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

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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<)t be applicable since we cannot disti n g u ish between "left socks" and '-�right socks." Nevertheless, the Axiom of Choice allows us to form the lIew s e t by m a king an arbitrary choice from each p ai r of s o cks A considerable number of mathelnaticians do not accept the val i d it y of a Ill'Ocedure of set formation that depends on such nonconstructive me thods ;IS the Axiom of Choice a nd they thus are able to sidestep the paradox (;111 hou g h some paradoxical special cases still a rise in the absence of this ;I.\iorn). T he majority of mathematicians� however, accept Choice and It' .��ard the present p a ra d o x and its n ear relations as pr o o f of the lIonexistence of the re]evant fi niteJy additive measures. Much In;llhelnaticai a ct ivi ty has r e v o l v e d around fi n d i ng a p p ro p r i a t e I ( III paradoxical measures.

out

,

�

.

(

-

READING

Wagon, Stan . The Banaclz- Tarski Paradox. Cambridge: Cambridge

U P,

1985.

B;\I{I�ER PARADOX, THE .. T his paradox is a humorous illustration

III Ihe

I n or

e technical

II'O,'ululation..

RUSSELL'S PARADOX.

"In a c e rtain

v i l lag e the barber shaves alL and only� the

18

Dictionary

of Paradox

men w ho do not shave themselves. Who shaves the barber?" (RusseH, Principia Mathematica, I 60).

Explanation. If the barber shaves hirnself, he must be one of those men who do not shave themseJves. B ut i f the barber does not shave hi mself�

he must be shaved by the barber ; that is, the barber must shave himself. In e i t he r case, we get a contradicti on. Resolution. Unl ess, of course, t he barber turns out to be a woman, there cannot be any barber who satisfies the p rem i s es of the paradox ; and, in fact, deny i ng that any sllch barber could exi st is the standard response to the p a ra d o x It i s e v id ent that thi s response is sati sfactory since� somewhat like a s q ua re circle� the ba rber has been defined by c ontradi ctory properties. The set theoretic anaJ ogue to the Barber Paradox i s not so easily d isp osed of, however, because it i s not i ntuitively evident what the source of the paradox is. .

BARBER SHOP PARADOX, THE. Formulated by Lewi s Carrol l, thi s story involving three barbers was i ntended by Carroll to be an "ornamental" presentati on of a problem i n the interpretation of conditi onal propositions. Formulation. AIJ en, Brown, and Carr are three barbers. They cannot all

b e out at the same ti m e becau s e someone has to m i nd the store . Furthermore, Al l en i s nervous about going out alone and always bri ngs Brown along w i th him. If Carr w ere to go out, then B rown would have to stay in whenever Al l en went out. Thi s last, however, i s impossible since B rown has to tag along with Al len. Hence, Carr can never leave the shopo Explanation. We are gi ven two premi ses: (1)

(2)

Allen, Brown, or Carr is in. If Allen is not in� then B rown is not in.

The argument proceeds by contradi cti on: (3)

( 4)

(5)

(hypothesis) Carr is not in . (from ( 1 ) and (3)) If Allen i s not in, then Brown is in. Propositions (2) and (4) are contradictory.

Therefore , (6)

The hypothesis (3) is false and Carr is in.

The justification for proposition (5) is that "Allen is not in" is a sufficient condition for two contradictory results and, thus, both (2) and (4) cannot be true. I{esolution. The i nterpretati on of conditi onal proposi tions has been

19

Dictionary of Paradox

problemati cal since the Stoics tried thei r hand at logic i n Classical Antiquity. 'rhus, it i s not surprizing that the various responses to the present paradox have i nvoked different i nterpretations of conditionals. Nevertheless, the overw helming majority of respondents to the paradox have argued that it is fal s i di cal since a contradiction only arises when , i n conj uncti on with hypothesi s (3), we assume that Al1en i s not in. Thus we are alJ owed to conclude only that whenever Carr is not i n, Allen must be i n. See the PARADOXES OF MATERIAL IMPLICATION. READ ING S

B urks , Arthur W. , and Irving M . CopL "Lewi s Carroll 's Barber Shop Paradox . " l'v1ind ns 5 9 ( 1 950): 2 1 9-222. Lewi s Carroll I Charles Dodgson 1 . " A Logical Paradox." Mind ns 3 ( 1 894) : 436-43 8 . Jones, E. E. C . "'Lewis Carroll 's Log ical Paradox. " Mind ns 1 4 ( 1 905) : 1 46-8, 576-578. Johnson, W. E. '"A Logical Paradox . " Mind ns 3 (1894): 583. HHypotheticals in a Conte xt. " Mind ns 4 ( 1 895) : 1 43 - 1 44. Sidgwick , A. "A Logical Paradox ." Mind ns 3 ( 1 894) : 5 82. - - . "Hypotheticals in a Context." Mind n s 4 (1895) : 1 43 . "W." " Lewis Carroll 's Logical Paradox ." Mi n d ns 1 4 ( 1 905) : 292-293 . -

.

I�EAKER PARADOX, THE. Cal l ed the wine and water probl em by

Ri chard Von Mises, thi s paradox is essentiaJ ly the same as

BERTRAND ' S

PARADOX.

Gi ven a mi xture of water and w ine that contai ns at least as Ill uch w ater as w i ne, there i s a 50% probabi l ity that the water to w i ne rati o i s between 1 and 3/2. B ut, by cal cuJ ating the inverse ratio fi rst, we fi nd that there is also a 50% probabi l ity that the water to w i ne ratio i s hetween 1 and 4/3 . The two resuJ ts are clearly i ncompati ble. Explanation. Let r be the ratio of the alTIOunt of water to the amount of w i ne in the given mixture. Clearly, if there i s as much water as there i s wine, r= 1 and, if there is twice as much water as wine, r=2. Hence, we have 1 � r � 2. I t i s, however, equal ly probable that r be at any poi nt in the stated i nterval a s at any other point in the i nterval . Thus, there i s a 50% chance of r heing i n the upper half of the interval and a 50% chance of r bei ng in the l ower half of the interval . That i s, there is a 500/0 chance that r is betw een I and 3/2. Consider, now, the ratio of wine to water� that i s , l lr. Clearly, i f there i s twice as much water as w i ne, l lr 1 /2 and, if there are equal a lnounts of water and w ine, l lr = 1 . Hence, we have Formulation.

=

1 /2 :s; l lr � 1 .

20

Dictionary oj' Pa rado)t

Once aga i n it i s e q u al l y pr o ba b l e that l /r be at a ny p o i n t in the stated ,

interval. Thus, there i s a 50% c h a n c e of l /r be i n g i n th e u pper half of th e i nterval ; that i s , be twe en 3 /4 and 1 . The refo re by t ak i ng t h e re c i procal s ,

th ere i s a 50% chance of r bei n g betw een 1 a n d 4/3 , which c o n tra di c t s the p re v i ou s r e s u l t

.

Resolution. T he kin d of p ro b l e lTI i l l ustrated by the pre sent paradox may occ u r w hene ver co nti nuous v a r i a b l e s are used to cha ra ct e r i ze a given pro p e rt y

.

Accordi n g to von M i s e s , however� the root p r o b l e m of these

pa rad o x e s i s t h e concept of eq u al l y po s s i b l e aJ tern ati yes b a s ed on i nt u i t i v e ( subjecti ve) j ll d gnle nts or on a pri n ci pl e of i n d i fferenc e due to symmetry

co n s i d erati o n s . T h u s , von l\l1 i ses argues that i t i s ne c e ssary to estabJish

the freq u e ncy d i st r i bllti on o f the d i stri b u ti on i s to be re garded as

an d i s thus independent of t h e

a l te rn a ti v e s

by e m pi r i c a l method s ; thi s

the i n i t i al conditions of any g iven p ro b l em p ro b ab i l i t y cal cul u s itself. In p rac t ic e the ,

di stri buti on i s ofte n determi ned by lon g-ran ge frequency counts, that i s ,

the e x p eri m e n t i s iterated a l arge n umber of ti ln e s and the re sults are t h en org ani z ed statisti cal l y. In c a se s such as the p re s e nt probl em, v on Mises warns , i t may be necessary to sp ec ify the actual

d e t e rm i n i n g th e gi v e n c ol l ec t i v e s

.

procedure i n vol ved i n

See B ER'r RAND ' S PARADOX an d the

INFI NITE S ERIES PARADOX . READING

Von Mises , Richard. Probabilitv. Statistics and Truth. New York: MacmiU an. ] 957. Ch. 1. '

.

BELL -EP R PAR ADOX, THE.

See the EI NSTEI N -PODO L S Ky -ROS EN

PARADOX .

BERRY 'S PARADOX. T h i s paradox w as fi rst descri bed b y Bertrand Russel I , w ho attri buted

it to the Cam bridg e I i b rarian G . G . Ben)'.

A ] og i c a J

p ar ado x rel ated to RI CHARD ' S PARADOX an d ulti mately to the LIAR, i t has

also been characteri zed as a semantic paradox .

Formulation. Co n s i d er the n u m ber sp e cifi e d b y the phrase "the le a s t natural num ber not specifi a b l e b y a p h ra s e c o nta i n i n g fewer than fifteen w ords ," which phrase u s e s fewer than fi fteen w o rd s . Vari ants i nclude re p l acing "word s " by

"

sy I J a b l e s

j u st m en t to the nu mb e r i n vol ved.

ad

"

or

"

l e t t e r s " w i th a co rre s p o n d i n g

Sti l l other v a ri ant s inc l ude Max B l ack's

Hthe l east i nt e ger not named in thi s b oo k and Robe11 E. Kirk ' s "the l arge st "

I l l i in b er

d efi n ab l e by an Engl i sh ph ra s e

c o nta i n i n g

. . . or fe wer w ord s . "

2/

Dictionary of Paradox

Explanation. A few facts about the natural n umbers and a natural l anguage s uc h as Engl i sh make the paradox formal l y deri vable. First, there i s an

i nfi n i te nu mber of n atural n umbe rs . Seco nd, any noneJnpty set of naturaJ

num bers has a l east el ement. Th i rd, the number of di sti n ct words ( lette rs, s y l l ab l e s )

i n Eng l i sh i s fi n ite . T hus, there is only a finite number of p hrases

restri cted to a certai n l e n gth w hich can be used to s pecify some member

of the n atural n umber sequence. Cons equentl y, there w i l l al ways be some n u m bers

that are not specifi e d by any phrase restri cted to a gi ven length.

Yet th i s number i s then specifi ed by Berry ' s p h rase , w hich i s w i thi n the

gi ven l ength.

i�esolutions. At fi rst si ght the paradox woul d seem to be avoi dabl e by o utlaw i ng p h rases contai n ing negati ve s . That such an approach does not s uffi ce, how ever� i s shown by Ki rk's vari ant w h i c h gene rates the paradox

w i thout the use of negati ves . Dorothy G rover attempts to give an acc o un t of r efe re nc e according to which phrase s such as Berry ' s fai l to refer� but

w h i le p rom i s i n g this approach does not e x p l ain w hy phrases such as ,

Berry 's generate the paradox w hereas phrases suc h as "the l east nUlnber

n ot specifi able in Engli sh i n l e s s than th ree w ords" do not. Sti l l another

; I pproach w ould be to order spec ifyi ng phrases i nto a hierarchy of types ( see the

LiAR) :

w here i n an atteTnpt to state the paradox wou l d invo l ve

s pecifi c ation of the l east number n ot speci fi able i n few er than n words 2 1 � l l l ci n o paradox w ould ari se.

READINGS

Black, Max .

The Nature of Ma thell1atics . London :

Ro u t led ge & Kegan

Paul , 1 93 3 . 97- 1 0 1 . Brady, Ros s T.

Quarterly

Repl y to Pri es t on the B erry Parado x

.

"

Ph ilosoph ical

34 ( 1 984) : 1 57- ] 62.

Grov er� Do ro thy .

"

Berry ' s Paradox . " Analys is 43 .4 ( 1 983 ) : 1 70- 176.

Ki rk� Robert E. "A Negation-Free Vers ion of the Berry Paradox." Analysis

4 1 .4

( ] 98 1 ): 223-224.

Pri est, Graham. "The Logical Paradoxes and the Law of the Excluded Middl e . " Philosophical Qua rterly 3 3 ( 1 983 ) : 160- 1 65. Di scussion: 34 ( 1 984) : 1 57-63 .

R usselJ , Bertrand.

"1£s Paradoxes de la Log ique." Revue de Metaphys ique

et de Morale 1 4 ( 1 906) : 627-650.

B I�: RT R A N D ' S PA R A D O X .

Fi r s t d e s c r i be d b y t h e

I l l a l hematician J. L. F. B ertrand, thi s is a

Fre n c h

PARADOX OF PROBABI LITY TH EORY .

I �'ormulationo Classical probab i l ity theory� s uch as Lap l ace 's� defi nes the p robabi l i ty of an event as the rati o of the n umber of time s that an event

22

Dictionary of Paradox

occurs to the number of " equally likely outcomes." Consider the probab il i ty that a chord� drawn at random in a circle, i s longer than the side of the equilatera] triangle i nscri bed in the ci rcl e. Since the midpoint of the chord is on the radi us of the circle, the required probabi lity mu st be 1/2. But since the chord i s contained by two sides of th e inscribed triangle, the prob abil ity Inust be 1 /3 . Moreover, since the In i d poi n t of the chord is contained in the circle inscri bed i n the tri angJ e, the pro babil i ty must be 1 /4. Explanation. G i ve n a c h o rd c , d raw n at random i n a circle, there are (at l east) three different w ays of cal c u l ati n g the probabi lity that the l ength of c wi l l be g r eate r than the l e ng t h of t he si de, s, of the inscribed equilateral t r i a n gl e Fi rst, d ra w t h e radi us, r, c o n ta i n i n g t he midp oi nt M, of c and o ri e n t the i n scri bed tri angl e so that one side is paral lel to c (fig. 1 ) . Let 0 be the center of the ci rcl e and A be the i nte r s e ct i on of the radius w ith the side of the triangle. Cl early, c>s if, and o n l y if, M i s between 0 and A . B ut OA ( 1 /2)r ( sin c e OA i s 1 /3 of the alti tude of the triangle). B y construction, M must be on r. Si nce the l ikeli hood that M w i l l be at any point on r i s the same as that i t will be at any other poi nt on r and since half of these positi ons make c g r e ater than s , the required probability must be 1 /2. A n other way of looking at the problem, however, gives a different result. Draw a tangent to the circle at one of the endpoints of the chord and orient the triangle so that one vertex is coincident with that e ndp oi nt (fig. 2) . Clearly, c>s if, and only iC c i s in the 60 degree angle made by the two sides of the triangle at the tangent. By construction, the chord must fal l i n th e 1 80 degree angle on the same side of the tangent as the c i rcle. Since the l i kel i hood t h at c wi l 1 be in any position in the 180 degree angle i s the same as that it will be in any other positi on i n th at angle and si n ce a thi rd of these posi t i o n s make c greater th an s, the required p robability must be 1 /3 . A lthough the paradox has already been establi shed by the two contradictory results for the required probabi lity, still another way of l ooki ng at the probl em results i n yet anothe r val u e for thi s probabil i ty. Inscri be a c irc l e i n the tri an gJe (fig. 3). C l e a rly c>s if, and onl y if, M (the m i dpo i nt of the chord) i s contained i n the small er c i rcle ( c=s w he n c i s tangent to the smaller c i rcl e) By constructi on, M must fal l somew here in the l a rge r ci rcl e. Since the likel ihood that M w i ll be at any poi nt in the l a rg er ci rc1 e is the same as that i t will be at any other point in that circle and si nce a fourth of these points make c greater than s (the i n fact, radi us of the small er circJ e i s half the radius of th e l arger circle it is OA of Fig. 1 therefore, the area is a fourth), the required proba b i l i ty i s 1 /4. It i s evident that the resuJ ts of these calculations are i ncompati bl e, h llt a l l three ari se from a considerati on of the ratio of successful outcomes ,

,

.

=

,

.

-

Dictionary of Paradox

23

to

"equally l ikely outcomes. " Consequentl y, a mathemati cal theory of probabi lity cannot be based upon thi s intuitively appeal ing notion.

./

o -----

/

/

/

" M A

Fig.

l � l

Fig.

1

Fig . 3

2

I�esolution.

The problem here i s that the phrase "equalJy likely outcomes" i n the fOlmulation of classical probabiJ ity theory is not welJ enough defined to support u n am b ig u ous probability attributions. There are at l east three approaches to resolvi ng the parad ox . First of al l , Bertrand suggested that probabi l iti es could not be wel l defined w i th respect to i nfi nite sets of possi ble outcomes, so that proba bi l ity theory shoul d be restricted to situati ons i n w hich the number of possible outcomes is fi nite. Since, how ever, i t is often necessary to state probabi l ities for a range of values i n a contin uous di stribution, thi s suggestion i s too restrictive. Secondly, a IHore typical response i s to abandon classical probability theory i n favor of a l ong-range frequency count (see the B EAKER PARADOX) . On thi s account, probabilities may not be derived a priori , but only through actual o bservati on of the frequency of successful outcomes i n a given sequence of trial s (but see the I NFINITE S ERIES PARADOX for a difficulty with thi s v iew ). Thirdly, it has been suggested that eq ually l i kely outcomes may be j udged onl y relative to a particular procedure. There are, for example, pa rti c u l ar procedure s that prod uce each of the cal c u l ati on s i n the formulation of the paradox: the chord i n the first calculation could be generated, all other things being equal , by rolling a rod across the face of a circle ; the chord in the second cal c ulation couJ d be generated by a rod attached at one end but free to spi n around i t � and the chord in the thi rd, by sel ectin g its center at random . REA DING S

Bertrand , J . L. F.

Calcul des Probabilites.

New York : Chel sea � 1 972 . Th i rd

Lecture. von Mi ses. Richard. .

1 957. Ch. 1 .

Probabili ty �

,

Statistics and Truth . New York: Macmil l an, .

Dict;oflar}' of Paradox

24

BISECTION PAR AD OX, THE. See the DICHOTOf\1Y .

BOD HIS ATTVA P AR ADOX, 'fHE. T h i s ethi co r e l i gio u s par ad o x was -

add uced by A rthur Da n to . Formulation. The bodh isa ttva can n ot pass over i n to N i rvana. He cannot becau se, w ere h e

to do so

h e wou l d e x h i b i t a se l fi sh n e s s

can n ot have . I f he has the s e l fi s h n e ss

�

he i s n ot

a

that a bodh isattva

bodhisattva , and so

can not e n ter i nto N i rv a n a . if he l ac k s t he se l fi s h ne s s , agai n � he can not ente r N i rvana� for t hat wou l d

be a s e l fi s h ac t.

S o either w ay� the bodhisattva

i s i In potent to entcr N i rva n a . lj ke G od w h o , i n the Ch ri sti an teachi n g ,

can not do ev i l because i t i s

i n co n s i ste nt w i t h h i s n ature � the bodhisattva

ca n n ot pe rfonn t h e u l t i nlate l y s e l f i s h act . S o n o one can reach N i rvana: �ve

can not because w e are not hodh isattvas a n d the bodhisattva cannot

bec ause h e i s

a

bodh isa ttva" ( Danto , Mys,tic isln and Morali(v

82; but see

a l ternati ve form ulation i n h i s A n alytic Ph ilos'ophy of A ction 1 66). Explanation. A ccording to Da n to the bodhisattva i s generaJ ] y concei ved ,

of as someone who has

atta i ne d

enl i ghten ment an d can thus pas s ove r

i nto N i rvana ; he n evert h e l e ss postpone s doing so

unti l al l

mankind can be

b ro u ght to the same p oi nt so that everyone w i l l pass over together. Danto ' s d e s c r i pti o n of t h e b odh isattv a , h o w e v e r d o e s n ot s q u a r e w i th al l i n t e r pret ati ons of B u ddhi st doctri ne. A ccord ing to one mai n conception, the bodh isattva does n ot delay e nteri ng Nirvana unti l al l manki nd (bette r, aJ l sent i ent creatures) can enter w ith h i m , but forebears only until they are secure i n thi s pro spect. But grant i ng Danto hi s interpretati on of the doctri ne, the paradox w o uld be that the bodhisattva can not not enter N i rv ana until al l manki nd enters with him because, if he tri ed e nte r in g before t h e m hi s selfi shness woul d prevent h i m fro m e n te ring N i rvana a n d hence, he would not be a bodhisattva after al l . I( howe ver, h e were i ndeed a bodhisattva , hi s lack of se l fi shness w oul d p reve n t hi m fro m enteri ng Ni rvana ahead of ,

,

,

e veryone el se. Thus, w hether the bodhisattva i s selfi sh or not, he i s unabJ e to pass over i nto N i rvana. B ut thi s re s ul t c ontradi ct s the conventi onal view that the bodhisattva ' s delay i s a postpo nem ent. R e s o lution ..

P e r h a ps

Danto ' s ( a n d

our

c o n v e nt i o n a l ) i d e a of w h at

consti tute s s e lfi sh n ess is not app l i cable at the poi nt at w h i c h one i s p o i sed to enter Ni rvana, becau se fro m that s t an dpo i n t even the d i s t i ncti o n se lfi sh! u n se lfi sh i s i rr ed e e m a b ly fra u ght w ith egoi sti c i dentifi cati ons. It i s even u n c l ear w hether the bodhisattva can be c redi t e d w ith

a

vo l u nt ary

act

of

post pon i n g h i s entry i nto bl i s s, because h e w oul d have had to h ave passed beyond any sort of

wil lfuJ ness

to be a bodhisattva . Thus the deci sion to

Dictionar),' of Paradox

25

d e l ay enteri ng N i rvana i s n ot made once t he bodhisattva i s a bodhisattva , but w hen the one w ho m i ght be a bodhisattva enters i nto the path of t h e hodhisattva . Thi s poss i bi J ity, how ev er, onl y re solves Danto ' s paradox by p utti ng another paradox i n its pl ace

- namely, that of h o w an act of w i l l

can pl ace a bei ng bey ond w i l lful ness and i ts opposite altogether. REA D I NG S

Danto, Arthur. Analytic Ph ilosophy of A c tio n . Cambridge : Cambridge UP, 1 973. - - . MysticisfJ1 and Mo ra l ity : Oriental Thought and Moral Ph ilos op hy Harmondsworth , Eng . : Pengui n, 1 976. Perr e tt Ro y. "The Bodhi sattv a Paradox. " Ph i lo sop hy East and West 36 ( 1 986): 5 5 -59. .

,

MlOOTS PARADOX, THE.

The Russ i an geographer V. N. B ugrome nko

u rges that the ful l potenti al of sophi sti cated m athemati cal tech n i q ues i n geography, such as those p i oneered by Barry N- . B oots , are not being re a l i ze d b e ca u s e "the m e th o d o l o gi c a l i n n o v at i o n s t h at h av e b e e n d evel oped i n En gJ ish s peaki n g geo graphy often fi nd them se l ves i n a c o nceptual vacuum, so that i t i s diffi c ult, if not i mpossi bl e, to use the se i n n o v a t i o n s fo r t h e r e s o l u t i o n o f co n c r e t e re g i o n a l p r o b l e m s ' � ( B ug romenko ,

8). Bugromenko cal l s thi s situati on the B o ots Paradox and

s u gge sts that i t i s ( partly) due to the fact that the i n novati ons are d ue to u n i v e r s i ty p rofe s s o rs w h o h a v e b u t l i t t l e c o n c e r n w i th prac t i c a l ,} ppl i cati ons.

READINGS

Boots , B arry N. "Using Explanatory Spati a l Models in Planning Contexts. " Soviet Geography 28 ( 1 987): 1 0- 1 3 . Bugromenko, V. N. "'The Boots Paradox." Soviet Geography 28 ( 1987): 1 -9.

IilOTTLE IMP, THE.

Citi ng "T he B ottl e Im p," a short story by Robert

Loui s Step hen son, Ri chard S harvey proposed thi s paradox to the read ers of Philosophia l I srael J .

Formulationo "An i ndestruct i b l e bottle conta i n s a powerful i mp , and the ovv ner of the bottle m ay command the i mp to grant ( al most) any wish. B ut if a person d i es w h i l e he ow ns the bottl e, he goes to h e l l and burns forever.

An ow ner of the bottl e may get rid of it o nl y by sel l i ng i t to

someone el se for l es s than he pai d for i t h i mself. Fu rt hermore, he m ust i nform the buyer of al l these conditi ons, and the buyer m u st u nde rstand t hem and be rati onal (or el se the bottle j u st comes back) . "

26

Dictionary of Paradox

Explanation. Sharvey argues that, even disregarding the risk of accidental

death while i n possession of the bottl e, i t wou l d be i rrati onal to buy the bottl e for any fi nite amount of money. O b v i o us l y, it woul d be irrati onal to buy it for one cent si nce it coul d n ot then be resold, thereby condemning the buyer to burn i n hel l forever. (The i m p cannot grant eternal life, nor create smaller currency units, nor grant any wish that wouJ d alter the conditions of the probl em. ) Furthermore� it i s i rrational to buy the bottle for n+ 1 cents , if it i s i rrati onal to by it for Il ce n t s since, once again, it co u l d not be resold. B y inducti on, therefore, it is i rrational to buy the bottle for any fi n i t e alTIOunt. The probJ em i s to explain why such a bottl e seems l i ke a bargai n at the pri ce of, say, one thousand dollars. Resolution. Many comlnentators have re l ated the Bottle Imp to various other paradoxes. Roy A. S ore n se n for exa m pl e , argues that it i s a variation of the PREDICrION PARADOX . M ich ae l J . Wreen , who rel ates the present pa ra dox to various others, contends that the argulnent, as stated, i s i nval id since there are various scenarios that would seem to m ake buy i ng the bo ttl e for even one cent a rational act. Thus, a person who believes himself to be pred esti ned to go to hel l anyway may quite rat i ona l ly b u y the bottle in order to afford himself the telnporary p l easures that i t coul d provide � alternati vely, as is the case in Stephenson 's story� the bottl e could be bought for various altrui stic reasons without the buyer b e in g thought irrational . Wreen then argues that even if these invalidating scenarios are elimi nated� buy i ng the bottle for a thousand doll ars would indeed be rational , gi ven that peopl e are like they a re S uch an argument evidently di srespects S harvey 's premise that the buyer be rationa t but it also indicates that it is preci sely S harvey 's defi nition of rational ity that is responsibl e for the p arado x i c al result. Indeed, Saul Traiger observes that S h arve y s premi ses i mply that any attempt t o buy the bottl e i s a sufficient condi tion for irrational ity ; hence, no prospective buyer could effect the purchase since he would have to be rational to do so. Thus� accordi ng to Trai ger, the purported paradox cease s to be an interesti ng a rg u ment . ,

,

.

'

READINGS Margalit, A . , and M. Bar-Hil l el . " Expecti ng the Unexpected . "

Philosophia

1 3 ( 1 983 ): 263-289. Philosophia 1 2 ( 1 983): 40 1 . Sorensen� Roy A . "The B ottle Imp and the Prediction Paradox . " Philosophia 1 5 ( 1 986) : 424-424. - - . "The Bottl e Imp and the Predicti on Paradox, I I" Philosophia 1 7 ( 1 987): 35 1 -354. Trai ger, Saul . ��The Probl em of the Bottle Imp :' Philosoplzia 1 5 ( 1 986) : 425-426. Sharvey, Ri chard . "The Bottle Imp."

27

Dictionary of Paradox

Wreen , Michael J. "Passing the Bottle:' Philosophia 1 5

I lRADLEY'S PARADOX OF REL ATIONS

( 1 986): 427-444 .

Adduced by F. H. Bradley, f h i s paradox p roduces an i nfinite regress of re l ati ons in a manner rC ln iniscent of LEWIS CARROLL 'S PARADOX OF ENTAILMENT , which is indeed a speci al case of the present paradox. Formulation. S uppose that A and B are two terms that are connected by 1 he re lation C. Then, there must be a relation D connecting A and C and a re lation E connecting B and C. By continuing to reason i n thi s manner, a n i nfi nite regress of relations is generated. I�xplanation. If A and B are two terms that are connected, they are connected by some relation C. Moreover, in order to explain the connection between A and B, the relati on must be different from both of them (A and B) . Yet C can only rel ate A and B jf t here is some connecti on between A and C and between B and C; thus, there must be new rel ati ons D and E to expl ain these connections. It would seem, then, that each new relation creates a need for sti ll other relations i n order to explain the connections between itself and its re lata . Thus, we are propel led into an infinite regress, thereby leaving the original connection between A and B unexplai ned. Resolutiono B radley used this paradox i n d efense of hi s idealism by conel uding that external rel ations are not real , but only apparent. Eric Toms, however, argues that thi s conc1 usi on is not necessary since the regress can be closed off by refl exive relati ons. A reflexive rel ation in thi s sense is not a reflexive relation in the usual ascripti on of the term (every obj ect in the domain is related to itself by the relation). Rather a reflexive relation not only connects its relata to each other but also connects each of the re lata to itself, obviating the need for a new rel ation to make the connection. Toms suggests that in the absence of a reflexive rel ation, the universe would be i ncompl ete. .

REA DINGS

Bradley, F. H. Appearance and Reality. Oxford: Clarendon, 1 897. Ch. 2. Toms, Eric. Being, Negation and Logic. Oxford: B l ackwell, 1 962. 54-55 .

A bridge para dox i s one adduced as an intermediate between two other paradoxes in order to di splay thei r kinship. Roy A . S orensen, for example, argues that Sharvey 's BOTTLE IMP i s a vari ati on of the PREDICfION PARADOX by way of two " bridge paradoxes.," the Last Buyer and the Desi gnated S tudent.

BRIDGE PARADOXES .

28

Dictionary of Paradox

BUD DHIST P A R A DOXES . i n te l l ectual

B udd h i sln i s a re l i g i on notable for i ts

sophi stication, and the rol e of paradoxes in the B uddh i st

traditi on refl ects thi s Ci rCU1TIstance.

Perh aps the most i mportant use of

i n b oth B udd h i sm and Taoi sm i s to reveal the in herent i n a b i l i ty of l anguage to expre s s ul ti mate rea l i ty. As s u ch, the paradoxes are not th ere to be sol ved , b ut to be u sed as ste ppi ng stones to a better apprec i at i on paradoxes

of the nature of kn ow l edge and of real i ty. Once th i s better appreciation i s att ain e d the paradoxes are no longer nee d ed and may be di scarded. The �

fo l l ow i ng exam ple i s fro m the Chi nese Taoist Chuang Tzu ( se e Edward T. Ch ' i en ) :

The fi s h trap ex i sts because of the fj sh ; once you ' ve gotten the fi s h , you can forget the trap. The rabbit snare exi sts because of the rabbit; once you ' ve gotten the rabbi t, you can forget the snare. Words exi st because of m eani n g ; once you've gotten the meani ng, you can forget the words . When can I find a man who has forgotten words so I can have a word with hi m .

exampl e i s that attri b uted to Bhartr kari , a fi fth­ ce n t u ry grammarian ( se e Hans G. an d Radhika H e rzberger) :

A s l i ghtl y more techn i cal

Consider the proposition '"There are some things which are unnameable. " Any attempt to verify this proposition by exemplifi cati on would be naming the unnameable. Mark S i d eri ts and J . h ave a d d re s sed

D erv i n 0 ' B ri en for examp l e , as weJJ as 1. W. Mabbett, th e q u e sti on of the ki n s h i p of N agarj una and ZENO ' S ,

PA RA DO X ES � w h i l e Tyson Anderson compares Nagarj un a to Wittgenstein. Noteworthy, by the way, i s how few paradoxes are actual ly

di scussed in

thi s literature � t h e Ina i n to p i c seems to be paradox itself rathe r t h a n

paradoxes consi dered i ndivi dually, but see the PARADOX OF NIRVANA .

REA D I NG S Anderson, Tyson. �'Wittgenstei n and Nagarj una's Paradox :' Philosophy East and West 35 ( 1 985 ) : 1 57- 1 70. Chung-Ying Chang. "On Zen (Ch ' an) L an guage and Zen Paradoxes." Journal of Chinese Philosophy 1 ( 1973 ) : 77- 1 02. Ch' ien, Ed ward T. "The Concepti on of Lan guage and the Use of Paradox in Buddhism and Taoi sm. " Journal of Ch inese Philosophy 1 1 ( 1984): 375 399. Herman, Arthur L. " Hedonism and N irv ana: Paradoxes, Dilemmas and Solutions." Philosophica rrndiaJ 1 0 (Ja-D 1 98 1 ): 1 - 1 0. Herzberger, Hans G. and Radhi ka Herzberger. "Bhartrhari 's Paradox." Journal of Indian Philosophy 9 ( 198 1 ): 1 - 1 7.

Dictionary of Parado.>.:

29

King-Farlow, John. "Anglo-S axon Que stions for Chung- y i n g Cheng.�· Journal of Chin ese Philosophy 1 0 ( 1 98 3 ): 285 -298. Mabbett, I . W. "Nagarjuna and Zeno on Moti on." Philosophy East and West 34 ( 1 984): 40 1 -420. Ni shitani , Keij i . Religion and Nothingness. B erkeley: U of Cal iforni a P, 1 982. Sel lman, James . "A Pointing Fin ger Ki lls 'The Buddha' : A Response to C hung-Yi n g Cheng and J ohn K i n g- Farl ow. " J ournal of Ch inese Philosophy 1 2 ( 1 985): 223-228 . Siderits , Mark, and J . Derv in 0 ' Bri en. "Zena and Nagarjuna on Motion." Philosophy East and West 26 ( 1 976): 28 1 -299. S later, Robert. Paradox and Nirvana. Chicago: U of Chicago P, 1 95 1 . Suzuki , D. T. "Reason and Intuition i n Buddhist Philosophy.�· Essays in East- West Philosophy. Ed. Charles A. Moore . Honolulu : U of Haw aii P, 1 95 1 . 1 7-48. - - . "Basic Thoughts Underlying Eastern Ethi cal and Cultural Practice ." Philosophy and CuLture : East and West. Ed. Charl es A. Moore. Honolulu: U of Hawaii P, 1 968 . 428-447. Tucker, John . "An Anglo-Saxon Response to John Ki ng-FarJow 's Questions on Zen Language and Zen Paradoxes." Journal of Ch in ese Phil osophy 1 2 ( 1 985 ): 2 1 7-22 1 .

BURALI .. FORTI PARADOX, THE. A logical paradox ari sing in nai ve

set theory, the B urali-Forti paradox was at one time mistakenly attributed to Georg Cantor (Moore and Garc i ad iego ; Menzel , 1 984) . CANTOR ' S PARADOX i s its analogue for cardi nal numbers. Forlnulation. The set of all ordinal numbers i s weB ordered and thus defi nes an ordi nal number q. B ut since q is an ordinal number, q i s an e l ement of the set of all ordinal numbers and, therefore, q
•

•

•

k = { O , 1 , 2 , . . . , k- 1 } •

•

•

w = {O, 1 , 2, . . . } w+ I = { O, I , 2 , . . � w } .

•

•

30

Dictionary of Pa radox

As we can see from the examples l i sted above, every element of a set that defi nes an ordi nal i s l ess than the o rdi nal so defi ned. Thi s i s evident in the case of the so-cal led successor ordinal s (for example, 3), since each new ordi nal is the next number in the initial segment and thus is one more than the l argest element of the set. But it i s al so true of the so-cal J ed l i m i t ordinal s; w, for example, i s by definition greater than al l the nonnegative integers. Thus, (*)

if a and b are ordinal s suc h that a i s an el ement of th e s et b, t hen a i s l es s than h.

Consider, then , q the enti re sequence of ordinal s. This will itse lf be an ord i nal and , hence, by (*) q will be l ess than itself, which i s a contrad icti on . Resolution. The B urali -Forti paradox i s most commonly addressed by placi ng restrictions on the formation of sets so as to block the construction of the set of all ordi nal numbers. In set theoretic terms, the collection of aJ i ord inals i s not a set, but a proper class ; thus the purported ordinal q i s not defined and the paradox does not arise. This resolution i s closely related to that for RUSSELL ' S PARADOX , t hough, as Suppes points ont, it i s important to rea l i ze that the d e v i c e s for avoi d i ng Ru s sel l 's P arad o x d o n ot automatical ly work for the BuraIi-Forti . READINGS Dumitriu , Anton. "The Logi co-Mathematical Parado x es

.

"

History ofLogic.

Vol . 4. Tu n bri d g e Well s , Eng. : A bacus, 1 977 .

Grattan-Guiness , I. Hal l ett, Mi chael .

-

Dear Russell Dear Jourdain. Ne w York, 1 977. 24-5 1 . Cantoriall Set Theory and Limitation of Size. Oxford :

Oxford UP, 1 984. 176- 1 85. Hazen , A . " Logi cal Objects and the Parado x of B ural i -Forti . "

Erkenntnis

24 ( 1 986): 283 -29 1 .

Que stion of the Exi stence of Transfinite Numbers. " Proceedings of the London Mathematical Society 2 , IV ( 1 906-07) : 266-28 3 . Menzel , Ch ri sto p her. " Cantor and the B ural i -Forti Paradox." Monist 67 ( 1 984): 9 1 - 1 07. - - . "On the I terati ve Ex pI anati on of the Paradoxes. " PhilosophicaZ Studies 49 ( 1 986): 37-6 1 . Moore, G. H. , and A. Garciadie g o. " Burali -Forti ' s Paradox: A Reapprai s al Historica Mathematica 8 ( 1 98 1 ): 3 1 9-350. of Its Ori gi ns Suppes , Patrick. Axiomatic Set Theory. 2nd ed. New York: Dover, 1 g-J2. 8-9.

Jourdain, P. E. B . "On the

.

"

The medieval l ogician John Buridan is commonly credited wi t h thi s paradox.

BURI DAN' S ASS .

31

Dictionary oj Paradox

FOl"lnulation. An ass midway betwe en tw o identical bales of hay w i l l

starve to death, because, havi ng no reason to go toward one anymore than the other, and being incapable of goi n g to both at the same time, he will remai n where he i s. Explanation. In the event, the ass has no moti vation to choose between the two courses of action that are equaJ J y attracti ve to him, so the result w i J J be for him to chose neither and die. Resolution. The i dea that something is so evenly suspended between two alternati ves that it finds itself incapabl e of comi ng down on either side has h ad various appli cations i n th e hi story of t h oug ht especi aJ ly for the questi on of how psychological agents choose between conflicti ng courses of acti on. The d eadl ock betw een t h e tw o equaJ J y attractive aJ ternati ves can be broken by making an arbitrary choi ce or by relying on an arbitrary decisi on procedure such as the toss of a coin . ,

READING

Makin, Stephen. " Buridan's Ass. " Ratio 28 ( 1 986): 1 3 2- 1 48 .

BYSTANDER PARADOX, THEo As suggested to Paul Woodruff by

Bernard Katz, a theory of self-defence containing the fol l ow i ng tw o princi ples fal l s prey to an existential paradox: first, if one is under attack and cannot save one's 1 ife from that attack w i th o u t killing one's attacker, one i s j ustifi ed i n killing one's attacker; second� one i s not under any circumstances justified in kilJ i ng an innocent person. Formulation. "The paradox arises when, during a homicidal attack, an unanned bystander occupies a position whe re he wHI be killed if the person attacked defends hi mself . . . B ystander hopes that Defender will consider him innocent under the theory of self-defence and prefer abandoning his defence to ki l l i ng an i nnocent person . Defender know s al l thi s� and tries to appl y the theory of self defence to the case. Observing that Bystander's del i berate presence as an innocent blocks the defence, he conc] udes that Bystander is a direct supporter of the attack and may be j ustifiably killed as an attacker. Si nce Defender i s not daunted by the prospect of killing an extra attacker, however, he no l onger finds hi s defence obstructed by the presence of Bystander. So if he finds Bystander to be an attacker, then he must reconsi der that findi ng; for by reaching his deci sion he has changed the rel evant evidence i n such a way that he ought to fi nd Bystander innocent. But i n doing so, he wouJd alter the evidence once more, since be balks at ki l ling an innocent" (Woodruff, 74-75 ) . Explanation. The probl em i s that on this theory of self-defence, the Bystander i s an attacker only if he i s j ud ged to be an i nnocent, for it i s

32

Dictionary of Paradox

only as an i nnocent that he ca n aid the attac k by thw arting the defensi ve response. S i m i l arl y� the B y stander is an i nnocent onl y jf he is j udged to be an attacker, for then hi s presence i s i m material to both the attack and the defe n si v e response. The argument does not issue i n a formal contra d ictio n but it is p aradoxical i n that it places the Defender i n an i mpossible situation. Resolution. Deeming it i mpossible to circumvent the paradox by qualifying the theory's account of 'attacker ' and 'innocent ' , Woodruff suggests that the theorist either create a third category "for awkward victims like Bystander and declare that killing them is parti a l l y justified" (76-77) or else develop a theory of excuses that would exonerate Defender from the blame of ki lling Bystander. Dave Lovelace, however, argues that the paradox i s fal sidical in the first place, on the grounds that the B y stande r who intends his presence to protect the Attacker is guilty, independe ntl y of how the Defender classifies hi m, because of his in tention to further the attack. ,

READINGS Lovelace , Dave. "A Note on the " B ystander Parado x

'

. " Analysis 38.4 ( 1 978) :

1 99-200 . Woodruff, Paul . "Th e By stander Paradox .'�

Analysis 37.2 ( 1 977): 74-78.

CANTOR 'S P ARAD OX. A

l ogi cal paradox about cardi nal numbers i n set-theory discovered by Georg Cantor in 1 899, i ts anal o gue for the ordi nal numbers i s the BURALI-FoRTI PARADOX. Formulation. For any set S , let lS I be the cardinal number of S and P(S ) be the power set of S . Consider the set of aU sets, U . By Cantor' s Theorem, l UI < I P( U ) I B ut P (U) is an element of U. T h erefore , I P(U)I :s l U I , contradi cting the p revious resu1t. Explanation. The cardinal number of a set S is the set of al l sets which are the same size as S (two sets are the same size if their members can be put in one-to-one c orres po n dence ) The power set of S i s the set of al l subsets of S . Cantor proved that the cardina1 number of a set is strictly l ess than the cardinal number of its power set. Now consider the set of all sets � cal l it U. C l earl y the power set of U i s a proper subset of U, so that i ts cardinal number is less than or equal to the cardinal n umber of U . Yet thi s finding contradicts the result t hat the cardinal number of any set i s strictly less than the cardinal number of its power set. Resolution. As with various other logical parad oxes , analysi s of Cant or s Paradox l eads to the conclusion that there is somethi n g wrong with the notion of the universal set, or the set of aU sets. Various attempts have been made to formulate set theory in such a way that the logical paradoxes m i ght be avoi ded ; see RUS SELL ' S PARADOX for further di scussi on. .

,

.

'

Dictionary of Paradox

33

READINGS Men del son, Ell i ott. Introduction to Mathematical Logic. New York: Van Nostrand Rei nhold , 1 979. S u ppes , Patri ck. Ax i oma ti c Set Theory. 2 nd ed. New York: Dov er, L 972.

C ATCH 22. This colloquial name for one main kind of exi stential ..

dil emma, namel y the "damned-if-one-does-and-damned-if-one-doesn 't" kind , comes from Joseph Hel le r 's cl ass i c war novel, Catch-22 . The dilemma does not actually issue i n a contradicti on. Rather, Orr i s faced w ith the disagreeabl e pros pect of conti nui n g to fl y more mi ssions regardless of hi s m ental wel l -being. Thus, the poignancy of the dilemma i s not so m uch a matter of logi cal consi stency as of the sense of unfai rness and helplessness that it engenders. The sense of frustration is increased by the circumstance that certai n of the regulati ons seem to support Orr 's desires ; yet other regulati ons make the fi rst ones i noperable. Hence , the w hol e set of regulations on the relevant question is quite pointless. READING Hel ler, Joseph. Catch -22 . New York: S imon and Schuster, 1 96 1 .

CHAIN STORE PARADOX, THE. Proposed by Rei nhard Selten i n 1 97 8 , the Chain Store Paradox concerns rational decision theory and i s

closely rel ated to the PRISONER ' S DILEMMA . :Formulation. The game has twenty-one players. Pl ayer A owns a chai n of stores i n twenty cities. At the fi rst round of the game , Player 1 i s to decide whether to open a rival store i n city 1 . If he chooses not to do so (OUT) , the payoff to Pla ye r A for that round is fi ve poi nts while Player 1 gets one poi nt. If he chooses to open a rival store (IN ), then Player A must choo se either a c o-operati ve pri c i ng strate gy (CO-OP) or an aggressive one (AGG). If Pl ayer A seJ ects CO-OP, the payoff to both players is two points � otherwi se both get none. In the second round, the sel ection process i s repeated w ith Player 2 in city 2, and so on until the tw entieth round when the game is over. A ll players are compl etely rational and all choices are known to aU players. The object for each player is to maximize hi s payoff. Although game theory dictates that player A always respond to IN with CO-OP, Selten reports that even mathematically trained players shun the valid game theory concl usion to opt for another strategy. Explanation . The parad ox ari se s from cons i d e ring tw o diffe rent approaches to playing the game. The fi rst of these i s the game theoretical approach, wh ich Sel ten cal l s the inductive theory. On thi s vi ew, Player

34

Dictionary o/Paradox

20 should reason as fol l ow s: " if A i s rati onal , then he w i ] ] sel ect CO-OP, because there are no more players after me to be deterred by an AGG strategy, and because CO-OP provides him w ith a higher payoff than AGG . B ut then my payoff for IN is 2, whereas for OUT it i s only 1 . Hence I select IN." A� being rati onal , then choose s CO-OP. Player 1 9 should reason as foHow s: "Pl ayer 20 i s rational and should therefore reason as above. Hence, whatever passes betw een m e and Pl ayer A can have no effect on Pl ayer 20's behavi or. S o I should choose IN , because A , being rational and know i ng how Player 20 w i l l behave, wi ll choose CO-OP. " The other players reason i n a si milar manner. The second approach , which S elten cal l s the deterrence theory, has A reasoning as fol low s : "While nothi n g can be done to i nf] uence Play er 20, or perhaps Pl ayers 1 9 and 1 8, an AGO response to IN will deter others from choosing IN, resulti ng in a higher payoff w hen OllT i s chosen. " S elte n mai ntains that though it cannot be decided exactly w hen Pl ayer A should stop choosing A GG, the deterrence theory is more i ntuitively appeal i ng than the i nducti on theory. The paradox i s that rati onal players refuse to accept the valid conclusions of the inducti ve theory and opt for the deterrence theory as a guide to their practical behavior. Resolution. Law ren ce Davies argues that there i s no paradox because t h e gam e th e oreti cal approac h dep e n d s u p o n c e rtai n unj u s t i fi ed i n particular, on a bel i ef by Player A at round n that assumpti ons Player n + 1 cannot be deterred by hi s choice at n. Davi s poi nts out that thi s bel i ef presuppose s a hi erarchy of bel ie fs about beli efs about the rati onal ity of other players, beli efs that may not be rel iable. Gi ven the i nformati on available to aJ l players, it may i ndeed be rational for Player A to choose AG G. Walte r Trockel also suggests that AGG i s rational gi ven the l i mits to t he information possessed by the players . Selte n himsel f believes that the paradox forces us to posi t that there are different levels of rati onal ity in human deci s i on maki ng. Since the game theoretical approach i s no longer the uniquely rati onal one, decisi ons not made in accordance w i th it do not have the paradoxical characte r ori g i na l l y suggested by the game store paradox. READING S Dav ies, Lawrence. " N o Chain Store Paradox."

Theory and Decision 1 8

( 1 985): 1 39- 144. Sel ten , Reinhard.

"The Chain Store Paradox. "

Theory and Decision 9

( 1 978): 1 27- 159. Trockel, Wa] ter. "The Chain Store Paradox Revi sited." Theorv and Decision 2 1 ( 1 986): 1 63- 1 79. �

Dictionary of Paradox

35

( �HANGE , THE ANTIMO NY OF. T hi s puzzle, discus sed i n Antiq ui ty hy both Pl ato and Ari stotl e, chal l enges our concepti on of c hange . Fornlulation. Con s ider an obj ect that i s unchanging ( i n s ome respect) ror a period of ti me and then undergoes a c hange (in that respect) , At the precise i nstant that the change starts, i s the object unchanging or changing? Explanation. The formulation gi ven above i s stated i n general terms hecause, as Ari stotle pointed out, the paradox appl ies to all forIns of change. N evertheless , it i s probably most easi ly di scussed in tenns of moti on. ' r hus consi der an object that is ori ginally at rest and at some instant begi ns 1 0 move. At that i nstant, i s the object (sti l l ) at rest or (already) in motion? N either al ternati ve seems sati sfactory. Neverthe less, as Joseph Wayne S lnith observes, The real difficulty in our question i s as follows. At the point of change i t seems that the

object i s neither in

motion or at

rest. But if for all objects at all ti mes, they are either at rest or in motion, a contradiction follows ( 1 0 1 ) .

Resolution. One re sponse to t he p aradox w ouJ d b e to agree w ith Parmenides that the very concept of c hange i s contradi ctory and, thus, d oes not occur. Another response concedes the contradicti on� but refuses to denigrate sense knowl edge by denying change . In thi s vi ew, however, our theoreti c al descripti on of change i s necessari ly convol uted . Sti ll another respon se, pioneered by Ari stotle (Physics, VIII 8 ) , i s based on "cuts" i n the conti nuum. Thus, the point of change can be seen i n various ways as both (i n the case of motion) the l ast point of rest and the fi rst poi nt of motion, or as the first poi nt of motion w hose antecedent period of rest has no last point. Smith argues that none of these sol uti ons are sati sfactory and that the only way out of the paradox is to recogni ze that Hnature i s arbitrary" ( 1 05). That i s, i n each particular case, one or other state i s realized by chance, as it w ere. S mith claims further that thi s result i s an i ncipient indication of free will in man. Ari stotl e (Physics, VI 3 and 8), following Pl ato (Parmenides 1 5 6C- 1 57A), however, gives another solution: at the point of change � the object i s neither at rest nor in moti on . To avoi d the paradox, thi s view necessitates the denial of Smith ' s premi se that "for all objects at all ti mes, they are either at rest or i n moti on" ( 1 0 1). This denial is indeed intuitively sati sfying because chan ge i s a process ; that is, it i s something that can only happen over ti me. "Instantaneous change" w ould seem, therefore , to involve a category mi stake . We al so need ob serve, how ever, that this di s cussion doe s not i nval i d ate such mathemati cal c oncepts as "i nstantane ous

36

Dictionary of Paradox

velocity," which are technical concepts (involving l i mits) used to modeJ physical situations. READINGS Kretzman , Norman. HAri stotle on the Instant of Change." Aristotelian Society Supplement 50 ( 1 976) : 9 1 - 1 1 4. Smith, Joseph Wayne . Reason , Science and Paradox: Against Received Opinion in Science and Philosophy. London: Croom Helm, 1 986. Ch. S . Sorabj i , Richard. "Ari stotle on the Instant of Change." Aristotelian Society Supplement 50 ( 1 976) : 69-89.

CHISHOLM 'S PARADOX. See the SLI PPERY SLOPE PARADOX.

CLARK'S PARADOXo Introduced by Romane Clark in 1 978, thi s is an

alleged problem for Hector-Neri Castaneda's Gui se-Constantiation Theory. Since description of the paradox requires substantial background i n Castaneda 's theory of predicati on, we onl y note its existence here. READING S Clark, Romane. "Not Every Object of Thought Has B eing: A Paradox in Naive Predication 'Theory." Nous 1 2 ( 1 978 ): 1 8 1 - 188. Landini , Gregory. "Sal v aging ' The F-er i s F' : The Lessons of Clark's Paradox:' Philosophical Studies 48 ( 1 985): 1 29- 1 36.

The Clock Paradox i s si mply Al bert Ei nstein 's predi ction that clocks and other temporal processes run more slowly from the standpoint of an observer moving relati vel y to them than do simi l ar cl ocks and processes in the observer 's own frame of reference. Such a prediction does not viol ate the principle of causality or the laws of logic. Ei nstein's prediction has recei ved some empiri cal affi rmati on by experi ments of J. C. Hafele and R. E. Keati ng, who measured small differences, consi stent with the predictions of rel ati vi ty theory, between the tilnes i ndicated by clocks that had been fJ own around the worl d and those i ndi cated by clocks that had stayed in the laboratory. Nevertheless, these experi ments are not enti rely convi ncing because they i nvol ve considerations of gravi tational fields and thus transfer the paradox from its natural setting i n the special theory to the general theory of rel ati vity. The Clock Paradox is especiall y controversial in the narrativel y sharpened versi on call ed the TWI N PARADOX. Al so see PARADOXES OF TIME TRAVEL.

C LOCK PARADOX, THE .

37

Dictionary of Paradox

COGNITION, THE PARADOX OF. A l t h ough this parad o x i s eas i ly stated, fu l l d i s c us s i on w o uld req u i re an e xte nsi ve bac k g r oun d i n

connectionism . Hence� we w i J i only state the paradox here and observe that the connecti oni st ' s response to the dilemma i nvolved w i ll strongl y c o l o r hi s version of connectionism . A s gi ven by Smol ensky the paradox is as foHows: "In attemptin g to characteri ze the laws of co g nition we are pul led i n two different directions: when we focus on the rul es govern ing high-leve l co g n i t i v e com petence, we are pul l ed tow ards struct u re d s Y ln boli c representations and processes ; w hen we focus on the variance and complex detai l of real i ntel ligent performance, we are pu lled towards stati stical , numerical descri ptions" ( 1 38). ,

,

READINGS Fodor, J . A . , and Z. W. Pyl yshyn. " C onnecti oni sm and Cogn i ti v e Archi tecture: A Critical Analysis." Cognition 28 ( 1 988): 2-7 1 . Gerken, L. , and T. G . B ever. " Li ngui stic Intuitions are the Resul t of Interactions between Perceptua] Processes and Lin guistic Universal s." Cognitive Science 1 0 ( 1 986): 45 7-476 . Smolensky, Paul. HThe Constituent S tructure of Connectionist Mental States: A Repl y to Fodor and Pylyshyn. " Southern Journal of Philosophy 26 Sup. ( 1 987): 1 3 7- 1 61 .

COGNITIVE RELATIVISM, PARADOXES OF. Jack W. Meil and

identifi es a n umber of attacks on a view he cal ls cognitive relativi sm , aU of which reduce to the fol l ow i ng basic dilemma. It i s similar to a paradox identified by C. R. Kordi g. Formulation. Either relativism (subj ectivi sm) appl i es to itself i n which case i t i s onl y re l ativel y (or subjecti vely) true, or i t does not apply to itself, i n w hi c h c ase i t admits that s ome view s are abs ol u tely ( o r objectivel y) true. Explanation. Either horn of the dilemma would seem to be distasteful to the relati vi st. On the one hand, if relativism is onl y r el at i vely true, the But, on the other r ela tivi st s p os i t i on seems to be greatl y undermi ned. hand, if relati vi sm is absolutel y true, that in itsel f would seem to be a proof of the fal sity of relativism. Resolution. Mei l and thinks that the cognitive relativist can grasp either horn of the d i lem ma w ithout self-contradi ction. The relati vist can accept the relative c haracter of rel ati vi sm s o w n truth, or he can insist on its ab s o l ute c haracter by d i st i n gu i shi n g the ses of i t s ow n ki nd ( say, metap hi l osophi cal theses) , w hich are absolutely true, from the ki nd of theses it i s about (say, phi losophical theses) , which are relati vely true. '

'

38

Dictionary of Paradox

Nevertheless , the second horn of this di lemma makes relativism much less i nteresting, because it limits the scope of its val idity and requi res an account of how the special cl ass of absolutely true doctrines escapes the considerations that l eads to cognitive relativism in the first place. Sti ll , many relativists hold that their view is neither self-contradi ctory nor self­ defeating. A relevant metaphor, dating to Sextus Empi ricus, i s that of cli mbing a ladder to reach a point at which it mi ght be kicked away. In other words relativi sts often claim that thei r arguments provide a means to an end that, once attained, allows disposal of the means. T he proposed metaphor, however, is not very enlightening because accepting the apparent contradiction in relativism invol ves accepting just what the rel ativist is convinced i s fal se. READINGS

GaJ l e , Peter.

"Kordi g's Paradox Objection to Radical Meaning Variance

Theories . " Philo s op hy Mei l and , Jack W.

of Science 50 ( 1 98 3 ) : 494-7.

" O n the Par a d o x of C o gni ti v e R e l a ti v i s m . "

Metaph ilosophy 1 1 ( 1 980) : 1 1 5- 1 26.

CONDORCET 'S PARADOX. A paradox of voti ng adduced by the Marquis de Condorcet i n 1785. It is a precursor of the recent PARADOXES OF VOTING .

Formulation. Suppose three voters, V I ' V 2 ' V 3' rank choi ces among three al ternatives, A, B , C, as follows: V 1 ={A , B , C}; V2={C, A, B}; V3={B, C, A}. Then A i s preferred to B by a two-to-one majority, B is preferred to C by a tw o-to-one majority, and C is preferred to A by a two-to-one majority. Therefore, no simpl e pairwi se com pari son can determi ne a preference among the three alternatives. Explanation. Wil liam V. Gehrlein terms the descri bed result a " no­ winner" situation. The "no-wi nner" result depends on the existence of a "cyclical majority." Clearly, V I ' V 2' and V 3 may be equinumerous groups of voters i nstead of merely three i ndividual s . Resolution. Insofar as Condorcet 's paradox show s a l i mitati on of a particu l ar voting mechani sm, it i s potentiall y worri some for architects of democratic systems. The practi cal worry engendered by the possibility of cyclical majorities should be directly correlated to the likelihood of such an occurrence. Reviewing attempts to estimate the probabi lity of cyclical majoriti es, GehrJein finds that the probability of a cyclical majority i ncreases as the number of choices increases, but decreases as the number of voters i ncreases . He ultimately comes out with an estimate of from between one to twelve per cent probability for a "no-wi nner" situation.

39

Dictionary of Paradox

Perhaps thi s probabil ity i s low enoug h for technicians of democracy to i gnore, but see ARROW ' s PARADOX OF S O C IA L CHOICE for a formal statement of the principles at stake when acquiescing i n thi s situation. READINGS '

Condorcet, Marqui s de. Essai sur I 'Appication de l An a lys e a la Probailite des Decisions Rendues a la Pluralite des Voix. Ne w York: Chel sea, 1 973 . Gehrlein, Willi am V. "Condorcet's Paradox . " Theory and Decision 1 5 ( 1 98 3 ) : 1 6 1 - 1 97.

CONFIRMATION, PARADOXES O F. S ee H EM PE L ' S PARADOXES OF CONFIRMATION , HUME ' S PROBLEM OF I NDUCfION, and GOODMAN ' S PARADOXES OF CONFIRMATION .

CONFLICT-OF -DUTY PARADOX. See PARADOXES OF DEOTIC LoGIC , especiaJly PLATO ' S PARADOX and SARTRE ' S PA RADOX .

CONTINGENT LIAR CYCLES . Saul Kri pke u ses these probl elTIs to argue agai nst Tars ki 's hierarchical semantics. Formulation. Consi der the following pai r of statem ents: ( 1) (2 )

Most of Ni xon's assertions about Watergate are fal se. Everything lones say s about Watergate is true.

The joi nt affi nnation of ( 1 ) and (2) i s paradoxical or not depending on the ci rcum stances i n which they are uttered. Tarski 's semantics, therefore, i s too radi cal . Explanation. In most ci rcumstances, the joi nt affirmation of ( 1 ) and (2) would be unexceptional . If, however, ( 1 ) was the only asserti on about Watergate that Jones made and if Ni xon made asserti on (2) as wel l as an add itional number of assertions about Watergate, half of which were true and half fal se, a paradox does ari se. If (2) i s true, then lones's si n gle statement about Watergate is true. But that statement is ( 1 ), so ( 1 ) i s true. By ( 1 ) , therefore, Nixon spoke fal sel y about Watergate more than half the time, so his asserti on (2) must be fal se. Therefore, if (2) i s true, it must be fal se. S uppose, then, that (2) is fal se. T hen Jones must have asserted some fal sehood about Watergate. Hi s only assertion about the subject, however, w as ( 1 ) ; hence, ( 1 ) i s false. T hus, Ni xon could not have spoken false ly about Watergate more than half the ti me ; consequently, (2) must be true. Combining this result w ith the previous one, we find that in

40

Dictionary of Paradox

the circumstances considered (2) i s true if, and only if, it i s false. IZripke argues further that any semantics such as Tarski 's that classifies sentences as m ean ingfu l or not by some property of the sentences themsel ves cannot deaJ adequately with thi s type of paradox. The onJ y way for Tarski to avoid the paradox is to cl ai m that the joint assertion of ( 1 ) and (2) vi olates hi s hierarchical di stinctions. Nevertheless� the joint assertion of ( 1 ) and (2) is only paradoxical i n excepti onal situations. Thus Tarski 's approach invoJ ves� so to speak, too much overkiJ l to be a sati sfactory response to the paradox. Resolution. Kripke's ow n approach i nvol ves building up a seri es of rTIodels in a recursi ve fas hion unti l a fixed point i s reached w hen the next lTIodeJ does not differ from its predecessor. Since the detai l s are too techni cal for reproductio n here , we l i mi t o urselves to observin g that Kripke's approach has been qu ite i nfluentia l i n discussions of the LIAR and related paradoxes. READINGS B arwi se, Jon, and John Etchemendy.

The Liar, An Essay on Truth and

CircuLarity. Oxford: Oxford UP, 1 987. 85 -89. Kri pke, Saul . '" Outline of a Theory of Truth ." Journal of Ph ilosophy 72 ( 1 975): 690-7 1 6.

CONTRADICTION, THE. Thi s was Bertrand Russe) I 's name for what

i s now known as

RUS S ELL ' S PARADOX .

C O NT R A RY.. T O - D U T Y I M P E R AT I V E PAR A D O X , T HE . I ntroduced i n 1 963 by Roderi ck Chi sholm, thi s paradox poses problems

for attempts to elaborate deontic J ogics capa ble of handl ing contrary-to­ duty i mperati ves. The l atter are imperatives that tel l us w hat we should do w hen w e negJ ect certain duti es. They are analogous to contrary-to­ fact conditional s, whi ch tell us what would be true were certain facts other than they are. Contrary-to-d uty imperatives are important because we someti mes fal ter, thereby incurri ng new duties. Formulation. Deonti c logics , which are logics that contain a modal ope rator, 0 , for " ought" or "i s "obl i gatory," usual ly subscri be to the foJ lowing two principles:

(1)

If i t ought to be that some event, a, occurs and if it i s

obli gatory that if a occurs then b occurs , then i t ought to be that b occurs. That i s : If [0 (a) and 0 (if a, then bYI , then 0 (b).

Dictionary of Paradox (II) It i s not true to say, of any a , both t hat and that a ought not to occur. That is: Not [C) (a) and 0 (not a) ] .

41

a ought to OCCUT

The follo\ving four (schematic) sentences are not consi stent ,vi th principles ( I ) and

(II): (1) (2)

(3 ) (4)

lJ 0

(a) (if a, then b)

If not a, then () Not a.

(not b)

Explanation. T he sentences ( 1 ) - (4) capture contrary- ta-duty si tuati ons .

I n parti cul ar, if a i s an act that \ve are obl i ged to do, but \vhic h \ve forgo doi ng, sentence (3 ) is a con trary-to-duty i mperati ve tel l i ng us ,,,,hat ne,,, obl i gation has been i nc lllTed by the fai l u re to do a . Consi der the foll o\vi ng exalnple (the s entences are nun1bered according to the correspondi ng schelnatic sentences above) :

(3 )

I ought to visit my elderly mother. It ought to be that: if J go v isit my mothe r then I tell her that I an1 cOIning . If I do not go then I oug ht not to tel1 my mother I am

(4)

I do not go.

(1)

( 2)

,

comIng . •

According to princi ple I am coming. Yet,

(I), ( 1 ) and (2) entai l that I ought to tel l iny mother

(3 ) and (4) entail that I ought not to tel l her that I am

com i n g . What shoul d be done? Ought I , or ou ght I not tell my n10ther I am conl i n g ?

I ntuiti vel y, the ans\ver i s clear that

I ought not, since, as

Chi shol m puts it, " n10s t of us do neglect our duti es from ti me to ti n1c yet i t i s reasonable to bel ieve that \'le should make the best of bad situati ons to \\' hi ch our n1i sdeeds have led" (33 ) . I n any cas c , thc t\VO resul ts

I ought

to

and that

I ought not to tell my Inoth er that I ' m coming

that are

inColnpati bl e \'lith pri nciple ( I I ) . Thus , our i ntui tion i s at odds with the resul ts of forn1al system s of deontic logi c .

For eval uati ons of speci fic

forn1al systems in terms of this paradox , see Tonlberlin and Dece\\,. Reso lution. The paradox does not ari se i n von Wri ght 's ori gi nal version

of deonti c l ogic because it does not countenance sentences of the form " I f G,

then O(b) . " But such l ogics , as C h i shol m poi nts out , arc inadequate i n

that they cannot deal \\lith contrary- to-duty i mperati ves .

I n logics that

are ri ch enough to deal "v ith s uch irnperati ves , there i s at p re sent no accepted resolu tion to the Contrary- to-D uty hn perati ve Paradox. t\V O pri nc i p l es

(I)

The

an d ( I I ) hav e not , hov/cver, gone unquestioned .

42

Dictionary of Paradox

Chisholm observed that the sentence O(if a, then b) occasions paradoxical results parallel to the PARADOXES OF MATERIAL IMPLICATION . Also, it has been argued against principle (I I) by van Fraassen, for example that it is not unreasonable to have conflicting obligations. Nevertheless, thi s approach is not entirely satisfactory if one accepts the notion of an all-things-considered duty that overrides confl icting obli gations. READINGS Chisholm, Roderi c k NL "Contrary-to-Duty Imperatives and Deontic Logic. " A n a lys is 24 ( 1 963) : 33 -36. Decew, Judith Wagner. "Conditional Obligation and Counterfactuals . " Journal of Philosophical Logic 1 0 ( 1 98 1) : 55-72. l\1ott, Peter. "On Chisholm's Paradox. " Journal of Philosophical Logic 2 ( 1 973) : 1 97-2 1 1 . Tomberlin, James E. "Contrary-to-Duty I mperativ es and Conditional Obligation. H Nous 1 5 ( 1 98 1) : 357-376 . Von Fraassen, Bas. " Values and the Heart's C:ommands . " Journal of Philosophy 70( 1 973) : 5- 19.

COPERNI CUS ' PAR AD OX .

Perhaps the most notorious scientific paradox of the last millennium is that of Nicolaus Copernicus ( 1 473 1 543) to the effect that the earth revolves around the sun and not vice versa . A l though we no longer refer to the heli ocentri c theory as paradoxical because it is nearly universally accepted as true in scientifically tutored nati ons, there still appears to be something paradoxical about the earth revol ving around the sun because not only do we speak of the sun rising and setting but it actually appears to ri se and set from our terrestrial vantage. READING Boorstin, Daniel 1 . The Discoverers. New York: Random, 1 983. Ch. 38.

CROCODILE 'S DILEMMA, THE .

This ancient puzzle is closely related to the LIAR. A crocodile steals a child and upon the mother 's entreaties allows : "I will return your child if you guess correctly \vhether or not I will return your child". The mother replies: "You wil l not return my child. " What should the crocodile do? If it returns the child, the mother has guessed incorrectly, so the chi ld should not be returned. (The condition is understood to be both necessary and sufficient. ) But if the crocodile does not return the child, i t is bound to return it because the mother has guessed correctly. Perhaps Plutarch 's report that the Egyptians worship God symbolically in the crocodile, for it is the only animal \\Jithout

43

Dictionary of Paradox

a tongue, like Divine Reason, which does not need of speech (I sis and Osi ris 3 8 1 B ) bears on the si gnificance of thi s dilemma. READING

Suppes , Patrick. Axiomatic Set Theory. 2nd ed. New York: Dover, 1 972. Ch. 9. Plutarch 's De lside et Osiride. Ed. J. Gwyn Griffiths. Cambridge: U of \\Tales P, 1 970.

CURRY PARAD OX, THE. This paradox rai ses difficul ties for the so­ cal led axiom of abs traction (al so cal l ed the comprehen sion princi pl e) i ncorporated into early, nai ve versions of set theory.

Formulation. A ny proposi tion, p, can be deduced from the axiom of abs tracti on .

In particul ar, p may be a contradiction ; thus , set theory

contai ning an unrestricted axi om of abs traction i s i nconsi stent.

Explanationo T he axiom of abs tracti on hol ds that for any concept there i s a set of things fall i ng under that concept. I n tenns of set membershi p, an obj ect belongs to the set determined by a gi ven concept ( property) if,

and

onl y i f, the object fall s under the concept (has the property) .

An

obj ect bel ongs to the set of artichokes, for exampl e, if, and onl y if, i t i s an arti choke . Cons ider, then, the fol l owing complex concept (about sets ) :

(*) P is true whenever x e x . B y the axi om of abstraction, there i s s ome set, C, that i s determi ned by the concept ( * ) :

( 1 ) C is the set of sets x such that p is true whenever x e x. Cl early, if C e C , then C falls under the concept ( * ) ; hence ,

(2) P is true \-"henever C e C. [Note that in the formal proof of (2) , the antecedent is conditi onal ized so that (2) only depends on defi ni tion ( * ) and the axi om of abstraction. ] Yet si nce C fal ls under the concept ( * ) , C must belong to the set determ ined by the concept ( * )

\vhich, accordi ng to

(3) C e C. B ut (2) and (3 ) , by modus ponens, entai l

(4) p.

( 1 ) , i s the set C i tself. Thus ,

44

Dictionary oj Paradox

Thus, since p is

p rop osi ti on \v hatever, sct theory \vith the axiom of abstraction i s i n c on sis te nt Resolution. T h e artichoke example above sho\vs the very great i ntui ti ve plausibi l i ty of the axiom of abstraction . Hence, there has been son1e rel u c tance to give it up. Nevertheless , the most successful attempts to resolve this paradox focus on vvays to abandon or 10 restrict the abs trac tion princi pl e so that the offending sets cannot be formed. Some c om me n ta tors poi nt out the self-referential elelnent in its formul ation, common to both the LIAR and RUSSELL' S PARi\.DOX. The latter is a separate but related attack on the ab s trac ti on principle. For further discussion, see RL7SSELL' S PARADOX. any

.

REAl)IKGS C-:urry, II. B . , 1 . R. Hindley, and 1. P. Seldin. j-\ msterdam: North I Iolland,

Cotrzb inatory Logic, Vol. 2 .

1 972. (�h. 1 2 .

Fitch, Frederick B . "A. Iviethod for Avoiding the Curry Paradox . " Essays in

Honor ofC�arl G. Helnpel : A Tribute on the Occasion of His SiXly-F'ifth Birthdav. Ed. Nicholas Rcscher. Dordrecht: Reidel, 1 970. 255-265 . Goldstein, Laurence. "Dpimenides and Curry. " AnaZvsts 46. 3 ( 1986): 1 17- 121 . Meyer, Robert K. , Richard Routley, and 1. I\1ichael Dunn . "Cuny 's Paradox. " .'

Analysis 39. 3 ( 1979): 124- 1 28.

p ar adox of i rra ti onality is a problem described by D onal d Davidson for his ovvn th e ory of i n te rpre tat i on (or tran s lati o n ) and appl ies onl y to Davidson 's theory or to others sharing its central features Formulation. " The under lyi n g paradox of i rrat i o nali ty from \vhich no theory can e n tir ely escape is th i s : if \ve expl ain [i rrati onal i ty] too \vell , "ve turn i t i nto a con c e al e d form of rati onal ity � whi le i f we assign i nc o h ere nce too glibly, we me rel y comp ro m i s e our abi l ity to di ag n os e irrationality by withdra\ving the background of rati onal ity needed to j ustify any d i a g n os i s at al l" (Davidson 3 03 ) . Explanation. Davi d s oni an vie\\' s of i n te rpre tati on require the assumption that the s p eakers of a l anguag e \v hich is to be translated (the so u rce lan guage) are both rati on al and, on the w h ol e , s peaker s of truth. The assumption is de e me d necessary because the basic data for a translator consist i n re ac ti on s of agreement and di sagreement to specific utterances o n t h e part of s p e akers of t h e s o u rc e l an g u a g e . To e s tabl i s h correspondences betvveen the truth conditions of the source l anguage and the language into vv hich the translation is made ( the targ et lan g uag e) such agreement and disagreement must be compared \vith the reactions of the speakers of the target l ang uage i n the same environmental conditions . D AVIDSON 'S PARADOX OF IRRATION ALI TY.

The

.

,

,

45

Dictionary of Paradox

According to this conception, too many irrational and/or false attributions vv i ll underm i ne the legiti macy of the proposed translati on manual . Nevertheless, some all o\vance must be n1ade for errors on the part of the source language speakers , and David son finds such errors acceptable insofar as they can be explained. Yet the consequence of thi s allo\vance ) is that the only kind of irrationality that is acceptable i s that which can bel : explained and is hence rational after all . Resolution. Davidson believes that some types of error can be allowed for vvithi n the process of translation, but that others absolutely undermine confidence in a given translation manual . Errors of logical consistency and of agreement vvith statements about di rectly observable features of the \vorld would seriously j eopardize translation ; but those arising from , say, shortcomings in perceptual systems \vill be explainable through em piri cal psyc hol ogy. B y di stingui s hi n g betw een acceptable and unacceptab1 e error, Davidson believes that the paradox of irrationality may be largely accommodated. Nevertheless, Davidson 's paradox mi ght be avoi ded al toge ther if expl anati on can be d i s ti n g u i s hed from rational i zation. Thus David Henderson suggests that the paradox ari ses because Davidson 's theory only allo\vs for explanation of the speaker 's beliefs and actions "in terms of the rationality of those beliefs and actions" (365) . Were, say, causal explanations allo\vcd to support the translation manual , the paradox of irrationality might not threaten the translation project at all. READINGS Dav id�on, Donald. Essays on Actions and E'vents.

()xford: Clarendon�

1 980. Henderson, Dav id. "A Solution to Davidson's Paradox of Irrationality. " Erkenntnis 27 ( 1 987) : 359-69.

DEMOCRITUS ' DILE MMA.

'This mathematical paradox \vas attributed

to Democri tus by Plutarch. Formulation. I n Heath 's translation the paradox runs as follo\vs. "If a cone \vere cut by a plane parallel to the base [by which i s clearly lneant a plane i ndefinitely near to the base] , \vhat must \ve think of the surfaces forming the sections? A re they equal or unequal ? For, if they are unequal , they vvill inake the cone irregular as having many indentations, like steps, and unevennesses � but, if they are equal, the sections \vill be equal , and the cone \vill appear to have the property of the cylinder and to be made up of equal , not unequal , circles, \vhich is very absurd" ( 179- 1 80) .

Dictionary of Paradox

46

Explanation. Democri tus seems to think of the cone as being made up

of many thi n circular disks piled on top of each other. Whether he thought there were an infinite number of them i s not altogether clear. In any case, i f \ve choose two disks that are i ndefinitely close to each other, they must be either equal or unequal . If they are unequa] , the cone would become tiered , like a wedding cake. But if they are equal , the cone would become a cylinder. Resolution. Dcmocri tu s ' res ponse to this problem i s not known. Perhaps more importantly, hov;ever, Archimedes attri butes to Democritus t\VO theorems about VOI UlTICS: ( 1 ) the volume of a cone i s a third of that of the cyli nder on the same base and of the same hei ght and (2) the volume of a pyramid i s a third of that of the corresponding prism. According to Heath, it is probable that Democri tus used the idea of a solid being composed of an infinite number of plane sections in order to obtain these results, thereby anticipating Cavalieri 's Principle. READING Heath, Thomas L. A History afGreekMathematics. Oxford: Clarendon, 192 1 .

DEONTIC LOGIC, PARADOXES OF.

Deonti c logics are those that contain a modal operator for "ought" or "is obliged to," so these paradoxes concern the project of fonnalizing ethical thinking. See A QUIST' S PARADOX, C LAR KE S PARADOX , the CONFLICf-OF-DuTY PARADOX , the CONTRARY -TO­ DUTY I MPERATIVE PA R AD OX the EPI STEMIC OB L I G AT I O N PARADOX , the EUALTHUS , the PARADOX OF GENTLE MURDER, the GoOD SAMARITAN PARADOX, the JEPHTA DILE�1MA , the PARADOX OF THE KNOWER (IN DEONT I C LoGIC) , PRIOR ' S PA RADOXES OF DERIVED OBLIGATION , the ROB BER S PARADOX , Ross ' s A NTINOMY , SARTRE ' S PARA D OX and the V rCfIM ' s PARADOX. '

,

'

,

READINGS A.quist, Lennart. Introduction to Deontic Logic and the Theory a/Normative SystenlS. Naples: Bibliopoli s, 1 987. AI-Hibri, Azizah. Deontic Logic: A Comprehensive Appraisal and a New Proposal. Washington: lJP of America, 1 978. Castaneda, Hector-Neri . "Aspectual Action and Dav idson 's T heory of Ev ents . " In Actions and Events . Ed. Ernest LePore and Brian P. McLaughlin. Oxford: Blackwell , 1 985. 294-3 10. - - . "Tomberlin, Fre ge and Guise Theory: A Note on the Methodology of Dia-Philosophical Comparisons. " Synthese 6 1 ( 1984) : 135- 148. l-lilpinen, Risto, ed. Deontic Logic: Introductory and Sys tema t ic Readings. Dordrecht: Reidel, 1 97 1 . ,

Dictionary of Paradox I H�:TE RRENCE, THE PARADOX OF.

47

This i s an old quandary of

I l l lcl ear deterrence strategy, expressed by David Gauthi er as a paradox of

.� ·,a l ne theory, i n \v hich al l actors are assumed to be ful l y rational . Fo rmulation . I n order to di ssuade the enemy from l aunching a nuclear

I i rst stri ke, i t is best for the agent to adopt the policy of respondi ng to a I i rst stri ke \vith a retal iatory nuclear strike. Neverthel ess , i f the enemy d ( lCS launch the n uclear fi rst stri ke , the di sastrous consequence s of a re taliatory strike make i t better for the agent not to retali ate. Yet, if it i s

hetter not to retali ate , the enemy may presuppose that the agent \\'ill follow j he

rational course and not retal i ate , \-vhi ch makes it more rational for the

e nemy to i nitiate a nucl ear fi rs t stri ke. Explanation. A s lon g as no enemy attack has occurred, the policy of l aunchin g a retali atory nuclear stri ke i n res ponse to a fi rst s trike i s the hest pol icy because i t most reduces the enemy 's reasons for launchi ng such a stri ke . B u t once the enemy has attacked and, thus , the s trategy for deterrin g the nucl ear fi rst strike has [ai led , the retal iatory pol icy i s no l on ger rational . I n [act, i f the age nt were to execute his pol icy of a nuclear retal iati on, al l that he \vould accomp1 i sh \-v ould be to con1plete the J estructi on of the worl d . Yet , i f it would be irrati onal for the agent to execute hi s retal i atory pol icy and i f the agent may be expected to act rati onal l y, the p utati ve threat of retal i ation ceases to be a deterrent. Moreover, by symmetry of reasoni ng , i n order to avoid bein g attacked hi m self and consequentl y put i n a positi on \v here his ow n pol icy of retal iation would be irrati onal , the enemy has incentive to l aunch a first strike. Hence, the deterrent may actually cause an attack. Resolution. Gauthier resolves his own paradox by denying the premise that it i s irrational for the agent to launch a retali atory strike should the policy of

nuclear deterrence faiL He argues that the costs and benefits of the retaliatory strike had already been evaluated by the agent at the ti me of the adoption of the retaliatory policy and, thus, need not be reconsidered after suffering the

first strike. I ndeed, according to Gauthier, one cannot evaluate one 's activities from the perspective of the present moment, but must do so from as \-vide a perspective as possible.

Hence, in order to insure the credibility of the

deterrent, implementation of the retal iatory strike is rational . David Luban counters, however, that the rational agent cannot ignore the ne\\' state of affairs occasioned by the first strike. I t i s in fact counterintuitive that the agent would remain rational by perfonning the suboptimal act of completing the destruction of the world when he has better options available. Luban concludes that the paradox has no resolution and that either the utility of deterrence or the rationality of the agent has to be given up.

I I

48

Dictionary of Paradox

RE�;\DI NGS

Gauthier, Dav id. "Deterrence, Maximization, and Rationality. " Ethics

94

( 1984) : 474-495. l-luban, Dav id. "T he Paradox of Deterrence Reviv ed. " Philosophical Studies 50 ( 1 986) : 1 29- 14 1 .

DEVIL'S OFFER, THE . Proposed by Edvvard J. Gracely, thi s exi stential di l e mm a i nvol v e s a p aradox of the i nfi n i te \v i th son1 e the ological . . pre sUpposItI ons . Formulation.

dies

Ms . C

and unfortunately goes to hel l , but a devil

approaches her \vith the opportunity to pl ay a game of chance .

I f she

\vins the game she can go to he av e n and if she l oses she must stay forever in hell , but she can onl y play the game once.

If she plays on the fi rst day,

she has one hal f chance of \vi nni ng, i f she plays on the second day she has t\VO thirds chance of � i nning , on the thi rd day three quarters chance , and '

so on. The q u es tio n i s : When i s i t m ost rational for her to play?

Explanation . If she waits for a year be fore playing, her c hances of \vinn i ng are . 997268 , but i f she \vai ts for a year and a day her chanc es \vi l l increase by . 000007. Al though \vai ting thi s extra day i ncreases the likel ihood of \vi nning v e ry I i ule, the re\vard of \vinning i s thought to be i nfini te o Indeed, a finitely 1 arge extra chance of ""'inni ng son1cthing infi ni te i s worth more than the presumably fini te suffering of Ms . C 's spendi ng one more day in

hell . (More techni cally, the uti lity of \v ai ti ng one more day chance of \\linni ng times the infi ni te payoff

the extra

\vi l l al\vays be infi ni te. )

Yet if i t is ah;vays \vorth\vhil e V\7ai ti ng one lnore day, then there \v o uld be no l i n1it to hovv' l ong she should \Nai t and Ms. C \vould end up re111aining

in hell forever i n order to i ncrease her chances of leav i n g i t ! .

Resolution . I f hope of heaven lessens the suffering in hell , then \vhatcver the odds it might be fool ish to play the game and risk l osing that hop e and heaven too . Compared to \vaiting i n hell forever \vi thout hope of escape,

deferring the playi ng of the gan1c forever mi ght be the \visest course.

Heaven

i tself n1ay have no plca�ure comparabl e to that of antici pating leaving hel l

for heaven. Given the enormous probability of \vinning that is eventually built up, ho\vever, such a soluti on seems l ike a perverse fonn of self-torment. The root of the problem is that \ve are unable to discrimi nate among the chances of \vinni ng because the infinite payoff Inakes each increase in our chances infinitel y desirabl e. COlnpare PASCAL' S WAGER.

REA.DINer GraceI y, f:ihvard 1.

"Playing Games vvith Eternity: The j )cv il ' s ()ffcr. "

Ana lysis 48 ( 1 988): 1 1 3 .

Dictionary a/Paradox

49

DICHOTOMY, THEo A l so cal led the Race Course and the B i section

Zeno 's first arg lun e nt agai n st motion. I t \vas menti oned by Ari sto tl e at Physics 239b l l - 13 and 263a4-6 as \vell as by Si m p l ici u s Formulation. In order for a n10ving object to reach a di stant po i nt , it must al \vays transverse half the rem ai n i n g di stance . Moti on i s thus i m possi ble because i t requi res trav e r si n g an i nfi nite seri es of di s tan c es one at a time. Expl anation o Consider a runner he adi n g to\vard the fi nish l i ne on a race course. I n order to fini sh the race , he ln u s t fi rst reach a point In i d \v ay between hi s present posi tion and the fi n i sh line. T he runner, once he has reached the mi d\\'ay poi nt, still has hal f the race to com pl ete ; but he cannot do so \vi thout reaching the t h ree - q u arte r poi n t, \vhich i s haU\vay behveen the In i dpoint and the finish line. T he process is repeated vv i thout end and , hence , there will always be a residual interval bet\vcen the lunner �s po s i ti on and the fi nish l in e m aki n g it i 1l1 poss i bl e for hin1 to compl ete the race. T h e parad ox i s often forn1 u l ate d as a re g re s si on in the fo ll ovv i n g manner: the ru nn e r m us t first reach the midpoint ; but, in order to do so, he m u s t firs t reach the q u arter po i n t ; and so on \\ i thou t end . The conclusion "", o u l d then be not that the runner c an n o t fi ni sh the race , but that he cannot even start. GTegory Vlastos has argued effecti vel y that the ori g i n al parado x \vas most l i ke l y to have been a progressi on, as we presented it above . Formul ated as a p ro g ressi o n i ts bas i c affinity to the A chilles Paradox i s readi l y seen . The arg ument can be sUlnn1ari zed as f01 10\\1 8 : Paradox, thi s i s

.

,

'

,

( 1) (2) ( 3)

Fi nisbing any Dl0ti on requires crossing an infi ni te sequence of successive di stances. The crossing of an i nfinite sequence of successiv e di stances cannot be cOlnpleted. Therefore no motion can be finished.

Resolution. A s may be expected,

the responses to the paradox have been, in the main, silnilar to the responses to the A c hi l l e s A ri s totl e di s ti n guish i ng behveen ac tual infinite and poten ti al infi nity, denied premi se ( 1 ) . Charles Chihara advances similar considerati ons. I n contrast, the rnaj ori ty of modern interpreters deny premise (2) , \\rhich the y take to mean one of the follo\ving: .

( a) (b)

,

crossing the infinite sequence cannot be done because it \vould take an infinite amount of tinle; crossing the infinite sequence cannot be done because the sequence has no final lnelnber.

I n either case , the e x i st e n c e of convergent seri es ( see the ACHILLES ) i s used to

falsi fy

the prem i se . Nev ertheless, Leo

Groarke argue s that si nce

50

Dictionary of Paradox

the runner must traverse the infi nite sequence of di stances one by one that is, he traverses them a fi nite number at a time he i s unable to get beyond a finite n umber of interval s, regardl ess of the fact that the seri es is convergent. Hence, according to Groarke, (b) has not been fal sified and the paradox remains a genui ne puzzle. See ZENO ' S PARADOXE S . READINGS Aristotle. Physics. VI IX. Chl hara Charles . "On the Possi bility of Completing an Infinite Process . " Ph ilosophical Review ( 1 965 ): 74-87. Groarke, Leo. "Zeno's Dichotomy: Undenni ning the Modem Response.�' Auslegung 9 ( 1 982): 67-75. Thomson, James. "Infinity in Mathematics and Logic." Encyclopedia of Philosophy. Ed. Paul Edwards. London: MacmiBan 1 967. Vol. 4, 1 831 90. Vlastos , Gregory. "Zeno's Race Course. " Studies in Presocratic Philosophy. Ed . R. E. Allen and D. 1. Furl ey. London: Routledge, 1 974. Vo] . 2, 20 1 220. ,

,

DILEMMA. A l em m a i s somethi ng taken for granted (from Greek lambane in,

to take), and a dilemma is a "doubl e lelnma," or rather a ki nd of "doubl e-take" between two alternatives each of w hi ch i ssues in equally undesirabl e consequences The di l emma has the following logical form: Either A or

B.

If A, then C. If B , then D. Th e refore C or D. ,

When C and D are identical , the argument is even more striki ng from a psychological poi nt of view. In Medieval phi losophy, di l emmas were called argumentuln cornuturn, because they are l ike bul l s that w i l l toss one w hi chever horn one seizes; and we still speak of the alternatives A and B as the " horns of a di l emma. " A l though modern logi ci ans tend to defi ne the dilemma as any argument of the given form, regardless of the repugnancy or not of the conclusi on, the traditional conception of the dilemma reveal s its close connection with rhetoric and, l i ke quandaries and predicaments, requires us to choose between equally repugnant courses of acti on. When one is caught i n a dilemma, it i s sai d that he i s "impaled on the horns" of the dilemma. "Escaping between the horns" destroys the dilemma by show i n g that the altenlatives propounded in the first p remi se are not exhaustive; the thi rd alternat iv e , naturally, w il l be helpful

Dictionary of Paradox

5]

onl y in so far as i t does not issue i n undesi rable consequences. "Taki ng the dilemma by the horns" i s show i ng that at least one of the alternati ves does not J ead to the purported conseq uence ; thus, it amounts to fal sifying either the second or the third premise . FinaU y, the most rhetorical l y sati sfying way to counter the di lemma i s to "rebut" i t (for an interesti ng example see THE EUALTH US ) . Thi s consi sts in constructing a new di lemma using the same alternatives, but i ssui ng i n consequences acceptable to the speaker and repugnant to his adversary (who had propounded the dil emma being rebutted) . Many paradoxes are expressible as dilemmas and vice versa; i n thi s regard. See B URIDAN' S Ass, DEMOCRITUS ' DILEMMA, the PARADOX OF COGNITIVE RELATIVISM, the CROCODI LE' S DILEMMA , the DEVIL' S OFFER, the JEPHTA DILEMMA, the PARADOX OF NAMING, PASCAL ' S WAGER , the SANCHO PANZA , and ZENO ' S DILEMMA . READING Eaton, Ral ph M. General Logic. New

York:

Scri bner, 1 93 1 . 1 9 1 -200.

DOGMATISM PARADOX, THE. This epistemol ogical paradox was

first formali zed by Gil bert Harman . Formulation. "If I know that h i s true, I know that any evidence agai nst h i s evidence agai nst something that is true ; so I know that such evidence i s mi sleading. But I shoul d disregard evidence that I know i s mi sleadi ng. So, once I know that h i s true, I am i n a position to di sregard any future evidence that seems to ten against h. (Harman, 148) Explanation. It is certainly a reasonable policy to disregard mi sleading evi dence. Again, evidence against something that i s true is bound to be mi sleadi ng. Furthermore, if I know something to be true, i t m ust be true. Hence al l evi dence against what I know to be true ought to be di sregarded. Yet i s seems overly dogmatic to i gnore all evidence that runs counter to what one knows to be true. Resolution. Both Norman Malcolm and Peter Unger support arguments of thi s kind , but with differing results. For whereas Malcolm thinks there are thi n g s so w e l l known th at countervai l in g evi d e nce s h ould b e d i sre garded, Unger believes that the argument shows that clai ms to knowledge are i ntrinsicall y dogmatic. Since, however, dogmati sm i s irrational , so are claims to knowledge. Other phil osophers try to resol ve the paradox by establishing that there are flaw s in Harman's argument. Carl Gi net, for example, uses the phenomenon of the acqui sition of a piece of knowledge resulting in an overaJ l loss of knowledge to show that i n a non-deductive argument we may come to know the antecedent of a

52

Dictionar)' of Paradox

conditional and yet be unable to detach the consequent by Inodus ponen s . Thi s happens when the antecedent and the conditional are so rel ated that not knowing the truth or fa] sity of the antecedent i s a condition for assertin g the conditional . Simp l ifying a bit Ginet's clai m i s that the conditional If I know that Ii i s true , I kno'rv that any evi dence against lz i s evidence agai nst sOlneth i ng that i s true.

can on l y be asserted if we do not know t h at h i s true. Thus, by asserting the antecedent, we l ose the conditi onal and , therefore, cannot progress by the fam i l i ar In odus pone!}s argulnent. rrhe very famili arity of the argument, h o w e ve r, l eads LI S to d i sre gard the e xcepti onal character of the cited cond i tional and accept Hannan 's paradox . Roy A. Sorensen uses Frank J ackson 's theory of the indicati ve conditi onal to argue that thi s type of "excepti onal" conditi onal i s in fact quite common . READING S Ginet , Carl .

" Know ing Les s by Knowing More. '�

Ph ilosophy

V:

Studies in Episternology.

Midwest Studies in

Ed . Peter A . French, Theodore

E. Uehl ing , Jr. , and Howard K . Wettstei n . Minneapolis: U of Minnesota

P, 1 980. 1 5 1 - 1 6 1 . Harman , Gi l bert.

Thought

Pri nceton: Pri nceton UP, 1 97 3 .

Malcolm, Norman . " Knowl edge and B el i ef. "

Knowledge and Certainty.

Engle\vood Cl iffs, NJ : Prentise-Hall , 1 963 . 67-68.

Sore)) , Tom . " Harman ' s Parad o x . " Mind ns 90 ( 1 98 1 ) : 5 57-575. Sorensen , Roy A .

" Dogmati sm, J unk Kno w l ed g e , and Conditi on al s . "

Philosophical Quarterly 3 8 ( 1 988): 43 3-454. Unger, Peter. Ignorance. Oxford: Cl arendon, 1 975 . 1 05- 1 14.

DONNE 'S PARADOX OF THE HOLY GHOST. The schol arship that

takes Donne 's paradoxi caJ thi nki ng abo ut reJ i gi on as a theme i s that extremely rare sort of l iterary study that uses parado x as a key word and actual ly identifies somethi ng l i ke a paradox i n i t. Formulation. In hi s study of paradox in Don ne 's thought, Jerome S . Dees cites four paradoxical formulations. There i s no s al v ati on but by fai th � nor fai th but by hearing, nor hearing b u t by preachi ng (Donne , VII 3 20). (2) H ere w as a true Tra n s u b s t a n ti at i on , an d a n e w Sacrament. These few word s , Saul, Saul, why persecutest thou me, are w ords of Consecrati on ; After these words , Saul was no 1 0nger Sau l b ut he w as Christ. . . (VI 209) . Knowledge cannot sav e us , but \ve cannot be saved (3) wi thou t Knowl edge (II I 359).

(1)

53

Dictionar.\' of Parada.).'

(4)

The Hol y Ghost falls, through US� upon yo u al so, so, as that you may, so , as that you must fi nd it in yoursel v es (VIII 267 ) . Explanationo

T he p ar ado x in the first statement l ies i n the ci rc u m s tan c e l hat the p r e ac h e r i s necessary for sal vati on because the w ord can only be I l card t h ro u gh p reach i n g Central to the reformed Protestant doctrine ( ' x postulated by Donne , how e ve r i s the idea that sal vation is effected by t h e unme d i ated g i ft of grace g i ve n d i r ec t l y to the i n d i v i dual . Thus, a c c o rdi n g to Dees, th e pre ache r i s absolutely n ec e ssary for salvation , whi le h e i n g ut te r l y useless as a causal a g e n t i n atta i n i n g that sal vati on . With r e ga rd to the second state ment, Dees remarks: "What exerci ses Don ne ' s i 1 l1 a g i n a t i on themati cal ly and st ruc t u ra l l y i s th e pa r a do x that the a c c u s a t i o n i s i t s e l f t h e tra n s u b s t a n t i ati o n , that t he l an g u a ge of condemnati on i s i tself the very means and seal of acceptance and uni on" ( X 1 ) . Donne resolves the paradox i n the third statement by fo llow i n g A ug usti ne 's postulate of natural reason and regenerate reason . Fi n aU y, t he l ast statelnent, whose paradoxi cal ity su bsu mes that of the fi rst three statelnents, asserts that altho ugh the H ol y Ghost is present i n e a ch Chri stian by vi rtue of the sacrament of b a pti s m it i s on] y t h ro u gh the m i n i st rat i on s of the p r ea ch e r that the Holy Ghost 's presence can be recal l ed by t h e .

,

,

,

,

fai t hfu l . l�esolution.

From the standpoi nt of faith , t h i s i s a veri di cal paradox . T he preacher, as bearer or vehicle of the S p i rit, d oe s not merely provi de doctrine to be und erstood by reason but engages i n a dialogue w i th h i s p a ri shio n e r s by which the memory of Ho l y G h o s t re s i di n g in them s i n ce bapti sm i s activated. READINGS Dees, Jerome S . "Logic an d Paradox in the Structure of Donne ' s S ermons . " South Central Review 4 ( 1 987) : 78 -92. Donne, J ohn . The Sennons of John Donne. 1 0 vols. Eel. Georg e R. Potter and Evelyn M. Si mpson. B erke1 ey : U of Cal ifornia P, 1 95 3- 1 962 .

DUMITRIU'S ANTIMONY OF T HE THEORY OF TYPES . The

The o ry of Types was elaborated in order to avoid c e rta i n parado xes, foremost among w h i c h was RUSSELL' S PARADOX . The theory was proven c o n s i s t e n t ori ginal ly by J ac q u e s Herbrand and l ater by G erhard G en tzen. E ve n so, A nt o n Dumitri u constructed a paradox i n the T h eory of Ty pes , t h e r e b y p u rp ort i n g to show it to be inconsistent. S ub se q u e n tl y how ever, D u mitri u ar g u e d that hi s p ara d o x was not due to an in c o n s i s t e n c y in the fh e o ry of Types, but rather to its i nco m p l e t e n e s s and th at t h e paradox ,

,

'

54

Dictionary of Paradox

could be avoided by i nvoking some well known conditions on definiti ons. It is perhaps thi s aspect of the paradox that is Inost i nteresting since the Theory of Types itself does not seem to be among the most favored ways of avoiding the original paradoxes such as Russell 's. READING S Church , A. "Review on A . Dumitri u ' s Anti nomy of the Theory of Types."

Journal of SYfnbolic Logic 37 ( 1 972): 1 94. Dumi tri u, A . "The Anti m on y of the Theory of Types . "

Rev iew 2 ( I 97 1 ) : 5 1 -54.

International Logic

- - . "The Anti mony of the Theory of Ty p es and the Sol uti on of Logico­

International Logic Re view 5 ( 1974): 8 3 - 1 02. Feher, Marta, " Is T here an Anti nomy in the Theory of Types?" International Logic Revie w 3 ( 1 972 ) : 1 26- 1 28. Mathemati cal Parad o xes . "

EDEN, THE PARADOX OF. Thi s reJ i gious paradox concerning the

fall of man was suggested by Richard R. La Croix. Formulation. Before eati ng the fruit of the tree of the knowJ edge of good and evi l , Adam and Eve either knew that obedience to God i s good and di sobedience is evi l or they did not know. If they did not know, they cannot be blamed for disobeying God by eating of the tree of the know ledge of good and evi l , and God should not have puni shed them by casting them out of Eden. If they knew, then they al ready possessed knowledge of good and evi l , and there w ould have been no temptation for them to eat the forbidden fruit. Furthermore, God would have known they possessed knowl edge of good and evi l , and he shoul d not have made not eati ng the fruit and gaining what they already possessed a test of their righteousness. I t was in fact unj ust for hi m to put them under such an idl e prohi biti on. Therefore, whether or not Adam and Eve knew that obeying him was good and di sobeying was bad, God acted unjustl y i n the episode. Explanation. By applying legal standards to the first two chapters of Genesis , i t i s argued that j ustice i s not a necessary attribute of God. Resolution. In response to La Croi x, Allen Howard Podet makes three main points. First, the interpretation assumes that the expul sion from Eden was a punishment for disobedience, but the language of the text does not a l l ow us this presupposition. Second, it assumes that know i ng that it i s good to obey God and evil to di sobey Hi m i s all that there i s to know about good and evil, but there is much el se that a knowl edge of good and evil might invol ve. Third, the i nterpretation assumes that God was testing Adam and Eve 's righteousness, but He mi ght have been testi ng something else, such as their obedi ence, or even nothing at all .

55

Dictionary of Paradox

READINGS I ,a

Croi x, Richard R. "The Paradox of Eden." International Journal for the Philosophy of Religion 1 5 ( 1 984) : 1 7 1 .

Podet, All en Howard . "La Croi x ' s Paradox: An Analy sis. "

International

.lournalfor the Philosophy of Religion 1 8 . 1 ( 1985 ): 69-72 .

11� I NSTEIN-PODOLSKY-ROSEN PARADOX, THE. First described

paper w ritten jointly by Albert Einstei n, B . Podol sky and Nathan Rosen, I h i s i s al so cal led the EPR paradox and the Bell-EPR paradox. Thi s p < I radox of q uantum mechani c s � li ke S CHRbDI NGER ' S CAT, conce rns di ffi culties i n the concept of wave fun cti ons. Formulation. "In a complete theory there i s an eJ ement corresponding h ) each element of reality. A sufficient condition for the reality of a physical q uantity i s the possibility of predicting it with certainty, without di sturbi ng I h e system. In quantum mechanics in the case of two physical quantities descri bed by non-commuting operators, the know l ed ge of one prec] udes j h e know ledge of the other. Then either ( 1 ) the descripti on of reality g i ven by the wave function i n quantum mechanics i s not complete or (2) I h e quantities cannot have simultaneous reality. Consi deration of the p roblem of making predictions concerning a sy stem on the basis of Illeasurements made on another system that had previ ousl y i nteracted with i t l eads to the result that if ( 1 ) i s fal se then (2) i s also false. One i s thus led ( 0 concl ude that the description of real ity as gi ven by a wave function i s not compl ete." (Abstract, Einstein , Podol sky, and Rosen) . Explanation. In quantum mechanics the fundamental mode of description for a physi cal system i s the "wave function" w hich gives probabilities for certain physi cal parameters of the system lying w ithin specific ranges. Certai n parameters (those described by "non-commun icati ng operators," i . e . , those for which the order of thei r application may affect the resul t) are not i ndependent and, according to Heisenburg ' s Unceltainty Princi ple� cannot be jointly measured without a combined degree of certainty. For i nstance, the velocity and position of a parti cle are not i ndependent. Increased certainty about the position of a particle comes only at the cost of decreased ce11ainty about its vel ocity, and vice versa. The EPR paradox considers two systems that interact with each other for a li mited peri od. If the wave function of each individual system before the interaction is known, the wave functi on of the combi ned system can be cal c u l ated without further measurement. Thus, after separation, Jneasurelnents of one or the other of the component systems enables the wave functi on for the other component system to be specified. S i nce the two systems are I II

56

Dictionar"v of Paradox

no longer interacti ng w i th each other, however, the measurements made on one syste m can not effect a change i n the other. Therefore, the derived wave function for the unmeas ured system must descri be the actual state of that system. In thi s w ay, w e may obtain a complete descri pti on of the systems ; hence, propositi on ( 1 ) i n the EPR abstract i s fal se. But, then , we have j o i nt certa i nty a b o ut q u a n t i ti es t h at are de scri bed by n o n ­ c o m m u n i cat i n g o p e rators , w h i c h re s u l t i s i n com pati b l e w i t h the Uncertai nty Pri nc i pI e ; h e n c e , proposition (2) i s also fal se. B ut, si nce the fal sity of (2) fo l l ow s fro ln the fal sity of ( 1 ) and si nce at l east one of the pai r lTI ll st be trlle� p ro po s i t i o n ( I ) m ll st be true. Thus, the quan tu m mec han i cal w av e fu ncti o n s do not cOln p l ete)y describe reality. Resolution . N i el s Boh r p ro v i d e d the fi rst and probably most i nfluential atteIn pt to resol ve the paradox. He argued for rejecting the pos sibi lity of predicting a physical q u anti ty, with certai nty and without di sturbing its syste m� as a sufficient condi tion fo r i ts real ity. Real ity at the quantum level j ust cannot be con si dered i ndependently of measurelnent. Bohr sees thi s sort of consequence of quantum theory as compel l i n g us to revise our naive vi ews about the relati on shi p of t h e o ry and real i ty. Instru mentali st vi ews of scientific th e o ry deny that theoretical terms must correspond to el ements of real ity. Al l that quantuJn theory need do i s to Jnake good predictions. Thus, n o paradox ari ses from th e a ssu mp t i on of in coln pati bl e wave functions, so l ong as each yields appropri ate predictions. C. D . Cantrel J and Marl an O. Scul ly mai ntain that EPR were correct to argue that wave function descri ption is unacceptable, but they al so suggest that the argument can not be reformul ated agai nst quantum mechani cal descripti ons usi ng "reduced density matrices," a technique used to descri be systems whose i nitial conditi ons are not compl etely specifi ed. REA DINGS

Al bert� Dav i d Z. "A Quantum- Mechanical Automaton. " Philosophy of Science 5 4 ( 1 987) : 557-585 .

Aerts, Dirk. The Mi ssing Element of Reali ty i n the Description of Quantum Mechani cs of the E. P. R. Paradox Si tuati on. "

Helvetia Physica 57 ( 1 984):

42 1 . Bohr, N . "Can Quantum-Mechanical Des cri ption of Physi cal Reali ty B e Consi dered Complete?" Physical Review 48 ( J 93 5): 696-702. Can trel l , C. D . , and M ari a n O. Scully. " The EPR Paradox Rev i s i ted . " Wallace R. Lalnh, Jr. : A Festschrift on th e Occasion of His Sixty-fifth Birthday. Physi cs Report Book S eries 3 . Amsterdam : North-Holl and , 1 978. 499-508. D ' Espagnat, Bernard. lIZ Search of Reality. New York : Spri n ger, 1 983. Eberhard, P. H. "The EPR Paradox : Roots and Ram ifications . " QuantuIIl Theory and

Pictures of Reality:

FoundationalisJrl. Interpreta tions and

57

Dictionary of Paradox

Ne �v

Aspects.

Ed. W. Schommers. Berl i n: Springer-Verlag, 1 989.

A . , B . Podol sky, and N . Rosen. " Can Q uantum - Mechani cal Descri ption of Phy si cal Real i ty B e Con si dered Complete?" Phy s ical Review 47 ( 1 93 5 ): 777-780. 1 ' ( ) rster, Malcolm R. " Countetfactual Reasoning and the Bell-EPR Paradox. " Philosophy of Science 5 3 : 1 3 3- 1 44. I !a l pin, John F. " EPR Res uscitated : A Reply to Wessell s . " Ph ilosoph ical S tu dies 40 ( 1 98 1 ): 1 1 1 - 1 14. I ; nn mers , M. Th e Ph ilosophy of Quantll111 Me ch a n ics . New York: Wiley, 1 974. M cGrath, James . "A Formal Statement of the Einstei n- Podo l sky-Rosen Argument. International Journal of Theoretical Physics 1 7( 1 978 ): 557. I I I l ste i n , ',

H

,(.� choch, DanieL "On the Fonnal Connection of the Einstein-Podol sky-Rosen A rg ument to Quantum Mechani cs and Real i ty. "

Erkenntnis 29 ( 1 988 ) :

269-278 .

Quantlun Mechanics Versus Local R e aii. nn : The Einstein -PodoLskv-Rosen Paradox. New Yo rk : Pl en u m , 1 98 8 .

,�) e l l eri , Franco , ed.

•

Wessell s , Li n da . "'The ' EPR' A rgument: A Post- Mortum . " Ph ilosophical

Studies 40 ( 1 98 1 ): 3- 30. - . " EPR Resuscitated? A Reply to H-al pi n. " Ph ilosophical Studies 47 ( 1 985 ) : 1 2 1 - 1 3 0.

I < I NSTEIN "S CL OCK. See the PARADOX OF THE CLOCK .

I I�Li�CTRA, THE. See EUBll LI DES ' PARADOXES .

I I:N�rAILMENT, PARADOX OF. Thi s paradox of modal logic was I n t ro d uced by C. I. Le w i s It i s related to LEWI S CARROLL ' S PARADOX OF

,

•

,

.

,

, ,

i

I ': N I A I LMENT ,

i nvol ving an i nfi nite regress of entai lments. I t'o;9Inulation. An impossibl e proposi tion entai ls anythi n g (everyth i n g ) . I <xplanation. Dissati sfi ed with the PA RADOXES OF MATERIAL I MPLICATION, I ,l' \v i s i nt e n ded to construct a modal l ogi c formal izi ng th e noti on of ( ' n t ai l me nt , w h i ch w o u l d n ot be he i r to the menti oned p a r ado x e s . . , Possi bl e" i s taken as a pri miti ve term and "P entai I s Q" i s defi n ed as �'(P ; 1 1 )(.1 Q) is impossible." Since Lew is was concerned w ith logical possi bi l ity, - po s si b J e i s i ntended to be understood as "se lf-consi stent" and thu s i t l u rns out that " P i s possi b l e" is e q uiv ale n t to "not: P entai I s its o- wn n e gation . " B ut, then, the paradox, enti rely ana) ogous to that of Material I u l pi ication, fol low s almost immediately :

I

1

,

i,

,

i ;

"

\

!

"

58

Dictionary of Paradox

(1) (2) (3) (4) (5)

(6)

P i s impossi ble P and not-P P P or Q

not-P Q

equi valent to ( I ) from (2) from ( 3 ) from (2) from (4) and (5).

Each l i ne of the above proof (except the first) i s entail ed by previ ous l i n e s and , si nce entai l ment i s tran siti ve Q , w h i c h i s an arbitrary proposition, i s entai l ed by the impossible proposition P. Resolution. There have been many attempts to sidestep the pa radox by attacki ng the vari ous steps in the given proof. For a summary of some of these attempts, as well as a cogent defense of the proof, see B ennett . One should observe, however, that the paradox i cal ity of Lew i s ' s result is i n no way due to a contradiction. Rather it i s only paradoxical in that it does not accord w i th our i nt u itions about valid argumentation. Yet since we do not usual ly countenance (expl i c i tl y) i m possible proposit i on s as pr emises counterintuitive results should not be d i sconcerting, which is i ndeed the vi ew that Lewi s eventually adopted. ,

,

READINGS Bennett. HEntail ment." Philosophical Revievv ( 1 969): 1 97-236. Lewis , C. I. and C. H. Langford. Symbolic Logic. New York: Dover, 1 959. Tennant, N. "Entailment and Proofs . " Proceedings of the Aristotelian Society 79 ( 1 978) : 167- 1 89. �

EPIMENIDES , THE . According to Henry Alford (cited by Anderson ,

2), the clai m that Cretans are liars i s attested to by a number of anci ent sources i ncl udi n g Pol ybi us, Di ogen i an u s Psel l us, S uidas, Diogenes Laerti us and St. Paul . Formulation. The King James version of Hol y Wri t gi ves St. Paul 's report as foH ows: "One of them, even a prophet of their own said, The Cretans are al ways l i ars, evi l beasts, sl ow bel li es. This witness i s true ( Titus 1 : 1 2- 1 3). Explanation. Some exegetes suggest that St. Paul was not cogni zant of the possi bil ity of any paradox i n his report. Indeed, that the deni gration of the Cretans attri buted to their prophet constitutes a paradox i s not at al l apparent For it i s perrectly possi ble to be a liar without al ways l ying. Indeed, as Kant pointed out, a necessary conditi on for lying i s that te J l i ng the truth be the common practice. This i s because lying i s te l l i n g a falsehood w i th the intent to deceive and one could hard l y i ntend to deceive a n y one if n o t hin g ever i ss ued from one's mo uth but untruths. On thi s ,

,

"

.

Dictionary of Paradox

59

account of what it i s to be a l i ar, the cl ai m that Cretans are always l i ars i s just true if al l of them practice verbal deceit sufficiently often to deserve the rebuke contained in the titl e of l iar. That, of course, coul d be true even if the Cretan prophet had not lied on thi s one occasion. To generate a paradox� one has to take the tenn "liar" in the very strong sense according to which a l iar never tel l s the truth about anything. Now thi s clai m can be just straightforwardly that all Cretans always lie about everythin g false, if, for exam ple, one Cretan ever told the truth about someth i ng. Indeed, if thi s i s the thesis to which the Apostl e to the Gentiles means to testify, hi s witness i s undoubtedly fal se, for surely at least one member of the fi rst civi l i zed nation i n Europe tol d the truth at least once. To generate the paradox, it has to be assumed as known that no Cretan ever told the truth about anythin g up to the time that the Cretan prophet whose name Diogenes Laerti us gives as Epimeni des - clai med that al l Cretans always l ie. Now there i s clearly a paradox. For if Epi menides's statement is true, then he told the truth when he sai d it, and hence at least one Cretan (Epimenides himself) tol d the truth once, so the statement i s false ; and if it i s false, then it i s true because al l Cretans, i ncl udi ng Epimenides i n hi s own statement, have always lied. On th is constructi on, the Epimenides i s just a versi on of the LIAR paradox credi ted to the Megarian philosopher Eubi l ides: suppose that a man said, "I am l yi ng" and nothing m ore. And indeed , medieval phil osophers treated the Epimenides as equ ival ent to the l iar. Even contemporary philosophers use the name "the Epimenides" for eloquent variation on "the Liar." B ut perhaps we have moved too fast, because, as AIonzo Church pointed out, there is something paradoxical about the Epi menides even if we do not assume that all Cretans had al ways l ied u p unti l the time Epimenides made h i s famous pronouncement. For if the statement is treated as true, then it fol l ows immediately that it i s fal se because at least one Cretan 's statement - namel y, Epimenides's i s true. In contrast, however, if we treat it as false, there i s no way own of showi ng it to be true because we are no l onger assum ing that a ll Cretans had up to that ti me l ied. Neverthel ess, to treat it as false is to suppose that at l east one statement by a Cretan other than Epimeni des 's own exi sts . A. N. Prior observes, "We thus reach the peculiar concl usion that if any Cretan does assert that nothing asserted by a Cretan is true, then thi s cannot possi bly be the onJy assertion made by a Cretan - there must al so be, beside thi s false Cretan assertion , some true one. Yet how can there be a l ogical impossi bility i n supposing that some Cretan asserts that no Cretan ever says anything true, and that thi s i s the only assertion ever made by a Cretan?" (26 1 ).

,

,

I

I

I , ,

!

, I

II I,

i

,, I '

, I

!

I

I

,

,

60

Dictionary of Paradox

Resolution. Recent interest in the Epi me n i d es ari ses from L. J onathan

Cohen 's cl ai m t hat it i s not equi valent to the Li ar� bec a u s e unl i ke the simple Li ar it t h re atens the fo r mal i z ati on of indirect rather tha n d irect di scourse. Prior argues that Cohen ' s statement, "If t he po l i ceman testifies that anythi n g which the pri sone r de poses i s fal se , and the pri son er d i spo s e s that s omet hi n g whi ch the po li ce m an testifi es i s true then something which the poli celnan testifies i s false� and so m ething which the p ri so n e r says is true ," i s l o g ical ly true o n l y if another statement. "lf the pol i ceman testifies that anything w hich the p ri sone r di s po se s i s fal se, and the pri so n er disposes that so m e th i n g w h i c h t h e po l i cema n testifies i s tr ue, then either the po l i celnan or the pri soner mllst have sai d s ometh i ng else , " i s al so true. The reason for thi s is that unless the pol iceman or the pri soner says something further, their testilDony wi l l not count as a p ropo s iti on at al l . On Pri or's view, if E pi menid es said , "Nothing asserted by a Cretan is true ,'� and no Cretan had ever said anything before, what he said woul d be neither true nor false, because he w ould not be say i n g anythi ng at all . S ee a l so PRIOR ' S FAMILY OF PARADOXES . �

READING S Anderson , Al an Ros s.

Liar. Ed .

" St. Paul 's Epi stl e to Titus . "

The Paradox of the

Robert L. Matin . New Haven: Yal e UP, 1 970. 1 - 1 l .

Ch urch, A . Re v i e w of

The Liar, by A.

Koyre.

Journal of Symbolic Logic

1 2 ( 1 946 ): 1 3 l. Cohen, L. Jonathan. " Can the Logi c of Indi rect Di scourse Be Formalized?" Journal of Symbolic Logic 22 ( 1 957): 225-232. Goldstei n, Laurence . "Epimeni des and Curry." Analysis 46. 3 ( 1 986): 1 1 7-

121. Mackie, 1. L. Truth, Probability and Paradox: Studies in Philosophical Logic. Oxford: Cl arendon, 1 973 . Prior, A. N. " Epimenides the Cretan . " Journal ofSYlnbolic Logic 23 ( 1958). 26 1 -266. - - "On a Fami ly of Paradoxes . " Notre Dafne Journal ofForrrull Log ic 2 ( 1 96 1 ) : 1 6-3 2 . - - . '"'A B udget of Paradoxes. " Objects of Thought. Oxfo rd : Oxford UP, 1 97 1 .

EPISTEMIC OBLIGATION PARADOX, THE. This paradox, adduced by Lennart A qvi st, ari ses in ep i ste m i c extensi ons of deontic logic. The paradox consists i n the fact that the follow ing two i ntu i t i vely consi stent proposition s ac t u al ly l ead to a contradiction: (1)

S mith ought not rob Jo n es .

61

Dictionary of Paradox

( 2)

W ri g h t. ought to know that S mith ro bs Jones .

' r he paradox i s very similar to (

the

GOOD S AMARITAN PARA D OX

and the

'( )NTRARY -TO I ) , r l ' y I MPERATIVE PARADOX . READING S

A.qvist� Len H a rt .

" Good Samari tans, Contrary-to-Duty Imperatives , and

Epi stemic ( ) hl i gations . "

Nous 1 ( 1 967) : 3 6 1 -379.

porn, Ingmar. " A N ote on Lennart Aqv ist's Epi stemic Obligation Paradox. "

Ph ilo.\ ()!JI!ical Essays Dedicated to Lennart Aqvist on his F�ftieth Birthda_v . I ':d . Tom Pauli . Uppsala: U of Uppsala P, 1 982. 296-298. In

EPISTEMI(� I'ARADOXES. E pi ste m i c paradoxes are those having to do with know l ed ge, bel i ef an d other related concepts. These include the PA RADOX O F ( ( ) ( j N I T IO N , the P A R A D O X O F CO G NIT IV E RELAT I V I S M , t h e PARADOXES OJ , ( 'o N F r RMATIoN , the DOGMATISM PA RADOX � THE PA RAD OX OF '

TilE

KNOW E R, t h e PA R A DOX OF KNOWA B I LITY , H U S S E RL ' S PA RADOX OF

S U BJ ECT I V IT Y , M ENO ' S PA RADOX , MOORE ' S PARA DOX , the PA RA DOX OF NAM ING , the PA RA DOX OF THE PREFACE, the PARADOX OF S ELF-REFEREN CE I N S KEPT I C I S M ., S ( )( ' R AT I e PA RA DOX � P R I O R ' S F A M I LY O F PA R A D O X E S A N D WOLGA ST ' S PJ\ I\ !\ ( )OXES OF KNOWLEDG E.

EPR PAR AD( )X, THE. See the EINSTEIN-PODOLSKy -RoSEN PARADOX .

ETHICAL PARADOXES . On the view of the Utilitarian th eori st , Hen ry S i dgw i c k mo ra l paradox ari ses whenever there are opi nions that run counter to COl ll lTIOn sen se moral ity (Marcus S i ng e r "Common Sense and Paradox in S i d gw i ck ' s Ethics," History of Philosophy Quarterly 3 l 1 986J ;74). ll uman want i n the midst of plenty i s a paradox of this k i nd : " U n e m p l o y e d p u r c h a s i n g p o w e r m e a n s u n e m pl oy e d l a b o r a n d unemployed l abor means human want in the midst of plenty. Thi s i s the most chal l e ng i ng paradox of o ur times" (Hen ry Agard Wal lace, Address 1 934). I n another sense , a paradox i s a proposition that appears absurd but i s actually VERI DICAL. Common sense has its own paradoxes i n th i s sense, as, for exampl e, the one according to which someti rnes w e must be cruel in order to be kind. In a third sense, a paradox i s a contrad i ction that i s su bt) e e n o u g h to be i nt ri g ui ng , for exampl e: "There i s that gl ori olls Epi curean parad ox uttered by my fri end the Hi storian [ J ohn Lothrop Motley ( 1 8 1 4·- l 877) 1 , i n one of his flashi ng mome nt s : 'G ive us the luxuries ,

Dictionary of Paradox

62

of life, and w e w i l l di spense with its necessiti es'" (Oli ver Wendel1 Holmes, The Autocrat oj'the Breakfast Table Ch . 6). I nsofar it sounds as faceti ous as Marie Antoi nettes 's "Let them eat cake," seeking l ife 's l u x uri es i n l i eu of its necessiti es i s one of these. Our treatment of moral paradox, however, restri cts itself to those ari sing i n ethi cal and metaethical th eory. These i nclude the By staNDER PARADOX , Danto ' s B ODHISATTVA PARADOX, the PARADOXES OF DEONTIC LOGIC, the PARADOX OF EXTREME UTILITARIANISM , the PARADOX OF FUTURE INDlVI DUALS , the PARADOX OF LOYALTY , the MERE ADDITION PARADOX, the PARADOX OF NIRVANA, the PARADOX OF PROMISING, the PARADOX OF UNSUCCESSFUL INTERVENTION, AND the UTILITARIAN PARADOX. READINGS C unningham , Stanley B . "The Courageous Villain: A Need less Paradox."

Modern Schoo/man 62 ( 1 98 5 ) : 97- 1 1 0. Lebacqz, Karen . Professional Ethics: Power and Paradox.

Nash v i l l e :

Abi ngton , 1 98 5 .

EUALTHUS, THE. Also cal led the Protagoras , this paradox of deonti c logi c harkens from ancient Greece and i s i ndeed typical of the l ove of di sputation and l ogical conundrums of that society. ForlDulation. "Protagoras agreed to teach Eualthus rhetoric, on conditi on that Euathl u s w ould pay h i m a certain SUln of money w hen he w on hi s first court case. B ut after compl eting the course , Eualthus did not engage in any lawsuits. Growing impatient, Protagoras sued Eualthus for payment of h i s fee. He argued: ' If I w i n thi s case Eualthus w i l l be bound to pay me, for the court will hav e so decided � if I l ose it, Euathl us wi l l sti ll be bound to pay, by our agreement, for he w i l l have won hi s fi rst case. So w hatever happens Euathlus will be bound to pay; the court shoul d therefore fin d i n my favor ' . B ut Euathl us, havi ng l earnt hi s lesson w el l , replied: 4 If Protagoras w ins thi s case, I shal l not be bound to pay, for I need not pay u ntil I w i n a case ; but if Protagoras l oses, thi s court wi l l itself h ave decided that I need not pay ; the court should therefore find for me. ' What should the court have done?" (Mackie 297-298). Explanation. Protagoras confronts Eualthu s w ith a d i lem ma. Eualthus wi l l either w i n the present suit or l ose it. But, i n either case, he i s bound to pay : on the one hand, by hi s contractual obli gation, or, on the other hand, by decision of the court. Eualthus rebuts the dilemma w ith one of his own. Protagoras win either w i n the present suit or lose it. But, in either case, he cannot col lect payment: on the one hand, by h i s contractual obl i gation, or, on the other hand , by decision of the court.

i ,

l

Dictionary of Paradox

63

Resolution. Li ke the SANCHO PANZA, the Eua1 thus i s an exi stential version

of the l iar paradox. Also note that the di lemmas are so set up that the court 's deci sion wi l ] become self-referential and , thus, the court becomes a LIAR. Hence , if the court i s w i l l i ng to disaJ I ow the self-referential impl ications of the ori gi nal contract - by arguing, for examp] e, that no pri vate citizen can i mpose such conditions on the court Protagoras's case woul d fal l apart. Should the court decide not to avai l itseJf of this argument, there may still be a practi cal sol ution. Unl ike the Sancho Panza w here it i s a q uestion of hanging and there i s no room for compromi se, here Eualthus can be made to pay half hi s putative debt. Such a resolution, in which the l itigants split the difference of the conundrum, however, does not resol ve the logical question of whether or not Euathl us should pay. READING Mackie , 1. L.

Truth, Probability and Paradox: Studies in Philosoph ical Log ic. Oxford: Cl arendon, 1 973 . Ch. 6.

EUBULIDES 'S PARADOXES . Eubulides (c. 430-360 Be) was a student

of and successor to Euclides, the sl i ghtl y older contemporary of Plato who founded the Megarian school of philosophy. Along with Zeno, Eubulides \-v as one of history's paramount paradoxers, Diogenes Laerti us credits him with i nventing seven paradoxes (iL I 08), w hich Wil l iam and Martha Kneale reduce to four basic kinds (The Development ofLogic, 1 14). Fi rst, there i s the Li ar: "A man says that he i s lyi ng. Is what he says true or false?" (Ci cero, De Divinatione, iL l l ; Academica, i i.96). Thi s ki nd of paradox (see the EPIMENIDES , INSOLUBILIA, and the LIAR) plays on the strangeness of propositi ons that make claims about their ow n falsity. It i s the archetypical semanti cal variety of logical paradox. Second, there i s the Hooded Man , a] so called the Unnoti ced Man and the El ectra: "You say you know your own brother. B ut that man w ho came i n j ust now w ith hi s head covered i s your brother, and you d i d not know him" (Lucian, Vitarum A uctio, 22) , Thi s kind of paradox not only points out that the word " know" has more than one sense, but aJ so suggests that just because tw o thi ngs are the same thing i s not always reason to assume that what can be sai d truly about one of them can be said about the other. This suggestion i s the basi s of the sense reference di stinction invented independe ntly in the late nineteenth century by the German logicians Edmund Husserl and Gottlob Frege. Thi rd, there i s the Bal d Man (sorites) , or the Heap (jalakros) : "WouJ d you say that a man was bal d if he had onl y one hair? Yes. Would

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you say that a man was bald if he had onl y two hai rs? Yes. Would you . etc. Then where d o you draw the l i ne?" (Di ogenes Laerti us, vii . 82 ; Cicero, Academica, ii.49; Horace, Epistu lae , ii. l .45). Thi s kind of paradox establi shes the vagueness of everyday w ords. For the recent di scussion of the Bal d Man, see SORITES , RECENT and TRADITIONAL. Fourth, there i s the Horned Man: "What you have not l ost you stil l have. But you sti l l have horns" (Diogenes Laertius, vii . 1 87). Thi s type of paradox re veal s that statements that presuppose something may be negated either in a restricted man ne r that accepts the presu pposition, or in an unrestri cted manner that rej ects the pres upposition. Consider the old questi on, "Do you sti l l beat your wife ?" Whoever says either yes or no accepts the presuppositi on that he had formerly beaten her. Onl y if he says, "No , and I never di d," or "Yes, I do now, but it i s a quite recent practice with me," does one deny the presupposition that he had been wont to do so. .

. �

EXCLUDED MIDDLE, THE PARADOX OF THE. This purported

paradox, adduced by M . A. Makinde, attempts to show that formal logic cannot be compJ etely separated from real ity. Formulation. Professor Popper i s either a U-boat or not a U-boat. But, we know that he i s a phiJosopher and not a U-boat. The fonnal logician , however, i s only entitl ed to assert that Professor Popper i s a non-U-boat thing. Thus, Professor Popper could just as welJ be, say, a kangaroo, for that val i dates Professor Popper 's non-U-boatness. Explanation. The purported paradoxical e lement of Maki n de 's argument i s that a kan garoo can val idate the truth of the Law of Excl uded Middle w hen thi s i s appJ i ed to Professor Poppe r and , hence , any tal k about kangaroos would seem i rrel evant. Resolution. In his expositi on of the paradox, Makinde asserts that "No P i s U" follows from our knowl edge that Popper i s a phil osopher and the convention that P stands for Popper and U stands for U-boat. Hence, he i s usi ng P and U as class terms. He then asserts "U or not-U ," whi ch i s either meani ngless if U conti nues to be a class term , or no l onger can simply stand for the class U-boat if it i s now a propositional term . Thus, he seems to have made a category mi stake, wh i ch mi ght account for the confusion. In any case, the question i s what would count as a verification of "(some) Popper is a non-U-boat thing." Clearly, if w e are using ' Popper' as a class term , we must find an instance of the class Popper that i s not a U-boat. N ow, w i th apologies to S i r Karl , shoul d we find a kangaroo that

Dictionary �f Paradox

65

i s a Popper (and given that no kangaroo is a U-boat ! ) � that woul d indeed

suffi ce. B ut note, any old kangaroo won 't do - it Inllst be a Popper ! But there i s nothing paradoxi cal , or even counterintuiti ve, i n al l thi s. Just as clearly� if we are using ' Popper' as a singular term to refer to a gi ven i nd i vi d ual , then no reference to a kangaroo will verify the proposition �' Popper is a non-U-boat thi ng." So once again no paradox arises. Makinde i s nevertheless correct in pointing out that the proposition "Popper i s a non-U-boat thing" tells us virtuaU y nothing about what Poppe r is. Th us, if w e can fi nd a predicate attributabl e to Popper and exclusive of U-boats, the proposition w ould be verifi ed. T h us, once again, if we know that no kangaroos are U-boats and if we can ascertain the truth of "Popper i s a kangaroo," then we have the desi red verificati on . B ut the i ntermedi ate l i nks b etween Popper, kangaroos, and U-boats g uarantee the logical relevance of kangaroos i n the argument, so again there i s no paradox.

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READI N G Makinde , M . A . " Fonnal Logic and the Paradox of the Excl uded Midd l e . "

International

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Log ic Review 8 ( 1 977): 40-52.

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paradoxes are more often cal led DILEMMAS, quandaries, or predicaments. They place the agent i n an i mpossi bl e situati on by making all of his opti ons result i n unacceptable conseq uences or by making it i m possible for him to choose among the vari ous al ternatives . See B URIDAN ' s A ss , the BY STANDER PARADOX , CATCH22 , the CROCADILE' S DILEMMA, EUATHLUS, SMULLYAN ' s PARADOX, the SANC HO PANZA , and the PARADOX Of THE S EEKER � also see PARADOXES OF G AME

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:EXIS TENTIAL PARAD OXES . Exi stential

T HEORY .

Proposed by Morri s L. Shames, this paradox deals w ith the experi menter expectancy effect i n psyc hological research. The experi menter expectancy effect, fi rst noticed by R. Rosenthal and often referred to as the Pygmal i on Effect, is that the experimenter 's expectancies about the outcome of an experiment can i nfl uence that outcome . Formulation. Experi mentation on the experimenter expectancy effe ct i s itself subj ect to the effect and , therefore, i ndetermi nate. Explanation. Suppose a researcher desi gns an experi ment to try to detect the experi menter expectancy effect. If such an effect exi sts, then it wou l d be operationa] i n thi s experiment. Hence, the resu l ts w i l l be determi ned -EXPERIMENTER EXPECTANCY PARADO X, TH E.

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Dictionary of Paradox

66

by the researcher 's own expectations and , therefore, wi l l be biased and not constitute scientific evidence for the existence or non-exi stence of the effect. On the contrary assumption that the effect does not exi st, it cannot, of course, be detected by scientific methods. Hence , i n either case, the effect cannot be detected. Resolution. S hames argues that the paradox i s a matter of degree. The m o re p e rva s i v e a n d the m o re inexorabl e i s the evid ence for the experi menter e xpectancy effect, the more pernicious is the paradox . In the "li mi ti ng case," i n wh ich the paradox i s completely established, " i nvesti gations i nto the research process itself are impossi bly hampered by the seri ous factor of i ndeterminacy" (S hames, 383). I n reviewi n g the l iterature o n the effect, however, S hames happily found that the size of the effect i s usually rel atively smaJl and that i t i s not generali zed to a l arge extent (apparently a large number of results ori ginal ly thought to be i nstances of the experimenter expectancy effect were actuaJJy due to improper data analysis). Thus, argues Shames , the li mited action of the effect reduces the i ndeterminacy invol ved and extricates us from the paradox . Shames warns, however, that a seriou s weakening of the evidence for the effect also carries a ri sk, namely that we may di sregard the very real consequences of the effect and tri vial ize the paradox. We may al so observe here that the effect i s someti mes said to occur even when such pre c a u t i o n s as " d o u b l e - bl i n d st u d i e s " a re take n , so that t h e se methodol o gi cal tec hn i q ues cannot be u sed to avoid the paradox. Presumably, Shames woul d argue that they lessen the effect of the effect and hence, make it more detectable. D. Primeaux, however, argues that Shames's procedure rests on a category mistake since scientific data cannot resolve the paradox. Indeed, given the paradox, the very studies that S hames cites would be indeterminate and could gi ve us no scientific evidence as to the pervasi veness or to the i nexorability of the effect. Moreover, according to Primeaux, the paradox can be generali zed to any and experiment not just those undertaking to study the effect itseJf reveal s that "the naive presumption that the existence of these effects is sci entifi call y demonstrable and their extent measurable" (636) i s not j ustified. READINGS Primeaux, D. "On Shames ' Experimenter Expectancy Paradox. "

of Science 47 ( 1 980) : 63 4-637 .

S hames , Morri s L.

Philosophy

" On the Metamethodol ogi cal Dimension of the

Expectancy Paradox."

Philosophy of Science 46 ( 1 979): 3 82- 388.

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Dictionary of Paradox

EXTREME UTILITARIANISM, THE PARADOX OF. Related to

Kroon 's UTILITARIAN PARADOX , this paradox, i ntroduced by Marcus G. Singer, appears to be a genuine paradox i n one form of Uti l itarianism. Formulation. According to one main version of extreme, or act, uti litari ani sm, the ri ghtness or wrongness of an acti on is a matter of the consequences of the acti on. Yet since j udgi ng an acti on to be right or wrong i s itself an acti on , the act of j udgin g an action right or w rong is a matter of the consequences ofjudgi ng that action right or wrong. Therefore it is right to judge some ri ght actions to be wrong actions. Explanation. Central to the theoretical elegance of uti l itari ani sm i s that it reduces the question of the approbation and d i sapprobati on of agents to the question of the rightness and wrongness of approving or di sapproving of them. Yet this maneuver puts the advocates of act utilitariani sm in the following predicament. If the consequences of j udging that an action i s right (wrong) i s better than the consequences of j udging that an action i s wrong ( ri ght), then the act utilitarian i s obl i ged to j udge the act right (wrong) even if the consequences of the action bei ng judged are worse (better) than one of i ts avai labl e alternati ves. It is to be judged a right (wrong) action even though by the terms of the theory itself, it is a wrong (right) action. Resolutiono The extreme util itarianist w ould be abl e to avoid the paradox by claiming that j udging an action to be ri ght or w rong i s not itself an action. Si nger, however, counters that the consequences of doing so would be overwhel mi ngly hi gh and that a more reasonabl e response w ould be to move away from extreme utilitarianism toward a more pragmatic stance, such as that of John Dewey, where the paradox does not ari se.

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REA DING

Singer, Marcus G .

"The Paradox of Extreme Utilitari anism."

Philosophical Quarterly 64 ( 1 983): 242-248 .

Pacific

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FALSIDICAL PARADOX. See VERIDICAL AND FALS IDICAL PARADOX.

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FORTUNATE FALL, PARADOX OF THE. Named by Arthur O. Lovejoy, the paradox of the fortunate fal l (felix culpa) has been traced by Lovej oy and Herbert Wei singer from John Milton 's Paradise Lost back

to ancient times. The paradox i s that if Adam had not fal l en and been expel J ed from Eden , there woul d not have been the good fortune of the

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story w i th i ts c o n c l u s i o n i n the c om i n g of the New J erusal em. I n other w ord s, without man ' s fai l ure the w o r l d woul d not be saved. As W. B . Ye at s puts it i n Leda and the Svvan, "Nothing can be sale or w hole t h at h a s n ot b ee n re nt A l th o u g h the skepti c may scoff th at had man not stumbl ed the w orld w ould not need sal vati on , from t he sta n d p oin t of Judeo­ Chri sti an faith, thi s i s on e of the most fund am e nt a l and veri di cal paradoxes and i s cl earl y r e l ate d to m a n s finitude and fa l l i b i l i t y as w e I J as God 's i nfi nite m e rc y . re d e m p ti o n

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REA DING S Johnson, Courtney. "John Marcher and the Paradox of the � Unfortunate ' FalL " Studies in Short Fiction 6 ( 1 969): 1 2 1 - 1 3 5 . Kauffman, Corinne E. "Adam i n Paradox." The Arlington Quarterly j ( 1 968): 1 1 1 - 1 1 7. Lovejoy, Arth ur O. " Mi l ton and the Paradox of the Fortunate Fal l . " Essays ill th e History of Ideas. Ed. Arthur O. Lov ejoy. New York: George Braziller, 1 955 . 277-95 . We i s inge r Herbert. T ra g e dy alld the Paradox of the Fortunate Fall. London: 1 95 3 . ,

FREEDOM, PARADOX OF. A c c o r d i n g to Han s JUrg e n Ey s enk, the "

action i s difficult to give any me an i n g to , and i s probab l y not w hat most p e o p l e mean w h en they tal k about ' freedom of w i l L ' or fre e d o m of cho i ce . ' But if we are n ot tal ki n g about a cOlnpl ete l a c k of m o t i v a t i o n t h e n s u re l y the acti o n i s dete rm i ned b y t h e pre po n de r a nce of m ot i v at i ng factors one way or th e other, an d t here i s n o freedom i nvolved�' (369). S i nc e the noti on of freedom is thus paradoxical , Eysen ck c o nti nu e s i t m ust be rej ected for s c i e nt i fi c determi ni sm . It i s not c l e ar how ever. that "motive" and "cause" are synonymous term s � i n fact, w e g en e ra l l y c l aim to be abl e to w eigh co nfl i ct i n g moti ve s and choose among them. Hence� Ey se n c k s clailTI that a moti v ated action i s dete nn i ne d by the m o ti ve s i n v o l v e d i s do u b tfu l and, u n l e s s thi s cl ai m c an be substantiated, t h e para do x fai l s. noti on of a

m ot iv e l e ss -

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READING Eysenck, Hans J Urgen. '�The Paradox of � Freedom ' and the Soci al Fu n cti on of Psychiatry_ " Metollledicine 3 ( 1 982): 367-374.

�-'UTURE INDIVIDUALS , PARADOX OFe Di sco vered i n d e pe n d e ntl y

by Robert M. Adams. Derek P alfi t and Thomas Schwarz, t h i s paradox was fi rst nam e d by Michael D . Bayl e s . I t seeks to sh o w that w e have no ,

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moral o b l i gati o n s to fu t u r e i n di v i d u a l s b e y o n d , say, t h e next fe w

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Formulation. Let us ass ume that i n a few generat i o n s from now a ce rtai n set of i n d i v i dual s w i l l exi st if we ad opt no control J ed growth pol i c i e s . Let us ass u me that th i s set of i nd i v i dual s w i J l be no w orse off than if th ey d i d not exi st at al L Do w e have an o bJ i gation to i m prove the q ual i ty of thei r li ves by ad opti n g a pol i cy of control J ed growth ? It appears that we do not on the fo l low i n g as s u m pti o n s . Fi rst , an i n di vi dual i s w ho h e or she

j s partly because of the i r specific g en eti c endow men t.

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broad - scaJ e pol i c i e s, s llch as control J ed growth pol i ci es, have the power of bringi ng abo ut m as s cha nges i n fut u re events t h r o u gh ri ppl e effects. S i nce controJ J ed growth pol ici es have the effect of changi ng th e specifi c conditi ons under w h i c h a very l arge n umber of peop l e are con cei ved , th e result of adopti ng s u ch pol i c i es w ou l d n ot be to i lnprove the J ot of the future generati ons of i nd i v i d ual s that w e r e to c ome any way, but rathe r to s ubstitute for those g e ne rati ons generati ons fi l l ed w ith othe r i nd i v i dua l s.

Explanation. If o b l i gations are al w ays to i n d i vidual peo pl e a n d not to

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peop l e under some generi c desc ri ption � c e rtai n o bl i gati ons toward fu ture i ndi v i dual s can never be ful fi l led because i n try i ng to fulfi l l t h em w e w i l l bri n g about the appearance of other i ndivid uals than those to w holn w e

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Resolution. The paradox i s m ore th reate n i n g to an i n d i vi duaJ i s t i c constructi o n of obl i gati on than t o obl i gati ons t o future generati ons .

It

w oul d seem that the membershi p of a fam i ly or a nati on or a s p ec i e s at one phase of i ts u nfolding mi ght have obl i gat i o n s to its Inembe rs h i p at a l ater phase w i th o ut those obl i gat i on s bei n g owed to s pec i fi c i ndi v i d ual s mem bers . Altern ati v e l y, the obl i gati on m i ght be o w ed to each i nd i vidllaI � b ut not de p end on h i s i ndi vi dual ity. READI NGS

Adams , Robert M . E x i stenc e Self-Interest, and the Problem of Ev il . " NOllS 1 3 ( 1 979) : 5 3 -65 . K av ka Grego ry S . "The Paradox of Future Indi vid uals . " Ph ilosophy & Public Affa irs 1 1 ( 1 98 1 ) : 93- 1 1 2. Parfi t, Derek . '"On Doi ng the Best for Our Children. Eth ics and Population . Ed . Michael B ay l e s . Cambri dge , MA: S chen kman, 1 976. 1 00- 1 1 5 . - - . " Future G enerations : Further Problems . " Philosophy & Public Aflairs 1 1 ( 1 98 1 ) : 1 1 3 - 17 2. Schwarz , Thomas. " Obli gations to Posterity. " Obligations to Fu ture Generations. Ed. Richard Si kora and Brian Barry. Phil adelphia: Temple UP, 1 978. 3 - 1 3 . "

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GAME THEORY, PARADOXES OF.

Game theory i s the study of games and, more specificall y, the strategies i nvolved i n playing them. As e lsew here i n mathematics, game theory considers formali zed models which are i deali zations of concrete situations. A mong the simpl ifying assumptions usually imposed on these models is that all players are rational and are completely informed about the rul es and about the other players "moves. " Thus all players are expected to make optimal choices , that i s they choose the move that wil l maximize their own positions. Game theory gives rise to many paradoxes, among which see the BACKWARD INDUCTION PA RA DOX , the BOTT LE I M P, the C H A I N STO RE PARADOX , the PARADOX OF DEfERENCE, the DEVIL ' S OFFER, GIDEON ' S PARADOX, the LoTTERY PA RA DOX , N EWCOM B E ' s PA RADOX, PA S CA L ' S WAGER , the P R EDICT ION PARADOX , the PRISONER ' S DI LEMMA , and the ST. PETERSB URG PARADOX. READING Owen , Guillermo. Game Theory. 2nd ed. New York: Academic, 1 982.

GEACH' S PARADOX.

A semantic paradox that apparently all ow s the vali d deri vation of any statement whatsoever. First announced by P. T. Geach as an example of negation-free INSOLUBILIA , Allen Hazen later noted i t i ndependently. We follow Hazen ' 8 presentation here. Formulation. Let q be an arbitrary statement. Consider the sentence

(*):

If ( * ) i s true� then q.

S i nce ( * ) asserts its own truth, ( * ) cannot be fal se. Hence q follows by modus ponens.

Explanation.

If we i nterpret conditional propositi ons as the materi al impli cation of classical logic, the only way that the conditional can be fal se i s for the antecedent to be true and the consequent false (see the PARA DO X E S OF MATERIAL IMPLICATION ) . B ut the antecedent of ( * ) i s just ( * ) - alternatively (although many l ogici ans clailn that this alTIOunts to the same thing), the antecedent of ( * ) asserts the truth of ( * ). In either case, if we assert the antecedent, we can assert the whole conditional (si nce the antecedent asserts the whol e conditional ) and, hence, w e may detach the consequent q by modus ponens. What w i ll happen on the hypothesi s that the antecedent i s false? The negation of the antecedent i s that the whole conditional ( * ) i s false; but we already know that ( * ) can only be fal se if its antecedent i s true, which contradi cts the hypothesi s. Hence, according to cJ assical logic, the antecedent must be true. The paradox, however,

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not depend on classical logic. Hagen cites, for example, the follow i ng t k ri vati on i n Intuitioni st Logi c, due to Raymond S muJJ yan:

j

(1)

(2 )

( * ) i s true If ( * ) is true, q.

(3)

q

(4 ) (5) ( 6)

If ( * ) i s true, q. ( * ) i s true. q

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premI se (2) is j ust ( * ) , which we may as sert from ( 1 ) ( 1 ), (2), modus ponens ( 1 ), ( 3 ), the Deduction Theorem restatement of ( 4) •

(4), (5), modus ponens.

( ) bserve that J ines (2) and (3) are dependent on the premise ( 1 ) they but line (4) eliminates the premi se h a ve been i ndented to reflect thi s hy incorporati ng it i nto a conditional . S i nce (5 ) and (6) only depend on ( 4 ) , they too are independent of ( 1 ) . Al so observe that, since q i s an :u-bitrary propositi on, we apparently have a val id deductive method for t- stabl i sh i n g any conc l u s i on w hatsoever� i ncl u d i ng fal se or e ven 'o ntradictory ones. Thi s result chall enges the usefulness of deductive l ogic as a truth-preserving inference system. I{esolution. The Smullyan proof reveal s that thi s paradox i s similar to I he CURRY PARADOX and, thus, admits of si milar responses. We merely note here that Geach, foll owing Moh Shaw-kwei, w ould focus on the I nference allowing the discharge of ass umption ( * ) at step (4), w hile Hazen fa vors an approach, suggested by John Myhil l , using level s of i mplication. ' r he self-referentiali ty of thi s paradox also makes it simiJ ar to the LIAR and to RUSS ELL ' S PARADOX . -

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READINGS Geach , P. T. "On Insolubilia." Analysis 1 5 .3 ( 1 955 ): 209-2 1 1 . Hazen , Allen. "A Variation on a Paradox . " Analysis 5 0 . 1 ( 1 990): 7-8. Myhill , John. "'Levels of Implication. " The Logical Enterprise. Ed. A. R. Anderson, R. B . Marcus, and R. M. Martin. New Haven: Yale UP, 1 975. 1 79- 1 8 5 . - - . "Paradoxes. " Svnthese 60 ( 1 984). l 29- 143 . Shaw-kwei , Moh. " Lo gi cal Paradoxes for Many-Valued Systems. " Journal of Symbolic Logic 1 9 ( 1 954) : 37-40. ,

(;EACH' S PARADOX OF THE 1,00 1 CATS . Remi ni scent of the S HIP

thi s puzzl e abo ut identi ty was formulated by P. T. Geach. Formulation. S uppose that there i s exactly one cat sitti ng on a mat. T h e cat has at l east 1 ,000 hairs. But for each one of these hai rs, t he re i s a cat that differs from the first cat by f xactl y that hair. Therefore� there are I , 00 1 cats sitti ng on the mat.

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Explanation. Cons ider a s i ngle hai r of the cat sitting on the mat. Clear]y,

the ani mal that cons i sts of all the parts of the cat, except the desi gnated hair, i s a cat. J ust as cl early� thi s second cat is di fferent from the first one since we can di stingui sh the two by the criterion of the designated hair (one has that hair, while the other d oes not) . Hence, there are two different cats sitting on the mat. This i s already paradoxical for there cannot be only one cat sitti ng on the mat and two cats sitting on the mat at the same time. Resolution. Geach argues that the paradox reveal s that the relati on of ab s o l ute i d e n t i ty c a n n ot b e m a i n t a i n e d w i thout c o n t rad i ct i o n . Nevertheless , each of the 1 ,00 1 cats i s the same cat as the ori gi nal one, accordi ng to Geach, where "is the same cat as" expresses an equivalence reJ ati on weaker than ident i ty. E. 1 . Lowe , however, counters that none of Geach's 1 , 00 1 " cats" i s reall y a cat at a l l because, in essence , havi ng identical physical parts i s not a criteri on for id entity i n cats. READINGS

Ge ach , P. T. "Repl y to Lowe. Analysis 42. 1 ( 1 982): 3 0. - - . '"Repl y to Lowe ' s Rep1 y. " Analysis 42. 1 ( 1 982 ) : 32. Lo\v e , E. J . '"Th e Paradox of th e 1 ,001 Cats. " Analvsis 42. 1 ( 1 982): 27-30. - - . " Repl y to Geach . " Analvsis 42. 1 ( 1 982): 3 1 . H

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··On Bei ng a C at. " Analysis 42 . 3 ( 1 982) 1 74- ] 77 .

GENTLE MURDER , PARADOX OF. Al so known as the Adverbial

Samaritan, thi s stren gthened version of the GOOD SAMARITAN PARADOX of deontic logi c was formul ated by James William Forrester. We gi ve a vari ant of the versi on gi ven by Barry Loewer and Marv i n Bel ger. Form u lation. C ertai n 1 y , ArabeJ I a o u ght n ot murder B arbare l l a. Nevertheless � i t ought to be that if she does lTIurder Barbarell a, she should do so gently. As a Inatte r of fact, she does commi t the murder. Hence, she ought to murder Barbarella gently. But thi s l ast implies that she ought to murder Barbare ll a. Therefore, Arabella ought to murder BarbareJ l a and s he ought not mu rder Barbarell a . Exp lanation. T he prelnises in the above story do not seem exceptional - i n parti cular, a g ruesome murder is general l y considered to be more shocking and mor e reprehen sib1 e than a "gentle'� murder. B ut, more i mportantly. the rea soning seems intuiti vely COITect. I n parti cul ar, if one ought to do somethi ng gently, the adverb can be separated i n the fol lowi ng manner: we ought to perform the act and we ought to do so gently. Hence, it follow s that we ought to pe Iform the act. In fact, the story is formali zable i n most systems of deo ntic l ogic and the conclusion is a valid consequen ce. Nevelthel ess , i n most systems the conc] usi on i s regarded as contradi ctory.

Dictionary of Pa radox

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Resolution. Hector-N eri Castane da o b s erve s that th e cru c i al step permi tti n g the ded uction of the co ntradi ction i s the reJnoval of an antecedent contai n i ng no free vari ables from th e scope of the o u g ht operator. T hus� whenever a deonti c systelll asserts that " ought (X does A � if p)" i s equi val ent to "i f jJ , the n (X ought to do A)�" the paradox w i l l ari se. Neverthel ess � Castaneda thi nks the e qu i v al e nce so p l au si b l e that he i s unwi l l ing to g i v e i t u p . Wal ter Si nnott-Armstrong, however, argues t h at a m o re d e tai l e d con s i derati on of t h e l og i cal stru ctu re of the proposi ti on s i nvol ved i n the p ar ad o x (us i n g Davi d son ' s theory of action sentences) revea1 s that the contradiction i s due to problems with the scope of the i ntensional operator, j ust as in the case of the GOOD SAMARlTAN .

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REA D I N G S :

Forrester, James Wi ll iam. " Gentle M urde r, Or the Adverbial Samaritan."

Journal of Philosoph.v 8 1 ( 1 984): 1 93 - 1 97. Loe\ver, B arry and Marvi n Bel zer. " Help for the Good Samari tan Paradox. " Ph ilosophica l Studies 50 ( 1 986): 1 1 7 - 1 27. Si nnott-Armstrong , Wal ter. "A Soluti on to Forrester �s Paradox of G e n t l e Murder. " Journal a/ Philosophy 8 2 ( 1 98 5 ) : 1 63- 1 68 . ,

GIBBS ' S PARADOX. Thi s i s a paradox i n the theory of chemical thermodyn ami cs. When two substances are mi xed , the entropy remai ns the same thro u g ho u t the proc e s s a n d c ol l aps e s to z e ro onl y at i t s com pl eti on . The paradox i s that no matter how cl ose the s ubstances come to bei ng an i dea] m ixture, no attendant change i n the entropy occurs unti l the mi x i ng i s completed. The responses to this paradox are too excessi vel y techni caJ to be presented here. READ ING Denbi gh, K. G. and M. L. G. Redhead. "Gi bbs ' Paradox and Non-Unifonn

C o nv e rge nce .

"

Synthese 8 1 ( 1 989): 283-3 1 3 .

GIDEON' S PARADOX. Attri buted by Maya Bar-Hi l l el and Av i shai M a rga l it to their friend Gideon Schw arz, this p arad o x concerns the concept of econom i c rati onal i ty. Formulation. Mary i s gi ven a choice between two gifts, say $ 1 000 and $5 . A bystander promi ses to rew ard Mary w i th $ 1 ,000,000 if she chooses i rrational ly. Thi s second offe r throws Mary i nto a quandary. S he w ou l d p r efe r $ 1 000 to $5 � and so shoul d seJ ect the former. Yet she wou l d then be acting rational ly and woul d l ose out on th e $ 1 ,000 ,000 . In order to recei ve the $ 1 ,000,000 payoff, she should se l ect the $5 . B ut then her

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selection of the smaller gift i s the rational choice with regard to the ultimate payoff and wi l l not be rewarded after all . Explanation. Mary 's probl em is that any attempt to make an irrational choice will itself be rati onal because she i s tryi ng to be irrational i n order to secure the biggest payoff. Should she try to make a rational choice, however, she would be i rrationally foregoing the largest payoff. Hence, her choi ce wil l be irrational if, and onl y if, i t i s rat i ona l Resolution. Perhaps we should first observe that it i s not impossi ble for Mary to act irrati onally i n the given situation. She could stomp her feet i n genuine frustration and j u st refuse to choose , thereby foregoing even the minor gift of $5 (depending on how we fi l l in the details, even thi s could be consi dered rati onal � but we simpl y suppose that it i s an i rrational , emotional reaction). M ary could even choose iITationally: she may choose the $ 1 000 because she took an immediate i rrati ona l dislike to the bystander, or she may choose the $5 because she superstitiously beli eves five to be a lucky number. When, however, we assume that Mary i s completely rati o nal and has as a g oal the maxi m i zati on of her payoff (see the PARADOXES OF G AME T H EORY ) , the paradox does ari se. B ar-Hillel and Margali t argue that the paradox ari ses because the attributi on of rati onality or i rrati onal i ty to Mary 's choi ce i s only made subsequently to her act of choice and , hence, any theory of rationality cannot make rational ity depend on the outcome of the act to be j udged. Nevertheless, the temporal aspects of the situation seem but i ncidental to the story as told. The real logical problem, al so noted by Bar-Hil lel and Margal it, w o u l d seem to be rather the self-referentiality of the conditi ons i mposed by the bystander. Thus, perhaps surpri singly, the present paradox i s akin to the Liar and RUSSELL ' S .

PARADOX. READING Bar-Hi l l el, M aya, and Avi shai Margal it. "Gi deon 's Paradox of Rationa1 ity. "

-

A Paradox

Synthese 63 ( J 985): 1 3 9- 1 55 .

G oDEL'S PARADOX OF UNDECIDABILITY. Attempts to prove

the consi stency of mathematics by reducing i t to l ogic were frustrated by the di scovery of the PARA DOXES OF S ET T H EORY . Especial l y notable i n thi s reg a rd was RUS S ELL ' S PARA DOX . David Hil bert thus set up the goal , caJ led Metamathematics, of s h ow i n g the consi stency of a fi rst order system of l og i c enti rely by finite means. Godel 's results are generally regarded as entail i ng the futi l ity of Metamathematics.

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Formulation. Godel demonstrated the follow i ng two theorems for certai n

logical systems:

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( 1 ) Consistency implies incompleteness. (2) Consistency implies that there is no proof of consistency formalizable within the system.

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G6de l ' s argument w a s ori ginal ly about the system elaborated by Alfred North Whitehead and Bertrand Russell i n the ir Principia Mathematica , but it i s generalJy con sidered to hold in any system rich enough to express elementary arithmetic. He proceeded by associating a unique natural n umber to all the expressions of the sy stem and then enumerated the formul as of the system. By using Cantor 's diagonal method , G bdel was abl e to fi nd a formula, G , that asserts i ts o w n unprovability. Simplifyi ng a bit, w e may assert that if the system i s consi stent, G cannot be false , for if it were, then G would be a false, provable formula. Theorem (2) i s a corollary of ( 1 ). Resolution. Godel 's results are extremely paradoxical in that they were hi ghly unexpected and counterintuitive. S ome logicians, however, claim that hi s results are paradoxical i n a more pernicious sense. Godel h imself observed that hi s argument was cl osely related to the LIAR, and Graham Pri e st argue s that w hat t h e arg u m e n t rea] ] y s how s i s that some contradi cti ons are true, thereby making l ogic paraconsi stent; indeed, for Priest, it i s exactly the l ogical paradoxes that are these true contradictions. Explanation.

READI NGS Broyles, James E. " Paradox and Argument." International Logic Review 8 ( 1 977): 1 60- 1 69. Chihara� Charles S . " Pri est the Liar, and Godel . " Journal of Philosophical Logic 1 3 ( 1 984): 1 1 7- 1 24. Dowden , Bradley H. A ccepting Inconsi stencies fonn the Paradoxes. " Journal of Philosophical Logic 1 3 ( 1 984): 1 25- 1 30. Priest, Graham. "The Logic of Paradox." Journal of Philosophical Logic 8 ( 1 979): 2 1 9- 24 1 . - - . "The Logic of Paradox Revisited. " Journal of Philosophical Logic 1 3 ( 1 984): 1 5 3 - 1 79. Smullyan, Raymond. Forever Undecided: A Puzzle Guide to Gb'del. Oxford: Oxford UP, 1 987. ,

1 954, Nelson Goodman posed "a new riddle of inducti on" as a successor to HUME' S PROBLEM OF INDUCTION and HEM PEL ' S PARADOXES OF CONFIRMATION .

GOODMAN ' S P ARADOXES OF C ONFI RMATION.

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Goodman ' s ri ddl e concerns the question of w hat predi cates or properti es of objects are suitab J e subjects for i nd ucti ve general i zati on . Formulation. Defi ne the predicate ' grue ' as ' green i f observed before ti ln e t, and bl ue otherwi se ' . Now if t is some ti me i n the future. al ] emeral d s observed thus far are both grue and green. S o either hypothesi s that are equal ly we l l all elnerald s are grue� and that all emeraJd s are green supported. Explanation The diffic ulty is how to detenn i ne which predicates are l egitimate tenns for inducti ve general ization ; or, as Goodman puts it, which are the "proj ecti bl e" predi cates. Whi 1 e i t i s intui tivel y obvi ous that ' grue ' i s l ess projecti bl e than ' green ' , Goodman � s argu ment challenges u s to fi nd an objecti ve bas i s for Ollr preference. Resolution. The present paradox has been widely d i sc u ssed s ince the t i me of i ts i ntroduction. Many p h i l osophers have fol l o w ed Rudolf Carnap 's suggesti on that ' grue' i s less accepta ble than ' green ' because it Inakes essenti al reference to a poi nt i n ti ln e . G oodman ' s response to thi s suggesti on i s to show how to make " green ' and ' blue' rel ati ve to ti me . He defines ' bJ een' as ' bl ue if observed before ti me t, and green otherwi se. ' He then argues that our tenns ' green ' and ' bl ue' can be defi ned respecti vely as ' grue if observed before time t, bleen otherw i se' and ' bleen if observed before time t, grue otherwi se. ' Michael Anthony Slote proposes that ' green' i s a differenti al property i n the sense that if sOlnethi ng i s green then i t can be d i stingui shed on thi s bas i s frOIn something that i s not green . B ut i t i s not always l ogically possi ble to di stingui sh rel evant objects using the property ' grue ' . John O ' Connor has challenged thi s sol uti on by recalJ i ng the i nterdefi nab il ity of th e bl ue-green and the grue-bl een pai rs and R. G. S w i n burne cJ ailTIS that Sl ate 's solution i s vague. Goodman ' s own sol uti on to the probleln i s to argue that ' green ' i s preferable to 'grue ' because the fanner term i s better "entrenched," that i s , has featured in more successful i nductions than the l atter. It does not, however, seem sati sfactory to hol d that a practice is justified just because it i s w hat has been practiced. Nathan Stemmer claims that the paradox can onl y be resol ved by adducing a further premi se. He suggests the foU ow i ng : "The nature of the worl d w i ) I conti n ue to be i ntuiti vely uniform" ( 1 8 1 ), recogni zi n g that h i s proposal i s onl y a parti al sol ution to the paradox because the new premi se i s itself only parti al l y justifi ed . Nevertheless , we may observe that a premi se of th i s kind i s aki n to our nai ve j udgment that we have direct empi ri cal evi dence for the proposition "al l emeral ds are green ," whereas we have no direct empi ricaJ evi d ence that they wi l l some day be percei ved as bl ue. 0

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READING S

Bunch, B . L. '" Rescher on GoodInan � s Paradox. "

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( 1 980):

] 19- 1 23 . C-hao-Tien L. " Solutions to the Paradoxes of Confirmation , Goodman 's

� Paradox and Two New Theories of Confirmation."

Ph ilosophy of Science

45 ( 1 978 ): 4 1 5 -4 1 9. Gol dsti ck, D. "The Meaning of " Grue' . '� Erken 11.tnis 3 1 : 1 3 9- 1 4 1 .

Goodman, Ne] son . Fact, Fiction, and Forecast. Cambridge: Harv ard UP, 1 98 3 . Ch . 3 , S ec. 4, 73 -80. Konynd yk, Jr. , Kenneth . " Sol v i n g G oodman 's Paradox : A Rep l y to Stemmer. " Ph ilosoph ical Studies 37 ( 1 980): 297-305 . 0 ' Connor, John. "- Differential Properti es and Goodlnan �s Riddl e." A l1a(vsis

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28.2 ( 1 967) : 59.

Schlesin ger, George N. "'Is It True What Cicero Said About Philosophers ?" Metaph ilosophy 1 9 ( 1 988): 282-293 .

S lote, MichaeJ A n tho n y . "Some Thoughts on Goodman 's Riddle. " Ana lysis 27.4 ( 1 967): 1 28 - 1 3 2. - - . "'A

General Sol ution to Goodman 's RiddleT'

An a lvs is 29.2 ( 1 968) : •

5 5 -5 8 .

Ste m lner, Nathan.

" A Partial S ol utio n to the Goodman Parado x . "

Ph ilosoph ica l Studies 3 4 ( 1 978) : 1 77- 1 85 .

Swi nburne, R. G. "Grue. "

Ana lysis 2 8 .4 ( 1 968) : 1 23 - 1 28.

GOOD S AMARITAN PARADOX, THE Thi s i s one of the PARA DOXES

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Formulation8 "S uppose that Arabel la (the good Samaritan) ought to help Barbarel l a who has asked her to spare a dime . Now� Arabel l a w i ll ki l l her husband's Ini stress next week and thi s person � unbeknownst to A rabel l a, happens to be Barbarel la. BarbareUa i s the only person Arabe l l a w i l l ki ll next week. The paradox i s that when these sentences are paraphrased into standard deonti c ] ogic . . . they seem to entai l that Arabella ought to ki l l someone'� (Loewer and Bel zer 1 1 7). Explanation. The paradox� of course, i s fi rst of al l that Arabe l J a be obli ged to ki ll anyone at al l , w h i ch goes agai nst a l l our moral intuition s . S econdly. the premi ses Arabella ought to hel p Barbarella. Arabe l la w i ll ki 1 J someone.

do not seem to w arrant the concl usion that she ought to ki l l anybody. Resolution. T h i s i s a fal s i d i caJ paradox d ue to the fa i l u re of the

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substitutivity of identity i n contexts governed by an intensi onal operator (here, the operator "ought"), which makes the present paradox resembl e the MORNING STAR PA RADOX in quantifi ed modal logic. Thus, the paradox is di spelJed by observing proper scope distinctions on the logical operators. But that i s not the end of the story, because Forrester has gone on to invent the PARADOX OF TH E GENTLE MURDER, a strengthened version of the Good Samaritan. READING S A qvist, Lennart.

" G ood Samari tans, Contrary to Duty Imperati ves and

Epi stemic Ob li gations . "

Nous 1 .4 ( 1 967) .

Loewer, B arry, and Marvin Belzer. "Help for the Good Samaritan Paradox . "

Philosophical Studies 5 0 ( 1 986): 1 1 7- 1 27.

GRELLING PARADOX, THE. See the HETEROLOGICAL PARADOX .

HEAP, THE. See EUB ULI DES ' PARADOXES and the S ORITES .

HEDONISTIC PARADOX, THE. That a person w ho insistently seeks

pl easure for himself will not find it, but that the person who hel ps others find pl easure w i ] ] in the end find pleasure hi mself (or has a greater chance of fi ndi ng it), is sometimes cal l ed the hedoni sti c paradox. Related to thi s paradox is that fact that pleasure is not to be sought for itself, that it is not an end in itself separate from the activity or experience of which it i s an aspect. Pl easure i s attai nable only as an attitude or feeling accompanying other things. For a related paradox i n B uddhi sm, see the PARADOX OF NIRVANA .

HEGELIAN PARADOXES . Among the many philosophers w ho have

a penchant for paradox, Hegel is one of the foremost. There are many specific paradoxes i n Hegel 's work, such as his contention that the Master i s hel d in th ral l by his dependence on the Slave while the latter 's work sets him free. Beyond these individual paradoxes, however, Hegel 's w hole dial ectical logic, which incorporates contradiction as an essential element, is paradoxical. According to Howard P. Kainz, "Hegel 's system as a whole i s a true paradox, not only i nvol vi ng a complex network of simultaneous rec u rs i on s , b u t s i mul taneous] y h ol d i n g to gether the o ve rarchi n g , architectonic oppositions of being and thought, along with numerous

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corollary and subsidi ary oppositions� whose dial ectical connections w ith the ori gi nary paradox w i lJ be . . . necessitated in the unfolding of the overaJ l system-paradox" (97). The "ori gi nary paradox" to which Kai nz refers i s the experience of se lf-consciousness , w hi ch Hegel general izes to the " unity-in-distinction of being and thought" (Kainz, 84).

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REA DING

Kainz, Howard P. Paradox, Dialectic, and System. Uni versi ty Park: Penn S tate UP, 1 988 .

HEMPEL' S PARADOXES OF CONFIRMATION. Rel ated to other

problems of i nducti ve logic, such

HUME ' S

PROB LEM OF I NDUCT ION and GOODMAN ' S PARADOXES OF CONFIRMATION, these paradoxes are al so call ed the Ravens Paradox. We w i ) ) consider only a single version of the paradoxes here. Formulation. S cientifi c hypotheses general ly take the form

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All Fs are Gs.

An hypothesi s i s confi rmed whenever we observe an instance of an F with the property G. We may, however, rewrite the hypothesis in the foll ow i ng, l ogi cally equival ent form:

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Therefore, the observation of an instance of a non-G , no matter how i rrel evant it may seem to the ori gi nal hypothesi s , w i l l confirm the hypothesis if it has the property non-F. Explanation. Consider, for example, the hypothesi s All rav ens are black.

Each time we observe a raven and find it to be black, we count that as a confi rmation of the hypothesis. When w e happen upon a green shamrock, how ever� we beli eve that w e are l eavi ng zoolo gy behind for botany. Nevertheless, the green shamrock is a non-black thing that has the property of bei ng a non-raven. Thus, it confi rms the hypothesis All non-black things are non-ravens.

But, since thi s hypothesis i s logicalJy equivalent to the first hypothesi s, we seem forced to accept the intuiti vely repugnant conclusion that the

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observ ati o n of a gree n s h am ro c k i s a c onfi rm ati on i n s t an c e of the hy poth e s i s " Al l ravens are black ." Hempel ' s paradoxes th us bri n g i n to question the e n ti re p r a ct ic e of confi nni n g sc i e nti fi c hypotheses. Resolution. T he present parad ox i s among the most w i del y d i s cussed para do x e s of the twenti eth century, yet n o commonl y accepted s o l ution h a s been found . One ap proac h i s to d eny th at al l ravens are black" i s equi valent to "al l non-black obj ect s are non-rav e n s . " S uc h a po si t i o n seelns too r ad i c al how e v er, and R. G. Sw i nb u rne arg ues that it w ould be contrary to ac c e p ted s c i e nt ifi c p ro c e d u re A n other approach i s to atte m p t to deve l o p a n o t i o n of r ele v a n c e of e v i dence t o hypotheses. Even though i t w ou l d c ompl i cate the underl y i ng l og i c o f sc i ence , th i s c o u l d be a p ro m i si n g ap p ro a ch ; nevertheless , it has p r o ve n extremel y difficult to formulate a de q u ate generalized rel evance c o nd iti o n s . A thi rd suggestion, due to Karl Po pper, is to abandon the g o al of confi rmati on a l tog et h e r � on thi s view, the onl y l egiti mate scientific approach to a hy pothesis is to attelnpt to di sc onfi rm it, by s e a r c h i n g for counter-i nstances. Po pp e r s p o s i ti o n has been e xtre m ely i nfluenti al and perhap s w i ll be i n c o rp orate d in any final solution to the p ro b l e m Even so, it seems a l m o s t perverse to deny that our vast e x pe r i en ce w ith bl ack ravens has no beari ng on the confi rmati on of the h y p o th e sis that "all ra v e n s are bl ack." Sti l l another response i s to accept the p aradox by ag re e i n g that the o bs e r v at i o n of green s h a mr o ck s does confi nn the stated h y p o t h es i s and t hen try t o d o w n pl ay i n some manner, t he i m p oltan c e or the d e g r ee of c o nfi r mati on obtai ned "

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the re by. READING S A l exand er� H. G. "The Paradoxes of Confi rmati on . "

British Journal for

the Ph ilosophy of Science. 9 ( 1 958): 227-23 3 . Aronson, Jerrol d L.

"The B ayes i ans and the Rav e n Paradox. "

Nous 23

( 1 989) : 22 1 -240. Black, M. "Notes on the Paradoxes of Confirmation� Aspects ofInductive Logic. Ed . J . Hi nti kka and P. Supp e s . Am sterdafl1 : 1 966. 1 75 - 1 97 . Gai fman, H . " Subj e ct i ve Probabi lity, N at ura l Predicates and Hempel ' s Rav ens . " Erken ntnis 1 4 ( 1 979) : 1 05 - 1 47. Cooke, Roger M. "A P aradox in Hempel ' s Criterion of Maximal S pecifi city. " Philosophy of Science 48 ( 1 98 1 ) : 327- 3 28 . ,'I Eri ksen, Leif. "Confi rmation, Paradox� and Logi c . " Ph ilosophy o.lScience 5 6 ( 1 989) : 68 1 -687. Fisch, Menachem "Hempel 's Ravens, the N atural Cl as sification of Hypotheses and the Growth of Know led ge . " Erkellntnis 2 1 ( 1 984): 45 -62. Hempel , C. G . '" Studies i n the Logi c of Confi rmati on ( I ) . " Mind ns 54 ( 1 945) : 1 -26. ."

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- - . "Studies in the Logic of COnfi lTIlati on Hinti kka, J. ,

(II).�' Mind ns 54 ( 1 945): 97- 1 2 1 .

"Inducti ve Inferences and the Paradoxes of Confi rmation . "

Es s ays in Honor of Carl

G.

Henlpel: A Tribute on

the

Occasion of His

Sixty- Fifth Birthday. Ed. N icholas Rescher. Dordrecht: Rei del, 1 970 .

24-46. Huggett, W. 1 . HOn Not Being Gul led b y the Ravens . " Australasian Jo u rn a l a/ Ph ilosoph)' 3 8 ( 1 960) : 48-50. Hum b urg Jurgen . "The Sol uti on of Hempel ' s Raven Paradox in Rudolf Carnap ' s System of Inducti ve Lo g i c. " Erkenntnis 24 ( 1 986): 57-72. ,

Law s on , Tony.

" T he C on text of Pred i cati o n ( a nd the Paradox of

Confi nnati on ) . " British

Jo u rn a l

for th e P h ilosop h}' of Sc ience 3 6

( 1 98 5 ) : 393 -407. Mackie , J. L. "'The Paradox of Confirmation. " British Journal for the Ph i los op hy of Scien ce 1 3 ( 1 963 ) : 265 -277. Rody, Phillip 1. "(C) I nstances , the ReJ evance Cri teri on , and th e Paradoxes of Confirmati on ." Ph i lo sophy of Science 45 ( 1 978): 289-302 . Sch\vartz , R . "Paradox and Projection . " Ph ilosophy of Sc ie n c e 3 9 ( 1 972 ) : 245-248. Stemmer, Nathan . '" "Justificatory ' SoJ utions to Hem p el 's Raven Paradox and Goodman ' s " New Ri ddle of Ind uction' . " Ph ilosoph ical Studies 34 ( 1 978): 1 77 - 1 8 5 . Stove, D. " Popperi an Confinnation and N ot the Paradox of the Ravens. "

Australasian lo u rnal of Ph ilosophy 37 ( 1959): 1 49- 1 5 1 . - - . "A Reply to Watki n s . " Australasian Journal of Philosophy 38 ( ] 960):

5 1 -54. S uppes , P.

" A Bayesi an Approach to the Paradoxes of C o nfi rrri ati on . � '

Aspects of Inductive Logic. Ed. J . Hi ntikka and P. Suppes. Amsterdam :

1 966. 1 98 -207. Sw i nb urne, R. G. "The Paradoxes of Confinnation: A S urv ey. " American Ph ilosophica l Qu a r te r ly 8 .4 ( 1 97 1 ) : 3 1 8- 329. Watkins , 1 . W. N. "Confi rmation , Paradoxes and Pos i ti vi sm. " Th e Critical App roa ch. Ed. M . Bun ge. New York: 1 964. 92- 1 1 5 . - - . " Repl y to S to v e. Australasian Jo urnal of Ph ilosophy 37 ( 1959): 240-24 1 . - - . "Reply to Mr. Stove ' s Repl y. " Australasian Journal of Ph ilosophy 3 8 ( 1 960 ): 54-58 Wei ntraub, R u th . "A Paradox of Confirmation. " Erkenntn is 29 ( 1 98 8 ) : 1 69- 1 80. Whitely, C. H. "' HempeJ ' s Paradoxes of Confirmation." Mind ns 54 ( 1 945): 1 5 6- 1 5 8 . Aspects (?f Indllc!ive von Wri ght, G . H. "The Paradoxes of C onfi rmati o n "

.

"

Log ic. Ed . J . H i ntikka and P. S uppes. Am sterdam : 1 966. 208 -2 1 8 .

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HERACLITUS ' PARADOXICAL APHORISMS. A native of Ephesus, Heracl itus di ed someti me between 478 and 470 B . C. He spoke in such

an obscure manner that the anci ents cal 1 ed hi m "the dark." Sometimes th e fra g m e nt s of h i s b oo k are paradoxi cal because they attrib u te contradictory attri butes to thi ngs, or because they identify opposites. Here are several paradoxes representative of Heraclitus. (1) God i s day, he i s ni ght; wi nter and summer, w ar and peace, sati ety and hun ger; he changes fonn ev en as fire when mi xed w i th various incen ses i s nam ed accordi n g to the pleasant perfume of each. (2 ) Immortals are mortal , m ortal s immortal , each l i ving and dyi n g the 1 ife of the other. ( 3 ) The way u p and the way down are one and the same . (4) The l iv ing and the d ead, the waking and the sleepi ng, the young and the old , these are the same; the former are mov ed about and become the l atter, the l atter in turn become the fOlIner.

In these remarks Heraclitus i s not trying to confuse us but merely to state the truth as he understands it. Someti mes he explains the contradiction in such a manner that it is cl early only apparent. S ea water i s the purest and the foulest. For fish i t is drinkab1e and life-preserving� for men it i s undrinkable and deadly.

(5 )

other times his remarks are paradoxi cal because they contradi ct accepted notions.

At

(7)

You could not step twi ce in the same ri vers ; for other and yet other waters are ever flowing on. (8) In the same ri vers we step and we do not step. We are and are not.

Plato characteri zes Heracl itus ' i dea that all things are i n perpetual change by the phrase "Everythi n g flow s." We do not step i nto the same river twi ce not only because the river changes between the steps, but because we oursel ves change i n a si m i l ar manner. The circumstance that not only does the river change the very movement we step i nto it, but that we do also, was expressed by a student of Heracl i tus, "You cannot step into the same river once.�' READINGS Ki rk, G. S .

The Cosmic Fragrnents.

Cambridge:

1 954.

Mackenzie, Mary Margaret. "The M-oving Posset Stands Still: Heracli tus

83

Dictionary of Paradox

Fr.

1 25 . " A ,ne rican Jo urnal of P h i lology 1 07 ( 1 98 6 ) : 5 4 2 -5 5 l .

Di scus sion: 1 09 ( 1 988): 397-4O l . Will i ams, Howard. "Heraclitus' Philosophy and Hegel 's Dialectic." History

of Political Thought 6 ( 1 985 ): 3 8 1 -404.

I-IE1'EROLOGICAL PARADOX, THE. Also called the Grelling, or

( i reJli ng-NeJ son, Paradox, thi s i s a set-theoretical paradox, related to RUSSELL ' S PARADOX . Formulation. "Some adjectives have meanings whi ch are predicates of

the adjecti ve word itself; thus the word ' short' is short, but the word ' l ong' i s not l ong. Let us call adjecti ves whose meani ngs are predi cates of them, l i ke ' short ' , autological ; others heterol ogical . Now i s ' heteroJ ogicaJ ' heterologi cal ? If it is, its meaning is not a predicate of it ; that i s , i t is not heterologi cal . If it i s not, its meaning is not a predicate of it, and therefore i t i s heterological . " (Ramsey, Foundations of Mathematics 27 ) Explanation. Since we can easily fi nd many exatnples of autologi ca) and heteroJ ogical adj ecti ves , it would seem that the cl assifi cation of adjectives using these concepts would be quite natural. But ' heterol ogical ' i s itself an adj ective and so should be heterologica) or not. Nevertheless, thi s apparentl y i nnocuous predi cate turns out to apply to itself if� and only if, i t does not apply to itself. Resolution. Russel l thought that the heterologi cal paradox w as further support for the hierarchy approach to the logical paradoxes ( see the LIAR and RUSSELL' S PARADOX ) . According to thi s view, which i s the basis for Russel l ' s Theory of Types , the proper formu l ati on of the statement, " � Heterol ogi cal ' i s hetero] ogical ," woul d include a device such as s ubscripts to show that the second u s e of ' heterol ogical ' make s a metal i ng u i stic c laim about the fi rst. " Hierarchy" sol uti ons remai n , however, controversial because there seems l ittle reason for them except for the purpose of avoiding paradox; in other words, such so] utions appear irredeemably ad hoc. Other attempted solutions have focused on the Ineaningful ness of the claim that " ' Heterological ' i s heterological ." For " example, Gilbert Ryl e argued that statements such as '' 'x' is y can only be meaningful when there are establi shed criteria for determi ning when the predicate y applies to 'x ' . In the sentence, ' ' ' Long ' i s heterol ogical ," such criteria are availabl e, since we know what it i s for a word to be long or short. But in the paradoxical sentence, there are no such criteria since we do not know what it i s for a w ord to be heterological i n the absence of any other relevant properties. That is to say, ' heterol ogical ' is appropriately ,

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, ;

I I.

8- 1

Dictiol1(Jr�v (�l Paradox

to w ord s o n l y w hen t h o s e w o rds i n d i cate t h e appropri ate phi l o lo gi ca l properties b y which they shoul d be j ud g e d Attempts to re s o J ve t h e parad ox b y c l a i m in g that the pa r a d o x i c a l sentence i s meani ngless, however, can be met by constructi ng the fol J o w i n g paradox. Define M-heterological as any "adjective w hose mean i n g i s not a pred i c ate of itself, or w hich is meaningless when a pp l i e d to itse lf ' . Then consi der the sentence , '' ' M- heterological ' is M-heterological ." a pp l i ed

.

READINGS

Bowden , Leon. �'Heterologicality. H Ana lysis 1 2 .4 ( 1 952): 77-8 l . Fitzpatri ck, P. J. H " Heterol ogical and Namely-Riders." Analysis 22. 1 ( 196 1 ): 1 8-22. G eac h P. T. "Ryle on Namely-Ri ders . " Ana lys is 2 1 . 3 ( 1 96 J ) : 64-67. - . "Namely Riders Agai n . " Ana lys is 22 . 3 ( 1 962 ) : 92-94. Goldstein, Lawrence. "Categories of Li ngui stic Paradox and Grelling's Paradox. " Linguistics and P h i lo sophy 4 ( 1 98 1 ): 405 -42 l . - - . "Lingui stic Aspects , MeaningJ essness and Paradox: A Rejoinder to John David Stone . " Linguistics and Philosophy 4 ( 1 982) : 579-592. Lawren ce, Nathaniel . "Heterology an d Hierarchy." Analysis 1 0.4 ( 1950): 77-83 . Mackie , J. L. , and J . J. C. Smart. '"A Vari ant of the ' Heterolo gi cal Paradox ' - A Further Note, " Analys is 1 4.6 ( 1 954) : 1 46- 148 . - - , "A Variant of the 'Heterological ' Paradox . " Analysis 1 3 . 3 ( 1 95 3 ) : 6 1 -65 Martin, Robert L. " O n Grel l i ng ' s Paradox . " Ph ilosop hical Review 77 ( 1 968) : 3 2 1 -330. Meager, Ruby. "Heterologicality and the Li ar. " Anal.vs is 1 6.6 ( 1 956): 1 3 1 1 38 . Ramsey, F. P. "The Foundations of M_athemati cs ." In Th e Foundations of Mathell'latics. Ed . R. B . Braith\vaite . London : Routledge, 1 93 1 . 1 -6 1 . Ryle, Gi l bert. "Heterologicali ty. " Analysis 1 1 . 3 ( 1 95 1 ) : 6 1 -69. Stone, John David. " Meaningl essness and Paradox: Some Remarks on Goldstein's Paper. " Lingll isties and Philosophy 4 ( 1 98 1 ) : 423 -429. von Wright, G. H. Ph ilosoph ical Papers: Philosoph ical Logic 11. Ithaca: Cornell UP, 1 983 . 1 -24 �

,

-

HILBERT ' S HOTEL .

Ever since the days of ancient BethlehelTI , travelers have been frustrated by ar ri vi n g at their desti nations on l y to fi nd that all the hotel s have been b oo ke d sol id. The management of Hilbert 's Hotel has taken a b i g step tow ard e l i m i nat i n g thi s probl em : the hote l has an infi nite number of rooms. G iven the great increase of late in the number of travelers, even this vast number of rooms i s comp letely occupi ed d ur i n g peak travel ti mes. The beauty of Hi lbert's Hotel, however, i s that it i s

I,

l)ictioJ1ary of Parado.r

85

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,, ,

. !I

, 1 1 , l e t o defy

our fi nitistic expectation s. For example, should a guest arrive \ \ ' lIe l l al l the rooms are occ upied , he would not have to be turned aw ay. I ' l l e' Inanagement w ould simply ask each of the guests to move i nto the H l l w ith the next highest number to that of their present room. The P l I l� st i n room 1 woul d move i nto room 2; the guest in room 2 wou l d I l l o v e into room 3 ; and so o n . Now, clearl y, each guest woul d sti ll be ; I t 'Cornmodated since there is no end to the number of rooms different I rUin room 1 . In fact, if we knew a guest 's old room number, we could I q!, lI re out exactly w hat his new room wouJd be. Yet just as c l early, room I has been freed so that the new arri val can al so be accommodated. Should ; 1 g roup of n new arri vals appear at the hotel , the first n rooms could be I n� ed by reassign i ng each guest to room number k+n , where k was the ori g i n al assi gnlnent. The capabilities of Hilbe11's Hotel , however, have v e t to be really tested. Even if an i nfinite number of new arrivals were to d iTi ve at the al ready ful l hotel.. the o l d guests cou l d be transferred to the , ' v e n numbered rooms (since there are an i nfinite number of even nu mbers, a l l these guests would sti l l be accommodated ), w hich w ou l d free the I nfi nite number of odd n umbered rooms for the i nfi nite number of new a rri val s . In fact, supposing that there are an i nfinite number of pl anets, I l i l bert's Hotel could host a convention, to which each pl anet sent an i nfi nite number of del egates !

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U ·i OLLI8 '8 PARADOX. This is a PREDICTION PARADOX . Formulation. Two people, A and B , thin k of positive integers and whisper the nUlnbers chosen in a third 's, C's. ear. C then asserts that neither A nor B can work out w hose number i s greater. He al so adds that they did not both choose the same number. A reason s as fol lows: "Cl early, B did not choose 1 si nce, i f he had, he could ded uce that my number would be greater than hi s. Further, B can reason simil arly about me, which is correct since I pi cked 1 57. Hence, 1 was not pi c ked by either of us. B ut then B di d not choose 2 si nce, now that 1 has been elimi nated, B cou l d again deduce that my number was the greater. Once agai n , B can reason analogously about me. Therefore, 2 coul d not have been pi cked by either of us. By continuing thi s chain of reasoning. i t follows that 1 5 7 was not chosen by either of us, even though I chose it ! " Explanation. Holl i s claims that the asserti on that C was w rong about A and B 's abi lity to deci de w hich number is greater is al so paradoxi cal , since, if he was, A must conclude that both numbers are greater than each other, whi ch i s obv iou sly absurd. C ' s asserti on th us el ilni nates the

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86

Dictionary of Paradox

possibility that 1 was chosen by either A or B since , i t bei ng the smal lest number, whoever chose 1 would know that the other 's number was larger. B ut, then 2 becomes the smallest number that can be chosen and so the same argument applies to it. After 1 57 steps, A i s forced to concl ude that he could not have chosen the very number ( 1 57) that he did choose and an inducti on argument results in the conclusion that no number could have been chosen by either A or B . Resolution. Dori s Oli n suggests that, even if B had picked 1 , he could only infer that A's n umber was smaller if he was sure of the correctness of C 's statement. Hollis, however, argues that a slight modifi cati on in the story line (substitute "at least equal" for "greater") w ould save the paradox. Michael Kingham argues, among sundry other things, that the paradox depends on the fallacy of epi stemic di stribution: "A know s that (if p, then q)" implies (fallaciously) that " if p, then A knows q. " George Rea claims that Olin and Ki ngham are correct if by "w ork out whose number is greater," we mean "infer from premises known to be true. " But, Rea also clai ms that the paradox can be reformul ated to avoid these objections, although the reformulation i tself - which invol ves i nference w ithout knowledge or conditional i nference - aJ so eventually succ umbs to an analysis reveal ing that C's statement cannot be true. READINGS Hollis, Martin. "A Paradoxi cal Train of Thought. " Analysis 44.4 ( 1 984) : 205-6. - - . " More Paradoxical Epistemics. " Analysis 46.4 ( 1 986): 2 17-2 1 8. Kingham , Michael . "A Paradox Derai1 ed: Reply to Hollis". Analysis 46 . 1 ( 1 986): 20-24. Olin, Doris . " On a Paradoxical Train of Thought." Analysis 46. 1 ( 1 986): 1 8-20. - - . "'On an Epj stemic Paradox . " Analysis 47.4 ( 1 987): 2 1 6- 17. Rea, George. "A Variation of Holl i s's Paradox." Analysis 47.4 ( 1 987) : 2 1 8-220.

EUBULI DES ' s PARADOXES .

HOODED MAN, THE.

See

HORNED MAN, THE.

See EUBULIDES ' s

PARADOXES .

In Chapter 3 3 of the Tsu appear the ten paradoxes of Hui Shi h (c. 3 7 0 - c. 3 1 0 B e) , al l

HUI SHIH'S PARADOXICAL APHORISMS . Chuang

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Dictionary of Paradox

I h at remains of the fi ve cart l oads of books thi s Ch i nese l ogician i s reputed 1 0 h av e written . Some of these paradoxes resembl e Heracl itus' . We cite the fol l owing from Vi n cen t Shi h's translation:

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( 1 ) The heavens are as low as the earth� and mountains are as l evel as marshes. (2) The moment the sun reaches i ts zenith it declines � and the moment a thing i s born i t dies." ( 3 ) The south has no l imit and has a limit. (4 ) I go to Ytieh today and arri v ed there yesterday.

' rhese oracular utterances are meant to suggest the mutability and relativity of all things.

': , ,' , ,

READING Shih, Vincent Y. C. " Hui Shih. " The Encyclopedia of Philosophy. Ed . Paul

Edwards. New York: Macmi1 lan, 1 967. Vol . 4, 69.

ver i d i c al paradox of cul ture criti ci sm was anal y zed by the l i te ra ry critic Joseph Wood Krutch. i1"'ormulation. "It w o u l d appear, then , that the compJ ex of ideas and preferences w hich passes current under the name of h umani sm may be separated i nto two di stinct parts. The complex i ncludes on the one hand a tendency to stress the i mportance of the social virtues set up in opposition to the d e stru cti v e anarchic tendencies of what it mistakenly call s the � natural ' man, and it i ncludes on the other hand a sympathy with the attempt to create human, as opposed to natural , val ues. Yet these soci al v i rt ues a re , as w e have seen, themselves ani mal , and among men they flouri sh Inost i n those soci eties w he re the genui nel y human tendencies - both virtuous and vicious - are l east prominent. ' Humanism ' i n thi s popular sense i s thus obvi ously at war with i tsel f, for the si mple reason that th e closer it comes to realizati on of one half i ts ideal the further it i s bound to be from the possibi lity of achieving the other� since the second demands a detachment from the ai ms of nature and the first a harmony w ith them" (Krutch 34-35 ). Explanation. I n the c h a pt er of w hich thi s passage i s the concl usion, Krutch exposes two contradictory te n d e nc i e s in the popu lar concepti o n of humani s m : h u man i sm is seen to favor "communitarian" val ues (in the sense now associated with philosophers such as Alasdai r MacI nty re) and "expressive i ndi viduali sm" (i n Robert N. Bellah ' 8 sense ). Yet, as MacInty re and Bell ah 's group argue so compe l J i ngly, the i ncompati bi l ity of the c ommun i ta ri a n and indi vidual i stic orientation s i n A merica and in

II

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HUMANISM, THE PARADOX OF. T hi s

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Dictionary of Paradox

88

the Occident i n gen eral , has grow n even more acu te si nce Krutch i dentified it. Resolution. The popu 1 ar concepti on of humani sm must be revi sed in order to better recognize the dramati c tensi ons that it masks. Just how thi s i s to be done, however, and w hether the tensi on s can be adequatel y resol ved are sti l l open q uesti ons. READINGS BeJ lah, Ro bert N. , and others . Habits of th e Heart: Individualisln and COlJuniflnen t in A,nerican Life. New York: Harper, 1 98 5 . Krutch, Joseph Wood . "The Paradox of Humanism ." The Modern Tefnper: A Study alld a CO/�less ioJl. New York : Harcourt, 1 929. 27-55 . MacIntyre , AJ asdai r. After Virtlle: A Stud)1 in Moral Theory. 2nd ed. Notre Dame: U of Notre Dame P, 1 98 3 .

HUMAN SCIENCES, PARADOXES IN THE. This heading comprises

paradoxes having to do with the character of human being and and its scientifi c investigation . Psychological paradoxes are especially prominent i n our J i st: the A BILENE PA RAD OX the PARADOX OF ACTING , the BODHISSATTVA PARADOX , B URIDA N ' S A s s , the EXPERIMENTER EXPECfANCY PA RA D O X the PARADOX OF FREEDOM , the HEDONISTIC PARADOX, the PARADOX OF L OG I CA L PSYCHOLOG ISM, M EEH L S METHODOLOGICA L PA RA DOX the PA RA DOX OF PAIN , PARADOXICAL SLEEP, PE R C EPT U A L PARADOXES, the PRODUCT IVITY PA RA D OX the SEEKER PA RADOX, the PARADOX OF S ELF- DECEPTION , the PARADOX OF S I NCER ITY , the PA RA D OX OF T H E THI NKING B EHAVIORIST , the PA RA D OX OF T RUST , and the PARADOX OF U N S U C C E S S FU L I NTERV ENTIO N . -

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HITME 'S PROBLEM OF INDUCTION. Rai sed by David Hume in 1 777, thi s is the orig i n of the problem of justifying non-deductive reasoning

and i s the p redec e s s or of

H EMPEL' S

and

GOODMA N ' S PARADOX ES O F

CONFI RMATION .

Formulation. "We have said that all arguments concerning existence are

founded on the rel ati on of cause and effect; that our knowledge of that relation i s d eri ved e nti rely from experience ; and that all our experimental c o n c l usions proceed upon the s u p positi o n that t h e fut u re w i l l b e conformabl e to the past. To endeavor, therefore, the proof of this last supposition by probabl e arguments, or arguments regard i ng exi stence, must be evidently going i n a ci rcle, and taki ng that for granted, w hi ch i s the very point i n question.'� CRume sec. 30)

Dictionary of Parado.\-

89

Val i d deductive arguments guarantee the truth of thei r conclusion s , given the truth of their premi ses � the rel i abi l ity of val i d ( Ieducti ve arguments can be readily establi s hed. The reliabi lity of inducti ve a rg uments i s � however� more probl emati cal . Suppose one establ ishes that : 1 1 I crow s observed thus far are b lack� and concl ude that the next crow to he observed wi l l al so be bl ack. The conel usion i s not guaranteed by the observational prelni se, because the color of the next crow i s l ogi call y i n d ep en d ent of the color of the previous ones. Yet the fonn of the argument ,<-; eem s i n herentl y reliable and it i s tempting to argue that the concl usion i s j u stifi ed i n virtue of the previ ous reliab i l ity of arguments j ust l i ke it. Yet w hy shoul d the next argument of th i s form be rel i able , j ust because previous examp l es have been? H u m e points out that thi s justification of i nducti ve reasoni n g i tself rel ies on i nducti ve reasoni ng, and is therefore Explanation .

I ,

c i rc ul ar.

ll{esolution. The most accepted response to Hume i s to point out that he a pparently requires a deducti ve standard of reJ iabi 1 ity for what it i s, after a l l � not a ded uctive argument. Inducti ve argumentation should perhaps he

held to a different standard . There have been i n n u me rable s ubsequent attempts to devel op suitabl e "i nducti ve logics. " Prominent among early schemes i s that held by Joh n Stuart Mi ll. REA D I N G Hume , David.

Enquiries Concern ing

HUnUlll Understanding. Oxford :

Oxford lIP, 1 975 .

»-I USSERL'S PARADOX OF S UBJECTIVITYo Edmund HusserJ , the roundecof the phenomenological movement� poses "the paradox of human

subjectivity": How can the o bjecti ve w orld depend for its character on fi nite human subjectivity? Formulation. "Uni versal intersu bjecti vi ty, into which aU objecti vity� everything that exi sts at aU , i s resolved, can obvi ousl y be nothing other than ll1an kind; and the latter is undoubtedl y a component part of the world. I-low can a component part of the world, its human subjecti vity, consti tute t he who l e w o rld, namely, constitute i t as i ts i nternal formation, one w hi c h has alw ay s already become w hat it i s and conti nues to devel o p, formed by th e u n i v e r s a l i nterc o n ne c ti on of i nte n ti o n a l l y acc o lll p l i s h i n g subjectivity, while the latter, the subjects accompl i shi ng i n cooperat i o n , are themsel yes only a partial formation within the total accoln pl i s lllnent?" ( H usserl 1 79) . Explanation. H umanity i s h e l d to b e a col l ecti v e s u bj e c t i vi ty that constitutes (not i n a causal sense b ut through the projecti on of Ineaning-

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Dictionary of Paradox

structure s ) the w orl d as the w hole of bei n gs, and y e t h uman i ty is i ncorporated i n the world, which seems abs urd. Resolution . Husserl tries to resol ve the paradox of human subj ecti vity by clarify ing an equivocation in its fonnulation. Humanity as a component part of the w orld must be sharply d i stinguished from the extramundane subjecti vity that i s the constituti ng ground of the world. The col lecti ve subjectivity of manki nd i s part of the world, and i s bracketed, along with the re st of the w o rl d i n the transcendental re ducti on that reveals transcendental sUbjecti vity. After Husserl , there was a fundamental break w ith h i s tran scende ntal phen ome nol ogy by thi nke rs , such as Marti n Hei degger, Maurice M e rJ eau-Po n ty, and Hans-G eorg Gadamer, who deni ed that the reducti on to transcendental subjectivity was possible. For the se so-cal J ed exi stenti al phenomenolo g i sts the paradox of h uman s u bjecti vity remains a paradox un l ess the phenomenol ogi st envi sions something, such as Heidegger 's Dasein or Merl eau-Ponty 's Ch iasma, that i s more fundamental than the di sti nction betwe e n subj ecti v i ty and o bj ecti vi ty. READING S Hopkins, B urt C. '"Husserl 's Account of Phenomenologi cal Reflection and Four Paradoxes of Reflexivity. " Research in Phenomenology 1 9 ( 1 989):

1 80- 1 94. Husserl , Edmund. The Crisis of European Philosophy and Transcendental Phenomenology. Trans. David Carr. Evanston: Northwestern UP, 1 970.

Contai ners of different shapes and si zes , connected with each other at the bottom are known as Pascal 's vases. If we (partial ly) fill a set of Pascal 's vases w ith a l i qui d� the height of the J i quid w i J i be the same i n aJ ] the containers:

HYDROSTATIC PARADOX, THE.

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' I ' hi s implies that the pressure at the bottom of each container will be the saIne even though the wei ght of the l i quid in each contai ner may be d ifferent. That different wei ghts of the same 1 iqui d can cause the same pressure i s known as the Hydrostatic Paradox. It i s resol ved by observi ng t hat the greater weight is distributed over a greater area, so that the pressure per unit area of the base i s the same i n each contai ner. .

I NCARNATION, THE PA RADOX OF THE.

S�ren Ki erkegaard c l ai ms that one must face thi s absolute paradox i n order to arri ve at true ( 'hri sti an belief. Formulation. One of the central tenets of the Christian faith is the i ncarnation. B ut the concept of the God-man is i nherently absurd. Explanation. Lui s Poj man i nterprets Ki erkegaard 's paradox as founded o n the fact that the meaning of God as an infinite, unchangi ng being is radically i ncompatible with the fi nite, constantly changing nature of man . I l ence, "God-man" i s an oxymoron, a contradicti on in terms. Resolution. There seem to be two basic responses (other, of course, than 1 he denial of the i ncarnation) to this paradox. The fi rst, and perhaps most \v idespread, is that Kierkegaard is claiming that Christian belief is opposed 1 0 reason and that the latter must be abandoned i n order that the believer em brace on faith alone w hat reason teJI s him i s i mpossible. C. S . Evans, however, argues that Kierkegaard does not mean to i mp l y that the i ncarnation i nvolves a logical paradox because that would i mply that it i s Ineani ngless. The absurdity of the incarnation does not stem from Ineani ngless ness, however, but rather from our i nabi lity to comprehend t he nature of God and, for that matter, our inabil ity to comprehend the nature of man. This interpretation, accordi ng to Evans, is supported by the fact that Kierkegaard argues that it is impossible to remain neutral i n face of the paradox : faith and taking offense are the onl y two options. A purel y rational reaction i s real ly naught but a di sgui se for offense, w hich i s grounded in the pride of the supposedly unli mited powers of hurnan reason. READINGS Evans, C. S . HIs Kierkegaard an Irrational ist? Reason, Paradox and Fai th. " Religious Studies 25 ( 1 989): 347-362. Gurrey, C. S . "Paradox, Wi ll and Reli gious Beli ef. " Philosoph}' 66 ( ] 99 1 ) : 503 -5 1 1 . Pojman, Louis. The Logic of Subjectivity. University, A L: l J of A l abalna P,

1 984.

I

,

,

, ,

,

I ,I

, I, I , "

,

I

I

I

Dictionary of Paradox

92

S u gge ste d by Lu d w i g \V itt ge n s t e i n s p e d a g o g i c al re marks i n On Certa inty, the p a r a d o x of i n d oc t r inati on was formulated by C . J . B . Ma c m il l an i n 1 983 . Formulation. �'T h e probl e m of i ndoctri nati on i s thi s : in a m odern d e m oc rat i c s o c iet y the desired goal of education i s that each s tud en t d e v el o p a s et of bel iefs that are rati onaJ J y g rou n de d and open to c h an g e w hen ch a l l e n g e d by better-grou nded bel i efs. In order to d e v elop such students, however, it wou l d seem t h at they Inust ac q ui re a bel i ef i n rat io n al m ethods of know ing w hi ch In ll st itsel f be be y o n d c ha l l e n g e i . e . , held in a m a n n e r i n co n s i stent w i t h its ow n c o n te n t T h u s , stu d ents m u st be i n doctri nated i n o r d e r n ot to be i n d o ctri n ate d : a p e d a gog i cal di l emma or

INDOCTRINATION, THE PARADOX OF. '

,

,

.

" parad ox ( Macmi l l an 370) .

Explanation.

A ccord i n g

to Wittgen ste i n ,

i n tel l e c t u a l d e vel op ment

depend s upon i mparti ng some bel i efs that "stand fast w i th re g ard to all ot h e r s (370) ; unless the c h i l d trusts the context of l e a r n i ng , i t can not l e a r n a n y th i n g . Yet i f that trust is not merited, the c h ao s of doubt" should foll ow (37 1 ) . Resolution. Macmi l l an suggests that the w ay o ut of the paradox i s to r e c o g n i z e that the basi c sy stem of be l i efs an d p rac ti c e s imparted through i nd octri nation makes p o s s i ble a lnore advanced l e arn i n g throu g h evi dence and re aso n i n g and that these critical acti vities p r o vide way s of av o i di n g indoctrination" (37 1 ). Ja me s E. Garri son d e ve l o p s thi s answe r more fu l l y. I nd o ct r i n ati on i s necessary, even desirable, both for the w eI l - bei ng of the i nd i v i dual and t h e c O lnm u n i t y , but there i s a w ay ar o u nd the p e rn i c i o u s aspects of the p a rad o x if there i s � in the words of John Dewey, "at l e ast a s e l f c o rrecti n g i ndoctri nation� not one that demand s the subord i nati o n of c ri t i ca l discri minati on and c o m p ar i s on (267). This program of Ii beraJ e d u c ati o n i s e x p r e s s i b l e i n the s l o g a n "no indoctrina tio n with o u t inoculation " (268) by the i ntroduction of dou bt directed at the basic systelTI of bel iefs. Thus, alt ho u g h a certai n amount of indoctrination i s needed to i nsti l l a rati ona l outlook i n OlIr children and make t h e m autonomous, functional members of society, they must e ventu a l l y be bro ught to confront the fact that rati ona l i ty i s not s elf g ro u n d i ng Pe rh ap s an effe ct i ve w ay of d o i n g so, i s to com e to gri ps w ith the present parado x i tself. "

"

,

"

-

"

-

.

READIN GS Garrison , James W. "The Paradox of Indoctri nation: A Sol ution." Synthese 68 ( 1 986): 26 1 -273 . Macmillan, C. J. B . " On Certainty and Indoctrination." Synthese 56 ( 198 3 ) :

363 - 372.

, I

I

,

I

93

Dictionary of Paradox

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,

, !

,

:I i

! I

I NDUCTlVE PROBABILITY, THE PARADOX OF. T h i s paradox of p r o b abi l i ty t he o ry w a s p r op o un d e d b y

Stephen Spiehnan. Formulation. Given a set of st atements one of w h i ch m u st be t r ue , and gi ven that the proba bil i ty of some c l ai m given one member of t h e set i s t he s am e as the probabi J i t y of that cl ai m gi ven any other member of th e set, a contradi ction ari ses because the probability of the c l aim is not unique. More formal ly, let P(al b) stand for "the p robab i l ity of a , given b . " T hen, .

b P( alb) P(alb)

. . or

11

,

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,

,

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,

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,I ,

I,

I '

I I I I = =

=

.

.

.

I,

,

=

P(alb" ) P(alb,,) Pea) � P(al hI or b2 o r .

,

.

.

,I ! ,

bn) ;;t. P(al b t ).

I

Explanation. Consider the p robabi l i ty that the rol l of two d i c e wil l total

7 . We know t hat each one of the numbers from 1 to 6 have an equ a l probabi l i ty of b ei n g thrown on the fi rst d i e and that one, and only one, of these six n umbers w i l l be thrown . Fu rt herm ore no m atter w hat number i s r o l l e d on the fi rst die, there i s only one fa v ora b l e resul t on th e rol l of the second die. Hence t h e prob ab i l i ty th at the d i c e w i l l total 7 , gi ve n that 1 i s rol l e d o n the fi rst die, is equaJ to the pr ob a bi l i t y that the d i ce w i l l total 7, gi ven that 2 i s rol l ed on the fi rst die, i s e q u a l to etc. T h u s the fi nat result is in d epend e n t of the nUln ber rol l ed on the fi rst die because the p robabil ity of tot a l i n g 7 d oes not change for different throws of the first di e and, h en c e th i s pro b a b i l i ty i s 2/ 1 1 . That i s , ,

,

P(a) ::: P(alb l ) =

2/ 1 1 .

=

P(alb)

=

P(alb)

=

P(alb4)

=

P(al b5)

=

T h us

,

p roba b i l i ty t heory implies two

diffe rent

i

,

, I

I I I

,

'

P(alb6)

S i nce one of the six numbers mu st be ro l l e d on the fi rst di e, however, the probab i l i ty t h at 1 or 2 or 3 or 4 or 5 or 6 is rol l ed o n that die is c e rt ai n ty (= 1 ) . Thus, the proba bi l i t y of total i n g 7 , gi ven that one of the s i x numbers i s ro l l e d on the first die� i s j u st t h e p ro b a bi l ity that the tw o dice wil l total 7 , w h i ch i s 1 /6. That i s ,

resul ts for the

same

event,

w hi ch i s , of course , contradi ctory.

Resolution. Spi e l m an argues that the p aradox i s due to the pri nci ple that

the probab i li ty of a clai m, b a s e d on the total ity of avai lable ev i d e nc e determines the odds of th e claim. Kenneth S. Fri e d ma n , however, counters t h at the princi ple i s i ndeed val i d and that the s o u rc e of the paradox i s that, ,

'

,., ! iI, ;

,

b 1 or b2 or and imply that

�

I,

94

Dictionary of Paradox

even when the d i sj u nc t s are m utual l y exhausti ve, assuming any one of t h e m does ac t ual l y c o n sti tute n e w i nfo rm ati on and that affe ct s the probabil ity of the ori gi nal clai m . READING S

Friedman, Kenneth S. "Resol v i ng a Paradox of Inductive Probability. "

Analysis 3 5 .6

( 1 975 ): 1 8 3 - 1 85 .

Spei lman, Stephen. HAs s u m i n g, Ascertaining, and Inductiv e Probability. " AlnericanPh ilosoplz ic Q u a rte rly Mo n o graph #3 : Studies in th e

Ph ilosophy o/Science.

1 969. 1 43 - 1 6 1 .

INFINITE, PARADOXES OF THE. A l on g w i th semanti caI and set­ the oret i c al paradoxes, parado x e s of t h e i nfi n i te are o n e of the most i mportant classe s of pa rad ox for t h e deve l opment of logic and mathematics. M any of them are due to the fact that our m ental habits, thought strategi es, and i ntuitions, aU of wh i c h w ere devel oped i n fi nite setti ngs, d o not carry over to the i nfinite. I nc] uded i n t h i s class are , among others , the DEVIL ' S OFFER, the PARADOX OF ENTAILM ENT , HI LBERT ' S HOTEL, the INRNITE S ERI ES PARADOX , KANT ' S COSMOLOGICAL A NTI N OM I E S , the LAMP PARADOX, LEWIS CARROLL ' S PARADOX OF E NTAILM ENT OLBER ' S PARADOX , PASCAL' S WAGER , ,

THE THIRD MAN, and ZENO ' S PARADOXES .

READINGS Bal zano, Bernard. Paradoxes ofthe Infinite. Trans. Fr. Prihonsky. London : Routl edge and Kegan Paul , 1 950. Hughes, Patrick, and George Brecht. Vicious Circles and Infinity : A Panoply of Paradoxes. Garden City, NY: D oub J eday, 1 975 . Sm i th , Josep h Wayne. Reason, Science and Paradox: Against Rece iv ed Opinion in Science and Philosophy. London: Helm, 1 986. Ch. 3 . Thomas, Iv a . "A Twelfth-Century Paradox of the Infinite." Journal of Symbolic Logic. 23 ( 1 958 ) : 1 3 3- 1 34.

I NFINI T E SERIES PARADOX, THE .

T h i s p ar a d o x

,

ori g i nally

adduced by Bertrand Russel l , i s proffe red to show that the von Mi ses­ Re i chenbach theory of probabil ity i s i nconsi stent. Let n be n on p r i m e and Pi pri me and arrange the natural i n u m bers i n a seq uence i n the fol l ow i ng two way s:

Forlll ulation

-

..

( 1 ), the probabi l ity that an i nteger chosen at random 1 /2� by u sing sequence (2) , thi s probabi l i ty i s zero .

B y usi ng sequence w i 1 1 be pri me i s

Dictionary of Paradox

95

I��xplanation. The von Mi ses-Reichenbach theory of probabil ity i s based

,I

,

on the

relative frequency of the favorable event occurri ng i n a large number of tri als. Any specific number of trial s , however, can give onl y an approxi mation of the probabi l i ty. In order to get the exact probabi l i ty, as \lv e) ] as to i nsure the uniqueness of the resul t, it is necessary to fi nd the l i mit of these favorable occurrences as the number of trial s approaches i nfinity. Hence, the tri a] s are arranged as a sequence and the requi red I i m i t i s found. In seq uence ( 1 ) each non-pri me is fol lowed by a pri me, so t hat the relati ve frequency of the primes for each initial segment endi ng i n a pri me i s 1 /2. B ut the l i mit of a con stant i s just that constant, so the probabi l ity of picking a prime at random i s 1 /2. In sequence (2), the ith prime is preceded by i non-pri mes. Hence, each i niti al segment ending i n a prime will contai n i primes and 1 /2 i(i+ 1 ) non-primes. Thus, the relative frequency of the primes will be il r l /2 i(i+ 1 )+i l which reduces to 2/(3 i+ l ). 'rhe limit of this ratio as i approaches infinity i s zero. Hence, the probability of pi cki ng a prime at random i s zero. Now, the absol ute number of both primes and non-primes is the same i n both sequences and, hence, the probability must be uni que. Thus, 1 /2 0, which i s a contradi ction. Resolution. Joseph Wayne Slnith argues that the paradox reveal s that the von Mi ses-Rei chenbach theory of probabi lity i s inconsi stent. Arthur Pap, however, has shown that the two sequences used i n deducing the paradox are not random sequences and thus vi olate a presupposition of the theory ; hence, no paradox is engendered. =

READING S

Pap, Arthur. An Introduction to the Philosophy of Science. New York : Free , � P, 1 962. 1 80- l 8 I . Smi th, Joseph Wayne. Reason, Science and Paradox: Against Received Opinion in Science and Philosophy. London: Croom Hel m , 1 986. 1 241 27 .

Davi d Mi ller formulated thi s paradox to show that Rei chenbach 's " strai ght rule" of inducti on is inconsi stent. Formulation. Let pea) be the frequency-based probability of a and P(A ,B) the logical probabil i ty of the statement A , gi ven the statelnent B. Let -a be the event that occurs when a does not occur. M i l l e r ' s deri vati on is then as foll ow s, where a is the event of rol l i ng a 5 on a fair die:

INFORMAT I ON , PARADOX OF.

"

"

I ,; ,

I

96

Dictionary of Pa radox pe A , pea) r) r pea) 1 /2 if, and only if, p ea ) p( -a) peA, pea ) 1 / 2) P(A� pea ) p ( -a» lI2 pe A , p e a ) 1 /2) p ( -a ) peA , pea ) = p ( - a » pC-a) 1 /2 pea ) 1 /2 1 /2 = 1 /6 =

(1) (2) (3 ) (4) (5 ) ( 6) (7 ) (8)

=

=

=

=

=

=

=

=

=

=

=

Explanation. Li ne ( 1 ) i s a statement of Reichenbach 's "straight rule" and may be t ra n s l ated as "if the frequency of roll i n g a 5 has been found to " be r, then the probabi l i ty of ro) J ing a 5 on the next roJ l of the die i s al so r. Gi ven that the frequency has been establi shed by extensi ve investi gation s,

thi s is a reasonable assumption. (For the purposes of the paradox, the existence of smal l margi ns of error are immaterial . ) Line (2) results from the fact that if pea) 1 /2, then p( -a) 1 - pea) 1 /2 and (3) is the result of the substitution of equi val ent propo s i t i o n s Lines (4) and (5) are instances of ( 1 ) . Line (6) foHow s from (4) and (5), w hile (7) foJ J ows from (6) by arithmeti c (as i n (2) . B ut we know from stati sti cs that pea) p(ro l l ing 5 on a fai r die ) 1 /6; hence� thi s fact and (7) gi ve ll S (8). Resolution. Miller 's notati on i s poorly formulated. It led J. L. Macki e to object that Mill er ' s deri vati on depends on an i ll i cit generalization over the variable a. Miller countered t h at a is a constant, standi ng for "5 is rolled on the next ro1 1 of the die" and thus Mackie 's obj ection was not pertinent. Even if the exact fonnulation of Mackie � s objection was fl awed, however, hi s poi nt is w elJ taken because l i ne (4) of the derivation, if it is to be regarded as true, i s a conditional probability and this condition i s l ost in Miller 's notation [si milar remarks apply to (5 )"J . The condition is that the stati stics has given p(a) 1 /2 , but l ater Lline (8)J Mi ller asserts that the stati stics gives pea) 1/6. Thus, the source of the contradiction would seem to be Miller's equi vocati on on the stati stics, rather than Reichenbach's rule. =

=

=

.

=

=

=

=

READINGS Bub, J. , and M. Radner. "Miller 's Paradox of Information. " British Journal for the Ph i lo soph y of Sc ien ce 1 9 ( 1968): 63-67. Mackie, J. L. " Miller's So-Cal1 ed Paradox of Information. " British Journal for the Philosophy of Science 1 7 ( 1 966): 1 44- 1 47. Mill er, David. "' A Paradox of Inform ation. " British Journal for the Philosophy of Science 1 7 ( 1 966): 59-6 1 . - - . " On a So-Called Paradox: A Reply to Professor J . L. Mackie. " British Journal for the Philosophy of Science 1 7 ( 1 966): 1 47- 1 49. - - . "The Strai ght and Narrow Rules of Induction: A Reply to Dr. B ub and Mr. Radner." British Journalfor the Philosophy of Science 1 9 ( 1 968): ] 45- 157 .

97

Dictionary of Paradox

Popper, Karl .

" A Comment on Miller ' s New Paradox of Informati on. "

British Journal for the Philosopizy of Science 1 7 ( 1 966) : 6 1 -69. - - . HA Paradox of Zero I nformation. " British Jou rna ljor the Philosoph.v of Science 1 7 ( 1 966) : 14 1 - 143 . S m i th� J oseph Wayne. Reason, Science alld Paradox: Against Received Op inion in Science and Philosophy. London : Helm, 1 986. Ch . 7 .

lnsolubilia, or "Unsolvables," are for the m o s t part medi eval v a r i a t i o n s of t h e L I A R p a r a d o x called the Mentiens . Med i eval l o g i c i a n s coll ected at least fourteen different versi ons of the paradox, as i n Albert of S ax o ny s Perutilis logica ( 1 5 22) and devel oped over a d o z e n different sol uti ons to it, as i n Paul of Venice's Logica magna ( 1 499) . One version goes as fo llows:

iNS OLUBILIA.

The

,

I I,

'

I !

I

'

PJ ato says : What Ari stotle i s about to say i s false. Ari stotle say s : What Pl ato j ust said i s true .

Assume that Pl ato has spoken tru ly. What Plato said was that Aristotl e w i l l speak an untruth, so as per Plato's dictum - Ari stotl e's statement is false. But A ri stotl e s statement was that Pl a to had i n de ed spoken truly and the o nl y w ay that thi s can be fa l se is for Plato 's statement to be faJ s e . Hence� if Plato 's statem ent i s true, it i s fal s e . Assume, then , that P l at o ' s statement is fa] se. This means that AristotJ e must h a ve s p oke n trul y. But hi s statement was that Plato had spoken tru l y which must i ndeed be true. Hence, if Pl ato ' s statement i s false, i t i s true. The ups hot is th at Plato 's s tate m ent i s true if, and o n ly if, it is false, which, of c ours e , i s a contradi ction. A. N. Prior 's solution to this ki nd of parad ox is discussed i n PRIOR' S FAMILY OF PARADOXES . Joseph R. Jones o utl i nes Paul of V e ni ce s pre se ntati on of another version of the Liar, called the B ri dge as foU ows ( 1 87): -

'

,

'

,

Let us establ i sh as a true principle that those who speak truly w i lJ cross the bri dge and those who speak fal sely wil1 not. Now Sortes say s, ' Sortes wi l l not cross the bri dge. ' If thi s is a true statement� then according to the above principle , he wi l l cross ; but he say s that he will not cros s ; hence ei ther the pri n ­ ci ple (that al l truth-say ers cross) i s fal se, or Sortes is lyi n g : i n which case he will n ot cross the bri dge , as he (truthfully ) says.

On Jones's view, the Bridge i s the s o urc e of the Sancho PanzA � and the saine resoluti ons ap p l y to both. See al so EUB ULIDES ' PARAD()X r�S and the LIAR. READINGS

De Ruk , L. M.

" Some Notes on the Med i eva l Tract IJe IJlso/ubiliblls. "

Viva rilun 4 ( 1 9 66 ) : 84.

•

Dictionary of Pa radox

98

Herz berger, H . G . "Truth and M odal ity i n Semantically Cl osed Languages." The Paradox of the Liar. Ed. R. Marti n. New Haven : Yale U P, 1 970. 25 -46 . Moody, E. A .

Trutlz

and

North-Holland, 1 973 . Pri or, A . N .

Consequence in Medieval Logic. Amsterdam : 1 03 - 1 1 0.

" S ome Pro bl e m s of Self-Reference i n J o hn B urri dan. "

Proceedings of the British AcadeJny 48

Spade, Paul Vi ncen t.

( 1 962) : 28 1 -296.

"The O ri gi ns of Medi eval Insolubilia-Li terature. "

Franciscan Studies 3 3 ( 1 97 3 ) : 292-3 09.

- - . The Medieval Liar: A Ca ta log u e afthe Insolubifia ,

Literature.

Toronto:

1 975 .

- - . " Ockham on Terms of Fi r st and Second Imposition and I nte n ti on , w i th Remarks on the Li ar Parado x

--.

,

"

Vivariurn 1 9 ( 1 98 1 ) : 47-5 5 .

" Fi ve Earl y T h e ori e s i n the M edi e v al I nsolubilia - Literature . "

Vi variunz 25 ( 1 987): 24-46.

- - , and Gordon Anthony Wilson . Jo/zannis

Wyclif' Sumlna In solubil ium.

B in g h am ton , NY: Medi ev al and Ren ai ssance Texts, 1 986.

INTERESTING NlTMBERS, PARADOX OF. Let the set of positive

i ntegers be divided i nto two d i sjoint, nonempty s ubsets - the i nteresting n umbers and the uni nteresti ng numbers - using any c riteri on w e w i sh . Then cons i der the set of uni nteresting numbers. It m ust have a l east element, say x. But, since x has thi s unique property of being the least unintere sting number, x i s indeed interesting after all , so w e move i t into the interesting set. The re sulting uni nteresti n g set will agai n have a l east e lement, w h i ch i n turn makes it i nteresti ng. By reiterating the arg ument a suffici ent number of ti mes , w e fi nd that there are no uni nteresti ng numbers ! Thi s is, of course, a fal s i di cal paradox, depending as it does for i ts plau sibility on the ambiguous use of "i nteresti ng. "

IRRATIONAL NUMBERS, THE PARADOX OF. To the man on the

street, it i s the Pythagorean be lief that the w orl d exhi bits an intrinsical l y rational structure that i s paradoxical , not the discovery of irrational number that so scandal ized the Pythagorean school . The earl y Pythagoreans thought that "al l i s n u mber and harmony. " B y numbers they meant the positive i ntegers; by harmonies, the rati os and proportions betw een these i ntegers . Yet not all geometrical rel ationsh i ps can b e expressed as ratio s between whole num b ers. For example, the isosceles ri ght triangle w i th the two equal sides of one un it has a hypothenuse e q ual to the square root of tw o . I n d e e d � t h i s b e c a m e t h e s t an d a rd e xa m p l e o f a n

, I!

, I , ,

,

Dictionary of Paradox

I

99

·�i nco m m e n s u rabl e" q u an t it y ( m ore p re c i se l y : t w o q u a n t i ti e s are i ncolnmensurable if they cannot be measured by a common un it), a l though the ori ginal di scovery may have been made i n th e regul ar pentago n . In a n y c a s e , geome try pl ay e d a cen tral role , s i nce l i ne s e g m e nts are conti nuous, as opposed to the di screte system of natural numbers. Hence � the Pythagoreans w ere faced with an examp1 e of something that (for th em) rea l l y exi ste d , but w hi c h c o u l d not be expl a i ne d by num bers and harmonies. Thus the di scovery of such �'irrati onaJ num bers'� seri ously threatened the Pythagorean concepti on of the cosmos, the ir existence being

i

I

,

I I

,

a f1 aw in an otherw i se elegant uni verse. Report has i t that w hen one of

thei r members made the exi stence of i rrati onal numbers publ i c. he was taken out to sea by the brothers and d rowned. REA DI N G von Fritz, Kurt. "The Di scovery of Incommensurabi lity by Hi ppasus of Met apon tu m. Annals of Mathematics 46 ( 1 945 ) : 242-264. "

Taken from the Boo/( oj'Judges by Georg Henr i k von Wrig ht as a test case for deo nti c logic, thi s paradox i nvol ves

JEPHTA PARAD OX, THE.

the substitution of i dentity i n deonti c contexts and, thus� i s s i mi l ar to th e MORNI NG STAR PARADOX i n modal logi c : Jephta promises God that he will sacrifi ce the first being that he meets on his way home. The fi rst being that he meets� however, i s his daughter Miriam. Therefore, he ou ght to sac­ rifice h i s d au ghte r.

Since the present paradox i s structurally s i mi l ar to the Good SAMARITAN, we will not discuss it further here. See also the PA RADOX OF GENTLE MURDER. READINGS van Eck, Job A. "A System of Temporally Relative Modal land Deonti c Predicate Logic and its Philosophical Applications . " wgique et Analyse 24 ( 1 982) : 249-290 and 3 39-38 1 . von Wri ght, Georg Henrik. "A Correction to a Ne\-v System of Deontic Logic. " Dan ish Yearbook of Philosophy 2 ( 1 965): 1 03 - 107.

KANT ' S ANTINOMI E S .

I mman u e l Kant ( 1 724- 1 804) w a s th e propounder of a series of paradoxes that he sty l ed "anti nolll i es." T hese i n c l u d e d h i s fo u r C O S M O LO G I CA L A NT I N O M I ES , h i s A N T I N O M Y O F T ELEOLOGICA L J UDGMENT , his antinomy of taste (see t he PARADOX O F TASTE ) , hi s A NTI NOMY OF PRACT ICAL REA SON and h i s anti nolny of bi po l ar dual i ty. ,

,

l Oa

Dictionary of Paradox

S alnuel Fl e i schacker adduces y et a "fifth an t i n o m y

be Kan t i an

i n s ty l e

,"

w hi ch h e c l ai ms to

.

REA DIN G Fleischacker, S amue l . " ' A Fifth A nti nomy. "

Ph ilosophia I Israel i 1 9 ( 1989):

23-27.

KANT ' S ANTINOMY OF PRACTICAL REAS ON. Thi s i s a r e l i gi o u s

and an ethi cal paradox. Formulation. T h e SUln nl Z-lJ11 hOn U 111 , or h i g h e st go o d h a s two el elnents, th e supreme good , w h i c h i s virtue, an d the com pl ete good , w h i ch i nc l u des not on l y v i rt u e but al so h a pp i ne s s . The h i g h e s t g o o d demands then th e u n i on of v i li u e and h a pp i n e ss but how i s t h i s union to be e ffec te d ? Either the de s i re for h app i n e s s must m oti vate the maxi m s of vi rtue, or el se the maxi ms of v i rtue m u s t be the effi ci ent cause of h a p p i ne s s . Yet neither of these aJternati ves i s p o s s i b l e , because, on the on e hand, i f h app i n e s s i s m a d e the end of acti on, it destroy s the moral i ty of acti on� and beca u s e on t h e other. v i rtue i s n e i th e r a n e c e s sary n o r a s uffi c i e nt condit i o n of h ap p i n e s s but o n l y the c on d i t i on for d e s e rv i n g h appi n e s s Exp Lanation. It ap p e a r s that if t h e w orld had a c o h e re n t structure, happ i n e s s and virtue w ould be rel ated to e ac h other in some fundamental ma n n e r e ither i nsofar a s acti ng fro m d uty must he l p to b ri n g ab o u t h a p pi n e s s or i n sofar as the p u r s u it of h app i n e ss facilitates acti ng from d uty . Ye t acti n g from d uty does not i ss ue i n h app i n e s s b ut only in be i n g w Ol1hy of h ap pi ne s s , an d act i n g from p ru d en c e is i n c o mp atibl e w ith acti ng from d uty. Thus, t h e h i ghe st g oo d does not seem to be a w el l w ro u g h t w ho l e . Resolution. AJ tho u g h no fi nite , and h e n c e se n s i b l e be i n g has suffi c i ent know l edge an d pow er to s e c u r e an absol ute h app in e ss con si stent w ith its duty, there is no reas o n to presume th at t h e w o rl d of sen si ble e x p e r i e nc e i s t h e the o n l y one there i s . In a - that i s , the w o r l d of ph e no ln e n a v irt u e n onsensi bJ e (that i s , i ntel l i gi b l e ) w orld - a noumenal w orl d a n d h app i ne s s m i g h t be united. The demand of p r a c ti ca l reason for a u n i on of v i rtue and happ i n es s m i ght be sati sfi ed then by th e i deal of an i ntel l i ge nt A u t h o r of nature , w ho w o ul d rew ard v i rt u e w i t h e t e r n a l h app i n e s s ,

,

,

,

.

�

,

,

,

.

Cas si rer, H. Barnes ,

READING S

W. A COfJunenta ry on Kan t 's Critique ofJudgnlen t. Ne\v York: 1 938 . Sec. 70.

Kant, Immanue l .

Th e Critiq ue of Practica l Reasoll .

Abbott. London : Longmans, Green, 1 909. 209-2 l 6.

Trans. Thomas K.

Dictionary of Paradox

1 01

KANT ' S ANTI NOMY OF T E L E O L O G I C AL JUDGME NTo I<.an t ' s d a y

,

s ci e n c e

In

e s p e c i a l l y p h y s i cal s c i e n c e - had al re ady

-

·

.

prop o u n d e d the goal of i de n ti fyi n g c a u s al i ty \\ri th e ffi c i e nt causali ty. Kant nevertheless proposed that understandi n g t h i n gs th ro u gh fi nal causal i ty does not in pri nc i pl e conflict \v i th m echani sti c e x pl anati o n

parad ox s upports t hi s c ontention Forlnulation.

The present

.

.

The thes i s i s that al l produc ti o n or material obj ects i s

possible in accordance \v i th purel y efficient c au s al i ty

.

The anti thesi s i s

that so m e p ro d u cti o n of m ateri al obj ects i s not po s s i bl e i n accordance

The t h e s i s and an tithes i s a re ge n e ra t e d by t a ki ng a s n atu ral

,vorI d t\\'O n1axims that pl ay a rol e

in

guiding

the e n1pi ri cal i nvesti gati on of nature i n a systematic m anne r The thesi s .

above resul ts from taking as a pri n c i pl e consti tuti ve of nature the regulati ve pr i n c i pl e that all m ateri al o bj e c t s should be j udged in accord an ce \vi th purely effi cient c au s a l it y

.

I n turn , th e

antithesi s issues

consti tuti ve the re g u l at i v e p ri nci pl e that not al l In atcri al

from taki ng as

obj ects

c a n be

j udged i n accordance \vith purely effi ci en t causaJ i ty, but req ui re i nstead to be j u d ge d i n ac cordan c e ''''i th tel eol ogical , or fi nal , c aus al i ty Resolution . The anti mony of te l e ol ogi c al as c ri p ti on ari ses fron1 taki n g as d e te rm i n ati v e of the n atu ral ,vorld pri nci ples that ari se from and have thei r val i dity i n the e m p i ri cal i n v e s ti g ati on of nature. So long as these p ri n c i p I e s are not absolutized in thi s ",'ay there i s no c on tradi c tio n The pli nc i ple of detem1i nant j ud gme nt which seeks explanati on thro u gh p ure ly mechanical lavv' s , n e e d not contradict the principle of reflecti ve j udgement , \vhi ch tri es to u nderstand thi ngs i n thei r pu rp o si v e o rgani z at i o n For i n the end i t n1i ght b e detern1 i ned that the v al idi ty of both p ri n c ipl e s i s to be cOlnprehended th rough a si n g1 e u n d e rl y i ng pri nci pl e. .

.

,

.

Cassirer, H . \'l. A Barnes,

York:

READINGS

Conunentary on Kant 's Critique ojJudgnzent.

Ne\v 'York:

1938. Sec. 70.

Kant, Immanuel.

The Critique of Judgnu!nt.

Hafner, 1 95 1 . Pt.

Trans . J. H. Bernard.

Nc\\'

2, ])iv. 2 .

KANT ' S COSM OLOGI C AL ANTINOMIES. I n th e Tran scend e n tal D i al e c ti c of the H

c o s nl 0 l o

g i c al

"

Critique oj

Pu re Re aso n , Kan t p u t fo r\v a rd fo u r

a n ti n om i e s .

T h e fi rs t t ,v o of t h e s e \v c rc c a l 1 e d

111athen1ati cal an ti n o nl i e s the last ,

I

•

I' i

..

I

,

i. .

con sti t u ti v e of the

I

, •

vv ith pu re l y e ffi c i e n t c au sal i t y Exp lanati o n .

I I

1\\/0 d y na111 i ca l

an t.i n oill ics � and each

con si s ts of a rati o n al i s t thesis and an enlpiri ci st an ti thes i s .

I n the First

Dictionary of Paradox

1 02

A n ti no m y the thesis i s that the \vorI d has a b e gi nning i n time, and i n ,

re sp e ct

of space i t i s enclosed i n li m i ts . T he antithesis i s t hat the \v orl d

has no temporal beginni ng, and no l i m i ts i n space ; i t i s i nfi ni te in respect to both space and time. In the Second A nti nomy the thesis i s that every ,

composi te thing i n the \vorId con s i sts of simple parts and nothing any" h e re r

exi sts except the si m pl e or that \v hi ch i s formed by composition of i t. The anti thesi s is that no composite thing i n the ,vorld consi sts of s i mp l e parts and there exi sts noth i ng s i m p l e any\v hc re in the world . I n the Third

A n t ino m y the thesi s is that c aus al i t y in accordance \\li th the la\v s of nature ,

i s not the onl y causal i ty from \\'hi ch the worl d as a \vhole i s derivable , but

i t i s necessary to accept a causal ity by freedom as an explanation of them. T he anti thesi s is that there i s no freedom happens accordi ng to the l a\vs of nature.

everythi ng i n the vvorld I n the Fourth An tinom y the ,

the si s i s that there belongs to the "vorld either as part of i t or as its cause

an abs o l u tel y nece ssary bei ng The an ti thes i s i s that there nowhere exi sts .

an absol ute being either in the "vorId or outside it as its cause. Explanation.

Together \vith the paralogi sms, the four anti nom i es are

meant to establish the Pri nc ipl e of Critical Phi l oso p hy namely, that ,

co nc ep ts are validly employed on l y i nsofar as they are applied to po s s i bl e experi ence. Appl i cati on of ideas beyond possible experi ence ine v i tab l y

resul ts i n the pernicious contradictions of Transcendent Metaphysics.

Resolution . The details of the arguments for the anti nomies are hard to

[01 10\\/, and commentators rarely fi nd then1 conv i ncing. For e xam pl e i n ,

The lVorld as Will and Representa tion , A rthur S chopenhaue r fi nds i n each

case the argulnents for the theses i nvalid and those for the an ti the s es v alid ( see Vol . I ,

"

Cri ti ci sm of the Kanti an P hi l o sop h y

"

).

Most phi l o sophy

si nce Kant has j oi ned i n hi s desire to 1iIni t the s\vay of m etaphysi cs , the common complai nt bei ng that he did not go far enough i n thi s di recti on. READINGS Baldacchino, Lewis. 97. E\\'ing, A. C .

"Stra\vson on the Antimony. '� A1ind 93 ( 1 984) : 9 1 -

A Short Cornmentary

on Kant s Critique of Pure Reason.

Chicago: U of (�hicago P, 1 938.

Gram� M. S .

"Kant 's First A ntin10ny. " Kant Studies Today.

Ed. Le\vis

\Vhite Beck. La S alle, IL: Open Court, 1 969 . 2 1 0-229.

(}re en\vood, John D. "Kant' s T hird A ntimony: A g ency a nd C ausal Explanation. " International Phi losophica l Quarterly 30 ( 1 990) : 43-57.

I(ant, Immanuel. hnnranue l Kant s Critique ofPure Reason. Trans . Norman l(emp S mith. New r7ork: St. Nlartin ' s P, 1 929.

384-484.

11oore, j\ . 'vV. "Aspects of the Infinite in Kant" �1ind ns 97 ( 1 988) : 205223 .

�

!

,

I

Dictionary of Paradox

1 03

,

I I

,

.. ,. i

,

i"( I

'i "

Schopenhauer, Arthur. The World as Will and Representation. 2 vals. Trans. E. F. J . Payne. New 'York: Dover, 1 %9. Stra\vson. The Bounds ofSense: An Essay on Kant � Critique ofPure Reason. London: Methuen, 1 9 66 34-36, 133 - 140. Welton, T. D. Kant � Critique of Pure Reason. Oxford: Clarendon, 1 958.

I

I ,

I' , I, I ,I I

i

.

,

I' , I

203-2 1 7.

Wolfe, Julian. "On the Impossibility of an Infinite Past: A Reply to Craig. " International Jo urnal for Phi losoph)' ofRe Iigion 1 8 ( 1 985) : 9 1 .

,

, , ,I

. I.

,

i! Ii , i

KNOWABILITY, PARADOX OF. Frederick B . Fitch presented this

purporting to show that the idea of a truth that is unknown is 1 )Y'oblematical . Formulation . The two principles upon which the arg ument rests are: If a propositi on i s true, then it is knowable. (1) (2) There i s at least one truth, say p, that is never known. Applying principle (2) to ( 1 ) , we obtai n (3 ) It is kno\v n that: p an d (p i s not kno\vn) . Yet si nce the kno\vledge of a conj unction entai l s kno\vl edge of each c.onj unct, we have P is known and it is known that p i s not known. (4) tJ earl y, if somethi ng is known, it is true ; hence \ve may drop the " it is known th at operator from the second conj unct, resulting in p i s known and p is not known. (5) But (5) i s a contradiction. For ease of exposi tion, we have i gnored the l110dal operator that appears in the ori ginal argument. Hence, \ve no\v refonnulate ( 1 ) - (5) , using 'P' to stand for logical possibility and ' k ' for H i t i s kno\vn by someone at sometime that. " For all q: if q, then PKq. (1) (2) P and not-(Kp) . PK(p and not-(Kp)) . letting q (p and not-(Kp) i n ( 1 ) (3 ) P[Kp and K(not-(Kp) ) ] . (4) ( 5) P( Kp an d n ot- ( Kp) ) . Line ( 5) nO\\f asserts that a certain contradiction i s logically possible. Explanation. Principle ( 1 ) asserts that all truths are knowable. This assertion is actually relatively vveak for it merely asserts that there i s nothing self-contradictory about having knowledge of a true proposi ti on . Principle (2) is obviously meant to be relativized to human kno\vers . B oth seem intuitivel y plausible. Nevertheless , \ve are led to concl ude that a certai n contradiction is not contradictory (logical possibi l i ty i s the absence of contradiction) , which is itself a contradicti on. argument

"

=

!I

,I

I

I� I ; ,

I

,

I

I

1 04

Dictional'.}, of Paradox

Resolution . Of the p ri nc i p l e s ( 1 ) and (2),

the second seems m ore secure ; thus, the first has recei ved n10re atte n tion Do ro th y Ed gington prop o s e s a r e fo rm u la ti o n of ( 1 ) to blo c k the paradox . She hol ds that the correct ex pre s si on of ( 1) is that for any actual truth it i s possible to kno,v that it is actually true. I n other words, there exists some \v orl d lV, s uch that someone in lV know s that the proposition �p and not- (Kp) , i s tr u e in the actual world. No contradicti on i s deri v able from thi s expre s s i o n provided that w i s distinct from the actual \\l orId. A probl e m \\lith t hi s sol uti on i s that the requisi te knowledge of the actual \vorld can only exist i n a di fferent pos sible \vorld, \vhich requirement s ee m s to run counter to the i ntui tion un derl y i n g ( 1 ) . nam e ly that there are no truths about our o\vn \vorld or si tuati on \vhich , i n principle, \ve are bl ocked from kno\ving . Another approach is to que stion the appl i cabi l ity of p ri nci pl e ( 1 ) to truths such as (2) . Support for ( 1 ) is derived from co ns i d erati on of si lnpl e facts about the \\lorld, together with the idea t hat \vith some hard 'Nark and serendi p ity \v e can di scover those fac ts Of c ou rs e the re may be facts that are suffici ently remote i n s pace and ti me to m ake them undi scoverable in p racti ce but \\lhich could have been di scovered by an agent i n th e ri ght place i n the ri ght ti m e . Sentences such as (2) , \vhich bri n g i n kn o 'A- l e d ge as th ei r ovvn subj ect matter, are not the stuff on vv hi ch such i ntuitions are .

,

,

,

.

,

'

built. READINGS Edgington, Dorothy. ' �T he Paradox of Kno\vability. '� A1ind ns 94 ( 1 985) �

557-568. Fitch, Frederick. B . HA Logical A.nalysis of Some Value (;oncepts. " Journal

of SYJ'nbolic Logic 28 ( 1 963): 135- 142. Williamson, Timothy. "On the Paradox of Know ability.

"

Mind ns 96 ( 1987) :

256-26 1 .

KNOWER, THE PARADOX OF THE.

Not to be mi staken for the

PARADOX OF THE KNO\VER IN DEONTIC LOGIC, thi s episten1 i c

version of the

LIAR \vas first adduced by Richard Montague and David Kapl an . I t seems

to j eopardize the status of the concept of omni science, for if the paradoxical statement S were in fac t true, there could be no all -knowing being. The re as oni n g is reminiscent of th at of the Paradox or KNO\VABILITY . Formulation . T he sentence No one knows

( S)

that this

senten ce i s true

cannot be fal se. Explanation . true.

Yet

Suppose

S to be false. Then someone

if someone kn o\v s

it

to be tru e then ,

it

is

does knO\\' i t to be true. Therefore the

1 05

Dictionary of Paradox

,

·,

,

supposi tion that i t i s fal se 11l ust be incorrect. Hence, \ve oursel ves no\\/

kno\v S

to

be true, since \ve ha\Tc j u s t demonstrated i t. Ye t i f SOlneone

,I

I' · ,

kno\\l s S to be true , i t must be fal se because S asserts that nobody kno\vs i t. Therefore , S ln ust be both true and fal se, \vhich i s contradictory. -'

Resolution . Perhaps the Inost common response to the pre sent paradox

i s to claim that S i s meani ngles s . S can be branded as m eani ngles s by asserti n g that no reasonable sentence can include self- reference, making

· I

,

.

I

I.

� 'I

, , .I

II

,

i t nothing more than a tri vial variant of the Li ar. A nother approach i s to c1 aim that propositi ons and not sentences are proper objects of knowledge �

thus , S vvoul d be ill -fanned because i t presu pposes that sentence s are obj ects of kno\vledge . S till another approac h \vould be to affirm that the pred i c ate "'kno\vs" i s meani ngles s . Thomas Tymoczko, ho\vever, argues

that al l these al ternati ves are too extreme . According to Tynl0czko ,

,, i, ,

I

I

a

c l o s e r e xa m i n a ti o n o f the paradox re v e al s that i t re s ts o n fo u r epi s ten10l ogical pri nciples :

The \Teracity Principle: If sonlc-o ne kno\vs a sente nce it i s true. T he Conlpletcness Princi pIe: If someone denlonstrates a se n ­ tence, he knovvs it. The Extension Principle: If someone kno\vs both a condi ­ tional and its antecedent and uses modus ponens to detach the consequent, then he kno\vs the consequent . The J\;letaveracity Principle: \Ve kno\v the \leracity Principle . Although al l fOllr of these pri ncipl es are intuiti vel y pl ausi ble, they have al l been chal l enged on independent grounds .

,

REAJ)Il\�GS

,

A. nders oll C:. A.nthony. "The Paradox of the Kno\vcr. " ,

80 ( 1 983): 338-3 55.

lournal o/Philosophy

i\'1ontague, Richard, and Dav id Kaplan. " A Paradox Regained. " Notre Dame

Journal 0/ Fornw l Logic 1 .3 ( 1 %0) : 79-90. S mullyan, Raymond. A lice In Puzz le-Land. Ne\v 'York, 1 9 82 . T y m o c z ko , T ho m a s . "A. n lJn s o ] 'l e d Puz z l e about Kno\v l edge . " Philosophical Quarterly 34 ( 1 974) : 437-458 .

KNOWER (IN DEONTIC LOGIC) ,

THE

PARADOX O F.

Th i s

probl e n1 of deonti c l ogic i s som evv hat s i mi 1 ar to the Par(l] )( )� O J : T i l E

KNO\VER.

The exam pl e i s due to Jam es

Form ulati o n "

,, ,

E. Tomberl i n.

Oi'len certai n reas onabl e s y s tenlati c pri n c i p i c s , the

foI l o\vi ng three sentence s are inconsistent:

Dictionary of Paradox

1 06

(1)

(2 )

X set fire to the store duri ng Y ' s watch.

If X set fire to the store duri ng V ' s \vatch, Y o ug ht to

know about it. (3) X ought not set fire to the store .

Explanation. From (4)

( 1 ) , (2) ,

an d modus ponens , \ve get

Y ought to knO\N that X set fire to the store.

Moreover, gi ven that kno\vledgc entai l s truth (see al so the PARADOX OF KNO\VAR ILITY and the PARADOX OF THE KNO\VER) , \ve concl ude that (5) (17 kno\vs that X set fire to the store) entails to thcstore) .

(X set fire

Fi nal ly given the pri nci ple that whatever is entai led by an obl i gatory ,

statement is i tsel f ob l i g ato ry \ve obtai n ,

(6) Lines

(3)

X ought to set fire to the store.

and (6) , ho\vev er, are contradi ctory.

Resolution. Fred Fe l d m an argues that the paradox can be avoided by rel ati v i zing the ought operator to temporal s tates and takin g certain prec a u tions in formulati ng condi tional ob li g at i on s

.

REJ-\ DINGS Feldman, Fred. 'The Paradox of the Kno\ver. "

Phi losophical Studies 55

( 1 989) : 93 - 1 00. Tomberlin, Jrunes E. "Obligation, Conditionals, and the Logic of Conditional

Phi losophical Studies 55 ( 1 989) :

Obligation. "

KRIPKE 'S PIERRE" See the PARADOX

8 1 -92.

OF NAlvIING .

LAMP PARADOX, THEo Propos e d by J . F. T ho m p s on thi s puzzle i s a ,

spi n-off of considerations re l ati n g to ZENO ' S PARADOXES .

Formulation. A l am p ori ginally off, i s tu rne d on and off an i n fi ni te ,

num ber of ti m e s

.

Wi ll i t then be on or off?

Explanation. I t is as sumed that the i nfinite number of s w i tc hi n g s can be

done i n a fini te amount of ti me. T his can be done as fo1 1o\\' s : the svvitch

is throvvn at tilne zero , agai n at one minute, again at one mi nute and thi rty

seconds , and so on. After t\VO m i n u te s have el apsed , th e s\vitch \vi l l have

Dictionary of Paradox

1 07

been th row n an i nfinite number of times. When the tw o minutes are up, t he 1 amp i s certai nly either on or off. I t cannot� however, be on because

time it was turned on it was subsequent1 y turned off. Li kew i se, it cannot be off because each time it was tu rned off i t was subsequently t urned on. Resolution. J . F. Tompson concl udes that the task, which is one e xa mp l e of what i s often called a "super-task,�' i s logicaJJ y impossibl e. Another approach i s to deny that the final state of the l amp i s determi ned by the i nfinite number of switchings effected.

each

I

,

I

READINGS Berresford, Geoffrey C. �'A Note on Thomp son 's Lamp " Paradox ' . " Analysis

4 1 . 1 ( 1 98 1 ): 1 -3 . Thompson, J. F. "'Tasks and Super-Tasks." Analysis 1 5 . 1 ( 1 954): 1 - 1 3 . Ray� Chri s topher. " Paradoxical Tasks . " Analysis 50.2 ( 1 990): 7 1 -74.

� .JAWYER, THE.

See the

LEGAL PARADOXES.

EUALTHUS .

Legal paradoxes may occur w hen different,

hut equally b i n d i ng laws i mpose conflicting obli gati ons upon the citi zen. A g e ne r a l l y more interesti ng type of legaJ paradox, however, results whe n a n apparentl y i nnocuous legal pri nciple p uts the citizen (or the State) in a n i mpossible position. Legal paradoxes are often Catch-22s or DILEMMAS, and may be consi dered to be a subspecies of DEONTIC PA RA DOXES , al though the l atter d esi g nati o n usual l y i mp l ie s that the paradox is being c onsidered i n the framework of deontic logic. See, for example, the EUATHLU S , the PARADOX OF NATURA L RIGHTS , S A NCHO PANZA , the PA RADOX OF S ELF­ AMEND MENT , and S M LJ LLYAN ' S PARADOX .

LEWIS CARROLL' S PARADOX OF ENTAILMENT.

This problern purports to be a serious chaJ Ienge to the n ot i on of entailment. Formulation. Let (A and B) entail C. But one may refuse to ac c e pt C unl e ss the propositi on "If A and B , then C" i s true. Addi n g th i s a s a pre m i se, w e o btain (A, B , and (If A and B � then C)) entai l C. Proceed i n g i n thi s manner, an i nfinite regress i s gen e r at e d . Explanation. Let A and B be the premi ses of a val id arg u me n t t h at has C as i ts conc l usion. It i s possible that someone accept both A a nd B and yet n ot accept C if he does not accept the propositi on � ' Jf A and B , then C." l Ienee, the l atter must be added as a premise to the arg u m e nt . The same

, ,

,

Diction ar}' of Paradox

J 08

reasonin g can be applied to the new, three-premised argument; and� i ndeed, each new argument gene rates a n eed for a new prem i se th ereb y generati ng an i nfi nite regress. Resolution. W. 1. Rees argues that "If A and B , then C" i s a meta- premi se and, thus, i s not on the same l evel as the premi ses A and B . Thus, i t should not be added as a new premi se to the argument and the regre ss i s avoi ded. The i mpOt1 of Ree 's argument i s w ell taken, though i t seems to l ose its pl ausi bi l i ty in the s i lnpl ifi ed form of the paradox gi ven above. (Carrol l ori gi nal J y llsed EucJ i d 's fi rst theorem as an exampl e . ) In any case , the pl allsi bi l i ty of Carrol l 's paradox seems to rest on view i ng logic in t er m s of psyc h o l ogy, that i s i n terms of an i ndi vi dual 's "acceptance" of the propositions i n vol ved i n the argu ment. For the logician, if the argument j s vaJ i d � that 's the end of the story. If someone accepts the premi ses but not t h e concl usion , either he i s i mpl i citly de nyi n g the val idity of the argume nt or has not understood that the argument i s v al i d . I n neither case i s it necessary to add a new premi se to the argument to i ns ure i ts val idity and� hence, no regress ari ses. We tnay al so observe th at, at l east i n cl assi cal l ogi c, w e have a result that states that (A a n d B) entai l C if, and on) y if, · ' If A and B � then C" i s a theorem. READING S Carrol l , Le'rvis l Charles D odgson ] . " What the Tortoi se S aid to Achi ll es. "

Mind n s

4 ( 1 895 ) :

278 -28 0.

Ree s , W. J. " What A c hi lles S aid to th e Tortoi se . " Mind n s 60 ( 1 95 1 ) : 24 1 -

2 46

LIAR,

.

THEo Known si nce anti quity., th i s i s perhaps the paradox that has

Inost e xe rci sed the occidenta1 tnind. Formulation. There are many versi ons of thi s paradox , i ncludi ng: What I am now sayi ng i s fal se . i The sentence pri nted in brackets on thi s page i s fal se. ] ' Yi elds a fa1 sehood when applied to i ts own quotation ' yiel d s a fal sehood w hen appl i ed to i ts own quotati on. (Qui ne, The ffilYS (�f Paradox� 9 . )

There are al so i ndirect versions, such as Pl ato s ays : What Ari stotle i s about to s ay is false. Ari stotle s ays : What Pl ato j us t said i s true.

T here are even imperati ve vari ants, for exampl e,

Dictionary of Paradox

1 09

,

Di sobey thi s order !

A p p a r e n t ly

the inhabitants of the i sl e of Crete were once know n for the i r ski l l at p r eva r i c ati o n for one of the most widel y known l i terary sources of the paradox affirms that �

,

I,

:,

,

"

One of them , even a prophet of thei r own , s ai d , The Cretan s are al ways li ars , evil beasts , slow bell ies . Thi s witness i s true . ( The Ep istle to Titus , 1 : 1 2- 1 3 . )

,

ExplanationD Perhaps one of the sources of the perennial popularity of this p ar ado x i s t hat it i s very easy to understand si nce it re q u i re s no spec i al i zed know l edge for i ts formul ati on. Consi der, for exampl e� the straightforward versi on What I am s ayin g now is fal se. If th i s statement i s t r ue , it

w hat i t s ta tes

itself is false . Thus, if it i s true,

it i s

Inust be the case .

But i t

fal se. On the c on t ra ry

states t h at

ass ll ln pt i o n �

i f the statement i s false� then w h at i t states m ust 11 0t be t h e case

a n d th us, i t i s true . Therefore, t he statement is true if, an d only if, i t i s fal se . One of the es s en ti al elements in thi s paradox i s self-reference, be it d i rec t or i ndirect. In t h e present version, the pa ra d o x i c a l proposition predicates fal s i ty of itself. In i ndirect versi ons, the self-reference i s effected by a ,

closed c hain of propositi ons th at circles back on itself. Observe, however, that self-reference does not seem to be a suffi cient condition for g e n erati n g t he Liar since the proposition What I am now s aying i s true

does not seem n ot paradoxical (but see the T RUTH-TELLER V AR1 ATIONS ( ) F T H E LOG ICA L PARADOXES) . Neverthe l ess, logi cians ge n e r al ly con s i d e r p aradoxe s that are generated by self- reference to be examples of the Li a r. w h i c h thus becomes the archetype of thi s ki nd of paradox. It w ou l d be too ted i o u s to l i st here aJ I the paradoxes considered i n thi s vol H ln e w h i c h are exampl es of the Li ar, but speciaJ menti on shoul d b e Ina de of the EPIMEN I DES a n d of RUSS ELL' S PA RA DOX. R.esolution. S i n c e o u r i ntuiti ons tel l us that the source of t h e Li ar Pa ra d o x is its self-refe rence what Be11rand Russell has referre d to as the �'v ici oliS c i rcle p r i n c i p l e most responses center on try i n g to b l oc k t h i s selfreference. It i s not suffi ci ent, however, to j ust b ran d se l f- refe rence as "

" ,

;

1

1 10

Dictionary of Paradox

Ineani ngless si nce, as we hav e al ready seen, not al l fOnTIS of se lf-reference are paradoxi cal . One way of b l ocki ng the self-reference i s through the use of "hi erarchy" theori es such as those pioneered by Russell i n set theory and A l fred Tarski i n semantics. I n this approach, "true ,'� "fal se �" and kindred pre di cates are not al l ow e d to be applied i nd i scrimi nately to sentences containi ng these expressions. They are typi cal l y suppl i ed w ith i nd exe s to sh ow th e i r rank. "True o" i s only predi cated of sentences containi n g no ranked ex pressi ons. If "true o" appears i n a statement, the truth or falsity of that statement can only be asserted by using the predi cates "true l " or "fal se l . " I n general , if n i s the h i ghest rank t hat appears i n a sentence, any statement about that sentence must be of rank n+ 1 . This approach does seem to e1 i lTIi nate the paradox and it has some plausibi l i ty s i n ce, w h en w e talk about a sentence, w e seem to be doin g so i n a rnetalanguage ; talk about a sentence i n the metalanguage would be done i n a meta-metalan guage, and so on. T hese series of metalan guages may serve as the basis of the rankings. Neverthe l ess, not al l l ogician s accept such a sol ution because ( 1 ) i t does not seem intuitively sat i sfying, (2) a metalanguage need not be d i sti nct from the object lan guage, and (3) most predicates can be used in simil ar contexts w ithout generating paradoxical results. Thi s l ast reason seems to i nd i cate t hat the hierarchy approach i s too radi cal and al so that it does not explain the source of the paradox . A nother suggestion, obvi ously related to the previous one� i s to clai m that the troublesome sentences are i l l-folmed or meaningless or v i ti ated by alTIbi guity. The basic problem w ith thi s approach i s that, as soon as the rel evant disti nctions are made, a new version of the paradox, respecting those disti nctions , i s adduced (see M-VARIANTS OF THE LoGICAL PARADOXES). Another i nteresting approach i s to use semantics w i th truth-value gap s and gluts t o resolve the paradox, whi l e sti l l another suggestion i s that w e shou ld just accept that the Liar i s a contradiction. Don S. Levi , for example, clai m s that the Liar i s j ust an example of a " strange loop," but that it does not i nterfere with any i mportant project that we are i nterested i n . G i ven the structural simil arity of the Liar w ith other paradoxes, such as RUSSELL' S PARADOX i n set theory, whi c h do i nterfere w ith impoliant proj ects, most thinkers w i l l undoubtedl y conti nue to hold that more has yet to be sai d. See PARADOXES OF S ELF-REFERENCE. READING S B arwi se, John, and John Etchemendy.

Circularity.

New York: Oxford UP,

B uckner� Dean, and Peter Smith.

The Liar:

1 987.

An

Essay

on Truth and

"Quotation and the Liar Paradox. " Analysis

1 11

Dictionary of Paradox

46. 2 ( 1986): 65-68. Dumitri u, Anton . "The Antimony of the Li ar. " Interna tional Logic Re view 1 1 ( 1 980): 1 07- 1 1 8. G oldstei n , Laurence. "The Paradox of the Liar: A Case of Mi staken Identi ty. " Analvs is 45 . 1 ( l 985 ): 9- ] 3 . Gupta, Anil. Tru th and Paradox. " The Journal of Ph ilosophy 78 ( 1 98 1 ): 73 5-736. Huggett, W. J. HParadox Lost. " Analysis 1 9.2 ( 1 958): 2 1 -23. Hugly, Phi lip, and Charles Sayward. " Paradox and Semantic Correctness . " Analysis 3 9.4 ( 1 979) : 1 66- 1 69. Levi , Don S . "The Liar Parody. " Ph ilos ophy 63 ( 1 988): 43-62. Martin, Robert L. "'Toward a Solution to the Liar Paradox. " Philosophical Review 76 ( 1 967): 279-3 1 1 . - - , ed. The Paradox of the Liar. New Haven : Yale UP, 1 970. - - . "Reply to Hugly and Sayward." Analysis 39.4 ( 1 979): 1 69- 174. - - , ed. Recent Essays on Truth and the L ia r Parado.x. Oxford: Clarendon, 1 984. - - , and P. W. Woodruff. " On Representi n g ' True i n L' in L. " P/Z ilosophia 5 ( 1 975): 2 1 3 -2 1 7. Mates, Benson. Scep tica l Essays. Chicago: U of Chicago P, j 98 1 . 1 5 -42. Parsons , Terence. "As sertion, Denia l , and the Li ar Parado x . Journal of Ph ilo soph i ca l Lo gic 1 3 ( 1984): 1 3 7- 1 52 . Rozeboom, William W. HIs Epi menides S till Lying?" Analysis 1 8 . 1 ( 1 957): 1 05 - 1 1 3 . Slater, B . H. "The Liar. " In ternational Logic Review 4 ( 1 97 3 ) : 8 6-89. S lezak, Peter. �' Descartes ' Diagonal Deduction . " British Journal for the Philosophy of Scien ce 34 ( 1 983): 1 3 - 3 6. Urs i e , M arko . "A Modest Proposal Concerning Paradoxes." Syn Phil 1 ( 1 986) : 1 93 -203. Vi sser, Albert. "Four Val ue d S emantics and the Li ar. " Jo ur n a l of Philosophical Logic 1 3 ( 1 984): 1 8 1 -2 12. Whi tely, C. H. " Let Epimenides Lie ! " Analysis 1 9.2 ( 1 958 ) : 23 -24. �

"

"

OF. Li bertarianism i s a pol itical phi losophy that pri oriti zes i ndivi dual freedom and seeks to l i Jn it the rol e of the state to a mi nimum. S uc h a positi on , however, may fai l to protect the ri ghts of a significant porti on of the citizenry. Harol d J . Joh nson i dentifi es four aspects of l i bertariani sm whi ch contribute to its potenti al for becomi ng tyrannical : 1 ) I n seeking the furtherance of his own i n terests, the

LIBERTARIANISM, PARADOX

indi vi dual 's actions have consequences that may affect others � in particular, acquired a d v antag es may l ead to w i despread bul lying and vio1 ence.

I

I

il

"

112

Dictionary of Paradox

There i s a natural te n den c y i n competi ti ve m arkets towards monopol i es. ( 3 ) M any of the i n s ti tuti o n s that ope rate i n th e fre e marketplace are themsel v es highl y authoritarian i n character. (4) Individua1 s can take on voluntary obli gations , but have l i tt l e o p portuni ty to re v i s e the s e i n l i g h t of chan g i n g circumstances and con seq uently some persons may attai n domi nan ce over others by this mean s .

(2)

A paral lel paradox i s that of Soci al i sm . Social i sm seeks to guarantee not only the i ndi vi d ua l 's ri ghts, but aJ so hi s w eB -bei ng, t hrough the acti ve i n terventi on of the state . B ut once t h e state i s gi ven the pow er to e nforce i ts decision arbitrari J y, the i ndi vi d ual 's r i ghts become subj ect to the whim of col l ecti ve d e c i si on s The two paradoxes are si milar i n that, i n each case , the soci a) instituti ons may actuaHy frustrate the real i zation of the soci al e nds that they w ere i nsti t uted to sec ure. Johnson argues t hat the paradoxes are real i n that the undesirabl e resul ts d o not come about by e xternal c i rc u mstan ce s , b u t are tendenc i e s i nhere nt i n the sy ste ms themsel ves. .

John son , IIarold 1.

READING " Extrem ism i n the Defe n se of Li berty i s a Vice . "

Univers ity of Ottal,va Quarterly 5 6 ( 1 986): 1 9-27.

LITERARY PARADOXESo T he w ord �'paradox" has been u nderstood v ari o u s l y as a l o gi cal contradi ction� a b s u r d i ty� eni gma, or see m i ng contradicti on . In The Well Wrought Urn � CJ eanth Brooks sees paradox as a fundamental element of poetic language � the l anguage of science depends essenti ally on the princi ple of noncontrad icti on, whereas poetry depends on the antithetical pri nciple of contradi ction, e xpressed as the l i terary trop e of paradox. Thus, paradox both teases and chal J enges the mind by te st i n g the l i mits of la n guage It is a uni que w ay of bi-pol ar t h i n k i ng ; "more t h an a verbal artifi ce, the paradox be spea ks the dial ectica1 tension i ntrinsi c to everyday, personal exi stence" ( S l aatte 3 3 ) and i n volves both the oppositi on and reci procation of ideas. Si nce paradox in one sense i s a phrase or statement that whi l e seemingly c o ntrad icto ry or absurd may actual l y be well-founded or true, paradox may be used as a rhetori ca1 and poetical dev i ce to attract attenti on or sec ure emphas i s , to expre ss i nsi ghts that are not as fully stated i n unequivocal l an guage , and to c halJenge rece i ved opinions or val ues. Alternati vel y, paradox i n the sense of real contradi cti on may be used to express the a b s u r d i ty of l ife or the absurdity that man makes out of i t. How ard P. Kai n z understand s literary paradox to differ from phi losophi caJ paradox i n that the y depend on i ns i g ht and .

"

Dictionary of Paradox

I l3 :, i '

i r nagi nati on rathe r than argum e nt," for t h ough "l i terary paradoxes could he arg ued for . . . they wou l d l ose thei r ae stheti c appeal and l i terary val u e i f th i s happe n e d

"

( 40).

Kai n z ad m i t s , h ow ever, th at certai n l itera ry

paradoxes, suc h as those i n the w orks of G . K. Chesterton

I'

Ii

,

( 1 874- 1 936) ,

i n S i r T homas More ' s U top ia , S ¢ re n Kierkeg aard 's Either/ Or, and

Fri edri ch N i etzsche ' s Thus Spake Zarath ustra , recei ve e xte nded argument and are hence l i terary -phiJ osophical hybrids. Writers ce1 ebrated for thei r use of paradox i ncl ude Erasmus

( 1 467- 1 5 36),

Joh n Donne ( 1 572- 1 62 1 ) ,

J o s e p h HaJJ ( 1 5 74- 1 65 6), La R o u ch efo u cau l d

( 1 6 1 3 - 1 680), G e org e

( 1 856- 1 950), Jorge Luis B orges ( 1 899- ] 986) , and S alTIuel Beckett ( 1 906- 1 990) . S ee al so Catch 22, DONNE' S PARADOX OF THE HOLY Bernard S haw

, ,I

, , , , ,I ,.

G HOST , OSCAR WILDE ' S PARADOXICA L EPIGRAMS , RENAISSANCE PARADOX, the SANCH O PA NZA , an d the S EEK ER PARADOX .

REA DING S

Brooks , Cl eanth. The Well Wrough t Urn: Studies in th e Structure of Poetry. N e\v York: Harcourt, J 947. Slaatte , Howard A . The Pertin ence o.ltlze Paradox. N ew York : H u lnan i ti es p� 1 968 .

LOGICAL PARADOXES.

Sorneti mes a lo gi cal paradox is taken to be ,

the apparent or real contrad i cti on betw een two statements , both of w hi ch seem to have good sup porti ng

evidence.

Lo gi cal paradoxes may res u l t

from a m i sapprehensi on of the rules of l ogi c (perhap s because they are not cl earl y expressed or recogni zed) , a

v i o l at i on

of those rul e s , or the

i napp l i cabi l i ty of the ru les of a l ogi c to the s ituati on in que sti on. T here i s also a more re stri cted sense of the term "l ogical parad o x . " Fol l ow i n g F.

P. Ramsey� i t i s custo mary to d i sti ngui sh betw een 10g i caJ paradoxes and semantic paradoxe s .

Logi cal parado xe s ari se i n th e obj e ct l an g uage

becau s e they contai n only the us ual logi cal and set-theoreti cal sy rnb o l s ;

they i n cl ude the BURA LI -FoRTI PA RADOX� CANTOR' S PARADOX and RUSSELL ' S PA RA DOX .

The semanti c paradoxes ari s e i n the metal anguage because

they i n v o l ve semanti c c oncepts such as truth and meani ng ; they i nc l u d e B ERRY ' S PARADOX, the EPI MENIDES , the H ETEROLOGI CAL PARADOX , the LIAR� and RICHARD ' S PARADOX . T here i s anothe r school of th i nki n g that refu ses to make Ramsey 's di stincti on and w o ul d cal l all of these parad oxes l ogical . See al s o M -V ARIANT S

OF

T H E LOGICAL PARADOX ES a n d

V ARIANTS O F THE LOGICAL PARADOXES .

READING S

T R l I T I--J -T E LLER

Chi hara, Charles S. "The Semantic Paradoxes: A I )i agnos ti c I nvesti gation. " Ph ilosophical R e vie w 88 ( 1 979): 590-6 1 8.

,

i�

1

,

I

1 14

D i c ti on a ry

of Paradox

- - . "The Semantic Paradoxes: Some Second Thoughts. '� Ph ilosophical S tu di e s 45 ( 1 984): 223-230. Gard ner, Martin . HLo gical Parad ox e s . Ant io ch Review 23 ( 1 963): J 721 78 . Grel ling, Kurt. "The Logical Paradoxes. " Mind ns 45 ( 1 936) : 48 1 -486. Hannson, B regt. Pa r a d o x e s i n a Semantic Perspecti ve. " Essays Oil Mathelna fical and Ph ilosoph ical Logic: Proceedings of th e Fourth B

"

Scandinavian Logic SYlrlposiunl and of the First Soviet-Finnish Logic CO/�ference. l.v vask�v la. Finland, Ju n e 29-July 6, 1 9 76 . Ed. laakko Hi n tikka , Il kka N i i ni l u oto, and Esa S aarinen . Dordrecht: Re i del , 1 976. 37 1 -385.

H eij enoort, John v an . '" Log i cal Parado x es . " Th e En cy c lop e d ia of Philosophy. Ed . Paul Edwards. New York : Macmill an, 1 967 . Vol . 5 ; 45 -5 1 . Mackie, J . L. Truth, Probability and Paradox: Studies in Philosophical Logic. Oxford: Clarendon , 1 973 . 237-30 1 . Perelman, M. Ch. "Les Paradoxes de l a logique." Mind ns 45 ( 1 936). 204-208. Quine, W V. O. " Paradox." Scien tific Alnerican 2 17 ( 1962): 84-96. - - . TIle Ways of Paradox: And Other Essays. Cambridge: Harvard lIP, 1976.

LOGICAL PSYCHOLOGISM , THE PARADOX OF . T his paradox

ari ses on the vi ew that l ogi c is ul t i m ate l y a department of psy c ho l og y Formulation. The truths of logic are about certa i n mental activities of h uman subj ects � but do not re l y on these acti viti es for j us ti fi cati o n Explanation Logi cal psyc h o l o gi s m affi nns that l ogic i s grounded i n the p arti cu l ar tho ugh t processes of i ndividual persons. While it i s i nd eed true that a given i n d i v i dua l 's thought� and espe c ia l ly hi s acti v i ti es of drawi ng inferences, are amenable to logical treat men t the evi dence for the acceptance of l ogical p r i nc i p l es does n ot depend on the examination of these activiti es. As Dallas WilJ ard affi rms� " this seems paradoxi cal . For how can clai ms abo ut a certain sort of t hi n g fai l to draw t he i r evidence from the examination of thi n gs of that so rt (43)1 T hat i s, how can logic be about our acti v it i es of drawi ng inferences an d yet transcend those very activities for its ju sti fi cation ? Resolution. Accord i ng to Wi l l ard the only plausible way of a v oi d i ng the paradox i s that formulated by Edm und Husse rl . I n pa rt i cu l ar the notion of a "propos iti on" as the abstrac t , but objective referent of s en te n ces and as the bearer of truth is rej ected as being both mystical and supelfluous. The basic relati on of m i nd to the p ro p os i t i o n i s instantiation� that is, an act of speech or a thought is a n instance of a c omple x of characteri stics of such act s These characteri stics themselves have determinati ons, among .

.

..

,

"

,

,

.

1 15

Dictionary o/ Paradox

'.tV hi c h

are truth and fal sity. Thu s , the pri ncipl es of logic appl y to i nd i v i dual

acts of i nfe r e n c e b e c au s e th e I atte r are i n stan c e s of t h e fo rm e r. N eve rthel e ss, the v al i dity of the i nference i s not deri ved from the particu l ar acts thelll se l ve s , but from the characteri sti cs of the acts . T hese l atte r are Hni versal s.

READINGS Husserl , Edmund. " A reply to a Cri ti c of my Refutation of Lo gical P s y c hol o g i s m . " I n R e ad i n g s o n Edln u n d Hu s s e r l 's L o g ica l In vestigations. Ed. J . N. Mohanty. The Ha gue : M arti n u s N ij h off , 1 977. 3 3 -42. Wi l l ard, DaHas. "The Paradox of Logical Psychologi sm: Husserr s Way Out." I n Readings O i l Edrnund Husserl s Log ical Investigations. Ed. J . N . Mohanty. The Hague: Martinus Nij hoff, 1 977. 43 -54.

LOSCHMIDT ' S PARADOX. T h i s p aradox pu rports to show th at the second l aw of t h e rmodynami cs is i n val i d by con s i d e ri n g the sudden rev ersal of the vel ocities of al l the part i c l es i n a c l osed sys teln.

One

con sequence of thi s paradox i s that phy s i c s wou l d be unab l e to ac c o u n t for the i rreversi bility of certai n macroscop i c events. The present paradox

i s e s senti al l y s i mi l ar to that engendered by MAXWELL' S D EMON .

I�OTTERY PAR ADOX, THE. Fi rst adduced by Henry Ky burg, thi s paradox puts the c onfi dence-thre shold theory of acceptance i n question.

Formulation. Let us agree to accept any propositi on whose de gree of confi rmati on i s greater than 0 . 9 . We are given , ex hypothese (hence w i th degree of confi dence

=

1 ) that there i s a fai r l ottery w i th 1 000 ti ckets i n

which one, and only one , ticket w i l l w i n . Let Pi' for i from one to be the propos i tion that the ith ti cket w i l l l o se. confi d e nce for each Pi i s

0.999,

1 000.

Hence , the de gree of

which i s greater th an

0.9.

T h erefore , w e

accept eac h Pi and, i n consequence, w e accept that no ti cket w i l l w i n , w h i c h contrad i cts our acceptance of the proposition t hat on e ti c ket w i l l WIn. •

Exp lanation.

The choice of

0. 9

as o ur thre sho l d of acc e pt a n c e i s

i mmateri al . The paradox w i l l fol l o w for any threshol d stri ctl y l e ss than

1,

although shoul d the threshold be very c] ose to ] , the l ot t e ry w ou l d

have to consi st of a greater n u mber of tickets. T he ] 000 t i cke t: l otte ry w i U suffi ce, how ever, for most stati sti cal stu d i e s , w h i ch req u i re a 0.9, occas i onal ly a

0. 98

0.95 , or

confidence level. T he d egree of co nfi d e n ce for each

Pi i s measured by the standard probabiJ i ty fu nct i on .

It i s a l so assumed

, ,

I I :

I ,

1 16

Dictionary of Paradox

that the subj ect is completely rational , which is taken to entai l that he

accepts all the 1 0gicaJ consequences of any accepted propositi ons. T hus� the subj ect m ust accept that no ticket w i l l w i n because thi s is a logical consequence of the 1 000 propositions P i that he accepts. Resolution. R. P Loui underli nes the fact that the Lottery Paradox can be avo i d ed by i mposi n g a diachronic conditi o n on acceptance, but that most theorists are u nw i l l i ng to acce pt thi s counteri ntuit i ve sol ution . Another approach i s suggested by Ell ery Eells. Using Mark Kap l an ' s notion of arbitrariness for a chain of reasoning, Ee lls argues that the paradox couJ d be avoi ded by rel ativizi ng the noti on of rati onal acceptance to a gi ven (formal ) l anguage. Eel l s neverthe l ess cauti ons that hi s i s but a parti al soJ ution to the probJ em because ( 1 ) i t l eaves open the j ustifi cation of the sp eci fic relativi zati on to be adopted i n any given case and (2) it does not g i ve us a compl etely sati sfactory account of w hy "arbitrari ness" should be the source of the paradox . READING S

Eell s, Ell ery. '"On a Recent Theory of Rational Acceptance. " Philosophica l Studies 44 ( 1 98 3 ) : 3 3 1 -344. Harman , G i l bert. Change in Vievv. Cambridge: MIT P, 1 98 6 . 2 1 -24 and 70-72. Loui, R. P. "Nozick's Acceptance Rule and the Lottery Paradox." Analysis 47.4 ( 1 987): 2 1 3 -2 1 6. Stalnaker, Robert. Inqu iry. Cambridge: M IT P, 1 985 . 90-99.

LOYALTY, THE PARADOX OF. This metaethical paradox, concerned

with the coh erence of the notion of 10yaJty, w as formul ated by Phi l i p Pettit. Formulation. "Yet the ideal of l oyal ty generates a parad ox , for i t requires that the J oyal agent sati sfy tw o apparently confl i cting ass u mptions. He must be sensitive to considerations of a sort that we can a ll fi nd compeH i ng; that i s why it is an ideal . And he must be sensiti ve to considerati ons rel ating di stincti vely to the welfare of a particu l ar person; that i s w hy it i s an i de a l of l oy a l ty. T he fi rst a s s u mpti on c asts the l oy a l agent a s prai seworthy from an i mpartial poi nt of view ; the second presents him as the very exemplar of partial concern . . . To be l oyal i s to be dedicated to a particul ar indi vi dual 's welfare and that seems to confl ict with the idea that the l oyal agent i s ideal i sti c or dutiful" (Pettit 1 63 ). Explanation. Loyalty is a virtue that transcends any particular situation s i n c e � i f i t w ere b o u n d to any g i v e n part i c u l ars � i t w o u l d not b e universal i zabl e - that i s it w ould not be an ideal w hi ch everyone can ,

Dictionary of Paradox

1/7

w ay of contrast anyone who i s l oy a l to anot h e r must be c onc e r n e d w i th th e par t i cul ar s of the person to w hom he i s l oy a l ; but the paJ1 ic u i ar situati on of any i n d i vi d u a l is not lIni versal i zable. Petti t com pares the parad ox of l o y al t y to two cJ oseJy rel ated p a rad o x es i n Chri stian be l ief and Kanti an ethi c s : fi rst, that no one i s l ov ed j u st for Chri st 's sake because to be l oved one m u st be loved for one ' s o w n s ake � and second , that the fri e n d who i s l oyal to an other out of a sense of d uty fai l s to di spl ay any sen s e of fr i e nd s h i p . Resolution. I d e nt i fy i ng the source of the pa rad o x i n the d i ffi cu l ty of i d e n t ify i n g m o t i v a t i n g re asons th at are at once pa rti c u l ar i z e d and uni versal izabJ e , Pettit suggest that loyalty ari ses from context-bound agent­ re lati ve reasons that d o not prevent uni versaJ i zati on. We m ay observ e , how ever, that the very formulation of t h e paradox se e m s to rest on an am b i guity. Lo y al ty as such , i s an ab stract co ncept and i s thus a un i v ersa1 . What w e a c t li a H y fi nd i n o u r w orl d , h o w eve r are p a l1 i c u l ar i n stances of loyal ty U u st as we percei ve green thi n gs, but not "'green" as such) . These take the fonn l oyal to X , " w here X i s a parti c u l ar person , c a u s e etc . Thus� be i n g loyal to X entai l s conce rn w i th the i n d i v i d u a l X� but i t d oe s not precl ude our abstractin g froin t h e part i c ul ar s i tuati on to th e general c o n ce p t . emul ate . B ut , by

,

,

,

"

�

READING Petti t, Philip. "'The Paradox of Loyalty.

I'

Aln erican

,

Philosoph ical Quarterl}'

25 ( 1 988 ): 1 63 - 1 7 1 .

MATERIAL IMPLICATION, PARADOXES OF. T hese are paradox es

of p ropositi onal l og i c that i nvol ve co nd i t i on al proposi tions. Formulationo A fal se p roposi t i o n i m p l i e s a n yt h i ng ; a true propos i t i o n i s impl ied by anythi n g. Explanation. Cond i ti onal propositi on s i n the l o g i c a l formal i s lll are s uppo sed to represent "if- t hen" statement s i n Eng l i sh. B ut the conditi o n a l (al s o c al l e d m a t e r i a l i m p l i c a t i on ) i s d efi ned t r u th fu nc t i o n a l l y i n t h e fol l o w i ng manner: The cond i ti onal i s fal se if, and o n l y i f� bot h i ts antecede nt (the i f- clause) i s true and i t s c on s equ e n t (the then·-cl a use) i s fa l s e Hence, if t h e antecedent i s fa] se the req u i red c OTl1 b i nat i on for the fa l si t y of t h e w hol e condi ti onal cannot occur; hence, the con d i t i o n a l i s tr u e re gard l e ss of th e truth val u e of th e c o n seq u e n t . S i mi l arJ y, t h e cond i t i ona l i s true whenever i t s consequent i s true, re ga rdl ess of the truth .

,

118

Dictionary oj Paradox

value of the antecedent . Thus the conditi onal does not seem to adequately represent Engli sh if-then statements. The paradoxi cal ity i s even more poi nted w hen we use if-then statements to express n ecessary or sufficient conditions. For example, the mere fact that the moon is not made of green cheese i s not usuaUy considered reason enough for us to call a green cheese moon a sufficient cause of Socrates' taking of the hemlock. Yet since "the moon i s made of green cheese" i s fal se, it materi ally i mplies "Socrates drank the hem l ock. " Resolution. The present paradoxes were a major concern i n the creation of modern modal and relevance l ogics. Many logi cians, however, have defended mate ri al i mp l i cati on by claimi ng that the conditional d oes adeq uately represent the i nfonnati onal content of if-then statements as they are normally used since there i s usually a presumption that the antecedent i s true or because we are i nterested in w hat w ould be the case were it true. In any case, the source of the paradoxes i s that, whereas (cl assi cal) propositional logic is truth functi onal , natural languages such as English are not. See a] so the Paradox OF ENTAILMENT. R EA D I NGS Brandom, Robert. "Semantic Paradox of Material Implication. "

Notre Darne

Journal of Forlnal Logic 22 ( 1 98 1 ): 1 29- 1 32. B ueno , Ani bal A. " Ari stotl e ; The Fall acy of Acci dent, and the Nature of Prediction: A Hi storic-al Enquiry. "

Journal oJthe History of Philosophy

26 ( 1 988) : 5 -24.

MATHEMATICAL PARADOXES. As Edward Kasner and James R.

Newman remark, "perhaps the greatest paradox of al l is that there are paradoxes i n mathematics" ( 1 936). Indeed, those not actively engaged in general l y regard mathematical thought - and some of those who are ! mathematics as an abso l utely certain and ri gorous science. Thus, it shoul d not be the abode of paradox. Nevertheless, follow ing Kasner and New man, we may di sti ngui sh at least three types of paradox found in mathemati cs. The first type is real ly naught but fall acy. It usual l y results from some error or overs i ght such as the i l l icit di vi sion by zero i n a widely know n purported proof that 1 =2. Recall the proof that a l ive man equal s a dead man, by arguing that a man who is half al i ve i s half dead, so that 1 12 dead man 1 /2 l i ve man, and then multi ply through by 2. These "paradoxes ," therefore, are not theoretical l y problematical , although they may be useful as a pedagogi cal tool if used skillfully. The second type of paradox encountered i n mathe mati cs i s t h e co unteri ntu it i v e re sult. S i nce mathemati cs often uses common terms in a narrow techni caJ sense , some =

1 19

Dictionary of Paradox

theorems may appear to violate common sense. A more i nteresti ng case� however, occurs w hen mathematics, as the science of the pos s i b l e , goes bey ond our l i m i ted e x p e r ie nc e and fi n d s p e rrectl y v al i d but hi gh l y co unteri ntu itive theorems. Thi s ty pe of paradox i s virtual l y ubi q uitous i n math e mati c s and al most invari ably occurs, for example, w h en w e leave o ur nonnal fi nite real m for the rei gn of the i nfi ni te, as i n the B URALI-FoRTI PARADOX.

Fi nal ly, w e sometimes fi nd that antinomi e s are ded uced i n a

mathemati cal theory. This, of course , means that the theory i s i n c on s istent

which i s general l y di s co n c e rt ing

.

,

F i nd i n g thi s ty p e of p arad o x reveal s

that there i s s o m eth i n g i n th e fonnulati o n of th e t h e ory that was p re v i ou sly

t ho u g ht to be i nn o c u ou s but w h i c h is i n nee d of reformul ati on and w hich

often J eads to the de velo pme nt of whole new fi el d s of m a th e m ati c a l

research ; see, for exampl e, BertraND ' s PARADOX. S ee PA RA D OX E S OF SET

T HEORY.

S e e al so the BANACH-TARS KY PA RA D OX DEM OCRITUS ' DILEMMA , ,

GOED EL ' S PA RADOX O F U N DECI DA B I LITY , the PA RADOX O F I NT ER EST I NG

N UMBERS , the PARADOX OF I RRATIONAL NUT\1BERS � the PA R A DOX OF Rl l LE FOLLOWI NG, WANG ' S PARADOX, AN D ZENO ' S PA RADOXES . READI NG S Dumitriu, Anton . "The Logico- Mathclnatical Paradoxes . "

J-Jistor)) (�fL()gic.

Vol . 4. Tunbridge Wells , Eng . : Abacus, 1 977. 1 1 3 - 1 1 7. Kneale, William, and Martha Kneale. The Phi losophy of Mathemati cs After Frege. " The DeveLoptnent

of La g ic.

O xford : Cl arendon, 1 962 . Ch. 1 1 .

Kasn er, Edw ard , and James R. Newman. " Paradox Lost and Paradox

Regained.'� The World of Mathematics. 4 vol s . Eel. James R. Newman. New York: S imon, 1 956. 1 936- 1 95 5 . Northrop, Eugene P. Riddles in

Mathenwtics : A Book of Paradoxes.

New

York: Van N os trand 1 944. .

,

MAXWELL'S DEMON. J. C. Maxw el l c onceived of the demon in 1 87 1 as a bei ng who coul d follow the movements of every lTIolecul e of a gas.

Formulation. S up p ose that a contai ner i s fi l l ed w i th a gas and p arti ti o n e d i nto tw o chambers , A and B . A demon, w h o can see the moveme nts of t h e mo l ecul es, opens and closes a smal l "door'� in t he partiti on in such a w ay as to a l l ow onl y the sw ifter mol ecules to p ass from A to B and on ly the slow er on e s to pas s from B t o A . T h e temperature of B w i l l conse q u e n t ly be r a i s e d and that of

A

l owered w i thout the expend iture of any w o r k ,

w h i ch contradicts the second l aw of thermodynam ics.

Explanation. Th e second l aw of thermody namics i ln p l i es t hat work be ex p end ed i n ord e r to chan ge the temperatu re of a c l osed sy stc nl .

T he

demon, how ever i s abl e to real i ze hi s u ne x pe ct ed re s u l ts by a st rateg i c ,

1 20

Dictionary of Paradox

sorti n g of the mol ecul e s i nto the tw o chambers.

In doi n g so, he

merely

of t he mol ecules ' s o w n v e l ociti e s (he doe s not , for example, push any m o l e c u 1 e from its course) and t h u s h e pu rpo rted l y does not d i sturb the system by i n tro d u c i n g new so urce s of energy. Resolution. A c c ord i n g to one approach, L. S zi J ard fo u nd the key t o t h i s p a rad ox by ob servi n g that the demon was chan gi ng the i nformati o n that he obtai n e d a bout the sy stem i nto n e g at i v e entropy. 'Th us, Leon B ri lloui n s howed that it would be i Inposs i b l e for the deInon even to see the i n di vi d u a l ato ms w i thout d i st u r bi n g the sy ste m . T h i s resuJt not only d i sarm s the p a r a d o x b u t , accord i n g to B ri l l o u i n , p rese nts u s w i t h the fol l owing i In p o rt a n t ph y s i c aJ J a w : " e v e ry phy s i c a l m e a s u r e m e n t r e q u i r e s a correspondi n g e n t ro p y i ncrease, a n d there i s a l ow er l i mi t below w hi ch the meas urement becomes i rn po s s i b l e " ( 1 68 ) . A c c ord i n g to a n other a p p r oac h however, B r i l l o u i n has completel y misconcei ved the nature of t h e parad ox. Joseph Wa y ne S mi th con te n d s that the demon i s naught b ut a fanciful embodi ment of the worki ngs of chance . In th i s view, there i s a c e rtai n , th o u g h p e r h a ps s m a l l , p r o b a b i l i t y th at on l y (mostl y ) swi ft molecules w oul d p a s s from A to B through a sIna l l hol e i n t h e part i ti on , w h i l e on l y (mostl y) s l ow m o l ecules w o u l d g o t h r o u g h i n t h e other m akes use

,

,

d i recti on, and the paradox goes t h r ou g h

as above.

READINGS Bril loui n, L. Science and injornlatiol1 Theory. 2nd ed. New York: Academic P, 1 962. ] 62- 1 82. Smith, J oseph Wayne. Reason , Science and Paradox: Against Rece ived Opin ion in Science and Ph ilosophy. London : Croom Helm� 1 986. 1 3- 1 6.

McTAGGART 'S PARADOX. J. Ell i s McTaggart w as the author of an argument t h at seems to e s ta b l i s h the paradoxi cal res u l t that time is un r e al Formulation. There are two separate te m po ral orders, the A-seri es that runs fro m the past thro ugh the p re se n t to t h e future and the B - seri es that run s from ear] i e r to l at e r The A -seri e s cannot be reduced to the B - series ; an d the B - se r ies can exi st o nl y if the A-seri es exi sts. Yet si nce th e A­ se ri e s i n vo l ve s an i n es c a pabJ e contradiction , i t can n ot e x i s t Thus the B­ s e ri e s cannot e x i s t either. And since nei th er order i nto w h i ch ti me i s .

.

.

an alyzabl e

is

reaJ , t i me i s unreal as w eJ ] .

Explanation The A-seri es cannot be reduced to th e B - series because it i m p l i e s a po i nt of vi ew with respect to w hich e v ents are pa st, pre sent or future w h i l e the B - seri e s does not. The B -seri es can exi st o n ly if the A­ seri e s exi sts bec ause it i s definable i n terms of the A- seri es. The A - seri e s invol yes an unavoi dabl e contradiction be c a u s e the characteri sti cs past, ..

121

Dictionary of Paradox

p resent and future are i n c o m pati b l e , but every e v e nt h a s t hem al l . lV1 cTaggart a n ti cipate s the i mmedi ate response that no obj ect has al l thre e c haracteri stics simultaneously b y rephras i ng the tensed p ro po si t i o n s as t e ns e l e s s state ments with an expli c i t te mporal phr a s e a n d t h en addu c i n g a n i nfi n i t e regress.

�{esolution. Few are convi nced by M c Tagg art s argument ; its success l i e s in c o mpe l l i n g so many phi l osop he r s to refute it. The di sp u t e amon g l he c ri ti c s i s chi efl y ov e r whether one can r ef u t e hi s argu ment by poi nti n g o ut a few petty confusions or w hether h i s errors at least touch upon the deepest a s p e c t s of real i ty. N e arl y every part of McTaggart's an a l y s i s has '

another, but the favorite target is the so­ c alJ ed "bas i c paradox" to the effect that the A-series i n vo l v e s an i nt e r n al contrad icti on . C . D . Broad , for e x am p l e clai ms that the tensed struct u re of o u r l anguage i nv o l ves no contradiction at al l and hi s clai m i s based on the very reason that McTaggart an t i ci p at ed : no object is past, present� and future sim ultaneously. Furthermore, si n ce the re i s no con t rad icti o n at this l ev e l of ev e ry d ay speech, there i s no re a s o n to e tnbark on M cl a gg ali s been

attac ked by

one criti c or

�

'

'

i nfi nite regre ss. I n contrast to B road , Da v i d Zei l i covi c i argues that the paradox does revea l

d e e p ph i l oso p h i cal

probl ems i n o u r concepti o n

of

He rejects McTaggart ' s concl u s i on that tilne i s un real , however, con te n d i n g that the paradox onl y s how s that ti m e i s not a "fl owi n g" or a becom i n g , but rather a st at i c ph e nom e n on time.

.

READING S B ro ad, C . D . A n ExalninatioJ1 of Mc Taggart 's P h ilo s op hy . 2 v o l s . Cambridge : Cambrid ge UP, 1 938 . Pt T. Chri stensen, Ferrel . M cTag g art s Paradox and the Notion of Time. " Ph ilosophica l Quarterly 24 ( 1 974). Dummett, M i ch ael . "A Defense of McTaggert's Proof of the Unreal ity of Time . " Philosophical Review 69 ( 1 960): 4 1 7-504. Gale, R i ch ard M. , ed . The Philosophy of TiJne: A Collection of Essays. London : Macmi 1 1 an, 1 968 . 65-85 . G-otshalk, D. W. " McTaggart on Ti m e . Mind ns 3 9 ( 1 93 0): 26-42. McTaggart, J . Ell i s. "The Unreality of Ti me . Mind ns 17 ( 1 908 ): 457474. - - . Th e Nature of Existence. 2 vols. Cambri dge: Cambridge U P, 1 927. 9-3 1 . Mi nk , Lou i s O . "Ti m e , M cTag gart, and Pi ckwi cki an Lan g u a g e . " Philosophical Quarte rly 1 0 ( 1 960): 252-263 . Ranki n, Kenneth. "McTaggart's Paradox : Two Parodies . n Ph ilosophy 56 ( 1 98 1 ) : 33 3-348. Roberts , J oy H. "Statements , Sentences and S tates of Affai rs i n McTaggart and in Gen eral . Erkennlll is 1 5 ( 1 9RO): 73-89. Sanford , Dav i d H. " McTaggart on Ti me." Ph ilosophy 43 ( 1 968) : 37 1 -378 . "

'

"

"

"

1 22

Dictionary of Paradox

Schl esinger, George .

Aspects of Time.

Indi anapoli s: Hackett,

1 980.

Ch. 2.

Zei j icovici , Dav i d. "A ( Di s )solution of McTaggart� s Paradox . " Ratio

28

( 1 986): 1 75- 1 95.

MEEHL' S METHODOLOGICAL PARADOX.

T hi s i s a puzzle

i nvolving stati stics in the soci al sci ences. F orm ulation. " I n t h e phy s i ca l s ci en c e s , the u s ual re s ul t of an improvement in experi mental desi gn, instrumentation, or numerical mass of data� i s to i ncrease the difficulty of the ' observational hurdle� which the physical theory of interest must successfully surmount ; whereas, i n psycho l ogy and some of the aU ied behavi or sci ences, the usual effect of sllch improvement in experi rnental preci sion is to provide an easier hurdle for the theory to surmount" (Meeh l 1 03). Explanation Stati sti cal reasoni ng in science i s based on the formulation of a nu l l hypothesi s which the i nvesti gation tri es to refute. The results of a stati sti cal study are subj ect to e rrors i n measurement and errors of sampl i ng and the stati stician 's job i s to decide whether these errors are suffici ently small for accepting the results as si gnificant. Rej ecti on of the nul l hypothesi s w hen i t i s true i s cal led a Type I error; retention of the null hypothesi s when it i s false i s cal l ed a Type II error. These two types of error are inversely related. Further, if p is the probability of a Type II error, I -p i s called the power of the test. By improving the experimental desi gn� the power i s i ncreased. The social sciences tend to use directional hypotheses, as opposed to point hypotheses , w ith the implication that if w e obtai n perfect powe r (false null hypotheses wilJ always be detected), our hypothesis w i l l be correct at l east half the time. This is because even if there i s no relation between the hypothesis and the sampling, it will be correct half the ti me j ust by chance - if there i s a relation, it w i l l be correct more often. Hence, by i ncreas i ng the power to perfection, we have an a priori lower bound of 1/2 that the test will confinn our hypothesis (rej ect the n ul l hypothesis). Therefore, by improving the experi mentaJ desi gn, we make it as l i kely to confinn as i t i s to fal sify an hypothesi s , even if that hypothesi s i s complete l y w ithout merit. Resolution. A ccordi ng to Meehl , the paradox results from the fact that the sociaJ sciences are not preci se enough to make point predi ctions . Although the paradox is i nevitable, Meehl al so cl aims that the s ituation i s made even worse by certai n soci al forces or i nte l lectual habits among researchers, one of which i s the inadequate appreci ati on of the present paradox . .

Dictionary of Paradox

1 23

REA DING

M e e h l , Paul E . "' T h eory -Te s ti n g i n P s y c ho l o g y a n d P h y s i c s : A Methodological Paradox. " Ph ilosophy of Science 34 ( 1 967) : 1 03 - 1 1 5 .

MENO'S PARADOX. Thi s epistemic paradox i s generally used as an

cTudite all usion for those who want to expand on the topic of knowJ edge. ,� ee al so PLATONIC PARADOX and SOCRATIC PARADOX. Formulation. How can one i nqui re i nto what he does not know ? What wi ll be the top i c of investi gati on? And if one di d find what he w ants, how ' night he know that this i s the matter w hich he did not know ? Explanation. One cannot search for what he knows or for what he does f l at know. He cannot look for what he knows, because he al ready knows i t � one who already possesses something cannot come i nto posses sion of i t . And he cannot search for w hat he does n ot know, because he d oes n ot k now what to l ook for (Meno 80e). Resolution. This i s ori ginal ly a debater 's puzzl e that Socrates treats w i t h scorn in the Euthydemus, (275d-278b) . Yet i n the Meno Pl at o treats it as a genuine quandary to be solved by hi s theory of know ledge as recoJ lection. l' here seems to be no great diffi culty, however, because an i nqui rer does know part of the topic he i s i nvesti gatin g and something about the form of acceptable answ ers to the kinds of questions he has, so h e know s something of where to start l ooki ng and i n a general way what he needs to discover to sati sfy h i s enquiry. T hat the inqui rer can ask a meaningful question seems to testify to the correctness of thi s response . Nevertheless� some commentators have asserted that the present paradox ex presses a genuine problem. Rosemary Desjardins� for example, assimi l ates it to the problem of the "hermeneutical ci rcle": in order to know the w hole one must first piece the parts together; but one cannot even recogni ze the parts as parts unless one already knows the whole. READINGS B luck, R. S . Plato 's Meno. Cambridge : Cambri dge UP, 1 96 1 . 8 - 1 7 . Arti culati n g Ou r I gnorance: Hopeful Skepti c i s m an d Bri skman , Larry. the Meno Paradox. " Et Cetera 42 ( 1 98 5 ) : 20 1 -227. Desj ardins , Rosemary. "Know1 edge and Virtue: Paradox in Pl ato 's Meno. " "

Re vie'w of Metaphys ics 3 9 ( 1 985 ) : 26 1 -28 1 .

Grene, Marjorie. HThe Legacy of the Meno . " The Knower (lnd flte Kl lo\1'n . New York: Basic Books, 1 966. Ch 1 . Pol anyi , Michael. The Tacit Dilnension. Garde n City, N Y: Do ubl eday, ] 966. Ch . 1 . Poppe r, Karl . " On the S ources of Know l e d g e a n d o f I g n o ra n ce . " Conjectures and RejzJtations. New York: B as i c B ooks , 1 962. 3 - 30.

I

,I ,

Dictionary of Paradox

1 24

Sanford , Dav i d H. " McTag gart on Ti me. '� Ph ilosophy 43 ( 1 968): 37 1 -378 . Thompson, E. Seymour. The Mello of Plato. ] 90 1 . Garland Press , NY: GarJ and P 1 980. Wel bourn e, Mi chael . "Meno' s Paradox . " Ph ilosophy 6 1 ( 1 986 ): 229-243 .

Related to the PARADOX OF FUTURE I NDI V I Dl 1A LS� th i s paradox was adduced by Derek Partit. Formulation. Let A be a population aJ ! of whose i ndivi dual s have a very h i gh q ual i ty of l ife. Let B be A pl us a group of i ndi vidual s whose q ual ity of life i s l ess t h an that in A � but sti l ) moderate ly hi gh ; further� suppose the newcomers do not affect the q ual ity of I ife of the origi nal group. Fi n all y let C result froln B by the worse-off i ndi viduaJ s gaining more than the better-off i n d i v i d u al s l ose. Then , B i s not worse than A and C i s better than A . Neverthel ess, C is worse t han A and our beli efs about the rel ati ve val ues of these societi es are i nconsi stent. Explanation. Observe that B res ul ts from A by the mere additi on of a group of i nd ividual s with a moderately h i gh quali ty of l i fe. If B were worse than A, then it woul d be better for thi s group not to have existed ; but thi s would be clearl y counter to all our i ntui tions about the val ue of thi s grou p. Hence, B m ust be no worse than A. Furthermore, C results from B by a kind of level i ng action ; the fi nal result, however, is not a mere average of those who are better-off w ith those who are worse-off, but rather the latter make substanti al gains whi le the fOlmer suffer relati vely minor l osses. Hence, we would conclude that C i s better than B . B ut if C i s better than B, w h i ch is no worse than A , then C certai nly cannot be worse than A . The fol lowing argument, however� show s that C i s worse than A. I n relati on to A, C comes about by i ncreasi ng the popul ation and l owering the qual i ty of l ife (a little). The same process can be iterated call until we arri ve at an enormous populati on l iving i n abject mi sery i t M . If C i s no worse than A� then the same can be said about each step i n the process . Thus, eventual ly we w ould conclude that M i s no worse than A , which i s absurd. Therefore, C must be worse than A, w h i ch contradi cts our previ ous resul t. Resolutiono Larry S . Temk i n denies the validity of the sub-argument purporti ng to show th at C i s worse than A by deny i ng the transitivity of " i s no worse than" for the concept i n question of populations with different l evels of q ual ity of l ife. T his can happen because not al l pairs of relevant po pulations lnay be com parable. Even the inference MERE ADDITION PARADOX, THE.

,

C i s better than B .

1 25

Dictionary of Paradox B is no worse than A.

Therefore, C i s no worse than A .

I llay be i nvalid, if we cannot cO ln pare C and A. Neverthel ess, even though we may have trouble d e c id i n g w h i ch soci ety i s better, it i s hard to see \-v hy any two of theln woul d be in pri nciple i ncomparable. Rather, t he plausibi lity of the paradox depends on the assumpti on that q ua] ity of life can be q u anti fi ed in a strai ghtforward m an ne r � once thi s assumpti on is di sallowed, however, i t is not cJ ear whether the pa radox can be g e ne rated .

R EA DINGS Parfit, Derek. ' "Future Generati ons : Further Probl em s . "

Philosophy

and

Public Affairs 1 1 ( 1 982): 1 1 3 - 172. " ' Intransiti v i ty an d the Mere Additi on Paradox . "

Te mki n , Larry S .

Ph ilosophy and Public Affairs 1 6 ( 1 987): 1 3 8- 1 87.

META P H Y S I C A L PA R A D O X E S . M e t a p h y s i c s atte m p t s t o comprehend b e i n g as a whole and i n its e s se n ce through the sy st em at i c deploy ment of c on c e pt s and thereby gen e rate s a ri ch vei n of paradox . . M e t a phys i c a l paradoxes i ncl ude B RA D L EY ' S PA RA DOX O F RELATIONS � G EACH ' S PARADOX OF 1 ,00 1 CATS , PARADOXES OF O M NIPOT ENCE, PARADOX ES ,

OF O M N IS CIENCE� HUME' S P R O B L E M OF I NDUCTION . KANT ' S ANTINOMIES� McTAGGART ' S PARADOX,

the

PARADOX OF PRI M E MATTER, REC ENT AND

TRA DITIONA L S O RI T E S the SHIP OF T HES EUS , the SOLIPSISTIC PARADOX , ZENO ' S ,

PARADOXES

and many others.

One of Zeno's paradoxes, Inentioned by Ari stotle at Physics 250a 1 9-2 1 and by Si mplicius at In Physica 1 1 08 , 1 8-28 , the Mi 1 1 et S eed i s a fal sidical paradox or even a pseudo-paradox. Formulation. If a gross of mi l l et seeds makes a noise when it fa l l s a s i n g l e m i ll et seed w i ] ] tnake a propol1i onate l y smal ler sound when i t fa l l s . Explanation. Moti vated by an El eatic prej u di ce agai n st percepti o n � Zen o i s ap pa re ntl y suggesti ng th at jf our senses fai l us in the case of the fainter sound, we should thi nk tw ice about believing w hat ou r sen ses re gi ste r i n the case of the louder sound. Resolution. Zen o 's argument presumes that we shou l d hear d i sturba n c es i n the ai r as sounds n o matter how sl i g h t they are, but t h e re i s n o spec i al d i ffi c u l ty in the circumstance that our audi tory powers are I i In i ted i n t h i s particular m an n e r. S ee ZENO ' S PARADOXES .

MILLET SEED, THE .

.

1 26

Dictionary of Paradox

THE. Da n i e l Cohen suggests p re s en ti n g Keith Campbell 's treatment of the m i nd bo dy probl e m as a paradox. Formulation. T he re are four p roposi tion s that e x pre ss comInon bel i efs or i n t u i ti ons but which are not ITIutuaHy consistent. Fi rst, the m i nd i s i mmateri al. Second, the body is material . T hi rd, the materi al and th e i mmaterial cannot interact callsaHy. And fo u rth the mind and the body do interact c a usa l l y . Explanation. Thi s i s j ust the cl assic m i nd b od y probJ em set out as a pa rad o x T he Ini nd � as Descaltes has it, i s not spatial l y extended, but the body i s. It appears t hat an extend ed substance cannot i nteract causaJ Iy w i th a unextended substance, but thoughts and physical objects do seem to infl u ence each other in p e rcepti o n and the wil l . Therefore, one of the fo u r pr o p o s i tion s has to g o Resolution. The four strai ghtforward responses each rej ect one of the fou r propositions. Spi noza, for example, denies the fi rst p ro p o s i ti on by claimi ng that the mind i s ultimateJ y i mmateri al . BerkeJey rejects that the body i s mate ri al , w hi l e Descartes deni e s that the material and the immateri al cannot interact c au s a l l y Fi nally, Mal branche claims that the mind and the bo dy do not i nte r act causalJy. An entirely different approach i s to deny that the mi n d body distinction can be drawn in the first place.

MIND .. BODY PROBLEM,

-

,

,

-

.

.

.

-

READINGS Campbel l � Kei th. Body and

Mind.

New York: Doub1 eday, 1 970 .

Cohen , Daniel . "Putting Paradoxes into Pedagogi cal Use i n Phil osophy. " Teach ing Philosophy 8 ( 1 985 ): 3 09-3 1 7.

pa r adox modeJ l ed on PRIOR ' S FAMI LY OF PARADOXES , was present ed by T h o ln as Storer. Formulation. To bui l d the world's smaI J est electroni c brain, MINIAC, wri te the words 'Y ES ' and ' NO ' on two pi eces of paper and g l u e one to each side of a penny. To operate the machine, ask MINIAC a question, fl ip the c oi n and r e co r d the answer; then ask MINI AC if its answer to the present question w i Jl have the same truth vaJ ue as its previ ous an sw er and flip the coin again. The answer to the second question w i l l determi ne the correct answer to the first question. Storer adds that the el ectronic nature of MI N IAC i s assured by the fact that the copper atom has tw o free electron s i n the outer shel l . Explanation. MINIAC ' S l ogi cal gearw ork can be e x h i b ite d i n the fol l owing table: MINIAC. This

,

,

r

Dictionary of Paradox

127

Answer to

Answer to

Second Questi on

Truth Val ue of First Questi on

Final Answer to First Questi on

NO

YES

Second Questi on

Truth Value of

NO

YES

Fi rst Question

NO

YES

t

f

t

f

t

f

t

f

t

t

f

f

t

t

f

f

YES

NO

NO

,

YES

The first three l ines of the table merely exh i bi t the various possi bJ e combi nations. T he fourth l i ne i s cal cul ated from l i nes two and three i n the fol l owi ng manner: if M INIAC answered that the truth val ues are the same (Y ES ) and i t "spoke" truly, then both answers have the same truth val ue ; but if MINIAC answered Y ES and i t spoke falsel y, then the answers have opposite truth values; if MI N lAC answered that the truth va l ues are no t the same ( N O) and i t spoke truJy� then the answers have oppos i te truth val ues ; final l y, if MI N I A C answ ered NO and i t spoke fal se l y� the answers have the same truth value. Comparing l i nes two and fou r of t h e table, we see that MIN IAC answers Y ES to the second q uest i on w he n � and only when, the truth val ue of the first question i s true ( be t h at an s w e r either YES or NO) ; further, MI NIAC answers NO to th e second questi on w hen, and only w hen, the truth value of the fi rst q uest i on ( agai n � be that answ er either Y ES or NO) i s fai se. T he last l i ne gi ves t h e fi nal answer by

1 28

Dictionary of Paradox

comparing l i nes one and fOUf. J . L. Mackie concludes that " w hat is puzzling about this is that w e appear to get l ogicall y guaranteed answers to al l yes­ no questions by a method which common sense te lls us has no rel i ability at aJ I" (280-28 1 ) . Resolution. Perhaps the easiest way to see the fal l acy i nvol ved is for the reader, if he be willing� to pl ay the part of the worl d 's smal lest el ectronic brain. We now ask you if Meei -Ling - a person unknow n to you - is ni ne years o l d . Pl aying the odds� perhaps , you answer NO. N ow we ask you if the truth val ue of the present question i s the same as that of the previous questi on. J ust to be contrary, you answ er YES . 'Thus, from the tabl e g i v e n above, your fi n al answer is that Meei -Ling i s not nine years ol d . As it turns out, ho"vever, she i s indeed nine years old. What went w rong? Observe that your fi nal answer was fal se . Was your second answ er true or false? If i t w ere true, i t would have to have the same truth value as your fi rst answer; that i s, it would be false. B ut, if it were false, i t w ould have to have the opposi te truth value as your first answ er; that is it would be true. Thus, your second answ er i s true if, and only if, it i s false. Thus, it is a LIAR-type sentence. (See Mackie for an anal ysi s of al l the possib l e combinations of answers. ) Indeed, i f w e recall how the table w as constructed, we see that the fourth l i ne was cal cul ated from the previous two lines, both of whi ch have to d o with the second questi on. T his process imputes a truth value to the first question independently of the state of affairs i n the external w orl d through the self-reference present in the second question. The process i s thus typical of Li ar-type paradoxes. READINGS Macki e , J . L.

Truth, Probability and Paradox : Studies in Ph ilosoph ical

Log ic . Oxford :

Cl arendon, 1 973. 28 1 -28 3 .

Storer, Thomas. " Mi ni ac." Analvsis 22.6 ( 1 962) : 1 5 1 - 1 52. �

MODAL LIAR, THE . For thi s formul ation of the LIAR i n modal l ogi c,

see the POSSIBLE

LIAR.

MOORE 'S PARADOX. This paradox about knowl edge and bel ief w as di scussed e x te n s i v ely by G. E. Moore. Formulation. " I w ent to the pi ctures last Tuesday, but I don ' t bel i eve that I did" ( Moore, Rej)l)' 543 ). Explanation. T he paradox does not invol ve a contradiction si nce it i s

possibl e to i rnagi ne ci rcumstances in whi ch someone mi ght naturally utter sentence s l i ke the one above. Thus� someone lni ght sheepi shly ad mit to

129

Dictionary of Paradox

h i s fri e nd :

"I

know that th i s rab bit 's foot i sn ' t magi cal , but

I

real ly don � t

bel ieve i t ' s n ot. " Neve rthel ess, sentence s of th i s sort seem to i nvol ve us

i n a puzzl e b ecause the speaker purports to disbe l i eve w hat he know s to be true .

Resolution. Moore hi mself responds that w he n the aforementi oned ki nd of statem ents are asse rted, there i s a strong presu mption that the speaker does know the facts of the matter and i s tel 1 i ng the truth. If e ither of the se ass umpti ons shoul d not hol d , how ever, the paradox vani shes .

A nother

approach to the parad ox i s to clai m that the asserti on is not completely expre ssed i n the statement itself, but that context, backgro und know ledge, or

shared presuppositi ons m ust al so be taken into account. John S earl e ' s

th eory of speech acts , for e xample, hol ds that speaki ng i s governed by a set of i mpl icit rul e s , some of w h i ch are broken by Moore ' s troubl esome statement. Approaches of thi s type tend to fi nd a real contradi ction i n the l arge r context posite d . 'T he rabbit ' s foot exampl e , however, does n ot see m

to be ful ly addre ssed by either of these approac hes ; rather, i t seelll S lllore a ki n to the

PARADOX OF

SELF-DECEPTION .

C harl es E. Caton argues t hat

there may be e pi stemi c quantifi ers of d i fferent strength s. I n t hi s w ay, the epi stemic strength of an asserti on w o u l d be greate r than that of a be l i ef state ment; thus, the as serti on of a statement and the bel i ef of i ts n e gati on w o u l d not be countenanced as a w el J -fonned propo sition.

READINGS Almstron g, D. M . "Does Knowledge Entail Bel ief?" Proceedings of the Aristotelian Society 70 ( 1 969-70). 2 1 -36. B l ack, Max. "Saying and Di s bel iev i ng. " Analysis 1 3 . 2 ( 1 952). 25-3 3 . Caton, Charles E. "Moore ' s Paradox, S incerity Conditions , and Epi s temi c Qualification. " On Be ing and Saying : Essays for Richard Carnvright. Eel. Judith J arvi s Thompson. Cambridge: MIT P, 1 987. 1 3 3- 1 50 . Deutcher, -M . "A Note on Sayi ng and Di sbel i ev ing. " Analysis 25 . 3 ( 1 9 65 ) . 5 3 -5 7 . - - . "'Bonney on Saying and Disbelieving." Annlysis 27.6 ( 1 967). 1 84- 1 86. Gombay, Andre. "Some Paradoxes of Counterprivacy." Ph ilosophy 63 ( 1 988): 1 9 1 -2 1 O. Harni s h , Robert M . " Se arle and the Log i c of Moore ' s Parad ox . " International Logic Review 1 1 ( ] 980): 72-76. Hi nti kka, J . Knowledge and Belief It h ac a : Cornell UP, 1 962. 64. Li nsky, Bernard. "Factives, B lindspots and SOIne Paradoxes. " Analysis 46. 1 ( 1 986): 1 0- 1 5 . M arti n i ch , A . P. "Con versati onal Maxi ms an d S orn e P h i l o s o p h i cal Problems. " Philosophical Quarterly 3 0 ( 1 98 0 ) : 2 1 5-228 . Moore, G . E. "A Reply to My Critics." The Philosophy (�l (J . H. Moore. Ed . P. A . Schilpp. Evanston: Northwestern U P, 1 942. 540-5 43 , 660-677.

,

Dictionary of Paradox

1 30

Ph ilosophy of Bertrand Schilpp. Evan ston: N orthwes tern UP. 1 944. 205-206.

- - . " Rus sell ' s -Theory of Des cripti ons' ." The

Russell. Ed . P. A Whi te, A . R. Modal Thinking. Ithaca: Cornell UP, 1 97 5 1 72. Wil l iams, J . N. " Moore �s Paradox: One or Two . " Analysis 39. 3 ( 1 979) : .

1 4 1 - 1 42 .

MORNING S TAR PARADOX, THE. W. V. O. Q ui ne adduced thi s

oda l l og i c (w ith i dentity). Formulation. The M orning Star and the E v eni n g Star are naught but the pl anet Venus but they a re no t n ecessari J y i dentical for i f our so l ar system had evol v e d d i ffe re n tl y M ercury, for example , cou l d have always appeared i n the morn in g and Venus aJ ways at n i ght Thus, "the clai m can be made that the Morning Star has a pro perty or attribute no t possessed by the E v e n i ng Star, name l y the property of bei ng necessarily i de n tica l w i th the M orn i n g Star. Thi s result l eads to the conel usion that the M o rn i n g Star i s a different indivi dual from t h e Evening Star, si nce otherw i se i t wouJd h ave exactly the same prope rti es B ut of course there i s good e mpi ri cal evi dence t h at the E v e ning Star and the Morning Star are one and the same individual , known as the planet Ven us. Hence � the parado x ( Fi tch paradox

i n q u anti fi ed

m

,

.

.

"

27 3 -274) . Explanation. As indi cated above it i s n ot l ogical ly necessary that the M o rn i n g Star be i dentical to the Even i ng Star. The Mor ni ng Star is, �

however, nece ss aril y identical to the Morn i n g Star. Hence, the Morni n g Star has a property that the Eve ni ng Star does not have ; the two can thus be d i st i n g u i s hed by usi n g thi s property, so that the M orn i n g Star i s not the same obj ect as the Evenin g Star. Resolution. Qu i n e c onte n d s that th i s p aradoxi c a l result reveals that quantifi cati on over modal conte x ts i s an i n he re n tl y i r rat i o n al_ p rocedure w hen i d entity i s i nvol ved. Hence, he proposes to replace i denti ty i n m odal contexts by a weaker cong r uence rel ation . Frederi c B . Fi t ch (foHowi ng Raymond Smu l l yan) argues that there are at l east two ways of avoiding the paradox. Fi rst ' Morni ng Star ' and Ev e ni n g Star ' may be regarded as p roper names of the same i n di vidual ; h ence, it would indeed be necessarily true that the Eveni ng Star i s i dentical with the M orn i n g Star. S econd, ' the M o rn i n g Star ' and ' the Eveni ng Star ' may be regar d ed as definite descri ptions according to Bertrand Russe1 l 's defi n i t i o n ; i n thi s case, it i s not n ece s sary that the Morni ng Star be identical with the M orn i ng Star (si nce that would i mply that the Morni n g Star exi sts necessarily) . Thus, i n either case, Q u i n e s argument w oul d fai l . One of the pi oneers of "

"

'

'

1 31

Dictionary of Pa radox

quantifi ed

modaJ l ogi c, Ruth Barcan Marcus, opts for the fi rst alternative i n her response to Quine. Note that the present paradox was hi storical ly i lnportant in that i t seemed to becorne a paradi gm case of the ki nd of problems that arose elsewhere i n q uantifi ed i ntensi onal l ogics, such as deonti c and epi stemic l ogic. READING S C op i , Irving M . , and James A. Gould, ed.

Theory.

Contefnporary Readings in Logical

New York : Macmil ] an, 1 967. The Problem of the Morni ng Star and the Evenin g Fi tch , Fred e ri c B . Star. " In Irvi n g M . Copi and James A . G oul d , ed . , Contemporary "

Readings ill Logical Theory.

273 -278 .

Kanger, S ti g . "'The Morning S tar Paradox . "

Theoria

23 ( 1 957) : 1 - 1 1 .

Marcus, Ruth B arean. '" Modaliti es and Intensi onal Lan g u ag e s

.

"

In Irv i n g

M. Copi an d James A . Gould , ed. , Contenzporary Readings in Logical Theory. 278 -293. Q u ine , W. V. O. "The Prob1 em of Interpreti ng Modal Logi c." I n J rv i n g M . Copi and James A . Gould, ed. , ContenlpOrar}' Readings ill Logical

Theory.

267-273 .

- - . ·'Reply to Profe s s o r Marcus . " In Irv i n g M . Copi and J ames A . Gould , eds. Contemporary Readings in LogicaL Th eory. 293 -299.

,

MOVING BLOCKS, THE. The fourth and l ast of Zeno 's arguments

agai nst motion , thi s paradox i s also called the Stadium. Formulation. OUf source for this paradox is Aristotle 's Physics , 239b3 3240a 1 7. �'The foulth argument is that concerning the two row s of bodies, each set being composed of an equaJ number of bodi e s of equal si ze, passing each other on a race-course as they proceed w ith equal vel ocity in opposite directions, the one row origi naJ Iy occupying the space between the goal and the mi ddl e poin t of the course and the other that between the midd l e point and the starting-post. Thi s, he [Zeno J thin ks, invol yes the concl usion that half a gi ven time i s eq ual to double that ti me . " Explanation. Ari stotl e 's tel egraphic style makes the paradox diffi cul t to deci pher. Nevertheless, the basic structure of the acti on in the parad o x i s easil y seen . Body A moves its own J ength (say, a unit) to the ri g ht, w hi l e Body B equal i n length and vertically ali gned w ith Body A � Inoves i ts ow n length to the left (see fi gure) . ,

1 32

Dictionary of Paradox

�!

•

D

D

D

D

D

D

F i rst I ns t a nt

N e xt I n st a n t

Rel ative to the stati onary grou nd, Body B has moved a unit i n a given amount of ti lne I. Re l ati ve to Body .A. � however, Body B has moved a di stance of two units i n the same ti me 1 and, therefore, i t has moved a si ngJe unit i n ti me t12. S i nce both t and 11 2 are the times that _B ody _8 took to move a single unit of distance, we have that t 112 ; that i s, "half a gi ven time i s equal to double that time . " Resolution. The fal lacy i nvolved in thi s paradox i s, at fi rst si ght ai l too obvious. _A s Ari stotle explains, "The faJJ acy of the reasoning l i es in the ass u mpti on that a body occupies an equal time i n pas sing with equal velocity a body that i s i n motion and a body of eq uaJ si ze that i s at rest. " T here i s, however, an al ternative i nterpretati on first proposed by Paul Tannery, accord i ng whom, the time t i s an atomi c quantu m of tilne. T he arg ument then proves that Body B � relative to Body A , would traverse the unit length (perhaps itself an atomic quantum) in one-half the atomi c quantum of time , thereby divi ding a supposed i ndivi si bl e . A l th ough Tannery 's i nterpretation makes the paradox veridical i nstead of ridiculous, Gregory Vlastos rejects i t on the grounds that i t has not been documented in h i storical sources. See ZENO ' S PARADOX ES . =

Ari stotle , Ph vsics -

R EA D I NGS

V I . IX. '

lannery, Paul . "Le Concept Scientifique de Continu: Zenon d ' Elee et George Cantor. �' Re v ue

Philosophique de la France et de L 'Estranger 20 ( 1 885):

3 85-4 1 0. Vl astos, Gregory. "Zeno of El ea. "

The Enc}'clopedia of Ph ilosoph�v . Ed.

Paul Edwards. New York: Macmillan, 1 967. Vol . 8 � 3 75 .

M VARIANTS OF THE I--IOGICAL P ARADOXES .

One of the lnost common strategies for di sarming the logi cal paradoxes is to try to show that the paradoxical sentence i s somehow IneaningJ ess . Th i s i s especial ly true of Li ar-l i ke and other selnantical paradoxes. Take the strai ghtforward versi on of the LiAR, for example� ..

1 33

Dictionary of Paradox

I I, I

,i , ,

(1)

This sentence i s false.

The claim i s that sentence ( 1 ) fa i l s to e x p re s s a

proposi t i on and , thus, i s

mean i n g l ess. The M-vari ant ( t h e term i s due to J. L. M_ac ki e) of the paradox, h o w e v er accepts the chal l en ge of the charge of m ean i n gl essness and turns it to ac c o u nt i n add u c i ng a new pa ra d ox : �

(2)

This sentence is false or meaningl es s .

If (2) i s tru e , then what i t says i s the cas e . H e nc e (2) is either fal se or m ea ni ngl es s (in either case i t i s not t r ue ) in contrast� i f (2) is fal se, then w h at i t a s se rt s i s not the case an d, conseq uentl y, i t i s true. S i m i larl y, if ( 2) i s mea n i ng l e s s th en what i t asserts i s agai n the case (i. e . , t h at i t i s mean i n gl ess) and t h u s i t is a g a i n true. T h e refo re ( 2 ) i s true if, and only if, it i s either fal s e or meani ngless . Thi s res u l t, al though a b i t more cumbersome than the original version , i s j u st as paradoxical as the ori gi n al . "Hi e rarchy ' ap p ro a c h e s ( see the LIA R and R U S S EL L ' S PARA DOX ) to t he resol uti on of the pa rado x e s how ever� see ln to be i m m u n e to M - vari ants because rn ea ni n g l e s s i s a se manti cal concept and w o u l d t h u s re c e i ve a rank, j ust as Htrue�� an d "'fal se" d o . S ee t h e PA RADO X OF T H E N O N ­ ,

.

,

,

,

"

"

COMMUNICATOR .

READINGS

Mackie� J. L. Truth, Probability and Paradox: Studies ill Ph ilosoph ical Logic. Oxford : Cl arendon , 1 973 . 242 -247, 290-295 . - - , an d J. J . C. Smart. "A Vari ant of the " Heterological ' Paradox. ' � Analvsis 1 3 . 3 ( 1 95 3 ) : 6 1 -66 . .

'

NAMING, THE PARADOX OF. P ro p o sed by Sau l Kri pke, th i s paradox i s a lso cal J ed K ri pke 's Pierre. Formulation. Pierre, a mono l i n gui sti cal French s peaker w ho has never l eft France, has heard e no u g h about London to agree that Londres est jolie . On t h i s basi s, we are i nclined to say : (1)

Pierre bel ieves t h at London i s pretty.

Pierre ' s circumstances c h an g e and he ends up l i vi n g i n a c i ty� i n ha b i ted by peopJ e speakin g a forei gn language, which he learn s t h ro u g h d ay-to­ day i nteraction w i th those p e o pl e . Pierre has n o o p port u n i ty to travel from the neighborhood i n which he fi n d s hiITISe lf. hut fi n d s t h at i n the nati ve l anguage the c i ty i s cal l ed "'London . " The pa rt of t h e c i ty w here

1 34

Dictionary of Pa radox

Pi erre l i ves is di rty and run down, and he forms the opi n i on i s not p retty . We are i ncl i ned to say :

that

London

Pi erre bel ieves that London i s not pretty.

( 2)

Pi erre never associ ates London with the city he know s as Londres, and i s stil l prepared to agree w i t h the statement, Londres est jolie. Thus, not on l y does Pi erre hol d i nconsi stent beliefs but, i n tryi ng to descri be Pi erre 's beli efs, we are forced into a contradiction. Explanation. G i ve n Pi erre ' s behavior as a French speaker, we are forced to concl ude that he bel i e ves that the city of London i s pretty. Nevertheless, given h i s beh a v i or as an Engl i sh speaker, we are forced to conclude the exact opposite. Thus, Pierre 's bel iefs seem to be i nconsi stent. Yet it seems unfair to Pierre to brand hi s bel i efs as i nconsi stent since he lacks the know l edge that ' London ' and ' Londres' refer to the same object. The chal l en ge i s to gi ve a rational acc-ount of Pierre 's be1 i efs which re s pects our i nt uitions about the facts i nvolved i n the story. Resolution. We should observe that the palt of the problem that ascri bes an i n c o n si stency to Pi erre 's bel iefs i s si milar to many paradoxes i n m odal deonti c , and epi stemi c l ogic, i n which the substituti on of i de ntity fail s in contexts gove rn e d by an i ntensional operator. An i nteresting aspect of the w i der part of the p roblem i s that i t i nvol ves a tr an sl at i on fr01TI one language to an other. Phi l osophi cal refl ecti on on the n ature of meaning and truth has l ed to the concept of a proposi ti on as the m ea n i ng of a sentence partial ly i n response to the question of how two sentences i n different l an g u ages can have the same mean i ng: they are supposed to express the same p ropositi on Kri pke 's Pi erre, however, seems to show that, at l east i n i nte n si onal contexts, this propositi onal analysi s of meaning i s i n ad equat e ,

.

.

READINGS Biro, J ohn . "What's In a Bel ief?"

Logique et A n alyse

1 07 ( 1 984) : 267-282 .

Elugardo, Reinal do. a Le\vi s ' s Puzzle about S i n g u lar B e l ief-Attribution. "

Philosoplzia

1 7 ( 1 987): 46 1 -476.

Kri pke, Saul . " A Puzzle About B el i ef. "

Meaning and Use. Ed .

A bv i s hai

Margal it. Dordrecht: Rei del , 1 979. 239-283 . K vart, I gal . Kripke's Belief Puzzl e . " Ph ilos ophy Research Archives 9 ( 1 984): 3 69-4 1 2 .

Le w i s , D av i d . W hat Puzzl i n g Pi erre Doe s n ' t B el i e v e . "

Allstra lasian Journal of Ph ilosopJzy 59 ( 1 98 1 ) : 283-289. Lycan , Wi l l i am G . "The Paradox of Nami ng . " Comparative

Perspective.

Rei del , 1 98 5 . 8 1 - 1 02 .

Analytic P h i lo s op hy in

Eel. B . K. Mati1 a] and 1 . L. S ha "v Dordrecht: .

1 35

Dictionary of Paradox

NATURAL RIGHTS, PAR ADOX OF. T hi s is a paradox i n John Locke"s T h e o ry of N atural ri ghts. Formulation.. In leffrie G. M u rph y s formul ation, . . the prelegal i nfl i ct i o n of hann for felt w rongs is mo r a l J y w rong. Thu s w e n e e d t he rule of law in order to acquire the ri ght to puni sh" (270). Explanation. Accord i ng to Locke 's p o l i t ica l the ory, the i nd i vidual abro g at es h i s nat ur al ri g h t to puni sh those who wrong him and establi shes '

a

",

government that wiJJ , among other t hing s , ful fi l l thi s functi on. Murphy, how ever, argues that the notion of a natural ri ght to p un i s h i s paradoxi cal because i t i nvolves i nfl ic ti n g harm on another. But del i berately i nfl i cting harm on another, even i n respon se to harm recei ved , i s moral ly \vrong and one cannot have a natu ral r i ght to do what is moral l y wrong. Hence, it i s only after civil government has been created and the rule of law establ ished that the i n d i v i dua l acquire s the ri ght of puni shment (which i s t h e n exerci sed through the agency of the ci v i l government), Resolution. According to Murphy, the paradox i s reso l v e d by Kant ' s po l i t i c al theory, for w hom the i n d i v i d u a l has n o prelegal n at u ral r i g h t to p u n i s h wrongdoers� but has a naturaJ need to d o so. By establ i s hi n g a civil government and acqui ring the l egal ri ght of e xt racti n g puni shrnent th ro u gh the a genc y of the government, the i ndi vi dual 's need i s fu l fi l l e d an d h i s freedom i s e xpanded, because h e can now do w hat wou l d not have been perm i s s i ble i n the state of nature. ci vi l

READIN G

Murp hy� leffrie G .

"A Paradox in Locke 's Theory of Natural Ri ghts . "

Dialog ue 8 ( 1 969) : 257-27 1 .

NATITRAL S CIEN CE, PARADOXES IN. The physicaJ sciences have al w ay s been a source of paradox because the rati onal , detai led s t u dy of phy sical nature has often resul ted in concepti ons that not on ly seem trul y marvel ous to the l ay m an but al so confl i ct s h arpl y with our pre-sci entific or non sc i e n t i fi c or extra-sci entific concepti ons of th e world about lI S . COPERNICUS ' PA RADOX i s of thi s sort. Recent advances in s c i e n c e h owever� have given ri se to a pl ethora of parad o xe s that, in th e aggregate . See lTI to challenge the presupposition of the rati onality of p hy s i c a l nature (that i s, that i t can be comprehended by rational means). This i s e s pe c i a l l y true of the pa rad o x es ari sing from relativity theo ry and q uan tu lTI theory. See the -

�

BOOTS PARADOX , the CLOCK PARA DOX, COPERNICUS ' P}\ RA I )( )X � t h e EI N STEI N ­

PO DO L S KY - ROS EN PA RADOX, GIBBS ' PARADOX,

HY DI{( ) STAT I < ' PA RADOX , LOSCHM I DT ' S PARADOX , MAXWELL' S DEMON , OLBER ' S PA RA DOX . PI-I I LOTAS ' THE

1 36

Dictionary of Paradox

PARADOX ,

the POTENTIAL ENERGY B A RRI ER PARA DOX , the PREC ESS ION

OF THE

EQ U I NOXES PA RA DOX , SCH RC)EDI NG ER ' S CAT , the S HA DOW PARA DOX , PARADOXES OF T I M E T RAV EL, ZERMELO ' S PARADOX.

th e

TWIN PA RA D O X ZENO ' S PARADOX ES ,

the and

R EA DINGS

Nature of Ph)'sical Reality. Ne"v York: Lang, 1 986. S . The Structure of Scien t�fic Revolutions: . 2nd ed . rev.

-Kak, S u bash . The

Kuhn, Thomas Chi cago: 1 970. Nathan, Amos. " Fal s e Expectations . " Philosophy of Science 5 1 ( 1 984): 1 28 - 1 36. RorJ ich , Fri tz. FrOlJ1 Paradox to Reality : Our Ne�J Concepts o/the Physical World. Cambri dge: Cam bri dge UP, 1 987. Smith, Jose p h Wayne . Reason, Science and Paradox: A g ai n s t Rece ived Op in ion in Science and Ph ilosophy. London : Croom Helm , 1 986. Whi te he ad , A lfred N orth . Scie n ce and the Modern Wo rld. New York: Macmillan, 1 925 . Wohlm uth , Paul C . " Ne sted Real iti es and Hu man Consci ousnes s : T he Paradoxi cal Ex pression of Evo1 uti onary Processes." World Futures 25.3-

4 ( 1 988): 1 99-23 5 .

NEGATION, PARADOX OF.

A

general i zation of the Paradox of Nonexi stent Objects, thi s paradox i s termed "scandalous" by Eri c To ms Formulation" T h e propositi on .

Uni corns do not e x i st

.

predicates nonexi stence of uni corns and , thus� is contradictory. Explanation. The resuJ t i s paradoxi cal because i t woul d seem that one cannot predicate a n y th i n g of that w hich does not exi st. In fact, all tal k about nonexi stent obj ects seems paradoxicaL Toms maintains that all theori e s of negati on are based on one or other of the fol low ing fo ur c o ncepti on s : (1 )

(2) (3)

(4)

the negati on of a p ro p osi ti on (or possibi li ty ) opp osi tion difference nonexistence.

He then

goes on to argue that the first three of these concepti ons do not provide an adequate theory of negati on and, hence, it i s necessary to resort to nonexi stence in order to e xpl ai n negat i o n I n consequence , "every negative proposition and every negative fact i s self-contrad ictory'� (96). Resolution. T here h ave been many attempts to g i ve an intuitively .

137

D ictionary of Paradox

sati sfactory characteri zati on of negati o n , i ncl u d i n g Bertran d Russel l ' s theory of desc r i p ti o n s and Teren ce Parson ' s resurrecti on of M e i non gi an nonexi stent obj ects .

If Toms

i s co rrect however, i n affi rnl i n g that o n l y

none xi stence tru l y expl ai n s n egat i on th e p ar ad o x w o u l d be i ne v i table. ,

T h u s the ad herent of one of the other co n ce p t i o n s of n egat i on w oul d have to show

i t d oe s not i n v o l ve the c i rc u l a r i t y or i nfi n ite re gr es s that

that

Toms argues that i t exhi bits. A lternati vel y, o ne coul d arg ue that n egati on i s a fu n da m ental mod e of thou gh t

that

i s i ntuiti vely clear and, th erefo re ,

n ot i n need of fu r th e r expl an ati on. N on e x i ste n c e couJd then be e x p l i cate d i n t e r m s of negati on. READING S

and Logic.

Tom s , Eric. Being. Negation

Oxford : B J ackweH , 1 962

Parson s , Terence. IVon existent Objects. New Haven : Yal e UP, 1 980.

NEWCOMB 'S PARADOX Thi s i s a PARADOX OF G A M E T H EORY re l ated to t h e PRI SON ER ' S DI LEMtvIA . Formulation A n agent i s to make a c h oi c e betw een re ce i v i ng th e c onte n t s of an opaque box and the c on t ent s of bot h the opaq ue box a n d a trans parent box. He can s ee that there i s $ 1 ,000 in the t r a n s p ar en t box . T he con t ent of the opaque box i s d ete r mi n e d by an e xtraordinarily abl e p re d i c t or (he h as yet to err i n s i m i lar s i t uat i ons ! ) i n the fo l l ow i n g manner: i f he pred i cts ..

..

that the agen t w i ll take j us t the opaque box , he p uts

$ 1 ,000,000

i n i t ; if,

howe ver, h e p r e d i cts that the agent w i l l opt fo r both boxes, he l eave s the

opaque b ox empty. How sho u l d the age nt c ho ose?

Explanation. On the one hand, it seem s e v i dent that the agent's c ho i ce has been acc u rate l y predicted by the pre d i c to r and, hence , he should choose the opaque box ,

thereby receiving the $ 1 ,000,000.

On the other hand , th e

age nt ' s c hoice cannot d e t e r m i n e the p r e d i c t o r s pri or acti on (as i t see m s '

to do o n the pre vi o us account) and, hence , the agent shoul d choose both boxes, the r e by guaranteeing at l east th e $ 1 �OOO and po s s i b l y res u l t i n g i n

$ 1 ,00 1 .000. Resolution.. G . S chlesinger, for exampl e, arg u e s that there are only two p o s s i b i l iti e s : t he p re d i ctor predi cts co rre c t l y that the agent w i l l c h oose the opaque box or h e p r e d ict s cor re ct l y that the ag e n t w i l l c h oo se both boxes. T h u s c h oo s i n g the opaq u e box w i l l be the rati onal c h o i c e . Don Locke , how ever� c ontends that thi s an al y s i s depends on t h e i nfal l i b i l i ty of the p red i ct o r S i nce th i s i nfal l i bi lity has not bee n e sta b l i s hed , t h e argument fai l s . Locke suggests t hat if a t h i r d pa rty had pee ked i n t o the ,

.

opaq u e box and cou l d advi se the agent � h i s a d v i c e

--

re gard l ess of the

,

]38

Dictionary of Paradox

contents of the opaque box - w ould be to take both: if the opaque box contained $ 1 ,000,000 the agent w ould gai n the max i m um payoff (an d the predictor would get hi s first setback) ; if the box were empty� the agent would get $ 1 ,000 ( rather than n oth i ng) . Andre Galloi s� however, argues that the i nvalidity of Locke 's argument can be shown by modifyi ng the situation so that the predictor places the rnoney (or not) i n the opaque box only after the agent makes his choice ; thi s al so removes the suggesti on that the agent's choi ce causes the prior action of the predi ctor. We may observe here that Newcomb's Paradox i s a rich philosophical puzzle, l inked as it i s to such perennial pro b l em s as those of countertactuaJ s and divi ne oln nlsclence . •

•

READING S

Ben - Menahem � Ye m i n a . " Ne w comb s Paradox and Com p ati bi l i s m . " Erkenntnis 25 ( 1 986): 1 97-220. B ar- Hi ll e l M . , and A. Margalit. "Newcom b's Paradox Revi sited . " Br itis h Journalfor the Philosophy of Scie nc e 23 ( 1 972): 295-304. Cargjle, J . "N-e wcom b 's Paradox.�' British Journal for the Ph ilosophy of Science 26 ( 1 975): 234-239. C rai g, William Lane. "Divine Foreknowledge and Newcomb 's Paradox." Plz i lo s ophia [Israel l 1 7 ( 1 987): 3 3 1 -5 0. Newco m b s Many Sol utions ." Theory and De ci sio n 1 6 Eel ls, Ellery. ( 1 984): 59- 1 06. Galloi s , Andre. "How Not to Make a Ne\vcomb Choice." A n a lysis 39. 1 ( 1 979): 49-5 3. Horgan, Terence. "Counterfactuals and Newcomb's Problem, " The Journal of Philosophy 78 ( 1 98 1 ): 3 3 1 -356. Horn e , James R . " Ne \v c omb ' s Pro b l e m as a Thei s t i c Probl e m . " International Journal for Philosophy of Religion 1 4 ( 1 983): 2 1 7-22 3 . Kavka, Gregory S . " What I s Newcom b ' s Problem A bout?" Anzerican Ph ilosophical Quarterly 1 7 ( 1 980): 272. Levi , I. "Newcomb ' s Many Problem s . " Theor.y and D e c i s ion 6 ( 1 975 ): 1 6 1 - 1 75 . Locke, Don. " How to Make a Ne\v comb Choi ce. " Analysis 3 8 . 1 ( 1 978 ): I

,

"

'

1 7-23 .

Macki e, J. L. " Newcomb's Paradox and the D i recti on of Causation. " Canadian Journal of Ph ilosophy 7 ( l 977): 2 1 3 -224. Nozick, R. "Newcomb 's Problem and Two Principles of Choice . " Essa)'s in Honor of Carl C. Henlpel. Ed. N. Rescher. Dordrecht: Reidel , 1 969 . 1 1 4- 1 46 .

Schlesinger, G. "'The Unpredictabil i ty of Free Choi ces." British Journal for the Phi lo sop hy of Science 25 ( 1 974): 209-22 1 . Sasieni, Maurice. "Newcomb ' s Paradox. " Theory and Decision 1 6 ( 1984) : 2 17-223 .

139

Dictionary of Paradox

Smith, J os eph Wayne

.

Reason , Science and Paradox: Against Received

Opinion in Science and Ph ilosophy. Lond on : Croom Helm, 1 986. Ch. 8.

Sorensen, Roy A .

" N ew com b s Problem: Recalculations for the One- Boxer. " '

Theory and De c is i on 1 5 ( 1 983): 399-404.

l�EW MEMBERS , THE PARADOX OF. Political w i sdom has it that

an effecti ve way of re d u c in g the voting power of a domi nant member i n a wei ghted b o dy of voters i s to i ncrease the size of t h e body. It t u r n s out, how ever, that the addition of one or s e veral new mem b er s may actual ly i n c re a s e the voti ng power of some of th e old Inembers. Amnon Rap opo rt and Ari e l C o h e n an a l y ze computati onal results that show that, althou gh the p ara d o x i s indeed inevitabl e in certain si tuations, it has l ittl e practi caJ si gnificance for groups th at have five or more members. The detai l s� i n vo l v i ng indi ces of voting power i n game t h e o ry are too e x ce ss i vely technical to be recounted here. �

READING S Raanan, J . '"The Inevitabi l i ty of the Paradox of New Mem bers . " Tech n ical Rep ort No . 3 1 1 .

S c h ool of Ope ra t i o n s Re searc h a n d I n d u s t r i a l

Engineering, Cornell U, 1 976. Rapoport, Amnon, and Ariel Cohen. "Expected Frequency and Mean Size of

the Paradox of New Mem bers . " Th eory and Decision 1 7 ( 1 984): 29-46.

NEW RIDDLE OF INDUCTION, THE. See GOODMAN ' S PARADOXES OF

CON FI RMATION.

NICHOLAS OF CUZA' S COINCIDENCE OF OPPOSITES . In De Docta Ignorantia , N i c h ol as of Cuza ( 1 40 1 - 1 464) a pp l i e s to God the p ar a d o x i c al l og i c of the c o i nc i d e n c e of o p p o s i te s ( co in c ide n tia oppositoruI11). S ince HERACITUS ' PARADOXI CAL APHORI SMS , th i n k e r s have

a n t i th et i ca l attri butes to God i n an attempt to w rap the h U ln a n mi nd a rou n d divine i neffability: He is X and not X at one and the saIne ti me an d i n one and the same respect, at once Y an d not Y, both Z and not Z. The parad o x however, i s not a l w a y s so bol d1 y stated . Fo r c x (un p l e� G od i s often supposed to be both infinitely just and i nfi n i te l y lne rc i fu l . The j o i nt a s s u m p t i o n of these p re d ica t e s is n e ve rt h e l e s s pa rad o x i c a l because th e _ for mer i mpl ie s that H e wi l l pun i sh w ro n gdoers , w h i l e the l atter i mp l i e s that He w i ] I forgive them. A l t h ou g h t h i s res u l t i s Il ()t stri ctly speaki ng , a l o g ical contradicti on, it d o e s h i g h l i g h t Inan 's i na bi l i ty to attri buted

,

1 40

Dictionarv of Paradox �

..,

compre hend God. Tn fa ct paradoxi cal di scourse about God i s frequently desi gned t o d rive home the l esson of the Voice from t h e Whi rl w i nd i n Job, n a m e l y that the Lord moves i n my steri o u s way s , that G od i s i nscrutable to h uma nki n d Compare V IA NEGATIVA . ,

,

.

NIRVA N A , THE PARADOX O F. There are at l east t w o separate p a ra d o xe s i n B uddhi st t h ou g h t about n i r va n a The first asserts t h at the .

very concept of ni rv ana i s paradoxical because it entail s both the ultilnate anni h i l ation of the individual and i ndi v i d u a l fulfi l lment i n eternal l ife. Robert S l at e r adduces two other rel ated paradoxes and affi lms th at n i rv ana ·'i s a p a r ad ox set w ithi n a co nt e x t of paradox" (4) . T he second Paradox of N i rv a n a , w hi ch w i J J be d i scu ssed here , i s re la t ed to the HEDONISTIC PARADOX . Forlnulation. 'Th os e w ho pursue or desire nirvana, or desirelessness, will n e v e r ( l og i c al l y ) or are least l i kely to (practi cal ly), get it" ( H e rm an 5). Explanation Nirvana i s not the ki nd of thing that one j ust h a pp en s upon or w hich j ust happen s to one� but rather a state that requires quite a bit of effo rt o n the part of anyone w h o i s to achieve it and, thus, w e must wa nt to a c hi e v e i t i n order to do those things necessary for its acquisition. Hence, we can never achieve ni rvana, a state of c om p le te desi relessness, because i n order to obtain it w e must desire to do so . Stated differently, we c annot make n i rv ana the end of our actions because that would defeat the very purpose of our u nd e rt aki ng � but w ithout maki ng nirv an a the end of our acti ons� we c a nn ot do w h at i s necessary to achieve i t . Resolution. The B uddhist sol ution to the parad o x is to recognize that there i s n othi n g that one c an do i n o rd er to escape the paradox and thus the d evotee " lets go" of the desi red goaL Up o n do i ng so, nirvana i s obtained. Observe , how e ve r, that nirvana is not just a "l ai d back�' attitude i n t h e face of a practi cal conundrum : it i s founded i n some sense on the insight gained from the st r u g gl e to come to terms w it h the parado x A s Arthur L. H e rm an expresses i t, "the p h i l o s oph i c argument was nec ess ary before the rational insi ght was p o s si ble, and that rational insi ght, namely, that t h e r e i s no way o u t was necessary before "letting go" could occur, and "'l etting go" w as necessary before ni rvana w a s p o s s i bl e (7). Western responses g e n era Hy e m p ha s i z e an i ndirect ap proach al t ho ug h at l east one ficti onal character reasons N i ch 0 1 a s i n John Fowles's The Magus to a " l etting go�' remini scent of thi s de script i o n of attai ni n g n i rv ana. �

,

,

..

.

,

"

,

DictionarJ' of Paradox

14 1

READING S Fossa, John A . "Through S eeking to Mystery :

A Reapprai sal of John Fowles '

The Magus. " Orbis Litterarllnz 44 ( 1 989): 1 6 1 - 1 80. Herman, Arthur L. Sol utions . " Slater, Robert.

"

H ed oni sm and Ni rvana: Paradoxes , Dil eInmas and

Ph ilosophica rlnd i a } 1 0 ( 1 98 1 ) : 1 - 1 0. Paradox and Nirvana. Chicago: lJ of Cru gago P, 195 1 .

3-4.

NON.. COMMUNICATOR, THE PARADOX OF THE. Adduced by

Theo dore Orange and rel ated to the M-V A R I A NT S O F T H E LO G I CAL PARADOXES , thi s paradox purports to show that al l hierarchi cal theories are contradi ctory. Formulation. Consi der the sentence ( 1 ) i s meani ngless.

( 1)

Accord i ng to hierarchical theories, ( 1 ) i s t r u e . T h u s , i t i s bot h 1l1ean i ngl ess and not meaning1 ess, whi ch i s a contrad icti o n . Explanation. Hi erarchi cal theories (see RUSS ELL ' S PARADOX ) brand a statement as meaningl ess w henever the hi erarchy c on d i t i o n s are v i o l ated . I n parti c u l ar, the th eo ry i s used to deal w i th sentences suc h as ( 1 ) which refer to themsel ves. Hence, ( 1 ) i s meani ngless. Yet ( 1 ) asserts that it itself i s meaningless andq thus, i s in accord with what is the case. Therefore, ( 1 ) i s true. But a true sentence cannot be mean i n gl ess. Therefore, ( .1 ) i s not meani ngl ess and the contradiction i s establ i shed. Resolution.. Robe11 W. Beard formu l ate s tw o bas ic obje ctions to the purported parad ox Fi rst, Drange admits (1) as a p rem i se and thus either assumes the truth of ( 1 ) i n ord er to prove the truth of ( 1 ), maki ng the argument ci rcular, or presu ppose s that truth and meaninglessness are not excJ usi ve, vitiating the validity of the argument (by fal sifying the princi p i e that true sentences cannot be meani ngless). Presumably, Orange w o u l d reply that ( 1 ) i s not a premise at aU, but rather a consequence of hie rarch ica l theory ; nevertheless� no hierarchical theori st w oul d be w i l l ing to acce pt ( 1 ). B eard ' s secon d objecti on i s that Tarski ' s defi nition of trut h i s relativ ized to the l anguage be ing used but Orange e lnpl oy s i t i n a n unrestricted manner. .

..

REA D £ NG S B eard� Robert W.

'"

S e m a n t i c T heory a n d the Parado x o f t h e N o n ··

Ph ilosophical Studies 1 7 ( 1 966): 44-45 . D ra n g e , T h e od o r e . " T h e P ar a d o x of th e N o n - C O l l l l n u n i c a to r. " Ph ilosophical Studies 1 5 ( 1 964): 92-96. Communi cator. "

142

Dictionary of Parado.x

NONEXISTENT OBJECTS, PARADOX OF. NEGATION .

OLBER ' S PARADOX.

See the

PARADOX OF

Fo rmul ated by H . W. M. O l bers i n 1 826

of earl ier formulations (most notably one by 1 . P. L. de Cheseaux in 1744) , thi s i s a problem for various cosmo] ogies� incl ud i n g N ewton s . Formulation. "An i nfi n ite n umber of stars di stributed eve n l y in infi ni te space -woul d l ogi cal ly create a sky bla zing al l over with extraordinary bri l l iance i n obvi ous contradicti on to the factual darkness of the n ight sky " (Jal i , 1 0). Explanation. Accord ing to N ewtonian cos mology space i s Euc l i dean and ba s i cal ly static. There are supposed to be an infinite number of stars, essentially simi lar to the sun, evenly distributed throughout thi s space. Since the area of the sky i s fi nite, the evenly di stributed, infinite number of stars should fill up the whole sky, thereby compl etely i l l u m i n ati ng it. Resolution.. Early attempts to avoid the paradox centered on tink ering w ith the inverse square lavlJ of the propagation of light and on the possibility of li ght energy bei ng absorbed by the mediu m th ro u gh w hich i t passes. T he latter hypothesis w ould not seem to eliminate the paradox� h owever , becau se, accord i n g t o then ac know ledged pri ncipl es� the co n tin u al adsorption of energy w ould cause the medium itself to glow. Eventuall y h owever, most of the New toni an hypotheses w e re abandoned . For exampl e, although space is stil l thought of as being homogeneous on the l arge scale, stars are not compl etely evenly di stributed on a smaller scale: t h ey form cl usters (gal axies) and clusters of clusters. N everthele ss, an i nfi nite number of stars is not actually needed to generate the paradox ; i n fact� a uni verse of very modest dimensions w i l l do so as well, dep e ndi n g on the density of matter in, say, the observable part of the universe. The observed density in fact seems compati ble with the Ol bers Paradox. Once the uni verse was seen to be expanding, however, rather than static, the consequent shift of l i ght towards the red end of the spectrum (aki n to the Doppl er effect for sound waves p roduced by a moving source) reduces the energy of l i ght from di stant stars so that the Olbers Paradox d o es n ot arise. The paradox nevertheless is sti l l trou bl esome for some forms of fi nite model s of the universe. i n d ependently '

,

READINGS

Di ckson , F. P.

Thought.

Cam bridge : MIT P, 1 968.

Jal i , Stanl ey L.

Thought.

The Bowl of Night: The Physical Universe and Scientific The

Paradox of OIber s Paradox: A Case of Scientific

New York: Herder, 1 969.

Dicfionary of Parado.r

1 43

OMNIPOT ENCE, PARADOXES OFe Thi s i s a class of ReliGIous

PARADOX. By God ' s omni potence i s meant that He i s ai l powetfuL i n the sense that nothi ng i s as pow etful as or more powerful than G od, and/or that He is i nfinitely powerful, i n the sense that His power i s i ndefinitely great. Someti mes God's omnipotence i s take n to mean that God can do anything. This i nterpretati on of Hi s omnipotence inspires a whol e set of tradi tional paradoxes. If God can do anything� can He square a circl e? Can He make a stone so heavy that He cannot l ift i t (see the PARADOX OF THE STONE) ? In general , can He invent problems that He cannot solve. In order to avoid these difficulties, God' s omnipotence i s often taken as a more modest attainment to the effect that He can do anything that i s l ogicall y possible. It is sometimes asked, moreover, if God can do such thi ngs as annihilate H i m self and never return to being, or deny Hi s own essence. These q uandaries - and more importantly the PROBLEM O F EV I L l ead i n turn to further restrictions of H i s pow ers , sllch as : God can do anything that i s w orthwhi le doing or (what i s nearly the same ) He can do anythi ng that expresses H i s necessary ess e n ce as God . Si nce logi cal consi stency i s typical ly taken to be pa11 of G od 's esse nce� thi s fornl ulati on usuall y i mpl ies that He cannot do anythi ng l ogi cal ly i mpossible. -

REA D I NG

Flint, Thomas P. and Alfred 1. Freddoso. '" Maximal Power. " The Existence and Nature oj God. Ed. Alfred J . Freddoso. Notre Dame: Notre Dame UP. 1 98 3. 8 1 - 1 I 3 . ,

OMNIP RESENCE AND TIMELESSNESS, THE PARADOX OF.

Ri chard La Croix i dentifi es thi s paradox i n the theoJ ogy of St. Thomas Aqui nas. Aq uinas holds that God is necessari ly omni present and eternal ; that i s, both of these properties bel ong to God's very essence and are indeed part of w hat it means to be God. Eternity, however� ca n be understood either as temporal i nfinity or as timeJ essness. Temporal infinity i s the claim that God somehow stretches throughout all times past, present, and future, whereas timelessnes s asserts that God is somehow olltsi de of time and not bound by telnporal considerations. Aquinas arg ues for the view that G od i s timeless. B ut, then, argues La Croix, if God is i ndeed omnipresent then it would appear that he must have been in the United Nations B ui lding yesterday as well as the day before yesterday. And if God was i n t he United Nations bui lding both yesterday alld the day be fore, then it would appear that he is in time and that tc�n poral predicates do apply to him. So, it woul d appear that Ciod

1 44

Dictionary of Paradox

i s 110t a timeless being if he i s omnipresent and that two doctri nes crucial to the th eology of Thomas Aquinas are 10gi caJ 1 y i ncompatib1 e (39 1 ).

more sop h i sti cated u n d e r stan d i n g of om n i presence w ould resolve the paradox� but h e al so argues that the c o n c eption used in his ar gu ln e nt i s true to that of Aqu i nas.

La Croix a d mits that a

READING La Croi x, Richard R. "Aqui nas on God's Omnipresence and Timelessness. " Ph ilosophy and Phenonlenoiogicai Research 42 ( 1 982) : 39 1 -399.

OMNIS CIENCE, P AR ADOXES OF. These are RELIGIOUS PARADOXES d eri v i n g from the view that God i s omn i scient. God h a s been sai d to be

omni scient i n thre e inter- related sense s . First., God perceives all thi ngs as they h ap p e n and hence know s of their occurrence. Second, God know s e v e ryth i n g i n the past, present and future. T hi rd� God know s everyth i n g that it i s possi bl e to know. It fol l ow s from these powers of G od that al l truth i s eternal , that al l truths are known by God, an d t h at no t h i n g can o c c ur unl ess it accords with these ete r n al truths. Yet if thi s is true, i n w hat sense can eith e r man or God have free wil l ? M an cannot have it b e c au se there i s noth i n g that he mi ght do t h at i s not a lre ady known b y God. God too c an n o t be sai d to have w i l led to do thi ngs i n one w ay rath er t h an another w hen He already knew w hat He w ou1 d will to be done. READ ING

Stenner� Alfred 1. "A Paradox of Omni science and Some Attempts at a Sol ution . " Fa ith and Ph ilosophy 6 ( 1 989) : 3 03 - 3 1 9.

OSCAR WILDE ' S EPIGRAMMATIC PAR ADOXES Here i s a bri ef ..

sel ecti on of paradoxi cal bons mots from O scar [ Fingal 0 ' Fl ahertie Wi l l s ]

Wilde ( 1 854- 1 900 ) :

( 1 ) There is only one thing in the \vorl d worse than being talked about, and that i s not being talked about (The Picture of Dorian Grey, Ch. l ). (2) In this worl d there are only two tragedies. One i s not gettin g what he ,vants, and the other i s getting it (Lady Wind­ ermere 's Fan , Act 3 ). (3 ) I suppose society i s wonderfully delightful . To be i n it is merely a bore. But to be out of it simpl y a tragedy (A WOlnan of No Importance, Act III). (4) A nybody can make hi story. Only a great man can wri te i t (Aphorisrns).

1 45

Dictionary of Paradox

(5 )

Yet each man kill s the th ing he lov es , By ea c h let thi s be heard, Some do it with a bitter look, Some with a fl attering word. The coward does i t wi th a ki ss , The brave man w i th a sword. (The Ballad of Reading Gaol I, st. 7).

OSTROGORSKI ' S PARADOX. Thi s i s on e of the PA RADOX ES O F VOTING , i nvoJ ving competi tion of two pal1i es i n a maj ority ruJ e context. Formulation.. A party can win an el ecti on even though it s o p pon e nt s '

pos i ti ons are preferred by a maj o ri ty of voters on every i ssue i n questi on. Explanation. C onside r an election dispu t e d by p a rty X and party Y, w hose positi ons on the i ssues are x an d y, respectively. The following chart tabulates the res ults , where V I -V 5 are the five voters of the co n sti t u e nc y and 1 1 - 13 are t h e t h re e i ss ues i n v o l v ed : Issues I

VI (j) Cl)

V2

1

x

X

I2

I

\

Bailot

X

y

X

Y

X

X

r

-

0

......

>

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

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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 "

"

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,"

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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 ,

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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. "

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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 . ,

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I I ,

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I

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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. "

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"'

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Dictionary of Paradox

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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

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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

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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

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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.

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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 .

.

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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. .

"

"

,

"

"

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'

'

'

. "

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

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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

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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 .

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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
We can then use thi s sym bol order to defi ne an order on E and , h en c e on ,

D ( this is kno\lvn as the l exi cographical order) :

bm a1a2 an < b1b 2 n=ln and th e re i s some i5:n such that for all j
if ( 1 ) n
(2)

)

•

.

•

.

•

}

The order on 0 detc m1 i n e s an enumeration of D . Let

h be the first el e ment

of the enumeration , 12 the second el ement, and so on. Then, th e func t i on

(, as defi ned abo v e , i s f(k)

\v a s

defined usi ng only

T hus

a

=

fk(k)

+

1,

for each n atural number

k. S i n ce j�

finite numbe r of �lords, f i s an element of D .

,

j= f

n

for s om e n. B ut fi:n)

=

f (n) n

+

1,

from !, for the argument n. Therefore, ! -:f-: in' whi ch i s a l contradiction. Thus, the noti on of finite defi nability s e em s paradoxical .

so f differs

Resolution" As in most S Etv1ANTICAL PARADOXES , it seems that t he root of

the trouble i s the el ement of sel f- reference t hat i ss ues i n a v ici ous circl e.

I n particular, the defini tion of.t is impredica ti ve. That i s j i s a member of the set D , but the very d efi ni ti on of J refers to D in an essential �lay. The ,

responses to this paradox , as for the other s e manti c al p arad oxe s have ,

centered on \lvays to block the fo nn a ti on of the circular defini tion.

Dictionary a/Paradox

Hazen,

1 73

I{EADINGS

Allen.

"Prcdicativ e Logics. " Handbook of Philosophical Logic .

Vo lurne One : Elen�ents

of (,'lassica l

Logic .

Guenthner. Dordrec ht : Reidel ,

Ed. D. Ciabbray and F

1 983 . 33 1 -407. Kleene, S te p he n (� . Introduction to J.\,1etanlathernatics. A.msterdan1: North­ H oll and 1 952. 38-39. .

,

ROBBER ' S PARADOX� THE. Thi s is a version of the Victim 's PARADOX in ,,\lhi c h the paradoxical concl usion i s that i t i s forbi dden [or the thi ef to repent his deeds.

ROSS ' S ANTIMONY. Thi s p arad o x res ul ts from James Ross 's modal arg ument for God 's exi stence .

Formu lation

Ross deduces a contradi ction ('rOITI

God does not exi st and the foll o\vi ng p re m i s e s :

( 1) ( 2) ( 3)

the

supposi ti on that

It is possible that there i s a l l ex pl a nat iol l for t he non ­ existence of C}od . T he e xistence of ( Jod is not i ncol l si st e n t . It i s i lnpossibl c t hat anythi ng could prev e n t ( l od rn Hn ex i sti ng .

He thus concl udes that God 's exi stence i s neces sary. YC L

a

siln i l ar proof

of the necessi ty of God 's non-exi stence follo\v s from the suppos i ti on that He does exi st and the premi ses :

(4)

( 5) (6)

It is possible that there is an explanation for the existence

of God.

The no n exi ste nce of God is not inconsistent. It is impossible that anything could cause C lod to exist. -

Explan ation. Ross 's ori gi nal argument i s an indirect proof: starti n g from

the s uppo s iti o n that God does not exist, he deduces a contradiction, thereby

proving God 's non- exi stence i s i m possi bl e. The premises that he uses i n the deduction seem i nnocuous.

Pren1i se

(1)

is a \veakened form of the

Pr i n c i p l e of S u ffi cient Reas on ; pre m i se ( 2) m e re l y s tate s that the

affi rmation of God 's exi stence i s not self- contradi ctory ; and p remi s e

(3 )

seClns to [ollo\v [rotn the very noti on of God . I f, hO\Ncvcr, \ve grant ( 1 ) ,

i t \voul d seem th at \ve shoul d al so grant (4) . Similarly, to stand or fall together, \vhil e

B ut, o nce w e hav e ( 4) - ( 6) ,

(6)

and

(5)

seenl

fo1 1o\\'s fron1 the very notion of God .

\v e c a n

i m pos s i bi l ity of God 's exi s tence.

(2)

c ons truc t an arg ument for the

Hence , \v e have pro ved

that

God 's

exi stence i s both necess ary and impossible, \v h i ch i s a contradicti on .

Dictionary of Paradox

1 74

Resolution. John Zeis points out that the antinomy depends on premises

(2) and (5) either standing or falling together. Thi s will be the case if the statement " God exi sts" is contingent. If, however, "God exists" is either necessarily true or contradictory, (2) and (5) will have opposite truth values and the antinomy fai ls. Interestingly, then, the antinomy reveals that "God exists" is not a contingent statement. Thus , either God necessarily exists or Hi s existence is itnpossible. Thus, a demonstration of either (2) or (5) would entail respecti vely either Ros s 's proof of God 's existence or the parallel proof of His non-existence. But Zeis argues further that a proof of (2) or (5) would already have to be a proof of God 's existence or non­ existence , thereby obviating the need for the purported demonstrations. All modal proofs (or disproofs) of God 's existence, according to Zeis, share this fla",'. READING Zei s , John. "Ross's �A.ntimony and Moclal Argunlents for God's Exi stence. "

International Journal/or Philosophy oj Religion 20 ( 1 986) : 15 9- 164.

Thi s paradox w as supposedly adv anced by Ludwi g Wi ttgenstein as an argument for scepticism. Formulation. "When I respond in one \vay rather than another to such a problem as '68+57, ' I can have no j ustification for one response rather than another. Since the sceptic, who supposes that I meant quus [instead of plus] , cannot be answered, there is no fact about me that dis tinguishes " bet\veen my meaning plus and my meaning gUllS (Kripke, 2 1 ) . Exp lanation. Consider the function "quus ," defined as [ol1o\\'s: RUL E FOL L O WING, T HE PAR AD O X OF.

x quus =

def

y

=

clef

x pl us Y if x and y are both less than 57

5 otherwise.

Now suppose that Jones has never had occasion to add numbers greater than 56. The sceptic then argues that there is no evidence in any of Jones 's past actions that would count against the hypothesis that he actually meant "x quus y" when he said "x plus y. " This is because Jones has never reached the point at which the two functions diverge and, thus, any action that supports the hypothesis that Jones means plus also supports the hypothesis that he really means quus . Even Jones himself, according to Wi ttgenstein, is in the same quandary, for it i s precisel y the notion of intentional meaning that the sceptic i s attacking , due to the lack of independent evidence for it.

Dictionary oj Paradox

1 75

Resolution. According to Saul Kripkc 's interpretation, Wittgenstein is

commi tted to an anti -realist solution to the paradox, in �'hich our practice of communicati on enabl es us to establish conditions for legitimate assertability of such statements as '�Jones is following this or that rule. " Roger Scruton, however, contends that both Kripke's interpretation of Wittgenstein and his (Kripke 's) response to the paradox are flawed. For Scruton, the paradox does imply that meaning is socially generated, but rules may nevertheless "be understood by the participants as defini ng modes of 'correspondence ' vvi th an independent reali ty" (598) and, thus, we are not forced to adopt an anti-realist position. In contrast, Barry Allen argues that the paradox reveals the inconsi stency of the notion of such a thing as a proposition as that vvhich grounds meaning. READTNC}S AJlen� Barry. "Gruesome Arilhnletic : Kri pke' s Sceptic Replies. " Dia logue

28 ( 1 989) : 257-264. A.nscombe , CT. E. M . "Rev iew of Wi ttg e n s t e i n on Rules and Private J.language, by Saul Kri pke C-'anadian Journal q( Philosophy l5 ( 1 9R5) : .

"

1 03- 109.

Baker,

J. , and P' I lacker. Scepticisnz, Rues, and A1eaning.

()xford: R lack�'cll,

1 984 . Goldfarb, W. "Kripke on \Vittgenstein on Rules. " Journal of Philosophy

82 ( 1 985) : 47 1 -488. Kripke, Saul .

Wiz tgenstein on Ru les and Private Language. Cambridge ;

Harv ard UP,

McDo\vell , 1.

1982.

"Wittgenstein on Followi ng a Rule . " Synthese

58 ( 1 984) :

325-363 . Rudebusch, George.

HHoffman on Kripke �s Wittgenstein. "

Phi losophy

Research Archives 1 2 ( 1 986-87): 1 77- 1 82. Scruton, Roger. "Wittgenstein on Rule

and Priv ate Language. " Mind ns 93

( 1 984) : 592-602. Stock, Guy. Quarterly Tait, W. W.

HLiebniz and Kripkc 's Sceptical Paradox. "

Phi losophica l

38 ( 1 988) : 326-329. "'\Vittgenstei n and the Sceptical Paradoxes. "

Philosophy 83 ( 1 986) : 475-88.

Wittgenstein, Lud\vig. The Phi losophica l Investigations. Anscombc. New York: Macmillan,

Journal of

Trans. (1 . E. Iv!.

1 953 .

RUSSELL'S PARADOX. Called Russell 's Class Paradox and named

by Bertrand Russell himself "the Contradiction," thi s parad ox i n set t he ory \vas hi storical ly i mportant because it put Frege 's attc lnpt to reduce mathematics to logic in check.

,

Dictionary of Paradox

1 76

Formulation . C ons i d e r the set of al l s e ts that do not contai n th em s el ve s as a member ; thi s s e t contains i tself as a m em b er

and on l y if, i t does

if,

not contai n itself.

Explanation. S ome sets are not members of themselves , \vhile other sets do contain themselves .

The

set of small se t s for ex am pl e i s not smal l . ,

,

The s e t of large sets , ho\vever, i s itself large and th us the set of large sets

is itself an element of the sct of l arg e sets . Defi ne the set

I

to be the set of

all sets that are not members of th emsel ves :

(1)

-,

{X / X

:=

ce

X} .

There arc many elements of -, nlost of \vhich are not problematical. Like the

set of small sets gi ven above , the set of tea kettles , for example, i s not itself a tea kettle and thus belong s to I . FroBl the defi n i ti on of -' the criterion for ,

membe rshi p i n

(2)

is that a set not be an el elnen t of itself; that i s

I

X

e

-,

if, and only if, X

re

X.

We may th e n ask if -, i tself i s a nlem ber of -, ; thi s anl0unts to i nstantiating the v ari abl e X in (2) to " vvhereupon vve obtain the fol1 o\vi ng contradiction :

(3 )

-,

e

-,

if, and only

iC

, re '.

T hi s i s Russel l 's Paradox. I t is a formal contradi cti on in Fre gi an logi c : if -,

i s an e l em en t of i tself, then, by defin i t i on i t i s not an elem e n t of i t s el f ; ,

but i f -, i s not an el ement of i tsel f, it s ati sfi e s the defini tion of -, and thus

i s an el eme n t of i tsel f. Resolutiono

Ru s s el l hi msel f argued that the paradox \vas due to the

element of self- reference that introduces a viciou s circl e into l og i c

.

He

thus p ropo s ed hi s T heory of Types to eli minate the problem . According of Types , typc o i s as s i g ned to i ndividual s , typ e 1 to sets of i ndi vidual s, type2 to sets of sets of i ndividual s, and s o on. Each type \\'as

to the

T h eory

als o divided into orders (Ramified T heory of Types) , but a S i mpl e T heo ry of

Type s

\vas presented by F. P Rams ey th at seems to be sufficient for

avoidi n g the paradox . The theory wou l d block the p arad ox by nlaki ng it i mp ossibl e to affirm "-, e

-, "

since the second occurrence of -, \voul d have

to be of a hi gher type than the fi rs t occurrence . W. V. O. Qui ne si m plified the theory con s i derabl y by eli minating the types in favor of bui lding the

hi erarchy i nto the defi ni tion of "veIl-formed fo rmula

.

Russel l w as of the

opi ni on that the sol uti on to the LoGICAL PARADOXES s ho u ld al so gi ve us

the key to the s olu ti on or the SEl\1 A.NTIC AL PARADOX ES and, i n d eed a ,

hierarchi cal semantical theory \vas devel ope d by A lfred Tarski . E,vcn the

1 77

Dictionary of Paradox

Simple Theory of Types , ho\�/ever, has undesirable consequences , such as the apparent reproducti on of the vvhole of logic for each type . Some logicians al so obj ect that i t is too radical i n that i t di sallovvs al l sel f­

reference, ev en though not all self- reference i s paradoxical .

S ince i t

became evident that one of the roots of the paradox \vas an axi om of

unrestri cted abstraction

that for any property, there i s a set of all and

onl y those things hav i ng that property

another response to the paradox

\NaS to develop a consistent axi omatizati on of set theory. I n the standard Zermelo-Fraenkel axiomati zation , the strategy is to lilni t the size of sets so that such trouble some sets as the Russel l set -, above cannot be formed . This is done by replacing unrestricted abstraction \\'ith an axiom that al lo\vs

us to affirm the exi stence of a set correspondi ng to a gi ven property onl y i f that set can be separated off from an al ready given set. A l though sets having a transfinite nun1 ber of el ements are countenanced i n Zermel o­ Fraenkel set theory, one counteri ntui tive resul t i s that there i s no uni versal set ( that i s , a set of al l sets) .

A nothe r approac h to thc parad ox \v as

devel oped by S tani sla\v Lesni c\vski , \v ho arg ueu t h a t R ussel l 's Paradox ari s e s because of an alnbi gui ty i n th e te nn s u s cd lo tal k a bo u t se ts .

A ccording to Lcsnie\vski the re l ation bc t\v ccn part and \v hol e

he nce

al so the term s "an cl cln en t of" and 'G thc s c t of" - can be though t of e i the r di stri buti ve l y or c o l l ecti vely.

When Russel l 's Paradox is form ulatcd ,

consistentl y ei ther in d i s tributi ve term s or i n collecti ve terms , certai n presuppositi ons turn out to b e fal se, thereby i nvalidati ng the paradox .

Sec Rl rSSELL ' s PARADOX

IN

MANY VALUED LOGIC. RE}\DINCiS

C:astafieda, Hector- Neri . "( )ntology and Grammar: I Ru ssell s Paradox and '

the General Theory of Properties in Natural Languages. "

Theoria 42

( 1 976) : #-92.

T.

Reflexive Paradoxes.

London : Ro utledge, 1 988. Hart, W. D. "Russell and I-lart. " PaCific Phi losophical Quarterly 64 ( 1983 ) :

Champl in,

S.

1 93-2 1 0. Jackson� Joseph E. "Shav ing Russell 's Paradox \vith Occanl's Razor. " ETC

22 ( 1 965) : 1 1 4- 1 1 6.

Rodrigues (�onsuergra , Francisco A. .

"Russell 's Theory of

Types , 1 90 1

-

1 9 1 0: Its C:omplex Origins in the lJnpubli shed I\ifanuscripts. " Histoly of Phi losophica l Log ic 1 0 ( 1 989) : 1 3 1 - 1 64.

RUSSELL' S PAR ADOX IN MANY VALlJED LOGICS. Many-vaJ ued

l ogic has often been seen as a promi si ng \vay of e scapi ng fron1 the l ogi cal paradoxes . If, beside truth and fal si ty, vve hav e a t hi rd tru th val ue - sa:y,

Dictionary of Paradox

1 78

" i ndetermi nate"

then the set of al l s ets that are not mem bers of

e xamp l e can be assi g n ed thi s thi rd tru th val ue and RUSSELL' S PARADOX will be avoided . In various man y-v a lue d l og ic s ho\vever, the paradox can be resurrected b y m odi fyi ng the defi nition of the troublesome "Russell set. " In set theory b as ed on three mutu al l y excl usive truth v al u e s , the third of whi ch corresponds to "'the po s si b i l i t y that a set has a fu zzy penumbra, so that a gi ven el ement l y i ng in th i s penumbra can be regarded nei ther as i nsi de the set nor as outside it" ( Rescher 208) , the troublesome set becomes then1sel v es , for

,

,

,

the set of all sets that are not members of themselves the fuzzy penumbra of themsel yes.

nor i n

In order to avoi d the p arado x i n a fi nitel y -valued logic, very strong i ndetelminacy condi tions mus t be i mposed. Alternati vel y, one may have recourse to i nfinitely-val ued logi c. RE�t\DING

Rescher, Nicholas . Many- Valued Logic. Ne\v York: McGra\v I--lill , 1 969. 207-209.

SANCHO PANZA, THE . A l s o cal led the G all o ",' s , t h i s l o g i c al paradox

poses an exi stential di lemma. I n the Middle A ges, thi s paradox in vari ous v ers i on s was

ri ghtl y considered as a form of the Liar (Jones ) . Formulation. "A certai n m anor was divided by a river upon which was a bri dge. The l ord of the manor had e re cted a ga l lows at one end of the bridge and had enac te d a law that whoever would cross the bridg e must first s wear whi ther he ", ere g oi n g and on what business ; if he swore trul y, he should be al lowed to pass fre el y ; but if he swore fal sel y and did then cross the bridge he sho u l d be han ged forthwith upon the gal l ows. One man . swore ' 1 go to be hang ed on yonder gallo",'s ' , and thereu po n crossed the bridge. The vexed que s tio n \vhether the man should be han g ed is brou ght to Sancho Panza as governor of B arataria" (Cervantes, Don Quixo te , Pt. 2 Ch. 5 1 , quoted by A . Church, I ntroduction to Mathenlatical '

.

.

Logic 1 05) .

Explanation. I f the man w ere hang ed he w ou l d have sworn hi s busi ness ,

tru ly and therefore would not have meri ted hanging. If the man were not han ged he w oul d have sworn his business falsely and must, according to th e po l i cy in force , be h an g ed . Resolution . On e could, of course, have the paradoxer h an g e d on general principles. Sancho Panz a s first s ol u ti on is j u st as deci sive: to hav e the ,

,

'

Dictionary of Paradox

fel l o\\J cut in t\vai n , h an g i ng h al f an d l e tt i n g

1 79

th e othe r go

free . ] n

that \v h c n th c re Sancho Panza ap pli e s a Quix oti c pri nciple justice on both sides of a case, the j udge shoul d favor 111crc y

-

the fello\\7 go. Jones Jo seph R. ,

the end , i s cqual and l ets

I

READING

"T he Liar Paradox in Don Quixo te II, 5 1 . "

HLc.;pallic

Review 54 ( 1 986) : 1 83- 1 93 .

SARTRE 'S PARADOX. Construed by J ean-Paul S artre and often call ed th e C on fl i c t o f D u t y P a r a do x t h i s p r o b l e m i n v ol v e s confl i ct i n g -

,

-

obli gati ons. Compare PLATO ' S PARADOX .

Formulation. One of S artre 's pupils wan ted to j oi n the Free French Forc e s

to ave n g e his brother, who vVas killed by th e GeIman army of o ccupati o n B ut he also wante d to stay b y hi s m othe r, \vho \vas deeply wounded by the loss of her el d e s t son. W hat should he do? Explanation. S artre s student has t\V O o p ti()ns � both arc cOln pc l h ng, b u t neith er absolutely binding. Th ey are i ncon1 p ati bl c , ho\vevcr, i n that th e student c ann ot do both and thi s i s refl ected i n the fonn al i zati on or th c tn in deontic l o gi c . Resol ution. Azizah al -Hi bri sugges ts that \v hi l e th e very sin1ilar Plato ' s Paradox may be re sol v ed by re s orti n g to e thi cal hi erarch i es , the present paradox resists thi s kind of sol u ti on becaus e neithe r option see m s to hav e prece d en ce over the o ther, t here by maki n g t h e configuration of the atte nd an t circumstances crucial to the arg u me nt . Thu s , acc ordi n g to al­ Hibri , the p arad ox cannot be us ed to und e rm i ne the principle of s tandard d eon ti c logic \vhi ch asserts that obli gati ons shou l d not confli ct. T his i s ' becau se the global conte x t does not dec i d e the issue so that the student s duty \vould not be e xpre s sed as confl icti ng obl i gati ons but as a di sju nctive obli g ati on : the s tudent should either j oi n the fi ght or stay by his moth er. T hus the e rst\vhile moral quan dary is re d u c ed to the student ' s perso na l choice. A I-Hi bri 's contention, from a l og i cal p oint of view, is th at a dec i s ion on whether to allow con f1i cts of duty i s prior to the pre s ent paradox si nce the latter do es not force the issue .

i n o rd er

.

'

,

READINGS Al-'Hibri, Azizah. Deontic Logic. \Vas hington: UP of America. 1978. 25 ,

69-72. Forrester, James W.

Why

You Shou ld, The Pragma t ics of Deontic Speech .

Hanov er, NH: Brown UP, 1 989. 1 5 - 17.

i

'

Dictionary afParadox

1 80

SCHRoEDINGER'S CAT. D e scrib e d by Er\vin S c h rL1e d i n ger i n ] 93 5, thl_ s

v e ri dic al

paradox concerns th e d i ffi c u l t y of u n d e rs tan di n g the ' wave

function ' col l aps e and the rol e of the re gis tra t i on or lTI c as u ri n g system or

of a con s ci o u s observer.

I t i s related to the EINSTEIN-PODOLSKy -RoSE0J

PARADOX and the POTENTIAL ENERGY B ARRIER PARADOX .

"A cat i s p en ned up in a steel cham ber, al on g \vith the

Formulation.

fo l l o w i n g d i a b o l i c a l d e v i c e ( \v h i c h m u s t be s e cu red ag a i n s t direct interference by the cat) : i n a Ge ig e r counter there is a tiny bit of radi oactive

sub s tanc e so ,

s m al 1

,

that perhaps i n the course of one hour o ne of the

atom s d e cay s but al so \vi th equal probability, p erh ap s none � if i t happe ns ,

,

,

the cou n ter tube di scharges a nd throug h a re l ay shat ters a snl al l fl ask of

hydrocyanic ac id

.

I f one has l eft thi s e nti re sys tenl to itsel f for one h o u r

,

one \vould s ay that the cat s ti l l l i v es �f mean\v h i l e no atom has decayed.

The fi rst atomi c decay \voul d have poi son ed i t. The [ \v av e- l functi on of the enti re systenl ,,,"oul d express thi s by hav i n g i n i t the l i v i n g and th e dead cat (pardon the e x p re s s i on) mixed or smeared out i n e q ual parts

"

( S chroedi nger 328) .

E x pl an at i on The \vave function d e s c ri p ti on of qu an tum lnechani cs does .

not al l 0\\' for p reci s e specification or al l variables simul taneously ( see

the EIN STEIN-PODOLSKy-RoSEN PARADOX) . Accordi ng to Schroedi nger, thi s ci rC u ln s t anc e ref1ects either a limitation on knO\v] edge or an indetenni nate

b l urri n g of th e v ari ables i n

re al

i ty

.

S chrbedinger arg ues that quantun1

mechani cs i s i n com pati bl e vvi th the first of the se o pti on s bu t that hi s Cat ,

Paradox sh o�ls that it i s "ri di cul ous " to accept the second .

T hu s the ,

coun teri ntuiti ve con sequences of quantunl me c h ani cs are not l i mi ted to systems i nv ol v' ing only very smal l parti c le s Resolution$ A s

\v i th

.

the EPR Paradox , an i nstrum ental i st attitude tovvard

qu antum mechanical 'N av e functi on s i s c ap able of

re s o l v in g

the paradox .

The i nstrumenta1 ist app roac h i s to res i st the te m p t ati o n to in t e r p re t \\lave

functi ons real i sti cal ly by i nsi sti ng that they are merely devi ces al l o \\ i n g l

the calcul ation of probabi l i ti es of certai n obs erv ati o ns

.

S c h roe d i n g e r

makes th i s su gge sti o n hi m sel f: "Reality resi sts ilni tation th ro u g h a m odel . So one l et s go of nai ve real i s m and leans di rectly on the i ndu bi tabl e

proposi ti on that is

onl

actually (for the phys i ci s t)

after al l i s sai d an d done th ere

y o b s e r v ati o n , ln ea s u re ln e n t " ( 3 2 8 )

.

N e v e rth e l e s s , s u c h

instrumental ism remai ns controversi al , and vari ous

"

h i d d e n variabl e"

theori e s have been devel oped to resto re a real i s t model of q uan tu m i nterac t i ons ( s ee the POT ENT I AL ENERGY B ARRI ER PARADOX ) .

So far,

ho \v ev e r physici sts have found no di rect evi dence of hidden variabl es . ,

1 81

Dictionar.-v a/Paradox

REA.DINGS

lY Es pa g nat B . Conceptual Foundations of Quantz.an A1echanics. 2nd ed. ,

Rcading� �'1A:

I3enj amin,

I-Ioltzman , J ack Iv1.

1 976. eh. 8 .

"A. Note on SchrOedinger 's Cat and

I-Ianging Paradox , "

the lJnexpectcd

B ritish lourna l for the Phi losophy of Sc ience

39( 1 988) : 397-40 1 . Loesser, 1. G. 'i'hrce Perspectives on Schoedinger 's Cat. " Anlen:can Journal of Physics 52. 1 2 ( 1 984) : 1 089- 1 093 . Primas , H . C-'h emistry, Quantum A1echanics, and Reductionisnt. 2nd. ed. Berlin: S pri nger, 1 983 . Schrbedinger, Envin. ' The Present Situation in QuantlU11 �1echanics: A Translation of Schr6edinger 's ' Cat Paradox' Paper. " Trans, John 'rrimmer. Plvceedings o! lhe Arnencan Philosophical Society 124 ( 1980): 323-338.

SEEKER PARAD OX,) THEs Thi s i s a parad ox i n th e fi ction or Pa tri c k

Whi te� \v i nner of the 1 973 Nobel Pri 7.c i n Li terat urc, \v h i c h i s an al yzed by Carol yn Bliss.

Formulation. " One of Whi te's m os t baffl i n g an d i nl portan t paradoxes l i s] that th e sel f n1ust be sou ght and fou nd onl y to be rel inqui shed , that the indivi d ua l i ty s o p o\v e r f u l l y exprc s sed by h i s Inaj or c haracte rs paradoxically enab les them to seek a s tate of understandi ng i n vv hich

sel fhood i s fi nal l y s ubsumed. I n

a

further permutati on of the paradox , the

surrender of the s e lf \v hi c h th i s unders tanding demands s Olne ho vv functions to permit the the character 's ful l es t realizati on of t he essenti al s elf; t h at i s , he beconles most himself \vhen he leas t seeks to be . I n term s of the

I

I. ,

related Chri stian paradox, he finds h i s life by l osing it, or as Em erson put it, 'The man \vho re nou n ce s himsel f comes to himsel f ' " ( B l i s s 8) . Explanation. Whi te has been g reat 1 y i nfl uen ced by the Judeo-Christi an concept of th e rel i gi o u s quest and i ts i nevi tabl e fa i l u re due to man 's i nabi l i ty to com prehend the i nfi ni te In particular, m an s search for an understanding of his own nature canno t be s ucces sful by an i nves ti gati on of himself. Resolution True understanding of h um an nature cannot be obtai ned 1'ro111 self-exami nation b e cau s e it entai l s nl an s relati on to th e A bsol u te. This observ ation doe s not, perhaps, resol ve t he parad ox ; rather, the paradox ,

'

.

..

'

re veal s the i neffable nature of the Judeo- Chri stian b e l i ef that " the center

of g rav i ty of exi s te nc e i s outside e xi s te nce

"

( A di n

S tci nsal tJ'. , ci led i n

B l i ss � 206) . REA I)I NG

Pa tr ic k Whi te s Fiction : The Paradox Failure. New York: St. Martin ' s, 1 986.

B liss , Carolyn.

qj' the Fortunate

I

,

1 82

Dictionary of Paradox

SELF-AMENDMENT, THE PARADOX OF. This is a Legal Paradox,

first adduced by Alf Ross. Formul ation. Suppose that we have a constitution that specifies a method for making amendments . Should \ve try to change the method of amendment to one that is incompatible with the original method, we find that we are involved in a contradiction. Hence, the original method of lnaking amendments i s immutable. Explanation. Any true change in the method of making amendments will result in a meth od inconsistent with the original method "'in the sense that each permits something that the other forbids and forbids something that the other permits" (Suber 39) . Further, if the new method is to be legally valid, i t must be the conclusion of a valid deductive argument that models the act of self-amendment. Since one of the premises of this argument must be the original method of making amendments and since the hvo methods are by hypothesis inconsistent, the conclusion (the new method) and the premise (the original method) are contradictory� therefore, the argument is invalid. Hence, it is impossible to make valid changes i n the original method of making amendlnents (except, of course, for merely cosmetic changes). This purported immutability, as Suber observes, makes the present paradox resemble that of divine OMNIPOT ENCE. Resolution. One response to the paradox is to observe that the original method and the new method do not hold at the same time and, hence, a temporal logic might validate the modeling argument. Suber, following Ross, rejects this solution because it is the ori ginal method that must validate the nevv method yet the t\\'O methods , considered abstractly without regard to time , are contradictory. Since the whole point of appeali ng to a tClnporal logi c, hovvever, was to eliminate the logical contradiction i n the model ing argument, S uber 's objection does not seem pertinent. Another response to the paradox would be to deny that deducti ve inference is an adequate model for legal amendment. To do so would be to countenance the i nconsi stency of s el f-amendment which would necessitate a compelling explanation. Suber, however, argues that it \vould in fact shift the foundations of the legal system to one based on authority. �

S uber, Peter.

READING The Paradox of Se lf-Amendment:

A

Study of Logic, La1rv

Omnipotence, and Change. New York: Lang, 1990.

Dictionary ofParadox

1 83

SELF-DECEPTION, THE PARADOX O F. "A man can love a parado x

"vithout either losing hi s \vit or his hones ty (Ralph Waldo Em erson Walter Savage Landor XI I ) . This ps y c h o logi c a l parado x aro s e from the difficul ties of the concept of self-decepti on. Formulation . De c epti on is c onv in ci n g s ome on e of so m eth i n g one knows to be false, but if one knows something to be false, hovv can one deceive oneself abou t it? Explanation . 'There are two clos el y related probl em s of self-decepti on, both g enerated by ap pl y i n g th e l og ic of i nterpe rsonal dece pti on to cases i n which one dec ei v es oneself. In order for someone to dec ei v e another person into believing some propos i ti on P, i t is necessary for the deceiver to bel iev e that P is false and to i n te nd to make the other person believe that P is true. One contradiction seems to be that in o rder to deceive m yself I m us t intend to fool myself into bel i e ving that P is true w he n I already bel ieve that P is false. The other apparent contradiction is that after I have deceived m ysel f I must bel i eve both that P is true and that P is raise. Resolution. M. R. Haight arg u e s tha t th e p aradox reveal s that se lf­ deceptio n is impossible s inc e i t \vo u1 d req ui re not on l y that the sel f­ deceiver si m u l taneously hol ds contradictory b el i e fs but al s o that he both knows and docs not know the s ame proposition. Roy A . Soren s en argues that s el f decepti on is possible because i t occurs over a p e ri od of time, not instantaneously, and thus the perpetrator of the s el f d e cepti on need not hol d con tradi c tory bel iefs sim ul tan eo us l y Raphael Demos claims th at there are two lev el s of aw are ness one bei ng j u s t a si m pl e awareness and the other an awareness \-vith attenti on Thus , one can s i mu l tan eous l y entertain contradictory beliefs by having one 's attention di stracted from c ertain aspects of the questi on entertain ed. D emos s app roach is typi c al of those responses that argue that s elf dece p tion is possible because o n e or another ty pe of m ental frag me ntation al l ows i n trap e r s onal d ece pti on to parallel i n te rpe rs onal d ec ep tion . Still another respons e is to argue that s e lf de ception does not paral lel i n terp ersonal decepti on at all, but rather operates in some other m an ner H. Fi ng are tte , for example, clai m s that the parad ox arises from an undue em ph asi s on the co g n i ti v e aspects o f our self-deceptive activities. A n analysi s of the i nte r ac ti on of vol i ti on an d action would reveal, according to Fi n g are tte that the basi c feature of the refus al to i den ti fy oneself as a certai n s elf dece pti on is disavo\val kind of pers on or wi th a certai n proj ec t ,

"

-

-

.

,

.

'

-

-

.

,

-

.

REA.DINGS Canfield� John, and Patrick �;[cNal1y.

Analysis 2 1 .6 ( 1 96 1 ) : 1 40- 144.

"Paradoxes of Sclf-l )cccption. "

1 84

Dicfionary afParadox

Charnplin, T. S . "Self-Deception: l\ Reflexiv e Dilemlna. " Phi losop hy 52

( 1 977) . 28 ]

-

299.

Demos , Raphael .

588-595.

Fingarette , H. Foss , Je ffrey.

" Lying to ()neself. "

Self-J)eception.

Jo urn a l of Philosoph}'

57 ( 1 960) :

London: Routledge, 1 969.

"Rethinking Self- Dece ption. "

237-243. A S'tudy o/Se(lDeceplion.

l\meri can Philosophical

Quarterly 1 7 ( 1 980) . Haight, �t R.

�e\v Jersey: IIumaniti es P, 1 980.

Martin , 1v1. , ed. S elf-f)eception and Se lf- Understanding. Lavvrence: IT of

Kansas P,

1 985.

IvIel c , Alfred . "Sel f- I)eception.

Phi l o sophi c a l Quarferly 33 ( 1 983) : 365-

,>

377. - - . " Incontinent Believ ing. "

Phi losophical Quarterly 36 ( 1 986) :

2 12-

222. - - . "Recent "Vorle on Self-Deception. " Arnerican Philosophical Quarterly

24 ( 1987) : 1 - 1 7. S orensen, Roy A .

"Self-I)eception and Scattered Ev ent s . "

A1ind

ns 94

( 1 985) : 64-69. - - . "1)ogmatici sn1 , J unk Kno\vledge , and Conditi onals . Phi losophica l

Quarterly 38 ( 1 988): 433-454.

SELF ... REFERENCE AND CIRCUL ARITY, PARADOXES OF. T"vo main c ycl e s of paradox, those related to the LIAR and to RUSSELL' S PA RADOX

,

in\Tolvc probl ems of self-reference and ci rcul ari ty. A mong entries that rel ate to thi s feature of di scourse and thou g h t arc : the BARB ER PARA.DOX, the PARADOX OF C OGNIT IVE RELATIVIS M, the CROCO DILE' S DILE�1r.1A , the EUAlTHUS , EUBILIDES ' s PARADOXES , the EPIl'AENIDES , GEACH ' S PARADOX , the HETEROLOGICAL PARADOX , I N SOLU B I LIA , the PARADOX OF THE KNOWER , MENO ' s PARADOX , the POSSIBLE LIAR, P!0\GMATIC PARADOXES , the PROBAB LE

LIAR, PRIOR ' S FAIvlILY OF PAI0\.DOXES , RUSSELL ' s P ARA D OX IN MANY V ALUED

LoGI C , the SANCHO PANZA , the PA RADOX OF SELl�-AMENDr\:IENT , the PARADOX OF SELF-REFERENCE IN SCEPTICIS tv1, AND ZALCivfAN ' S PARADOX.

SELF- REFERENCE IN SCEPTICISM, THE PARADOX OFo Thi s i s a cl assi cal versi on of the paradox of rel ati v i sm .

Formulation. "Just as , for exan1pl e, fi re after consumi ng the fuel destroys al so itself, and l i ke as purgati ves after dri nki ng the fl uids out of the bodi es expel them s el ves as vvel l , so too the argument agai nst proof, after abol ishi ng every proof, can cancel i tsel f al so . And again, j ust as it i s not impos si ble for the man \vho has ascended to a hi gh place on a ladder to

Z)ic'lionar.v of Paradox

1 85

o'verturn the ladder \v i th hi s foot after hi s ascent, so al so i t is not u n l ike l y

that the Scepti c after he has arri ved at the den10nstrati on of hi s thesi s by

means

of

a rg u m ent proving the non-exi stence of p roo f as it \vere by ,

a

step- l adder, shoul d then abol i sh thi s yery argulnent" ( Scxtus Empi ri c lls,

Against the Logicians , 480-48 1 ) . E xplanation .

I f the s ce p t i c deni e s that p h i l oso p h i c al knovvl edge i s

poss i bl e then hi s o\v n vic\vs must not cou n t as philosophical knovvledge ,

e i ther.

Resolution . Avner Cohen argues that the nobl e skepti c lnust, for the sake of consi stency, abandon phi l os op hy alto ge ther but fi nd s that s ce p t i c s such ,

as S extus Empiricus and Lud\\'i g Wi t tge ns t e i n are of t\vo minds about

q u i tti n g p hi l os o ph y

.

Once one ac ce pts the p o s i ti on that phi l osophi cal

kno\vlcdge i s unattai nable, ho \ e ve r pe rhap s cons i s te n c y ceases to be an v

,

ovcniding concern ! REA. I)ING

(--:ohen � Av ner. "Sextus Enlpiricus : Sceptici sDl Forunl 1 5

( 1 984) : 405-424.

as

Therapy. " Ph i losophica l

SE MANT I C A L PA R A D O X E S . T h e s e parad o x e s i n v ol v e s ti c h se nl an t i cal concepts as nl can i n g and tru th , and i ncl ude the PARADOX OF ANALYSIS and DAV IDSON ' S PARADOX OF

I RRATIONALITY .

For a techni cal

di sti nction \\li th logical parado x e s see LOGICAL PARADOXES . ,

SENIOR SNEAK

WEEK..

Thi s i s a version of the PREDICTION PA RADOX .

SEN 'S PARADOX. A d duced by Amartya K. Sen, thi s politi cal p arad ox pu rports to s how that the " Pare ti an l i be ral " concept of s o c i e t y i s i nconsi stent.

Formulation. A n y soci ety that i nco rp orate s the Pareto Princi pl e and a

mini nlal d e g ree of per s o nal freedom cannot have a reasonablc deci sion

p roce du re .

Explanation . The Paretian l i beral vie\v of soc ie ty affirms that any soc i ety will be tolerable onl y in so far as

it e mploye s rea son abl e ( rati o n al ) decisi on

p ro ce dure s and allo \v s i ts members a l arg e degree of personal frccdo111 .

T he requirement that de c i s i on procedures be rational i nl p l i cs that the so c i e ty have a means of consistently ran kin g

pol icy opti ons. I t al so i lnplies

the Pareto Principle: vvhatever i s prefe rred by al l the lTIcmbcrs of the soc ie ty

,

1 86

Dictionary of Paradox

should be selected by the society 's decision procedure. The req uirem en t of person al freedom implies that some options are completely detennined by the i ndi vidual s pre fere nce As a minimal condition, the society should respect A's choice betw ee n the pair (x ,y) and B 's choice between the pair (t,u) . Jonathan Barnes gives an example of a society (club) in which the onl y two members are A and B . A's preferences are ranked in the fo l low ing decreas i ng order: '

.

t, y, x, U,

whereas B 's ran ki ng i s x,

Thus , the

u, t, y.

society must make the foll owing ranki n g s :

( 1)

Y i s preferred to

(2)

x

(3 )

( 4)

x

is preferred to u II i s preferred to t t is preferred to y

deci si veness

of A Pareto Princi pIe deci si veness of B Pareto Princi pIe.

Hence, the society cannot establish a linear order among the options and thus its decision procedure is not reasonable. The fore goi ng i s , of course, only an exampl e, but i t displays the spiri t of Sen 's proof. Resolution. The model assumes that all the choices must be ranked. Should A and B in th e above example abstain from m aki n g some pairwise rankings, the cycle might be broken. Ultimately, however, thi s is perhaps a minor matter (al though Barnes gi ves i t as the solution to the paradox) . The decision proc edure i s no t rational because it cannot handle all the possible configurations of pe rs o nal p re fere nc es I n any parti c ul ar society, it may be that the trou bl es ome c onfi g uration s j ust never ari se . Thus Sen observes .

,

The ultimate guarantee for indiv idual liberty may rest not on rules for social choice but on developing indiv idual v al ues that respect each other 's personal choices. The conflict posed here is concerned with societies �'here such a condition does not hold and where pain.vise choice based on liberal v alues may conflict with those based on the Pareto Principle. ( 1 551 56)

Indeed, the delineation of the spheres of individual l i berty and collective responsibility has been and continues to be a maj or concern of politi cal theory. Nevertheless , other responses to the present paradox inc l ude

Dictionary of Paradox

1 87

attempts to invalidate one or other of the premises, for example, by only attributing rights to individuals conditionally. READINGS Barnes, Jonathan. "Freedom, Rationality, and Paradox. " Canadian Journal of Philosophy 1 0 ( 1 980) : 545-565. Kruger, Lorenz, and Wulf Gaertner. "Alternativ e Libertarian Claims and Sen's Paradox. " Theory and Decision 1 5 ( 1 983): 2 1 1 -229. Sen, Amartya K. "The Impossibility of a Paretian Liberal . " Journal oj Po litical Economy 78 ( 1 970) : 152- 1 57. - - . Co llective Choice and Social Welfare. San Francisco: Holden Day, 1 970. Ch. 7. "Liberty, Unanimity and Rights.H Economica 43 ( 1 976) : 2 17-245. -

_

.

SET THEORY, PARADOXES OF. Various LoGICAL PARADOXES have

emerged involving Set Theory, of which RUSSELL ' S PARADOX is perhaps the paradigmatical example. Others incl udc B ARRY ' S PARADOX, the B URALI­ FORTI PARADOX , CANTO R ' S PARADOX , the C U RRY PARADOX , DUMITRIU ' S A NTINOMY OF THE THEORY or TYPES, the HETEROLOGICAL PARADOX , the PARADOX OF THE NON - COMMUNICATOR, RICHARD ' s PARADOX , the SKOLEM PARADOX , and the ZERMELO-KONIG PARADOX. ,

SHADOW PARADOX, THE .

Daniel Cohen attri butes this didactic paradox to J. Tienson. It is what mi ght be called a "toy" paradox. Formulation. There are four principles in which we mi ght summarize what we know about s hadows: ( 1 ) S hadows are dark spots on the surfaces of opaque objects caused by the interruption of a beam of light. (2) All shadows are the shadows of something. (3 ) A shadow cannot be cast through an opaque object. (4) An object must be illuminated to cast a shadow.

Now consider the following case . There is a dark spot on a wall caused by the i nterruption of a beam of light hence, by the first pri nciple, there is a shadow on the wall. Between the shadow and the l i g ht source are two disks either of which by itself would cast the exact shadow on the wall . Yet the disk closer to the li ght source cannot be that of which the shadow is the shadow because, if it were, the shadow on the wall would have to be cast through the second disk� but this is di sallowed by the thi rd principle. Moreover, the disk closer to the wall cannot be that of which the shadow is the shadow because it is not i l l uminatcd, as is required by

I ,

I

,

Dictionar.v of Paradox

1 88

the fourth pri nciple. S o "v h at i s the sOlnething of \v hi ch the s hado \\' i s the shado\v, as demanded by the second pri nci ple? Explanation. The pa radox assumes that the li ght source has no di lnensi ons

i n order to lul e out the q ues ti on o f a penulnbra. The probl eln is that, on the pri ncipl es enulnerated , neither di sk can cast the shadow i n question.

The fol l o vv i ng diagram , adap ted from Cohen, expli cates the paradox:

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Resolution . Si nce Cohen cites the present paradox as an i nstance of how

paradoxes lnay be used to sti mulate th ou g ht the reader i s left to w ork out ,

hi s o\vn response . READING Cohen, Daniel . " Putting Paradoxes into Pedagogical Use in Philosophy "

Teaching Philosophy 8 ( 1 985) : 309-3 17.

SHIP OF THESEUS, THE . T h i s p arado x concerns the criteri a for the id e nti t v of arti facts . ..I

the sole ancient source for thi s paradox (IJ�fe oj Di llon, 4 1 1 ) : "The ship [the l/zeoris] on w hi ch

Formulation. Plutarch i s

l'heseus �

taken from

Theseus sai l ed \vi th the youths and returned safel y, the thirty-oared galley, v·/as prese r e d by the A thenians do\vn to the ti me of Demetrius of Ph ale ron v

[regent of A thens ,

3 17-307 B . C . ] .

T hey took a\v ay the old ti m bers from

time to ti lne, and put ne\v and sound ones

in

their places , so that the

ves sel became a standin g il l ustration for the philosophers i n the much -

Dictionary oj Paradox d e bated A u gm entat i on A rgun1ent ' '

[auxolnenos logos],

that it rC ln ai n e d the s aIn e others that i t \vas ,

Explanation. There i s a s h i p

1 89

some decl ari ng

n ot the sam e vessel . "

A that i s enti rely overhauled by g rad uall y

repl acing all of i t s old parts with new ones . Call the sh i p at the end of thi s

process s hi p B . At the san1e ti m e the old parts hav e been scavenged and ,

u sed to bui l d another ship,

C,

\vi th the same confi guration as A .

q u e s t i o n i s \vhic h of the later s hi p s B or ,

The

C, is i d en t i c al \vith ship A ? B o th

B and C h av e good claims to bei n g i denti cal \vi th A . On the one han d

,

there i s a s patio-temporal co n t in u i t y of A and B . The t rans it ion from A to

B i s achieved in a series of s ubsti tutions of one ol d part for one new part, rand there i s no reason to say th at A c ease s to be A \vhen a si ngl e part i s replaced. On the ot he r hand , shi p C i s n1aterial ly and formally identical

\vi th ship A . If one left shi p A and came back to fi n d ship C , there \vould

be n o question that C \vas the same as A , and even if one \vere told (\vithout ln e n tio n i ng

s hi p B ) that A h ad been taken apart and reassembled as C ,

one \vo u i d sti l l say that they \vere the same shi p.

Resolution . There are various argulncnts for both B and C as i dentical

\vi th A , \vith the i n tu i ti ons of v ari ous phi l osop hers di ffering. Perhaps the Wi ttgenstei ni an l i n e that the son1e p h i l o s o p h i c a l questi ons hav e the status

of a ci viI di spute i s ri ght: \v h at we should say in these cases d e pe nds upon \v hat i s fi nal l y at s take i n the d is p u te condi ti ons abs tractl v consi dered . 01

CATS .

,

because there are no identi ty

See GEACH' s PARADOX OF THE 1 ,00 1

REA.l)INC}S

Davies , Laurence .

"

S tnart on C:onditions of Identity. "

,

Analysis 33 .3 ( 1973) :

1 09- 1 1 0 . Dauer, Francis .

" Ho\v � ot to Rei dentify the Parthenon. "

Analysis 33 . 2

( 1 973) : 63-64. IJi llon, John. The A·1iddle Platonists: 80 B . C. to A . D. 220. Ithaca: Cornell LTP, 1 977. 4 1 1 -� 1 2. Garrett, B . J. " Noonan, ' B est Ca ndidate T heories and the Shi p of T heseus. " i\nalvsis 45.4 ( 1 985) : 2 1 2-224. Heller, Iv'1ark. "T he B est C:andidatc A.pproach to Diachronic Identity. " Australasian Journal of Philosophy 65 ( 1 987) : 434-45 1 . Noonan, Harold. "\Viggins, /\rtifact Identity and " Best Candidate ' Theories. " Analysis 45. 1 ( 1 987): 4-8. Scaltsas� T heodore . "T he· S hi p of T heseus . " Analysis 40. 3 ( 1 980) : 1 52] 57. Snlart , B ri an '�Ho\v to Reidenti fy the Ship o f Theseus. " A na ly.\'is 32.5 ( 1 972) : 145- 148 . - - . "The S hip of Theseus, the Parthenon, and L)i sassclnblcd (Jbjects. " Analysis 34. 1 ( 1 973 ) : 24-27 . '

�

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I! I

1 90

Dictionary of Paradox

SIMPSON ' S PARADOX. This paradox i n probability theory has been

used to undermine the probabilistic theory of causality. P(A I B ) peA ) ; P(A I B ) Formulation. �'The fact is thi s: any association > P(A) ; P(A I B ) < peA) behveen t�'o variables which holds in a given population can be reversed in the subpopulations by finding a thi rd variable which is correlated with both" (Cart""Ti ght, 422) . E x p l a nati o n . Let a g i v e n popul a ti on be parti ti oned into t\\lO sUbpopulations. In each of these subpopulations A and B might be hi ghly correlated and yet not be correlated on the entire population. Cartwright gives the fol lowing example. It may be that smoking and heart disease are not correlated very highly in a given society because a large number of smokers exercise regularly, thereby offsetting some of the harm done by smoking. Nevertheless, the society can be split into t\\'o distinct groups: those 'Alho exercise and those who do not. Those \-v ho exercise and smoke may have a significantly higher rate of heart disease than those who exercise but forgo smoking. Similarly, heart disease may be higher among those who do not exercise and smoke than those \-v ho do not exercise but abstain from smoking. Hence , smoking and heart disease are highl y correlated in both subpopulations, but not in the entire popUlation. There is also a connection with causality: if we are to give a probabilistic account of causality, the probability of the effect, given the cause, should be higher than the probability of the effect in the absence of the cause. The present paradox, however, suggests that this expected increase of the probabili ty of the cause in the presence of the effect does not always occur. Resolution . Observe that Simpson's Paradox is not in itself contradictory (though it may become so i n conjunction with certain kinds of rules for combining probabilities) . Hence, the paradoxicality of the result is i ts unexpectedness. Once a clear example of how the paradox arises is understood, however, it seems to lose its bi te. Nevertheless, it remains a problem for those who hold a probabili stic theory of causality. T hese latter argue that the paradox ari ses when the population is partitioned according to a variable that has a causal relation with the effect being studied. For example, in the situation descri bed above exercising was used to partition the popUlation, but it al so has a causal relation to heart disease, the effect in question. This fact is then used to devise appropriate restrictions on the definition of causality. =

READINGS Cartwright, Nancy. "Causal Laws and Effective Strategies. " Nous 1 3 ( 1979): 4 1 9-437. Eells, Ellery "Cart\vright and OUe on S impson's Paradox. " Philosophy oj Science 54 ( 1987) : 233-243 .

1 91

Dictionary ofParadox

Otte, Richard. "Probabilistic Causality and Simpson's Paradox. " Philosophy 0/ Science 52 ( 1 985) : 1 1 0- 1 25.

S uppes, P. A Probabilistic Theory a/Causality . .Amsterdam: North Holland,

1 970.

SINCERITY, PARADOX OF. Pe o pl e such as salesmen, \v hos e j ob s ,

d e pe nd on con vinc i n g others

of thei r sinceri ty, face th e fol l o w i ng practi cal p aradox : by de libe rately ac ti ng in such a way as to appear sincere, sincerity i tself is lost. In so far a s i t i s th e agent s intentional actions that are sel f­ defeati ng, thi s paradox i s si milar to that of the BODHISATTVA. Observe that the paradox ari ses be ca u s e the salesman cannot afford to j us t be s ince re and hope that his prospecti v e cli ents "vill perceive h is sinceri ty. I n orde r to maxi mize hi s c han c es to close the sale, he must rather take steps to appear sincere. As s tated above, thi s results in a l ac k of s i n ce rity which may be perceived by the client. I n any case, most clients are aw are of the [act that the salesman must "seJ l hi mself" and arc thus re ti ce n t abo u t cre d i ti ng the salesman as sincere. For another, silnilar p rob l e m in the art of s al e s m an s h i p see the PaRADox OF TRUST . '

,

READING

Oake s , Guy. "The Sales Process and the Paradoxes of Trust. " Journa l of Business Ethics 9( 1 990) : 67 1 -679. ,

SKOLEM PARADOX, THE. T his purported paradox arises from certain

results in m odel theory when th ey are appl i ed to set theory. Formulati on. The Low enheim - S kol em theo re m states that, if an enumerab l e set of fi rst order formul as h as a model , it has an enumerable The theorem can be applied to set theory even though certai n set theoretic theorems assert the existence of non enum erabl l y many sets. Consequently, set theoretic notions like c ard i n al i ty are rel ative. Explanation. A n en u m e rab l e set i s one that i s fi ni te or that can be put into a on e one correspondence \vith the natural numbers. The LOwenheim­ Skolem theorem shows that if a ce rtai n set of fi rs t order formulas has a m o de l then it has a m od e 1 with an enumerable domain. Set th e ory, however, can be formulated as a first order theory using an enum e rabl e number of axioms. Hence, set theory (since i t has non-enumerable models) has an enumerable model despi te the fact that the theory affinns that there are a non-enumerable number of sets. This result i s ca1 1 ed Skolem 's Paradox, a l t h o ug h Skol e m him self d i d n o t bel ieve i t p a ra d o x ic a l .

m od el

.

-

-

,

I, I

I

,,

1 92

Dictionary

of Paradox

Nevertheless , he did believe that i t entail ed certain notevvorthy resul ts such as the relativity of concepts like cardinality. According to this resul t, the cardinality of a set is not an inherent property of the set, but rather depends on other parameters. Resolution. Skolem 's Paradox is onl y paradoxical in the sense that it is a surpri sing result. There is no contradiction invol ved. The relativity of set theoretic notions , often called the "Skol ernite position ," is lTIOre controversial. On the one hand, concepts such as cardinality (in particular, non-enumerability) may be thought of as having an objective content; in that case, no enumerable set of axioms can adequately characterize this content. On the other hand, the concepts themselves may be thought of as bei ng relative to the axiom systelns in \'/hich they are defined. The latter vie\v seems more consistent \vi th the modern view of set theory as a fonnal system , suscepti ble of various interpretations. RE.ADINGS I'vlcl ntosht Clifton. ··Skolem's Criticisms of Set Theory " Nous 1 3 ( 1 979) :

3 13-334. 1vloore, 1\ . -V-V. " Set Theo ry, Skolem' s Paradox and the Traclatus. " Analysis 45. 1 ( 1 985): 1 3 -20. Resnick, IVI . D. " On Skolem's Paradox. " Journa l ofPhilosophy 63 ( 1966): 425-437. In FroJn Skolem, T horalf. "Some Re·mark ' s on Axionlized Set Theory. Frege to Gode l: A Source Book in A1atheJrtatical Logic. Ed. Jean v an Heijenoort. Cambridge: I-Iarv ard lJPt 1 967. 290-30 1 . Thomas , \Villiam J. "Platonism and the Skolem Paradox. " Analysis 28.6 ( 1 968) : 1 93- 1 96. - - . ' "C) n Behalf of the Skolemitc. " Anal\'sis 3 1 . 6 ( 1 97 1 ) : 1 77- 186. "

-'

SLIPPERY SLOPE PARADOX, THE.

A lso called Chishohn 's Paradox, this paradox of n10dal logic was adduced by Roderick C hisholn1. The term "slippery slope" i s often appl ied to any SORITEs-like argument in \vhich, once the first step is granted, a series of similar steps must also be granted because they rely on the same principle. Formulation. Consider, say, the Elnpire State Building. It is composed of a nUlnber of parts , girders, stones, pieces of glass, and so forth . Presumably it could have been made of a slightly different set of parts and have still been the Empire State Building ; imagine, for instance, that a stone had been cracked before it \vas pl aced and \vas replaced by another. Suppose that saying that the Empire State B uilding could have been cOlll posed of a slightly di fferent set of parts nleans that in S0111e other pos si b l e \\lorl d the Empire State Building is Inade of slightly different

IJictionary of Paradox

1 93

parts , di fferin g by one stone only We can no\v i m agi ne another possible .

world �'hich differs from thi s first possibl e world j us t by havi n g another stone different i n the constructi on of the Empi re S tate B uilding . I ndeed, there i s a sequence of possibl e \vorids in \vhich the Empire S tate B uildi ng differs i n each one by j us t part from the one before ; at the end of the sequence the Empire State B ui lding i s made of none of the parts of the actual one ! Explanation. In the fonnulation given above, the modal nature of the

parad ox i s onl y reflecte d i n the use of po s s i ble \v o rl d s e m anti c s . Chi sho l m ' 8 argun1ent, ho\vever, i s di recte d agai n s t the unre s tri cted acceptance of the following axi om of (S4) modal l ogic: i f a proposition i s neces saril y true, then i t i s necessary that i t i s necessari l y true. The resul t is paradoxical because i t i s usual l y assumed that i f t�'o physical obj ects are made of different parts , then they are not the same physical obj ect.

Yet Chishohn 's paradox seelns to force us to the conclusion

that thc i denti ty

of a physical obj ect does not depend upon i ts parts. Resolution. One vvay of responding to the parauox i s to i nsi s t that the

identity of a physi cal

object

i s dc tcrnl i neu by i t s parts, t hereby d c n y i n g

that the 1\vo �'Empire State B u i l d i ngs, " only di fferi ng by a s tone, Llrc t h e same .

A

s e c ond

fi rst Enl pi rc i s identical to

approach \vo uld be to cl ai ln th at \vh i l c the

State B ui l di ng i s i dentical to the second and that the second

the thi r d the first i s neverthel ess different from the thi rd . I n other vv o rd s ,

,

one could deny the transi ti vi ty of i dentity across possible worl ds . A s in most Sorites, ho\vever, sati sfying manner.

it i s

di ffi cul t to j ustify thi s step in an i ntui tively

Still another approach vvould be to eschc\v talk of

trans-worl d i dentity altogether. The puzzle, hovvever, can be reformul ated in such a m anner as to avoi d the use of poss ible vv orld semantics . I n fact, the paradox has occasioned son1e debate over \Nhether there i s a ten1poral analogue \�/here changes occur to a physical obj ect over ti n1e i nstead of across \vorlds . REJ-\ DINGS Chisholm, R. "Identity Through Possible. \Vorlds: Sonle Questions.

n

Nous

1 ( 1 %7): 1 -8. Lo\ve, E. J.

" On a Supposed Temporal/�vrodal Paral lel .

"

Ana lysis 46.4

( 1 986) : 1 95- 1 97. - - . "Reply to Over. " Analysis 46 .4 ( 1 986): 200. Ov er, D. E. "On a Temporal Slippery Slope Paradox. � ' Analysis 46. 1 ( 1 986) : 1 5- 1 8. - - . "Is There a Temporal Slippery Slope Paradox?" Analysis 46.4 ( 1986) : 1 97-200. - - . ·'Reply to J--O\ve . " Analysis 46 .4 ( 1 986) : 20 1 . Salmon, Nathan. "Ilnpossible \Vorlds . " AnalysiS 44.3 ( 1 984) : 1 1 4- 1 1 7.

','

Dictionary ofParadox

1 94

SMULLYAN ' S PARADOX

.

A l egal paradox adduced by Raymond

Smull yan i n 1 978. Formulation. "Three men, conveniently named A , B and C, converge for a ni ght at an oasi s in the desert, parting company the next day. For whatever reasons, A decides to murder C and sneaks some poi son i nto his canteen late that ni ght. Shortly afterwards, B , acting quite independently, also decides to murder C and punches a smal l hole in C 's canteen. The next day, C dies of thirst. I t all comes out in the tri al . The question i s : who ki lled C?" ( Cohen 3 1 1 ) . Explanation. Defendant A argues that he i s not responsible for C 's death because C never drank any of the poison. Defendant B al so claims that he i s not responsible for C 's death because he did not deprive C of potable w ater, but only of poi soned water. Nevertheless, someone must be responsible for C 's demise. Resolution . Cohen suggests unless it is deci ded that the case is not one of murder, the paradox might be resol ved by revising our concepts of legal and moral guilt or of causality. Defendant B , ho\vever, seems to have the weaker case because his acti on was intended to cause C to die of thi rst in the desert and, in fact, C 's i ntention was realized. Defendant A performed an act that w as intended to lead to C 's death by poi son � hi s intentions, through no fault of his own, were not realized. I t would thus seem that A is at least gui lty of attempted murder. READINGS Cohen, I)aniel . "Putting Paradoxes into Pedagogical U-se i n Philosophy. " Teaching Philosophy 8 ( 1 985) : 309-3 1 7. Smullyan, Raymond. What Is the Name a/This Book? New York: Prentice, 1 978.

SOCIALI SM, PARADOX OF. This paradox is discussed together with

the

PARADOX OF LIB ERTARIANI SM

S OCRATI C PARAD O XES.

T hese are the paradoxes of Socrati c optin1ism and Socratic rationalism. See also MENO ' S PARADOX. Formulati on. Some of the more paradoxical doctrines espoused by Plato 's Socrates include: ( 1) ( 2) (3 )

No on e does ev il willingly. Ev i l is the result of ignorance. To know the good is to do the good.

Statements ( 1 ) and (2) are closely related: no one does evi l of one 's own free will, but rather evil is done by those w ho lack knowledge of the

1 95

Dictionary of Paradox

good . I f one kne\v the good , one woul d not do evil. Someth i n g of thi s sort is connected with Socrates '8 demon, which warned him to desist when he was on the verge of doing somethi n g wrong. S tatement (3 ) i s a some\vhat stronger claim: he \vho know s the good would not only avoid evi l but \vould moreover do the good unhe si tati n gl y Explan ati on. S tatements ( 1 ) - (3 ) fl y in the face of our everyday experiences of peo ple doi ng evil of their own free will , and of knowing what is ri ght but doi ng what i s wrong. Resolution. There are basically two ways to respond to these fundamental Socratic paradoxes . On the one hand, Socrates 's claims can be reconciled \vi th our i ntuitions by seeing the m as an attempt to s pec i fy a very stron g sense of the word "knovvledge," one i n vvhich acting from one 's knowledge is a cri terion of having the knowledge in the fi rs t place. On the other hand, one can attribute his claims to his pre-Christian intuitions which did not envi sion the possibility of daemonic despair, or of knowi ngly and \villingly contradicting the will of God. For thi s second al ternative, the paradoxical character of S ocrate s s cl ai ms i s the result of hi s relati vel y impoverished understanding of the conce pt of ev i l . .

'

READINCi S

King, James . "Elenchus, Self-Blame and the Soc rati c Paradox . '� Review of

J\lletaphysics 4 1 ( 1 987): 105- 1 26.

Vlastos, Gregory. "The Paradox of Socrates. " Socrates. Notre Dame UP, 1980.

SOLIPSISTIC PARADOX, THE.

"

Notre Dame:

T hi s \vas our paradox: no course of

action could be determined by a rule, because every course of action can be made out to accord with a rule. The answer \-v as: if ev e rythi ng can be made out to accord with the rul e, then i t can also be made out to conflict wi th i t . A nd so there would be ne i ther accord nor conflict here " (Wi ttgenstein, Philosophical Investigations, 20 1). See the PARADOX OF RULE FOLLOWING.

SOPHISM. A sophism is defined as a plau s i bl e but nevertheless fallacious

Sophi sms are thus a type of paradox (see DEFINITION OF PARADOX) . Even i n the appropri ately restricted sense of paradox, however, we \vould hesitate to affirm the compl ete synonymi ty of sophi s m" and "pa radox Thi s is b ecause so p h i s m connotes dup l i c ity whereas "paradox" tends to connote wonder at the marv elous Of, at least, at surpri sing results . argument.

"

"

.

"

"

,

!)ictionary ofParadox

1 96

SORITES, RECENT

..

Recent sorites are somevvhat more nan�o",' i n focus

than tradi ti onal sori tes , insofar as they bear principal l y on questi ons of the i denti ty of be i ngs rather than on causal rel ati ons. The j ury i s sti l l out on \vhether certai n recent SOI1tes are veri di cal or fal sidical paradoxes s o

be genuine anti nomies i n Quine 's sense . Formulation . There i s a vari ety of Sorites arguments in the recent l i terature. A mong the p red i cate s that have been argued to be i ncoherent are : beauty (Gol dstein) , gi zzle gi cks (Gri m , "I s T hi s a S\vizzl e S ti ck I S ee B efore Me ?") , j u s tice ( Q ues t , thus challe ngi ng Nozi ck's t heory of j usti c e ; see al s o Sorensen ) , people ( Unger, " Why There Are No Peopl e") , Peter they may \veJ l

Unger ( Unger, HI Do Not Exist") , redness (Travis , " Vagueness , Observation

and Sori tes") , s"" i zzle sti cks (Unger, "There Are No Ordinary Thi ngs") , s tones ( U ng e r "There A re N o Ordi nary Things") , tables (Unger, "There ,

Are No Ordinary Thi ngs") .

so-and-so if i t has n such-and-suches , then i t i s al so so-and- so i f i t has n 1 (or n + 1 ) such-and-suches, an d so , by i ndu ctio n i t \v i ll al so be a so- and- so \vhen i t has one or no (or n + rn , '.vhere lfl i s as l arge as one likes) such-and -suche s , but that i s absurd ! Yet si nce " such-and- such" i s such an c ln balTas s i ng l y vague predicate , i t j us t Explanation. I f a s o me th i n g i s a

-

,

'Non ' t do to say that there are any such-and-suches i n the fi rst place . Resolution .

The S ori tes argume nt has l ead p h i l o sophe rs to drasti c

conclusions : that language

is incohe re nt (Dumrnett) , that the "",orId i s vague

(Peacocke) , that fnodus ponens i s an acceptabl e pri nci p l.e onl y in s m al l doses (Parikh). There are vari ous responses to the challenge o f the S o ri te s Fi rs t , one can take them as veridi cal an d hence as reveal ing a fundamental vagueness or un d erl y i n g i ncoherence i n the predicate i n questi on (e.g. , U nge r ; Wheel er) . Second , one can as s i gn the pri nci pl e of bi v al en ce to sp e c i al i z e d languages and believe that i t doe s not apply to everyday d i s c o u r s e i n a s y s ten1 at i c n1 an n e r ( Ro l f) . Thi rd , o n e c an r ej e c t .

math clnati cal i n d ucti on, or restri c t it to special i zed contexts (Wei s s ; Napoli ; Trapp) .

Fourth, one can concl ude that the Sori tes reveal truth

ad o pt supervaluati onal logic, thus rej ecti n g bi val ence i n metal ogic ( Fi ne) . Fi fth , one can ado p t a In an y - v a l u ed or a fuzzy logic approach accordi ng to \v hi ch each appl icati on of i n du cti o n has a high d e g ree of tru th but there i s enoug h doubt i n troduced to c i rcu ill v e n t val ue gaps and

counteli ntui tive concl usi ons (Ki ng � Sanford) , S i xth , one could adopt the e p i s teln i c ap proac h a c c o r d i n g to \\t' h i ch t he re

i s an u n k n o \v ab l e

counterexalTIpl e to the i nducti on step (Cargi l e � Campbel l ) . See al so the S LIPPERY SLOPE

PARADOX and WANG ' S Pj,\RADOX .

Dictionary ofParadox

1 97

RE�ADING S

A.bbott. '}.,,T. R. "}\ Note on Griln �s Sorites A.rg uInent. " A JUl lys is 43 .4 ( 1983) :

1 6 1 - 164. Barnes, .T .

'�j\'1edicine, Ex pe ri ence and Logic. " Science

Ed. J . Banles ef

al.

and Speculation .

Camb ri d ge : C ambridge lIP, 1 9 82 : 24-68.

Black, Max. "Reasoning with Loose Concepts. " Dialogue 2 ( 1963) : 1 - 1 2 . Burnyeat, lv1 . "(rods and Heaps . " Language and I.o gic . Ed. J . L . Schofield and Martha Nuss baum. Cambridge : Canl b r i dge lTP, 3 1 5-338.

Campbell, R.

"T he Sorites Paradox. " Phi losophical Studies 26

( 1 974) :

1 75- 1 9 1 .

Cargile, JaBles . "The S o ri te s Paradox. " British Journalfor the Phi losophy of Science 20 ( 1 969) : 1 93-202.

Fine, Ie.

"

Vague ne s s Truth and ,

(J·oldstein, Laurence .

Logic .

"The Sorites

as

Synthese 30

( 1 975) :

265-300. a Les son in S e ma nti c s . l\ lind 97

"

"

( 1 988) : 447-455. Grim, Patrick.

"'\That Won 't Escape Sorites }\rguillents. "

( 1 982) : 38-43 . "Is This a Swizzle Stick \V hich I Sec Before [\i1 c'? "

- -.

Ana lvsis 42. 1 Ana(vsis

43 .4

( 1 983) : 1 64- 1 65. Jacquette , Dale .

"T he IIi ddcn Lo gi c of S l i ppery S l ope i\ rglllne nt s . "

( 1 989) : 59-70. "T he Paradox of the Heap. " Aspects of Phi losophica l Log ic .

Philosophy and RhetoriC' 22

Kamp, I Ians .

Ed Uwe I'vl o n ni c h J)ordrecht: Reidel, 1 98 1 . 225-277 . .

King, J. L .

B i v al e n ce and the Sorites Paradox. " Arnerican Philosophical

"

( 1 %9) . 17-25. Lederkramer, Dav id. "Quest on Enti tle m ent Theory. " Quarterly 1 6

Analysis 39.4 ( 1979 ) :

2 1 9-222. Moline, Jon.

'"Aristotle , Eu bul i des and the Sorites . "

393-407. Napoli, Ernesto. " I s \lagueness a Lo gi c al

Eniglna?"

A1ind ns 78 ( 1 %9) :

Erkenntnis 23 ( 1 985) :

1 15- 1 2 1 .

Pari kh

,

R.

"T he Problem of \Tague Pre di c ate s

.

Boston S tudi es in the

"

Phi losophy of Science : Lang uage, Logic and Science. \lol . 3 1 . Dordrecht: Reidel, ] 983 . 241 -26 1 . Peacocke, (�hristopher. "Are Vague Predicates I ncoherent? " Synlhese 46 ( 1 98 1 ) : 1 2 1 - 14 1 . Putnalll, lIilary. "Vagueness and Alternativ e Log i c. Erkenntnis 1 9 ( 1983 ) : 297-3 1 5. - - . "A Quick Read Is a \Vron g Wright. Analysis 45.4 ( 1 985) : 203 . Quest, Edward. "Whatev er Ari ses from a Just Distribution by J1L�t Steps i s Itself Just" Analysis 37.4 ( 1 977) . 204-208. Qui ne, 'vV. \7. "\Vhat Price Bi v alence ? Journal of Ph ilo,)'ophy 77 ( 1 9R I ) :

I

'I

1

, ,

n

,

n

"

90-95.

Read, S . , and C . \Vright. "Hairier Than Putnan1 T hou ght . "

( 1 985) : 56-58.

Ana lV,'·;is 45. 1

,I ,

1 98

Dictionary of Paradox

Rolf, Bertil . "Sorites. " Synthese 58 ( 1 984) : 2 1 9-250. S anford, D. H. "Nostalgia for the Ordinary: Comments on Papers by Unger and Wheeler. " Synthese 4 1 ( 1 979) : 1 75- 184. Schwartz, Stephen P

"Intuitionism and S orites. "

Analysis 47.4 ( 1 987) :

1 79- 183 .

Sorensen, Roy A .

"Slipping off the Slippery Slope: A Reply to Professor

Jacquette. " Philosophy and Rhetoric 22 ( 1 989) : 195-202. - - . "Nozick, Justice and the Sorites. " Analysis 46. 2 ( 1 986) : 1 02- 106. Thorpe, Dale A. "The Sorites Paradox. " Synthese 6 1 ( 1 984): 3 9 1 -42 1 . Trapp , Rainer W.

"

S i nki n g into the S and: T he Falsity of All S orites-

Arguments. " Erkenntnis 23 ( 1985) : 1 23- 1 25.

Travis , Charles. "Vagueness, Observ atioll, and Sorites. " Mind ns 94 ( 1985): 345-366. - - . "On W hat Is Strictly S peaking T rue . " Cana dian Jo u rnal of Phi losophy 1 5 ( 1985): 1 87-229. Unger, Peter. "I Do Not Exist. " Perception and Identity . Ed. G. F. MacDonald. Ithaca: Cornell UP, 1 979. 235-25 1 . - - . "T here A re No Ordinary T hings . " Synthese 4 1 ( 1979) : 1 17- 154. - - . "Why There Are No People. " Midvvest Studies in Philosophy 4 ( 1979):

1 77-222. - - . "Skeptici sm and Nihilism. " Nous 14 ( 1 980) : 5 17-545. Wheeler, Samuel C. "Reference and Vagueness. " Synthese 30 ( 1 975): 367-

379. - - . "O n That Which Is Not. " Synthese 4 1 ( 1 979): 155- 1 73 . Weiss , S. E. ''T he Sorites Fallacy: What Difference Does a Peanut Make't' Synthese 33 ( 1 976) : 253-272.

Wri ght, Crispin. "On the Coherence of \Tague Predicates . " Synthese 3 0 ( 1 975) : 325-365. - - . "Language Mastery and the Sorites Paradox. " Truth and Meaning.

Ed. G. Ev ans and J. McDowell . New York: Oxford UP, 1 976. 223-247. - - . "Further Reflections on the S orites Paradox. " Phi losophical Topics

1 5 ( 1 987): 227-290.

SORITES, TRADITIONAL.

Also known as the Heap, this family of paradoxes was introduced by EUBULIDES. Long of scant philosophic i nterest, there has been a cottage industry in sorites arguments since 1975, for which see RECENT SORITES . The famous Sorites of T hemostocles, to the effect that his baby son ruled the whole world, goes as follows. My infant son rules hi s mother. Hi s mother rules me. I rule the Athenians. The Athenians rule the Greeks .

Dictionary of Paradox The ( i rec ks rule the Europe, A nd I '� l I r( ) p (' ru les the world.

A sorites i s an arg u l l H ' n l i l l \v h i ch a series of incomplete syl l ( l l ', I � : I I I " I ' , arranged so that the p rcd l c; l l e ( )j " each premise forms the subj ec t ( .1 l i l t subsequent one, unti I the S l I h l ( 'c I ul the fi rst i s joined wi th the prcd i c� l t \ ' I I the last in the conclus i on . I l e 1 ( : I S a l lot her old example. (

All men who be l i e v e �, ; h ; d l l l( ' :-;: 1 \' ('< 1 , All who are say cd n 1 1 t� 1 I )( ' I I I T ( . 1 :-; i 1 1 , All who are free of si n I l I u : � I I It " 1 I 1 I 1( )( T H I i ll the eyes of God. All who are innocent in t I H' �� I J ', h I ( I I ( ; . )( I ; I I ( S I I i 1;\ hie for heav en. All who are suitable for hea v e n \\' 1 1 1 he ; Id l l l l l l e d 1 1 1 1 0 heav en. Therefore� all who believe will be ; U I I l I l I t C( 1 1 1 1 1 , 1 I W . I \' . ' I I "

There is another causal sorites in the ol d savv that t ( )1 \\'; 1 1 1 1 • I I . 1 1 1 . 1 1 1 t l shoe was lost, for want of a shoe the horse \vas l ost , I ( )I \\ ' ; 1 1 1 1 ( , I : 1 h ( I I : , , ' I l u t l ! I t I .' . the rider was lost, for want of a ri der the battle "vas I ( )s t " concluded that for want of the nail the empire was l os t. I } LI I �';( · 1 '; 1 .'« - ; 1 1 presents a truncated versi on of this argument called Cleopa t r� l � s I I ( ),'-.;c " I I the nose of Cleopatra had been shorter, the whol e face of the ea r t h \\ ' ( l l i l d have been changed" (Pensees, vi i i , 29) . The paradoxi cal i lY o f I h e conclusi on of a Sorites i s usually that a seemi ngly insignificant cau se has enormous effects ; other Sorites make one class a subclass of another i n contrast to our expectations. ..

.

_

STADIlTM, THE . See the MOVING B LOCKS .

STONE, THE PARADOX

OF THE . A particul arized version of the

PARADOXES OF OMNIPOT ENCE, thi s medieval paradox seems to threaten the

characteri zation of God as all -powerful . Formulation . Everyone has asked himsel f what would happen i f an irresisti ble force meet an i mmovable object. I n Greek myth, Zeus himself once posed such a question when a hound that was fated always to catch whatever i t pursued was sent out after a fox that \vas fated never to be caught. I n the event, Zeus turned both the hound and the fox to stone. T he question of the title paradox is \vhether, when God makes a stone, He can make one too heavv for Hi mself to lift. -'

,

200

Dictionary ofParadox

Expl anation. I f God can create such a s tone , His potency i s limited by

Hi s inability to l i ft i t ; and if He cannot that limi tation does no t s peak \vel l of Hi s po\vers either. In e i th er cas e, it \v Q u l d seem that God i s not all ­ po\verful . Resolution . J . L. Mac ki e i ni ti ated the modem vogue i n thi s problem, suggesting that the p aradox requires u s to di spense e i ther \vith the concept of olnni potence or \vi th that of God . G. B . Keene, ho\vever, argues that "God c ann ot create a s t o ne that He cannot l i ft i s equi valent to �'Every A ccordi ng to Keene, the s tone that God can create , He can l i f t eq ui valence di s sol ves the second horn of t he d i l emma and actual l y rei n forc e s the c onc e pti on of God as al l -po\verful since He can i n fact lift all s to n e s that can be created. B ernard Mayo contends that Ke e n e s anal y s i s i m pl ie s that no canno t statelnent could be construed as a ss e rti n g a lin1 i tation and suggests that the parad ox m i g h t be resol ved by treati n g it as one V\ioul d the probl elTI that G od cannot create a square ci rcle : s i n c e there i s no such thing as a s q u are circle, there i s nothing i n th e case that God cannot create . A no the r approach i s that taken by Alfred R. M el e and M. P S m i th yvho recast the paradox as a competition bet\veen a pair of omn i p ote nt beings and concl ude that the om n i p otenc e of th es e bei ngs does not guarantee that they \vi l l be able to do al l they i ntend to do For Mel e and Smith , ho\veve r the problem i s not i n the be i n g s s omni potence, ,

"

.

H

'

"

"

.

'

,

but i n their intentions .

,

READINGS Anderson, C . A.nthony.

Di v i ne ()mnipotence and Imposs i ble Tasks : An

"

Intensional Analysis. " International '!ournal for Phi losophy ofReligion

1 5.3 ( 1 984): 1 09- 1 24. Chisholm, Roderi ck . Person and Object. La Salle : Open Court, 1 976 . Chapter 3 . (�o'van, J. L. ' The Parddox of OmniJX>tence. " Ana�ysis 2'5. 1 ( 1964): 102- 108. - - . "The Paradox of Onlnipotence Revisited. " Canadian Journal oj Phi losophy 3 ( 1 974) : 35-45. Hoffman, ]OSU3 , and Gary Rosenkranz. "\Vhat an Omnipotent A.gent (:an Do. " International lournalfor Phi losophy ofReligion 1 1 ( 1 980) : 1 - 1 9. - - . "Omnipotence Redux. " Philosophy and Phenomenological Research 49 ( 1 988) : 283-3 0 1 . I(eene, G. B . "A S inlpler Solution to the Pa radox of Omnipotence. " A1ind ns 69 ( 1 960) : 74- 75. Londey, TJav id, B arry NIiller, and John King-Farlo\v.

"God and the Stone

Paradox : Three Co mments. " Sophia 1 0 ( 1 97 1 ) : 23-33.

1v1ackie, J. L. "Ev i l and Omnipotence . n A1ind ns 64 ( 1 955): 200-2 1 2 .

fv1av rodes, G. 1 . "Sonle Puzzles Concerning O mni pote nce .

Review 72 ( 1 963): 22 1 -223 .

"

Philosophical

,

I I

201

Dictionary of Paradox

Mayo, Bernard. "�jfr. Keene on Omni potence. " A1ind ns 70 ( 1 96 1 ) : 249250. 1\·1 elc, Alfred R. , and M . P. S n1ith. "T he New Paradox of the Stone. �' Faith and Ph i losophy 5 ( 1 988) : 283 -290. Sav age, G . \Vade. "T he Paradox of the Stone. " Phi losophica l Revie'w 76 ( 1 %7): 74-79. Schrade.r, David E. "1\ Solution to the Paradox of the Stone. " Synthese 42 ( 1 979) : 255-264. \Vi e r e n g a , Edw a r d .

" O m n ipote n c e l)e fi n e d . "

Phenornenolo g ical R es ea rch

Wolfe, Julian. "Omnipotence. 245-247.

P h i losophy a n d

43 ( 1 983) : 363-375 . Canadian Journal afPhilosophy 1 ( 1 97 1 ) :

H

ST PETERSBUR G PARADOX, THE. ..

This parad ox of proba b i l i t y

theory i s c o nce rn ed \v ith a c oi n fl i ppi ng galn e.

Formulation .

"

S uppo s e tha t a man tosses an idea1 coi n un ti l heads

appears . Then the game i s ov er. He recei ves X

=

2n

ce nts ,

\v hcrc

II

i s the

nUlnber of tosses of the coi n . T h u s , i f hcaJ s appeared for th e fi rst ti n1C on

the third tos s , he col l ects 8 cen ts. I t i s to h i s advantage to get a very lon g run of tai l s b e fore h ead s appears .

Wi th probabil i ty onc , heads \v i l l

ev e ntu al l y appear an d he vv i ll col lect SOlne lll oney What i s the expectati on .

of the amount of money he \vill receive?

In o th e r \\lords, ho\v much

should he pay for the privilege of playi ng thi s game to make i t a ' fair ' game" ( C hernoff and Moses , 1 04) .

Explanation I f \ve toss a c o i n there are two equi probable results : heads ..

and tai l s . Hence , the probabi lity of obtai ni ng tail s i s 1 /2. I f the coi n is to ss ed t\vicc, the pos sibil ities are ( h e ad s , heads) , ( heads , tail s) , ( tai l s ,

beads) , and ( h e ads , head s ) . T bus, the probabi lity of obtai n i ng t\VO tai l s i s

1/4.

Similarly, i f the coi n i s tossed

i

ti mes, th e re will be 2i poss i bil i ties ,

on l y one of \vhich consi sts of al l tails. Hence, the pro b abi l i t y of obtain in g

i tails o n i tosses of the

coin is 1 /2i . The expectation i s the pro d u ct of the

payoff by the probabi l ity of o b tai ni n g that payoff. T hus , t he expectation of re ce i v i ng t\VO c e n ts i s

2( 1 /2)

=

1 , si nce the probabi lity of obt ai ning

tails on o ne toss i s 1 /2. Si milar] y, the expectation of recei vi ng fo u r ce n ts is

4( 1 14)

=

1 and the expectati on of re cei v i n g 2i cents i s 2i( 1 /2') 1 . T he

total expectation of the game i s the sum of al l these p arti al expec:tations : E

=

=

2( 1 /2) + 4( 1 /4) + 1

+

infinity.

1

+

.

.

.

.

. . +

+

2i( 1 !2i) + 1

+ . . .

. . .

I

I

I

I! I .I

,

202

Dictionary of Paradox

The game i s quite favorable to the pl ayer since he can always \vi n more than any fi nite amount that he has to pay for playing. The paradox is that few would be willing to pay more than a few doll ars to pl ay the game. Resolution. The infinite expectation of the game is predicated on the assumption that the pl ayer can play an i ndefinite num ber of times. In fact, one can only be reasonably sure of averagi ng more than twenty cents a game by playing a very large number of games. READING

Chernotf, I-Ierman, and Lincoln E. Moses. Elemental Decision Theory. Ne w York: Wiley, 1959. 1 04- 106.

SUBJE CTIVISM, PARAD OX OF.

See PARADOXES OF COGN ITIVE

RELATIVISM, AND HUSSERL ' S PARDOX OF SUBJEC1IVITY .

SURPRISE AIR RAID DRILL, THE. See the PREDICTION PARADOX .

SURPRISE QUI Z, THE. See the PREDICTION PARADOX.

TAOIST PARADOXES. T hese are historically related to, and thus are

treated with, the B UDDHIST PARADOXES ; but see also HuI SHIH ' S PARADOXICAL APHORIMS . .

TASTE , THE PARADOX OF. This paradox goes back to Kant and

Hume before him. We follow the Kantian version. Formulation . The thesis is that judgments of taste are not based on conceptions, because i f so, judgments of taste would eli ci t controversy and allow for proof. T he antithesis is that the judgments of taste are based on concepti ons , because if not , there would be none of the controversy that ari ses from the our insi stence that other agree with our j udgments. Explanation. On the one hand, one cannot establish the judgment, say, that an object is beautiful through an argument based on concepts, so i t appears that the judgment of taste does not have the objective universality of any judgment that is based on rules . On the other hand, the judgment that something is beauti ful is unlike the j udgment, based on i ndivi dual

203

Dictionary of Paradox

incl ination, that something is pleasing, for while we accept that others do not find the same things pleasing as w e do, the j udgment of beauty is universal in the sense that it demands the accent of others. Resolution. Kant resol ves the paradox of taste by distinguishing between two kinds of conception. There are detenni nate conceptions , such as the concepts of the understandi ng, which ground knowledge of obj ects, and there are indeterminate conceptions, such as the i dea of the supersensi ble, that have objective validity. The i dea of the supersensible, whi ch underlies both the obj ect j udged and the j udgi ng subj ect, has neverthele s s a subj ective v al i di ty . The universality of the j udgment of taste is based precisely on this i dea of the supersensible, \vhich i s that of "purposiveness without a purpose. " READINGS

A Comlnentary on Kant 's Critique ofJudgment. New York: Barnes, 1938. S ec. 70. Kant, Em m anue L The Critique of.ludgment. Trans . J . I I. Bernard. Nevv York: Hafner, 195 1 . Part 1 , Di v ision 2. l\1othersill , Mary. "l-Iume and the the Paradox of Taste . " Aesthetics : A Critical Antho logy. Ed. (reorge I)ickie, Richard Scalfani, and Ronald Cassier, H. W.

Roblin. 2nd ed. New "York: St. Nlartin's, 1 989. 269-286.

TERTULLIAN ' S PARADOX. Quintus Septimus Florens Tertullian (c.

, I

1 60-220) was an African patriarch of Chri stianity who loved paradox and contradiction. In his "On the Body of Christ," Tertullian argued that the I ncarnation of Chri st "is certain because impossible." He was evidently pushi ng to an absurd extreme A ristotle 's i dea (Rhetoric, II , xxiii , 22) that it is likely that unlikely things should happen. Various Christian apologists, such as O . K. Chesterton, have rei terated the argument by claiming that the whole idea of God becoming man i n order to be crucified i s so absurd that it could never be the product of the human- imagi nation and so must be true. READING Williams, Bernard.

Tertullian �' Paradox.

L ondo n: SCM,

1955. •

THINKING BEHAVIORIST, THE PARADOX OF THE.

By thi s

paradox, Arthur O. Lovejoy seeks to s ho\v that the behaviori sm o r ..T . B . Watson i s internally inconsi stent. Formulation. The behaviorist makes cogniti ve claims, but vvhen he makes

204

Dictionary of Paradox

the se clai m s he contradicts him sel f u nl es s he denies that he kno\v s anythi n g . Explanation. Wats on i denti fies the a\vareness of thi ngs distant in space

and

time w i th certain m uscular or bodily m ovem e nts,

but no de scri pti on

of internal bodi l y events can expli cate intentional reference to external

o bj ects .

Resolution.

I n response to

Lovej oy,

Gil bert Ryl e

recommends

th a t

kno\vledge of external real ities be understood di spositionally: that someone

is R\Vare of son1ethi ng in hi s environn1ent j ust means that if that s om ethi n g

i s present i n his env i ronm ent and if such and such condi tions obtai n, then

he \v i l l behave i n such and such manner \v ith regard to it. I n thi s \vay, \ve can refer to external obj ects \vhi l e l i n1 iti n g these obj ects to behavioral processes .

READINGS

Lov �joy, A . () . "Paradox of the T mnking Rehaviorist. �' 3 1 ( 1922) : 1 3 5- 1 47. Ryl e, G-i lbert. The Concept ofA1ind.

Phi losophical Revie'w

London: I-Iutchinson,

1 949.

THE TIDRD MAN ARGUMENT. This p arado x arises in Plato 's Theory of

Forms.

Formulation.

"

W he n several thi ngs seem large to

you it seems perhaps ,

that there i s a single Form \vhich is the sam e in your vi e\\' of al l of them. Hence you bel ieve that Largeness is a singl e thing . .

.

What then if

y ou

si m i l arly vie vv mental ly Largeness i tsel f and the other large things? Wi ll not a single Largeness appear once agai n, in virtue of which al l these

(sc

Largeness and the other l arge th in g s) appear l arge?" (Pl ato, Parnlenides 132a l -b2 ; Vlastos 's transl ati on 320-32 1 ) .

Explanation. Accordi ng to Plato 's Theory of Forms, l arge things, for example, are large by virtue of their participation i n the Fonn of Largeness � al so ,it is b e caus e of thi s that the single \vord "large" can be appl i ed to the many l arge thi ngs . The Form of Largeness i s , so to speak, the es sence

of

largeness and so must i tsel f be l arge . Hence, there must be another Fonn in \vhich both t h e

Fonn

of Largeness and al l l arge thi ngs partici pate and

that i s , we can p redi cate the one \vord "large" of of Largenes s and large things only if there i s a si ngle n e w

which makes both l arge ;

both the FOIID

Fonn that both the Form of Largeness and large thi n gs parti e] pate in.

This argument, ho\vever, can be iterated indefinitely, res ulting in an i nfi nite regres s .

Even wors e , l arge t h i n g s are l arge not by v i rtue of their

pa r t i ci pati on in a s i ngl e Fonn , but by virtue of their participation in an 1 I 1 l " i l l i ( c n u nl bcr of Form s.

Diclionary q/Panuh n Resolution. A ccordi n g to Gregory Vlastos, PJ ato h i rn sc l l

'/

c(

J •,

� I J l d I It ) I I t - I I I I <

t he argument but nevertheless formulated i t w i th grea t t "( H C ( ' ,

tIH·1

t

-

h\

revealing hi s i ntell ectual honesty. A ri stotle thought the arg u l l l c l I l \ ; d I d and used it to argue against the Theory of Forms . Vlastos con lCI H I .� ; I h . l 1 the argument i s actual l y inval i d because it depends on the foi l < )\\" 1 I t ". contradictory presuppositions:

( 1) ( 2)

The Form h as the property that it ilnparts to things

(Largeness is l arge) ; That \vruch has a property cannot be that by virtue of which it h as the property (Largeness cannot be large because of Largeness).

K. W. Rankin argues that Plato could not have disti nguished the extensional

and the i ntensi onal aspects of the Forms and, thus, "i t i s unl i kel y that he could have at any stage used the Forms as expl anati ons i n an uncq ui vocal

\vay" ( 1 96) . Thus , Pl ato \voul d not be forced to d i s t.i ngu i sh properti es fron1 Forms, which vvould dcfusc t.he pa radox. . A ccord i ng to Ranki n, Plato 's understand i n g of the logical stru c t u re ( ){" t h e arg u lllc n l vv' as C( lnfu scd

but, in fairness

to

Pl ato� he

"front-line" today_

\v as

\v rc s t t i n g vv i l h probl cll1S that

R I �A J ) I N(

is

Rankin, K. \V. ""I ' he J )upl ici t Y (">{" 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
Phi losophi cal Re vic vv

()��

( 1 954) :

sti l l

"

3 1 9-349.

TII\I{E., PARADOXES OF. Paradoxes i n v ol v i n g the charac ter of time incl ude the ACHILLES , th c A RRO\V , the (� l J )( 'K PARADOX, the DEVIL ' S OFFER,

GOOD1\'IAN ' S PARADOX OF CONFIR�'L\TJ( )N , KANT ' s COSl\.'1 0LOGICAL ANTINOlvlIES , t he LA MP PARA DOX , McT A ( J ( J El{I" S PA RA D OX , PA S CAL ' S W AGER , the

PRECESSION OF TIlE EQUINOXES PAl{i\DOX , PARADOXES OF T IME T RAVEL, the

TWIN PARADOX .

TIME TR AVEL , PARADOXES OF Not to be confused \�/i th the T\VIN ..

PARADOX,

th ese paradoxes seem to con fute the conceptual pos sibil ity of

travel i n g back\vard in ti me . Like the T\v in, they ari se i n the context of Ei nstei n 's Special Theory of Rel ati v i ty.

Formulation. Paul Hor\vich fonnul atcs five paradoxes resul ti ng from the cl ai n1 that it i s possi ble to travel back\vard i n tin1e:

(1)

A. temporal interv al \vould hav e- to be trav ersed in an amount of time less than that of the interv al .

,

206

Dictionary of Paradox

( 2) (3 ) (4)

(5)

Back\vards time trav el is incompatible with Leibniz's law of the Identity of Indiscemibles. The past cannot beundone� but back'A'ards time travel would make it possible to undo what i s already done. The time traveler would be forced to hold inconsistent beliefs stemming from conflicts in his memories and in his intentions. A rocket is programed to fire a probe unless a safety switch is on. But the switch is on if, and only if, the probe 's return is detected by a sensing device. Hence, the probe will be fired if, and only if, it i s not fired (see Earman 23 1 -232) .

Explanation. According to Horvvich, the first four of these paradoxes

are easily resolved sophisms. The first, for example, relies on an illicit absolutist conception of ti me. Thus the paradoxicality of the argument disappears once we real ize that the time i n the reference frame of the time traveler is not the same time as that in the reference frame of the interval to be traversed. We will therefore refrain from further consideration of these arguments here. The fi fth argument, however, seems to be a genui ne paradox. It was fonnulated by lohn Earman entirely in terms of mechanical operations in order to avoid considerations of human agency which viti ated earlier versions of the paradox. Thus a rocket is equi ped with a probe that will be sent back\\'ards i n ti me and al so with a device with which to detect the probe's return to the rocket's present. Furthermore, the probe \vill be fired whenever a safety swi tch is off; it cannot be fi red when the switch is on. Yet the necessary and sufficient condition for the switch bei ng on i s that the rocket has detected the probe 's return. Suppose, then, that the probe has been fired. Then, the switch must be off and, hence, the return has not been detected. B ut since the probe returns to the rocket 's present, the only way that the return could not be detected i s for the probe not to have been fired, which contradicts our supposition that it has been fired. On the contrary suppositi on that the probe was not fi red , the switch must be on. Hence, the return of the probe has been detected. Yet if the probe has returned , it m ust have been fired, which agai n contrad icts the supposi tion. Resolution . Earman argues that it follows from the paradox that we must give up one of t\VO things: ( 1 ) backvvards time travel or (2) the supposition that the rocket can be programed in the manner deli neated above. There SeelTIS to be no logical reason to favor one over the other of these two options but we do have solid empirical evidence for the possibil ity of executing the desired program in the existence of similar programs. Hence , we have empirical evidence for denyi ng the possibil ity of backwards time

207

Dictionar.y of Paradox

travel . P. Horwich, however, argues that Eannan has committed a subtle, but fatal , modal fal lacy. Perhaps more importantly, he sug gests that the paradox may be avoi ded by observing that if backward ti me travel i s possible , i t would sti ll be the case that not al l causal chains would be possible. One cri terion for the existence of such causal chains vvould be consis tency and thi s i s enough to eliminate the paradox . READINGS Earman, John. "On Going Backwards in Time.

"

Philosophy of Science 34

( 1 967) : 2 1 1 -222. --.

"I mplications of Causal Propagati on O utside the N"ull Cone . "

Australasian Journal of Philosophy 50 ( 1972) : 222-237. Harris on, J. "Dr. Who and the Philosophers or Time-Travel for Beginners . " Aristo te lian

Horwich, P.

Society Supplement 45 ( 197 1 ) : 1 -24.

HOn Some Alleged Paradoxes of Time Trav el . " Journal

of

Philosophy 72 ( 1 975): 43 2-444.

Lewis, Dav id. ''T he Paradoxes of Time Trav eL " Quarterly

A rnerican Phi losophical

13 ( 1 976) : 145- 1 52 .

NIeiland, J . W. "A Two-Dimensional Passage Iv1odc1 for Time Trav eL "

Philosophical Studies 26 ( 1 974) : 1 53- 1 73 .

Smith, Joseph Wayne. Reason, S('ience and Paradox: Against Recei ved

Opinion in Science and Ph i losophy

.

I .london: lIelm, 1 986. Ch. 2.

Stein, H. "On the Paradoxical Time-Structures of G6del . "

Phi losophy oj

Science 37 ( 1 970) : 589-60 1 .

TRAGEDY, THE PARADOX OF.

T he paradox of tragedy i s an

AESTHETIC PARADOX like the PARADOX OF THE UGLY . Like other works of

art, tragedy provides us with aesthetic beauty and pleasure, but tragedy depicts happenings that are not pleasurabl e to \vitnes s , that are pai nful and traum ati c to witness, and which may even border on the ugly.

TRINITY, THE PARADOX OF THE. The paradox is expressed by the A thanasian Creed: " We worship one God i n Trini ty, and Trini ty in Unity ; Nei ther confoundi ng the Persons , nor dividing the Substance . " Various manners of avoidi ng the paradox of three persons in one substance have led to the fol low i n g heresies : A pollinari anism ( Christ's di vinity and humani ty are the same and, hence, there could be no moral development i n Hi s life) , Arian i sm (Christ is not div i ne but created by God expressly

for the creati on of the world) , Euthychi ani sm (the deni al of Chri st's humani ty) , lacobi ti sm (a type of Monophys i ti s m ) , Monophys i ti s m

i

.,

q !

!,

Dictionary oj Paradox

208

(Chri st's nature i s \vholly divine) , Nestorianism (there are t\VO separate persons in Chri st) , Sabellianism ( the persons of the Trinity can onl y be di sti ngui shed by their functions) , and Unitarianism (the rej ection of both the di vini ty of Christ and the Trinity).

Thi s i s a prac tical pro b l e m i n salesn1anship, simi lar to the PARADOX OF SINCERITY. I n order to maximize his chances to com plete most sal es , a sal esman must establi s h hi s trust\vorthiness to the prospective client ; his assertions and his actions must be consistent. The sales process, ho\vever, requires that the salesman do \vhatever may be necessary in order to control the situation and close the sale. These necessities often i nclude submission to indignities and the performance of acts th at violate good taste and/or moral princi ples. Since the client is a\vare of these exi gencies , he is un li ke l y to believe that the salesman i s trust\vorthy. While not formally contradictory, the salesman does find h i m self i n an impossible situation since, in order to make the sale, he lTIUst be trushvorthy and he must also perform acts that impugn thi s trust\vorthiness.

T RU S T, T H E PAR AD OX OF.

READINCJ

Oakes, Guy. "The S ales Process and the Par adox of Trust. " Business Ethics 9( 1 990) : 67 1 -679.

Journa l of

TRUTH .. TELLER VARIANTS OF THE LOGIC AL PARADOXES.

One of the lnost conspicuous aspects of the LIAR and other paradoxes of the san1e ilk is that it predicates fals i ty of itself. The result is paradoxical in the strong sense of being contradictory. What happens, howev er, w hen a statement predicates tru th of itself? Mackie (298) collects the follo\ving list of examples of truth-teller variants :

(1) (2) (3) (4) (5)

(6) (7) ( 8)

(9)

"Vhat I am no '0.' sayi ng i s true. S = d f ' S ' is truc. e [The sentence printed in brackets on thi s page is true . ] " V'iclds a truth \vhen appended to it� own quotation' yields a truth vv-hen appended to its own quotation. Obey this o rder. Carry out the next instruction: carry out the prev ious i nstructi on. I s ' autologicaJ ' autological? T he largest nunlber named in this book. [T he only success fully referring phrase printed i n brackets on this page. ]

,I ,

Dictionary of Paradox

209

( 1 0) If the instruction in Richard' s paradox had read ' If the nth figure in the nth decimal i s p let the nth figure in N be p \\'ould N hav e been a nlcmber of F.,' ? ( 1 1 ) ' J am not goi ng to be han ged on yonder gallovvs . ' ( 1 2) Is the c las s of all classes that are menlbers of the ms elv e s a member of itself? ( 1 3) In a certain village the barber shav es all, and only, the men \vho do s have themselv es; i s the barber bearded or cleanshav en? ,

"

S defined above. I f \ve suppose that S i s true no contradi ction results � hke\vise, i f we sup po s e that S is false. Nevertheless , many logicians still consider S p arad oxi cal because it fai l s to assert any thi n g at all. T hi s i s p e rh aps more obvi ous i n the i mpe rati v e v ari an t Consider the sentence

,

Obey this order !

Here \Jve have a sentence that purports to C0111111 a n d us to do sot11cthi n g , but there is n ot any thi ng that it C0111 nlands u s to 00. Wc d< ) n o t knov\' \v hat actions \vould count as con1pl iancc (or noncoln pi iancc) to the ordcr and , hence, it seems that the order cannot place any obl i gati on on us . :t\1ackie, .T. L . Truth,

Lagic

.

HJ �j\])IN(JS

Prohability and

0 xford: ( :iarel1don, 1 973 .

Paradox : St udies in Ph i losoph ica l

l\:1ortensOll, C. , and G. Priest. "The Truth Teller Paradox. " Logique e[ Analyse

24 ( 1 98 1 ) : 38 1 -388. yTablo, Stephen. �Truth, Definite Truth, and Paradox. "

Journal ofPhi losophy

86 ( 1 989 ) ; 539-54l .

TWIN PARADOX, THE

A PARADOX

T II\1E T RAVEL,

like the C LOCK PAPv\DOX , of \v h i c h it i s a graphic illustration, the T\v i n Paradox is a p u zzl e in rel ati v i ty theory about asymmetry i n an appare ntl y symnletri cal ..

OF

situati on. Formulation . A n a stronaut says

fare\vell to his t\vin brother before leaving i n a space ship for a star S0111e l i g ht years away. T h e sp ace s hi p travel s near t he speed of light, and \v h e n it reaches its destination, turns around and returns to earth at the s ame spee d On returning , t he astronaut fi nds that, \\t'hi l e his clocks show the journey to ha v e ta.ken a certain amount of tin1e, the c l ocks on earth s ho\v a rathe r large r elapsed tin1c , and his t\vin brother is no\v considerably aged. Explanation. The result i s paradoxical because the n1otion of the earth relative to the spacecraft i s j ust the same as the motion of the spacecraft .

•

210

Dictionary of Paradox

rel ati ve to the earth. From the astronaut ' s perspective, the t\vin on earth went a\vay and came back at a veloci ty close to the s peed of l i ght

.

So

\vhy has the as tronau t not aged more than his terrestrial sibling?

Resolution. A cco rdi n g to the theory of re l ativi ty the astronaut is firs t ,

accel erated, then turned and accelerated i n the opp o si te directi on. This "non-inertial" motion is thus in two directions . In contrast, the earthbound twi n has no signi ficant motion o ther than that provided by the i nertial pas s age of the earth around the sun .

I t i s th i s as y mme try, when

incorporated into the equati ons of the s pe cial theory of re lativi ty that resul ts i n less time measured and experienced by the astronau t

.

READINGS Kroes, Peter. ' T he Clock Paradox, Or How to Get Rid of Absolute Time. " Phi losophy afScience 50 ( 1 983) : 1 59- 1 63 . Little, E. M. " Two Simpler Relativ ity Twin Paradoxes. " American Journal of Physics 33

( 1965):

747-748.

Lowry, E. S . 'The Clock Paradox. " Anzerican Journal afPhysics 3 1

( 1 963): 59.

iVlarder, L. Time and the Space Traveller. London: 1 97 1 .

Muller, R. A . "The 1\vin Paradox in Special Relativ ity.

"

American Jaurnal

of Physics 40 ( 1 972) : 966-97 1 .

Schlegel ,

R.

'The Clock Paradox : Some New T houghts. �' Phi losophy of

Science 44 ( 1 977) : 306-3 12.

UGLY, THE PARADOX OF THE . Like the PARADOX OF TRAGEDY , this i s an AESTHETIC PARADOX . The ugly, whether found in works of art or i n ordinary experience, is presumably di stasteful and u nap peal ing but under ,

certain circumstances , the ugly is assoc iated with an appe al i ng aestheti c fasci nation and pl easure

.

UNEXPECTED E GG, THE.

T h i s i s a v e rs i o n of the PREDICTION

PARADOX.

UNEXPECTED HANGING, THE. See the PREDICfION PARADOX . UNNOTICED MAN, THE .. See EUBULUIDES ' PARADOXES .

UNSUCCESSFUL I NTERVENTION, THE PARADOX OF. Proposed in

1 986 by Robert M. Gorden, this i s a paradox ab ou t an individual fi ghting

agai nst hi s own desires.

,, ,

Dictionary a/Paradox Formulation.

I 1 /'

211

An agent "intervenes against hi s desi re to X and yet,

unsuccessful, X 's all the same. Assuming that there ",, ere desires (or reasons, or whatever) that motivated him to intervene because his i ntervention is '

against his desire to

, , ,, ,

X, \vhy were they not sufficient to m ov e him simpl y

.

not to X, maki ng interventi on unnecessary ?" ( Gorden 222) . Explanation. Suppose that an individual who has a craving for cream pies al so has a desire to lose wei ght and, therefore , a des i re to abstain from eati ng the pi e his wife j ust baked. The i ndi vi dual i s equiped wi th a machi ne that will quell hi s desire for the pie

all he has to do is to press

a button. Unfortunately, the machine's battery i s dead , so that, when he presses the button, the machine does not work and he ends up eating the pie .

S i nce he pressed the button

,

however,

hi s desire not to eat the pie

must be stronger than his desire to eat it (otherwise, he would j ust have eaten the pie and left the button al one)

,

But if his d e s ire not to eat the pie

,I

, J' I , II II

I

,

.

i ,

,

was stronger than hi s desire to eat it, why did he need to push the button i n the fi rst pl ace ?

Observ e that the s t ory I i nc ad m i ts of vari oll s

�'psychological" or temporal explanations � these exp1 an ations do not, howev er, addre s s the paradox, which foc u s e s o n

the

i n d i v i d u al ' s

i' /

moti vation .

Resolution. Gorden argues that in addi tion to our normal desi res ,

there

are second order desi res that one or other of our conflicting (primary) desires win ou t over the competition .

The second order d es i res are

generated by pri mary desires, but not all primary desires generate second order one s

.

In parti cular, onl y stable de s i res

roughly, those that are

based on general i zable principl es - motivate second order desi res. Thus, according to Gorden, the i ndividual 's desire to eat the pie i s actual ly the stronger of the competing desi res, but i t i s i nherently instable and , thus, generates no second order desire. The desire not to eat the pie, however, i s real l y a des i re for good health, which i s a stable desire ; thi s desi re generates a second order desi re which moti vates the agent to intervene against hi s desire to eat the pi e

.

This i ntervention , then, may be either

successful or unsuccessful .

READING

Gorden, Robert M, "Desire and Self-Intervention, " Nous 20 ( 1 986) : 22 1 -

238.

UTILITARIANISM, A PARADOX OF.

Cl osel y related to Singer 's

PARADOX OF EXTREME UTILITARIANISM, thi s ethical parado x , proposed by Frede ri c k Kroon, turns on the introducti on of telepathic students to moral eval uati on.

,, '

I

I

, I

IJictionary of Paradox

212

Formulation.

som eone .

C al l

" A person

I am observ i n g Jones, is i nfl i cti n g p ai n on ,

thi s ac ti on ' A ' . I , a c on fi rm e d utilitarian , am about to pass

moral j ud gmen t on ac ti on A . Under normal circumstances I \\loul d class i fy

A

a\vare of al so kno\v th at

as te rrib l y \vrong, but on this occasion I kno\v that others are

the fact that I am about to pass III oral j udgment o n A , and I

these others , \\'ho are i n s ome \vays superi or to me ( they can d i rec t l y

i ntui t, by a ki nd of m ental telepathy, a pe rson '8 emoti onal responses ,

i nc l ud i ng hi s moral ap proval or disapproval of actions ) , will l e arn frc>ln

nly moral stan ce: if I eval uate A as \vrong , that \\'ill rei nfo rc e thei r hatred of such acts (a very good thi n g as there are many of the s e ,

'

\vi t n ess c s

'

to

my act of ev al ua t i on ) vvhile i f I ev al uate A as ri ght, th at will rei nforce ,

thei r admi ration of such ac ts

(a

very bad th i n g ) .

Suppose

I

kno\v

th at

there are no other moral ly relevant cons e quences and suppose that I c anno t ,

communi cate \vi th thes e ' vvi tnesses ' . fin al

m oral

'

S uppose in fact, that th i s i s the ,

trai ni n g that I , or anyone else, can provide for these vvitnesscs , '

who are due to s p end the rest of the i r l ives on a de sert i sl and , ap p l yi n g the

moral l e s so n t he y \\/ i l l p re s e n tl y l e arn from m e ( \v i th ho r r e n d o u s

consequences , shoul d I approve of c ru e l ty l i ke Jones ' ) .

do? "

What should I

( Kroon 107) .

Expl anation . S i nc e on the ac t utili tarian vie\v, an acti on i s ri gh t or \vrong

d e pe ndi n g on the cons equenc es of th e ac tion, and s i nce no a v ai l ab l e

al ternati ve \vi l l h av e better cons equences than A

( bec a u s e

o n l y by

obs e rv i n g A , together \vith n1 y d i s approv al of A , \vill the \vi tnesses recei ve an e x tre me l y valuabl e lesson) , A m ust be j ud ged a ri ght action. Yet i f Kroon j udg e s A to be ri ght, i t \v i l l have very un fav orabl e consequences

(in the c rue l ty the \vi tnesses vvi ll i n f1i c t on each other on the desert i sland) , and hence A will not be ri ght after al l . Nor can Kroon rem ain neutral or

noncommi ttal \v i th regard to A because thi s reacti on (a further sti p u lati on ) will al s o encourage the \vi tnesses to perform ac ti on s sim i l ar to A . Reso l ution .

Kroon rej ect s as inadequate several Inanners of

"

d r a\v ing

the paradox 's sti ng": first, deni al of the exi s tence of b ei n g s such as the \vitnesses \vho re cog ni ze approval and disapproval \vi thout un d erstan di ng

the i ntel lectual bac k grou nd of these acti ons ; second, asserti on that the ac t

of evaluati on should not c ount as a casual consequence of the act ev a l u ated ; third, restri ction of the eval uation of an action to its intended consequences ; and fo urth , devel opi ng a theory of the context of m oral evaluation . Unl i ke i n S i nger ' s di sc u s s i o n of his qui te similar paradox , Kroon suggests that thi s sort of p a rado x appli e s eq u a ll y to at leas t SOlne versions of rul e

utilitariani sm .

We In ay ob serv e here that there does seem to be an

ambi gui ty i n Kroon 's forn1 u l at i o l1 of the paradox.

Let B be Kroon 's

Dictionary of Paradox

2 13

di sapproval of A . Then i t i s not A alone \v hich has the good consequ ences, but rather the conj unction of A and B . Thus, i t vvould seem that Kroon could j udge A to be vvrong and yet j udge the conj unc tion of A and B to be rig ht.

The paradox then fades into the non-utili tarian trui sm that good

consequences often [ol1o\v from bad actions (it i s indeed this ci rcumstance that has ahvays been troublesome for uti l itari anism) . Perh aps the paradox can be

resurrected

by sti pulating that Kroon have the po\ver to stop A

before i t occurs . On the one hand , since i t i s better for A not to occur than

prevent A from happening. On the other hand , if he stops A from occurri ng, the c onj unction of A and B \v ill not occur� but i t i s better for thi s conj unc ti on to occur than n o t to occur, so for it to occur, Kroon should

Kroon shoul d all o\v A to occur. Presumably, the utili tarian vvoul d have to decide whether the nonoccurrence of A or the j oint occurrence of A and has the preponderance of desirabl e consequences ,

B

but thi s c al c ul us \vould

seem to be confounded if the \vi tncsscs unders tand K roon 's al lo\vancc or

A as a mitigation of B

Kroon, Frederick.

(his

d i s ap p rova l or A ) .

REA 1 )INU

"A. lJtilitari an Paradox . "

VERBAL PARAD OXES.

Ana!.vsis 4 1 . 2 ( 1 ()X ] ) :

J 07 1 1 2 .

See LITERARY PARADOXES , OXYMORON , and

RHETORICAL PARADOXES.

VERIDICAL AND FALSI DI C AL PARADOXES.

This

d i c h oto m y i s 01

due to W. V. o . Quine. A ccording to Quine, a veri dical p aradox sustains "prin1a faci e absurdities by conclusi ve argu ment" (2) , vvhereas a falsidical

paradox "i s one \v hose propositi on not onl y seems at first absurd but i s

al so fal se, there bei n g a fal lacy in the purported proor �

(3 ) .

Whereas a

fal s i di cal p aradox al w ay s contai n s a fal l acy, not al l fal l ac i e s are paradoxi cal .

As exalnples of veridical paradoxes , Qui ne ci tes the fact

that a man m ay be tvventy y ears ol d and yet have had only five bi rthdays ( i f he was born on February 29th) and the B ARBER PARADOX. Fal si dical

paradoxes are exen1pli fi ed by a pu rpo rte d proof of "2

=

1 , " in which

div ision by zero surreptitiously occurs . Qui ne 's noti on of paradox as an argum ent purportedly sustaini ng an at l east seemi n gly absurd conclu sion i s especial l y appropri ate to the LoGICAL PARADOXES � but does not quite capture all the n uances of the term . See DEl'l NITION OF PARADOX . RE�L\DING Quine, W. \T. ()

"Ways of Paradox . " The

Ways of Paradox: And

Essays. Rev. en! . ed . Cambridge: I-Iarv ard UP, 1 976.

Other

214

Dictionary of Paradox

VERIFICATION PRINCIPLE, THE PARADOX OF THE. Logical

positivi sm sought to establ ish a cl ear demarcation between true sci ence and nonsense-that-might-otherwise-pass-itself-off-as-science. The means of achievi ng this not ignoble goal was to be the so-cal led "verification principle," to the effect that statements that were not associated, at l east in theory, with a rule or a procedure for their verification were to be accorded nonsense. I n the i s sue , the u n v eri fi abi l ity ( and hence nonsensicalness) of thi s very principle proved the petard upon which logical positi vism hoist itself. READING Ayer, A. J. ed. Logical Positivism. Glencoe, Ill . : Free P, 1 959 .

From Dionysus the Areopagi te to Meis ter Eckhart, Chri stian mystics have used the via negativa, or apophati c path, to affi rm God 's divinity as a superlogical reality. By sayi ng what God is not, the apophati c path awakens our appreciation of God '8 radi cal othern e s s . I n ev ery day l i fe , m en l o s e thei r s en s e of G o d ' s incomprehensi bility. In order to lead men to confront the inadequacies of the conventional formul as for the divine essence, St. John of the Cross once described God as "dung. " This "negative theology, " which seeks the "unknown God," is the closest the occidental mind reaches to the sense of nothi ngness found in eastern mystici sm and especially in Zen B uddhi sm with its paradoxical koans (see B UDDH I ST PARADOXES) . VIA NEGATIVA , THE.

READING I-Iopkins, Jasper. Nicho las ofCuza s On Learned Ignorance: A Translation and an Appraisal of De Docta ignorantia. Minneapolis: A. J. Benning P,

1 98 1 .

Thi s PARADOX OF D EO NTIC LOG IC contravenes the i ntui tive princi ple that the consequences of obli gations shoul d al so be obl igatory. Formulation. I f a victim bemoans hi s fate at being robbed, a robbery has occurred. But robbery is forbi dden. Therefore , the victim i s forbidden to bemoan hi s fate. Explanation. Given the two premises V I C TI M ' S PARAD O X , T HE .

(1)

If a v ictim bemoans hi s fate at being robbed, a robbery has occurred.

Dictionary of Paradox (2)

215

A robbery has occurred. ,

we may concl ude by modus tallens that (3 )

T he v ictim does not bemoan his fate.

Moreov er ro bbery i s forbi dde n ; that is, (2) i s o b lig atory

Since (3) i s a by the principle cited above , (3) and the fact ( 1 ) con se q uence of (2) i s also obligatory ; thus , the victim is forbidden to be moan his fate. Resolution . Azizah al -Hibri argues that this paradox is easil y resol ved by i ntroduci ng a d y adi c obl i gati on operator into the deonti c formalism . She al so points out the essenti al similarity between the present paradox and that of the GOOD SAMARITAN . ,

.

-

READING

AI-Hibri , Azizah. Deontic Logic. Washington: UP o[ America� 1978.

VOTER 'S PARADOX. See ARRO\V ' s PARA DOX

O} ;

24, 43.

SO( ,I A L C : 1 I0J( ' L .

VOTING, PARADOXES OF. T h i s class of paradox

co nce r n s 1l 1 a ttcrs

of democrati c procedure . 'The paradox es ari s e w hen one or anot. he r tech nique of de tennining the outcome of a vote or el ecti o n i ssues i n res u l ts that are i ntui tively unfair. The class i n cl ude s A NSCOMB E ' S PARA DOX,

A R R O W ' S PARA D O X O F S O C I A L C H O I C E , C O ND ORCET ' S PARA DOX , OSTROGORSKI ' S PARADOX ,

an d

the PARADOX OF NEW MEMB ERS . READINGS

Fi scher, D , and A . S chotter. Redistribution"

"T he Inev itability of the ' Paradox of

in the Allocation of Voting Weights. " Public Choice 3 1

( 1 978) : 49-67. Fishburn, P. C . " Paradoxes of \Toting. " American Po litical Science Review 68 ( 1 974) : 537-.546. Niemi , R. G. , and W. H. Ri ker. " The Choice of Voting S yste ms Scientific American 234 ( 1 976): 2 1 -27. Petit, 1. L. , and E. Terouanne . "A. Stormy Assembly: Electoral Paradoxes. " Theory and Decision 22 ( 1 987) : 27 1 -284. .

"

WANG 'S PARADOX. One of the best kno\vn RECENT SORITES i s Wang 's Parado x proposed by Hao Wang. Here the predicate " l arge" or "small" is

app l i ed to ever decreasing or increasing numbers : "Consi der being

a small

Dictionary of Paradox

216

number. 1 i s s u re l y s mal l . I n fact, since it cannot make much di fference to size, if n i s s ln al l then so is n+ 1 , for any n . B ut then m athematical i nducti on app l i e s . Concl usion : all numbers are small" (Travi s 3 47) , ,

RE.A.DINGS

Dumlnett, j\/Iichael . " \Vang ' s Paradox . "

5)nlhese 30 ( 1 975) :

30 1 -3 24.

Trav is, C: harl e s " \Tagueness , () bserv ation , and Sorites. " Alind TIS 94 ( 1 985) : .

345-366.

WINE AND WATER PROBLE M,} THE

..

See the B EAKER PARADOX .

WOL GAST 9 S PARADOXES O-F KNOWLE DGEo Elizabeth Hankins

has formulated seven paradoxes of kno\vledge, \vhich she takes to spri ng [rom several un tenabl e ass ump li ons about kno\vledge. Si nce n10st of these are equi valent or si mi lar to paradoxes addressed el sevvhere in this volume (one of the s e v e n for exan1 pl e is Moore 's Paradox) , we \vill n ot consider them further here. Wol gast

,

,

REA.DI NGS

Fetzer) James I-I.

\Vol ga st s Paradoxes of Kno\vledge.

"

'

1 2 ( 1 983 ) : 403 -42 1 .

"

Philosophia [I srael ]

Paradoxes of Kno'rvledge.

'Nolgast , El i zabeth l-Iankins .

Ithaca : Cornel l

UP, 1 977.

LavvTence Z alc m an ob serves that m o s t an s w ers but then asks

ZALCMA N ' S PARADOX

questions

are

not

their o\vn

..

,

I s the question "vVhat is a n example of a question which is n ot its own ans\\lcr?" its O\Vll ans vve r? ( 1 60) .

Thi s i s a PaRADOX OF SELF-REFERENCE silnilar to the LIAR. REAI)ING Zalcman, J ..a\vrence. "I ' m Glad You A.sked Me That 48.3 ( 1 988) : 1 60 .

Question.

n

ZENO ' S ARGUMENTS AGAINST PLURALITYo Zeno

Ana(ysis

authored a n umber of paradoxes against pl ural i ty. The argument discussed belo\v i s cited by S i mpl i c i us In Physica 1 4 1 , 6-8, and \vas p ropou n ded i n support of the Eleatic ph ilo so ph y ,

.

21 7

Dictionary of Paradox

Formulation. I f t here are many thi ngs, they must be both i nfi n i te l y smal l and infini tel y l arge. Explanatio n .

Z eno seeln s to have arg ued that \vhatever has size i s

d i v i s ib l e i nto parts and, th u s does not pos sess the unity characteri sti c of ,

a si ngl e thi ng, as oppos ed to nl any things . Hence , none of the s u ppo s ed p l ural i ty of things can have any size . I n contrast, no amoun t of si zeless

thin gs could produce a body with any size . Hence , the parts of any g iven body m ust have some size.

No\v, since each part i s si zable , cach part

lTIUst be di visi ble into parts havi ng some size. B y rei terating the arg un1cnt for each new part, the body i s found to be composed of an infi nite number of parts having some size and, therefore, the body n1ust be infini tely large Resolution .

.

Gregory V l astos points out that th ou g h the pre m i se that a

thing cannot be a si ngle thi n g if d i v i s i bl e i nto parts i s undoubtabl e frolTI

the standpoi nt of El eati c doctrine, it is in fact fal se. A n app l e, for exaJn p l e, has many parts - skin, core, seeds , and \v hut not

Ii I: II "

,I

I I I i I , I ; ,

,

,

I ,,

bu t rClllai n s for al l

that but a single thing. Thus , Zeno 's c( )nc l u s i ()n that th e l11a n y (\ rc i n I 'j n i tc I y small does not follo\v.

Vlas tos then i n v al i datcs the cO lle l lJ s i on t h�l t t he

m an y are i n finitely l arge b y appeal i n g to the 11lod ern n o t i on or co n v erge n t seri es

Thus the sum of an i nfi n i te num ber of nonzcro s i zes In ay be a

fi ni te s ize and agai n Zeno ' s con cl usion docs not fol l o\\'.

See ZLN( )

PARADOXES .

,

s I' ,

READING

\11astos, Gregory. "Zeno of Elea. " Encyclopedia of Phi losophy. FA. Paul Edwards. New York: Macmillan, 1 967. Vol. 8, 369-37 2 .

ZENO 'S DILEMMA. One of Zeno 's Paradoxes, mentioned by A ristotle

at Physics 209a23 -25 and ag ai n at 2 1 0b22- 24. Formulation. Ei the r place does not exi st or, exi sting, m u st have a place

i n which i t exi sts . Explanation.

Zeno was presumably m otivated by a desi re to show that

the very concept of pl ace i s contradictory. He arg ue d that vvhatcver exi sts

must exi st i n some pl ace .

Hence, i f pl ac e exi sts , pl ace exi sts in some

place . By rei teratin g the argument for each ne\vfound pl ace, an, infi nite regress i s obtai ned. Resolution . A ri stotle resolved th e paradox by d i s tingui shing among the

various s ense s of '4to be" ( there was no separate \vord for "to exist") .

Grego ry Vlastos, however, p oi nts out that A ri stotl e 's sol uti on could not

have been advanced i n Zcno's ti me because the re q ui red di stinctions had

not yet been nl ade. T h us th e paradox nl ust hav e bee n genuinely puzzl i n g to Zeno s contemporari es. See ZENO ' S PA R A ! )( )'\ I � S . '

Ii ; ,,

21 8

Dictionary of Paradox

REA DING S

Aristotle. Physics. IV, VI . Vlastos, Gregory. "Zeno of Elea. " The Encyclopedia of Philosophy. Ed. Paul Edwards. New York: Macmillan, 1 967. Vol. 8, 369-372.

ZENO 'S PARADOXES. Zeno of Elea (born c. 490 B C) , a disci ple of

Parmenides , is the most famous paradoxer of all time. Zeno 's basic strategy was to di sprove posi tions contrary to the Parmenidean insi ght that si nce only the unchanging One exists nothing in reality can change. He proposes a series of arguments purporti ng to show that the concepts of the many ( pl ural i ty ) , m oti on ( ch an ge) , and p l ace ( space) w ere inhere ntl y contradictory and led to absurdi ties. A ristotle '8 Physics attributes to Zeno four mind-boggling paradoxes agai nst motion: the Arrow, the Moving Blocks (the Stadium) , the Achilles and the Dichotomy (the Race Course) . T he A rro\v and the Mov ing Blocks are generally interpreted to be predicated upon the assumption that space and time are composed of fi nitely sized indivi sibles or atom ic quanta. The Achi l l es and the Dichotomy, in contrast, presuppose that space and time are composed of infinitesimal "points . " Thus , whatever may be one 's pleasure in regard to the philosophy of space and ti me, the very concept of motion will purportedly sti11 1and one in inextricable difficulties. Less well known than the paradoxes against motion are Zeno 's paradoxes against pI urali ty and against place. The latter is called Zeno 's Dilemma by A ristotle. Zeno 's Paradox of the Mi llet Seed, which attempts to discredit sense know ledge, is, i n contrast to hi s more subtile paradoxes, an easil y controverted pseudo­ paradox. It was because of these arguments that Aristotle calls Zeno the inventor of dialecti cs. See the ACHILLES, the ARROW , the DICHOTOMY , the MILLET SEED , the MOVING B LOC KS ZENO' S ARGUMENTS A GA IN ST PLURALITY , ,

AND ZENO ' S DILEMMA.

READINGS Aristotle . Physics. Bk. IV, 'VI. Bendegem, Jean Paul van. "Zeno's Paradoxes and the Tile Argument. " Philosophy of Science 54 ( 1 987): 295-302. Booth, N. B . "Zeno's Paradoxes. " Journal ofHellenic Studies 77 ( 1957) : 1 87-20 1 . Bouwsma, O . K. Toward a New Sensibility . Lincoln: U of Nebraska P, 1 982. 2 1 3-240. B rochard, Victor. Etudes de Phi losophie Ancienne et de Phi losophie Moderne. Paris: 1 9 1 2. 7-9. Caj ori , F "The History of Zeno 's Arguments on Motion. " Arnerican

l)icliOflt.lI)' oj Paradox

219

22 ( 19 1 5): I · (), 39-47, 77· -82, 109- 1 1 5, 143- 149, 179- 186, 253-258, 292-297.

Mathematical Monlhly Dej nozka, Jan.

"

Z eno ' s Paradoxes and thc Cosll1ologica l Argument. "

Phi losophy of Religion

25 ( 1 989) : 65-8 1 .

Gale, Richard M. , ed. The Philosophy of Time. New York: Anc hor, I 96R.

387-494. Grunbaum, A .

"

Mes srs. Black and Taylor on Temporal Paradoxes. " Analysis

1 2. 6 ( 1 952): 144- 148. "Modem Science and the Refutation of the Paradoxes of Zeno. "

--.

Scientific Month ly

8 1 .5 ( 1955). 234-239.

- - . Modern Science and Zeno s Paradoxes . Middletown, CN : Wes leyan

UP, 1967. Knorr, Wilbur R. "Zeno's Paradoxes Still in Motion. " Ancient Philosophy

3 ( 1 983): 55-66. King, H. R.

"

Ari s totle and the Paradoxes of Zeno. " Jo urna l (�r Ph ilosophy

2 1 ( 1 949): 657-670. Lee , Harold N. "Are Zeno's Paradoxes Based

( 1 %5) 563-70. Lee, H. P. D. Zeno of Efea :

on

a I\1 i st a ke '!" /I. ·1ind

A text wilh Trans/litioN AJld Nol es.

tiS

7- �

( ': l I n h r i dgt'

UP, 1 936. McKie, John R.

"The Persuasiv eness of /,cno 's Paradoxe s. "

and Phenomenological Research Mill, 1. S . System of Log ic.

O w e n G. E. L .

5th ed.

47 ( 1 987) : G� I ·-(>:)!) 389-390.

I )lI i !o.\"0e'1 v

"Zeno and the Mathematicians . " Proceedings

,

of Ihe

Aristo te lian Society Ross, W. D.

58 ( 1 957-58) : 1 99-222. Ari stotle's Physics. Oxford: 1 936. 8 1 -82.

Rossetti, Liv ia. 'The Rhetoric of Zeno's Paradox. �' Philosophy and Rhetoric

2 1 ( 1988): 145- 1 52.

Russell , Be rtrand .

1929. Wes ley.

Norton, Salmon,

Our Knowledge of the External World.

Lectures V and VII. Space, Time and A1otion. Encino, CA : Dickenson,

- - , ed. Zeno s Paradoxes. Indianapoli s : Bobbs , Sherry, Dav id M. Science

New York:

1 975.

1 970.

"Zeno ' s Metrical Paradox Rev isited. "

Phi losophy oj

55 ( 1988) : 58-73 .

Simplicius. In Physica, IV, VI. Szekely, Laszlo. "Motion and the Di alecti c al Vie\v of the World.

H

in So v ie t Tho ught 39 ( 1990) : 24 1 -255.

Tan ne ry Paul. La Geometrie Grecque. Paris : Gauthier-Villars, ,

Studies

1877. 124-

1 25. - -.

Pour I 'Histoire de La Science Hellene.

Paris: Felix Al can ,

1887.

247-2() 1 . - -.

"Le ( � ( )nccpt Sci cntifique de Continu: Zenon d' Elee et Georg e

Cantor. " Re vue J 'hi /osophiq ue de fa France et de l 'Estranger 20

385-4 1 0.

( 1 885) :

220

Dictionar)'

of Paradox

Te I-Iennep e, Eu ge ne "Language Reform and Philosophical �lateria1isnl: A.nother Round With Zeno. �' Analysis 23 [Suppl . ] ( 1 %3): 43-49. .

lJshenko, A . "Zeno 's Paradoxes. " i\4ind 55 ( 1 946) : 1 5 1 - 1 65. Vlastos, Gregory. "Zeno of Elea. " Enc'yc lopedia of Philosophy. Ed. Paul E
ZERMELO K()NIG PARADOX, THE. Accordi ng to Zermelo 's Well­ ..

Ordering Theorem , every set can be ,veil-ordered . This result \vas itself highly surp ri zin g thoug h i t turns out to be equivalent to the AxiolTI of Choice. B y definiti on, two ordered sets are similar if the re is a one-to­ one order preserving function from one of them to the other. The Well ­ Ordering Theorem then al l ow s the fom1ation of the fan1ily of sets simi l ar to a given set and the derivation of a result that c on tr adic ts Cantor 's Theoreln, namely: the cardinality of the po\ver set of this family is less than or equal to the cardinali ty of t h e mentioned family. The parado x may be blocked by i mposing certain conditions on sct formation. See ,

RUSSELL ' S PARADOX .

ZERMELO ' S PARADOX. This paradox in statistical them10dynamics

is essenti al l y similar to that of MAXWELL ' S ..

DEMON .

GLENN W. ERICKSO N , PH. D. Profe ssor Adj unto Departamento de Fi l osofia

Uni vers i dade Fed e ra l do Ri o Grande do Norte, RN '\ B rasi l JOHN A. FOSSA, PH. D. Professor Adj unto Departamento de Matemati ca

U n i versi d ade Federal d o Rio Grande d o Norte , RN , Brasi l

DTP Beth Camara