Dictionary of Paradox

Glenn W. Erickson John A. Fossa

l lniversity Press of America, ® Inc. l-1anham· New York· Oxford

Copyright © 1998

University Press of America,® Inc. 4720 Boston Way Lanham, Maryland 20706 12 Hid's Copse Rd. Cummor Hill, Oxford OX2 9JJ All rights reserved Printed in the United States of America British Library Cataloging in P u b l i cat i on Infonnation Available Library of Congress Cataloging-in-Publication Data

Erickson, Glenn Dictionary of paradox

/ Glenn W.

W.

Erickson, John A. Fossa.

cm. p. Includes bibliographical references.

L

Paradox-Dictionaries. I. Fossa, John A. II. Title. BC199.P2E75

1998

165

dc21

98-10526 elP

ISBN 0-7618-1065-X (cloth: alk. ppr.) ISBN 0-7618-1066-8 (pbk : alk. ppr.)

8™The paper used in this publ ication meets the minimum

requirements of American National Standard for infonnation Sciences

Pennanence of Paper for Pr inted Library Materials, ANSI Z39.48

1984

PREFACE

We have tried to w rite a diction ary that i s both useful and intri guing, ful l of i ndications for further thought and providing the reader w ith the bibliographic resource s to pursue those s uggestion s . OUf search for material has been painstaking but hardly exhaustive, and we mi ght, w ith some ingen uity and patience, have doubled the three to four hundred odd paradoxes named in this work. From Anscombe 's and Arrow's to Zeno 's and Zermelo 's we have col lected and summarized paradoxes as we found them; that i s to say, the i ncl usion of a entry in thi s dictionary i s not a si gn of the i mpo11ance or even coherence of the paradox reported. To be sure, we have not abstained from adding our own views when we saw fit. We have i ncluded cross-references and generic entries to gui de the reader through the landscape of paradox. We are indebted to many people for their i nvaluable help in thi s work, and since there w il l inevitably be some Ini stakes in a book of thi s nature, it goes without saying that we and not they are responsible for them.

O n e of a s e rie s of para d o xic a l observations about management propounded by Jerry B. Harvey, its name deri ves froln the exampl e with with he i l l ustrated it. Formulation. G roups of people frequently act contrary to the desires of all of their members, thereby fai l i ng to achi eve thei r goal s . Explanation. A family i s relaxi ng on a hot S unday afternoon when one of them suggests that they dri ve i nto Abil ene, Texas , for din ner; no one i n the family actually wants to face the four-hour round tri p trek to Abi lene, but each yie lds to his (mi s)percepti on of the general wi l l . It is important to see that the paradox does not arise from compromi ses that the group's members make in order to accomlnodate differi ng vi ew points. Rather. all the Inelnbers implicitl y agree on someth i ng (that i s, not to go to Abilene), but the agreement i s not recognized due to poor communications in the group. Indeed , by suppressing his own desires , eac h mem ber contributes to the general misperception. In the corporate setti ng, the situation is further compl icated by the multifarious power relati onshi ps i nvolved. Accordi ng to Harvey, the paradox reveal s that the most serious obstacl e to good mallagelnent i s not the inabi lity to manage di sagreement but the i nability to manage agreement. Resolution. In order to overcome the paradox, it i s necessary to recognize i ts psychological roots: action anxiety� l eading to a refusal to act at all; negati ve fantasies , depicting exaggerated consequences of proposing the desi red action to the group; and fear of ostraci sm or separation from the group, i ncl udin g, in the corporate setting, loss of one 's job. Once one'i : reali zes that the consequences of i naction may be just as momentous as those of acti on, however, these psychol ogi cal roots may be undercut. Thus Harvey suggests two steps i n coping with the paradox. Fi rst one shoul d Inake a real istic eval uation of the ri sks consequent upon both rnaking one 's desires know n to the group and continui ng to go along with the undesired proposal . l�everthel ess, since by hypothesis the other me mbers of the group are acti ng in such a manner as to enhance the undesired proposal, it may be extremely difficult to make this evaluation. Second, since other melnbers of the group may i n fact be acti ng contrary to their own desires, one shou l d refrai n from attributing any given feelings to the others and expl ai n one 's own positi on i n a group setting. Thi s step may expose the underlying agreelnent i n the group and thereby prevent an unnecessary trek to Abi lene. ABILENE PARADOX, THE.

,

READING

Harvey, Jerry B .

Th e Abile n e Paradox and Other Meditations on

Management.

Lexington, MA : Heath,

1988.

6

Dictionary of Paradox

ACHILLES, THE. Al so know n as Achilles and the Tortoi se, the paradox

is the second of Zeno's arguments against motion . Formulation."In a race the quickest runner can never overtake the slowest. Since the pursuer must first reach the point whence the pursuit began, the slower must always hold a lead" (Ari stotl e, Physics, 239b 1 5 - 1 8). Explanation. A fast runner (say, brave Achil les) may be set the task of overtaking a s l ower one ( say, a tortoi se) that is given a head-start. I n order to catch up to the tortoi se� however, Achil les must fi rst reach the poi nt whi ch the tortoi se had attai ned when Achi lles started after it. Yet duri ng the i nterval it takes Achi lies to reach thi s point, the tortoise will have advanced to a farther point. Th us Achilles must reach thi s new point, but by the ti lne he does so the tortoi se wilJ have again moved to a new positi on. Hence there i s alway s a resi d ual interval between Achi l l es and the tortoise and the hero never catc hes the slow beast. Resolution. Since we know from experi ence that Ach i l l es catches the tortoise in short order, the point of resol ving thi s paradox is to identify the fallacy that makes the paradox plausible. Aristotle di stinguishes potential infi nity from actual infinity. By the former, he means something like 'infi nitely divisible'. Since space and time are only potentially infi nite, the paradox, according to Ari stotle, does not arise. The concept of infinity w as troublesome to ancient Greek mathemati cians because, i n part, they did not have the concept of a l imit. Rene Descartes, C. S . Peirce, and A . N. Whitehead are among those who have u sed the idea of a limit to resolve the paradox. According to this view, Achilles' predicament can be reduced to a convergent infinite series, such as

T he limit of thi s series is simpl y one. Max B l ack argues, how ever, that the existence of a finite l imit of an i nfinite series i s not suffici ent to resolve the paradox because it doe s not obvi ate the nece s s i ty of Achi l l es performing an i nfinite number of acts. J. M. Hinton and C. B. Martin contend that the paradox depends on an ambiguity. On the one hand, the description of the paradox merely requi res Achi lles to pass through all the i ntermediary points between hi s starti ng poi nt and the poi nt at whi ch he overtakes the tortoi se and thi s causes no further complications. On the other hand, the paradox may be interpreted as a series of commands restri cti ng Achilles' movements in certain way s. The fi rst interpretation, according to Hinton and Martin, verifies the premi se of the paradox but i Ilval idates the argument. In contrast, the second i nterpretation val idates 1hc arglilnent but fal sifies the premise. See ZENO'S PARADOXES.

7

Dictionary of Paradox

READINGS Aristotle.

Physics. VI-IX. Analysis 11.5 (1951): 91-101. A chi l les and the To rto i se. Analysis 14.3

Hlack, Max. "Achilles and the Tortoise."

linton, J. M, and C. B. (1953): 56-68. TayJor, Richard. "Mr. I

Martin.

"

"

Wi sd om on Tem po ral Paradoxes." Analysis 14.4

(1952): 15-17. Whitehead, W. N.

Process and Reality.

Cambridge: Cambridge UP,

1929.

10] -108. Wi s d om J. O. "Achilles on a Physica1 Racecourse." Analysis 12.5 (1951): 67-72. ,

A( �TING, PARADOX OF.

Written by the philosophe and encycl opedi st I )eni s Diderot ( 1713-1784)� The Paradox of Acting is still frequently I t'garded as the single most significant statement about acti ng. Formulation. T he paradox is si mply that i n order to move an audi ence I he actor must remain himself unmoved. I(�p]anation. If the stage actor is himself ful) of feeling, he will not be ;11 )Ie to play the same part repeatedly with the same spirit and effectiveness. lie woul d be emoti onally drained after a few performances. Diderot (lhserved " the unequal acti ng of players w ho play from the heart. Thei r playing is a lte ln atively strong and feebl e, fiery and cold, duU and sublime." I f is not that actors should feel nothing at al l ; rather, what is needed i s craftsmanshi p i n order to achieve the fulfi ll ment of auctorial design. lienee, the actor must strive for unity and consistency of peIformance regardless of hi s own emotional , mental, and even physical state. Further, I he most artful acting does not give i tself over to the passions being I )oltrayed, but maintains its contro] and reserve. Like great poets, great ;Ictors must be fabulous puppets. I{esolution. This plaradox is real enough. Whi l e actors who y i eJd to their e motions may manage single great performances, they are l i abl e to he uneven. They may have great expressive range and del icacy, but as a Illatter of nature rather than of art. They may thri ve on fi lm but fail to nouri s h on the more ri gorous stage. It was in response to the present I)aradox that the various techniques for promoting inspiration i n the actor were developed . READINGS Diderot, Denis. The Paradox of Acting. Trans. W. H. Pollock. New York: Hill, 1957. Vartanian, Aram.

"Djderot's Rhetoric of Para d o x , Of, The Conscious

Automaton Observed.�'

Eighteenth-Centur)' Studies 14 ( 1 9 8 1 ) : 379-405.

I )i( '/

iOI/(II"_V of Pa radox

ADVERBIAL SAMARITAN, THE Seethe PARADOX OF GENTLE MURDER. ..

AESTHETICS, PARADOXES IN. Aesthetic pa rado xes con cern qu est i on s of the b eautif u l and other aesthetic effects, or pro b l ems in the p h i l o soph y of

A f ew of the varieties of paradox in pa inting migh t be mentioned here. There are paradoxicaJ posit i on s and juxtapositions of bodily palts in pai nt ing from an ci e nt Egypt to Cubism, the paradoxical effects of Tro mpe d'OeiL the p aradox i ca l geometries of the graph i cs of M. C. Es cher Agai n, there are Marcel Du cha m ps s paradoxical ass i gnment of the status of art obj ect on his ready-mades such as Founta i n the urinal tu rned masterpiece; and Jasper lo hns s American flags that play on a PERCEPTUAL PARADOXES such a s retinal art.

.

'

"

,"

'

exaustion. See the PARADOX OF ANTI-ART, the PARADOX oFAcrING, LITERARY PARADOXES, the PARADOX OF TASTE, the PARADOX OF TRAGEDY , and the PARAOOX

OFTHE UGLY.

ANALYSIS, PARADOX OF. Acco rdin g to G. E. Moore, the premi ses of philosophical analysis

that an infonnative ana l ysi s is

i mply

non-informative tautology. This

H. Langford

identica l to a paradox was a pparently first notic ed by C.

.

Formulation. "If you are to give an ana J ys is of a gi v en concept, which is the analysandun1, you must m ent i on as your analysans, a concep t such that: (a) nobody can know that the analysandum app l ie s to an objec t without knowing that the analysans app lies to it, (b ) nobody can verify that the analysandurn applies without verifying that the analysans applies (c) any e xpression which expresses the analysandum must be sy nonymo u s with any e x pre ssi o n which expresses the analysans" (Moore, 663). H e nce the meaning of "analysanduJn is alUllysans" is the same as "analysanduln is analysanduln. " Explanation. Co nside r the concept of' b eing a b rother. An adeq uate analysis '

'

,

,

,

'

of this concept, since it fulfils conditions (a)-(c) above, would

be

(1) To be a broth e r is to be a male sibling. Thus, according to Moore, b rothe r and 'male s i b l i n g have identical denotations and we cannot em ploy the concept 'brother' without al so kn ow ing that the concept ma l e si b l i ng app lies . Neither can we verify that 'brother' is applicabl e to any given object wi tho ut simuJtaneously ve rify ing that 'male si bli ng is also applicable, nor is there a s ynony m for the first term that i s not synonymous with the second. Thus, there seems to be no way to distin g ui s h the cog ni t i ve content of t he aforementioned analysis from that of '

'

'

'

'

'

9

Dictionary of Paradox

(2) To be a brother is to be a brother. The problelll is not that the two propositions are merely logically equivalent but that they have identical meanings. The second proposition, however, is plainly not an analysis of the concept brother'. Resolution. Moore himself suggests that a statement of an analysis is partly about the concept being analyzed and partly about the verbal expressions used to express it� but he was never wholly satisfied with this response because he could not see how the statement of an analysis could be about the expressions rather than about the concept being analyzed. Max Black argues that, whereas proposition (2) above is an identity, proposition (]) is not an identity but a three-termed relation. AccordingJy� the two propositions cannot be the same. Morton G. White, however, contends that Black�s proposed three-termed relation is really an identity after all and thus Black's analysis does not dissolve the paradox. Ernest Sosa argues that from the point of view of Fregean semantics the terms "brother' and 'male sibling' are co-designative but not synonymous; thus Sosa's proposed solution seems to contradict Moore's premise (c). SeelningJy harking back to Black's position, but explicitly invoking a Kantian viewpoint, T. W. Schick equates the analysis of a concept with its decomposition into component concepts, thereby making explicit what was only implicit in the original concept. Again, Richard A. Fumelion denies that the psychologicaJ significance of proposition (1), as cOlnpared to the triviality of proposition (2), is sufficient reason for concluding that the two propositions differ in meaning. READINGS Ackennan� Di ana F. "The Informa ti veness of Philosophical Ana lysi s. " Midwest Studies in Philosoph.Y 6 ( ] 981): 313-320. - - . "Two Pa rado xes of Anal ysi s." Journal of Philosophy 78 (] 981):

733-735. Anderson, C. Anthony. Philosophical

"B ealer's 'Quality and Concep t'." Journal of

Logic 16 (1987):

l] 5-164.

Bealer, George. "'Remark s on Classical Analy sis. " Journal of Philosophy 80 (1983): 7 L 1-712. - -. Quality and Concept. O xford: Cla rendon, 1982. 69-77. Mind ns 53 (J 944): 263-267. B l ack, Ma x. "The 'Paradox of Anal ysis. " A Reply. Mind ns 54 (1945): - - . "The ' Pa radox of Anal ysis Again: ".

272-273. Fumerton, Ri chard A. "'T he Paradox of Analy si s. " Philosophy and

Phenolnenological Research 43 (1983): 477-498. Langford, C. H. '"Moore's Notion of Anal ysi s." The Philosophy of C. E. Moore. Ed. P. A. Schjlpp. Evanston: Northwestern UP, 1942. 319-342. Moore , G. E. "Repl y to My C riti cs." The Ph ilosophy of G E. Mo ore . Ed. .

10

Dictionary of Paradox

P. A. Schilpp. Evanston: Northwestern lJP, 1 942 660-667. .

Schick, T. W., Jr. "Kant,Analyticity, and the Paradox of Analysis."

Idealistic

Studies 16 (1986): 125-131. Sellars, Wilfrid.

"The Paradox of Analysis: A Neo-Fregean Approach."

Ph ilosophical Papers.

1967. Journal of Ph ilosophy 80 (1983): 695-

Springfield, IL: Thomas,

Sosa, Ernest. "Classical Analysis."

710. Encyclopedia of Philosophy. Ed. Paul Edwards. New York: Macmillan, 1967. Vol. 1, 97-105. White, A. R. Moore: A Critical Erposifion. Oxford, 1958. White, Morton G. "A Note on the "'Paradox of Analysis." Alind ns 54 (1945): 7 1 -72. --. "Analysis and Identit y : A Rejoi nder." Mi n d ns 54 (1945): 357-361. Weitz, Morris.

"'Analysis, Philosophical."

Zalta, Edward N. "Meinongian Type Theory and Its Applications." Studies in Logic 41 (1982): 297-307.

ANSCOMBE 'S PARADOX. First noted by G. E. M. A n scoln be in 1976, this i s one of the PARADOXES OF VOTING. Formulation A set of pro p osal s selected by simple majority rule may result in outcomes with w h ic h a majority of voters di sagree in a m ajo r i ty of cases. Hence the m aj o rity may be frustrated by the majority'S will being fu l filled. Explanation. The fol l o w i n g chart of fi v e v ot ers d el i ber ati ng on three proposa l s i l l ustrates the po s s i bi J ity j ust id entified ( b ased on Gorman, 46), ..

I 1 2

Voters

3

4 5

p

R

0 P 0

S

A

L

I

S

A

B

C

Yes No No Yes Yes

Yes

No

No

No

Yes

Yes Yes Yes

No No

A and C are both to be implemented si nce a majority of voters approves

them. Yet the first three voters� who are al so a majority, di sagree with the results in a m aj o rity of cases: Voter One disagrees with the outcomes on proposal s Band C; Voter Two with those on the proposal s A and C; and Voter Three w ith those on proposaJ s A and B. Resolution.. The paradox cannot ari se w hen only a sin gl e proposal is c o n s i d ered . Whe n a seri e s of proposals (not necessari ly voted on simultaneously) are considered, however� the paradox may occur either

11

Dictionary of Paradox

fortuitous result or as the result of deliberate manipulation by a CCtyrant." In either case, the cumulative effect of the i mplemented proposals Illay well be a society repugnant to the majority of voters. M. P. T. Leahy 's that the total number of satisfied desires wil l always be in objection 1 he majority i s irrel evant to thi s poi nt. Consequently, two plausibJ e that it i s the best mode of decision-maki ng arguments for democracy because it i nsures the sati sfacti on of the wilJ of the majority and that it i s the fairest because it gives each person 's desires equal wei ght are lIndermi ned by the paradox. Carl Wagner poi nts out that the paradox does not ari se when the prevai l i ng vote i s at least three-fourths of the electorate on each proposal .

as a

READINGS

Anscombe, G. E. M. "On the Frustration of the Majority by Fulfillment of the Maj ority s Will." '

Analysis 36.4 (1976): 1 6 1- 168.

Gorman, J. L. "A Problem of the Justification of Democracy."

Analysis

38. 1 (1978): 46-50. Leahy, M. P. T. "Lies, Damned Lies, and Miss Anscombe.�' Analysis 3 7.2 ( 1 977): 80-81. Wagner, Carl. An s com be s Paradox and the RuJe of Three- Fourths." Theory and Decision 15 ( 1 983): 303-308. "

'

ANTI .. ART, THE PARADOX OF. Anti-art was the fundamental concept of

the Dada movement, w h i c h represe nted a radi cal repudiation of bourgeois art and cultu re . T he more recent Pop Art and Ki netic Art Inovements al so began as anti-art. Yet however shocking and destructi ve of traditional art anti-art i s, the i ne vitable historical paradox has caught up with it and now anti -art i s as cel ebrated, col lectable and integrated into the tradition of bourgeoi s culture as any other arti stic tendency.

ANTINOMIANISM. T hi s doctrine holds that Chri stians are not bound

obey the law of God, especial ly as represented in the Old Testament l egal system, but may continue i n sin so that divine grace, that is, God's forgi veness of sin, may abound. The term was first appl ied by Marti n Luther to John Agricola, and gi ven to a Chri stian sect that appeared i n Germany about 1 535, but the argument was put forward as a rationale for Inora11icence si nce early Chri stian times. In coloni al Massachusetts, Anne llutchi n son l ed anti nomians i n a controversy with John Wi n throp_ Antinomian doctrine has been seen in the sixteenth-century legend of the Ini nnesi nger Tannhauser, which Wagner made into an opera. Tannhauser

10

I

Dictionary of Paradox

12

spent a voluptuous year with Lady Venus at Venusburg , a magi cal land of sensuous deli ght accessibl e throug h a s ubterranean passage. Once he obtai ned perm i s s i on to le ave. he went strai ght to Pope Urban for absoJ ution; but the Pope refused� �'No. you can no more hope for atonement than thi s dry staff here can be expected to bud again. " Tannhauser left but three days later the staff burst into fu l l bJ oom. Urban sent in every direction to call Tannhallser bac k, but by that tilne the poet had returned to spend his rern aini ng days w i t h L,ady Venu s. The Iniracle remi nds the Pope that God's grace is not lilnited by the constraints of moral law.

ANTIN()MY.

An

anti nOlny

(ej reek

"'oppos i ng the l aw" ) is a logical contradiction. or extreJlle variety of paradox in w h ic h there is contradi ction betwe e n two p ri n cipl es or inferences each of w h i ch seems equa l l y necessary or reasonabl e but w h i c h can not both be true. A frequently occurring disti nction is that an antino111Y is inconsistent, whereas a paradox may be only seemingly so. See DUMITRIU'S ANTINOMY OF THE THEORY OF TYPES, KANT'S ANTINOMIES, and Ross's ANTINOMY.

AQUIS T'S PARADOXQ See the EPISTEMIC OBLIGATION PARADOX.

ARROW, THEG Menti oned by Aristotl e , Epi phanus, and D i ogenes

Laertius, thi s i s the third of Zeno's arguments agai nst motion. Formulation. "An arrow i n fl i ght is motionl ess because it is al ways occ upying a space equal to itse l f at each moment and because everything that occupies an equal space is at rest" (Ari stotle, Physics, 239b 5-7). Zeno �'argues thu s: what is moving moves either in the place in which it i s or in the pl ace in w hich it is not. And it moves neither in the place in w hich i t i s nor i n that w h i ch it i s not. T herefore nothi ng mov e s . " (Epi phanu s, Adversus Haereticos, 111.11; cited in Lear, n. l ). Explanation. Anyth i ng that occupies a space its own size i s at rest. An arrow i n flight, whi l e it is in flight, exists in a present moment. Yet i n that present moment the arrow occupies a space j ust its own size. Thus i n the present moment the arrow i s at rest. Yet that means that a movi ng arrow i s al so at rest, which i s absurd. Resolution. The present paradox is general ly consi dered to be predicated lIpon the assumption that ti me consi sts of n on-instantaneous atomic intervals. (For another view, see V l astos. ) Indeed, the paradox does not

13

Dictionary of Paradox

see m to ari se when time is conceived of as a successi on of instantaneous Inoments since movement i s dep e n de nt on ve l ocity. But ve loci t y is the rati o of distance (change i n pos it i o n ) to elapsed tilne and there is n o e l apsed time at a point l i k e moment. Hence it woul d seem that the very concepts, of rest and motion would n ot be applicabl e to an o bject at an instantaneous' moment. Jonathan Lear is probabl y correct in asse rting that the extension of the concept of velocity to i nstanta neous velocity by the concept of "]imit' i s irr elevant to the paradox. Ari stotle used the idea of velocity to argue that the concepts of rest and motion are n ot even applicable to atomic interval s si nce, by consideri ng obje c t s with different velociti es, it would be p o s s i b1e to divide the atomi c interva1. De n y i n g that motion i s inconsi stent w ith the atomi cit y of time, how ever woul d seem to be but a hol l ow vi ctory for, on the present view, that was Zeno's purpose anyway - moreover, Zeno is armed w ith other paradoxes purp o rting to show that inoti on is also inconsistent with instantaneous time. A possi bil ity apparently not di scus se d in the lite rature is that a ITIovi ng arrow might effecti vely fi ll a larger amount of space than an arrow at rest. ThllS� the paradox would fai l because it would contain a fal se (alnbi g uollS) prelnise; nevertheless, (n ot i o n and atomic ti me w ould not be contradictory. Lest thi s hypo thes i s be considered more paradoxi cal than the paradox itself, w e hasten to add that the Lorentz-Fitzgerald contracti on of Relativity Theory could be used to the same purpose . See ZENO'S PARADOXES. -

,

READINGS

Physics. Bk. VI, ch. IX. Hag er . Pa u l . "Russell and Zeno's Arrow Paradox." Russell 7 ( 1987) : 3- 1 0. Lear, Jonathan. HA Note on Zeno's Arrow." Phrollesis 26 (1981): 91-104. VJastos, Gregory. "A Note on Zeno's Arrow." Studies in Presocrafic Philosophy. Vol. 2. Ed. Allen, R. E., and D. 1. Furley. London: Routledge, 1974. 184-200. White, Michael 1. "The Spatial Arrow Paradox." Pac�fic Philosophical Quarterly 68 (1 987): 7 1-77.

Aristotle. ,

ARROW'S IMPOSSIBILITY T HEOREM. See ARROW'S PARADOX OF SOCJAL CHOICE.

ARROW'S PARADOX OF SOCIAL CHOICE economist Ke n n e th Arrow,

..

First proposed by the

Arrow's Paradox i s sOlnetimes called Arrow's Impossibility Theorem or the Voter's Paradox. Re1 ated to ANscoMBE's PARADOX� thi s paradox of social choice theory purports to show that no

14

Dictionary of Paradox

rational , democratic m echani sm i s avai l able for basi ng social pol i cy on ind ivi dual preferences. Formulation. Arrow proved that no mechani sm for instituting social choi ce on the basis of indi vidual preferences can simultaneousl y sati sfy the following four intu i ti vely pl ausible princi pIes: (1)

(2) (3) (4)

Collecti ve rati on a l i ty : the mechanism should be applicabl e to any 10gicaHy coherent finite set of individual preferences, o ver any fi ni te set of choices. Pareto pr inci ple: any preference unani mously shared by indiv idual v o ters sho uld be p re s e r v ed by the mechanism. Non-di ctato rship : a mechanism cannot ope rate so as a l w ay s to produce the choices of a single i nd i v i dual

.

Independence of ir relevant alternatives: only the i ndi vidual preferences with respect to the choices to be ord e red may be considered b y the mechanism.

Explanation. Arrow formal i zed pri nci p I e s (1 )-(4) u si ng fi rst order

predicate l ogi c and then deduced the consequence that a single i ndividual cannot determine the outcome of the social choice mechani s m for any gi ven pai r of alternatives without determini ng i t for all the choi ces and thereby being a dictator. But conditions ( 1), (2), and (4) i mply that some indi vidual must be deci si ve for some given pai r of choices. Thus, conditions ( 1 )-(4) are i nconsi stent. The following example captures the fJ avor of Arrow 's demonstration. Let V l' V 2 and V 3 be three voters who are to order the four soci al choices C1, C2, C3, and C4' Each orderi ng will be gi ven by {a, b, c, d}, w here each choice i s preferred to all those to the ri ght of it. Thi s notati on i s possible since the assumption of collecti ve rati onali ty i ncludes the provision that the orderings are logical ly wel l behaved ; in particular, the orderings are connected (gi ven any two choices, one is preferred to the other) and transiti ve (given any three choices, if the first i s preferred to the second and the second i s preferred to the thi rd, then the first i s preferred to the third). We now s how that any mechani sm sati sfying Arrow 's conditions will enforce one i ndi vidual's choice against those of aU the others for some pair of choi ces. By the pri nciple of coll ective rational ity, the mechanism should be applicable to the fol lowing preference orderi ng:

15

Dictionary of Paradox

Since C1 is preferred to C2 by al l the voters, t he Pareto pri nci ple demands that C be preferred to C2 by the social choi ce mechani sm . There are only 1 t wei ve possible orderings ranking C { to the l eft of C2 and for each one of these possible orderings there i s at least one voter whose choice for some pai r i s preserved w he n the others chose the opposite order. If the Inechani sm produces the order {C1, C3, C2, C4}, for example, then only V 1 agrees that C1 i s preferable to C4' The other eleven cases are similar and can be done by inspection. The result i s paradoxical because it i s hard to see how the apparently reasonable, and rel ati vely w eak, conditions (1 )- ( 4) are mutuall y incompati ble. l�esolution. Given the parad ox, a consi stent social choice mechani sm Blust give up one of Arrow 's four assumpti ons. S ince conditions (2) and (3) have seemed unassailable to most commentators, attention has focused on conditions ( 1 ) and (4). One possible attack on the principle of coll ective rationality i s that it cannot be taken for granted that indivi dual preferences are l ogi cally well behaved i n the requi site manner. In particular, AITOW'S proof depends on the transiti vity of the preference orderi ng. Another possi b i l i ty i s to deny that social preferences should be model ed on individual preferences, because the l atter have an i nherently psychol ogi cal character that need not be reflected on the social l evel . Arrow himself suggested that the best point of attack is conditi on (4): jf choices could be quantitati vely assessed in a particular way rather than nlereJ y ordered, the paradox could be avoi ded. READINGS A rro\\', Kenneth.

Social Choice and Individual Values. New H av en: Yale

UP, 1951; 2nd ed. 1963.

Davi s, Michae l . "Avoiding the Voter's Paradox Democratically."

Theory

and Decision 5 (1974): 295-311. MacKay, Al f re d F. Arrow's Theorem: The Paradox of Social Choice; A Case Study in the Philosophy of Economics. New Haven: Yale UP, 1980. Rusci ano, Frank Louis. Isolat ion and Paradox: Defining 'The Public' in Modern Political Analysis. New York: Green wood, ] 989. Tullock, Gordon. Comment."

"Av oi ding the Voter's Par adox Democratical1y:

Theory and Decision 6 (1975): 485-486.

Thi s is a case of the PARADOX OF NONEXISTENT OBJECT'S appl ied to fictional characters. S ee the PARADOX OF

AUTHOR'S PARADOX, THE. NEGATION.

READINGS Glannon, Walter. "The Author's Paradox."

28 (1988): 239- 247.

Br itish Journal Of Aesthetics

16

Dictionary of Paradox

Pollard, D. E. B. "Authors without Paradox."

British Journa l ofAesthetics

29 (1989): 363-366.

BA CKWARD INDU CTION PARADOX, THE.

Suppose that two

players are confronted with a finite number of PRISONER'S DILEMJ\1AS. By a backward induction., each player will defect in every round of play even though a cooperati ve strategy is lTIOre advantageous to both players. Philip Pettit and Robert Sugden, however, argue that the players are not in a position to make the backward induction because the initial assumptions of game theory - that the players are rational and at the start of the game beJieve that the other players are also rationaJ

"does not entitle him

r the first pJayer I to believe that in subsequent rounds his partner will still believe he is rational, irrespective of how he, the first player, has acted in the interim" (172). For a concrete example, see the CHAIN STORE PARADOX. READING

Pettit, Philip and Robert Sugden. "The Backward Induction Paradox."

The

Journal of Philosophy 86 (1989): 169-182.

BALD MAN, THE. See EUBUL1DES' PARADOXES and the SORITES.

BANA CH-TARSKY PARADOX, THE. One of a series of paradoxical

decoJnpositions in Ineasure theory� this paradox is related to the Hausdorlf Paradox and other similar resuJts. Formulation.

Any ball in R3 can be decomposed into a finite number of

pairwise disjoint subsets that can then be recombined to form two balls of the same radius as the original balL The result can be generalized to Rn for n�3. The paradox is frequently, but fancifully, given as "a pea rnay be taken apart into finitely many pieces that may be rearranged using rotations and translations to form a ball the size of the sun" (Wagon, 3-4). Explanation. Clearly, duplicating a ball or enlarging it are equivalent. In

either case, the ball is cut into a finite number of pieces and then these pieces are rearranged in such a way as to obtain a set with a greater measure ("volume") than the original baJJ. The actions pelformed in this process are limited to the group of isometries on the metric involved; that is, they are limited to one-one functions from the metric onto itself that preserve "distances (and, therefore, areas).

Intuitively� the pieces of the original

balJ are not stretched by the rearrangement. Although the paradox does

not arise in R2, a two dimensional example of a decomposition may be

l l

l,

Dictionar)) of Paradox

i

I

, , I

I

17

I I

!

• , • • ,

enl i ghteni ng. We can fi nd the area of a parallelogram, for exam p le by d ec om p os i n g it into two p i ece s and rea rrangi ng them to form a rectangle: ,

A i thou gh an an alo gou s p roce dur e

has been used since the time of ancient (ireece to find v ol u mes of solid fi gu res the present paradox shows that I he new fi gure may have a different vo] ume than the o ri g i n al ��e§olution. The proof of the Banach-Tarski Paradox de p ends on the ,

.

Axiom of Choice. This axiom al10ws one to form a new set by picking

an element from each set of a giv en collection of sets. A use f u l Illetaphor was adduced b y B ertrand Russell. Given a collection of pairs (11' s h o e s we can form a new set by picking th e left shoe of each pair. [f t he o ri gi nal collection were pairs of socks, howe ver the above rule would II

V3 V4 V,

Preferred Position

Y

X

X

X

Y

Y

Y

Y

Y

Y

Y

Y

y

y

y

row of t h e chart recor ds a voter 's preference on each issue an d the b allot cast by that voter. V l ' for e x a m ple pr efe r s p o s it i on x on i ssues [ I and 12 and posi t i on y on 1 3 ; he consequentl y votes for p a rt y X . The c hart show s that a majority of voters prefers p o s i ti on y on each of th e th ree i s s u es . Nevel1heless , pal1y X w i n s the e j e cti o n by a vot e of 3 to 2. Cl earl y, V 1 -V .s may be fi ve (equinumero us, or nearly so) groups of voters i n stead of fi ve i ndi viduals. Resolution. That configurations l i ke the one considered abo v e are possi ble is often taken to imply that democrati c p ro c e s ses are d ee p l y fl aw ed . Eac h

,

1 46

Dictionary of Paradox

READI NGS Rae , D. W. and H. Daudt. '"The Ostrogorski Paradox : A Peculiari ty of Compound Majority Decision . " European Journal of Political Research 4 ( 1 976 ) : 3 9 1 -398 . Shelley, Fred M. '" Notes on Ostrogorski ' s Paradox. " Theory and Decision 1 7 ( 1 984): 267-273 .

Paradoxi cal expre ssion s , such as s q u a re circ l e" o r " honest t h j ef�" are n ot c al J ed paradoxe s but rather oxy moron s , o r anti syzygi es. A lthough appearing contradi ctory, such phrases are, strictI y speaki ng, neither true nor fa1 se. Paradoxes, by contrast, admit of truth and fal sity. A paradox usually results when someone tri es, either d i rectly or i ndi rectly, to predicate one of these phrases of something, as in "A point is a square circle" or "He's an honest thief. " Observe that i n at least one sense "honest thi ef' i s not a contradiction because someone In i g h t make a profession of steal ing and yet always tell the truth ; even if stealing does n ot impl y lyi ng, however, the conj uncti on of the two terms may sti ) I be consi dered an oxymoron in that it i s., based on our expectations, an apparent contradiction. These expectati ons resu lt not only from our experience with thi eves, but al so from the sense of "di shonest" which includes thi every. OXYMORON.

"

PAIN, THE PARADOX OFo Bernard E. Rol lin adduces ten �'apparent

paradoxes" i n the medical sciences. T he follo\v ing i s perhaps the lnost interesti ng of these. Formulation. "Although a good deal of pain research has traditional J y been done on animal s, the scientifi c community has been l oath to affirm that one can know that animal s feel pain., rather than si mpJ y exhibit pain responses and mechani sms" ( Rol lin 2 1 1 ). Explanation. The resu lt i s not strictly contradictory since the feeling of p ai n may be associat ed with p ai n mechani sms in h umans, but not i n animal s even if the mechani sms themsel ves were identical . Nevertheless, this logi cal possibility is clearl y not a sati sfactory response since, were i t true, it would itself be paradoxicaJ to most people. Resolutiono The sci enti st i s supposed to l i mi t h i m self to physi cal principles and thus must remain agnostic about the subjective pain of ani mal s. Yet, as Rollin points out� the same wou l d apply to subjective human pai n (despite the fact that humans have J an g u a g e) Moreover, science i s supposed to be val ue-free and th us i gnores these subjecti ve .

Dictionary of Paradox

147

considerations unless forced to do so by s o ci etal pressures (which non-human animals are not in a position to app l y ) Therefore, argues Ro l li n science tends to invoke m e thodo logi cal fiats in order to avo i d moral questions that m i ght hinder business as usual . In p articu l ar " the ideology and value system of scientists contri bute to their very perception of pain (or J ack of s uc h perception) � and thus, such ideology and i ts attendant moral system (or lack thereof) has profound moral co nseq uen ce s (Rollin, 2 1 7 ) . ,

.

,

"

READING Rol1in, Bernard E. "Pain, Paradox, and Value." Bioethics 3 ( 1989): 2 1 1 -225.

PARADOX, DEFINITION OF. A ccording to the Oxford E n gl i sh

D i ct i o n a ry the word "paradox" i s used in several c l ose l y rel ated senses. For our purposes, only the two or three most i mpo rtant of these are relevant. As first sense, the OED gi ves: ,

A statement or tenet c on trary to recei ved opi ni on or expecta­ tion ; often with the implication that it is marvel l o us or j n ­ credible ; sometimes with unfavorable connotation, as bei ng discordant with what i s held to be establ ished truth, and hence absurd or fan ta s ti c ; sometimes with favorab1 e connotation , as a correcti on of vu] gar error.

The OED goes on to say that thi s sense is rare since the seventeenth century, but that l ater writers often i ns i st that thi s sense i s the proper one. Indeed, thi s i s the etymol ogi cal sen se of the w ord, which comes t hro u g h French an d Latin from the G-r eek para-, aga i n st p lus doxa, "the taken," hence "received o p ini o n, whi ch deri ves from an I ndo-European root meanin g "to take. " In Greek and Lati n the adj ecti ve was more common than the substanti ve� but i n French and Engli sh the substantive i s attested earlier and i s more i m po rta nt With regard to the d efi nition from the OED, w e would l ike to say that the question of w h e th e r the connotati on i s fa v o ra b l e or unfavorable i s pretty muc h irrel e vant, the p o i n t being rather that paradoxes are al l s o me w hat i ncredible. All paradox pa rt ake s of the character of the absurd, that i s to say that paradoxical utterances are those that tend to strike the hearers as ridicul ous, i n co n g ru o u s or u n reasonab l e . Paradoxes are i ncongruous statements i n that they are i nconsi stent w i th what is correct, proper or logical. In this first sense of parad o x , th e pa ra d o x stri kes the hearer as absurd beca u se it fl i e s i n the face of w hat i s cOll1 Inon l y accepted to be true. We would not general ly c l a s s i fy a statc lne nt as paradoxical, however, on the basi s of its abs urd ity a l one . l"' h ere s houl d also be some putatively cogent reason for acce pt i ng the asse rtion. T hus "

"

.

,"

,

Dictionary of Paradox

148

were w e to clai m, des p i t e the fact that everyone bel i eves the matter to be otherwi se, that the moon i s indeed mad e of green cheese , w e w o u l d be more li kely to be accused of uttering a falsehood th a n a paradox. If we were abl e to argue to thi s conclusion from, say, the di scovery of mil l i ons of mice on the moon ' s s urface, or should some space probe di scov er that the moon actual l y i s m ade of green cheese after a1 1 , then we would have a paradox. Of course, if e v e ry d ay people have no opi nion at aJ 1 about some thi ng there can be no question of c o n trad i cti n g their bel i efs, a nd i t suffi ces to Inake somethi ng a p arad o x that it stand i n opposition to learned opi nion. Thus i n A Budget oj'Paradoxes , the parad oxe s that Augustus De Morgan chronic1 es are cl ai ms to have achi eved the q uad rat ure of the circle or to have i nvented a perpetual moti on machine, technical ad vances about w hose i mpossi bil ity scientific gentlemen are convi nced, but which oth er people are prepared to a c c e p t if and w hen they are forthcomi ng. One of the most remarked paradoxes si nce the fal l of Byzanti um i s COPERNI CUS ' PA RA DOX . Cal l ing C o p ern i c u s o pi nion a paradox in thi s fi rst sense i ll u strate s that, w hen they are correcti ons of v ul gar error, such paradoxes may i ndeed be true despite thei r i n c ong ru o u s character. In a second sense of paradox, the OED has : ,

'

A statement or propo si ti on which on the face of it seems self­ c o nt rad i c t o ry, absurd, or at variance with common s e n s e , though , on investi gation or when exp l ained it may prove to be "veIl -founded (or, according to some, thoug h it i s essen­ ti al l y true).

It adds the foll owing sub-sense : Often applied to a statement or p rop os i ti o n that i s actuaJ ] y contradi ctory, o r contradi ctory to reason or ascertai ned truth, and so, essentially absurd and false ,

explai ning that som e hav e d eni ed statements to be paradoxes w hen t he y can be proved after aU to be true , or have cal l ed them ' app arent paradoxes ' when they are real paradoxes in sense 2.

The d ifferences between thi s second sense and the first are twofold. Fi r st in the the first sen se the i n c onsi s tency i s between the p arad ox and accepted opi ni o n whereas i n the second the paradox may be contrary to know n truth s or s i mply self- contrad ictory. The c J as s of paradoxes w hose appearance of absurd i ty l i es i n t h e poss i bi l ity of thei r c o n t ra d i cti n g themsel ves, which class i s recognized in the second sen se of paradox but ,

,

Dictionary of Paradox

149

not the fi r st i s cal l ed l ogical p a rado x . Second, whi l e i n th e fi rst sense a paradox cou l d be ei th e r true or fal se , in this second sense wri ters tend to differ about w h et h er p arad o x es are al ways true or al w ays fal se o r not a lways either one or the other. It seems to be a cl umsy use of the t e r m '" p a r ado x , however, to insi st that paradoxes are al ways t r u e or aJ w ays fa1 se. If al way s tr u e , p aradoxes are "crypto-truths" ; a nd if al ways fa l s e , �' p s e udo -t ruth s . " S o o n c e a g a i n . i t w o u J d s e e m rath e r t h at the para do x i cali t y of the par a d ox l i es essenti al l y i n th e q uestionabl e c h aracte r of its appearance, q ui t e apart from the rea l i ty of its t ruth val ue . We say thi s because if a statement i s too clearly fal se, i t would not be a p arado x Consider the classi cal example of an antil ogism, a triad of statements w here by if two are true the third can be seen as inconsistent, fal se or c on t r ad i c t ory , as i n the case of a s y llo g i s m i n w h i c h the stated c on cl u s ion i s the o p posi te of the vali d concl usion. I n the textbook an ti lo g i sm , " A l l men are m o rtal ; Socrates i s a man ; Socr a t es is i mmortaL" there i s noth i ng pa ra dox i c al because the faI J acy is too s t ra i gh tforw ard . T he statement i s self-contradi ctory but wi t h o u t anything that wo u l d l ead us to bel i eve otherw i se . A g ai n , when we use pa rado x o n l y i n th e case of true statelnents t h at seem fal se on the surface, we pl a c e the paradoxi cal ity in the wrong place, i n its success i n bei ng true, and not where i t would appear to b e long , n ame l y , in i t s success i n maki ng itself q uestionable. Let us l eave aside p arado x es that fl y in the face of a c cepted o p ini on � and consider only lo g i ca l p aradox e s � w h ose p u t ativ e falsity is a matter of their ap pe ari ng to contradict th e m s el v es. S uch parado x es would seem to be objects of th e science of log ic, which detenni nes q u e s ti on s of con s istency, and t h e te rm p a rado x" wouJ d seem to be a term of logi c I n truth, lo gi c does treat of paradoxes up to a p oi nt and " p arado x , " l ike A nt i nomy, "paralogi slTI," and so on, is so m et i mes treated as a techni cal term i n log i c Yet log i c is not th e o n l y sci ence that treats of logical parad o x es Prosody does as wel l , and rh e to r i c too. In both the se " li te rary s tudies ," p arado x is treated as a trope of thought rather than of s p e ech , e v e n th ou gh the trope i ne v i tab l y associ ates i tself with certai n fi g u re s of s p e e c h. T hat i s w h y log i c always has sorneth i ng to say about paradoxes. B ut even when l ogic has said its pi ece, the poet or the w ri te r sti J l must determine the use to w h i c h he mi ght p ut the pa rad o x . A g a i n , even if the statement is contra di c to ry, there are ci rC lI lll sta n c es i n w hich such s tat e m ent s are a p p ro pr iat e : to evoke a se n se of Iny ste ry, or path os , i n so phi sm, i n irony or hu m or T hat the OED i d e nti fi e s no se n s e in w hich p arad o x e s are al w ay s a m atte r of self-consi ste ncy i n d i cate s that logic i s only one of the special studies re le v ant to the c O l ll p re h e n s i o n of p ara d o x Al so re J e v ant are the studies of n at u re a n d of h U lnan society. "

"

"

.

"

.

.

.

.

.

150

Dictionary of Paradox

More important than any of these for the treatment of paradox w ould be rhetoric i n a robust (or medieval ) i nterpretati on . Rhetori c i n thi s sense does not restrict itself to the pol ite forms of speechifyi ng, but takes in the whole questi on of w hat w e say when, and i n so doing becomes the master science of l anguage. See also B RI DGE PARADOX , DILEMMA , OXYMORON, SOPHISM, and V ERI D I CA L AND FALS IDICAL PARADOX. READINGS

Col ie, Littell Paradoxia Epidemica. Princeton: Princeton U P, 1 966. De Morgan, A ugustus . A Budget afParadoxes. 2 v ol s. 2nd ed. Eel. David Eu g ene Smith. C hi c ag o : Open Court, 1 9 1 5 . Fa1 1 atta, Nichol as. The Paradoxicon. Garden City NY: Doubleday, 1 983 . Mi nton, Arthur J . , and Thomas J . Shipka, eds. Ph ilosophy: Paradox and Discovery. 2nd ed. New York: McGraw, 1 982. Sai nts bury. Paradoxes. Cambri d ge: Cambri dge UP, ( 1 988). .

,

P ARADOXICAL COLD . See PERCEPTUAL PARADOXES .

P ARADOXICAL HEAT . See PERCEPTUAL PARADOXES .

PAR'ADOXICAL SLEEP. Thi s term refers to the REM (rapid eye

movement) sleep in whi ch w e dream. Dream sJ eep is paradoxi cal becau se, whereas in normal sleep patterns of brain waves are rhythmic and subdued, i n dream sleep brain fi bers fi re rapidly and at random. Thus, dream sleep seems more akin to wakeful activity than to sl eep.

PASCAL'S WAGER. Invented by the ei ghteenth-century mathematic ian

Blai se Pascal , this argument takes the fonn of a dil emma for non-believers. Formulation. If God exi sts and one bel i eves i n Him, He will reward one's faith w ith etelllal happiness, and if He exists and one does not believe i n Him, He w i l l consi gn one 's soul to eternal damnation . If G od does not exi st and one beli eves i n Him, one loses l imited means inve sted i n an erroneous bel ief, and if He does not exist and one does not bel i eve, one has not lost these l imited means. In other words, the ri sk of finite means al l ow s one to win an i nfinite reward and the fai l ure to i nvest these fi nite means makes one liabl e to an i nfinite l oss. T herefore, it i s rationaJ to bel ieve i n G od even if the chances of His existence are very, very smal l � and onl y if i t is absol utel y certain that He does not exi st i s it rati onal not

151

Dictionary of Paradox

to b e l i eve i n H i m. Yet since there is no such absol ute c e rt ai nty one ought to bel ieve in God. Explanation. Pa s c al con structs a s i mp l e decision matri x. One can be l i e v e or not. If God exi sts, bel ievers win an i nfinite reward and non believe rs re c e i ve a n i nfi n i t e pu n i sh me nt . These payoffs are such that reason obl i ges o n e to risk the finite i n v e stm ent of b e l i ef even if t h ere i s no evi dence for God 's ex istence, so l on g as there i s at l e a s t a m i nimal (fi n ite ) chance for God to exi s1. Resolution . Three mai n o bj e ct i o n s are rai sed to the Wager. Fi rs t even if God exists He mi ght believe that rati on a l disbelief i s more to be rewarded than blind fai th, on the grounds that the pri nci pled atheist better employed the di v i n e gift of reason. Second, even if G od exists He might rew ard t he person whose l ife was more vi rtuous and b en e vol e nt than the person w ho Inere l y bel i eved . Third, and thi s i s the central o bjec ti o n from a spi r i tu a l po i n t of view, if o n e beli eves i n God on the b as i s of the l o gi c of Pascal ' s Wager, one has d i m i n i s h e d the value of on e ' s bel i ef so tn u c h that it is u n l i ke l y that it i s worth an y th in g . For to be l i e v e i n G od i n order to be rewarded i s to treat God in a calculati ng� e v e n Inercenary manner� the rewards of b e l i e f deri ve from loving and re specti ng Ci od . n o t from treating Hi m as a mere i n stru m e nt fo r o n e 's own advantage . Accordi ng t o Moses A . M ak i nde for exaln pl e � Pascal h i ln s e l f d i d not i ntend the Wager to be an a r g u m e nt for Inercenary bel i ef. but a n i ncentive for the nonbel i ever to begin to se e k for God . ,

, I

!

I I ,

,

I

,

,

,

R EADING S Duff, Antony. "Pascal 's Wager a nd Infinite Uti l i ties . " Analysis 46.2 ( 1 986) :

1 07- 109 .

-

Th e Hidden God: A Stud)' of the Trag ic Vision in th e Pensees of Pascal and tile Tragedies (�l Racine. Trans. Phili p Thody.

Gol dman, Lucien.

London: Routledge, 1 964. 28 3 -

109.

" Pa s ca l ' s Wa ger a n d the Athe i st ' s Di l em ma . " International Journal for Ph ilosophy (�l Religion 1 7 ( 1 985 ): 1 1 5 - 1 29.

M a k i nde , M o s e s A .

Pascal , Blai se. Thoughts. Trans. W. F. Trotter. Harv ard Clas sics ser. Ne\v York: Colli er, 1 9 1 0. Sec. 3 .

48.

PERCEPTUAL PARADOXES . There are a very l arg e ll u lll ber of perceptual i l l u s i o n s an d most of them have bee n tenned �'parado x i ca l " at one ti me or another. The n ature of the parado x i s not a l way s t h e s,une, h ow ever. S uch i l l usions seem to di vide thelnse l v e s i nto tw o Ina i n ki nds. One ki nd of i l l usory experience i s a re s u l t of d i stort i oll s i n the sti In u l u s rather th a n i n the percei ver. Som e vi sual i l l u s i o n s , fo r e xa nl p l e , are the

,

Dictionary afParadox

1 52

result of th e refracti on of l i g h t Th u s m i rage s , s u c h as the "pool s" of w ater that appear on a s p h a l t on a hot day, are co n s i d ered i l l usions , but th e se effects w o u l d be paradoxi cal onJ y if one w ere taken in by the j J l u s i o n . I n t h e same sense, rainbows a r e t h o ugh t to be i l l usi ons, though how we are taken i n by them is hard to say ( u n l e s s of c o u rs e one expects to fi nd a pot of goJ d at i ts e n d ) . A b ette r example of a paradoxi cal i l l us i on of t h i s ki nd i s the sti c k that appears b en t vvhen i t i s partly i mme rse d i n a pool of water, b e c a u se i n t h at case there i s a contradiction between what our eyes tel l u s (th at the sti ck i s bent) and w hat our hands tell us (that it i s not bent) . 'T h e re are v a r i o u s auditi ve i l l u si ons of thi s kind. The best known i s t h e Doppler effect , as seen i n the fact that the p i tc h of an automobile horn SeelTIS to d rop afte r it has p a s sed us and i s head i n g away from us. The wi nd also affe cts the pitch that we p erc e i v e T he p ara d o x here i s the c ontras t between the true and the a p p are nt pitch. A ga i n two nearby sounds of nearl y th e s am e fre q u e nc y can arrive at the ear in p h as e producing a combined l ouder sound. Other sti mulus-di storti on i l l us i o n s are the result of a p hy s i c a J apparatus, s uch as the eyeglasses that fl i p i m a g e s from top to bottom, or the e arp h o n e s that switch the sound s arri vi n g at t h e l eft and r i g h t ears. Such devices for r ev e rs i n g sti muli Inay b e di sori enti ngq but there i s l ittl e re a s o n to cal l them p aradox i c a L There is, however, a ni ce paradox of a btthroom c ab inet apparatus: .

�

,

.

,

,

Why does a mirror rev erse i mages from left to ri g ht but not fro m top to bottom?

The other mai n ki nd of experi ential i l l usion i s not a l re ady contai ned i n the i ncomi ng p e r c e p tual array, but ari s e s from the percei ver 's p e rcep tu a l apparatus. Opti c a1 iJJ usions, su c h as the figure-and-ground i l J usi on are good e x am pl e s I n the famous D uck-Rabbit i l l usi on, t h e fi gure appears fi rst as a d u ck and then as a rabbit, d epend i ng how one looks at it, but o n e cannot make the duck and the rab bi t appear to get h e r Other ambi gu o u s fi g u re s incl ud e the on e that ap pears to b e either two black faces i n pr ofi J e or a w hite vase , an d the Necker c ube that s w itches orientation as o n e o bserves i t. A m b i g u ous fi g ures of thi s sort are so s tri ki n g that i t i s hard not to appreciate their "paradoxical" c harac te r Sti mul i al s o affect the sen ses differently d e p end i n g u pon the i m med iatel y pre c ed i n g e x per i e nce s Thus if o ne stares long e n oug h at an American fl a g j n which the red, w hite and blue have been changed to g reen b lack and ye l l ow and then l ooks at a w h i t e wal L the true c o l o rs of the fl a g w i l l appear as after i,m ages on the s u rfac e . A g ai n , p e r c e p t i o n vari e s d e pendi n g upo n w h at i s b ei ng ,

.

.

.

.

,

�

1 53

Dictionary of Parado.'"

e x per i e n ced through the othe r senses (i n synesthesi a, i ntersensory ri v al ry, and i ntersensory facilitati on), or u pon the pe rce p t ua l conte x t (a gray paper appears l i ghter ag ainst a black bac kground than a w hite o n e ) . I n these case s the parad ox lies in th e contrast b e t wee n what i s experi enced and w h a t w oul d h ave b e e n expe ri e n ce d u n d e r no rm aJ c on d i ti o n s or circumstances . (N ote that Wittgenstein di scusses most of th e i ll u si on s mentioned in thi s paragraph at o ne pl ace or another. ) More clead y paradoxical i s the result of the experiment i n whi ch on e l eaves one hand i n col d water and the other i n warm water unti l th e y have become accll stolned to the temperature , and then puts them both i n a bowl of l ukewarm water. Here the paradox i s that the same sti m ul us ! p ro du c es conflicting sensations of w armth and col dness i n one ' s hand s. i It is i n co nj u n ct i o n with certai n tactil e i l l usi ons, however, that the term " p aradoxical" i s s ta n d ard J y appJied . The ski n contai ns numerous areas that are sensiti ve to either ho t or col d . S Olneti mes a warm stimulus w il l produce the sensation of col d when pl aced on an area sensiti ve to col d . Thi s occurrence i s termed the i l l usion of " parad oxi cal coJ d . � ' The i l l us i o n of "paradoxical heat" occurs w h en a warm sti lTI ltJ us affects warm and col d areas at the same ti Jlle; the In i x ture of wannth and " paradoxi cal cold" causes a n uneasy even p a i nfu l fe e l i ng of heat. One of the most i ntri gu i ng i 1 1 usi ons� both as percei v ed and i n t hi nk i n g it over, i s that moon appears l arger on the hori zon than it does hi gh i n the s k y Innumerabl e theories have been adduced to explai n this phenomenon, and t h e o r i sts are di v i ded that w hich arises from about which kinds of i J ] usory experi ence it i s the stimul u s or that whi ch ari ses from the percei ver. •

.

READING S Gombrich � Ernst H . Art and Illusion . N ew York : Pantheon , 1960 Robinson, Hel i er ] . " The Tw o Head Hypothesis and the Paradoxes of Perception . �' International Logic Revie�v 3 ( 1 972 ) : 99- 1 23 . Wittgenstein, Ludwi g. Philosophical Investigations. Trans. G. Anscolnbe. Oxford: B1 ackwel l , ] 95 3 .

PHILOTAS ' PARADOX. Plutarch rel ates the foll owing tal e, i n w hi c h Phi l otas use s a paradox to si l ence a b l ow hard ! "Phi l otas the p h y s i c i a n toJ d my gran dfath e r thi s tal e, and sai d moreover that it w as h i s c h a n ce Sh011i y after to serve the eJ dest son of t h e sai d A nton i us, w hOlll he h ad by h i s w ife Fu lvi a, and that he sat commonly at h i s tabl e w i th h i s oth er friend s, when he did not d i n e n or sup w ith his father. rt c h a n ced one day there came a phy s i cian that was so full of word s that h e Bl ad e e v e ry .nan weary of him at the board, but Philotas to stop h i s 1l1 o u t h p u t out a subtl e

..

154

Dictionary of Paradox

proposition to him . ' It i s g ood in some sOli to J et a man drink cold w ater that hath an ague : eve ry man that hath an ague hath it in some sort, ergo it i s good for a man that hath an ague to drink col d water. ' The physician was so gravel 1 ed an d amated withal that he had not a word Inore to say. Young A n to n i u s burst out i n s uch a-l aughing at hi m, and was so glad of it that he said unto hi m , Phi l ot a s take all that, I g i ve it thee, ' show i ng him h i s c up boa rd ful l of plate with great pots of gol d and s i l ver. " '

�

REA DIN GS

Plutarch of Chaeronea. The Lives of the Noble Grecians and Ro nUU l S. 2 vol s. Trans. Thomas North . Ne'rv York : Heritage, 1 94 1 ; 1 692- 1693.

PLATONIC PARADOXES. "If a pe rs on show s th at such thi n g s as w ood,

stone, and the l ike, bei ng many are also one, we adln it that he shows the coexi stence of the one and the many, but he does not s how that the many are one or the one m a n y ; he i s utte r i n g not a p ar ado x but a truism" (Plato, Parlnen ides , 1 29 ) . The re are many para do x e s i n Plato (BC c. 428-c . 248 ) ; fo r a few of the better known, see M ENO ' S PARA DOX , PLATO ' S PA RA DO X SOCRATlC PARADOXES � and the THI RD MAN . ,

READINGS Gri s\Nold, Charl es . "'Ph1 1osophy, Ed ucation, and Courage in Plato 's Laches. " Interpretation 1 4 ( 1 986): 1 77- 1 93 . Lentz , Tony M. "The Third Pl ac e from Truth: Plato ' s Paradoxi cal Attack on Wri ting. " Comlnun ication Quarterly 3 1 (Fall 1 983): 290- 3 0 1 . Mackenzie , Mary Margaret. " Putti n g the Cratylus in Its Pl ace. " Classical Quarterly n s 36 ( 1 986): 1 24-50. - - . "The Vi rtues of Socrati c I gnorance ." Classical Quarterl}' ns 3 8 ( 1 988): 3 3 1 -5 0. Ranki n , Kenneth . "The Dupli city of Plato' s Thi rd Man. H Mind ns ( 1 969). V l as to s , Gregory. Platonic Studies. Princeton: Princeton UP, n.d. Ch 2, 1 2.

PL.l\TO'S PARADOX. E. J . Lemmon att ri b utes th i s p arad o x also cal l ed the CON FLICf-OF-DuTY PA RA I�)X to Plato. In Lemmon's ve rsi on a neighbor ,

,

,

e n t rust s hi s fr i e n d with a gun after extracting his p r om i se to return it that eveni ng. A t the a pp o i n te d time, the n e i g h b o r demands the gun's ret u rn so that he c an ki l l h i s u nfaithfu l wife. The confl i ct of duties occurring i n thi s pa ra d o x can be resol ved by s u pp os i ng a hierarchy of moral values so that� i n the case at i ss u e the s a v i n g of human l ife takes precedence over keepi ng a prom i se. See, however, SARTRE' S PA RADOX for an oth e r Confl i ct­ of-Duty Paradox which i s not amenab le to thi s kind of resol ution. �

Dictionary of Paradox P O LITI CAL PARADOXES .

1 55

Pol i ti c al paradoxes occ u r w h en o u r

i n sti tuti ons of social re gul ati on gi ve ri se to confl i cti ng d uti e s or i mpossi ble si tuati ons for the c i ti zen or w hen those insti tutions i n h i bi t the very e nds S ee t h e PARADOX OF F UT U RE

that they w ere e stabl i s h e d to fac i l i tate .

G E N E R AT I O N S , t h e P A R A D O X OF I N D O CT R I N AT I O N , t h e P A R D O X O F

LI B ERTANIANISM � THE PA RADOX OF NATURAL RIGHTS � and S EN ' S PARADOX . Many pol iti cal pa rad ox e s Inay al so be categorized as ErHICAL PARA.DOXES ,

LEGAL PARADOXES o r PARADOXES OF V OTI NG .

POSSIBLE LIAR, THE Thi s paradox i s one of the many modal vari ants .

of the LIAR. The sente n ce

Thi s sentence is possi bly fal se. entai l s

con s iderations

e s sential l y s i m i lar to the PARADOX OF T H E PREFACE.

READING S

Post, John F. "Presupposition , B i valence an d the Possi ble Li ar. " Ph i/osoph ia 8 ( 1 979): 645-650. - - . "The Possi bl e Liar. NOlls 4 ( J 970): 4.05 -409. H

POTENTIAL ENERGY BARRIER PARADOX, THE. Fi rst descri bed by the Hi l l ary Putn am, thi s veri di c al paradox i s re lated to SCHROEDI NGER ' S

CAT.

Formulation.

" In qu a ntum p h y s i c s , i f w e u se the ( ori g i n a l ) B or n

i nte rpretati o n , then . . w e get (the fi gure .

random

i n the

10

p e r cent h as been i n s ert e d at

exampl e ) :

Every atom i n the popul ation P has the energy level e. 1 0 percent of the atoms i n the population P have v alues of D (the proton-electron separation \vhich exceeds d), Those statements are of course i n logical contradiction" ( Putnam 1 44).

Explanation. In c l a s s i c a l p hy si c s , the total energy of a systern i s equal to the sum of the potenti al energy of the sy stem and t h e ki neti c energy of the sy ste m. In the e x amp l e g i v e n by Putn am, eve ry atom is aSSll ll1ed to h av e

total e n e r g y

e.

T h e p o te n t i a l energy i s a functi on of t h e d i stance bet w ee n

t h e proton and the e l ectron ; the p ot e nti a ] energy i ncrease s as t h e d i stance i nc reases .

N ow, for ten per cent of the ato m s . the d i stance betw een the

proton and the el ect ron is suffi ci entl y l arge s o a s to Bl a ke t he pote nti al ene rgy gre ater than e . B ut, si nce the ki neti c e n ergy In t l s t be greater than

,

Dictionary of Parado.x

1 56

or

e q u a l to zero � the total energy

of

these atoms i s al so greater tha n e .

the tot a l energy of these atoms is both equal to e an d greater than e, w h i c h i s contradi ctory. Resolution. One reso) uti o n i s to abandon q uantum mech anics a l to geth er not a l ik e l y move in the absence of a l egiti mate com p eti n g theory. Putnam su ggests that a more plausi bl e re sol uti on w o u l d res ult by dropping w hat he calls th e pr i n c i p l e of no di sturbance. " T h i s pri n ci p l e affirms that that i s , the "measurement does not di sturb the observable measured observ ab l e has a hn o s t the same value an i nstant b efore the meas u rement as i t do e s at t h e Inoment the measurement i s taken" ( 1 3 8 ) . By d ro p p i ng t h e p r i n c i p l e of no di sturbance , the ori g i n a l conditions of the p r o bl e m be c o m e

Thus,

�

"

If an energy m e as u rem en t i s made on any atom i n P, then the value e i s obtai ned . If a D-measurement i s made on any atom in P, in ten percent of the cases a val ue greater than d \v i 11 be obtained ( 1 1 4) .

S i n c e a c c o rd i ng to quantum mechanics, measurements of the total energy and the d i stanc e cannot be made Si lTIul taneou s l y, th i s formul ati on i s con s i stent. The p ri n ci p l e of no d i s t u r b an c e i s i nt e g r a l to the B orn i n t e r p r e t at i o n of q u an t u m m e c h a n i c s , b u t n o t to t h e s u b s e q u e n t Copenhagen i nterpretati on. On th e l atter i nterpretati on, the fa ct that some of the atoms have a proton-el ectron s e p arat i o n g reater than d does not mean that they h a d a p p ro x i m at e l y t h e s a m e s e p a ra t i o n before the measurement w a s made a n d , hence , no contradi ction resul ts. Anothe r approach to the p aradox attempts to preserve t h e B orn i nterpretation by i n t ro du c ing the assumpti on of hidden-valiables. Arthur Fine, for example, suggests that energy m ea s u rement s on atoms are possible on l y for atolllS w here the s e p ar at i o n of proton and e l e c tr on i s l e s s than d ( perhaps due to a l i mitati on of energy measuri ng devi ces). If thi s be so, even g r anti n g that di stance can be ac c u r at e ly measured, in measuring t h e energy none of those atoms with s ep a rat i o n greater than d w i l l regi ster on the device and no contradiction ari ses. ,

REA DING Putnam, Hillary. " A Phi losopher Looks at Quantum Mechanics ." In Robert

G. Colodny� ed. , Beyond the Edge of Certainty. Engl ewood Cliffs , NJ : Prentice-H all , 1 96 5 . 75- 1 0 1 .

PRAGMATIC PARADOXES . A s Roy S orensen p o i n ts out , t here are v ari o u s p ra g m a ti c

p ara do x e s

.

We shal l examine

on l y tw o

vari eties:

Dictionary oj Paradox

157

paradoxical questions and p a ra doxic a l imperatives. Formulation. Pairs of paradoxicaJ questions a n d answers incl ude the foHowing: HAre you "Are you "Are you "Are you

there?"

asleep?" ali ve?" mute?"

"Are you deaf?,� "Can you hear meT' " Do you speak any English?"

"No� I am not here. " "V J es� I am as 1 eep. " "N-o, I am dead. " "'Yes, I am mute. " "Yes, I am deaf." "No, I cannot hear you. " '"No, I do not speak any Engl ish. "

Paradoxical i mperati ves are of thi s sort: A s i gn that reads : H Ignore this si gn . " A note that says only: "'Do not read thi s note."

" Don' t listen to thi s utterance. " A posted bil l that reads : "Post no bi l 1 s . "

Explanation.

the first case , th e quest i o n s seeIn to as k for either a n e g a ti v e or an affi rmative answer, but i n fa ct only one of th e s e an swers woul d make sense, b e cau s e one can not consi ste ntl y answer the questi on in t h e standard manner i f o ne i s not there, as l ee p , dead, deaf, d u m b, cannot hear, or not an ang l op hone. A n sw eri ng i n t he p a r adox i ca l tnanner i s often a witty, snappy o r sa rcast i c I n a n n e r of say i n g the opposite. U nl ike the questions, which ca n e l i c i t u s efu l i nfo r m at i o n the fi rst three example s of imperati ves not on l y h av e th e sce n t of paradox b ut tend to be pragmaticall y usel ess, unless they are m e a nt as sOlne ki nd of j oke The l ast exampl e of an i m perative paradox does fu ncti on to i nform peopl e of a pol icy of not p os ti n g bi B s i n a certai n pl ace, but i t does so by viol ati ng the policy in q u e s t i o n and mi ght stri ke peopl e a s a so m e ,,,, hat a b sur d manne r of i mplelnenti ng the poli cy in questi o n . Resolution. We sometimes avoid paradoxi cal responses by restating the questions in such a form t hat o n l y one res ponse is expected: "Tel l m e if you are there," or "Rai se yo u hand if y o u can hear me. " Someti mes we use a speci al fOlm of diction that aSSll ln e s t h e person cannot understand if he were to hear the more con venti onal form of the q u e sti o n : �' Do y o u savvy my lingo?" Such measures d o n ot guarantee su cc e s s however: does the other re a l l y not hear our st ate nl e n t or does he just not r a i s e hi s hand out of a perverse refusal to comply w ith our request? In

,

.

,

,

READING S Sorensen, Roy A. "Pragmatic Paradox Li abl e Quest. i on s." Pll i/osoj)h ica / Studies 3 9 ( 1 98 ] ): 1 5 5- 1 62. Cam pbell , C . A . H Common - S ense Pro p o s i t i o n s a n d P h i l oso ph i c al Paradoxes." Proceedings (�f the Aristotelian .)'ocicly 45 ( 1 945 ): 1 -26 .

1 58

Dictionary

of Paradox

PRECESSION OF THE EQUINOXES , THE PARADOX OFo The

confi guration of the ni ght sky was of consi derable i mportance to many peopJ es as a ti me -kee p i n g devi ce . 'T he re gul ari t y of astronomi cal movements was used in order to d eterm i ne the onset of spri ng and the consequent sow i n g of the fields. NaturaJly, not only most agricultural rhythms, but also most major ritual oc currences were determi ned by cel esti al signposts. T he regul arity of the yearly cycles was also considered evi dence for the stabi l i ty and wel l -being of the uni verse. Neverthel ess, the yearl y cyc l e, say as measured by the spri ng equinox i s not completely reg u l ar. That is, the once yearly occurrence of the spri ng e q uinox does not occur at e x actJ y the saIne spot in the sky each year. Thi s phenomenon i s known as the precessi on of the equi noxes. The difference from year to year is too smalJ to be noticed, but over time the famil iar celestial si gnposts began to lose thei r s i gnifi cance. T he Pl eiades, for exampl e, once marked the spri ng equinox� but due to the precession they seemed to sl i p ever l ower i nto the hori z on like someone w ho was fal l i ng into the sea or i nto the sun's glow. This was indeed a wondrously paradoxi cal occurrence, full of foreboding for the stability of the uni verse. Early astronorners such as H i pparchus (who reputedly d i scovere d the phenomenon of the precessi on) resolved thi s paradoxical situation by positing a "great year," the per i od i n which the precession would Inake a ful l cycle in the sky_ Earl ier resol uti ons, how ever, are found i n mythology as chan ges i n epoches. The sl ayi n g of the Mi thraic bull , for e x amp l e corresponded to the constel lation Taurus los i ng i ts place as regent of the spring equ i no x . Again, at about the be g i n n i n g of the Chri sti an era Pi sces became the new regent of this equinox and thus t he fish symbol i zes a new Christian era which many expect to culminate in an apocal yptical new age of Aquari us. ,

,

R EA DING Worthen, Thomas D. The Myth

1 99 1 .

of Replacelnent.

Tucson: U of Ari zona P,

P REDICTION PARADOX, THE. The predi cti on paradox ari ses when

the very abi l i ty to predi ct the occurrence or n on-occurrence of an event makes the reali zati on of that event i mpossi ble. Some of the most well known versions of the paradox are HOLLI S ' S PARADOX , the Unexpected Hanging, the Surpri se Air Raid Dri l l , and the SURPRISE QUIZ. In the S urpri se Qui z, for example, a teacher announces that he wi l l give a surpri se qui z next week. The student s know that it can not be on Friday, the last day of the week, because then it woul d not be a surprise. Thus, Thursday becomes the last d ay that it cou l d b e gi ven and, thus, if the teacher waits unti l

Dictionary oj Paradox

T h ursday, it will agai n be

no

159

Sim i l arly, each day can be e l i m i nated and the qui z can never be gi ven . S ee the the CHAIN STORE PARADOX � and the PRISONER' S DI LEMMA . s u r p ri

se

.

of the w eek B OTT LE I M P�

REA DING ,

'

A l exander Pete r � Pragmati c Paradoxes . " Mind ns 59 ( 1 950): 5 36-53 8 . Austin, A . K. "The U n e xpected Examination. " Ana(vsis 39. 1 ( 1 979): 63 -64. B l i nkJ ey, R. '"The S u rpri se Ex aminati on i n M odal Logi c . " Journal of Ph ilosophy 65 ( 1 968): 1 27- 1 36. B u rge, Tyler. " B uri dan and Ep i s tem i c Paradox . " Philosophical Studies 3 4 ( 1 978) : 2 1 -3 5 . Cohen L. J . " Mr O ' Connor 's ' Pragmatic Para d oxes ' " Mind ns 5 9 ( 1 95 0) : 8 5 -87. ,

.

.

.

Ferguson, Kenneth.

"Equivocati on in the Surprise Exam Paradox . "

The

Southern Journal of Philosophy 29 ( 1 99 1 ) : 29 1 -3 02 . Hol tzman , Jack M. "An

Undeci dabl e A s p ect of the Unexpected H a n gi n g Probl em." Ph ilosopiz ia 1 7 ( 1 987) : 1 95- 1 98 . - - . " S c h oedi nge r s Cat and the Un e x pe cte d H a n gi n g Paradox." British Journal /or the Ph ilosopizy of Sciellce 3 9 ( J 988 ) : 397-40 I . '

Janaway, Chri stophe r

.

" Knowi n g about S u rpri ses : A S u pposed A n ti nolny

Revi sited. " Mind ns 98 ( J 98 9): 3 9 ] -409. Kapl an, David, and Richard Montague . "A Paradox

R ega i ned . Notre Dallie "

Journal of FOrlnal Logic 1 ( 1 980): 79-90. Ki rkham , Ri chard . "The Two Paradoxes of the Unexpected Exami nation . " Ph ilosoph ical Studies 49 ( 1 986) : 1 9-26. Loesser, J . G . "Th ree Perspe cti ve s on Sch bed inger ' s Cat. " A,nerican Journal of Physics 52. 1 2 ( 1 984): 1 08 9-93 . Margalit, A. , and M . B ar- Hi l l e1 . "Expecti n g the Unexpected. " Philosoph ia

1 3 ( 1 983 ) : 263-289. O' Connor, D. J . "'Pragmati c Paradoxes . Mind ns 57 ( 1 948 ): 3 5 8 - 3 59. Oli n , Dori s. "The Predi cti on Paradox Resol ved. " Ph ilosoph ical Studies 44 ( 1 98 3 ) : 225-234. - - . "The Predi cti on Parad ox: Re s o l v i n g Recal citrant Vari ati on s . " Australasian Journal of Ph ilosophy 64 ( 1 986): 1 8 J - 1 89. Qui ne, W. V. "On a S o - Cal l ed Paradox . " Mind n s 62 ( 1 95 3 ) : 65 -67. Scri v en Mi chael . "Paradoxical Announcements. " Mind ns 60 ( 1 95 1 ): 403 407. S ore nsen, Roy A. C ondit i onal B l i ndspots and the Knowl edge Sq lleezc .�' Australasian Journal of Ph ilosophy 62 ( 1984): 1 26- 1 35 . - - . "A S tren gthened Pre d i c a t i o n Paradox. " Ph ilosophica l Quarterly 3 6 ( ] 986): 5 04-5 1 3 . Di scussion: 38: 1 1 1 - 1 5 . - - . " B l i ndspotti ng and Ch oi c e Variati ons of the Pred i cti on Parado x . '� AlnericanPlz ilosoph ical Quarterly 23 ( 1 986): 3 3 7-52. Wei ss, Paul . "The Prediction Para d ox " Mind n s 6 J ( 1 95 2 ) : 265 -269. "

,

"'

.

Dictionary of Paradox

1 60

PREFACE, THE PARADOX OF THE. Thi s l ogical paradox was fi rst adduced by D. C. Maki nson in 1 965 . Formulation. "It i s customary for authors of academic books to incl ude in their prefaces statements such as thi s: '1 am indebted to . . . for their

' i nval uabl e hel p ; h o w e v e r, an y errors w h i ch rema i n are m y s o l e responsibility. ' Occasi onal l y an author will go further. Rather than say that if there are any mi stakes then h e i s respons ible for them, h e w i l l say that there will i nevitably be some mi stakes and he i s responsibl e for them . . . If he has already written other books, and received con'ections from readers and review ers, he may al so beli eve that not everything he has w ritten in his l atest book i s true. Hi s approach i s emi nently rati onal ; he has l earnt from experience . . . Yet [ si nce he believes of each of the asserti ons i n his book that it i s true 1 he i s hold ing l ogi caJ 1 y incompatible beliefs. . . The man i s being rati onal though i nconsi stent" (Makinson 205 ) . Explanation. The author beli eves that each assertion i n hi s book i s true ; nevertheless , he al so bel ieves that at l east one of these assertions is fal se. Hence, he hold s contradictory beliefs. Even w orse, we j ud ge the author to be acting ratj onally, n ot in spite of, but because of h i s inconsi stency ! Resolutiono Makinson sugge sts that the author 's beliefs about each statement (that it i s true) as wel1 as hi s bel ief that at l east one statement i s faJ se are all indi vidually rational beliefs, but that the set of all those beli efs i s not rational . S uch a posi tion seems to give up too much� however, because the auth or presumably does not intend hi s book to be a mere coll ection of i ndependent assertions , but a coherent " positi on" on the question addressed . Bernard Linsky suggests that the bel ief that at l east one of the assel1ions i s fal se i s a meta-beli ef� so that a type theory would resolve the paradox. Li nsky 's suggestion is perhaps strengthened by the observation that there i s no one asserti on which the author bel i eves to be both true and fal se. I ndeed, he seems ready to abandon hi s beJ ief in any one of the statements should futu re events warrant it. Alternatively, we mi ght accept the paradox and claim that the author spoke too rashly : perhaps h e shoul d onl y affi rm the possi bil i ty of error. A . M. MacIver, however, has adduced similar paradoxes i nvolving modal ities. Other versions of the paradox are formulated in terms of rational acceptance using probabi l i ty theory. READING S Laraudogoi ti a, Jose Perez. "A Doxastic Paradox . " Analys is 50. 1 ( 1 990 ) : 47-48 . Linsky, B ernard .

"Facti ves, B lindspots and Sonle Parad oxes . l '

46. 1 ( 1 986): 1 0- 1 5 .

Analvsis v

1 61

Dictionary of Paradox

Maclver, A. M. " ' How Can I T hink it Possible that i Mi ght be Mi staken? ' " Ana lysis 17.2 ( 1 956): 25-30. Makinson , D. C.

205-207 .

"The Paradox of the Preface . "

Mos er, Pau l K . , and Je ffrey Tl umak . Acceptance. "

Analvsis 25 .6 ( 1 96 5 ) : �

"'Tw o Paradoxes of Rati onal

Erkelt ntnis 23 ( 1 985 ): 1 27- 1 42.

Poll ock, John L. "The Paradox of the Preface. "

Philosophy of Science 5 3

( 1 986) : 246-258 .

MATT E R, TH E PARADOX OF.. Thi s paradox purports to show the i ncoherence of Ari stotle ' s d o c tri n e of prime matter. Formulation . Pri me n1 at ter i s real b e ca u s e it i s the s u bstratum for elemental ch an g e . Nev e rthe ] ess, pri me matter has no c h a racter i st i c s of its ow n and, hence , i s n ot real . Explanation� AJ I change , for Aristot l e , re q u i re s a s u bsta nce t hat persists throughout the process!, whi c h i s understood as ·a n a l teration of some of its acci dental q u aJ i t i e s . T h u s, a statue of E u c l i d c a n be rncl ted down an d recast i nto a statue of A fc h i ln edes. I n t h i s c x a r n p l c , i t i s t h e bronze that is the substratum to which the c h a nge occ u rs . I n e l cIllc nla l c h a n ge , however, there i s a prob l e m s i nc� th e c l C lll e n ts ( fi rc , a i r, wate r, a n d earth) , whic h are transformed into o o e a n ot h e r, a rc not cOln posed of anythi ng more basi c t han themselves. �rh ll S, t here SeC ITI S to be n o s u b sta n c e that suppo11s , for exam p l e , the e v a p ora t i on o r water i nto ai r. A ri stotJ e, therefore, posits the exi stence of an alTI Of p ho liS stufl w i thout a n y characteri sti cs of its own to support e l eme ntal change. ' [' h i s i s pri lne rnatter. If it i s to serve as the sub st r at e of elemental c h ange, pri I n c (natter In u st be a substance and , hence, a real thing. But a l l s ll bs� a l lcc , for A ri stotle, i s d etermi nate. Si n c e prime Inatter, however ha s a bsol u t e l y 11 0 d etermi nations, it cannot be a substance, nor can i t be a re al t h i ng. Resolution. Ari stotle's resp on se w a s t h at si nce the elements have pai r s of characteri stics, one of these [TI LI st rC llla i n the same d u ri ng the c hange . Thus , when the cold" w et element i s tra n sformed into the warm we t element, a lum p of wet p ri me matte r i s the su bstratum for the change . Daniel W. G raham, h ow e v e r argues t h at t h e pe rsi stence of the w e t ne s s i s not relevant and thus Ari stotle 's s o l u t i o n fa i l s . Sheldon Cohe n sugg e sts that pri me In att e r coul d be e ssentially exten ded . Thi s would i ndee d re solve t he para d o x . G rahaln � however. poi nts out that AristotJe expl ici tly argues agai n st t h i s possi hi I i ty a nd l ocates the so u rc e of the p a r ad o x iIl Ari sto tle 's confl i cti n g c r i t e r i a of rea l i ty : On t h e one hand reality i s a fu nc ti on of determi n a cy a n d c o n c rete ne s s : to be a P RI ME

,

1 62

Dictionary of Paradox

' this,' a particular thing. On the other hand real i ty consi sts i n being a subj ect for predi cation, but never a predicate. As one approaches the l i mi ts of being i n descendi n g through the chai n of bei ng to si mpl e substance, the substances become more real or at least no l ess real as subjects ; at the same time they become less rea] as detenninate particulars . At the point where one meets prime matter the divergence has become complete. Prime matter i s both an ulti mately real substratum and an ulti matel y unreal particular (489) . READING S Cohen, Sheldon. " Aristotle's Doctrine of Material Substrate." Philosophical

Review 93 ( 1 984): 1 7 1 - 1 94. Graham , Danie1 W. "The Paradox of Prime Matter. " Journal o/the History of Philosophy 25 ( 1 987): 47 5 490 Sokolowski, Robert. " Matter, Elements and S ubstance i n Aristotle." Journal

-

.

of the History of Ph ilosophy 8 ( 1 970): 263 -277.

P RI O R ' S F A MI LY O F P A R A D O X E S .

\

)

B u i l d i n g o n c e rt a i n observations made by Alonzo Church, A . N . Prior adduced thi s set of paradoxes related to the LIAR. Formulation. "If i t is sai d by a Cretan that w hatever i s said by a Cretan i s not the case , then at l east two things are sai d-by-a-Cretan" ( Prior, 1 7). which is a Explanation. If the Cretan spoke the truth , his statement statement made by a Cretan must be fal se. If the Cretan spoke fal sely, no contradi cti on arises, but we must con c lude that something else i s said by a Cretan and that something el se must be true. Such a result is decidedly odd, however, because the Cretan ' s (fal se ! ) affirmation does not seem to be sufficient evi dence to conclude a substantive fact (that another Cretan affirmation must have been made). T he l atter w ould seem rather an empi ri cal result than a matter of l ogic. We certai nly seem to be abl e to i magine other scenari os. S uppose that only one Cretan ever l earned to talk and that he was very reticent, making only one assertion during his life. The paradox seems to i mply that it would be logical ly i mpossible for him to affirm "whatever i s sai d by a Cretan i s not the case . " (In fact, given the circumstance s as described, hi s statement woul d reduce to the LIAR) . Resolution. Pri or observes that the paradox i s a coroll ary of the obvious

trui sm that "If no fact i s asserted by a Cretan, then that very fact i s also not asserted by a Cretan." It i s hard to see how this observation ameliorates the paradoxical ity of the ori ginal statement. Nevertheless, Prior adduces

Dictionary of Paradox

1 63

further related examples that "appear as odd and unpleasant gaps" (20) i n our logi cal methods. I n discussing one of these examples, Tyler Burge contends that the paradox i s actually rooted in an ambi guity. J . L. Mackie. however, argues that the source of the paradox i s to be found in the fact that l ogic assumes that the Liar Paradox cannot occur. Thi s i s because if the Cretan were to affirm the utterance that logic says i s impossible (in the descri bed circumstances) , a formal contradiction wouJ d result. See al so MINIAC. READING S Burge, Tyler. "Epistemic Paradox. " Journal of Philosophy 8 1 ( 198 4): 5 -29. Mackie, J. L. Truth, Probability and Paradox: Studies in Philosophical Logic. Oxford: Clarendon, 1 973 . 276-285 . Prior, A. N. "On a Family of Paradoxes. " Notre Dame Journal of Fonnal La g ie 2 ( 1 96 1 ): 1 6-3 2 .

P RIOR ' S P ARADOXES OF DERIVE D OBLIGATION. PA RADOXES OF D EO NT I C LOG IC

para l l e l th e

T hese

PA RADOXES O F M AT ER I A L

I MPLICATION.

The fol l owing paradoxes were adduced by A .

Formulation.

(1) (2)

N.

Pri or:

A forbidden action makes all actions obl i gatory. Any action commits u s to what i s al re ady obli gatory.

J u st a s a fal se propo siti o n (materi a l l y ) i mpl i e s al l propositions, ( 1 ) states that a forbidden action makes any action obli gatory ; one wrong makes everything ri ght, so to speak. Agai n , j u st as a true proposi ti on i s (material ly) implied by any proposition, (2) states that if something i s obli gatory we are committed to it by any action at all even though it may seem irrelevant to the obl i gatory acti on. Prior derived the formal analogues of ( 1 ) and (2) in an early form of deonti c logi c. Resolution. The second paradox seems l ess pernicious; of the two i t is al so the more easily resolved. Georg Henri k von Wri ght suggests that we disallow si mpl e commitments. Thus no simpl e action woul d be obligatory of itself and the paradox cannot be form ulated. Von Wri ght resolves the first paradox by reformul ating commi tments as dyad i c obl i gation predi cates i n deontic logic.

Explanation.

READING S Pri or A . N . "The Paradoxes of Derived Obl igati on ." Mind ,

ns

6J ( 1 954):

64-65 . von Wright, Georg Henrik. An Essay in Deon tic Log ic and the General Theory of Action. Amsterdam: North- Hol l and , 1 968 . 76-78.

,

1 64

Dictionary of Paradox

In thi s clas s i c e xampl e of the PA RA DOXES Of GAME TH EORY , two pri soners must choose w hether to confess or not with variable payoffs. Form ulation. " If both prisoners confess, they w i l l both be convicted of their cri me but w i l l receive some l eniency i n l i ght of their confessions and recei ve 5 years i n pri son . If neither confesses, there w ill onl y be eno ugh evi dence to send each to pri son for one year. However, if one confesses w h i l e the other does not, the confessor w i J 1 be set free w hi J e the ' sucker ' recei ves tw enty years in pri son" (Sorensen 1 57) . Gi ven that nei ther pri soner knows w hat the other w i J I deci de, the most rational choice i s to confess. Explanation. Game theori sts tend to argue that confessing i s the rational strategy because i t m ax i m i z es one 's payoff. The fi rst pri soner reasons as fol l ows: PRISONE R ' S DILEMMA, T HE.

If the second prisoner confesses, I wi 1 1 get twenty years by not confessing, but only fi ve years by confessing . Hence , confessi ng is better in thi s case.

S imi l arl y, if the second pri soner d oes not confess, T wi ll get one year by not confes si ng , but I will go free by

confessing. Once agai n, confessing i s the better choice.

The second prisoner obviously reasons in a parallel fashion. Consequently, both pri soners confess. As Anatole Rapoport puts it� "the paradox i s t h at both l ose'� (5 1 ) . That i s, by if both pJ ayers Inake the rati onal choice, acti ng rationally and confessing� both get fi ve years ; if they had both acted irrational l y and refused to confess, they would both recei ve but a si n gl e year i n jai l. Yet, that i s not the end of the paradoxi cal ity of the present dilelnma, for Roy A . Sorensen presents an argument for the rati onal ity of not confessi ng. S i nce each pri soner knows that the other is rati onal , each know s that the other w i l l come to the same deci sion as he hiiTIse1 f does. Thus� they wiJ I either both confess or both not confess. Gi ven that these are the only two choices, it i s obviously better for both not to confess. Therefore, there are stron g arg uments for both the rationality of confessi ng and the rati onality of not confessi ng . Resolution. Perhaps t he most promi sing response to the present diJ em ma i s to use condi ti onal strategies to construct metagames. Ni gel Howard shows that two metagames are sufficie nt to determi ne j oi nt nonconfession as the rati onal choi ce� he al so shows that no further metagarnes wi l l alter thj s resu l t. We shoul d observe that the metagames should not be confused .

.

.

Dictionar:v of Parado.r

1 65

w i th the Iterated Pri sone r ' s Di l emma. In the l atter� the two pl ayers pl ay

the game vari o us ti mes i n a row ( t h e payoffs nl ay be regard ed as lTIOnetary rew ard s instead of years i n j ai l ) .

I n the

i terated versi on eac h p l ayer m u st

take i nto acc ount th e other p l ayer ' s pre vi o u s move s , b ut the u nde rJ yi n g re a s o n i n g i s othe rw i s e the same as i n

N EWCOMB ' S PARADOX .

the

s i m p l e vers i o n .

S e e a J so

READI N G S

A fneric(Jfl Davi es , Law rence. " Prisoners , Paradox � a nd R ati o n al i ty Philosophical Quarterl)} 1 4 ( 1 877): 3 1 9-3 27. Gordon . Dav i d. "Is the Prisoner 's Di lemma an Insolubl e Pro bl em ? lvtind ns 93 ( 1 984): 98- 1 00. G uiasu, Sil viu. P redicti o n Paradox Rev i s i ted. " Logical Analysis 30 ( ] 987): 1 47- 1 54. Howard , Ni geJ . T he Theory of Meta-Games . " General Systefns J 1 ( 1 966): 1 67- 1 86 . General S.'I,�steJJ1S I I ( 1 966) : - - . "'The M athemat i c s of Met a G am es 1 87-200. Le wi s , D. "Prisoner' s Di J em rna Is a Ne wco nl b Probl ern . " Ph ilosophy and Public Affairs 8 ( 1 979) : 235 ·-240. Ol i n , D ori s . " P redicti on s , Intentions and the P ri so ne r ' s Di l em rna. " Philosophical Quarterly 3 8 ( 1 98 8 ) : 1 1 1 - 1 1 6. Porter, Joseph Paul . " Rel evant Interest and the Pri s on er s Dilemma. " Mind 93 ( 1 984): 1 0 1 - 1 02. Rapoport, Anatol. '" Escape from Paradox." Scientific Anzerican 2 1 7 ( 1967) : 50-56. Snow, Pau l . "The Val ue of Infonnati on in Ne w c o m b s Probl em and the Prisoner 's Di lemma." Th e ory and Decision 1 8 ( 1 985 ): J 29- 1 3 3 . Sorensen, Roy A . "'The Iterated Versi ons of Newcomb 's Problem and the P ri son er s Dilemma. " S�v nthese 63 ( 1 98 5 ) : 1 57- 1 66. Mind ns 9 1 S teiner, Hi lle1 . " Prisoner 's Dil emma an Insoluble Pro bl em ( 1 982): 285-28 6. .

"

"

,

"

"

-

.

"

'

'

'

. "

PROBABILITY THEORY, PARADOXES B EAKER PARADOX , B ERTRAND ' S PA RADOX , the

OF�

PARDO X

See e s pe c i a l l y t h e OF I NF() R M ATI ()N � the

PA RA DOX OF I NDUCTI VE PROBABI LITY , the I NFIN IT E S ERIES PA R A DOX , the

PROBA B LE LIAR, S rMPSON ' S PARADOX and the ST. PETERSR l l R( , PA RA DOX .

PROBABLE LIAR, THE

G.

..

A vaIi ati on of the LIA R i ntrod uced

by Wi l l i aITI

Lycan .

Formulation. "T he probabil ity of the titl e of th i s paper, gi v e n i tself ( and

,

1 66

Dictionary of Paradox

the fact of its being a generalization), is less than 1/2. Yet the probability of any contingent statement given itself is 1 . So 1 is less than 1/2" (Lycan, 202). Explanation. The titl e of Lycan's paper i s "Most General i zations are Fal se. " Given that most general i zati on s are false, the probability that any given generalization i s true is less than 1 /2. The probabil ity i s here calculated by the frequency distribution of the favorable cases: s ince less than half of all generali zations are true (given the truth of the title of Lycan 's paper), the probability of any one of them being true is less than 1 /2. But the title of Lycan 's paper i s itself a generali zation and hence, given itself, must have probability less than 1 /2. Yet, if something i s true, i t i s true. lience, the probabi 1 ity of a statement, given itself, is 1 . Therefore, 1 i s the probabi lity of the title of Lycan's paper, given itse1f. And since thi s probability was already found to be less than 1 /2, 1 i s less than 1 /2 . Resolution. R. D. Boyd and S . K. Wertz argue that the paradox depends most generali zations are false on an equivocation. The first premise - i s , according to Boyd and Wertz, an esti mate and n ot the i ni ti a1 probabi1 ity that it purports to be. Thi s i s because initial probabilities range over single events and not general i zations. In contrast, the secon d premise i s a total probabi lity, - the probability of a statement, given itself, i s 1 describing a general situation. The tw o probabilities, therefore, cannot be d irectly compared. READING S Lycan, William G. "Most Generalizations are False." Pacific Philosophical Quarterly 65 ( 1 984) : 202. Boyd , R. D . and S . K. Wertz. "Probability and Lycan's Paradox." Southwest Ph ilosop hy Review 4 ( 1 988): 85.

PROBLEM OF EVIL, THE.

Traditionally, thi s paradox has played the

devil w ith Christian theology. Formulation. If God i s omni potent, omni scient and petfectly good, why i s there evi l in the w orld? Explanation. If God knows all , can do all , and sti l l lets evi l exi st, then i n what sense may he be characterized as having good w il l toward men? Resolution. There are several traditional responses to thi s question. That which enters into the mi nd of those who take the unnecessary suffering of the world to heart is often that there i s no God. Another answer i s that evi l i s the result of man 's fall from grace. He could have w ith stood temptati on and mai ntai ned hi mself i n an Edeni c state, but he exercised his free will and i s reapi ng the consequences. If it i s asked further why

167

Dictionary of Paradox

God gave man free w i l l , thus allow i ng his fal l , the answer i s that God mad e the world as penect, that i s, compl ete, as possi bl e, and the w orld i s more complete for having a being with free will such as man i n it. Thi s answer i s i nadequate because i t only explains the kinds of evi l that are a di rect result of man 's fall . Yet, if all evil i s explai ned in thi s manner, that i s to say if the w orl d 's evil comes to us by way of puni shment or test, then God does not seem particularly good after all , because either way, the torment i s excessive. Besides it i s the good rather than the deservi ng wicked who seem to suffer most i n this life. A more mystical answer to the questi on i s to deny that eviJ exists. It i s mere privati on, or distance from the fullness of God 's bei ng, and we suffer it because w e are, or became, distant from the divine wholeness. The problem with this response i s that i t i s whol ly counteri ntuitive that the pain and anguish we suffer is, from the divine standpoint, nothing at aB . Either He is constitutionally blind to our dolorous reali ty and hence not omniscient or el se Hi s poi nt of view i s so detached that there i s l ittJe of w hat w e mi ght recogni ze as "goodness" in it. More sati sfactory, perha ps , i s the M ani chean, and ultimately Zoroastrian, sol uti on to the effect that God i s not aU -powerfu l after al l , but competes with another G od , one of darkness, whose presence explains the worl d 's evi l. Thi s vi ew, banished in the West more by the sword than by argument, plays a si gnificant but surreptitious role i n those forms of vul gar Christianity i n whi ch the Devil i s a powerful ri val to God. The weight of occidental theological opinion, how ever, seems to be that a dual i sm of Jehovah and Satan i s too hi gh a price to pay for resolving the problem of evi l . •

READINGS Andre, S hane. "The Problem of Evi1 and the Paradox of Friendly Atheism .

"

International Journal/or Ph ilosophy of Religion 17 ( 1 985 ): 208-2 1 6. A n g l i n, B i ll , and Stewart Goetz. " Evil Is Privation.�' International Journal for Philosophy of Religion 1 3 ( 1 982): 3 1 2 . Clark, Kelly James. "Evil and Chri stian Belief. " International Philosophical Quarterly 29 ( 1 989): 175 - 1 89. Mackie, J . L. "Evil and Omni potence." Mind ns 64 ( 1 95 5 ) : 200-2 1 2. Wachterhauser, Brice R. "The Problem of Evi l and Moral Sceptic i sm. " International Journal for Philosophy of Religion 1 7 ( 1 985 ) : J 67- 1 74. -

Yandell , Keith E.

�'The Probl em of Evil and the Content of Moral i ty. H International Journalfor Philosophy of Religion 1 7 ( 1 985 ) : J 3 9- 1 65 .

Dictionary of Parado.,r

1 68

PROBLEM OF GOODNESSo Steven M . Cah n p o s e s the foll ow i n g probl em: "co u l d a worl d c o n ta i ni ng goo d n e ss hav e been c r e ate d b y an om n i pote n t omni sci ent, omn ima1 evolent being?" (69). T h e probi em� e x ac t ly p ara l l e l i n g the PROBLEM OF EV IL� p u rport s to sh ow that The Demon d oes not exi st. Cah n� l i ke Ed w a r d M_ad den an d Peter Hare b efo re h i m , c o n c l u de s that the co nj u n c t i o n of th e s e tw o p r o b l e m s m a ke s both d e lTIO n i s m and thei sm hi ghly i mprobabl e. One of the b as i c el ements in Cahn's argument i s that th e re i s a strict i s o m orp h i s m betw een theodicy and c ac o d ae m o n y (the demoni c a nalo g u e to t h e od i cy ) John King-Farl ow, however, c onte n d s that Cahn has m erel y demonstrated the i s o mo r p h i s m for one t h e o di c y and one cacod aemony ; s i nc e there are m any vari ati on s of t heodi cies, how ever, Cahn 's case i s not c o m p l e t e Ki ng-Farl ow al so a pp e a l s to PASCAL' S WAG ER and to the fact that th e r e has been a long hi story of p rofou nd n u mino u s experi ence s� w h e re as there has not been a s i m i J a rl y strong hi story of d i a bo l i c a l exp eri ence s . .

.

READIN G S Cahn , Stev en M . " Cacodaemony. �' Analys is 37.2 ( 1 977) : 69-73. Ki n g-Farl ow, John. "Cacodaemony an d Devi li sh Isomorphi sm ." Analysis 3 8 . l ( 1 978 ): 59-6 1 . Madden� Edward and Peter Hare. Evil a nd the Concept of God. Springfield, IL: Thomas , 1 968 .

PRODUCTIVITY PARADOX, THEo Despite s tre n u o u s efforts to i nc re a se productivity, many manufacturi ng concern s have e x p e r i e n c e d l i ttle return on t h i s effort and some have eve n e xpe ri e n ce d ne gat i v e re sults. Wi ckham S k i n n e r argues that ' ''the very w ay managers defi ne productivity i m p ro v e ment and the tool s they use to ac h i e v e it p ush their go al further out of reach" (56), th e reby e n g e n de ri n g the paradox. Accordi ng- to Skinner, the focus on co s t reducti on s, whic-h is the e s se n c e of m ost p ro d u ct i v i t y progra ms, i s harmful i n th at it h i n d e rs i nnovati on and c re at e s a need to i n sure effi c ie n cy in the workplace that both a lie nates the workers and i s excessively t i me con su m i n g for the m an age rs Av oi d i ng the pa rad ox ac c o rd i n g to Skinner, requi res the d ev e l o p m e nt and i mplementation of long - ra ng e manufacturi ng strategies. .

READING

Ski nner, Wi ckham . "The Productivity Paradox. '� Harvard Bus in ess Review 64 ( 1986): 5 5 -59.

,

Dictionary of Paradox PROMISING, THE

PARADOX OF.

1 69

T h i s paradox of ethi cal theory

w as i ntrod u ced by Ju l i a Dri ver.

Formulation. The fol l ow i n g three state ment s are i nd i v i d uaJ I y true yet j oi ntl y i nconsi stent :

( 1 ) When someone m ak e s a promise to do something, he thereby puts hi mself under an obli gation to do it. (2) If someone i s obli gated to do sOInething, he can do i t. In the phi losophical rubri c: ' " Ou ght impl ies can . (3 ) Peo ple someti mes make promi ses they cannot keep. H

Explanation. The fi rst state m ent i s unassai l abl e beca use i t j ust ex pl ai n s part of w hat i s i n volved i n the concept of pro mi si n g : i f one has promi sed to do something, one i s o bl i ged to do it. The second statement i s i ntuiti vely correct and has been arg ued for by a variety of philosophers, i ncl uding Kant� l-Iare and Von Fraasen.

And the third statement i s in l i n e with

common sense ; pol iti c i ans, for exampl e, are al w ay s be i n g charged w i th Inaki ng promi ses they cannot kee p . Neverthel ess � if prornj si n g p uts one under an obl i gation and one can not be u n der an obl i gati on unl e ss one can d o w hat one i s obl i ged to do� then one can not i n pri nc i p J e fll ake a pro ln i se

o ne cannot keep. Resolution .

S i nce the fi rst statement m ust stand , the paradox lTIllSt be

met by den y i n g either the second or the third statement and by e xpl ai n i ng w hy peopl e mi ght have found them com pel l i ng despi te their fal s e hood .

E. 1 . Lemmon rej ects the second statement.

One diffi culty w ith

(2) seems

to be that if ' ou ght impl i e s can' , the n, by the contrapositi ve , ' cannot impl i e s o ught not.

�

B ut j u st because , say, SOlneone i n a w heelchair cannot save a

drowni n g man , it see m s counteri ntu i ti ve to insist that he ought not save hi lTI. A more careful form ul ati on of the contrapositive, how ever, al levi ates the d i ffi c u Jty : ' cannot impl ies not obl i ged to ' (instead of ' obl i ged not to ' ) . Certai n l y, w e would not w ant to hol d that the person i n the w heel chai r w as obl i ged to refrai n from savi n g the drow ning man. Rather we wo uJ d o n l y cl aitll that he i s not obl i ged to save him. A . P Mal1ini c h

rej ects

the

thi rd statement, partly on the view that " pol i tician ' s promi ses" are not to be taken as real proln i ses unti l i nve sti gati on determi nes w hether they can be kept in the first p l ace.

U s i n g Gri c e �s theory about con ve rsati on �

Martin ich argues that (3 ) is intuiti vely pl ausible because it ce nters atte n tion on the promi ser ' s i ntentions rather than h i s i n abi l iti e s and o n e ' s i n te n t i on s are

thou ght to be c entral to the acti vity of promi si n g .

READING S

Dri ver� Ju1 i a. "Promi ses , Ob!igations� and Abi l iti es. " Phi/osoph ic(J/ S/udies 44

( 1 983):

22 1 -223 .

,

1 70

Dictionary of Paradox

Martinich, A . P. "A Solution of a Paradox of Promi sing . "

Philosophia 1 5

( 1 985): 1 1 7- 1 22. - - . " Obli gati ons , Ability and

Prilna Facie

Promisi n g . "

Philosoph ia 1 7

( 1 987): 3 23 -330. "A Resol ution of a Paradox of Prom i sing." Philosophia 1 7 ( 1 987): 77-83 .

Si nnott-Annstrong , Walter.

PROTAGORAS, THE. S ee the EUALTHUS .

PYGMALION E:FFECT, T HE.

See t h e

EXPERI MENTER EXPECTANCY

PARADOX.

RACE COURSE, THE. See the DICHOTOMY .

R AV E N ( S ) PA R A D O X , T H E ..

S e e H E M P E L ' S P A RA D O X E S O F

CONFIRMATION.

RELATIVISM, THE PARADOX OF See the PARADOXES OF COGNITIVE ..

RELATIVISM .

RE L I G I O US PA R A D O XE S .

paradoxe s d i ffe r fr om p h i l o s op hi c a l paradoxes insofar as religious paradoxes depend on mystical e x p er i e nce , scripture or o rthodo x systems of bel ief. One main cJ ass are those that ari se from the attri butes of God. There are hence paradoxes of OMNIPOT ENCE (such as the paradox of the S TONE) , of OMN ISCIENCE, of OMNIPRES ENCE AND T IMELES S NESS , of divine existence ( s uch as Ross ' s ANTIMONY), and of divi ne petfecton ( such as the PROBLEMS OF EV I L AND OF GOODN ESS ) . The m a i n class of paradoxes sp ec i fi c to Chr i st i an i ty are, of course, the Chri stol ogical paradoxes, such as the paradoxes of the TRINIT Y , the Immacul ate Concepti o n , the I NCARNATION, the Vi rgi n Birth, and of the R e s urr e ct i o n La C ro i x s ParADOx OF EDEN concerns the faJ l of man an d the paraclete appears i n DO N N E S PARADOX OF THE HOLY G HOST. Paul pronounces many p aradoxe s , particularly i n 2 Cori nth i an s as w hen he says, �'For when I am w eak, then I am stro ng Again, Chri stianity and .

Re l i g i ou s

'

'

,

.

"

Dictionary of Paradox

1 71

Judaism are founded on the paradox that the world will be saved by fail ure see the PARADOX OF THE FORTUNATE FALL. See also: ANTINOMIANIS�l , B ODHISATTVA PARADOX , B UDDHIST PARADOXES, the DEVIL ' S OFFER , NICHOLAS OF CUZA ' S COINCIDENCE OF OPPOSITES , PASCAL ' S WAGER, Ross ' s A NTIMONY , TERTULLIAN ' S PARADOX and the V IA NEGATIVA . READINGS Calhoun, Robert L. "The Language of Re li g i on . The Unity ofKnow ledge. Ed. LouisLeary. Garden City, NY: Doubleday, 1 955. 248-262. Ramsey, I . T. Paradox in Religion. In New Essays on Religious Language. Dall as M. High, ed. New York: Oxford UP, 1969. 138- 16 1 . S te ng e r 1\1 a r y Ann. "The S i g n i fi ca nc e of Paradox for T h e olog i c al Verification: Difficulties and Possibilities. " International Journal for Philosophy ojReligion 1 4.3 ( 1 983): 1 7 1 - 1 82. Yusa , Michiko. "Paradoxes and Riddles . " Encyclopedia o/Religion. M i rcea Eliade, ed. New York: Ivlacmillan, 1 987. Vol. 1 1 , 1 89- 1 95. "

"

"

,

RENAISSANCE PARAD OXES. Rosalie Colie sees Shakespeare 's King Lear

as expressing the range of stock Re nai ssance paradoxes. Following Lando, she gives these as fol lows: It is better to have no servant� than to have them. It is better to weep than to laugh It is better to be ignorant than learned It is better to be mad than wise. (5) It is not a bad thing for a prince to lose his state. (6) It is better to live in a cottage than a great palace. (7) It is better to be }XX>r than rich It is neither shameful nor odious to be a bastard. (8) (9) A frugal life is better than a splendid and sumptuous one. ( 10) It is better to be blind than have si ght. ( 1 1 ) It is better to have an ugly wife than a reautiful one. ( 12) It is better to be in prison than at liberty. ( 13) It is better to live in exile than to languish in one's native land

( 1) (2) (3) (4)

Colic,

READING

Ro s al ie . Paradoxia Epidemica : The Renaissance Tradi tio n of Paradox. Princeton: Pri nceton UP, 1 966.

RHETORICAL PARADOXES. I n rhetoric, paradox sonl e ti mes means

"a conclusion or apodosis contrary to \vhat the audience has been l ed up to expect" ( OED) . See also PARAOOX, DEFINITION OF and LJTER ARY PARAIX)X .

Dictionary of Paradox

1 72

Adduced bv lules Richard in 1 905, thi s

RI CHARD ' S PARADOX..

.,

paradox i s concerned \lvi th fi ni te defi nabi li ty.

Form ulation. Let D be the set of al l functi ons of one vari abl e fron1 th e

set of nat ural numbers into itsel f th at can be defi ned in a finite nu m be r of

\:vords . Given an e numeration of D , letfbe defi ned as "the function fro m

the set of natural nu m bers into i tsel f vvhose v alue for any gi ven argum e nt i s one Inore than the val ue , for that sam e argument, of th e function that

c orre s po n ds to the argun1ent in the enumeration of D . ' � B y the defi ni ti on of ,f, it must differ from e very elem ent of 0 and, th e refore

,

J doe s

not

bel ong to D . Nev erthel ess , f i s defi ned in a fi n ite number of words and

thus be l o n g s to D .

Explanation. Let E be the set of all ex pre s s i ons of the form

\vhere

n

of the

a.

I

a 1 a2 . . . an i s a natural number ( c aned the length of the expressi on) and each is

a

lette r of the Lati n alphabet, a semi -col on (used to separate

'N o rds ) or a com ma ,

.

The set 0 i s clearl y a s ubset of

E. We can o rder

these sym bol s by using an augmented fOlTIl of the natural alphabeti cal order, as fo11o\vs:

a{" 1 )1 ato ' s ' J ' Ilird t\·-ian. " A4ind ns

1 78- 197 . V lasto s , (J re gory.

.''

arc

78 ( 1 969) :

) ' he T h i r