of paradoxes of commitment: they
arise only if the troubling commitments are made. 'I promise not
to keep this promise' is puzzling, but, if nobody uses it to make a
promise, there is no paradox. If there is no commitment, then the
issue of its fulfillment does not arise; there is no commitment to be,
or not be, fulfilled. Thus, in (26), no contradiction results unless we
assume C[--q f[8]], that is, that a commitment has actually been made.
This vindicates Martinich's view: speech acts such as (1)-(7) must
somehow be defective, for the assumption that they give rise to the
commitments they purport to express leads to absurdity. Indeed, if we
take (1)-(7) to be performative, where a sentence rp is performative if
and only if it satisfies
(P)
A[~o] ~ ~o
for a sentence predicate A symbolizing 'is (successfully) asserted', then
the reasoning of (26) leads to the conclusion --7 A[0], that (1)-(7)
cannot be successfully asserted.
This reasoning reflects an important difference between paradoxes
of fulfillment and the liar paradox. The liar is a problem whether or
not anyone ever asserts, or even can assert, a liar sentence. Asserted
or not, the question of the truth or falsity of such sentences remains.
We could try to mimic the liar as a paradox of fulfillment based on a
sentence such as
(27)
I assert that I do not believe this assertion.
(28)
I claim that this claim is false.
or
But the result is not the liar at all. 'The sentence is false', 'This
sentence is not true', and the like do not involve assertion directly;
their evaluation is independent of their assertion.
One might object that, because (1)-(7), (27), and (28) are all unassertable, they express no propositions and thus cannot be evaluated. This,
however, reverses the order an analysis ought to have. We know that
238
DANIEL BONEVAC
asserting these sentences is problematic because we know what they
mean; we can analyze them as unassertable only because of a prior
semantic interpretation of them.
The reasoning of (26) and the analysis of (1)-(7) as unassertable
does not extend immediately to (8)-(11), which are not performative.
Uttering "I p r o m i s e . . . " under appropriate conditions constitutes a
promise; uttering "I intend . . . . " "I need . . . . " or "I w a n t . . . "
does not constitute an intention, need or desire. In these cases, (26)
seems to show that there cannot be intentions, needs, desires or obligations of the requisite sort. This conclusion, however, is problematic.
Sentences (8)-(11) are obviously self-referential; it may seem plausible
that there are no desires, needs, or intentions of that kind. But selfreference may be indirect and contingent. As Martinich indicates,
Kripke's (1975) examples adapt to ordering, obligation, and the like:
(29)
Major: Don't follow any of the Captain's orders.
Captain: Follow all the Major's orders.
(30)
Jones:
Smith:
(31)
You should not meet any obligation alleged
by Smith.
You ought to meet every obligation alleged by
Jones.
Gomez: I promise to do whatever McCoy promises
to do.
McCoy: I promise not to do anything Gomez promises
to do.
These cases do not preclude an unassertability analysis for performatives. Consider (31), assuming, for simplicity, that Gomez and
McCoy promise nothing else.
(32)
1. q~ ,-, C[-7f[~b]]
diagonalization (McCoy)
2. 0 ~ C[f[q~]]
diagonalization (Gomez)
3. f[tp] ~ f[C[--7 f[O]]]
1, (E)
4. f[O] *-*f[C[f[~ol]]
2, (E)
5. f[q)] ~ (C[-n f[O]] & -7 f[O])
3, (I')
6. f[O] '--' (C[f[~o]] &f[rp])
4, (I')
PARADOXES OF FULFILLMENT
239
7. f[~9] ~ (C[f[~0]] & C[~ f[~]] & -7 f[~])
5, 6
8 . - 7 f[~]
7
9. -7 C[f[cp]] v 7 C [ ~ f[O]]
7, 8
10. 7 ~
v ---7~0
1,2,9
If ~0 and ~ are performatives, then we can conclude -7 A[~] v
-7 A[~0]; either Gomez or McCoy has failed to make an assertion.
This seems a reasonable conclusion. If Gomez speaks first, then
McCoy's promise fails; if McCoy speaks first, Gomez fails. If they
speak simultaneously, perhaps, both fail.
The following circumstances, however, have the same form, but do
not involve performatives. (Assume the reports are true.)
(33)
Mother:
I want all my daughter's desires to be satisfied.
Daughter: I want most of my mother's desires to be
frustrated.
(Assume that, with the exception of the desire mentioned, exactly one half of mother's desires are
frustrated.)
(34)
Sniggerum: I want all Sniggeroo's desires to be frustrated.
Sniggeroo: I want some of Sniggerum's desires to be
satisfied.
(Assume that, except for that mentioned, none of
Sniggerum's desires are satisfied.)
The reasoning of (32) therefore applies. Here, however, it leads to the
conclusion that the reports cannot all be true; at least some of the
desires in each instance cannot be held. l'his seems wrong. The desires
could be fulfilled together, in fortuitous circumstances. In the circumstances described, however, at least one desire turns out, contingently,
to be paradoxical. The mother-daughter pair in (33), for example,
seems all too plausible. Only the unfortunate and contingent accident
that exactly half the mothers' other desires are frustrated makes the
mother's desire indirectly self-referential. At least for nonperformatives, then, we should not accept the reasoning of (32) at face value.
240
DANIEL BONEVAC
In keeping with the analysis of this section, relying on (I') rather
than (I), there are consequently two strategies for handling the paradoxes of fulfillment. (1) One can deny that the paradoxes are real,
holding instead that the troublesome diagonalization instances cannot
be asserted or, when no performatives are involved, that they cannot
be true. This strategy takes the reasoning of (26) and (32) at face
value. (2) One can hold that the paradoxes are real, because at least
some troublesome diagonalization instances can be true. This strategy
seeks a way to reject or, at least, explain in a semantically coherent
way the reasoning of (26) and (32). Note that, even on this latter
strategy, the paradoxes of fulfillment differ sharply from the liar and
its immediate relatives. Contradictions arise only on the assumption
that certain sentences are true. Strategy (1) is plausible chiefly for
performatives; strategy (2) can apply to performatives and nonperformatives alike, s
4. THE SEMANTICS OF FULFILLMENT
Strategy (1) permits a simple semantics for fulfillment predicates. We
can adopt a principle like (35):
(35)
v e [[f~n iff v has the form C[(p], [[v] = 1 & [[(p]] = 1.
Or, allowing truth value gaps, we can adopt (36):
(36)
If v has the form C[rp], then
v e ~fl] 1 iff [[v] = 1 & [[rp~ = 1.
If v has the form C[r then
ve~f]~
= 0 v ~o~ = 0.
If v does not have the form C[~o], then
v ~ ~f~l and v ~ [-f]]0.
On either approach, rC~ ~ _ ~f~0. This may destroy some of the
intuitive appeal of these clauses, since, if there is no commitment, the
question of fulfillment seems not to arise. Clauses (35) and (36) treat
f[C[~o]] as entailing C[q~], at least for unproblematic sentences; so, in
particular, -1 C[qq entails -1 f[C[~o]] for those sentences. Similarly,
within the same limitations, -7 q~ entails --1 f[C[~o]]. The following
PARADOXES OF FULFILLMENT
241
examples, valid according to (35) and (36), suggest that we might
reflect natural language more accurately by taking C[q~] as a presupposition of f[C[~o]]:
(37)
a. John didn't bet that Miami would win.
.. John lost his bet that Miami would win.
b. Today is not Friday.
.. Lynn's desire that today be Friday has been
frustrated.
A slight revision of (36) captures a presuppositional approach:
(38)
If v names a sentence of the form C[~o], then
v E [~f~l iff ~v~ = 1 and [[r = 1.
If v names a sentence of the form C[~0], then
v ~ ~f~0 iff ~v~ = 1 and [~o~ = O.
If v does not name a sentence of the form C[q)], then
v r ~f~' and v r ~f~0.
Strategy (2), however, requires a more sophisticated approach.
Clauses (35), (36), and (38) validate the reasoning of (26) and (32),
leading to the conclusion that the problematic sentences are false.
For desire and other nonperformatives, as we have seen, this seems
too harsh a conclusion. To take the paradoxical character of (1)-(11)
seriously, we must devise a different semantics for fulfillment predicates.
The theories of truth developed by Kripke, Gupta, Herzberger and
others account for paradoxes involving the truth predicate. In this
section I shall construct analogous theories for commitment and
fulfillment predicates. Throughout I shall focus on the account of section 3, using (I'); theories applying to section 2's account are easy to
derive and have largely the same character. Where possible, I shall
tailor the accounts so that f[C[~p]] presupposes C[~0], following the
pattern of (38). Say that L is a language without commitment or fulfillment predicates, and that ~e is L w {C, f}. Condition (I') requires
that f act as inverse to C within C's extension. This determines the
extension o f f only partially. When should f hold of other sentences
of the language? When is 'snow is white', for example, fulfilled? The
242
DANIEL BONEVAC
answer, "when snow is white," has a certain intuitive appeal, but it is
just as plausible to hold that 'snow is white' is not the sort of thing
that can be fulfilled. Here, to isolate issues involving fulfillment from
those involving truth, I shall construct a model in which f a c t s as C's
inverse in C's extension and, otherwise, holds of nothing at all.
I shall present first a construction following the central ideas of
Kripke (1975). Begin with a structure ( D , ~ ]]) for L, where D contains all sentences of 5r In fact, for convenience, assume that D is
just the set of sentences of 5r We can interpret ~ by a series of
models (D, ~ ~, ~C~ 1, ~C~~ ~ f ~ , ~f~0), where [[C~1 is the extension,
and [[C]~ the antiextension, of C, and ~f~] and ~f~0 are, similarly, the
extension and antiextension o f f at ordinal stage i of the construction.
I assume that the extension and antiextension of C are given in some
way once and for all; since commitment predicates themselves do not
lead to paradox without implausible assumptions, there is no need to
construct their interpretations explicitly. 6
Let M0 = (D, [[ ]], [[C~J , ~C~~ ~ , ~ ) . This determines the extensions of the predicates of L w {C}, but gives no information about
sentences containing f. To continue the series, we need clauses telling
us how to construct extensions and antiextensions for f at both
successor and limit ordinal stages.
(39)
If v names a sentence of the form C[~o], then
v ~ [[f]]l~+l iff [[q~ = 1 and []v~]~ = 1.
If v names a sentence of the form C[q~], then
v s ~f~0+~ iff [[q~]l = 0 and ~v~, = 1.
If v does not name a sentence of the form C[~p], then
v r [[fl]l,+l and v r ~f~]~ I .
(40)
If a is a limit ordinal,
~f~, = Ua<~ [ f ~ ; ~f~0 = Ua<~ [Ff~.
These revision rules characterize an operation K from models to
models: K(M,) = M,+~. The operation is monotone in the following
sense: for each ~, I f ~t ___ [Ff~l+l and ~-f~0 ___ ~f]]0+~. A structure Ms
is a fixed point just in case K(M~) = M~. It is possible to show that,
for any initial M0, there is such a fixed point. In fact, the construction
PARADOXES OF FULFILLMENT
243
always yields a minimal fixed point. If L is denumerable, then the
fixed point occurs at or before the first nondenumerable ordinal. At
any stage ~, ([~f]]l u ~f~0) ___ gc~]0l " Paradoxical sentences such as
(1)-(11) receive truth values based on the interpretation of C; sentences saying that they are fulfilled, however, receive no truth values
at any fixed point. If the relevant commitments exist, these sentences
lack truth values because they are ungrounded. If the commitments
do not exist, they lack truth values because fulfillment presupposes a
commitment to fulfill.
Converting this construction to follow the method of Gupta (1982,
1987) and Herzberger (1982a, 1982b) requires that we begin the construction process with a model M0 -- (D, ~ ~, ~C~, [~f~0) that expands M
by adding an appropriate extension for C and an arbitrary extension
for f. We then extend this model according to the rules
(41)
If v names a sentence of the form C[~p], then
V ff [[f~+l iff I-Opt, = 1 and [v~]~ = 1.
If v does not name a sentence of the form C[q)], then
v r [-f~+1-
(42)
If 9 is a limit ordinal,
~f~]~ = {~P: q7 < ~Vfl(7 ~< fl < ~ ~ q~ ~ ~f]l~)}.
The function H from models to models corresponding to this construction is not monotone. Some sentences, such as ones asserting the
fulfillment of (1)-(1 I), will flip back and forth between truth and
falsehood. A sentence is positively stable at M s iff it is true at every
ordinal stage after and including ~, and negatively stable iff it is false
at every such stage. ~0 stabilizes at ~ iff ~ is the first ordinal at which
~0 is stable. ~ is a stabilization ordinal iff every sentence true in M~
stabilizes in M s. The least stabilization ordinal is an enclosure point. It
is possible to prove that, for any initial structure, the construction
yields an enclosure point. Truth values of (1)-(11) are established by
WC~ in the initial model; truth values of sentences asserting the fulfillment of (1)-(11), in contrast, do not stabilize.
The construction also converts to the method developed by Robert
Koons (1987, 1988a), based on the "indexical-hierarchical" approaches
of Tyler Burge (1979) and Charles Parsons (1974). Koon's theory
244
DANIEL BONEVAC
reintroduces types, but without the restrictions on well-formedness
that limit Russellian type theories. Koons represents reflective reasoning on truth by assigning the truth predicate different "levels," corresponding to degrees of reflection. The theory thus consists of two
parts: (i) an algorithm assigning ordinal levels to occurrences of 'true',
and (ii) an interpretation of sentence tokens, given an assignment of
levels to such occurrences. The latter part itself divides into two components: (a) a method for generating an assignment to the language
from an initial assignment (here Koons takes over Kripke's construction) and (b) a method for generating an initial assignment at a level
from assignment at previous levels. Constructing a model consists
of a transfinite recursion, each stage of which is itself a transfinite
recursion.
To capture the assignment of ordinal levels to occurrences of the
fulfillment predicate, we must assign truth values, not to sentences,
but to pairs of the form (q~, h} consisting of a sentence together with
a function from its occurrences o f f into ordinals. Equivalently, we
may assign values to subscripted sentences, in which each occurrence
o f f receives an ordinal subscript. As in Kripke's construction, we
begin with a structure (D, [~ ]]) for L. We interpret ~ by a series of
models Mj. k of the form (D, ~ ~, [[C;', ~C]]~ [~f~', ~ f l 0 ) , where ~f~'
and ~f~0 are disjoint functions from pairs of ordinals into sets of sentences. ~f~l.~ is the extension, and [Ff~~162
the anti-extension, off~ at
ordinal stage fl of the a-th recursion.
The first structure in the series is Mo.0 = (D, [~ ~, ~C]]~, [~C~~
~ , ~ ) , yielding a partial characterization of truth in .L~~ In general,
for any ordinal 7, ~f~.o = ~f]]~.o = ~ , and, for each 7 > J, [-f~l..~ =
~f~]~.a = ~ . We proceed to apply the revision rules:
(43)
If v names a sentence of the form C[~o], then
v r [ f ~ . , + , iff ~q~;.~ = 1 and ~v~..,~ = 1.
If v names a sentence of the form C[~0], then
v ~ [~f]]~.~+l iff ~o~:..~ = 0 and ~v]~.., = 1.
If v does not name a sentence of the form C[~0], then
v ~ ~f~l.~+~ and v 6 [~f~.~+,.
PARADOXES OF FULFILLMENT
(44)
245
If ~ is a limit ordinal,
~f~,,~ = U~<, ~f~,~; ~f]]~.~ = U~<~ [[f~.~.
As above, the construction reaches a minimal fixed point, say, 6...
From M;..6., which, in effect, specifies an interpretation o f f . , we
derive an initial assignment tof.+l. Let M;.+l.O = (D, ~ ]l, ~C]]I. [~C]~
I
1
1
1 and [~f~;,+l.O
0
~f~,.~,
D - ~f~;.~;.);
that is, [f~,.+l,0
=
= ~f]]:..~;.,
D - Ff]]!.~.. (In Kripke's terms, this "closes off" the interpretation
off..) In general, for each 7 < J, ~f~;!,a ---- ~f]];.,a;,
l where 3;. is the
fixed point at level 7, [[f~.~ is its complement relative to D. Thus,
each model Mj, k includes a complete interpretation o f f for every
i < j; a partial interpretation offj; and an empty interpretation o f f
for every i > j. Sentences asserting the fulfillment of (1)-(11), in a
Koons-style approach, are truth-valueless 0 and either truej or falser,
depending on how they are stated. (For example, a sentence asserting
the fulfillment of (9), 'I want this desire to go unfulfilled', is true~, but
one asserting the fulfillment of 'I want this desire to be frustrated' is
falsel .)
I have tried to show that paradoxes of fulfillment arise from a combination of commitment and fulfillment predicates, and may be solved
in various ways, adapting already-developed solutions for the liar
paradox. Other theories of truth may be adapted similarly. The
analogy between the liar paradox and paradoxes of fulfillment thus
provides no argument for any particular approach. It does, however,
indicate that the problem of providing an adequate semantics for
truth is part of a larger and more general problem.
5. APPLICATIONS
The paradoxes of fulfillment indicate that certain standard philosophical ideas can lead to paradox. Here I shall focus on just one philosophical idea, utilitarianism. But similar points can be made concerning obeying commands, fulfilling obligations, following rules, and
acting on intentions.
In many modern forms of utilitarianism, an act counts as right if
and only if it maximizes the satisfaction of desires. The logic of desire
satisfaction, however, is not straightforward. In particular, specifying
246
DANIEL BONEVAC
the conditions under which a desire is satisfied requires a solution to
the paradoxes. Furthermore, the solutions I have discussed in this
paper, all of which share the structure of a transfinite recursion,
characterize those conditions quite differently.
Issues arising from the paradoxes of fulfillment may sound
recherch~ to a normative ethicist. But they do have some significance
for normative issues. People often have desires that have other people's
mental states, and even desires, as their objects. I may want my
friend's desires to be satisfied and my enemies' desires frustrated. If
my friend or my enemy has desires about my desires, then my desires
concerning the satisfaction or frustration of others may well land me
in paradox.
(33) and (34) above, for example, were cases of higher-order desires,
some of which turned out paradoxical given unfortunate contingent
circumstances. The more complex such cases become, the harder it is
to try to rule out or even recognize problematic desires. Consider this
soap-opera version of Carroll's three barbers:
(45)
Allen: I want most of Baker's desires to be satisfied;
ditto for Carr.
Baker: I want most of Allen's desires to be satisfied;
ditto for Carr.
Carr: I want all of the desires Allen and Baker have in
common to be frustrated.
If exactly one half of each person's desires, except for those mentioned,
are satisfied, and Allen happens to share at least half of Baker's
desires, then at least Baker's desire is viciously self-referential.
Evidently, a utilitarian can evade the fulfillment paradoxes only by
ignoring all higher-order desires. This would entail that a utilitarian
should, in effect, maximize the satisfaction of first-order, mostly selfinterested desires. Since many or even most people have desires concerning the happiness of others that are important to their own
happiness, this would often yield intuitively implausible results.
Indeed, many virtues and interpersonal relations - love, friendship,
charity, generosity, a concern for others - involve higher-order
desires essentially.
PARADOXES OF FULFILLMENT
247
When performing a utilitarian calculation, therefore, we need to
take higher-order desires into consideration. But developing a method
for doing so requires some solution to the paradoxes of fulfillment, as
well as a formula for assigning values to desires that are satisfied,
frustrated, or neither.
On a Kripke-style approach, sentence (9), 'I want this desire to go
unfulfilled', has a truth value assigned by an initial structure. The sentence asserting (9)'s fulfillment receives no truth value, in the initial
structure or at a minimal fixed point. Similar truth value gaps will
arise for sentences that are paradoxical only because of contingent
circumstances. This raises several important questions for utilitarianism. First, what notion of truth is relevant to evaluating desire satisfaction? In general, given a sentence q~, there are three possible
semantic outcomes for f[~0]: f[~0] may be true, false or truthvalueless.
(Say that, in these three cases, ~p is satisfied, frustrated or unresolved.)
Kripke defines grounded and intrinsic truth; corresponding to them
are grounded and intrinsic satisfaction and frustration. Other candidates are possible.
Second, how should a utilitarian calculus handle a nonbivalent
logic of desire satisfaction? Maximizing desire satisfaction differs from
minimizing desire frustration. Moreover, once bivalence is rejected,
additional options become attractive. A utilitarian might think of
satisfied desires as counting in a particular outcome's favor, while
frustrated desires count against it and unresolved desires are neutral.
One might thus want to define the "difference" between desire satisfaction and desire frustration in a state of affairs and maximize that.
Various definitions of "difference," of course, will lead to different
utilitarian theories.
On a Gupta-Herzberger analysis, utilitarianism faces similar issues.
A Gupta-Herzberger construction preserves bivalence at each ordinal
stage, but allows some sentences to flip between truth and falsehood.
Any attempt to include these in the moral calculation is likely to be
arbitrary. To eliminate them, we can focus on stable truth. This, however, raises the issues arising from the nonbivalence of a Kripke-style
approach: certain desires will be neither stably satisfied nor stably
frustrated.
248
DANIEL BONEVAC
On a Koons-style approach, nonbivalence also raises the questions
facing a Kripke-style theory. Some very different issues arise as well.
Koons's theory follows Parsons and Burge in analyzing truth as an
indexical, hierarchical notion. We thus need to distinguish levels of
satisfaction and frustration, not in the sense of quantities of these, but
in the sense of ordinal levels like those in the theories of truth that
Parsons, Burge and Koons offer. Should we maximize satisfaction0?
Satisfactions? For what n should we minimize frustration,? Recall
that sentences asserting the fulfillment of problematic sentences such
as (1)-(11) are truth-valueless0 and either true~ or falser, depending on
how they are stated. (9), in particular, is satisfied~. The satisfaction of
(9), therefore, should count in favor of a circumstance if we try to
maximize satisfactions, but not if we try to maximize satisfaction, for
n r 1. Perhaps satisfaction on any level should count in favor of an
outcome. If so, should all levels contribute equally? Or are some more
important than others? Finally, we must be very careful about allowing quantification over levels in a Koons-style analysis, for 'I want
this desire to be unsatisfied at every level' is paradoxical ;n a fashion
the theory cannot handle.
An example may make these considerations less abstract. A simple
paradox arises for utilitarianism that relates to a more complicated
paradox, involved in the argument that an act-utilitarian cannot sanction punishment (Hodgson 1967, Regan 1980, Koons 1987). The pain
of the guilty counts against an outcome, so punishment can be justified
only if the deterrence or other positive effects of the punishment
exceed this cost. If it is commonly believed that only a specified, finite
number of criminal opportunities are available, and if the act-utilitarian
is known as such, then punishment deters crime if and only if it is
irrational to believe so. Thus, the act-utilitarian should approve
punishment if and only if it is irrational for him to do so.
This is not, as it stands, a paradox of fulfillment. But there are such
paradoxes in the neighborhood. Suppose that we have a single agent,
an anti-utilitarian, with a single desire: that desire satisfaction not be
maximized. Evidently, in such a circumstance, we maximize desire satisfaction if and only if we do not. On a Kripke- or Gupta-Herzbergerstyle approach, the agent's desire is neither satisfied nor frustrated (in
PARADOXES
OF FULFILLMENT
249
a grounded or stable sense). A plausible treatment of such desires
would probably count the desire irrelevant to determining which
option is best. On a Koons-style approach, we must assign the
agent's desire a level. Say that the agent desires that we not maximize
desire satisfactionn. This desire is neither satisfiedn nor frustrated,.
Since it is the only desire we have to consider, all options are equal at
level n; assuming 'maximize desire satisfactionn' means 'choose
an option satisfying,, at least as many desires as are satisfied, by any
other option', all options maximize desire satisfaction,. Consequently,
the desire is bound to be frustratedn+l. The example indicates
how, on a Koons-style analysis, differences in level are extremely
important. At level n, all options are acceptable; at level n + 1,
none are.
ACKNOWLEDGEMENTS
I am grateful to Nicholas Asher, Robert Koons, A1 Martlnich and an a n o n y m o u s
referee for their very helpful comments on earlier drafts of this paper. I would also
hke to thank the Center for Cognitive Science of the University of Texas at Austin
and the Knowledge and Database Division of the National Science Foundation for
their research support.
NOTES
Not all performatives give rise to such paradoxical lllocutionary acts. Reszgn and
accept, for example, m the relevant illocutlonary sense do not accept clausal complements.
2 See, for example, Williams (1973), van Fraassen (1973), Marcus (1982), and Bonevac
and Seung (1988).
3 Other principles involved m deriving those paradoxes are also dubious. In the case of
promising, (TR) seems foolish: A promise is not automatically a promise to promise. A
proposal is not automatically a proposal to propose; a desire is not automaucally a
desire to desire. (SR), too, is troublesome. Could there be a world in which you keep
all the promises you make in this world, but in which you make additional promises
you fail to keep? If so - and it is hard to see why not - then (SR) fails for promising.
The same holds for commands. Surely there could be a world m which all your actual
c o m m a n d s are obeyed but in which you issue new orders that are disobeyed. Similarly,
there could be worlds in which all your actual needs are met, and m which you win all
your current bets, but in which you have new unmet needs and in which you place
new, losing bets. (AB), trivial for necessity or truth and plausible for justification, has
no plausibility when apphed to commitment predicates. That Sarah promised not to
promise anything does not mean that she did promise nothing.
250
DANIEL
BONEVAC
4 Looking briefly at the other paradoxical assertxons of the previous section' I f f never
applies to both a sentence and its negation, then ( cannot be fulfilled.
1.
( ~ C [ f [ 7 (11
diagonalization
2.
f[(] ~ f [ C [ f [ ~ (ll]
(E), 1
3.
f[(] ~ ( C [ f [ 7 (]] & f[-7 (])
(I'), 2
4.
-7 f[(]
3
Moreover, i f f never applies to both a sentence and its negation, then ~ is either true,
giving rise to no commitment, or cannot be fulfilled.
1.
r ~ 7 C[f[~]]
diagonalization
2.
f[~] , - , f [ 7 C[f[~]]]
(E), 1
3.
f[~] ~ 7 f[f[f[~]]]
2
4.
f[~] ,--, ~ (C[f[r
(I), 3
5.
~ f[r
6.
-7 f[~] v (~ & -7 C[f[~]])
&f[r
v 7 C[f[~]]
4
5, I
Using (I) to characterize fulfillment predicates, we could define truth as fulfillment of
a commitment. With (I'), that appears to be impossible. The natural options all fail:
a.
T[q~] ~--~f[C[~o]]
b.
T[~0] ,--, ( C H --' f [ C M ] )
c.
C[~o] --, (T[~o] ~ f [ C H ] )
These do not permit the derivation of liar-like paradoxes; from the appropriate dlagonalization instances, we can obtain, not contradictions, but either ( ~ q~ & ~ C[~0]) or
( 7 tp & 7 C [ 7 ~o]). This again affirms that the required commitments cannot be made.
Again taking ~o as a performatwe, we can derwe 7 A[~0]; the required instances cannot
be successfully asserted.
A final point: if we take both C and f to be the truth predicate, (I') and (E) both
hold. The analogue of paradoxes of fulfillment with the truth predicate appears to be
the following argument:
1.
,9 ~ T[TT[0]]
diagonalization
2.
T[0] ,---,T[T[TT[0]]]
(E), 1
(I'), 2
3.
T[0] ~ (T[7 T[Ol] & ~ T[g])
4.
7
5.
~ T[-7 T[O]]
3
6.
70
5,1
T[0]
3
Far from being the liar, this is not a paradox at all. The conjunction of 4 and 5 is
bizarre, but showing this requires a principle, ~o ~ T[q~], not available from the analysis
o f the fulfillment paradoxes
PARADOXES
OF FULFILLMENT
251
5 Strategy (1), implausible for nonperformatives, has other problems. Martinlch adopts
this strategy, attempting to solve the paradoxes implicit in (1)-(11) and (29)-(31) by
denying that they succeed as illocutionary acts; he maintains that the liar ~s an unsuccessful assertion, (1) an unsuccessful bet, and so on. He relies on Searle's idea of essential
conditions of speech acts: A person must be able to perform the required speech act,
know what it means, and intend the audience to react appropriately. It is not obvious
how such an approach can distingmsh paradoxical sentences from simply contradictory
speech acts, on the one hand, and harmless but ungrounded self-referential sentences
like the truthteller on the other. Arguably, in all three kinds of cases, the speaker
cannot perform the act, know what at means, and intend the audience to react in the
fashion appropriate to the speech act.
6 It is possible to make the construction of an interpretation of commitment predicates
a part of the process of model construction. The techniques Asher and Kamp (1986,
t988) use for knowledge and belief can apply to any commitment predicate, and the
model construction methods I shall describe can incorporate those techmques without
major changes.
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University o f Texas at Austin,
Department of Philosophy,
Austin, T X 78712,
U.S.A.