Before Refraining: Concepts for Agency Author(s): Nuel Belnap Source: Erkenntnis (1975-), Vol. 34, No. 2 (Mar., 1991), pp. 137-169 Published by: Springer Stable URL: http://www.jstor.org/stable/20012334 Accessed: 28/05/2009 14:40 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=springer. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact
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BELNAP
NUEL
A
ABSTRACT. a modal are are
CONCEPTS
REFRAINING:
BEFORE
structure
is described
construction
agentive
such
that can serve as "a
sees
FOR AGENCY
as a foundation
to it that Q"
([a
for
The
primitives this structure
choice). Eleven governing simple postulates a picture time consisting of branching Tree and < encode a time? Instant chains called "histories." into maximal imposes gathered moment to and each and each of choice consists agent Agent agents, assigns
=ss, Instant, Agent, (Tree, set forth and motivated.
of "moments"
for a semantics
sur.Q]).
like ordering. in Tree a set of "possible are of these ingredients
choice where each possible choices", to in the semantics referred suggested
is the motivation for part of the discussion complex a typically non-terminating chain of moments jointly a moment.
the definition to witness
is a set of histories. for
All
[a sur.Q]. The most it means of what for
the truth of
[a
m.
Q]
at
which The paper begins with an informal account of the "Refref says that conjecture", to something to refrain is to see to it. The the only way from refraining from seeing sense contem? of an argument of Prior's that in a certain paper ends with a consideration plation
and action
are
inconsistent.
it easy to think logic, makes logic of agency, like any modal of nesting agentive expressions.1 the expressive possibilities Thus tense logic work of Prior, and building on fundamental encouraged, a straightforward and Perloff 1989 proposes modal analysis of Belnap (an approximation to) refraining from an action. This analysis leads for instance to the "Refref conjecture", that the only way to refrain from refraining from an action is to carry out that action. This paper started out to be an account of what is known about the Refref conjecture, but in the event that story will have to be told on another occasion, for a preliminary elaboration required to talk sensibly about the conjec? ture has used up all the allotted time and space. The difficulty is that these ideas were only briefly and informally described in Belnap and Perloff 1989, and again in Belnap 1989; these papers instead concen? trated on motivation I therefore and applications. What offer here, the Refref conjecture, apart from a few informal paragraphs concerning
The modal about
is the laying out in much more substantial detail of some of the funda? mental ideas of the formal metaphysics and semantics of the modal of these ideas logic of agency, together with a condensed application to an argument of Prior's concerning the compossibility of contem Erkenntnis ?
34:
1991 Kluwer
137-169, Academic
1991. Publishers.
Printed
in the Netherlands.
138
NUEL
BELNAP
plation and action. You may well come paper with the feeling that a great many manufactured that as yet have not been so, I hope that you will turn to the other
from a tour through the have been pieces of machinery given important jobs to do; if above, and papers mentioned away
also accept what our admired and beloved Wilfrid Sellars, used to call a "promissory
1 . THE
REFREF
too-soon
departed
colleague,
note".
CONJECTURE
idea is that one good way of approaching is The background agency stu: as sees to to "a it via the modal be read that locution [a Q], to the thesis that the ?", where a is an agent and where, according of stit should be unrestricted, Q may take the place of any complement sentence
whatsoever.2
the locution [a stu:Q], it is easy to see that refraining when a for has just the form [a sm.~Q]; by non-agentive complemented Given
example,
comes
Autumn
Jane refrains
Autumn
Jane
from becoming
muddy
to
Accordingly, For example,
refraining
Autumn
sees
from
Jane
to it that she does an action
refrains
from
not become
has the form
seeing
muddy.
[a stir.~[a
sm.Q]].
to it that she becomes
muddy comes
to Autumn
Jane sees
becomes
muddy.
to it that it is false
that she sees
to it she
but the reflective eye The two forms are easy for the ear to confuse, can see that advice to refrain from seeing to it that one becomes muddy is much easier to follow than advice to refrain from becoming muddy. Parents and children alike doubtless rely on the ear's confusion when after the dress is splattered other matter each with they hash out the a by passing truck. is wanted. The form Give a modal logician a little nesting, and more
BEFORE REFRAINING! ~ [a st?: which may
~ [a mu
CONCEPTS
FOR AGENCY
139
[a sm:?]]],
be read as a refrains
from refraining
to it that Q,
from seeing
itself. In this language the nesting of refraining within as the can be noted above expressed question ~ ~ sm. sm. [a Q]]] is equivalent Refref conjecture, [a sm. [a
illustrates sample
the
to
[otstit'.Q].
Jane is true, then the only way that Autumn If the Refref conjecture can refrain from refraining from seeing to it that she becomes muddy is to see to it that she becomes muddy. in itself, though I confess is perhaps not so exciting The conjecture to a certain mud-pie fondness for it. In order to think about the conjec? ture with precision, it is apparent that we shall be lost without however, an effort intuit? a careful metaphysics and semantics. Neither making of space time to chunks in four-dimensional outline ively imagination nor are ourselves that from refrainings, with inci? regaling refrainings to I is therefore describe the suffice. dramatic stories, dent-filled going was a semantics of stit that indicated of choice and briefly metaphysics in Belnap the very generally
and Perloff
1989,
in the belief
to that it not only contributes the Refref but is conjecture,
of clarifying special problem on a right track toward an understanding
2.
THEORY
OF
BRANCHING
of agency.3
TIME
The metaphysics is based on branching time with agents and choices. I an structure (Tree, underlying postulate temporal-modal-agent-choice =^, Instant, Agent, choice) subject to certain constraints. These I present time (as it is often beginning with the theory of branching gradually, called). (Tl)
is a set whose
I call moments. members I use m as over moments. w, often with markings, ranging
Tree
be pictured as situation, where plete, really possible relativistic or as requiring some one tivistic sense.4 Tree, although a set of
Each moment
should
and
an instantaneous, spatially com? take this either as pre you may to make rela frame of reference moments,
may
perhaps
be taken
140
NUEL BELNAP
or supervene to represent upon a concrete (I am entirety or world matters it whether the of Lewis of course 1986). Probably thinking or or concrete is abstract "Tree" notion fundamental by represented as or be taken itself should the indeed whether distinction mixed, matter is the not What does for but intramundane, present purposes. this concept of "world" and that of Kripke 1959 key difference between concept (etc.), of Lewis 1986 (etc.), or of the standard four-dimensional and Minkowski: from Newton derived by way of Einstein (I suppose) to contain real possibilities both for what I take the world, our world, might be and for what might have been. If there are alternative worlds, then they, too, come with their real possibilities.5 (T2)
and ordered by ^ (reflexivity, transitivity, < for the associated use order I strict partial anti-symmetry). ^ mu which (i.e., ^ but not =). Picture the direction of m0 flows from past to future, as generally upwards from m0 to I read ^ and its converse with the unmarked mi. In English words "earlier/later", etc., "below/above", "lower/upper", and mark by the insertion of "proper" when I intend < or are comparable if ^ goes one way or its converse. Moments Tree
is partially
we the other, and a chain is a set of comparable moments; c over chains. let range the chain is outside chain in Tree (no moment A maximal the moment inside to every chain) is called a comparable over as and use I often histories, marked, h, ranging history. H as ranging over sets of histories. H(w) is the set of histories in which w lies or, to use a different geometrical phrase, w. the set of histories "passing through" should be taken to this sense of history (as a set of moments) Perhaps as a concrete whole. on an underlying notion of history supervene as a set matters whether it upon history supervenes history Probably does matter as a concrete whole, but not for present purposes. What is chock is that a history is not an entirety or world. First, our world with and with a-block with real possibilities, chances, certainly perhaps a a in home none which finds of Second, although actions, single history. the phrase "our world" makes sense, "our history" does not, unless - the be permanently determinism true, for if it is not, our moment - is context of this communication situation that is the (idealized) part no one of which in any way as is marked of many possible histories,
BEFORE REFRAINING!
CONCEPTS FOR AGENCY
141
can determine A context of utterance much, including a a an is moment auditor, given only (whose spatial spread speaker, a past, a focus of attention, etc., but relative, to a frame of reference), a unique history of which it is a part, for (unless it cannot determine be permanently determinism true) there are too many histories of which a artificial or rooted in it is device, whether part. No conversational "ours".6
can overcome than any this difficulty, anymore language, can use true make it that each of "our child" story about the couple has exactly one child. The point is not that we cannot take the trouble to uniquely specify some history or other; maybe we can, and if so we can certainly follow 1974 in calling it "Kronos". Nor is the point that the convers? Plantinga ational situation cannot render some one history "salient". The point can specify the (unique) history to which we instead is that nothing ordinary charitable
and
because of utterance belongs, no to is be had. true) uniqueness
this context
permanently Let us now
(T3)
(unless
determinism
be
look backwards.
Incomparable moments bound (No downward
in Tree never have a common
upper
branching).
I mean a By a past or a past history (the phrases are interchangeable) set moments of that contains bounded every moment upper nonempty it contains; and I let p range over pasts. Because below any moment of No downward branching, any past is a chain and thus can be extended to a history. The set of (either improper or proper) predecessors of each moment is a past. Thus, the phrase "the past" or "the past history of which the present moment is endowed is the last moment" by each context of utterance with a perfectly determinate Let p(m) be meaning. the past containing m and all its predecessors. a portion of a history, In contrast, if "the future" connotes then there is no sense to the phrase "the future" - unless determinism be - for the to true claim is false. For permanently uniqueness implied this reason we should distinguish between the phrases "a future" and "a future history" even though it is unnecessary to distinguish between set of "a past" and "a past history". By a future I mean a nonempty proper upper bounds of some past; each future will have the shape of a tree, and will be a subtree of Tree. By a future history I mean a nonempty
maximal
lower bounded
chain of moments
(no moment
out
142
NUEL
BELNAP
to every side the chain except one of its lower bounds is comparable moment inside the chain).7 at a given moment When Lee-Hamilton says "the past is stone, and stands forever fast", his use of the phrase "the past" safely refers to a context of utterance. past that is uniquely determined by the (idealized) this past is a portion of each history of which his utterance Furthermore, is a part: no matter the future history, that past stands forever fast. at a certain moment When, however, says "I dipt into the Tennyson the Vision of the world, and future, far as human eye could see,/Saw all the wonder that would be", we must be careful, especially when we There learn that the Vision involves a sea-battle. is no philosophical as we tree the if the of possibilities poet denoting problem, interpret in intricate arborescent that fans out from the moment of his utterance - a indeed, even if our anticipation proper source of wonder profusion is stopped at a distant horizon. Nor if the poet is using "the future" as relative not only to the moment of utterance but having its denotation as long as we do not forget also to a history8 is there a problem, and take proper care to disambiguate this relativization Tennyson's in the case of an occurrence just as we would accordingly, expression in ordinary discourse. of "the man" If, however, by the phrase "the future" the poet intends to denote a unique future history of which his moment is a part, then his intention cannot be carried out, for (unless as far as human eye can see) be true from that moment determinism there is no such unique future history. But more of this below. on about or a physicist maunder one hears a philosopher Sometimes it is and doubtless in a present moment, distinct pasts that coalesce I our that I limits be tested. that confess, however, good conceptual these fancies. That I face alternate future histories I am faced away from alternate pasts seems to that right; me wrong. That starting with the concrete situation yesterday morning there were alternative possible ways in which yesterday might (qua time slice) have been filled seems to me right; that any of these distinct situation alternate pasts might have finished up in this very concrete seems to me wrong. One of the reasons I think this is that I do not know how to make sense out of agency on such a theory. myself seems
cannot
follow
to me
time can be modeled Some of branching equally well on what might after be called a "T x W" theory (with T for times and W for worlds) a or well be called Thomason 1984, theory "divergence" might equally after Lewis 1986, or even just a "Kamp" theory after Kamp 1979. The
BEFORE REFRAINING:
CONCEPTS FOR AGENCY
143
are taken as (to use the present terminology) common and that each each from distinct (no parts), ontologically one to say is added to permit concepts enough in the way of additional that two histories "perfectly match up through" (Lewis) or "differ only key
idea is that histories
a particular two time. If, however, in what is future to" (Thomason9) can match before a certain time while failing to match after histories that time, it seems hard to deny that they can equally well match after a certain time while failing to match before that time; for this reason, to metaphysics, unlike branching such an approach time, can well en? time, incidentally, courage backwards babbling. Branching branching instead what it of distinct histories; does not deny forwards matching it may be that one of denies is their forwards overlap. For example, the histories on which I turn left becomes very like one of the histories on which I turn right at some point after the heat death of the sun; why not? But "very like" is not the same as sharing particular moments (and is not even very like). I insert a little more about pasts and histories. A For later reference or it is a superset of that is extends through a past just when history mean set of all histories that extend p. A moment, I the past. By Hfp) a m a if is an upper bound for for is minimal bound m, upper past, p, no for is such that other bound p p upper properly earlier than m. A or more no minimal than one minimal have upper bounds, past may a or in minimal the last case I speak upper bound; upper bound, unique if it contains any least I say that p is complete of a least upper bound. a it have. Observe that that bound may upper complete past may have no minimal one. or more I than upper bound, let/?* be the completion of p, that is, the smallest p, so that p is past extending complete = a case in is p. Evidently just p* /?(m) complete complete past. (T4)
have a lower bound (Historical connec? Every two moments not I do that they have a greatest lower suppose tion); but bound (Semi-lattice condition).
So every two histories share a common past: their nonempty set-theor? etical intersection. When in fact there is a least upper bound, u>, of the of two histories in a common past, I call w the moment intersection of intersect of the two histories, and also say that the two histories intersection in w. There may also be a greatest lower bound of those moments the moment
in the two histories that are not in their overlap, which I call and I say that the two histories split at such a of splitting,
144
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BELNAP
or the moment of intersection are how? identical. Because, they to verify the Semi-lattice enough there is a for each two histories at which they split.10 (The picture of a "moment of splitting" moment is good when one is tracing the two histories upwards, beginning before is good of intersection" they have split. The picture of a "moment one when is tracing the two histories downwards, beginning before they have intersected.) in Tree have a common lower It is not supposed that all the moments connection that It is Historical bound, which would imply a Big Bang. as Lewis 1986 of Tree "in suitable external relations", puts all moments either the moment splitting exists, they both exist, and ever, the world may not be kind we cannot be sure that condition,
moment.
When
of
says (p. 208). It gives content to exactly the sort of real possibility of stit.11 to an understanding I think pertinent 3.
If "histories" then Instant (T5)
THEORY
are a way of making is a kind of horizontal
OF
that
INSTANTS
a sort of vertical
division
of Tree,
counterpart.
is a set of that is, Instant of Tree; Instant is a partition moment in Tree moments that each sets such of nonempty one Instant. members The of to member exactly belongs of Instant are called instants', i, often marked, ranges over to which instant determined is the and instants, uniquely /(m) a moment m belongs.
I am not in this study because it is intuitively correct to think relying for "filling" the time of m. I of /(m) as the set of alternate possibilities I think that for the sense of stit I am after, in need instants because at a certain Jane stit she was muddy Autumn whether considering to consider what else might have been at the it is relevant moment, Instants
are perhaps on measures
not fully "times" or distances, but
inhabited by that moment.12 on Instant that are very likely too I enter a pair of conditions is that agency is already my justification strong (too oversimplifying); so that it won't hurt to try to see what it comes to hard to understand, - as realistic that are not altogether in circumstances long as we keep closer to track of what we are doing so that later we can try to move instant Next
reality.
BEFORE REFRAINING:
(T6)
Each
CONCEPTS FOR AGENCY
intersects
instant
let m^h) be the moment tersects with) history h. (T7)
145
I each history in a unique moment. in which instant / cuts across (in?
never
order: given two distinct distort historical instants ?i and i2 and two histories h and A', if the moment at which at which ?i intersects h is earlier than the moment the ?2 intersects h, then the same relation holds between Instants
moment i2
at which
intersects
*W(il,/i')
