AN INTRODUCTION TO LOGIC AND ITS PHILOSOPHY

§ 4 RRelattionns bbetweenn Prrooppoosittions 29

Coonnttrraarrieietyty is that sppecies of inconsistency which holds betweeenn two prrooppoositions when although
they cannnnQot both be trruuee, they neverrthelessss can both be false. Consider, forr instance, the relation
betweeenn the contingeenntt prroppoositions

(1.3)) The U.SS.. entered Woorrld Warr I in 11917
and

(1.199)) The U.SS.. entered Woorrld Warr I in 119914.

Plainly, if one member from this contrraarryy pairr is trruuee, the otherr is false. The truth of one eexcludes
the truth of the otherr. So if two prrooppoossittions arre contrraarriiees it must be that at least one is faallse.

u.s.Moreoveerr both may be false. Aftteerr alll,, we can conceive of its haavinng been the caase that the U.S.

entered World Warr I neitherr in 19144 nor in 19177 but rraatthheerr,, let us suppose, in 1916. Thus while
prrooppoossittions (1.3)) and (1.19)) cannot both be trruuee, they can both be false. There is some ppossible
world in which (1.3)) and (1.19)) arre false. Betweeenn themm, then, the prrooppoossittions of a contrraarryy pair
do not exhhaauustt all the possibilities. In short, contrraarriiees divide the sett of all possible worlds into two
mutuaally exclusive subbsets which arre not jointly exhaauussttiive.

Both members of the contrraarryy pairr justt considered arre contingenntt prrooppoossiittiioonnss. Caann noncontingent
prrooppoossittions also be contrraarriiees?? Caann a nonconntingeenntt prrooppoossiittiionn be a contrraarryy of a contingent
prroopposition? As we have here defined ""ccontrraarriieetty"",, the answerr is ""YYees"" to both qquueessttioionnss. I.919
Consider, firstt,, two prrooppoossittions which arre necessaarily false. Since there arre no possible worlds in
which eittherr is trruuee, there is no possible world in which both arre trruuee.. That is to say, since both are
necessaarily false, they cannot both be trruuee. But equaalllly, since both arre necessaarily false, they arre both
false in all possible worrlddss, and hence there is a possible world in which both arre false. Henncce, since
twoo neceessssaarrillyy faalssee pprrooppoositions caannnnoott bbotth bbe trruuee, bbuutt caan bbotthh bbe faalse,, theyy arre ccoonnttrraarriies.
Connsiiddeerr, seeccoonnddly,, a ppaaiirr off pprrooppoositions onnee off whhich is neceessssaarrillyy faalsee and the otthheerr off whhich is
ccoonnttiinnggeenntt.. SSiinnccee tthheerree aarree nnoo ppoossssiibbllee wwoorrllddss iinn wwhhiicchh tthhee nneecceessssaarriillyy ffaallssee pprrooppoossiittiioonn iiss ttrruuee,, tthheerree
ccaann bbee nnoo ppoossssiibbllee wwoorrllddss iinn wwhhiicchh bbootthh iitt aanndd tthhee ccoonnttiinnggeenntt pprrooppoossiittiioonn aarree ttrruuee.. TToo bbee ssuurree;; tthheerree
wwiillll bbee ssoommee ppoossssiibbllee wwoorrllddss iinn wwhhiicchh tthhee ccoonnttiinnggeenntt pprrooppoossiittiioonn iiss ttrruuee.. BBuutt iinn aallll tthhoossee ppoossssiibbllee
wwoorrllddss tthhee nneecceessssaarriillyy ffaallssee pprrooppoossiittiioonn wwiillll bbee ffaallssee.. HHeennccee,, eevveenn iinn tthhoossee ppoossssiibbllee wwoorrllddss iitt wwiillll nnoott
bbee tthhee ccaassee tthhaatt bbootthh aarree ttrruuee.. MMoorreeoovveerr,, bbootthh pprrooppoossiittiioonnss mmaayy bbee ffaallssee.. TThheeyy wwiillll bbootthh bbee ffaallssee iinn
aallll tthhoossee ppoossssiibbllee wwoorrllddss iinn wwhhiicchh tthhee ccoonnttiinnggeenntt pprrooppoossiittiioonn iiss ffaallssee.. HHeennccee,, ssiinnccee ttwwoo pprrooppoossiittiioonnss
oonnee ooff wwhhiicchh iiss nneecceessssaarriillyy ffaallssee aanndd tthhee ootthheerr ooff wwhhiicchh iiss ccoonnttiinnggeenntt ccaannnnoott bbootthh bbee ttrruuee,, bbuutt ccaann
bbootthh bbee ffaallssee,, tthheeyy aarree ccoonnttrraarriieess..

Necessaarily false prrooppoossiittiionnss, it is clearr, arre prroofligate sources of inconsistency. Every necessarily
false prrooppoossiittiionn is a contrraaddiiccttoorryy of, and hence inconsisstteenntt with, every necessaarily trruue pprroposition.
Every necessaarily false prrooppoossiittiionn is a contrraarryy of any and every contingenntt prrooppoossiittiioonn.. Anndd everry
necessaarily false prrooppoossiittiionn is a contrraarryy of every otherr necessaarily false prrooppoossiittiioonn.. Inddeed., we
need only addd that the term ""sself-iinnccoonnssiisstteenntt"" is a synonym forr the terrmm ""nneeccessarily false"", in
orderr to conclude that a necessaarily false prrooppoossittiionn is inconsisstteenntt with every prrooppoossiittiionn whatever,
including itself, i.e., that a necessaarily false prrooppoossiittiionn is s^e/l/f'-inconsistent.

From the fact that two prrooppoossittions arre inconsisstteenntt it follows that at least one is acttuuaalllyy faallse.
Foorr since, by the definition of ""iinncconsistenncyy"", there is no possible world in which both memberrs off
an inconsisstteenntt pairr of prrooppoossittions arre trruuee, every possible world -— includdiinngg the actuaal world -— is
a world inn whhichh aat least one of tthhem is faalse. IInncconnsistency, we mmaay saay, pprroovviiddeess aa guarraannttee of
falsity. Buut thhee coonnvveerrsee ddooes nott hold. Frromm thhee faactt thhaatt onnee orr bbotth off a ppaaiirr off pprrooppoossittions is

19. Historrically some logicians haave used the terrmms ""ccontrraaddiiction"" anndd ""cconttrraarriieettyy"" as if they apppplliieedd only
in caasseess inn whichh bothh prrooppoossittiionns arre contingent. Forr morree onn thhis, see the subsection entitled ""AA NNoottee oonn
History camndd Nomennccllaattuurree"",, ppp. 53--5544..

30 POSSSIIBBLLEE WWOORRLLDDS

aacttuuaallly faalse, it ddooeess nnot follow thhaat thhey aarree innconnsistent, i.e., thhaat inn everry ppoossibble world onne oorr
bbothh of thhemm is faallsee. Thus (11..44)) aanndd (11..1199)) aarree faalse inn thhe aaccttuuaall worrldd. Yeett thhey aarree not
innconnsistentt. This is fairrly eaassy ttoo shhoww. CConsider thhe faact thhaat (11..1199)) is contingent aanndd hhennce is ttrrue
in somme ppoossibble worrllddss,, inn paarrttiicularr, inn aallll tthhoossee ppoossibble worrllddss inn whhiicchh thhe U..SS.. ennterredd
World WWaarr II inn 1914. Buutt inn eaachh of tthheessee ppoossibble worrllddss,, it tuurrnnss out thhaat (11..44)) is aalso trruuee:: aannyy
ppoossibble world inn whhiicchh thhe U..SS.. enterredd World Waarr II inn 1914 is aalso aa world inn whhiicchh it is nnot the
caassee thhaat thhe U..SS.. enterredd World WWaarr II inn 1917. Anndd thhiis is juust to saay thhaat inn aallll tthhoossee possible
worrlddss inn whhiicchh (1.1199)) is trruuee,, (11..44)) is aalso trruuee.. HHeennccee thherre is aa ppoossibble world inn whhiicchh tthese
aacttuuaallly faalse pprrooppoossiittiioonnss aarree trruuee togetherr. IInn shhorrtt,, frrom thhe faactt thhaat thhey aarree faalse inn faactt it ddoes
nnot follow thhaat thhey aarree innconnsistentt. Beinngg faalse inn thhe aaccttuuaall worrldd, it tuurrnnss ouutt, pprroovviiddeess no
gguuaarraanntteeee ooff iinnccoonnssiisstteennccyy..

EEXXEERRCICSEISE

Can two propositions be contraries as well as contradictories oj one another? Explain your answer.

Can two propositions be contraries as well as contradictories of one another? Explain your answer.

*****

Consistency

WCohnastistdeonecsy it mean to say that two propositiOns are consistent with one another? Given that we

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propositions ((71..33)) TThhee UU..SS.. eenntteerreedd WWoorrlldd WWaarr II iinn 11991177
aanndd ((71..2200)) LLaazzaarruuss LLoonngg wwaass bboorrnn iinn KKaannssaass iinn 11991122..

WWhhaatteever other rreellaattiioonn mmaayy hold bbetweenn thheemm,, pplainly thhe rreellaattiioonn of connsistency ddooeess:: it nneedd not
bbe thhe caassee thhaat if onne is trruuee thhe other is faalse;; bbotthh caann bbee ttrruuee.. Noo mmaatttterr whhaat thhe faacts hhaapppenn
ttoo bbee aabboouutt thhe aaccttuuaall world (no mmaatttteerr,, thhaat is, whhetherr either (11..33)) orr (11..2200)) is aaccttuuaallllyy trruuee)),, it is
ppoossibble thhaat bbootthh of thhemm shhouulldd bbee ttrruuee -— whhiicchh is juust tto saay tthhaat tthherre is aat least onne possible
world inn whhiicchh bbotthh aarree ttrruuee..

Iff two contingent pprrooppoossiittiioonnss hhaappppeenn bbotthh to bbe trruuee inn thhe aaccttuuaall worrldd, thhenn since thhe aaccttuual
world is aalso aa ppoossibble worrld, thherre is aa ppoossibble world inn whhiicchh bbotthh aarree trruuee,, aanndd hhennce thhey aare
connsistent. Actual trruutthh, thhaat is to saayy, pprroovviiddeess aa guaarraanntteee of connsistenncy. Buutt thhe converrse ddooeess not
holdd. Frrom thhe faactt thhaat twwo pprrooppoossiittiioonnss aarree consisstteenntt it ddooeess nnott follow tthhaatt tthhey aarree bbootthh ttrruuee iinn
tthhe aaccttuuaall worrldd. What ddooeess.Jfoollooww is thhaat thherre is soommee ppoossibble world inn whhiicchh bbootthh aarree trruuee;; yet
thhaat ppoossibble world mmaayy bbee nnoonn--aaccttuuaall.. Thuus (1.33)) aanndd (11..2200)) aarree connsistent. Yeett tthheey aarree nnott bbotthh
trruue inn tthhe aaccttuuaall worrldd. (1..33)) is trruuee inn thhe aaccttuuaall world aanndd faalse inn somme nnoonn--aaccttuuaall worrllddss,, whhile
(71..2200)) is faalse inn thhe aaccttuuaall world aanndd trruuee inn somme nnoonn--aaccttuuaall worrllddss.. HHeennccee pprrooppoossiittiioonnss caann bbe
consisstteenntt even if onne orr bbootthh is faallsee. IInn shhorrtt,, connsistency ddooeess nnott pprroovviiddee aa guaarraanntteee of truuthh.

Sinnce aa nneecessarriily ttrruuee pprrooppoossiittiioonn is trruuee inn aallll ppoossibble worrllddss,, aa nneecessarriily trruuee pprropositioonn

§ 4 RRelations bbettwween Prrooppoossitions 31

will be consisstteennt witthh any prrooppoossiittIiOon which is true in at leeaast onne possiibbllee world. Itt will bbee
consisttent, thaatt is, witthh any coonnttiinnggeenntt prrooppoosition and witthh any necessssaarrillyy true prrooppoossittionn. Itt will
be inncoonnssiisstteenntt only witthh thhoose prroppoositionns which arre not true in any possiibbllee worldds, i.e.,, wwiitthh
necessssaarrillyy false oonnes.

Impliccaattiioonn

Off the fouur mmoddaal rrelations we arre currrreennttlly considerrinng, it is prroobbaabbllyy impplicattionn which is mostt
closellyy identtified, in mmostt perrsons'' mminnddss, witth the connceerrns of phhilosoopphy in gennerral andd of logic iinn
parrtticular. Phhilosophhy, abbove alll,, is connceerrned with the purrssuuit of truth; andd logic —- its hhaanndmaidenn
—- witth discoveerring neww trruuths onnce esttaabliisshheedd onnees arre within ouurr graaspp. To be surre, phhilossoopphheerrs
and logicciiaanns alike arre connceerrned to avoid inconsisteennccyy (since the inconsisstteennccyy of two prrooppoositions is
a suufficciieenntt conddittionn of tthhe falsittyy of aat leaastt one of tthhemm) aanndd tthhuus tto preservvee connssiisstteennccyy (s(sinincceeththee
consisstteennccyy of two prrooppoositions is a neceessssaarryy —- but not suffiicciieenntt —- condition of the truthh of both).
But it is in tracingg implicaattiioonnss that they mmostt obviously addvannce theirr commmmonn concerrn with the
discoveerry of new trruuthhs on the bbaassis of onneess already establisshheedd. Forr impplicattionn is the rrelation wwhhicchh
holds bbetween an ordderredd pairr of prrooppoositions whenn the firrst cannott be trruue witthhoouutt the seccoonndd aalso
bbeing trruuee, i.e., when the trruutthh of the first is aa sufficciieenntt conddition of the trruutthh of the second.

Like the rreelations of inconsisstteennccyy aanndd consisttenncyy,, the rreelaattion of immppllicaattionn cann bbe ddefinnedd inn
termms of ourr taalk of ppossible worrllddss. HHerre aarre thrree equuivaleenntt wwaayyss of so ddefining it:

(a) aa pprrooppoossittion PP immppllies aa pprrooppoossittion Q if aanndd only if Q is trruue inn
all tthhoossee ppossible worrlddss, if aannyy, in whhich PP is truue;

(b) aa pprrooppoossittion PP immppllies aa pprrooppoossittion Q if aanndd only if there is no

ppossible worrldd in whhich PP is trruue aanndd Q ffaalsles;e2;o20

(c) aa pprrooppoossittion PP immppllies aa pprrooppoossittion QQ if aanndd onnly if inn each ooff

all ppoossible worrldds if PP is ttrruue tthhen Q is aallssoo trruuee.

As aann exampple of tthhe rreelaattionn of immpplliicaattiioonn considderr tthhe rreelaattionn whhichh tthhe prroppoosition

(71.33)) TThhee U..SS.. entered World WWaarr II inn 11917

hhaas tto tthhe prroppoosition

(71.2271)) TThhee U..SS.. entered World WWaarr II bbeforre 1199220.

WWhhaatteever otherr rreelaattions mmaayy hhooldd bbeettwweeenn (71.33)) aanndd (71.271), pplaaiinnllyy tthhe rreelaattionn of iimmplicaation
ddooeess.. AAlll tthhrreee of tthhe aabboovve ddeefinitions aarree saattisfieedd inn tthhiis ccaassee.. TThhuuss:: [ddeefinition (aa)] (71..2271)) is trruue
in aallll tthhoossee ppoossible wworrllddss inn wwhhiichh (71..33)) is ttrruuee;; [ddeefinition (bb)] tthherre iss nnoo ppoossible wworrlldd inn which
(71..33)) is ttrruuee aanndd (71.-2271)) is faallse;; aanndd [ddeefinition (c)] inn eaachh of aallll ppoossible wworrllddss,, if (71..33)) is trruue
tthhenn (71.21/)) is trruuee.

To saay tthhaat aa pprrooppoossiittiionn Q followwss fromm aa pprrooppoossiittiionn PP is jjuust ttoo saay tthhaat PP implies QQ.. HHennce

tthhe rreellaattiionn of following frroomm,, likkee its connverrsse, caann bbee expplained inn tterrmms of ppoossible wworrllddss.. IInnddeeedd,

tthhe explanaattion caann bbee given bbyy tthhe simmpple expedieenntt of suubbsttituting tthhe wworrddss ""aa pprrooppoossiittiionn Q

2200.. DDeeffiinniittionn ((bb)),, iitt sshhoouulldd bbee nnootteedd,, aammoouunnttss ttoo ssaayyiinngg tthhaatt PP iimmpplliieess QQ iif aanndd oonnllyy iif tthhee ttrruutthh ooff PP is

innccoonnssiisstteenntt wwitithhtthheeffaalslsiittyy ooffQQ.. IImmpplilcicaatitoionn, ,ininsshhoorrt,t,isisddeeffiinnaabblele inintteerrmmssooffininccoonnssisistteennccyy..

3322 POSSIBBLE WWORRLLDSS

follows from a propositionn P"" for the words "aa propositionn P implies a propositionn Q" in each of tthe
definitions (a)),, (b),, and (c)) aabboovve.

In terms of the relatiioonn of implication (anndd hennce in terms of our talk of possible worlddss) we can
also throw light on another imporrttaanntt logical conceptt: that of the deedduuccttiivvee vaalliidditiyty of an inferennce or
argumenntt. True, we have not hitherto had occasion to use the words "deductivveellyy valid". YYeett it will
be eviddenntt to anyone who has even a superficial understaannddiinngg of the meanniinnggss of thhese words that
much of our discussionn has connssiisstteedd in marshalliinngg deductivellyy valid arguments and drawing
deducttiivveellyy vaalliidd infeerreenncceess..2211 Tiime anndd aggaaiinn wee have siiggnnaalleedd thhee ppresennce off arrgguummennts anndd
inferennces bby meanns off suuchh woorrddss ass "hhennce",, "ccoonnsseeqquueennttllyy"",, "tthherefore", anndd "iitt follows thhaatt"";
aanndd iimmpplliicciittllyy wwee hhaavvee bbeeeenn ccllaaiimmiinngg tthhaatt tthheessee aarrgguummeennttss aanndd iinnffeerreenncceess aarree ddeedduuccttiivveellyy vvaalliidd.. BBuutt
wwhhaatt ddooeess iitt mmeeaann ttoo ssaayy tthhaatt aann aarrgguummeenntt oorr iinnffeerreennccee iiss ddeedduuccttiivveellyy vvaalliidd?? AAss aa pprreelliimmiinnaarryy iitt mmaayy
hheellpp iiff wwee rreemmiinndd oouurrsseellvveess ooff ssoommee ffaammiilliiaarr ffaaccttss:: tthhaatt iitt iiss pprrooppoossiittiioonnss ffrroomm wwhhiicchh aanndd ttoo wwhhiicchh
iinnffeerreenncceess aarree ddrraawwnn.. CCoonnssiiddeerr,, tthheenn,, tthhee ssiimmpplleesstt ssoorrtt ooff aarrgguummeenntt ((oorr ccoorrrreessppoonnddiinngg iinnffeerreennccee))
wwhhiicchh ffeeaattuurreess jjuusstt oonnee pprrooppoossiittiioonn aass iittss pprreemmiissee aanndd jjuusstt oonnee pprrooppoossiittiioonn aass iittss ccoonncclluussiioonn;; aanndd lleett
uuss ddeessiiggnnaattee tthhee pprreemmiissee "" PP "" aanndd tthhee ccoonncclluussiioonn "" QQ"".. TThheenn wwee ccaann rreeffoorrmmuullaattee oouurr qquueessttiioonn bbyy
aasskkiinngg:: WWhhaatt ddooeess iitt mmeeaann ttoo ssaayy tthhaatt aann aarrgguummeenntt oorr iinnffeerreennccee ffrroomm PP ttoo QQ iiss ddeedduuccttiivveellyy vvaalliidd??
TT oo ssaayy tthhaatt aann aarrgguummeenntt oorr iinnffeerreennccee ffrroomm aa pprrooppoossiittiioonn PP ttoo aa pprrooppoossiittiioonn QQ iiss ddeedduuccttiivveellyy vvaalliidd iiss
jjuusstt ttoo ssaayy tthhaatt PP iimmpplliieess QQ,, oorr ((ccoonnvveerrsseellyy)) tthhaatt QQ ffoolllloowwss ffrroomm PP..2222 DDeedduuccttiivvee vvaalliiddiittyy,, tthheenn,, wwhhiicchh
iiss aa pprrooppeerrttyy ooff aarrgguummeennttss oorr iinnffeerreenncceess,, ccaann bbee eexxppllaaiinneedd iinn tteerrmmss ooff tthhee mmooddaall rreellaattiioonnss ooff
iimmpplliiccaattiioonn aanndd ffoolllloowwiinngg ffrroomm.. AAnndd,, lliikkee tthheemm,, iitt ccaann bbee eexxppllaaiinneedd iinn tteerrmmss ooff ppoossssiibbllee wwoorrllddss.. IInn
tthhiiss ccaassee,, wwee nneeeedd oonnllyy aaddoopptt tthhee eexxppeeddiieenntt ooff ssuubbssttiittuuttiinngg tthhee wwoorrddss ""aann aarrgguummeenntt oorr iinnffeerreennccee ffrroomm
PP ttoo QQ iiss ddeedduuccttiivveellyy vvaalliidd"" ffoorr tthhee wwoorrddss ""aa pprrooppoossiittiioonn PP iimmpplliieess aa pprrooppoossiittiioonn QQ"" iinn eeaacchh ooff tthhee
ddeeffiinniittiioonnss ((aa)),, ((bb)),, aanndd ((cc)) aabboovvee..

AA casual reading of our thhrree definitions of "implication" may suugggest to SsoOmlue thhat only true
proposittiioonnss cann have implications. After all,, we defined "implication", in (c)) for instanncce, as the
relatiioonn which holds between a propositionn P and a propositionn Q when in all possible worlds if P is

true thenn QQ is also true. Andd we illustraatteedd the relatiioonn of implication by citing a case where a true

proposition, viz..,, (1-.33)),, stoodd in that relatiioonn to another true proposition, viz..,, (1..2211)).. Does this
meeaann thhatt false pprooppoossiittiioonnss caannnnoott hhave impplicaattiioonnss?? Doeess itt meeaann,, too ppuutt thhee quessttiioonn in ootthheerr
wwoorrddss,, tthhaatt nnootthhiinngg ffoolllloowwss ffrroomm ffaallssee pprrooppoossiittiioonnss,, oorr tthhaatt ddeedduuccttiivveellyy vvaalliidd aarrgguummeennttss ccaannnnoott hhaavvee
ffaallssee pprreemmiisseess??

Not at alLl. On a more careeffuull reading of thhese definitions it will be seen that they say nothing
whhatever about the actual truthh-values of P or Q;; i.e., that they say nothing at all about whether P
or Q aree true in the actual world. Hencee they do not rule out the possibility of a propositionn P
implying a propositionn Q when P is not true but false. In (c)),, for instanncce, we merelyy said that

where P implies Q, in all possible worlds Q will be true iff P is true. We have not asserrttedd that P is

trruue in thhee acttuuaall worrlldd bbuutt meerreellyy ennterrttaaiinneedd thhee ssuuppppoosistiitoinon thhatt P is trruue in some worrlldd orr ootthher;
anndd thhatt is soommeetthhiinngg wee caann do evveenn in thhee caasse where P is false in thhee acttuuaall worrlldd orr evveenn wwhere
PP iiss ffaallssee iinn aallll ppoossssiibbllee wwoorrllddss..

211.. WWhheen persons draww a coonncclluussiioonn outt of a propoossitioon or a sett of propoossiittiioonnss they caan be correctly said
to be "inferring a proposition". Inferringg is sommeetthhiinngg persons doo;; it is noott a logical relaatioonn between
propoossiitioonss.. Itt is not only graammmmaattiiccaalllyy incorrect to speak of one propoossiitioon inferring another, it is logically
coonnffusseedd as weell. See H.WW. FFoowwlleerr,, A Diiccttioionnarayry ooff Mooddeernrn EEnngglilsihsh Usagee,, 2nndd EEddiittioion,n, revised by Sir
EErrnest Goowweers, Oxxford, Claarendon Press, 1996655, , p. 228822.

222.. NNoote that here we aree defininngg "deedduuccttiivvee validityy"", not "vvaalidityy"" ppeerr see. LLaater, in chapter 4, we shall
define the wider concept of vaalidityy.

§ 4 RReellaattiioonnss bbeettwweeeenn PPrrooppoossiittionns 33

MMorreeoverr, we mmight just aas eaasily hhaavve chosen to illuusttrrate thhe rreelaattiionn of immpplliicaattiioonn bbyy citinng aa
caassee where aa false pprrooppoossiittiionn immpplliies aannotherr pprrooppoossiittiioonn.. CCoonnssiddeerr suuchh aa caassee.. TThhee pprrooppoossiittiioonn

(71.7199)) TThhee uU..sS.. entered World WWaarr II inn 119914

is continngeenntt aanndd hhaappppenns to bbe faalse; it is false inn thhe aactuuaall worrlldd even thhouugh it is trruue inn aat leeaastt
sommee nnonn-aactuuaall worrllddss.. Does thhis (actuuaallly) false pprrooppoossiittiionn hhaavve aanny immppllicaattionnss?? ((EEqquivalently:
Do aanny otherr pprrooppoossiittionns follow from (1.19)?? CCaann aanny otherr pprrooppoossiittiionn bbe inferrreedd with ddedduuccttive
validity frrom (1.719)??)) OObbvviioouussllyy ennouughh, thhe aannswerr is:: Yeess.. A false pprrooppoossiittiioonn,, like aa trruue one,
will hhaavve counnttlesss immpplliiccaattiioonnss.. FFoorr instannce, thhe false pprrooppoossiittiionn (1i..19) immpplliies aallll thhe ccoouunnttlleess
pprrooppoossiittionns that we couuldd expprreessss bbyy uutttterriinngg aa senntteennccee of thhe form ""TThhee U..SS.. enterredd
World WWaarr II bbeforre ... "" aanndd filliinngg inn thhe bblannkk with thhe sppecificcaatioonn of aannyy ddaatte whaateveerr laatteerr
thann 1914, e.g., 1915, 1916, 1917, etc., etc. TThhee crruucial ddifferrence bbeettwweeenn thhe immppllicaattionnss of aa faalse
pprrooppoossittion aanndd thhe immppllicaattionnss of aa trruue pprrooppoossiittiionn lies inn thhe fact that onn thhe one hhaanndd,, aa faalse
pprrooppoossiittiioonn hhaass iimmpplliiccaattiioonnss ssoommee ooff wwhhiicchh aarree ffaallssee -— aass iiss tthhee pprrooppoossiittiioonn tthhaatt tthhee UU..SS.. eenntteerreedd
WWoorrlldd WWaarr II bbeeffoorree 11991155 -— aanndd ssoommee ooff wwhhiicchh aarree ttrruuee -— aass iiss tthhee pprrooppoossiittiioonn ((11..2211)) tthhaatt tthhee
UU..SS.. eenntteerreedd WWoorrlldd WWaarr II bbeeffoorree 11992200 -— wwhhiillee,, oonn tthhee ootthheerr hhaanndd,, aa ttrruuee pprrooppoossiittiioonn hhaass
iimmpplliiccaattiioonnss aallll ooff wwhhiicchh aarree ttrruuee..

HHerre, thheenn, aarree two impporrttaanntt logicaal facts aabbout thhe rreelaattiionn of immpplliiccaattiioonn., (i) Allll the
immppllicaattionns of aa trruue pprrooppoossiittiioonn hhaavve thhe same trruutthh--vvaalluuee aas that pprrooppoossiittiioonn,, i.e., thhey ''pprreeserrve'
its trruutthh. FForr thhis rreeaason immpplliicaattiioonn is saaidd to bbe aa truth-presseerrvviinngg rreellaattiioonn.. IInn trraacciinngg out the
immppllicaattionns of aa trruuee pprrooppoossiittiionn we cann bbe led onnly to furrtthherr trruuee pprrooppoossiittiioonnss, nnever to false ones.
Or, inn otherr worrddss,, thhe onnly pprrooppoossiittionns that follow frrom orr cann bbe inferrreedd with ddedduuccttiive vvaaliddity
frrom pprrooppoossiittionns whhichh aarree trruuee aarre pprrooppoossiittionns whhichh aarree aalso trruuee., (ii) TThhee immppllicaattionnss of aa faalse
pprrooppoossittion nneed nnot hhaavve thhe same trruutthh--vvaalluuee aas that pprrooppoossiittiioonn.. Somme of thhe immppllicaattionnss of a
false pprrooppoossiittiionn aarree tthheemmsseellvveess faalse; bbuut others aarree trruuee.. IImmpplliiccaattiioonn,, we mmaayy saay, is nnoot
falsity-prreserrvvinngg. Among thhe pprrooppoossiittionns whhichh follow frrom orr cann bbe inferrreedd with ddedduuccttive
validity frrom pprrooppoossiittionns whhichh aarree faalse, thherre aarree sommee trruuee pprrooppoossiittionns aas well aas sommee faalse
onneess..

TThheese simpple logicaal facts hhaavve impporrttaanntt pprraaccttiiccaall aanndd mmeetthhooddoollooggicaal aapppplliicaattionns whhen it ccoommes
to thhe ppuurrssuuiitt of trruutthh. Byy virttuuee of (i), it follows that one of thhe wwaayyss inn whhichh we cann aaddvvaanncce the
frronnttierrs of hhuummaann kknnoowwledge is simmpplly to rreeflectt uuppoonn,, orr rreeaason ouutt, thhe immppllicaattionnss off
pprrooppoossiittionns we aalrreaaddyy kknnoow to bbe trruuee.. Thhis is, ppaarraaddiiggmmaatitcicaalllyy,, thhe waay inn whhichh aaddvvaanncces are
mmaadde inn mmaatthheemmaattiics aanndd logic. Buutt it is aalso thhe waay inn whhichh uunnrreeccooggnniizedd trruutthhs cann be
ddiscovered inn otherr aarretas aas well. Many of thhe aaddvvaanncces mmaadde inn technnology aanndd thhe aapppplliieedd scieences,
forr instanncce, occurr bbecause sommeeonnee hhaas rreeaasoned out forr ppaarrttiiccuullaarr circummstaanncces thhe immppllicaattionnss ooff
uunniivveerrssaall pprrooppoossiittiioonnss aallrreeaaddyy aacccceepptteedd aass ttrruuee iinn tthhee ppuurree sscciieenncceess.. BByy vviirrttuuee ooff ((iiii)),, iitt ffoolllloowwss tthhaatt
wwee ccaann aallssoo aaddvvaannccee tthhee ffrroonnttiieerrss ooff hhuummaann kknnoowwlleeddggee,, nneeggaattiivveellyy aass iitt wweerree,, bbyy tteessttiinngg tthhee
iimmpplliiccaattiioonnss ooff hhyyppootthheesseess ,wwhhoossee ttrruutthh--vvaalluueess aarree aass yyeett uunnkknnoowwnn,, wweeeeddiinngg oouutt tthhee ffaallssee hhyyppootthheesseess,,
aanndd tthhuuss nnaarrrroowwiinngg ddoowwnn tthhee rraannggee ooff aalltteerrnnaattiivveess wwiitthhiinn wwhhiicchh ttrruutthh mmaayy yyeett bbee ffoouunndd.. AAnn
eexxpplloorraattoorryy hhyyppootthheessiiss iiss ppuutt ffoorrwwaarrdd aanndd tthheenn tteesstteedd bbyy sseeeeiinngg wwhheetthheerr iittss iimmpplliiccaattiioonnss ''hhoolldd uupp'' ((aass
wwee ssaayy)) IiInI tthhee lliigghhtt ooff eexxppeerriieennccee.. OOff ccoouurrssee,, iiff aa hhyyppootthheessiiss hhaass iimmpplliiccaattiioonnss wwhhiicchh eexxppeerriieennccee
sshhoowwss ttoo bbee ttrruuee,, tthhiiss ddooeess nnoott eennttiittllee uuss ttoo ccoonncclluuddee tthhaatt tthhee hhyyppootthheessiiss iittsseellff iiss ttrruuee.. FFoorr aass wwee hhaavvee
sseeeenn,, tthheerree aallwwaayyss aarree ssoommee iimmpplliiccaattiioonnss ooff aa pprrooppoossiittiioonn wwhhiicchh aarree ttrruuee eevveenn wwhheenn tthhee pprroopp--
oossiittiioonn iittsseellff iiss ffaallssee.. BBuutt iiff,, oonn tthhee ootthheerr hhaanndd,, aa hhyyppootthheessiiss hhaass aannyy iimmpplliiccaattiioonnss wwhhiicchh eexxppeerriieennccee
sshhoowwss ttoo bbee ffaallssee,, tthhiiss ddooeess eennttiittllee uuss ttoo ccoonncclluuddee tthhaatt tthhee hhyyppootthheessiiss iittsseellff iiss ffaallssee.. FFoorr aass
wwee hhaavvee sseeeenn,, tthheerree ccaann bbee nnoo ffaallssee iimmpplliiccaattiioonnss ooff aa pprrooppoossiittiioonn iinn tthhee ccaassee wwhheerree tthhaatt pprrooppoossiittiioonn
iittsseellff iiss ttrruuee.. HHeennccee iiff aannyy ooff tthhee iimmpplliiccaattiioonnss ooff aa hhyyppootthheessiiss ttuurrnn oouutt ttoo bbee ffaallssee,, wwee mmaayy vvaalliiddllyy
iinnffeerr tthhaatt tthhaatt hhyyppootthheessiiss iiss ffaallssee..

34 PPOOSSSSIIBBLLEE WWORLDS

Theessee two facttss about thhee relation of implicationn aree reflleecctteedd in thee staannddaarrdd methodoollooggyy (or
'logic' as it is oftteenn called) of scientific enquiry. Thee cost of ignoring them, when onee is conducting
scienttiiffiicc resseeaarrcchh orr when onee is pursuiinngg knowledge in anyy fieldd whatteevveerr,, is that thee discovery ooff
truutthh theenn beccoommeess a completely happhhaazzaarrdd matter.

TTwwoo furrtthheerr impoorrttaanntt logical facttss about thee relation of implicationn deserve notice andd ddiscussion.
Itt follows from thee definitions given of implicatioonn that: (iii)) a necessarily false proposition implies
any and every proposition; and (iv) a necessarilyy true proposition is implied by any and every
proposition whateveerr.. Conclusioonn (iii)) follows from thee fact that, if a proposition P is necessarily
false thenn there is no possible world in which P is true and a fortiioorrii no possible world such that in

itt both P is true and some otheerr proposition QQ is false; so that [by definition (b)] P mustt bbe said to
iimmppllyy QQ.. CCoonncclluussiioonn ((iivv)) ffoolllloowwss ffrroomm tthhee ffaacctt tthhaatt iiff aa pprrooppoossiittiioonn QQ iiss nneecceessssaarriillyy ttrruuee tthheenn tthheerree iiss nnoo

ppoossssiibbllee wwoorrlldd iinn wwhhiicchh QQ iiss ffaallssee aanndd aa ffoorrttiioorrii nnoo ppoossssiibbllee wwoorrlldd ssuucchh tthhaatt iinn iitt bbootthh QQ iiss ffaallssee aanndd

ssoommee pprrooppoossiittiioonn PP iiss ttrruuee;; ssoo tthhaatt [[aaggaaiinn bbyy ddeeffiinniittiioonn ((bb))]] QQ mmuusstt bbee ssaaiidd ttoo bbee iimmpplliieedd bbyy PP..

Thessee conclusions, however, strike many persons as counterintuitive. Surely, it would be said, the
necessarily false proposition

(1.6) The U.SS.. entered World War I in 1917 and it is not the case that the UU..S.
entered World War I in 1917
entered World War I in 1917

does not imply the proposition
does not imply the proposition

(1.2) Canada is south of Mexico.
(1.2) Canada is south of Mexico.

And surely, it again would be said, the necessarily true proposition
And surely, it again would be said, the necessarily true proposition

(1.7) If some thing is red then it is colored
(1.7) If some thing is red then it is colored

is not implied by the proposition
is not implied by the proposition

(1.20) Lazarus Long was born in Kansas in 1912.
(1.20) Lazarus Long was born in Kansas in 1912.

Foorr the propositions in the first pair have 'nothing to do with' each other; they are not in any sense
about the same things; on~e is 'irrelevant' to the other. Andd the same would be said for tthe
propositions in the second pair.

These admittedly counterintuitive results are ones to whichh we devote a good deal of discussion in
chaptteerr 4, section 6, pp. 224-30. For the present, justt three brief observatioonnss must suffice.

Inn the first place, (iiii)) and (iv)) ought not to be viewed solely as consequueenncceess of some recently
developed artificial definitions of impliiccaattiioonn.. They are consequeenncceess,, rather, of definitions which
philosophers have long been disposed to givvee;; indeed, comparable definitions were adopted by, and
the consequeenncceess recognized by, many logiciaannss in medieval times. Moreovveerr,, they are iimmmmeediate
(even if not immediately obvious) consequeenncceess of the definitions whicchh most of us wouldd nnaattuurally
be inclliinneedd to givvee:: as when we say that one proposition implieess another if the latter can't possibly be
false if the former iss true; or again, as when we say that one proposition implieess another just when iif
the former is true then necessaarriillyy the latter is trruuee..2233 Once we recognize this we may be more rready

23. For ddiscuusssionn of an ammbbiigguuiittyy,, and a ppoosssibbllee phhiilosophical confusion, lurking in tthheessee naturraal waayyss of
speakkiinngg,, see chapterr 6, sectionn 3.

§ 44 RReellaattiioonnss bbeettwweeeenn PPrrooppoossiittionns 35

tto eduucate ourr inttuuittions tto tthhe ppoint of rreeccooggnniizzinngg (iii) aanndd (iv) aas tthhe immpporrttaanntt logicaal truuthhs

whhich tthhey arre.

Seconnddllyy, it is nnot hard tto uunnddeerrssttaanndd whhy ourr uunnedduucaated inttuuittions ttend tto rreebbeell aat accepting
(iii) aanndd (iv). FFoorr tthhe pplaainn fact of tthhe mmaatttterr is tthhat mmoost of tthhe instances of immpplliiccaattiioonn whhichh we

aarree likkeelyy tto tthhinkk of inn connnection with tthhe inferrencess we pperrforrmm inn ddaaiillyy life, orr inn ssciennttific
enqquuirryy, aarree instances inn whhichh tthhe rreelaattiionn of immpplliiccaattiioonn hhooldds bbeettwweeenn connttiinnggeenntt prrooppoossitions;
andd one continngeenntt pprrooppoossiittiioonn,, aas it hhaappppeennss, immpplliies aannotherr onnly if tthherre iss aa cerrttaainn mmeeaasurre ooff
''rreellevannce' tto bbee founndd bbeettwweeenn tthhemm -— onnly if tthhey aarree,, inn sommee sseennssee,, ''aabboouut'' tthhe samme tthhiinnggss. Not
surrpprriissiinnggllyy, tthheenn, we aarree innclinedd tto indduullge ourr aall-too--ccoommmmon ddisppoosition tto generalize -— to
suuppppoose, that is, tthhat aallll ccaasseess of immpplliiccaattiioonn mmuust bbe like tthhe oonneess with whhichh we aarree mmoost fammiliarr.
Hadd we, frrom tthhe bbeeginnnniinngg,, aatttendedd bbothh tto tthhe conseqquuenncceess of ourr ddeefinitionns aanndd tto tthhe fact that
tthhey aallow of aapppplliiccaattiioonn tto nnonncontingeenntt pprrooppoossiittionns aas well aas continngeenntt oonneess,, we mmight neveerr
hhaavve coomme tto exxppeecctt tthhaatt aallll ccaasseess ooff iimmpplliiccaattiioonn wwoouulldd ssaattiissffyy tthhee aalllleeggeedd rreelleevvaannccee rreeqquuiirreemmeenntt
wwhheenn,, iinn tthhee nnaattuurree ooff tthhee ccaassee,, oonnllyy ssoommee ddoo..

Thirrddlly, inn chappter 4, sectioonn 6, pppp.. 222244--3300, we pprreess tthhe ccaassee furrtthherr forr aaccepptance of (iii) aanndd (iv)
by shhowwinng, aammoonngg otherr thhiinnggss, tthhat tthhoossee whho aarree ddisposed to rreejject thhemm aarree likkeely tto hhaavve othheerr
competing aanndd even mmoorree commppeellllinngg inttuuittions onn tthhe bbaassis of whhichh tthhey will bbe strroonnggly ddisposed,

aas well aas logicaally obblligedd,, tto aacceppt (iii) aanndd (iv). Buutt tthhe ddeettaaiiled aarrgguummeennt onn tthhat cann waait.

EEXXEERRCCISIESSES

11.. Exppllaaiinn thee differreennccee bettwweeeennaassseerrtitningg ((11)) ththaat tQQisisaafafalsleseimimppliliccaattiioonn ooffPP,, aannddaassseerrtitningg((22))
thaatt it is falsee thaatt P implies Q..

2.. Give ann exammppllee of two proploJssiittions succhh thaatt thee lattteerr iss a falsee implication of the formeerr..

3.. Give ann exammppllee of two propositions succhh thaatt it is falsee thaatt the former implies thee llaatteterr. .

** ** * * * * * *

Equivaleennccee

OOnncce we hhaavve tthhe conceppt of immppllication inn hhaanndd it is eaassyy tto give aann aaccount of tthhe mmooddaall rreelaattiionn ooff

eqquuiivvaalenncce. TToo say tthhat aa pprrooppoossiittiionn PP is eqquuivalennt tto aa pprrooppoossiittiioonn Q is just tto say tthhat thhey
impply one aannootthheerr, i.e., tthhat nnot onnly ddooeess Piimmppllyy Q bbuut aalso Q immpplliies PP,, i.e., tthhat tthhe rreelaattiionn off
mutual immpplliiccaattiioonn hhooldds bbeettwweeenn PP aanndd QQ..

Nooww tthhe rreelaattiionn of immppllicattionn, aas we hhaavve aalrreaaddyy seenn,, cann itselff bbee ddeefined inn tterrmms of ppoosssible
worrllddss.. IIt follows tthhat tthhe rreelaattiionn of eqquuivaalence is likewise ddeefinaabblle.

CConnsider onnce mmoorree hhow we ddeefined ""iimmpplliiccaattiioonn"".. Annyy of tthhe ddeefinnitions, (aa), (bb), orr (c), will do.

Lett uus chhoouossee (aa). TThheerree we saaid that aa pprrooppoossiittiioonn PP immpplliies aa pprrooppoossiittiioonn Q if aanndd onnly if Q is
ttrruuee inn aallll tthhoossee ppoossible worrllddss,, if aannyy,, inn whhichh PP is ttrruuee.. Suuppppoossee, nnooww, tthhat PP aanndd Q are

eqquuiivaalent, i.e., tthhat nnot onnly ddooeess Piimmppllyy Q bbuut aalso Q immpplliies PP.. Then nnot onnly will Q bbe ttrruuee inn

all tthhoossee ppoossible worrllddss,, if aannyy,, inn whhichh PP is ttrruuee,, bbuut aalso tthhe converse will hholdd, i.e., PP will be

trruue inn aallll tthhoossee ppoossible worrllddss,, if aannyy,, inn whhichh Q is ttrruuee.. IIt follows tthhat whhere ttwo prrooppoossitions

aarree eeqquuiivvaalleenntt,, iiff tthheerree aarree aannyy ppoossssiibbllee wwoorrllddss iinn wwhhiicchh oonnel^( ooff tthheemm iiss ttrruuee,, tthheenn iinn eexxaaccttllyy tthhee
ssaammee wwoorrllddss tthhee ootthheerr iiss aallssoo ttrruuee.. MMoorree bbrriieeffllyy,, ttwwoo pprrooppoossiittiioonnss aarree eeqquuiivvaalleenntt iiff aanndd oonnllyy iiff tthheeyy
hhaavvee tthhee ssaammee ttrruutthh--vvaalluuee iinn pprreecciisseellyy tthhee ssaammee sseettss ooff ppoossssiibbllee wwoorrllddss,, ii..ee..,, tthheerree aarree nnoo ppoossssiibbllee
wwoorrllddss iinn wwhhiicchh tthheeyy ddiiffffeerr iinn ttrruutthh--vvaalluuee..

36 PPOOSSSSIIBBLLEE WW ORLDS

AAs an examplee of thee relation of equivalencee consider thee relation whhichh thee contiinnggeenntt pprrooppoossiition

(17.2) Canadaa is southh of Mexico

hass to thee contiinnggeenntt pprrooppoossiition

(1.222)) MMexico is north of Canada.

Even if we were merely to rely on our untuttoorreedd intuitions most of us would find it naturall to say
that these two propositions aree equivalent. But now we can explain why. We can point out that not
only does (17.2) imply (17.22), butt also (17.222)) implies (17.2). Or, getting a little more sophisticated,
we can point out that in any possible world in which one is true the otheerr is also true and that in
any possible world in which one is false the otheerr is false. It matters not at all that both propositions
happpeenn to be false in thee actual world. As we have alreadyy seen, false propositions as wweell as true
ones can (and do) have implicatiioonnss.. Andd as we can now see, false propositions as wweell as true ones
can be equivalentt to one aannoother.

Noncontingent propositions as well as contingenntt ones can stand in relations of equivalence to one
another.. Indeed, if we attenndd carefully to the definition we have given for equivalence it is easy to
see: (i) that alll noncontingently true propositions form what is called an equivalence-class, i.e., a class
all of whose memberrss are equivalent to one anotthheerr;2;244 and (ii) that alll noncontingently false
propositions likewiissee form an equivalence-class. Conclusion (i) follows from the fact that if a
proposition is necessarily true it is true in alll possible worlds and hence is true in precisely the same
set of possible worlds as any otherr necessarily true propositionn.. Conclusion (ii)) follows from the fact
that if a proposition is necessarily false it is false in alll possible worlds and hence is false in precisely
the same set of possible worlds as any otherr necessarily false proposition.

These two conclusions strike many personnss as counterintuitive. Surely, it would be said, there is a
difference betweeenn the necessarily true proposition

(71.5) Either the U.S. entereedd World War I in 1917 or it is not the case that the UU.S.
entereedd World War I in 1917

and the necessarily true proposition

(1..2233)) Either Canada is south of Mexico or it is not the case that Canada is south off
Mexico.

Afterr alll,, the concepts involved are not the same. One of these propositions makes referennccee to an
item calledd "thee U.S." and to an event that occurred at a specific moment in time. The other makes
referenccee to two very differenntt items calledd "Canada" and "Mexico" and to the geographical location
of one with respect to the other. Howw,, then, can the two propositions be equivalent? Likewise, it
would be saidd,, there is a difference -— a conceptual difference, one might say -— between tthe
necessarily false proposition

244.. In this booookk wwee aare ussinngg the term ""eeqquuiivvaalleennccee--ccllaass" aass aa ssyynnoonnyymm foorr "a cclass oof equuivalent
prooppoossiittiioonnss"".. OOnn this reaaddiinngg,, it iss poossssiibbllee foorr aa prooppoossiittiioonn too beelonngg too sseevveerraal eeqquuiivvaaleennccee--cclasses. MMore
sstandardly, hhoowweevveerr,, the term ""eeqquuiivvaalleennccee--ccllaass" iss uused inn ssuucchh aa wwaayy tthhaat aa prooppoossiittiioonn ccaann bbee aa mmeemmber ooff
only oone eeqquuiivvaalleennccee--ccllaassss.. TThere sshhoouulldd be littttllee ccaauussee foor ccoonnffuussiioonn.. TThe mmoore uussuuaall ccoonncceeppttioonn ooff
eeqquuivvaalence-class ccaann ssimmppllyy be regarded aas the logical uunniioonn oof aallll the eeqquuivalence-classes (as hheerree ddeefined) ooff
a propossiittiioonn..

§ 4 Relattions bbettwween Prrooppoossittions 37

(1.6) The U.S.. entered World Warr I IiInI 19177 and it is not the caase that the UU.S.
entered Woorld Warr I in 11917

and the necesssaarrily false proposition

(1.244)) Canada is south of Mexico and it IiSs not the caase that Canada IiSs south of
Mexico

which prrevents us, in any ordinary sennse of the worrdd,, from saying that they arre ""eeqquuiivalennt".
The same prroobbllemm arrises in connection with contingeenntt prrooppoossittions. Let us see how.
Consider, for a start, the fact that any prrooppoosittion which asserrtts of two otherr prrooppoositions that

both arre trrue will be trrue in all and only thhoose possible worlds in which both arre trruuee. Aftteerr alll,, in
any possible world, if any, in which one werre trruue and the otherr false, the claim that both of them
arre trruue would be false. Suppose, now, that we wantt to asseerrtt of a contingeenntt prrooppoosittion that both it
and a noncontingently trrue prrooppoosittion arre trruuee. Then the prrooppoosittion in which we asseerrtt their jjooint
truth will be trruue in all and only thhoose possible worlds in which both arre trruue togetthheerr. But they will
be trrue togeetthheerr only in thhoose possible worlds in which the contingeenntt prrooppoosittion is trruuee. Hence the
prroopposition which asserrtts the joint trruutthh of two prrooppoosittions, one of which is contingeenntt and the otherr
of which is necesssaarrily trruuee, will be trruue only in thhoose possible worlds in which the contingent
prroopposition is trruuee. But this means that any prrooppoosittion which asserrtts the joint trruutthh of ttwo
pprrooppoositionns onnee off whhichh is coonnttiinnggeenntt and thhee otthheerr off whhichh is necceessssaarriillyy trruue wiillll ittseellff bbe
coonnttiinnggeenntt and eqquuivvaalleenntt to thhee coonnttiinnggeenntt pprrooppoossiittiioonn.. Forr exxaammple,, suuppppossee thhaatt wee have aa
pprrooppositionn whhich asseerrtts bbotth thhaatt a coonnttiinnggeenntt pprrooppoossiittiioonn,, leett uus saay

(1.2)) Canada is south of Mexico

and that a necesssaarrily trruue prrooppoossiittiioonn,, let us say

(1.5)) Eitherr the U.S.. entered Woorld Waarr I in 19177 or it is not the caase that the UU.S.
entered Woorld Warr I in 11917

arre trruuee. This will be the proposition

(1.255)) Canada is south of Mexico and eitthherr the U.S.. entered Woorld Warr I in 11917
or it is not the caase that the U.S.. entered Woorld Warr I in 119917.

Then it follows from what we have said that (1.255)) is trruue in all and only thhoose possible worlds in
which it is trruue that Canada is south of Mexiccoo,, i.e., in which (1.2)) is trruuee. But this means not
only that (1.255)) is contingeenntt but also that it is trruue in prreecisely the same sett of possible worlds as
(1..22)),, and hence that (1.2)) and (1-.25) form an equivalence-cllaassss,, i.e., arre equivalent.

But arre (1.2)) and (1.255)) ideennttiiccaal?l? Ourr intuitions arre likely to rrebbel at the very suggesttioonn.. AAnnd
this is for the very same sorts of rreasons which would lead them to rrebbel at the suggessttiioonn that the
necesssaarrily trruue prrooppositions (1.5) and (1.233)) arre identical.

Perhaps the first point that nneeds to be made in rrepply to thheese objections is that the sennse in which
we arre saying that two contingeenntt prrooppoositions mayy be equivalent, and that any two necesssaarrily trrue
prrooppoositions are equivalent, and that any two necesssaarrily false prrooppoositions are equivalenntt -— is
simply that which is conveyed in our definition, viz., that members of each of thheese settss of
prrooppoositions have the same trruutthh--vvaalue in the same sett of possible worldds. We arre not claiminng that
equivalenntt prrooppoositions arre identical with one anotherr. To be surre, there arre uuses of the terrm

38 POS S I B L EE WWOORRLLDDSS

"equivalent" in ordinary discourse which foster the idea that "equivalent" is a precise synonym for
"identical". For instance, someone who says that a temperature of zero degrees Celsiuuss is equivalent
to a temperature of thirty-two degrees Fahrenheiitt might just as well claim that the temperature as
measureedd on one scale is the same as, or is identicall with, the temperature as measureedd on the other
scale. But the claim that two propositions are equivalent is not to be construed in this way. Two
propositions can have identicall truth-valluueess in identicall sets of possible worlds without tthheemselves
being identiiccaall.. They can be identicall in thheesseerreessppeecctstswwitihthoouutt bbeeininggidideenntitcicaal lininaalllreressppeecctsts.. TThhaatt isis
to say, they can be equivalent without being one and the same propositioonn.. Draw a ddiistinction
betweenn equivalence and identity, and conclusions (i) and (ii)) no longer wwiill seem counterintuitive.
Similarly, we shall then be able, with consistency, to say that the two equivalent contingent
propositions (1.2) and (1.-2255)) are likewiissee non-identtiiccaall.. What wouldd be counterintuitive would be
tthhee ccllaaiimmss tthhaatt aallll nneecceessssaarriillyy ttrruuee pprrooppoossiittiioonnss aarree iiddeennttiiccaall wwiitthh oonnee aannootthheerr,, tthhaatt aallll nneecceessssaarriillyy
ffaallssee pprrooppoossiittiioonnss aarree iiddeennttiiccaall wwiitthh oonnee aannootthheerr,, aanndd tthhaatt aallll eeqquuiivvaalleenntt ccoonnttiinnggeenntt pprrooppoossiittiioonnss aarree
iiddeennttiiccaall wwiitthh oonnee aannootthheerr.. FFoorr tthheenn wwee sshhoouulldd hhaavvee ttoo ccoonncclluuddee tthhaatt tthheerree aarree oonnllyy ttwwoo
nnoonnccoonnttiinnggeenntt pprrooppoossiittiioonnss -— aa ssiinnggllee nneecceessssaarriillyy ttrruuee oonnee aanndd aa ssiinnggllee nneecceessssaarriillyy ffaallssee oonnee;; aanndd
tthhaatt aallll eeqquuiivvaalleenntt ccoonnttiinnggeenntt pprrooppoossiittiioonnss aarree iiddeennttiiccaall..

But precisely how is the distinctioonn betweenn propositional equivalence and propositional identity to
be drawn? More particularllyy,, since we have already said what it is for two propositions to be
equivalent, can we give an account of propositional identity which wwiill enable us to say that
propositions may be equivalent but non-identical?

It is worth noting, for a start, that discussions of identity -— whetheerr of the identity of prCoplpoossiittiions
or of people, of ships or of sealing wax -— are alll too often bedeviled by difficultieess even in posing
the problem coherently. We can say, without any sense of strain, that two propositions are
equivalent. But what would it mean to say that two propositions are identical? If they are iidentical
how can they be twwoo?? Indeed, how can we sensibly even use the plural pronoun "they" to referr to
that which we want to say is onnee?? One's head spins, and we seem to be hedged in between
inconsistency and futility. One of the greatest philosophers of the twentieth century, Ludwig
Wittgenstein, put it aapphhoristically:

Roughly speaking, to say of two things that they are identicall is nonsense,, whilee to say of one
thing that it is identicall with itselff is to say nothing at aallll..2255

One way out of this incipient dilemma lies in the recognition that on most, if not alll,, of the
occasions when we are tempteedd to say that two things are identiccaall,, we could equally well -— and a
lot more perspicuously -— say that two linguiissttiicc items symbolizzee (refeerr to, mean, or express) one
and the same thing.. Insteaadd of saying -— with alll its attendant awkwardnesss -— that two people, let
us say Tully and Cicero, are identiccaall,, we can say that the names "Tully" and "Cicero" referr to one
and the same person. Insteaadd of saying that two propositions, let us say that Vancouver is north ooff
Seattle and that Seattle is south of Vancouvveerr,, are identiccaall,, we can say that the sentences
""VVaannccouverr is north of Seaattttle"" and ""SSeeaattttlle is southh of Vancouverr"" expprreessss one and the sammee prrooppoosi-
tiioonn..2266 This essennttiiaallllyy, is the solution onnccee offerred, butt suubbseqquueennttly rreejjeecctteedd, by the grreat GGerman
pphhilosoopphherr and mmaatthheemmaattiicciaann, GGottttlob FFrreeggee. As he putt it:

25. Ludwig Wittgenstein, Tractatus Logico-Philosopphhiiccuuss,, trraannss. D.F.. Pearrs and B.F.. McGuinneessss,, London,
Routtleddge && Kegan Paul, 196611 (original edition published in German under the title LLooggiisscchh--PPhhilosophiche
Abhaannddlluunngg,, 1921), prrooppoosition 5.5303.

26. Furtheerr rreeaassonnss for adopting the distinction between senntteenncceess and prrooppoossittions will be developed at
length in chapter 2.

§§ 44 RReellaattiioonnss bbeettwweeeenn PPrrooppoossiittiioonnss 3399

WWhhaatt iiss iinntteennddeedd ttoo bbee ssaaiidd bbyy aa == 6b sseeeemmss ttoo bbee tthhaatt tthhee ssiiggnnss oorr nnaammeess ''aa'' aanndd ''b6'' ddeessiiggnnaattee

tthhe same thhiinngg. .2277

Although FFrreege''s suggeessttiioonn woorrks well ennouugghh whheenn wwee wwaanntt tto mmaakke ssppeecciiffiicc iiddeennttity-ccllaaiimmss,, iitt
ddooeess nnot enable us to aavoidd the dilemmaa when we trry to formulate the conditions off idenntitty ffoorr
tthhiinngs qquuitte generraalllly, thhings forr which there arre no linguistic symbols as well as for things ffoorr
which tthherre aarree.. Ass it stands, Frreege's claimm suuggests that we should be able to say sommeetthhiinngg along
tthheessee lines:: ""TTwwoo signns orr names 'a' and 'b' desiggnnaattee the same thing if and only if. ...... "" (wwhheerree the
bblank is ttoo bbe filledd inn bbyy the specificaationn of apprroopprriiaattee conditions of identity). But it is jusstt ppllaaiinn
faalse tthat tto maakke aann identtiittyy--claimm is, in general, to asseerrtt that two exppresssiioonnss have the same
rreefferreenncce. FFrreege's rreefforrmmuullaattiioonn works well enough in the case of items which happen to have beenn
nnaammeed orr rreeferrrreedd tto bbyy someone or otherr in some language or other. But are therre nott at leaastt ssoomme
uunnnnaammed items, forr which linguistic symbols have not yet and perhaps neveerr wwill be, devisseedd?? SSuurrely
tthherre mmuust bbe identtitty-connditionns forr tthheesse itteemmss as well. YYet iiff tthiss iiss ssoo,, hhooww ccaann wwee eevveenn bbeeggiinn?? AAss
wwee hhaavvee aallrreeaaddyy sseeeenn,, wwee ccaann hhaarrddllyy ssttaarrtt ooffff bbyy ssaayyiinngg ""TTwwoo tthhiinnggss aarree iiddeennttiiccaall iiff aanndd oonnllyy iiff ...... ""

In orrdderr tto give qquuite general identity-conditions for any items whateveerr, we would do well to, as
it werre, ''tturrn tthhe prroobblleemm arround' andd ask forr the conditions of non-identittyy. We would do well, thaatt
is, ttoo aasskk, ""UUnder what conditions should we feel compelled to say that therre are two itemmss rratthheerr
tthhaann just one?"" Not only iss. thhis way of puttting the question.-ppaarradox-frree, but it also avoids the
limitaattiions impplliicciitt in Frreege's ffoorrmulattion.

The aannswerr which commends itself to most thinkkiinngg people, philosoopphheerrs and nonphilossoopphheerrs
aalliikkee,, is essentiallyy tthat which has come to be known as Leibniz''ss Principle. Leibniz put itt this way:

There aarre never in naturre two beings which arre exactly alike in which it is nott possiibbllee to find
aan intterrnnaall ddifference........ 2288

IInn effect, Leibniz claimedd that it is impossible forr two items to have all theirr attrributes —- iinncclluuddiinng
rreellaattiional oonneess -— in commmoonn.. Puuttttiinng the point in still anotherr way:. therre are two itemmss rather tthhaann
one if aanndd onnly if one item has at least one attrribute which the otherr does not.

Armed with tthhis aaccount of identity, let us rreturrnn to the taskk of distinguishing bbetweeenn
pprrooppoossittional equuivalence anndd prrooppoossittiionnaall identity. Caann we give an accoouunntt of the connditionns ooff
pprrooppoossittional identity which will enable us to prreserve our intuitions thatt prropositions may bbe
equuivalent aanndd yet non-identical?

IIt sseemmss clear tthat what guides our intuitions when we insisstt that prroppositions, coonnttiinnggeenntt aanndd
nnonncontingent aallikkee,, need not be identical even when they are memberrs of the same
eqquuiivalence-cllaasssseess is something like Leibnniiz''ss Prriinncciple. We note that in each of the pprroblemmatic
ccaasseess pprreecceeddiinngg,, one prrooppoossiittiionn has at least one attrribute which the otheerr lacks, and so connclluude —-
in aacccorrddaannccee with Leibbnniizz''ss Prriinncciippllee -— that the two, though equivalennt, are not iiddeenntical.

Let uus call aanny aattttrriibbuutte which serrvvees to sort out and diffeerreennttiiaattee between two orr more ittemmss aa
difffeerreenntitaitaintigng attriibbuuttee.. Then we may say that what guides our intuitions as to the non-iddenntitty ooff

2277.. GGoottttlloobb FFrreeggee,, ""OOnn SSeennssee aanndd RReeference"", inn Translations fromm the Philloossoopphhiiccaall Writingss of GGoottlob
Freeggee,, eedd.. PP.. GGeeaacchh aanndd MM.. Black, Oxford, Basil Blaacckkwweelll,, 1952,, pp.. 56.

2288.. G.W.F. LLeeibbnniizz, Monadology, ttrraannss.. RR.. Latta, London, Oxforrd Univerrssittyy Prreess, 19655,, Sectioonn 9, p. 222.
TThhiiss pprriinncciippllee iss wwiddeely kknnoowwnn aass tthhe PPrriinncciippllee of IIddenttity of Innddiisscerrnniibbllees. More appttlly, it might be called tthhe
PPrriinncciippllee oof NNonn-IIddeennttiitty of Discerrnnibblles.

40 POSSIBB LE WWOORRLLDDSS

tthhe equivalent contingent prrooppOoSsIitIiOons (17..22)) andd (71.25)) iss tthhe fact tthatt tthheere iss at least oonnee
ddiiffffeerreennttiiaattiinngg aattttrriibbuuttee wwhhiicchh mmaakkeess tthheemm nnoonn--iiddeennttiiccaall.. IInnddeeeedd tthheerree aarree sseevveerraall.. TThhee aattttrriibbuuttee ooff
mmaakkiinngg rreeffeerreennccee ttoo tthhee UU..SS.. iiss oonnee ooff tthheemm:: iitt iiss aann aattttrriibbuuttee wwhhiicchh ((71..2255)) hhaass bbuutt ((17..22)) ddooeess nnoott..
AA nndd tthheerree aarree ssttiillll ffuurrtthheerr ddiiffffeerreennttiiaattiinngg aattttrriibbuutteess wwhhiicchh,, iinn tthhee ccaassee ooff tthheessee ttwwoo pprrooppoossiittiioonnss,, sseerrvvee
ttoo ddiiffffeerreennttiiaattee oonnee ffrroonmi tthhee ootthheerr:: ((11..2255)) mmaakkeess rreeffeerreennccee ttoo aann eevveenntt wwhhiicchh ((17..22)) ddooeess nnoott;; ((71..2255))
mmaakkeess rreeffeerreennccee ttoo aa ddaattee wwhhiicchh ((17..22)) ddooeess nnoott;; aanndd ssoo oonn.. IInn sshhoorrtt,, tthhee iitteemmss aanndd aattttrriibbuutteess ttoo
wwhhiicchh oonnee pprrooppoossiittiioonn mmaakkeess rreeffeerreennccee aarree nnoott eennttiirreellyy tthhee ssaammee aass tthhee iitteemmss aanndd aattttrriibbuutteess ttoo wwhhiicchh
tthhee ootthheerr mmaakkeess rreeffeerreennccee.. HHeennccee tthhee pprrooppoossiittiioonnss tthheemmsseellvveess aarree nnoott tthhee ssaammee bbuutt ddiiffffeerreenntt..

Similarly, by invvookkiinngg Leibniizz''ss Princciple we may distinguish the twwo equivalent nnooncontingent
propossiittions (11..55)) and (17..2233)):: (17..55)) referrs too the U.SS..,, (11..2233)) does not; (17..2233)) referrs too CCaannada,,
(17..55)) does not; etcc. Onncee agaiinn,, wee may coonncclluuddee that twwo equivaalent propossiittions are nott iiddenttiiccall..

To sum up, equivalent propositions cannot differ froom one another in respecct off the attttrriibbuuttee oof
having the same trruutthh--vvaalluuee in the same sets off possible worlds.. But theey can differ froom one another
in respecct off other attttrriibbuutteess.. Identtiiccaall proopositioons,, by way off commpparisoon, cannot differ froom oone
anottherr in respecctt off this orr any otthheerr attttrriibbuuttee.. They have allll off theirr attttrriibbuutteess in ccoommmmoon..

In chapter 2, section 2, we return too the prooblem off draawwiinngg a line betweeen ppropoossiitional
equivalence and propositioonall identiittyy and come up ww iit h a moorre precise statement (in section 3) of
the coonndditions for propositioonall iiddeenntityy.

EEXXEERCRISCEISSES

71.. Whiicchh prrooppoosistitoinosns (a. - e.)) aree incoonnsistent witthh whiicchh prrooppoosist1iotwnsns (i. - v..))?? WWhhiicch

prrooppoosistitoinosns (a. - e..)) aree coonnssiisstteent witthh whiicchh prrooppoosistitoinosns (i. - v..))?? Whiicchh prporpopoossiittiioons
(a. - e.)) implyy whiicchh prrooppoosistitoinosns (i. - v..))?? Anndd whiicchh prrooppoosistitoinosns (a. - e.)) aree eeqquuiivvaalleenntt
to whiicchh prrooppoosistitoinosns (i. - v..))??

a. Theree aree 8,009988,,778899,2,24433 starss.. i1. AAllll triaanngglleess have three sidees.

b. Allll squareess havee foouurr sides. lili. Theree aree feewweerr thann 771,75,6576,12,22244,,3389
stars..
cc.. Some squareess havee siixx sidees.
d. Therree arree 8,009988,7,78899,2,24343 stars oorr it is noott lll. Theree aree more thann 8,009988,,778899,2,24422 stars
m
thee casee that there aree 8,009988,,778899,2,24343 ssttaarrss. annddfeewweerr thann 8,009988,,778899,2,2444 ssttaarrss.

e. Thee UU.sS. enteerreedd Woorrldld Waarr I in 719177. lV. Theree aree 172244,,775599,3,323,25,15111 ssttaarrss.
W
v. Thee U.S. entereedd Woorrlldd Waarr I after
1791722..

((PPaarrtitaial l answeerr:: a. is inccoonnssisitsetnetnt wwiitthh iv.
c. is inccoonnssisitsetnetnt wwiitthh i., ii.,, Hiii., iv..,, anndd v.
a. is ccoonnssiisstteenntt wwiitthh i., ii.,, Hiii., anndd vv.)

2. a. Is pprrooppoosistitoinon AA,, deeffiinneedd beelooww, ccoonnssiisstteenntt oorr inccoonnssisitsetnetnt wwiitthh pprrooppoosistitoinon BB??

""AA" " == BBiilll is eexxaaccttllyy 66'' ttaall.

""BB"" == BBiilll is eexxaaccttllyy &6' 22/1"taatll..

§§ 44 Relations betweeenn PPrroopposittiioonnss 41

bb.. IIss propositionn C,, definneedd beellooww,,ccoonnsissitsetnetnot roirnicnocnosnisstiesnttenwtitwhitphropproospitoiosintioDn?D?

""CC"" == Soommeeoonneeisiseexxaactcltyly6'6t'altla. ll.
""DD"" =— Soommeeoonneeisiseexxaactcltyly66' '22""tatlal.ll.

cc.. IIss propositionn E,, definneedd beellooww,,sseelfl-fc-ocnosnisitestnetnotrosrelsfe-ilnf-cionncsoisntseinst?ent?

""E"" =— Soommeeoonneeisiseexxaactcltyly6'6t'atlal lal nadnd6'&2"2"taltla. ll.

3.. Expllaaiinn whhyy it iss misleading too saayy suucchh thhiinnggss aas:s:

""IInn thhee actuuaall woorrlldd,, Canada''ss beinngg norrtthh ojf Mexico IiSs inccoonnssiisstteenntt withh MMexico's
beingg noorrtthh ojf CCaannaaddaa"";;
orr
""IInn thhee world ojf Time Enough foorr LLoovvee, thhee propositioonn thhaatt LLaazzaarruussjfaalllss ininlolovveewwitihth
twwoo of hiss dauugghhtteerrss implies thhee propositionn thhaatt LLaazzaarruuss isisaajfaaththere.r"."

*$ *% *$ *$ %*

Symmbboolliizzaattiioonn

OO uurr reppeerrttooiirree off symbols caann nooww bee exteennddeedd too encoommppaassss hnoott only thhee modal pprrooppeerties
represented byy "0", ""~0"", , "\V7"", , anndd "/A:;."", , buutt also thhee modal relationnss off consistency, inconsistency,
imppllication, andd equivalence. Th e y arree standdaarrddllyy represented inn symbols ass ffoolllloowwss::

(1) Thee conncceepptt off consistency: ""o0"
(2) Thee conncceepptt off inconsistency: ""4.>p"
(3) Thee conncceepptt off impliccaattiioonn:: ""—_»"" [called arrrrooww]]
(4) Thee conncceepptt off equivalence: '"'<.—......•'"' [called doouubblele-a-arrrorwowJ2]929

Eacchh off these symbols maayy bee written betwweeeenn symbols which stanndd foorr propoossiitionnss too yield ffuurrtthher
propoossitional symbols (e.g., "PP ooQQ"",, ""PP—_Q»Q""))..

Eacchh off these symbols maayy bee definedd contextually ass ffoolllloowwss::

"" PP 0o Q"" = dd ff "" PP iss connssiisstteenntt with QQ"
"" PP .*p QQ"" = dd ff "" PP iss inconnssiisstteenntt with QQ"
"" PP -_QQ"" = ddff "" PP implies QQ"
"" PP iss equivvaalleenntt too QQ""
"P~Q" df

299.. These latteerr twwoo symbols aarree nnoott too bbee confuseedd with thhee twwoo symbols H"::o:>"" aanndd H" =e" with which some
readers mmaayy already bbee familiaarr.. Thhee twwoo symbols H" :D;>"" (called hhooookkororhohroserssheoseh) oaen)d aHned"" =(c"all(ecdaltlreidplter-ibpalre)-bar)

wwiill bbee introduced later inn this book anndd wwiill there bbee used too standd foorr thhee relations ooff material conditionality
and material biconditionalliittyy respectively. AAtt that time wwee shall take some pains too argue that thhee relations ooff
material conditionality anndd material biconditionalliittyy aarree distinctly differreenntt from annyy relations which have been
introduced inn this first chapter; inn particular, that they aarree distinct from implicattiioonn anndd equivalence.

42 POSSIBLE WORLLDDS

EEXXEERRCCISIESE
Refer too queessttiioonn 1 in thee preceddiinngg exerrcciissee.. Let the lettteerrss "A" - "E" stand for propositions

a.. - e.. and thee letttteerrss '"'JI'"' - "N" for thee propositions i., - v.. Re-doo queessttiioonn 1 expprreessssiinngg alll yyoour

answweerrss in thee symbolismm jjuusstt intrroodduucceedd.,

(Partiall ansswweerr:: A $~ M
C ~<>t],/ , CC~<>tK,K, CC~ <L>t , L,C ~C M4> ManadndCC ~ N<>t N
A o0]/ ,, A o0 K, A o0 L, A o0 N)

5. SETS OFF PROPOOSITIIOONNS

The trruutthh--vvaalluueess, mmooddaall pprrooppeerrttiieess, aanndd thhe mmooddaall rreelaattions whhichh mmaay bbe aascrribbedd to iinnddiivviidduual
pprrooppoossitions aanndd to ppaairrss of pprrooppoossiittionns, mmaayy,, with equuaal pprroopprriieettyy,, bbe aascrribbedd to seettss off
pprrooppoossitions aanndd to ppaairrss of sseettss of prrooppoossiittionns.

Truth-values of propositionn--sseettss
A set of pprrooppoossittions willl bbe said tto bbe truuee if every mmemmbber of that set is trruuee.. Anndd aa set off
pprrooppoossitions will bbe said to bbe falssee if nnot everry mmember of that sett is trruuee,, i.e., if aat leaast one
mmemmbber of that sett is faalse. Note caarreeffuullllyy:: aa sett of pprrooppoossitions mmaay bbe faallse evenn thhough nnot everry
mmemmbber of that set is faalse. A single faallssee mmemmbber in aa sett of pprrooppoossitions is sufficciieenntt to rrenndderr the
sett faalse. Anndd of courrse it follows frroomm thhis that if aa sett is faalse, we aarre nnot entitledd to inferr of any
parrtticulaar mmemmbber of that sett that it is false; we aarre entitledd to inferr onnly that aat least one mmemmbber is
faalse.

Exammppllee 11:: A truuee sett of propositions
(1.2266)) {Snow is whhite, Thhee U.SS.. enterredd World WWaarr II inn 19177}}.3o°

Exammppllee 2:: A falssee seett of propositions
(1.2277)) {SSnnoww is whhite, Thhee U..SS.. enterredd World Waarr IIiinn 11991144}}..

IIt shouuld bbe cleaar that the two exprressioonnss (1) ""aa set of faallsee pprrooppoossitions"" aanndd (2) ""aa faallssee sett ooff
pprrooppoossitions"", ddoo nott mmean thhe saammee thhiinngg.. A sett of faallsee pprrooppoossitions is aa set alll of wwhhoossee mmemmbberrs
aarre false; aa faallssee set of pprrooppoossitions is aa sett at leeaassttoonneeooffwwhhoosesemmemembebresrsisisfaflasels.e.

Modal properties of proposition-sseettss
A set of pprrooppoossittions willl bbe said to bbe posssiibbllyy truuee orr self--ccoonnssiisstteenntt if aanndd onnly if thherre eexxisttss a
ppossible worrlldd inn whhich every mmemmbber of that sett is trruue.

Self--ccoonnssiisstteennccyy is aa ''frraaggiile' pprrooppeerrttyy.. IIt is eaasily aanndd oftenn uunnwwittttinngly lost (seee chapterr 6, section
7). CCoonnssidderr thhe following exammpplle:

30. Here and suubbseqquuennttly in thhiis book we use a pair of bbrraacceess (i.e., ""{{"" and ""}}"")) as a mmeeaannss to ddeesignaatte a sset.

§ 5 Sets of PPropositions 4433

{{ AApprriill is taller than Bettyy,
BBeettty is taller than CCarol,
CCaarrooll is taller than Dawn,
DD aawwnn is taller than EEdith,
EE ddiitthh is taller than FFrancces,
FFrraanncceess is taller than AApprrill}}.

NNootice how eveeryy subsseett consistingg off annyy ffiivvee off these propositions is selff--ccoonnssiisstteenntt. RReemmoovvee the ffiirrsstt,,
orr the second, orr the third, etc., and the reemmaainininigng set is selff--ccoonnssiisstteenntt (which off course is noott to ssay
that it is true). BBuutt in reintrroodduucciinngg the removed propositiioonn and consequently enlarging the set to
wwhhaatt it wwaass,, selff--ccoonnssiisstteenccyy is lost. AA nd once selff--ccoonnssiisstteenccyy is lost, in this set as in any other, it can
neveerr be regained by adddiinngg more propositiioonnss.. SSoommee persoonss think that selff--ccoonnssiisstteenccyy can be
restored by insertiinngg a propositiioonn of the followwinngg kind into a selff--iinnccoonnssiisstteenntt set: TT hhee immmediately
ppreceeddiinng ppropositiioonn is faallssee.. BBuutt this deviiccee caann nevveerr ressttoorree sellff--ccoonnssiisstteennccyy.. (SSeeee the exeerrcciisseess on
pp..4444..))

AA seett off propositions is poossssiibblyly fallssee if and only i ff there exxiissttss a possible wwoorld in wwhich at least
one memmbbeer off that set is ffaallssee..

AA seett off propositions is necceessssaarriilyly true if and only if evveerryy meemmbbeerr off that seett is necesssaarillyy true,
i.e., if in eveeryy possible wwoorld eveeryy memmbbeer of that set is truee.

AA seett off propositions is necceessssaarriilyly fallssee or seellff--inincocnosnistiesntetnt if and only if there does not exxiisstt any
possible wwoorld in wwhich that set is true, i.e., if in eveeryy possible wwoorld at leaasstt one propositiioonn or
another in that set is ffaallssee..

AA nndd ffinalllyy,, a set of propositions is coonnttiinnggenetnt if and only if that set is neither necessarilyy true nor
necessarilyy falsee, i.e., if and only if there exxiissttss some possible wwoorld in wwhich eveeryy memmbbeer off that set
is true and there exxiissttss some possible wwoorld in wwhich at leaasstt one memmbbeer off that set is ffaallssee..

EEXXEERCRISCEISSES

PPaarrtt A

FFoorr each seett below tell wheetthheerr that seett is (11)) poossssiibbllyy true anndd//oorr poossssiibbllyy false;; anndd (22)) nneecceessssaarrily
true, neceessssaarriilyly false orr cocnontitninggeenntt..

zi. {CCaannaaddaa is noorrtthh off MMeexxicicoo, , Haawwaaiiii is in thee PPaaccifiifcic Occeeaann,, Cooppppeerr condduuccttss eleelcectrtriciciittyy}]
iziz. {Snnooww is whiittee,, PPiinnee is a softwood, Cooaall is rreedd}]
lHlli.. {Thheerree were exaaccttllyy twelvvee tribes off Israell,, Theree were exaaccttllyy foouurrtteeeenn tribes off IIssrraaeell}]
wiv. {AAlll sisters arree feemmaallee,, Allll triiaanngglelses have three sides, Allll squarreess have fouurr sides}]
v. {Some coffee cups arree blue, Some coffee cups arree greeeenn,, Some coffee cups arree yyeellllooww}]
VviI.. {Some triiaanngguulalrar hats arree blue, Alll triiaanngglelses have three sides, Some squarreess have ffiive sides}]
vziiz.. {AAlll triiaanngglelses have three sides, Some triiaanngguulalrar hats arree blue}]
VvliZiiZ.. {Soommeoonnee believeess that today is MMoonnddaay,y, Someoonnee believeess that today is WWeeddnneessddaayy}]
IiXx.. {GGrraassss is greeeenn,, Someoonnee believeess that graassss is greeeenn}]
x. {AAlll sisters arree feemmaalleess,, Allll feemmaalleess arree ssiistteerrs}]

444 POSSIBLE WW OOR LDS

PPaarrtt B

1.. EExxpplalainin whhyy seellff--ccoonnssisistteennccyy caann neveerr bee restoorreedd to a seellff--inincconosnisitsentetnt seet ooff pprrooppoosistitoinosns by tthhee
device ooff insseerrtitningg into that seett a pprrooppoosistitoinon ooff thee sort:: Thee immmeeddiaiatetleyly pprreeccedeidnigng pprrooppoosistitoinon is
false.

2. Thee exxaammppllee used above whhiicchh repoorrttss thee reellaattiivvee heigghhttss ooff AApprrilil anndd Beettttyy,, etcc..,, caann be made
seellff--ccoonnssisistetennt t by thee reemmoovvaall ooff thee last pprrooppoosistitoino.n. What arrgguummeennt t is to be used aggaaiinnsstt tthhee
cllaaiimm that it is thee last pprrooppoosistitoinon in thee above seett whhiicchh 'inducceess'' thee seellff--iinncconosnisitsentecnycy anndd heenncce
is ffaallssee??

3. A seellff--inincconosnistiesntetnt seett ooff three pprrooppoosistitoinosns ooff whhiicchh every prrooppeerr nonn--eemmpptyty subset is
seellff--ccoonnssisistetenntt is caalllleedd an anttiillooggiissmm. Ann exxaammppllee woouulldd bbee::

{{LLoorrnnaa hass three bbrrootthheerrs,
Syyllvviiaa has twoo bbrrootthheerrs,
SSyyllvviiaa hhaass ttwwiiccee aass mmaannyy bbrrootthheerrss aass LLoorrnnaa}]..
FFiinndd tthhrreeee eexxaammpplleess ooff anantitliologgiissmmss..
4. FFiinndd a seett ooff three contingeenntt pprrooppoosistitoinosns such that each ppaaiirr ooff pprrooppoosistitoinosns draawwnn froomm that sset
coonnssttiittuutteess a seellff--inincocnosnistiesntet nt seet.. ExEaxmampplele::

{{NNoorrmmanan is shoorrtteerr thann PPaauull,,
Noorrmmaann is thee same heigghhtt as PPaauull,,
Noorrmmaann is tallleerr thann PPaauul}l}..

5. EExxpplalainin whhyy onnee shoouulldd not adopt thee foollloowwinigng deeffiinniittiioonn ooff "necceessssaarryy falsseehhoooodd"" foorr a seet off
pprrooppoosistiitoinosn:s: A seett ooff pprrooppoosistitoinosns is neecceesssaarirliyly falsee iff anndd only iff every membeerr ooff that seett is
neecceesssaarirliyly ffaallssee..

** ** * * * ** *

MMooddaall reellaattiioonnss betweeenn proprpoopsoistiitoionn--sseets

Two sets of propositions will be said to stand in the relattion of coonnssiisstteennccyy if and only if there exists
some possible world in which all the propositions in both sets are jointly truue.

Two sets of propositions stand in the relattion of inccoonnssisistetnencycy if and only if there does not existt a
possible world in which all the propositions of both sets are jointly truue.

One set of propositions stands in the relattion of impplliiccaatitoionn ttoo anotherr set of propositions if and
only if all the propositions of the latter set are true in every possible world,, if any, in which all the
propositions of the formerr set are truue.

AAnnd two sets of propositions stand in the relattion of eqquuiivvaalelenncece if and only if all the prropositions
in one set are true in all and jusstt those possible worlds, if any, in which all the propositions of the
otthheerr seett arre ttruue.

To illustrate thhese definitions, we cite the following eexxaammpples.

Exxaammpplele 1:

The set of prropositions

(1.2288)) {{OOttttaawa is the capital of Canadaa,, AAll men are morrttaall}}

is coonnssiisstteenntt wiitth the seett off prroposittionns

(1.299)) {SSnnow is white, Today is Tuesday, Some dogs meeooww}}.

§ 5 SSeettss ooff PPrrooppoossititiioonnss 455

EExxaampllee 22:

The set of propositions

(1.3300)) {April iss older than Beetttty, Betty iiss older than Carol, Carol iiss oldder
than DDiiaannee}}

is innconsistent with the set ooff pprrooppoossiittiioonnss

(1.371)) {Diane is older than Edith, Edith is older than Aprriill}.

EExxaamppllee 33:

The set of propositions

(1.322)) {Mary innvitedd Brett to act inn the pplaay, Greshamm invitedd Sylvia to act iinn
the pplaayy}

immpplies the set of propositions

(1.333)) {Sylvia was innvitedd ttoo aacctt inn tthhee pplalayy,, MMaarryy ininvviitteeddssoommeeoonneetotoaaccttinin
the pplaayy}.

EExxaampllee 44:

The set ooff pprrooppoossiittiioonnss

(1.344)) {Toddaay is Wednneessddaayy}}

is eqquuivaaleenntt to the set ooff pprrooppoossiittiioonnss

(17.355)) {IItt iss later inn tthhee week than TTuueessddaayy,, IItt iiss earlieerr inn tthhee week thhann
TThhuurrssddaayy}}.

As wwee cann sseeee,, ssoomme,e,aaltlthhoouugghhnnoot taalll,l,oof fththeesseesseetstsoof fpproroppoosistiitoionnssccoonntataininmmoorereththaannoonneemmeemmbbeer,r,
i.e., mmoorre than onnee pprooppoossiittiioonn.. DDooeess thhis mmeeaann thhaat tthhee relatioonns ooff connssiisstteennccyy, incoonnssiisstteennccyy,, eettcc.,.,
are nnoot aalwaayss ddyaaddiicc,, oorr two-placed relatioonns? NNoott aatt alll.. FFoorr aa ddyaddiicc relation iiss aa relation wwhhicchh
hhoolddss bbetweenn two itemmss anndd eachh ooff the abboove sseettss mmaay bbee couunntteedd aass aa sinnggle itemm even iff sommee ooff
them hhaavvee two oorr mmoorre mmeemmbbeerrss.. HHeenncce aa relation whhichh hhoolddss bbetweenn two settss ooff pproppoossitionnss iiss
still aa ddyaaddiicc relation even if thheerree is mmoorre than onne pproppoossiittionn inn either or bboothh seettss.3.131

IInnssofarr aass mmooddaal relatioonns ccaann obbttaainn bbetweenn sseettss ooff pproppoossitionns aass well aass bbetweenn single,
inddividual pprooppoossiittiioonnss,, certain connseqquueennccees follow whhichh we wouuldd ddoo well too expplore.

31. Note that although (71.35) iss equivalleenntt too (71.34), thhee sseett (71.35)) does nnoott itselff constitute aann
equivalence-class, i.ee..,, nnoott aallll ittss members aarree equivalleenntt too oonnee another. One should bbee careful nnoott too suppose
that thhee relation oof equivalence can hold only betweeenn eeqquuiivvalence-classes.

4466 PPOOSSSSIIBBLLEE WW OO RR LL DD SS

It needs ttoo bbee pointed oouutt thaatt whheenneevveerr a mooddaal rreellaattion RR hhoollddss bbeettwweeeenn ttwwoo indiviiduuaall
propositions, PP and Q,3, 322 there ww iillll always bee ann inffiinniittyy ooff nnoonn--iiddeennttical pprrooppoossiittiioonnss bbeelloonnggiinngg tto tthhee
same equivalence-class aass PP,, aanndd ann infinniittyy ooff non-identical propoossitionnss belonging too thee same

equivalence-class aass Q,, aanndd that aannyy proposition belonging too thhee formmeerr class wwiillll sttaanndd inn tthhee
relation RR ttoo anyy proposition belonging too thhee llaatttteerr ccllaassss.. TThhiiss iisseeaassyy ttoopprroovvee..

Remember, ffirst, that ffoorr any proposition PP,, wheetthheerr contiinnggeenntt orr nonnccoonnttiinnggeenntt,, there iss a sett ooff
propositions each ooff which iss true inn precisely thhee same seett off possible worlds ass P;; thatt iss too saayy,, any
proposition P, of whatevveerr modal status, iss a memmbbeerr of an equivalence-class.

Secondly, tthhee equivalence-class too which annyy given proposition P belonnggss iss a sett off propositions
with aann infinitee numbbeerr ooff memberrss.. How may wwee establishh this latttteerr claim? FFoor a start, wee may
note that tthhee sseett ooff natural numbbeerrss iss aa seett wi t hh ann infinite nummbbeerr off mmeemmbbeerrss.. NNo w foorr eeaacchhooff
these natural numbeerrss there exists aa proposition whhicchh asserts that thhee nuummbbeerr haass a succcceessssoorr.. Hence
the numbeerr ooff such propositions iss itseellff infinite. Moreover,, eachh of these proppoossiittioonnss iss nnoott oonly true,
but necessarily true. IItt follows that there iss ann infinite nummbbeerr off necceessssaarryy truutthhss.. NNo w, as wee saw
before, since every necessaarryy truthh iss true inn precisely thhee same seett off possible wwoorlds ass evveerryy otherr
necessarryy truth, tthhee sseett ooff necesssaarryy truths forms ann equivalence-class. A nd,, ass wwee haavvee jjuusstt seeeenn,, this
equivalence-class must have aann infinite nummbbeerr off memmbbeerrss.. Inn shoorrtt,, eveerryy nneecceessssaarriillyy true
proposition belongss too aann equivallennccee--ccllaassss whhiicchh haass ann iinfinite nnuummbbeerr ooff mmeemmbbeerrss.. BBuutt iiff tthhiiss iiss ssoo,,
thenn tthheesame muusstt bbee true also ooff everryy necceessssaarriillyy fallssee pprroposition. FFoorr iitt iiss oobbvviioouuss tthhaatt tthheerree muusstt
bbee aann iinnffiinniittee nnuummbbeerr ooff nneecceessssaarriillyy ffaallssee pprrooppoossiittiioonnss:: ttoo eeaacchh nnaattuurraall nnuummbbeerr tthheerree ccaann bbee ppaaiirreedd ooffff
aa nneecceessssaarriillyy ffaallssee pprrooppoossiittiioonn,, ee..gg.,., tthhee pprrooppoossiittiioonn tthhaatt tthhaatt nnuummbbeerr hhaass nnoo ssuucccceessssoorr,, aanndd iitt iiss
eeqquuaallllyy oobbvviioouuss tthhaatt tthhiiss iinnffiinniittee sseett ooff pprrooppoossiittiioonnss ccoonnssttiittuutteess aann eeqquuiivvaalleennccee--ccllaassss wwiitthh aallll ootthheerr
pprrooppoossiittiioonnss wwhhiicchh aarree nneecceessssaarriillyy ffaallssee.. HHeennccee eevveerryy nneecceessssaarriillyy ffaallssee pprrooppoossiittiioonn bbeelloonnggss ttoo aann
eeqquuiivvaalleennccee--ccllaassss wwhhiicchh hhaass aann iinnffiinniittee nnuummbbeerr ooff mmeemmbbeerrss..

HH ooww aboutt contingeenntt propositions? Thhee same resuulltt holds foorr theemm toooo. . AAss wee saaww bbeeffoorree ((pp.
37)),, ffoorr aannyy contingeenntt proposition whatevveerr,, there exists anootthheerr non-identical buutt equivalenntt
proposition which asserts thhee joiinntt truutthh ooff bbootthh tthhaatt pprrooppoossiittion aanndd ssoommee nneecceessssaarriillyy ttrruuee
proposition. But there iss aann infinite nummbbeerr ooff necessarily true propossitionnss aannyyoonee off wwhhiicchh mmaayy bbee
asserteedd ttoo bbee true conjointly with aa given continnggeenntt proposition. HHeennccee foorr annyy contingenntt
proposition whatevveerr there exists aann infinite nummbbeerr off non-identical buutt equuiivvaalleenntt pprooppoossiittiioonnss each
of which asserts the joint truthh ooff tthhaatt pprrooppoossiittiioonn aanndd ssoommee nneecceessssaarriillyy ttrruuee pprrooppoossiittiioonn.. HHeennccee eevveerryy
contingentt proposition belongs too aann equivalence-claassss wwhhiicchh haass ann infinite nuummbbeerr off mmembers.

Consider, iinn tthhee light ooff alll this, two individuaal propositions, AA andd B, wwhhiicchh staanndd inn soommee mmooddaal
relation R.. For instance, let uuss suppoossee that AA iss thhee continnggeenntt proposition

((11..33)) The U.SS.. enterreedd WW oorrlldd WW aar I inn 1914

and BB iisstthheecontingenntt proposition

((11..2211)) The U.SS.. enterreedd WW oorrlldd WW aar I beffoorree 1920.

There iiss aann infinitee numbbeerr ooff non-identical propositions wwhhiicchh arree equivvaalleenntt too AA, anndd ann iinnffinite
numberr ooff non-identical propositions which arree equivaalleenntt too B.. TThhuuss itt followwss from thhee faacctt that
((11..33)) (i.ee..,, AA)) implieess ((11..2211)) (i.e., BB)) that there iss ann infinniitteellyy larggee nuummbbeerr off propositions

equivalent ttoo AA each of which impliess ann infiniteellyy large nummbbeerr off propoossitionnss equuiivvaalleenntt too BB..

32. Or,, we might equally say, "betweeenn two unit sets {P}} anndd {QQ}}.. ..... "

§ 5 Sets of Propositions 477

Parallel concllusions folllooww for eachh of the other modal relationns of consistency,, inconsistency, and
equivalennce.

In sum, the point may be put this way: Whenever two propositions, P and Q, stand inn any modal

relation R, all those propositions which are equivalent to P, of which thhere is necessaarriillyy an iinnfinite
number, will similarllyy stand in the modal relation R to eachh of the infinniittee number of propositions
which are equivalent to Q.

AAn interesting, neglectteedd corollary may be drawn from this prinncciippllee.. In sectionn 4 we argued that
thhere are only two speciess of the modal relation of inconsistency: eitherr two inconsisstteenntt propositions
(annd now we wouldd add "proposition-seettss"")) are conntrraarriieess or they are contradicttoorriieess.. Now while it
has long been acknowledged, indeed insisted uponn,, that no proposition has a uniquue (i.e., one and only
onne)) coonnttrraarryy,, it hass offtteenn beeenn ass sttrreennuuoouussllyy insistedd thhatt evveerryy pproposition doess havee a uunniiqque
coonnttrraaddiiccttoorryy,, i.e., thhatt thhere is onnee anndd only onnee pproposition which sttaanndds in thhee reellaattiioonn off
coonnttrraaddiiccttiioonn to a given pproposittiioonn.. Buutt in lighht off thhee distinnction bbettwweeeenn ppropositional-iiddenntity aanndd
pprrooppoossiittiioonnaall--eeqquuiivvaalleennccee aanndd iinn lliigghhtt ooff tthhee ffaacctt tthhaatt mmooddaall rreellaattiioonnss hhoolldd eeqquuaallllyy wweellll bbeettwweeeenn sseettss
ooff pprrooppoossiittiioonnss aass bbeettwweeeenn pprrooppoossiittiioonnss tthheemmsseellvveess,, tthheessee ccllaaiimmss nneeeedd ttoo bbee rree--eexxaammiinneedd.. LLeett uuss bbeeggiinn
wwiitthh ssoommee eexxaammpplleess.. CCoonnssiiddeerr tthhee pprrooppoossiittiioonn

(17.36)) Today is Wednesday.

Among its conntrraarriieess aare

(17.37)) Today is MMoonnday;

(17.38)) Today is Saturdday.

Now let's look at some of its contradicttoorriieess.. These will include

(17.39)) Today is not Wednesday;

(71.40) Today is not the day after Tuesday;

(17.417)) Today is not the day before Thursddaayy, eettcc.

What diffeerreennccee can we ddeettect betweenn the conntrraarriieess of the proposition (17.36)) and the ccoonnttrraaddiicctories
of that same proposition? Just thiss:: the conntraddiccttoorriieess of a given proposition form an
equivalence-ccllaassss (e.g., (17..3399)),, (17..4400)),, and (17.417)) are all equivalent to one another)),, while the
conntrraarriieess of a given proposition are not all equivalent to one another.. Thus while the claim that
evveerryy pproposition hass a uniquue coonnttrraaddiiccttoorryy caannnnoott bbe suupported, itt caann bbe suuperrsseeddeedd bby thhee ttrruue
cllaaiimm thhatt all thhee coonnttrraaddiiccttoorriieess off a pproposition arree logically eqquivaalleenntt to onnee annootthheerr,, i.e., thhatt the
seett off coonnttrraaddiiccttoorriieess off a pproposition is itseellff ann equivalennce-ccllaassss.. NNo suchh cllaaiim caann bbe maaddee for the
coonnttrraarriieess off a pproposittiioonn.. Thhe seett coonnssiissttiinngg off all thhee coonnttrraarriieess off a given pproposition is nott a seett ooff
eeqquuiivvaalleenntt pprrooppoossiittiioonnss..

Miinnddinigng ouurr "P"ss anndd ""QQ"s

Insofaarr as the kinddss of propertiess and relationns we are concceerrnneedd to ascrriibbe to single propositions may,
as we have justt seen, be ascribed to sets of propositions, we wouldd do welll to point out that both
propositions and sets of propositions may equally welll be represseenntteedd by the same sorts of symbols in
the concceeppttuuall notation we use. More specifically, when we write such things as "PP stands in the

48 POSSIBLE WW OORRLDS

relation R to Q if and only iff ...... "" ,, etc., we should be underrsttood to be referrrriinngg,, indiscrriimmiinnaatteellyy,, by
our use of "" P"" and "" QQ "" , both to single propositions and to sets of propositions.

IIn) the next section we shall intrroodduucce whatt we call ""wwoorrlds--ddiagraammss"" and will label parts of them
with "" P "" ss and "" QQ""ss.. For convenience and brevity we ofttenn treat these symbols as if they referred to
single prroposittiioonnss.. In fact they oughht to be thought to refer either to single propositions or to
proposittiionn-sets.

EEXXEERCRISCEISSES

1.. Whhiicchh prrooppoosistitoino-nse-tssets (a. - e..)) aree inccoonnssisistetnetnt witthh whiicchh prrooppoosistitoino-nse-tssets (i. - v..))?? WWhhiicch
prrooppoosistiitoino-nse-stsets (a. - e..)) aree coonnssiisstteenntt witthh whiicchh prrooppoosistiitoino-nse-stsets (i. - v..))?? WWhhiicch
prrooppoosistiitoino-nse-stsets (a. - e..)) imppllyy whiicchh prrooppoosistiitoino-nse-stsets (i. - v..))?? AAnndd whiicchh prporpoopsoistiitoionn--sseets
(a. - e..)) arree eqquuiivvaalelnent t to whhiicchh pprrooppoosistitoin?-nse-tssets (ii. - vv.).)??

a. {{ TTodaayy is Tueessddaayy,, BBiilll has mmiissssed 1i.. {{BBiilll has misseedd thee bbuuss}]
thee bus, BBiilll is late foorr wwoorrkk}}

b. {SSomeoonnee reettuurrnneedd thee wwaallleett,, 1ii1.. {SSomeoonnee whoo lostt his keys
Someoonnee lost hiss kkeeys}] reettuurrnneedd thee wwaallleett}}

c. {{TThhee PPrriimmee MMiinniissteter r is 6'' ttaallll,, ulizi. {{MMuusshhroroomom omeletts aree nnot
Thee PPrriimmee lMvlininisistetrer is eexxactly pooiissoonnoouus,s, Noo mmuusshhrroooom
5'' 2/"1 ttaallll}] omelet is ppooiissoonnoouuss}}

d. {SSome musshhrroooommss aree popiosiosonnoouuss, wiv. {{Jjohhnn is 15 years olld,, JJoohn
Some mushhrroooommss aree nott ppooiissoonnoouuss}} is 5'' 3/"1 ttaallll}}

e. {{Jjohhnn is 15 years olldd anndd is 5'' 3/"1 ttaallll}) v. {{AAlltthhoouugghh BBiilll has misseedd thee bus, hee is
noott late foorr wwoorrkk}]

2. Whiicchh onnee ooff thee ten setss ooffprrooppoosistitoinosns in exeerrcciissee 1 is a seett ooff eqquuiivvaalelnent t prporpoopsoistiitoionnss??

3. Coonnssttrruucct t an eqquuiivvaalleenncec-ec-lacslas ss ooff three prrooppoosistitoinosns onnee ooff whiicchh is thee prrooppoosistiitoinon that Syllvviiaa is
DDiiaannee's's mmootthheerr.

4. Coonnssttrruucct t an eqquuiivvaalleenncec-ec-lacslas ss ooff three prrooppoosistitoinosns onnee ooff whiicchh is thee prrooppoosistiitoinon that twoo pplluus
twoo eqquals ffoouurr..

6. MM OODDAALL PROO P ERR TTIIEESS AA ND RELAA TTIONN SS PICCTUREDD ON WWOORRLLDDSS--DDIAGRAAMMS

WWoorrllddss--ddiiaaggrrams have already been usedd:: figurree (ll..bb) ) intrroodduucceedd our basic conventions for
representing an infinite number of possible worlds, actual and non-actuuaall;; and figurreess (ll..dd),), (l.(eh)e),
and (1.fj)) gave graphhiic significance to our talk of the diffeerreenntt sorts of modal status that propositions
have accordiinngg to whettherr they are contingent, necessarily true or necessarily false, resppeeccttiivveellyy. So far
wee have giiveen thessee diagrrams meerrely an illlluussttrraatitvieve rollee:: ouurr tallkk off ppossssiibbllee woorrldds couuld have
suffiicceedd bby itself. Hoowweevveerr, thessee diaagrramss caan also bbe giiveen an immppoorrttaanntt heeuurriissttiicc rollee:: they ccaann
faacciilliittaatte ouurr discoveerry and pprooff off logical trutthhs whhich miigghhtt otthherrwiissee elluuddee uus.

§ 6 Modal Prroopperrttiiees andd Relations Picturreedd on WWoorrlds-Diiaagrrams 49

Inn order that we may bettteerr be able to use them heurriisttically we adopt the following ttwo
simplifying cconnventions:

(a) We usually omit from our diagrams any rreprresentationn of the distinction between the actual
world and otherr (non-acttuual)) possiblee worlds. When the need arrises to investigate the consequennces of
supposing some prrooppoosition to be actually trrue or actually false, that distinction can, of course, be
reintrroduced in the manner displayed in figures (1..dd)),, (lJ..ee)), , and (1Jj)),, or as we shall see soon, more

perrsppicuously, simply by placing an "xX"" on the diagramm to mark the location of the actual woorrld

among the sett of all possiblee worlds. But, for the mostt parrtt, we shall be concerrned primarily with
investtiggaattiinngg the rrelationships between prroppositions independentlyy of their trruth-statuss in the actual
world, and so shall have infrequenntt need to invoke the distinction between actual and non-actual
worlds.

(b) We omit from our diagrams any brraacckketting spanning thhose possiblee worlds, if any, in which a
given prrooppoosition or prrooppoossiittiioonn--sseett3333 is false. This means that everry brraacket that we use is to be
interprreted as spanning thhoose possiblee worlds only in which a given prrooppoosition (orr prroposition-sseett)) is
trruue. In the event that a given prrooppoosition is not trrue in any possiblee world, i.e., is false in all ppoossible
worlds, we ''locaatee'' that prrooppoosition by means of a point placed outsiddee (and to the rright of) the
rrectangle rreprreesenting the sett of all possiblee worlds. In effectt we thus ''locaatee'' any necesssaarrilyy false
pprrooppositionn among thhee immposssiibbllee wwoorrlds.

Inn light of thheese simplifying conventionns,, let us first rreconsttrruct the thrree basic worlds-diagrams
depicting the modal prroperrtties of contingency, necessssaarryy truth, and necessssaarryy falsity (figguurress (1.d.d)),
(1l..ee)), , and (1.j/)J),, and then consiiddeerr how they might be supplemeenntteedd in order to depict modal
rrelations.

Woorrllddss-d-diaigargarmasms for modall pproroppeerrttiies

A single prrooppoosition (or prropposition-sseett)) P, may be trrue in all possible worlds, justt some, or none.
There arre no otherr possibilities.. If, then, we depict the sett of all possiblee worlds by a single box, it
follows that we have need of thrree and only thrree basic worlds-diagrrams for the modal prroperrtties of a
prroopposition (orr prroposition-sseett)) P. They arre:

pp pp 3 pp

" 2 •

1

FIGUU RREE ((11..kh))
33. Seee the subsection "Minding our ''PP''s and 'Q's"",, pp.. 47-48.

50 POSSIBLE WWOORRLLDDS

Diagrram 1 inn figuure (7l..hh)),, pp.. 49, ddeppicts thhe contingenncyy of aa single pprrooppoossiittionn (or pprrooppoossiittiioonn--set)
P. Thhee pprrooppoossiittionn PP is continngeenntt bbeecaauussee it is trruue inn sommee ppossible worrldds bbuut faallssee in aallll the
othherrs. In effect, ddiagrraamm 1 is aa rreeconstrruuccttion of figuurree (1.dd)) mmaadde inn aaccorddaannce with thhe two
simpplifyinng conventioonnss sppecifieedd aabboovvee. OOuurr ddiagrraamm givveess thhe mmooddaall staattuuss of PP bbuut saayyss nothing
aabbout its aactuuaal trruutthh-status (i.e., trruutthh--vvaaluue in the aactuuaal worldd).

Diagrram 2 ddeppicts thhe nnecessssaarryy trruutthh of aa single pprrooppoossiittionn PP.. TThhee pprrooppoossiittionn PP is nneeccessarrily
trruuee, since it is trruue inn aallll ppossible worrllddss.. In effect, ddiagrraamm 2 is aa rreeconstrruuccttion of figuurree (11..ee)) made
in aaccorddaanncce with ourr simmppllifying cconnvvennttionns.

Diagrram 3 ddeppicts the nnecessssaarryy falsity of aa single pprrooppoossiittionn PP.. HHerree PP is nnecessaarilyy faallssee since it
is faallssee in aallll ppossible worrllddss.. IInn effect, ddiagrraamm 3 is aa simppllification of figuurre ((11f.f)).

Worlds-diagrams fjor moddaall rreellaattiioonnss

In orrdderr to ddeppict mmooddaall rreelaattions bbeettwweeenn two pprrooppoossitions (orr two pprrooppoossition-seettss)) PP aanndd Q,, we need
exactly ffiifftteenn worrldds-diagrraammss.. In tthheessee worrlds-diaagrramms (seee figuure (1h.ii)) onn pp.. 51), nno signifiiccaannccee is
to bbe aatttaached to thhe rreelaattive sizes of the varriiouus segmmeennttss.. Foorr ourr pprreesenntt ppuurrppoosses aallll we nneed attennd
to is thhe rreelaattive placemeenntt of thhe segmmeennttss,, orr aas mmaatthheemmaattiiciaans mmight saay, to thheir toppoology. FForr tthhee
ppuurrppoosses of the pprreesenntt ddiscussionn ourr ddiagraamms nneed onnly bbe qquuaalliittaattiivve, nnot qquuaanntittiatatitvivee.3.344

EEXXEERRCCISEISE

Reproduce ffiigguurree (1l..ii) and add braacckkeettss fjor "r^PvP" " and fjor ""r^vQQ" " too eaacchh ojf thee ffifjteen
worlds-diagrams.

*****

Interpretation ooJf worlds-diagrams

Diagrramms 1 to 4 ddeppict ccaasseess where bbothh pprrooppoossitions aarre nnonncontingent. Diagraammss 5 to 8 ddeppict ccaases
where one pprrooppoossiittionn is nnonncontinnggeenntt aanndd thhe otherr is contingennt. TThhee finaall sevenn ddiagraamms (9 to
15) ddeppict ccaasseess where bbothh pprrooppoossitions aarre cconnttinngent.

Now each of tthheesseeffiifftteeeennddiiaaggrraammss llooccaatteess ttwwoo pprrooppoossiittiioonnss,, PP aanndd QQ,, wwiitthh rreessppeecctt ttoo tthhee sseett ooff aallll

ppossible worrllddss, aanndd tthheennccee with rreespectt to one aannothherr, inn such aa way that we cann ddetermmine what
mmoddaal rreelaattions one pprrooppoossiittionn hhaas to the otherr. HHow cann we ddoo thhis?

Thee mmooddaall rreelaattions we hhaave singled out forr considderration so farr aarree tthhoossee of iinnccoonnssistenncy,
consisttenncy,, immpplliiccaattiioonn,, aanndd eqquuivaalence. Recaall, thhenn, hhow each of tthheessee fourr rreelaattions was ddefined:

P is innccoonnssiisstteenntt with Q_ if aanndd onnly if tthherre is nnoo ppoossible worrlldd inn whhichh bbothh aarree ttrruuee;; PP is
connssiisstteenntt with Q if aanndd onnly if thhere is aa ppossible worrlldd inn whhichh bbothh aarre trruuee;; PP implies Q if aannd
onnly if [ddefinitioonn (b)] thhere is nno ppossible worrlldd inn whhichh PP is trruue aanndd Q is false; aanndd PP is eeqquuiivvaalleenntt

to Q if aanndd onnly if inn each of aallll ppossible worrldds PP hhaas thhe saammee trruutthh--vvaaluue aas Q.. Recaall, furrtthheerr,, that

ouurr ddeevice forr ddeppicting aa pprrooppoossiittionn aas trruue inn aa ppossible worrlldd is to span that worrlldd bbyy mmeans of aa
bbrraacckket labbeled with aa symmbbool signifyinng that pproppoosition.

34. Later, when we come to discuss the conceppt of ""tthe contingeenntt connteenntt"" of a prrooppoossittiioonn,, we shall suuggestt
how one might want to rreinterprret thheessee worlds-diagrams so that the sizes of the segmennttss do take on
significance. (See chapter 6, section 1111.)

§§66 Mooddaall Propertiess and Relations Pictured on WWorlddss--Diagrams 51

PP , .P P

A 6 r

1 11

- .v.. ... .. I
QQ
QQ Q
..P
.PP P
2
7 A

QQ QQ 12

~

Q

P.

PP 13

38 .. -J

.... '---v--' Q
QQ Q
P...
.P.
14
4 PP/,QQ 9
.. ,

Q Q

P .P P

5 10 A

'---v----' .--J 1155

Q Q .,.....J
FIGURR EE (l(.hii)) Q

5522 POSSIBLE WORLDS

The rreeqquuiissitte rruulleess forr tthhe innterrpprreettaattiionn of ouurr worrlds-ddiaaggrramms follow immmmeddiaattely:

RRuullee 11:: PP iss inccoonnssiisstteenntt wwithh Q if aanndd onnly if thhere ddooeess nnot exist anny sett of possibblee worlds wwhhicchh
iss ssppaannnned bbootthh bbyy aa bbrraacckkeett forr PP aanndd bbyy aa bbrraacckket forr QQ;;

RRuullee 22:: PP iss ccoonnssiisstteenntt wwith QQ if aanndd onnly if thhere ddooeess exist aa sett of possibblee worlds which is
ssppaannnnedd bbootthh bbyy aa bbrraacckkeett forr PP aanndd bbyy aa bbrraacckket forr QQ;;

RRuullee 33:: PP implies QQ if aanndd onnly if tthherre ddooeess nnot exist aanny sett of possibblee worlds which is sppaannnneedd
by aa bbrraacckkeett forr PP aanndd wwhhichh is nnot spanned bby aa bbrraacckket for QQ (i.e., if and only if any seett

of ppoosssible wworrllddss sppaannnned bbyy aa bbrraacckkeet forr PP is aallsoo spannedd by a brraackket for QQ);)3;535

RRuullee 44:: PP iss eeqquuiivvaalleenntt ttoo QQ if aanndd onnly if thherre ddooeess nnot exist aanny sett of possibblee worlds which is

ssppaannnnedd bbyy tthhe bbrraacckkeett forr onne aanndd whhich is nnot spanned by the brraackket for the otthheerr ((ii..ee.,
tthhee bbrraacckkeettss forr PP aanndd forr Q span pprreecciisely the samme sett of worldds).

IIt iiss tthhe aaddddiittiioonn oof tthheessee rruullees of interrpprreettaattion that giveess our worrlds-ddiaaggrramm the heurristic vaaluuee
tthhaatt wwe eeaarrlliierr cclaaimmedd forr tthheemm.. BByy aappppllyyiinngg thhem we can pprroovve aa larrge nummbber of logical truths in aa
ssimmpplle aanndd ssttrraaiighhtforrwarrdd wwaayy. CCoonnssidderr sommee eexaammppllees:

((i) DDiaagrraammss 22, 33, 4, 7, aanndd 8 commpprriisse aallll thhe ccaasseess in whhich one or both of the prrooppoositioonns P aanndd

QQ iiss nneecceessarrily ffaallssee. IInn nnoonne oof tthheessee ccaasseess is tthherre aannyy set of ppoossible worrldds spannedd both bbyy
aa bbrraacckkeett ffoorr PP aanndd bbyy aa bbrraacckkeett fforr QQ.. HHeennccee,, bbyy RRuullee 11, wwe mmaayy vaalliddllyy inferr that in aall ooff
tthheessee ccaasseess PP is innconsisstteenntt with QQ.. IInn shorrtt, iiff onee orr bootthh of a paaiirr of propositions iiss

neecceessssaarriillyyffaalslseeththeennththooseseprporpoopsoistitoinosnsaraereinicnocnosnisitsetnetnwt witihthonoeneanaonthoethr.er.

((iii) DDiaagrraammss 11, 22, 33, 55, aanndd 6 commpprriisse aallll thhe ccaasseess in whhich one or both of the prrooppoositionns P aanndd
Q iiss nneecceessarriily ttrruuee.. IInn eeaachh oof tthheessee ccaasseess,, eexxcepptt 2 aanndd 3, tthherre is aa sett of ppossibblee worrlds
ssppaannnnedd bbootthh bbyy aa bbrraacckkeett forr PP aanndd bby aa bbrraacckket forr Q.. HHenncce, by RRuule 2, we may vvaalliiddllyy
innfer tthhaat inn eeaach oof tthheessee ccaasseess,, eexcept 2 aanndd 3, PP is connsisttennt with Q.. But diagrrams 2 and 33
aarree ccaasseess inn wwhhiichh onne orr otherr of tthhe two pprrooppoossitions is necesssaarrilyy false. WWe may coonnclluudde,
tthheerreefforree, tthhaat aa necceessssaarriillyy truuee propositionn iss connssiisstteenntt wwiitthh aannyypprrooppoossiittiioonn wwhhaatetevveerreexcxecpetpt
a nneecceessssaarriillyyfafalsleseoonen.e.

((iiiii) DDiaagrraammss 33, 44, aanndd 88 commpprriisse aallll tthhe ccaasseess inn whhich aa prrooppoossition P is necessssaarrillyy false. IInn
nnoonne oof tthheessee ccaasseess is tthherre aa set of ppoossible worrldds spanned by a brraackket for P. Hence in nonnee
of tthheessee ccaasseess is tthherre aa set of ppoossible worrldds whhich is spannedd by a brraackket forr P and nnoott
ssppaannnnedd bbyy aa bbrraacckkeett forr QQ.. HHeennccee,, bbyy RRuullee 3, we mmaay validdly infeerr that in eachh case in wwhhicchh
P iiss nneecceessarrily ffaallssee, PP immpplliiees QQ nnoo mmaatttterr wwhhethheerr Q is nnecessaarrilyy trruue (as in 3), nneecessaarilyy
ffaallssee ((aas inn 44), oorr contingeenntt (as inn 8). Byy aannaalogouus rreeaasoning concerrning diagrramms 2, 4, and 77

-— aallll tthhe ccaasseess inn wwhhiichh aa pprrooppoossiittionn Q is nnecesssaarrilyy faalse —- we can show thaatt in eaachh ccaassee
in wwhhiicchh Q is nneecessarrily faalse, Q immppllies PP nno mmaatterr whetthheerr PP is necessssaarrillyy true (aass in 2),

nneeccessarrily ffaallssee (aas inn 4), orr continngeenntt (as inn 7). In shorrtt, we mmaay concluudde thatt a neecceessssaarriillyy
ffaallssee pprrooppoossiittiioonn iimmpplliieess aannyy aanndd eevveerryy pprrooppoossiittiioonn nnoo mmaatttteerr wwhhaatt tthhee mmooddaall ssttaattuuss ooff tthhaatt
pprrooppoossiittiioonn..

((iivv) DDiaagrraammss 11, 33, aanndd 6 commpprriisse aallll tthhe ccaasseess inn whhich aa prrooppoossition QQ is necessssaarrillyy trruuee. IInn

nnoonne oof tthheessee ccaasseess is tthherre aa set of ppoossible worrldds whhich is not spannedd by a brraacket forr QQ.

HHeennccee,, bbyy RRuullee 33, wwe mmaayy vaalliddllyy inferr that inn each caassee in which QQ is necessssaarrillyy trruuee, QQ iiss

immpplliiedd bbyy aa pprrooppoossiittiionn PP nnoo mmaattterr whetthheerr PP is nnecesssaarrilyy trruue (as in 1), necessssaarrillyy ffaalsee

3355.. TThhiiss mmeeaannss tthhaatt PP iimmpplliieess QQ inn tthhrreee ccaasseess:: ((i) wwhherre tthhe bbrraacckkeett forr PP spans nnoo ppoossibblee worrldds at aallll
((ii..ee..,, PP iiss nneecceesssaarriily ffaallssee));; ((iii) wwhheerree tthhee bbrraacckkeett forr PP is inncluddeedd wwithhinn thhe bbrraacckkeett forr Q; aanndd (iii) wherre tthhee
bbrraacckkeett ffoorr PP iiss ccooeexxtteennssiivvee wwiitthh tthhee bbrraacckkeett forr QQ..

§ 66 MModal PPrrooppeerrttiieess aanndd RReelaattiionns PPiictuurreedd oonn WWoorrllddss-Diagramms 53

(as in 3), orr continngeenntt (as in 6). Byy aannaallogouus rreeaasonning concernninng ddiagraamms 1,,22, aanndd 5 -— aalll
thhe caasseess inn whhichh aa pprrooppoossiittionn PP is nnecessaarilyy trruue -— we cann shhoww that inn each caassee inn which

P is nneecessaarily ttrruuee,, PP is immpplliieedd bbyy aa pprrooppoossiittiionn Q nnoo mmaattter whhetherr Q is nneecessaarily trruue
(as inn 1), nnecessaarilyy faallsee (as inn 2), orr continngeenntt (as in 5). In shorrtt, we mmaay conclude that aa

necessssaarriillyy truuee proposition iss impliieedd by any and eveerryy proposition noo maatttteeTr whaatt thee mmooddaall
staattuussoofftthhaat tpprrooppoossitiitoionn. .

The sppecial hheuurriissttic aappppeeaal of tthheessee worrlds-diaagrramms lies inn thhe fact thhaatt, taakken ttogetherr with
certaain rruullees forr thheir interrpprreettaattiioonn,, we cann literraallly seeee immmeeddiaattely thhe trruutthh of tthheessee aanndd of many
otherr pprrooppoossitions aabbout thhe mmooddaall rreelaattions whhichh pprrooppoossitions hhaave to one aannothherr. TThhee aaddddiittiionn off
still furrtthherr ddefinitionns aanndd rruullees of interpprreettaattion laterr inn thhis bbook will enabble uus to pprroovviiddee more
pperrssppicuous pprroooofs of imporrtaanntt logical trruutthhs -— includdiinngg sommee whhichh aarre nnot aas well--kknnown aas
tthhoossee so farr mmennttioned.

A notee onn history and nnoommenclature

So farr we hhaave given nnaammes to onnly aa few of the mmooddaall rreelaattions whhich cann obtaain bbeettwweenn ttwo
pprrooppoossiittiionnss. WWee hhaave spoken of inconsiisstteennccyy (and its two sppeecciieess,, contrraaddiiccttiionn aanndd contrraarriieettyy)), off
conssiisstteennccyy,, of impliccaattiioonn,, aanndd of equivaalleennccee.. Within thhe pphhiilosophhiicaal trraaddiittiioonn,, hhoweverr, we find
logiciiaans taalkkinngg aallssoo of mmooddaall rreelaattions whhich thhey caallll ""ssuuppeerriimmpplliiccaattiionn"" (sommeettimmeess calledd
""ssuuppeerraalltterrnnaattion""), ""ssuubbiimmpplliiccaattion"" (sommeettimmeess caalled ""ssuubbaalltterrnnaattion""), ""ssuubbcconntrraarriiety"", aannd
""iinnddeppendennccee"" (sommeettimmeess caalled ""iinnddiifffferrennce"").

By ""ssuuppeerriimmpplliiccaattiionn"" aanndd ""ssuubbiimmpplliiccaattion"" (orr thheir terrmmiinnoollooggiicaal aalternnaattes), ttrraditional
logiciiaans mmeant simpply the rreelaattions of implicattiioonn aanndd of follloowwiinngg fromm rreessppeeccttiively. TToo say that P

staannddss inn thhe rreelaattionn of superimpplliiccaattiioonn to Q is simpply to say that PP implies Q,, whhile to say that P

staannddss inn the rreelaattionn of subimpliccaattiioonn to Q is simpply to say that P follows from Q (or that Q implies

P). In shorrtt, subbimmpplliicaattionn is the converrse of supperriimmpplliicaattionn, i.e., of immpplliiccaattiioonn..

By ""ssuubbcconntrraarriiety"" we mmean the rreelaattionn whhichh hholds bbeettwweenn PP aanndd QQ when PP aanndd Q caann bboothth

bee truee toggeetthheerr but cannnnoott bootthh bbee ffaalslsee. . TThhaatt isis ttoo ssaayy,, ssuubbccoonntrtraarrieietyty isis tthhee rreelalattioionn wwhhicichh hhoollddss

bbeettwweenn PP aanndd Q when thhere is aat least one ppossible worrlldd inn whhichh bbothh aarre trruue (P aanndd Q are
connssistenntt)) bbuut thhere is nno ppossible worrlldd in whhichh bbothh aarre faallse (~'V PP aanndd '~V Q aarre iinnccoonnsissitsetnent)t.3).636

By ""iinnddeeppendenncee"" we mmean thhe rreelaattionn whhichh hholds bbeettwweenn PP aanndd QQ when nno rreelaattionn otherr

than consisstteennccyy hholds bbeettwweenn thhe two. That is to say, inddeeppeennddeennccee is thhe rreelaattionn whhichh holds

bbeettwweenn PP aanndd Q when thhere is aat least one ppossible worrldd inn whhichh bboth aarre trruue (P aanndd Q are

connssistenntt)), thhere is aat least one ppossible worrlldd inn whhichh bbothh aarre faallssee (('~VPP aanndd Q'V aarre also
connssistenntt)), thhere is aat least one ppossible worrlldd in whhich PP is trruue aanndd Q is faalse, aanndd thhere is aat leeaastt
one ppossible worrlldd inn whhich PP is faallssee aanndd Q is trruue.

Of the varriiouus mmooddaall rreelaattions we hhaave ddistinguished, onnly one is uunniiqquueely ddeppicted bby aa single
worrlds-diagramm. Thhis is thhe rreelaattionn of inddependdennce. IInnddependenncee is ddeppicted bbyy ddiagrraamm 15 aanndd bby
that ddiagrraamm aalonne. Each of the otherr mmooddaall rreelaattions is exempplifieedd bbyy two orr mmorre of the fiftteeeenn
worrldds-diagrraammss.. HHiissttoorriiccaalllyy,, thhis fact hhaas nnot aalways bbeen rreeccoognniizedd. Within trraaddiittiioonnaall looggicic,3,737

36. The terrmm ""ssuubbcontrrarriiety"" rreflectts the fact, well known to trraaddiittiioonnaall logicians, that ssuubbccoonnttrrary
prrooppoositions arre contrraaddiictorriies of prrooppoositions which arre contrarriies andd stand in the rrelattion of ssuubblimmpplliccatlion

to thheessee contrarryy prrooppoosittions. Thuuss P is a subcontrarryy of QQ if andd only if (a) '-"v P andd '~" QQ arre contrraarriieess, aand

(b) the contrarryy prrooppoositions, '~" P anndd'~" Q, imppllyy Q andd P rreessppeeccttiivvely.

37. By ""ttrraaddiittiioonnaall logic"" we mean the logic which was founded bby Aristottllee (384-3222 B..CC.), enrriched bbyy the
Stoics andd Megarriiaannss,, andd effecttivveellyy canonized bby sixtteeenntthh-- andd sevveenntteeeenntthh--cceenntuturryy logicians.

54 POSSIBLE WWOORRLLDDS

thhe termms ""eeqquuiivalennccee"",, ""ccoonnttrraaddiiccttion"",, ""ssuuppeerriimmpplliicaattion"", ""ccoonnttrraarriiety"", ""ssuubbiimmpplliicaattion"", and

""ssuubbccontrraarriety"" wwerree oftteenn uused ass if each of thhemm, too, wwerree rreelations whhichh,, inn termms of ouurr

worrldds-diagrammss, we would uunniqquueely ddeppict bby aa sinnggle ddiaagrraamm.. Trraaddiittional logicciiaanns tendedd to thhinnkk

off equuivalennccee aas that rreelaattion whhich we can rreeconstruuct, bby mmeeaannss of our worrldds-diagrramms, aas holding

in ddiagraamm 9 aalone; of contrraaddiiccttion aas if it could bbe ddeppictedd inn ddiagraamm 10 aalone; of superrimmppllicationn

(and hheennccee of immppllication) aas if it could bbe ddeppictedd in ddiagraamm 111 aalone; of contrraarriiety aas if it couuldd
bbe ddeppictedd in ddiagraamm 12 aalone; of subbimmppllication aas if it couldd bbe ddeppictedd inn ddiagraamm 13 aalone; and
off subcontraarriety aas if it couldd bbe ddeppictedd inn ddiagraamm 14 aalonne. That is to say, tthheeyy tendedd to thhink off
tthheessee rreelations aas if tthheeyy could hholdd onnly bbeettwweeenn pprrooppoossitionns bboth of whhich aarre contingenntt. (IItt is eaasy
to see, bby simple inspection, that ddiagramms 9 thhrroouughh 15 ddeppict the onnly ccaasseess inn whhich both
pprrooppoossiittiioonnss aarree ccoonnttiinnggeenntt..)) OOnnee ccoonnsseeqquueennccee ooff tthhee ttrraaddiittiioonnaall pprreeooccccuuppaattiioonn wwiitthh rreellaattiioonnss bbeettwweeeenn
ccoonnttiinnggeenntt pprrooppoossiittiioonnss wwaass,, aanndd ssoommeettiimmeess ssttiillll iiss,, tthhaatt ssoommeeoonnee bbrroouugghhtt uupp iinn tthhaatt ttrraaddiittiioonn,, iiff aasskkeedd
wwhhaatt llooggiiccaall rreellaattiioonn hhoollddss bbeettwweeeenn aa ppaaiirr ooff ccoonnttiinnggeenntt pprrooppoossiittiioonnss ((aannyy ooff tthhoossee ddeeppiicctteedd bbyy
ddiiaaggrraammss 99 tthhrroouugghh 1155)),, hhaadd lliittttllee ddiiffffiiccuullttyy iinn iinnvvookkiinngg oonnee ooff tthhee ssttaannddaarrdd rreeppeerrttooiirree ooff
""eeqquuiivvaalleennccee"",, ""ccoonnttrraaddiiccttiioonn"",, ""ssuuppeerriimmpplliiccaattiioonn"",, ""ccoonnttrraarriieettyy"",, ""ssuubbiimmpplliiccaattiioonn"",, ""ssuubbccoonnttrraarrii--
eettyy"",, oorr ""iinnddeeppeennddeennccee"";; bbuutt iiff aasskkeedd wwhhaatt llooggiiccaall rreellaattiioonn hhoollddss bbeettwweeeenn aa ppaaiirr ooff pprrooppoossiittiioonnss oonnee oorr
bbootthh ooff wwhhiicchh iiss nnoonnccoonnttiinnggeenntt ((aannyy ooff tthhoossee ddeeppiicctteedd bbyy ddiiaaggrraammss 11 tthhrroouugghh 88)),, ssiimmppllyy wwoouulldd nnoott
kknnooww wwhhaatt ttoo ssaayy.3.838

A secoonndd aanndd mmorre signiffiiccaanntt connseqquueennccee of tthhiiss pprreeooccupaattion with rreelaattions bbeettwweeenn ccoonnttingennt

pprrooppoossitions waas, aanndd still is, that the ""iinnttuuiittionns"" of sommee (but nnot, of courrsse, aall) pperrsons uunndder the

influuenncee of tthhiiss trraaddiittiionn aarre nnot well aattunedd to the aannaallyysseess,, in termms of ppossibblee worrlddss, of rreellaattions

like equuivalennccee aanndd immppllicaattiionn;; tthheey tend to considdeerr tthheessee aannaallyysseess aas ""ccoouunntteerr--inntuuitivvee"", and
certaain connsequueenncce of tthheessee aannaallyysseess aas ""ppaarraaddooxxicicaal"l."3.939

Capsule descriptioonnss ofj moddaall rreellaattiioonnss

IIt mmaay bbe hhelpful to gatherr ttogetherr the ddescriptions we hhaave given inn varriouus ppllaacceess of tthhoossee mmodal
rreelations whhich aarre our pprriinncciippllee ccoonncceerrnn..

P is incoonnssiisstteenntt with QQ:: tthherre is nnoo ppoossible worrlldd inn whhichh bbothh aarree trruue

P is aa contradiccttoorryy of Q: tthherre is nnoo ppoossible worrlldd inn whhichh bbothh aarree trruue
andd nno ppossibblee worrldd inn whhich bboth aarre false

P is aa contraryy of Q: tthherre is nnoo ppoossible worrlldd inn whhichh bbothh aarree trruue
bbuut there is aa ppossibblee worrldd inn whhich bboth arre
faallssee

P is connssiisstteennttwwiitthhQQ: : tthheerreeisisaappoosssiibbllee wwoorrldldininwwhhicichhbbootthhaarreettrruuee

P implies (superimplies) QQ:: tthherre is nnoo ppoossible worrlldd inn whhichh PP is ttrruue and

QQ is faalse

P followss fromm ((ssuubbiimmpplliieess)) QQ:: tthherre iiss nnoo ppoosssible wwoorrlldd inn wwhhiicchh Q iiss ttrruuee and

P is faalse

38. Answers cann, howeverr, bbe given. WWorrllddss-ddiaagrraamm 8, forr example, depicts a rreelattion which saattiissfies the
descriptions given of twoo mmoddaall rreelattions, viz., impplication anndd contrraarriieettyy. That is, two prrooppoossitions, P anndd QQ,

which stand in the rreelattion depicted in diagraamm 8, arre such that (1)) P impplies QQ anndd (2) P anndd QQ are

contrraarriieess.

39. In chapterr 4 we developp thhiis point furrtherr andd suuggggestt aallsoo that sommee rreessisttenncce to tthheessee aannaallyysseess has its
source inn a failurre to distinguish bbettwweenn validd inferrencee anndd ddemmonnstrraabbiillitty.

§ 6 Modal..PPrrooppeerrttiieess and Relations Pictured on WWorlds-Diagrams 55

P iss eqquuiivvaalelennt t tto QQ:: in each possible worldd,, P and QQ have matching
P is aa subccoonntrtraaryry of QQ:: trruutthh--values

P is indeppeennddeenntt of QQ:: there is a possible world in which both are true
but there is no possible world in which both are
false

there is a possible world fhni which both are true,
a possible world inn whichh both arre false, a

possible world in which P is true and QQ is false,
and a ppossssiibbllee woorrld in whhich P is faallssee and QQ iis

true

EEXXEERCRISCEISSES

PPaarrtt A

1,.. Whhiicchh woorrllddss-d-diaiagrgarmams, s (in ffiigguree ({L\i.»i))) reprreesseenntt caseess in whiicchh PP is neecceessasarirliyl-y true (i.e., in
whiicchh oOPP obbttaaiinnss)) ?

2. Whhiicchh woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh Qisis neecceesssaarirliyly truuee? [O[•Q(?]]
3. Whhiicchh woorrllddss-d-diaigargarmasms reprreesseenntt caseess in whiicchh PP is coonnttiinnggeenntt?? [[\V7P]]
4. Whhiicchh woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh Q is coonnttiinnggeennt?t? [\V7QJ]

5. Whhiicchh three woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh wee maayy valliiddllyy infeerr that PP is accttuuaalllyy true
(i.e., true in the accttuuaall woorrldld) )??

6. Whhiicchh seven woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PP impplliieess Q?? [P-—--+>QQ]]

7. Whhiicchh seven woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh Q impplliieess PP?? [Q-—--+*P]\
8. Whhiicchh eighht woorrllddss-d-diaigargarmasms reprreesseenntt caseess in whiicchh PP is coonnssiisstteenntt wiitthh Q?? [PP 0o Q]

9. Whhiicchh seven woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PP is inccoonnssisistetnetnt wiitthh Q?? [PP <~t> QQ] _]
10. Whhiicchh three woorrllddss-d-diaigargarmasms reprreesseenntt caseess in whiicchh PPaandnd Qaarere cocnotnrtardaidcitcotorrieiess??
11. Whhiicchh foouurr woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PP anndd Q aree coconntrtararrieiess??

12. Whhiicchh foouurr woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PP anndd Qaarere susbucbocnontrtararrieiess??
13. Whhiicchh three woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PPaandnd Qaarere eqquuiivvaalelennt?t?[P[<P—......>QQ]]
14. Whhiicchh woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PP is poossssiibbllee anndd in whiicchh Q is poossssiibbllee?? [[Vr~P:p aand

<0>.0.0}]
15. Whhiicchh woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh PP is poossssiibblele, , Q is poossssiibblele, , anndd in whiicchh PP is

inccoonnssisistetnetnt wiitthh Q?? [[OOPPanadnd OOQQanadnd (PP <~t> QQ)j]]
16. Whhiicchh woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh neiitthheerr PP impplliieess Q noorr QQ impplliieess PP??

[[^'V(P(-P>--Q-+) Q) and a'nVd(^Q(-Q---+>P)]
17. Whhiicchh woorrllddss-d-diaiagrgarmams s reprreesseenntt caseess in whiicchh both PP impplliieess Q anndd PP is coonnssiisstteenntt witthh QQ??

[(P-->-+QQ) ) anndd (P (0PoQQ))]}

5566 POSSSSIIBBLE WWOORRLLDDSS

188.. Which worlds-diagrraammss repreesseenntt caasseessininwwhhicichhbbooththPPimimpplileiessQQaannddPPisisiinnccoonnssisistetenntt
with Q? [{((PP-_>QQ) and (P*4> Q)]]

199.. Which worlds-diaaggrraammss repreesseenntt ccaasessesininwwhihcihchbobtohthP PimipmlipelsieQs QanadndQ Qdodeosensont oimt ipmlyplPy?P?
[((PP-_+Q) anadn'"d^((Q-_>PP))}]

20.. Which worlds-diagrraammss reprreesseenntt caasseessininwwhhicichhPPaannddQQ aarereccoonnsissitsetnetntwwitihthoonneeaannotohtehrer
but in which P dooeessnnoot timimpplylyQQ??[[(P(P0 oQQ) )anandd'"^(P(P_—Q>) Q] ) ]

21. Which worlds-diagrraammss repreesseenntt ccaasessesininwwhihcihchPPanadndQQaraereinicnocnosnistiestnetnwt withithonoeneanaontohtehrer
and in which P dooeessnnoot timimpplylyQQ??([(P(P4> $QQ) )aanndd'"~(P(P-—Q>)]Q) ]

22. Which worlds-diagrammss repreesseenntt ccaasessesininwwhihcihchPPimimplpileisesPP? ?[P[P_—PJ>P]
23. Which worlds-diagrraammss repreesseenntt caasseessininwwhhicichhPPisisccoonntitninggenetn,t,QQisisnneceecsessasrairlyilytrutreu,ea,nadnd

in which P implies Q? [[VV'PP,, onQQ, and (PP—_Q>Q)]) ]
24. Which worlds-diagrraammss repreesseenntt caasseessininwwhhicichhPPisisnneeccesesasrairlyilytrtureu,eQ, Qisiscocnotnintignegnetn, ta,nadnd

in which P implies Q?? t[oUPP,, VvQ'Q,, and ((TP—-Q>Q)J] ]
25.. Which worlds-diagrraammss reprreesseenntt caasseessininwwhhicichhPPisiseeqquuivivaalelennt ttotoQQaannddininwwhhicichhPPaanndd

Q arree inconssiisstteenntt withh onee anotthheerr?? [[((PP...<....^Q->) Qan) dan(Pd 4(>PQ<t)>] QJ]
26.. Which worlds-diagrraammss repreesseenntt ccaasessesininwwhihcihchPPimimpplileiess'"^PP? ?[p[P--*'"~PJP]
27.. Which worlds-diagrraammss repreesseenntt ccaasessesininwwhihcihch'"^PPimimplpileisesPP? ?[ '[" P~_PP—]tP]
28.. Which worlds-diagrraammss repreesseenntt ccaasessesininwwhihcihchPPisiscocnosnistiesntetnwt iwthit"h, P^?P[?P[oP'o" P~JP]
29. Which worlds-diagrraammss reprreesseenntt caasseessininwwhhicichhbbooththPPimimpplileiess '"^QQ aannddPPisiscocnosnisitsetnetnt

with Q?? [/P>->",Q^QananddPPooQQ])
30. Which worlds-diagrraammss repreesseenntt caasseess ininwwhhicichhPPisisccoonntitninggenent taanndd '"~PPisisnenceecsessasrailryily

truuee?? [[VV'P and 0 d'"^PP]
31 - 45. For eaacchhooffththeeffifitfeteenenwwoorlrdlsd-sd-idaigargarmams sininfifgiugruere(1(.li).ifi)nfdinadnyantwy otwporopproospiotisointisonwshwichicshtasntdand

in the relation depicted by thhaatt ddiiaaggrraamm..

Exampllee:: diagraamm 114

Let "P" == Theerree are fewweerr tthhaann3300,0,0000gaglaalxaixesie. s.
Let "Q" = Theerree are morree tthhaann 1100,0,0000g0aglaaxlaiexsi.es.

PartBB

The expresssiioonn "It issffaallssee tthhaattPP iimmpplliieess QQ"" iissaammbbiigguuoouuss bbeetwtweeennsasyaiynigng"P"Pdodeosensont oitmipmlyplyQ"Q"anadnd
"P's being falsee implies Q" which may be symbolized unambiguously as "", (PP—_>QQ))"" and
"(( '~" PP—_*QQ))""rreessppeecctitviveelyly. .

46. Which worlds-diaaggrraammss repreesseenntt ccaasessesininwwhihcihchPPdodeosensont oimt ipmlyplQy ?Q[?'"[(~P-Q(P)^Q1 )}

47. Which worlds-diagrraammss repreesseenntt ccaasessesininwwhihcihchPP's'sbebienigngfaflaselseimimplpielisesQQ? ?[ '"[ P^P-—Q»l Q]
48. Which worlds-diaaggrraammss reprreesseenntt caasseessininwwhhicichh bbotohthPP's'sbbeeininggtrtureueimimpplileiessQQ.aarn>d-dP'Ps's

being falssee implies Q?? [((PP_->QQ)) aanndd ('" P(^_PQ^Q)J1}

§ 6 MModdaal PPrrooppeerrttiieess aand RReellaattiioonnss PPicttuurredd on WWoorrllddss-Diagrramms 57

Part C

The expression ""QQ is a falsee implicationn of P"" meeaannss "Q is falsee and Q is an implication of P."" In

order too reprreesseenntt thhiiss relation on a set of worlds-diagrams we shall haavvee too haavvee aaddeevviciecefoforrdedpeipcitcitningg

the actual world (i..ee..,, for depictinngg thhaatt Q is falsee)).. Rathheerr thaann persissttiinngg with the convention ooff

figures (1l..dd)),, (h1e.e)), , and (1If)j,), we shall hereinafter adoopptt the simplleerr convention of representinngg the
locattiioonn of the actual world among the set of possible worlds by "Xx"".. In worlds-diagrams 71 throuugghh 4,,
the "xX"" may be placed indiscrimiinnaatteellyy anywheerree withinn the rectangle.. Butt whheenn we coommee to
diagrams 5 throuugghh 7155 we are faceedd wwiitthh a nummbbeerr of alternatives. Too shhooww thheesseeaaltleterrnnaatitviveesswweewwilill
label eaacchh of the internall seggmmeennttss ooffththee rreecctatannggleleffrroomm leleftfttotorrigighhtt bbeeggiinnnniinngg wwitihth ththee lelettteterr""aa".".
TThhuuss,, ffoorr eexxaammppllee,, wwoorrllddss--ddiiaaggrraamm 7171ccoonnttaaiinniinngg aann ""xX"" iinn tthhee cceennttrraall sseeggmmeennt twwoouuldldbebeddeseisgignnaatetedd
""7171bb"".. AA wwoorrllddss--ddiiaaggrraamm oonn wwhhiicchh tthhee llooccaattiioonn ooff tthhee aaccttuuaall wwoorrlldd iiss eexxpplliicciittllyy mmaarrkkeedd bbyy aann ""Xx"",, wwee
sshhaallll ccaallll aa ""rreeaalliittyy--llooccaattiinngg wwoorrllddss--ddiiaaggrraamm""..

49. Takinngg accoouunntt of all the differeenntt waayyss the actual world may be depicted on a
worlds-diagraamm,, how maannyy distinct reality-locating worlds-diagrams are theerree for two
propositions?

50. Using the convention just discussed for describing a worlds-diagram on which the actual

world is deppiicctteedd,, which worlds-diagrams reprreesseenntt caasseessininwwhhicichhQQisisaafafalsleseimimpplilcicaatitoionn

ofPP?? [(P---^-Q.Q)J andd '~VQQ]

571. Assummee P to be the truuee proposition thhaatt theerree are exaccttllyy 71300 persoonnss in soommee particular
room, 2B. Let Q be the proposition thhaatt theerree are fewer thaann 71400 persoonnss in roooomm 2BB..
Draww a reality-locating worlds-diagram representiinngg the moddaall relationn beettwweeeennththeseesetwtowo
propositions.

(For queessttiioonnss5522--5555)) AAssssuummeetthhaat ttthheerreeaarereexeaxcatcltyly731030pepresrosnosnisninroroomom2B2. BL.eLt ePt bPe btheethperopprospiotisoitnion

thhaatt theerreeaarreeeexxaacctltyly 713355pepresrosnosnsininroroomom2B2B, ,ananddQQthteheprporpoopsoistitoinonthtahtatthtehreerearaeraetaltealestas7t3133p3erpseornssons

in rooomm 22BB..

52. Is QQ implieedd by P?

53. Is Q truuee??

54. Is P ttrrue?

55. Draww a reality-locating worlds-diagram represennttiinngg the modaall relationn beettwweeeenn the

propositions, P andd Q.

Apppeennddiixx too seccttiioonn 66 *****

It is an intterreestting qquueessttion to ask, ""HHoww mmaanny ddiffeerrenntt wwaayyss may thrreeee arrbbitrarrily chhoseenn

pprrooppoossiittiionns be aarrrraanngged on a worrlds-ddiiaaggrraamm?"" The aannsswer is ""2255"".

It is ppoossssibble to give a generraal forrmulaa forr the nnuummbber (WW'„n)) of worrllddSs--ddiiaaggrramms rrequirreedd to depict
all the ppoossssibble wwaayyss of aarrrraannggiinng any arrbitrrarry nnuummbber (n) of pprrooppoossiittiioonnss. That forrmulaa iiss4400

Wnn == 22 "22n -- 11

40. Alterrnnaattiivveelly, the forrmulaa may be given rreeccuurrssiivveellyy, ii..e.,
WIj == 3, and

+WW„n+r+ 7 =- (Wnn + 11))22-1

Using thhiiss latterr forrmulaa it is eaasy to shhow thhaat the nneext entrry in taabblle (71J.)j,),i.ie.e.,., WW7' ,wwoouuldldbbee3399 ddigiigtistsinin
lengthh.

58 POSSSIIBBLLEE WWOORRLLDDS
Wee mmaay calculate thhe value of Wnn forr thhe first few values of n..

n Wn
13

2 15
3 255
4 65,535
5 4,294,967,295
6 188~,44446,,774444,,007733,,770099,5~5151,6,16515

TABBLLEE ((1l.j)

7. IS A SINGLE THEOORRYY OOFF TRUTHH AD EQUATEE FOR B O TH
CONT ING E NNTT ANNDD NNOONNCCOONNTTIINNGGEENNTT PROPOSITIIOONS?

In thhis chapterr we hhaave intrroodduucceedd one thheory of trruutthh, thhe so-calledd CCorrrreessppoonnddeennccee Thheory of Truth
whhich hhaas it that aa pprrooppoossiittionn PP, whhich aascrribes aatttrribbuuttes FF to aann item a,, is trruue if aanndd onnly if a hhas
thhe aatttrribbuuttes FF.. Sometimmess thhis thheory is summmed up epigraammmmaattiiccaalllly bbyy saying that aa pprrooppoossiittionn P
is trruue if aanndd onnly if ''it fitts thhe faacts''.

Now sommee pphhilosoopphherrs who aarre pperrffectly willing to aacceppt thhis thheory aas thhe corrreect aaccouunt of the
way thhe trruutthh--vvaaluues of continngeenntt pprrooppoossitions come aabbouutt, hhaave felt thhis sammee thheory to be
inadequate orr inapppprroopprriiaattee in the caassee of nnoncontinnggeenntt pprrooppoossiittiionns. WWee would ddoo well to rreeview
thhe sort of thhinnkkiinngg whhichh mmight lead one to thhis ooppinion.

Suppppose one werree to chhoooossee aas onnee''ss first example aa continngeenntt pprrooppoossiittiioonn,, let uus say the
pprrooppoossittion

(1.4422)) Canada is nnorrtthh of MMexico.

WWhhen one aasskkss what mmaakkes (1.-4422)) aactuuaallly trruuee,, thhe aannswerr is obbvious enough:: thhis ppaarrticcuullaarr
pprrooppoossittion is aactuuaallly trruue bbeecaauussee of certaain ppeculiarr geogrrapphhicaal feaatures of thhe aactuuaal world,
feaatures whhich aarre sharred bbyy sommee otherr ppossible worrldds bbuut ddefinitely nnot bbyy aalll..

Following aalong inn thhis veinn,, we cann aask aa similarr qquueestion aabbout nnoncontinngeenntt pprrooppoossiittiionnss. LLet
uus taakke aas aann example th~e nnoncontinngeenntt prropposition

(1.4433)) Either Boothh aasssaassinnatedd Abraham Lincoln orr it is nnot thhe caassee that Boootth
aasssaassinnatedd Abraham Lincoln.

§ 77 IIss aa SSiinnggllee TThheeoorryy oof TTrruutth AAddeeqquuaattee fforr BBootthh CCoonnttiinnggenntt aanndd Noncontingennt PPrrooppoossiittionns?? 59

IIf wwee nnooww aasskk wwhhaat mmaakkees tthhiis pprrooppoossiittiioonn ttrruuee,, sommee pperrsons hhaave felt that the answeerr caannott be ooff
tthhee ssaamme ssorrtt aass tthhaat jjuust givenn inn ttnhe ccaassee of thhe exampple of aa continnggeenntt prrooppoossiittiioonn.. Forr tthhessee
ppeerrssoonns rriigghhttllyy nnoottee tthhaat wwhherreas tthhe ttrruutthh--vvaalluuee of (1.-4422)) varriies from possibblee worrldd to possibblee
wworrlldd,, tthhee ttrruutthh--vvaalluuee oof (11..4433)) ddooeess nnoott.. No mmaattter hhow aannotherr possible worrldd mmaay diffeerr frrom thhe
aaccttuuaall wworrlldd,, nnoo mmaatttterr hhoow ouuttlaannddiishh aanndd farrffetched that worrlldd mmight seemm to us in the aaccttuuaall
wworrlldd,, tthhee ttrruutthh--vvaalluuee oof (11..4433)) will bbee tthhe samme inn that worrlldd aas inn the actuuaal world. In the aaccttuuaall
wworrlldd,, BBooootthh ddiidd aassssaassssinnaattee LLinncoln aanndd (1.-4433)) is trruuee.. Buutt there aarre otherr possibblee worlds in wwhhicchh
LLiinnccoolnn ddiidd nnoott ggoo ttoo FForrdd''s TThheeaatterr onn April 14, 1865, aanndd livedd to be rree-eleecctteedd to a thirrd termm;; yyeett
inn tthhoossee wwoorrllddss,, (11..4433)) is ttrruuee.. TThhen, ttoooo, tthherre aarree tthhoossee ppossible worrldds in which Booth didd sshhoooott
LLiinnccoolnn,, bbuutt tthhee wwouunndd wwaas aa suuppeerrffiicciiaall onne aanndd Lincoln rreeccoverreedd;; yet in thhoose worrlddss, too, ((11..443)
iiss trruuee..

IInn tthhee eeyyeess oof ssoommee pphhiilosopherrs tthheessee latterr sorts of facts chaallennge the corrrrespondence theorry ooff
ttrruutthh.. IInn eefffect tthheessee pphhiilosopherrs aarrgguue tthhat thhe trruutthh off nnoncontinngeenntt prrooppoossitions caannott bbee
aaccccouunntteedd fforr bbyy ssaayyinngg tthhaat ''tthhey fit tthhe faacts' since thhey rreemmaaiinn trruue whateveerr the facts happpen to bbe.
HHoow ccoouulldd tthhe ttrruutthh oof (11.-4433)) inn onne ppoossible worrlldd bbe explained in termms of one sett of facts and itts
ttrruutthh iinn aannootthherr ppoosssibble wwoorrlldd inn tterrmms of aa ddiffeerrennt set of facts? Thhee very suggessttiioonn has seeemmed tto
tthheessee pphhiillosophherrss ttoo cconstittuute aa faattaall wweeaakknneessss inn thhe corrreesponddence theory of truth. In theirr vieeww
tthheerree jjuust ddooesnn't sseeemm ttoo bbee aannyy pprrooppeerr set of facts to whhichh trruue nonconntinnggeenntt pprroppositioonns
''ccoorrrreessppoonndd''.. AAnndd ssoo,, tthheey hhaavve bbeen ledd tto pprrooppoose additional thheorries of trruth, theoriess to aacccoouunntt
ffoorr tthhee ttrruutthh--vvaalluueess ooff nnoonnccoonnttiinnggeenntt pprrooppoossiittiioonnss..

OOnnee oof tthhee oolddest ssuuppppllemmennttaal tthheeorries innvokkeedd to aaccouunt forr the trruutthh of necessssaarryy trruuthhs hholdds
tthhaatt tthheeirr ttrruutthh ddeeppenndds uuppoonn wwhhat wwerre ttrraaddiittiioonnaallllyy caalled ""TThhee LLaawwss of TThhouughhtt"",, or whhaatt mmosstt
pphhiilloosophherrss nnoowwaaddaayys pprreefferr ttoo caall ""TThhee LLaawwss of Logic". PPrreecciissely what is encoommpassseedd wwiitthhiinn
tthheessee ssoo-caalled llaawwss oof tthhouught hhaas bbeen aann issuuee of lonng ddispute aammoonng phhilosopherrs. Neveerrthelesss,, iitt
sseeeemmss faairrllyy cclearr tthhaat wwhhaatteverr eellssee is tto bbe inncluuddeedd,, thhe lawws of thoughhtt do include the 'Laws'' ooff
IIddeennttiittyy, oof TThhee EExxcluded MMiddddlle, aanndd of Noncontrraaddiiccttionn:: rroouugghhllyy,, the ''laws'' that an itemm is whhaatt iitt
iiss,, tthhaatt eeither aann itemm hhaas aa cerrttaaiinn aattttrriibbuutte orr it ddooeess nnott, aanndd that ann itemm cannot both have aa
ccerrttaaiinn aattttrriibbuuttee aanndd faail ttoo hhaavve it.

BBuutt tthhiiss ssuuppppllemmeennttaal tthheeorryy ttuurrnnss ouutt, onn examminnaattiionn, to bbe whhoolly lackkinngg in expplaannaatorry ppowweerr.
FFor iif tthheerree iss cconnttrroovveerrssyy ooverr tthhe mmaatttter of what is to bbe includdedd aammonng the laws of thought, thherree
hhaass bbeeenn eevenn ggrreeaater connttrroovveerrssyy,, wwithh mmoorree farr--rreeaacchhiinngg immppllicaattiioonnss,, overr the mmaatterr of whhaatt soorrts
oof tthhiinnggs tthheessee sso-caalled ""llaawwss of tthhought"" arree.. TThhee aaccouunt whhichh seemmss to worrk bbest in eexxppllaaiinniinngg
wwhhaat tthhee llaawwss oof tthhoouught aarree,, is tthhaat whhiichh saayyss of thhem that thhey aarre thheemmsseellvvees nothing otthheerr tthhaann
nnoonnccoonnttiinnggennttly ttrruuee pprrooppoossiittiioonnss.. BBuutt if tthhis aaccount is aaddopptted —- aanndd it is by mmost ccoonntteemmppoorraarryy
pphhiilloosophherrss -— tthheenn it is verryy hharrd tto see hhow thhe laawwss of thoughht cann serrve as an expplaannationn ooff
wwhhaatt iitt iiss tthhaatt tthhe ttrruutthh--vvaalluuees of nnoonncontingenntt pprrooppoossiittions ddepend onn.. Thhe trrouubble is thatt if aa
ccerrttaaiinn ccllaassss oof nnoonnccoonnttiinnggennttly ttrruuee pprrooppoossiittiioonnss, thhe hhonnorreedd ""llaawws of thoughhtt"", arre to accoouunntt ffoorr
tthhee ttrruutthh ((orr ffaallssittyy) oof ootthheerr (tthhee rruunn--ooff--tthhee--mmiillll)) nnonncoonnttiinnggeenntt pprrooppoossiittionns, tthheenn ssoommee ffuurrtthheerr,,
pprreessuummaabbllyy ssttiillll mmoorree hhoonnoorreedd pprrooppoossiittiioonnss mmuusstt aaccccoouunntt ffoorr tthheeiirr ttrruutthh,, aanndd ssoo oonn aanndd oonn wwiitthhoouutt
eenndd.. IInn sshhoorrtt,, iinnvvookkiinngg ssoommee pprrooppoossiittiioonnss ttoo aaccccoouunntt ffoorr tthhee ttrruutthh oorr ffaallssiittyy ooff ootthheerrss lleeaaddss oonnee ttoo aann
iinnffiinniittee rreeggrreessss oorr llaannddss oonnee iinn aa cciirrccuullaarriittyy..

A sseecondd ssuuppppllemmeennttaall tthheeoorryy,, wwhhiichh wwe shhaallll call ""tthhe linguuistic thheory of necessssaarryy truth"", holdds
tthhaatt tthhee ttrruutthh--vvaalluueess oof nnoonncontingennt pprrooppoossiittionns come aabboouutt, nnot thrroouugghh corrrrespondence witthh tthhe
ffaaccttss,, bbuutt rraatthheerr aass aa rreessuult of cerrttaaiinn ''rruullees of lannguaage'. Caappiittaallizing on the very wordd ""cconntingeenntt"",
tthhiiss tthheeoorryy iiss ssoommeettiimmeess exprreessedd inn tthhe following waayy:: Thhee trruutthh--vvaaluues of conntinnggeenntt pprroppositioonns
aarree ccoonnttiinnggeenntt uuppoonn tthhe faaccttss, wwhhiile tthhe ttrruutthh--vvaalluuees of nnonncontinngeenntt prrooppoossitions arre not conntinnggeenntt
uuppoonn tthhee ffaaccttss bbuutt aarree ddeetteerrmmiinneedd bbyy rruulleess ooff llaanngguuaaggee.. OOrr ppuutt ssttiillll aannootthheerr wwaayy:: ccoonnttiinnggeenntt ttrruutthhss
((aanndd ffaallsseehhooooddss)) aarree ffaaccttuuaall;; nnoonnccoonnttiinnggeenntt ttrruutthhss ((aanndd ffaallsseehhooooddss)) aarree lliinngguuiissttiicc..

60 PPOOSSSSIIBBLLEE WW OO RR LDD S

The linguistiicc thheeoory off neceessssary truth does noot beloongg oonly to the ppreservee ooff pphiloosoopphheers. It iis
enshrined allssoo inn commmmoonnppllaaccee talk off ceertaiinn propoossiitions being true ((oorr ffalse) ""bbyy ddeffinittiioonn"".. TThhuus ,
it might be saidd, the propositioon

(1.44)) AA lll rectangles have foouur sides

owes ittss truth too the fact that we human beings have resolved to definee the term ""rrecctangle" as ""ppllane
closed figuree ww iith four strraaiigghtt sides allll off whose angles are equal"". F or, ggivven such a ssttiippuullaative
definitiioon, thhe truth off (1.44)) follows iimmmmeeddiiately.

This talk of "definittiioonnaall truutthhss"", and simmiillaar talk ooff "vveerbal truths", suggggeessttss that neecceessssaarryy ttrruuths
aree not grounded -— as are coonnttinggeennt truutthhss -— in coorrreesppoonnddeennccee bbeetwweeeenn pproppoossiittioonns and states ooff
affairs, but are grounded ratthheer in some sort off coorrresppoonnddeennccee bbeetwweeeenn pproppoossiittioonns and the arbitrary
connvveennttionnss orr ruulleess forr the use off woorrddss whhich wwee languuagee uusseerrs hhaapppeenn tto adopptt.

YYeet, despite itss initial plausibility, this theory seems to us ddefective.
Its plausibilityy, we suggeesst, derives -— at least in part —- from a ccoonnffuussioon. LLeet us ccoonncceeddee tthhaatt
when we inttrroodduucce a new teerrmm intoo ouur language by definingg it ((exxpplicitllyy oorr immpplliicciitly)) in terms off
alrreeaaddyy avvaiillaabbllee expressions, we make availaabbllee to oouurselves neeww ways ooff eexxpprreessisningg truths. BBuut this
dooes not mean that we theereebby make availaabbllee to ourselves new truthhs.
Once we havve inttrroodduucceedd the termm "rectanggle" by definitiioonn in the mmaannneer sketchedd abovve, the
sentencee

(1.45)) "" AA llll rectangles have foouur sides""

ww iill express a necessary truth in a way in ww hich wwee ccoouuld noott have exxpprreesssseedd it eeaarlieer. BBuut the
necessary truth whicch it exppresseess is just the necceessssaaryy truth

(1.46)) AA lll plane closed ffiguurreess ww iitthh foouur sides alll ooff whose
angles are equal have foouur sides

and thhis iss a proopositiioon whose truth is far less plaussiibbllyy attrriibbuuttaabbllee to ddeffinitiioonn.. FFoorr this
proopositioon, one is more inclined too say, wwoould be true evveen if human bbeeiinggss had neevveerr eexxiissted and
had thheeree neverr beenn anyy language whhaatteveerr in the wwoorrlld.

Talk abouutt neceessssaarry trruutthhss being "ttrue by definiittiioonn"" mmaay aalso ddeerrive ssommee ppllaauussiibbiilliittyy ffrromm tthe
ffaacct thatt ouurr knowwlleeddggee that ceerrtaiinn propoossiittionns are ttrue sommeettiimmeess ssttermnss ffrrom oouurr kknnoowwlleeddggee ooff tthe
mmeeaanniinnggss,, oorr ddeeffiinniittiioonnss,, ooff tthhee tteerrmmss iinn ww hhiicchh tthheeyy aarree eexxpprreesssseedd.. TThhuuss aa ppeerrssoonn wwhhoo ccoommeess ttoo kknnooww
tthhaatt tthhee eexxpprreessssiioonn "" ttrriiaacc"" mmeeaannss tthhee ssaammee aass tthhee eexxpprreessssiioonn ""bbiiddiirreeccttiioonnaall ttrriiooddee tthhyyrriissttoorr"" mmaayy
tthheerreebbyy ((eevveenn wwiitthhoouutt kknnoowwiinngg iinnddeeppeennddeennttllyy wwhhaatt eeiitthheerr mmeeaannss)) kknnooww tthhaatt tthhee sseenntteennccee

(71.47)) "" AA llll trriiaaccss are bidireeccttiioonnaall ttriioode tthhyyrriisstors""

eexxpprreesssseess aa nneecceessssaarriillyy ttrruuee pprrooppoossiittiioonn.. BB uutt tthhiiss ddooeess nnoott mmeeaann tthhaatt tthhee nneecceessssaarryy ttrruutthh ooff tthhee
pprrooppoossiittiioonn ssoo eexxpprreesssseedd iiss iittsseellff ssoommeetthhiinngg tthhaatt sstteermnss ffrroomm tthhee mmeeaanniinnggss ooff tthheessee eexxpprreessssiioonnss..

PPeerrhhaappss tthhee ggrraavveesstt ddeeffeecctt iinn tthhee lliinngguuiissttiicc tthheeoorryy ooff nneecceessssaarryy ttrruutthh hhaass ttoo ddoo wwiitthh iittss iinnaabbiilliittyy ttoo
eexxppllaaiinn hhooww iitt iiss tthhaatt nneecceessssaarryy ttrruutthhss,, ssuucchh aass tthhoossee ooff llooggiicc aanndd mmaatthheemmaattiiccss,, ccaann hhaavvee ssiiggnniiffiiccaanntt
pprraaccttiiccaall aapppplliiccaattiioonnss iinn tthhee rreeaall wwoorrlldd.. IInn ssuucchh aapppplliieedd sscciieenncceess aass aaeerroonnaauuttiiccss,, eennggiinneeeerriinngg,, aanndd tthhee
lliikkee,, aarriitthhmmeettiiccaall ttrruutthhss mmaayy bbee aapppplliieedd iinn wwaayyss ww hhiicchh yyiieelldd iimmppoorrttaanntt nneeww iinnffeerreennttiiaall ttrruutthhss aabboouutt
tthhee wwoorrlldd aarroouunndd uuss:: aabboouutt tthhee ssttrreessss ttoolleerraanncceess ooff bbrriiddggeess,, tthhee eeffffiicciieennccyy ooff aaiirrppllaannee pprrooppeelllleerrss,, eettcc..
BBuutt iiff tthheessee nneecceessssaarryy ttrruutthhss aarree mmeerreellyy tthhee rreessuulltt ooff aarrbbiittrraarryy hhuummaann ccoonnvveennttiioonnss ffoorr tthhee uussee ooff

§ 7 Is a Single Theory of Truth Adequaattee forr Bothh Conttinngent anndd Noncontingent Prrooppoossittiionns?? 6611

mathematical symbols, all this becomes a seeminng mirraaccllee.. Whyy should the worldd conform so
felicitiously to the conseeqquueenncceess of our linguistic stipulations?

The explanation of the success of logic and mathematics in their appplications to the world, we
suggest, lies elseewwhheerree:: in the possible--wwoorrldds anaalysis of necessssaarryy truth. Neceessssaarry trruutthhss, such as
thhoose of mathemattics, appply to the worldd becausee they arre trrue in all possible worlds;; and since the
actual worldd is a possible world, it foolllloowwss that they arre trrue in (i.e., appply to) the actual world. This
accouunt of the matter, it should be noted, does not rrequuirree a diffeerrenntt theory of trruutthh from that which
we have given for contingeenntt prrooppoosittions. To say that a prrooppoosition is trruue in the actual worldd is, we
have claimedd, to say that it fits the facts in the actual world, i.e., that staattees of affairs in the actual
world arre as the prrooppoosition assertts them to be. Andd likewise, to say that a prrooppoosittion is trrue in some
otherr possible worldd -— orr, for that matter, in all possible worlds -— is just to say that in that otherr
world -— or all possible worlds -— staattees of affairs arre as the prrooppoosittion asserrtts them to be. One and
thhee saammee theoorry off trruutthh suufffiicceess foorr all cases:: thee caase whheenn a pprrooppoosittion is trruue in all pposssiibbllee wwoorrlds
and thee case whheenn itt is trrue in jjuusstt soomme (ppeerrhhaappss thhee acttual onnee iinncclluudded).

In orderr to see how this single accouunt of trruutthh sufficeess for all cases, let us rreturrnn to the example of
the noncontingeenntt prrooppoosittion (1-.43) to see how this theory might be invoked to explain its truth.

But first, let us rreflectt for a moment on anotherr prrooppoossiittiioonn,, viz., the contiinnggeenntt proposition

(1.48)) Eitherr Georrge Washington was the first prresident of the Unitedd Stattes orr
Benjamin Franklin was the first prresident of the United States.

This prrooppoossiittiioonn,, being contingennt, is trruue in some possible worlds and false in all the others. What
feaature is it, in thhoose possible worlds in which (1.48)) is trruuee, which accounts forr its trruutthh?? The
answeerr is obvious: (1-.48) is trrue in all and only thhose possible worlds in which eittheerr Washington orr
Franklin was the first prresidennt of the United Stattees. But note -— as in the case of the nnonncontingent
prroopposition (71.43)) -•— how much varriaattiioonn occurs between the worlds in which the prrooppoosition is trruue.
In the actual world, for example, Washinnggttoonn,, not Franklin, was the ffiirstt prresident of the UUnnited
Staattees. Thus in the actual world, (1-.48) is trruuee. But in some otherr possible worlds, Franklin, not
Washington, was the firstt prresident of the Unitedd Stattees. But in that world, too, (1.48)) ffiitts the factts,
i.e., is trruue.

Here, then, we have a parraalllleell to the situation we discovered in the case of the nnonncontingent
prroopposition (1.43));,. that is, we have alrready noted that (1.43)) is trrue in some possible worldd in which
Booth assasssiinnaatteedd Lincoln and is trrue in some possible worldd in which he didd not. The rrelevant
difference, in the prresenntt contextt, between thheese two cases -— the noncontingeenntt prrooppoosittion ((11..43)
and the contingeenntt prrooppoosition (1-.48) -— lies in the fact that the formerr prrooppoosition is trruue in every
possible worldd in which Booth didd not assassinate Lincoln, while the latterr is trruue in only some of the
pposssiibbllee woorrlds in whhich Washington waass nott thee ffiirrst pprresiddenntt. Buutt this difffeerreennccee clleeaarrly has oonnly
ttoo ddoo wwiitthh wwhheetthheerr tthhee sseett ooff ppoossssiibbllee wwoorrllddss iinn wwhhiicchh eeaacchh pprrooppoossiittiioonn ffiittss tthhee ffaaccttss ccoommpprriisseess aallll,,
oonnllyy ssoommee,, oorr nnoonnee ooff tthhee ttoottaalliittyy ooff ppoossssiibbllee wwoorrllddss.. IItt iinn nnoo wwaayy cchhaalllleennggeess tthhee ccllaaiimm tthhaatt tthhee ttrruutthh
ooff bbootthh ((11..4433)) aanndd ((11..4488)) iiss ttoo bbee aaccccoouunntteedd ffoorr iinn tthhee ssaammee wwaayy,, vviizz..,, iinn tthheeiirr ffiittttiinngg tthhee ffaaccttss..

Summing up, we may say that there is no speecciiaall prroobblleemm about the trruutthh--vvaalues of nnonncontingent
prrooppoosittions. They arre trrue or false in exactllyy the same sort of way that contingeenntt prrooppoositions are
trrue or false; i.e., depending on whetthheerr or not they fit the facts. The supposition that there arre ttwo
kinds of truth, or that there is a need for two theories of truth, is misconceived. Prrooppoossitiitoionsns arre true
or false; also they arre contingeenntt or noncontingent. Andd thheese varrious prroperrttiies of prrooppoositions ((ttwo
for trruutthh-status and two for modal-sttaatus)) can combine in varrious ways (four, to be exact). But it
shhouuldd nott be thouugghhtt thhaatt in saayinng off a prrooppoosittion thhaatt itt is, foorr exxaammple,, nonncoonnttinnggeennttllyy trruuee, we
arre saayinng thhaatt itt exxeemmpplliiffiieess onnee off twoo typpes off trruutthh.. Rathherr,, this exxpprresssiioonn ouugghhtt to bbe ccoonnssttrruuedd
as ""nnoonnconnttiinnggeenntt anndd trruue"". Viewingg thhee maatttteerr in this faashion, itt is eaasy to seeee thhaatt thee quuestionn

6622 PPOOSSSSIIBBLLEE WW OO RR LDS

abboouutt thee specciial wwayy inn wwhhich thee truuthh ((oor ffalsityy)) ooff noonnccoonnttiinnggeenntt pprooppoossiittioonnss is supppoosseedd to coome
abboouutt does not eevveenn aarriissee.. LLooggiic dooees not reqquire mmoorree thaann oone theooryy ooff truth.

8. TTHH EE ""PPOO SSSSIBLE-WW OORRLLDDSS"" IIDDIIOOMM

TThhroouugghhoouutt this booook we speeaakk oofftteenn of ootthheerr ppoossssiibbllee wwoorrlds. WWhyy have we,, inn ccoommmmoonn wwiitthh many
ootthheerr pphiloosoopphheerrs anndd looggiicciians, adoopptteedd this idiom? T hee answeerr lies inn its enoorrmmoouuss heuristtiicc vvaalue.

MMaannyy ccoonntteemmppoorraarryy pphiloossoopphheerrs bbeellievvee that thhee ppoossssible-wwoorrlldds idioom pprovviiddees a single
theooreetical fframewoorrk ppoowweerrffuul enoouugghh to illummiinnaatee anndd resolve mmany off thhee pphilosophicall prooblems
surroundingg such mmaatters aas

1. T hee loggical nootioons off necceessssity, ccoonnttingencyy,, ppoosssibilittyy,, immpplliiccaatioonn, vvaalliiddiity, anndd tthhe
lli k ee

2. T hee distiinnccttiioonn bbeettwweeeenn loggical necceessssiityy anndd pphysical neceessity
3. T hee adequaccyy of a single theory of truth
4. T hee identittyy ccoonnddiitions of propoossiitions and of ccooncepts
5. Thee disambiguattiioonn of sseenntteenncces
6. Thee linkk beetwweeeenn thhee mmeeaning of a sentenccee anndd thhee truutthh--ccoonddiittiioonnss of tthe

propoossiitioonn((ss)) itt eexxpresses
7. Thee technique of refutattiioon by imaginarryy ccountereexxamples
8. T hee epistemic coonncceepptt of that wwhich iss humanllyy knoowable
9. T hee distiinnccttiioonn beetwweeeenn accciiddeental anndd essentiall propeerties
10. Thhee cooncceeppt of the coonnttinggeennt coonntteenntt of a propoosition
11. Thee cooncceeppt of pprroobbaabbiilliiffiiccaation

Each of these iss dealt ww iith, inn thhee ordeer given,, inn this book. Buutt ththeelilsistt ggooeess bbeeyyoonndd ththeeccoonnffiinneess ooff
this book. Reecceenntt woork coouucchheedd ww iit h i n thhee possible-woorrlds idiioomm includes woork iin

12. Ethhiiccs
13. Coouunnterfaacctuals
14. Epistteemmiicc loogicc
15. Prooposittiioonnaall aattittiutuddeess

and much more besides. Too cite some examples: C.BB.. Danieellss inn TThhee Evaalluuaattiioonn of Ethiiccaall TThheeoorriieess::

Foo r tthhee purposes of this book, ann ethhiiccaall thheeoory iss simmppllyy aa deterrmmiinnaattiioonn of aa unique sett of
ideal woorlds.. Ann ideal worldd,, relaattiivvee ttoo aa thheeoory that determines itt aass ideal, iss aa possible wwoorld,,
a fairyy worldd perhaappss,, inn whiicchh everyytthhiinngg tthhee tthheeooryy sayys iss iddeeaallllyy ttrruuee iss inn fact ttrruuee.. AAn
ideal woo rldd relaattiivvee ttoo aann ethhiiccaall thheeoory iss aa woo rldd inn whicch everythhiinngg tthhee thheeoory says ougghht too
be tthhee case iss tthhee ccaassee.4.141

4411.. HHaalliiffaaxx,, NNoovvaa SSccoottiiaa,, DDaallhhoouussiiee UUnniivveerrssiittyy PPrreessss,, 11997755,, pp.. 11..

§ 8 Thee ""PPoossssiibblle-Worrldds"" Idiom 63

David Lewis, in CoCuonutnetrefra/accttuuaallss:

'lIIff kangaroos hadd no tails,, they woulldd topplee over'' seemms to me to mean something like this: in
any possibblee state off affairs in which kangarroos have no tails, and which rresembless our actual
state of affairs as much as kangarroos having no tails permits it to, the kangarroos topple over. I
shall give a generral anaalysis of counterfactual conditions along thheese lines.

MM yy methods arre thhoose of much rrecenntt work in possible-wwoorrld semanticcss for intensional
loggiicc..4422

And ]Jaaaakkkkoo Hintikka, in ""OOnn the Logic of Perceptionn"":

Whhen ddooeess a kknnow (bbeellievvee, wish, pperrcceeive) mmorre than b?? TThhee onnly rreeaasonabble generaal aannswerr
seemmss to bbe that aa kknnoows mmorre thhaann b if aanndd onnly if thhe claassss of ppossible worrldds comppaattibble witth
what hhe kknnows is smmaallerr than thhe claassss of ppossible worrldds comppaattibble with what b kknnoowws.s4.343

The extraordinarryy success, and the veritable explosioonn of rresearch adopting the ppoossssibble-worlds
idiom was not the motivation or even the expeccttaattiioonn in its adopttion. As a matterr of fact, the idiom is
not especiaalllyy new; only its widesspprreaad adoption is. Talk of possibblee worlds can be found thrroughout
the writings of the great German mathematician and philosoppher, Gottfrriieedd Leibniz (1646-1716).
Leibniz was quite at home in the possibblee--wwoorrllddss idiomm.. He liked to philosophize in terms of that
idiom, asking such questioonnss as, ""Is this the bbest of all possiblee worlds?"" and ""WWhhyy didd Godd crreate
this parrttiicularr world rrather than some otherr possiblee world?"" Andd writing of thhoose trruths which we
(and he on occasioonn)) have called ""nnecessssaarryy"",, he penned thheese germinal lines:

These arre the eternal trruutthhss. They didd not obtain only while the world existteedd,, but they would
also obtain if GOODD had created a world with a diffeerreenntt plann.. But frromm thheesse, exisstteennttiiaall orr
continnggeenntt trruths differr ennttirireleyl.y4.444

Leibniz's style and idiom were in advance of their time. Otherr logiciaans, and perhaps Leibniz too, saw
no parrttiicularr advannttaaggee in this way of talkkinngg,, merrely an alternaattiive. It was nOoil. unttiil the earrly 1960s
that philosoopphheerrs such as Kripke and Hintikka rreturrnned to Leibniz''ss idiom and used it both to
illuminate the philosophical bbaasses of logic and to push the frontiers of logic and many otherr arreas of
philosophy in new dirreecttions.

Nonetheleessss,, it is important to keep this talk of ''otherr possiblee worlds' in its prroper philosophical
perrspective. We must neveerr allow ourrselvess to rregarrdd this manner of talk as if it were talk about
actually existtinngg parrts of this worrldd.. Whoever suppppoosses that otherr possiblee worlds arre basic enttitties of
the physical universe has failed to apprreeciate the point of our earrlier insistence that otherr ppoossible
worlds arre not ''out there' in physical space. Otherr possiblee worlds arre abstract enttitties like numbers

42. CCaammbbrriiddggee,, MMaassss..,, Harvard Unniverrssittyy PPrreesss, 1973, pp.. 1.

43. Models for Modalitiess:: Selecctteedd Essayss,, Dorrddrreecchhtt,, D.. RReeiiddeell,, 1969, pp.. 157.

44. ""NNeeccessarry. aanndd CConnttingent Trruths" trraannsslated frroomm Oppuussccuulleess ett frraaggmmeennttss ineiddiittss dee Leibbnniizz,, ed.
Courturraatt, PPaarriiss,, FFeelix Alcan, 1903, pppp.. 1166-22, rreepprriinntteedd inn From Dessccaarrtetess too Locke edd.. T.V. Smmithh aand
MM .. GGrreennee,, CChhiiccaaggoo,, Thhee Unniverrssittyy of CChhiiccaaggoo PPrreesss, 1957, pppp.. 3066-3122. WWee aarre grraatteeful to ourr ccoolleague,
David Copp, forr calling thhis ppaassssaaggee to ourr aatttteennttion.

64 PP OO SSSSIIBB LLEE WWOORRLLDDSS
anndd propoossiittiioonnss.4.545 We feel driven to posit them becauussee we seem unable to make ultimate sense ooff
logic without them. Positing the existence of these abstrraacctt entities allowwss us to unniiffyy logic, to rigorize
it, to expand it, and most importantly in our eyes to unnddeerrsstanndd it.

455. This distinnccttiioonn between abstracctt and non-abssttrraacctt (i.e.,, concrete) entities wwiilll be examined at ggreater
length in the next chapter.

2

Propositions

1. I N T R O D U C T I O N

In chapter 1 we undertook three main tasks: to introduce the concept of possible worlds; to introduce
the concept of propositions and their truth or falsity; and to show how various logically important
properties and relations of propositions can be explicated in terms of the ways in which the
truth-values of propositions are distributed across the set of all possible worlds.

In this chapter we invoke the concept of possible worlds in order to give an analysis of what
propositions are; to give an explanation as to why they need to be distinguished from the sentences
which may be used to express them; and to provide a method for identifying and referring to particular
propositions.

2. T H E B E A R E R S O F T R U T H - V A L U E S

When we first introduced propositions as the items which are the bearers of truth-values, we said that
propositions must be distinguished from the sentences which may be used to express them in much the
same way as numbers must be distinguished from the numerals which may be used to express them
(chapter 1, p. 10, footnote 8). But why must they be so distinguished? And what are propositions if
they are not to be identified with sentences? These are the questions which we wish to answer in this
section.

First, however, let us explore briefly the analogy we have drawn between propositions and numbers.
Why should we want to distinguish between numbers and numerals? The following reasons have
seemed to most mathematicians and philosophers to be compelling:1 (i) — Numerals have physical
existence as marks on paper, on blackboards, etc., as patterns of sound, distributions of molecules on
magnetic tape, or the like. Numbers do not. It makes sense, for instance, to speak of a numeral being
written in blue ink or white chalk, being large or small, decipherable or indecipherable. None of these

1. Exceptions are those who espouse the nominalist thesis that there are no abstract entities. See, for instance,
W.V.O. Quine and Nelson Goodman, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic,
vol. 12 (1947), pp. 105-122. Not surprisingly, Quine and Goodman are foremost among the proponents of the
equally nominalist thesis that sentences, not propositions, are the fundamental entities required by logic. For a
spirited criticism of nominalism in mathematics, logic, and physics see Hilary Putnam's short and eminently
readable Philosophy of Logic, New York, Harper Torchbooks, 1971.

65

66 P R O P O S I T I O N S

properties, by way of contrast, can sensibly be attributed to numbers. For numbers are abstract items,2
expressible by those physical items which we call numerals, but not identical with them. We can erase
a numeral — render it nonexistent as a physical entity — but in so doing we do nothing to render the
number which that numeral expresses nonexistent, (ii) — Numerals do not stand in a one-to-one
correspondence to numbers. On the one hand, for any given unique number — for example, the
number two — there are different numerals which may be used to express it. Corresponding to the
number two is not only the Arabic numeral " 2 " but also the Roman numeral "II". The numerals " 2 "
and "II" are items in different languages. But the number two is not. If it were an item in a language
then we should have to say, absurdly, that it belonged to one particular language, say Arabic, to the
exclusion of any other, (iii) — Numbers have arithmetical properties and stand in arithmetical
relations. Numerals do not. Mathematical operations of addition, subtraction, multiplication and
division can be carried out on numbers, but not on numerals. To be sure, there are some idioms which
suggest the contrary. We can and do speak of halving both numbers and concrete physical things. But
it is clear that "halving" is here ambiguous between a mathematical operation and a physical one.
Half the number two is the number one. Half the numeral " 2 " is, e.g., " ° " or "1". The mathematical
operation called "halving" can be applied to numbers but not to numerals, whereas the physical
operation called "halving" can be applied to numerals but not to numbers.

Each of these three arguments has its analogue for the distinction between propositions and
sentences, (i) — Sentences have physical existence as marks on paper, on blackboards, etc., as patterns
of sound, distributions of molecules on magnetic tape, or the like.3 Propositions do not. It makes sense,
for instance, to speak of a sentence being written in blue ink or white chalk, being large or small,
decipherable or indecipherable. None of these properties, by way of contrast, can sensibly be attributed
to propositions. For propositions, we shall argue, are abstract items, expressible by those physical items
which we call sentences, but not identical with them. We can erase a sentence — render it nonexis-
tent as a physical entity — but in so doing we do not deprive the proposition which it expresses
of existence, (ii) — Sentences do not stand in a one-to-one correspondence to propositions. On the one
hand, for any given unique proposition, such as two plus two equals four, many sentences may be used
to express it. Corresponding to the proposition that two plus two equals four is not only the English
sentence "Two plus two equals four" but also the French sentence "Deux et deux font quatre", the
German sentence "Zwei und zwei gleich vier", and so on. The sentences "Two plus two equals four"
and "Deux et deux font quatre" are items in different languages. But the proposition that two plus
two equals four is not. If it were an item in a language then we should have to say, absurdly, that it
belonged to one particular language, say English, to the exclusion of any other, (iii) — Propositions
have logically significant attributes: they have truth-values, have modal properties, and stand in modal
relations. Sentences do not. Logical operations, such as those of conjunction, disjunction, negation, and
the like, can be carried out on propositions but not on sentences. To be sure, some idioms suggest the
contrary. We can and do speak of conjoining propositions and of conjoining sentences. But it is clear
that "conjoining" is here ambiguous between a logical operation and a grammatical one. The logical
conjunction of the proposition that two plus two equals four with the proposition that two plus two

2. This claim is compatible both with conceptualism — the view that such abstract entities exist insofar as
they are created by the activity of human thinking — and with Platonic realism — the view that such abstract
entities exist in their own right, independently of human thinking. The philosopher Kant and the mathematician
L.E.J. Brouwer are foremost among the representatives of conceptualism regarding numbers. Plato, Frege, and
Russeli are notable exponents of this sort of realism. As will become obvious in this section, the present authors
believe that such realism is the only ultimately defensible theory regarding the fundamental items both of
arithmetic, viz., numbers, and of logic, viz., propositions.

3. In arguments (i) and (ii) we use the word "sentence" to refer to what we shall later call a "token" (as
distinct from a "type"). We did this also in the parallel arguments (i) and (ii) regarding numerals. For more on
the type/token distinction see p. 7 Iff.

§ 2 The Bearers of Truth-Values 67

equals four is logically equivalent to the proposition that two plus two equals four. The grammatical
conjunction of the sentence "Two plus two equals four" with the sentence "Two plus two equals four"
is the norcequivalent (because double-length) sentence "Two plus two equals four and two plus two
equals four." The logical operation can be applied to propositions but not sentences, whereas the
grammatical operation can be applied to sentences but not to propositions.

In terms of the analogy between numbers and propositions and of the attendant arguments for
distinguishing these items from numerals and sentences, respectively, we foreshadow some of the main
conclusions of the present section. But not all of them. In order to do justice to some of the other
arguments for identifying propositions as the bearers of truth-values — and, incidentally, in order to
do at least partial justice to some of the arguments against such an identification — we propose to set
these conclusions aside, provisionally at least, and pursue the question "What sorts of items have
truth-values?" with open minds.

Our approach in what follows is the time-honored one of dialectical enquiry. We advance a
tentative thesis, subject it to critical examination, repair its shortcomings by offering a new thesis,
subject it in turn to critical examination; and so on until we produce a thesis which, hopefully, is found
viable.

EXERCISES

Fill in the blanks with either "numeral" or "number" so as to allow the sentence to express a true
proposition.

1. Twice jour is the eight.

2. I just repainted the s on my mailbox.

3. The "25" is constructed by placing the "5" to the
right oj the "2 ".

4. Twelve s appear on the face oj a clock.

5. "Two" and "2" are two different symbols which may be used to rejer to the
which is twice one.

Fill in the blanks with "sentence" or "proposition" so as to make the following claims true.

6. "Two plus two equals jour" is a oj English.

7. "Two plus two equals jour" expresses a necessarily true __

8. The "Today is Monday" means the same as "Today is the day before
Tuesday."

9. The that today is Monday is equivalent to the that

today is the day after Sunday.

10. It is possible to erase a ; it is not possible to erase a

68 PROPOSITIONS

Thesis 1: Such things as beliefs, statements, assertions, remarks, hypotheses, and theories are the
bearers of truth and falsity.

Objection to Thesis 1: Each of the terms, "belief, "statement", etc., is ambiguous.

Now it is absolutely clear that we do, on occasion, attribute truth and falsity to beliefs, statements,
assertions, remarks, hypotheses, and theories. And it may plausibly be argued that each of these kinds
of things is a genuine bearer of truth-values. What is not quite so clear, however, is just what is meant
by "belief, "statement", "assertion", "remark", "hypothesis", and "theory" when truth and falsity
are at issue.

Each of these expressions is ambiguous. Each is ambiguous, in the first place, between:

(a) the state, act, or disposition of believing, stating, asserting, remarking,
hypothesizing, or theorizing;

and
(b) that which is believed, stated, asserted, remarked, hypothesized, or theorized.

Let us illustrate this distinction by considering a case of belief.
Suppose John Doe believes himself to be ill. Then there are two quite different sorts of questions —

corresponding to (a) and (b) above — that we might want to ask about this belief. On the one hand,
we might want to ask a question like "What brought about this belief of his?" or "When did he start
believing that?" In such a case we would be asking about John Doe's state of belief (or, as some would
say, his "act of believing"). His belief, in this sense of the word (that of (a) above), is something which
may arise at a specific moment of time and persist through time; it may be brought about or caused by
some other event or state of affairs, e.g., by his having eaten too much; and it may in turn bring about
or cause another event or state of affairs, e.g., his calling for the doctor.

On the other hand we may distinguish the content of his belief, that which he believes. It is this
latter feature which may occur in other persons' beliefs as well. Although no other persons can have
John Doe's belief in the sense that their acts of believing cannot be the same act as John Doe's, what
they believe, viz., that John Doe is ill, may be shared both by them and John Doe. In this second sense
of "belief, the sense in which we talk of what is believed (the sense (b) above), a belief may be shared
by many persons.

Thesis 2: Acts of believing (stating, asserting, etc.) are the bearers of truth-values.

Objections to Thesis 2: First, the class of truths and falsehoods vastly outnumbers the class of
belief-acts. Secondly, under this proposal some truths and falsehoods would be without
contradictories.

Acts of belief (assertion, etc.) are temporal entities. They begin at some point in time and end at some
later time. For example, when we were young many of us thought that the moon traveled along with
our moving car at night, but as adults we no longer believe this. Over the course of our lifetimes, we
will each entertain thousands, maybe even millions or billions of beliefs. For present purposes,
however, the exact number is of no importance. What is of importance is whether or not the total
number of belief-acts for all mankind is sufficiently large for them to be identified as the bearers of
truth and falsity.

Earlier, in chapter 1, we proved that the number of truths is infinite and that the number of
falsehoods is infinite: to each number within the infinite set of natural numbers there corresponds the
truth that that number has a successor, and to each natural number there corresponds the falsehood

§ 2 The Bearers of Truth-Values 69

that that number has no successor. It now needs to be observed that the number of truths and
falsehoods exceeds the number of natural numbers, i.e., is nondenumerably infinite. For the class of
real numbers is larger than the class of natural numbers and to each of the real numbers there
corresponds at least one unique truth and at least one unique falsehood about that real number. The
full import of this fact is not always appreciated. It is all too easy to conceive of an infinite class as
being just a very large finite class. But to say that the number of truths and falsehoods is infinite (let
alone nondenumerably infinite) is not just to make a claim of the sort that the number is a trillion or a
centillion4 or the like. It isn't even to say that a centillion is just a minute fraction of the number of
truths and falsehoods. So vast is the class of noncontingent truths and falsehoods that a centillion truths
or falsehoods — or any otherfinitenumber — does not constitute any fraction whatever of the whole
class.

And not only is the class of noncontingent propositions nondenumerably infinite. So, too, is the class
of contingent propositions. The number of points of physical space is nondenumerably infinite, and to
each of these points may be paired off, as the case may be, the contingent truth or falsehood which
ascribes the presence of physical matter to that place.

With this background, we now come to the critical question: Is the class of belief-acts large enough
for them to be identified with the nondenumerably infinite number of members of the class of truths
and falsehoods? As a matter of fact, there does not seem to be even an infinite number of belief-acts (let
alone a nondenumerably infinite number). This is not to say that there could not be, that in some
possible world there isn't an infinite number of belief-acts; but it is to say that in this, the actual world,
there seem to be far too few belief-acts for them to be reasonably identified as those things which are
true and false. Just reflect on how many possible beliefs about the actual world are never entertained
by us. Who, for example, has had or ever will have beliefs about the exact position of each atom of the
sun's interior at exactly 6:01 A.M. on June 18, 1893, etc. etc.? Is it reasonable to argue that although no
one has yet had any such beliefs, someday someone will ? We think not.

If we are right in thinking that the number of belief-acts is smaller than the number of truths and
falsehoods, have we not found a fatal objection to the theory that belief-acts are those things which are
true and false? Obviously we have. For it is logically impossible that truths and falsehoods should be
belief-acts if there are more truths and falsehoods than there are belief-acts.

However, the matter does not quite end here. For it is open to the proposer of the thesis that
belief-acts are those things which are true and false to take objection to the idea that we have some
independent way of ascertaining the number of truths and falsehoods. We can imagine him arguing in
this way:

You seem to know that there are an infinite number of truths and an
infinite number of falsehoods. But this is question-begging: your belief is
a result of your theory that truths and falsehoods are propositions. If you
put that theory out of your mind, and assume nothing about the number
of truths and falsehoods, the proposal that belief-acts are those things
which are true and false, works.

But does it? Let us see.
Presumably the person who argues that belief-acts are those things which are true and false will

allow the possibility, indeed the actuality, of mankind's having neglected collectively to believe
everything there is to believe. But this concession has odd implications. Imagine a person walking
along in a forest one day. He glances at a tree and believes (correctly) that it is a birch. Suppose,
however, that he is the only person ever to have seen that tree and that a forest fire destroys it that

4. One (British) centillion = 10600.

70 PROPOSITIONS

night without anyone else ever taking any notice of it. Suppose, further, that no one ever entertains the
belief that it is false that that tree is a birch. It will then follow that if the class of bearers of
truth-values were to be identified with acts of belief, a certain act would be true but would have no
contradictory, since no person ever believes that it is false that that tree is a birch. In short, if we were
to identify the class of truths and falsehoods with the class of belief-acts we would have to give up the
claim that to every truth there corresponds a non-empty class of falsehoods each of which is a
contradictory of that truth. Under this proposal, some truths and some falsehoods would be without
contradictories. Not only would this make a shambles of logic; it is thoroughly counterintuitive as well.
We have a strong disposition to insist that even if no one were to believe of the birch tree that it is not
a birch, then were anyone to believe it, what he would believe is false.

Note carefully how we just expressed ourselves. In effect we said that it is possible that even though
no one believes some particular thing, that thing which could be believed is false. We did not say that
that thing which could be believed would be false if it were to be believed. That is, our common,
hard-won conception of truth and falsehood has it that truth and falsehood may exist independent of
human belief, that truth and falsehood do not as it were 'come into existence' with correct or mistaken
human belief.

In short, it is not our holding to the theory that propositions are the bearers of truth and falsity
which prompts our belief in an infinite number of truths and of falsehoods; it is quite the other way
around. It is the widespread and strong belief that there are unexpressed and unbelieved truths and
falsehoods which prompts us to look for a truth-value vehicle of prodigious number.5

It follows that insofar as a belief can properly be said to be true or false, we must understand that
we are using the term "belief" in sense (b), not in sense (a). And by similar reasoning we may
conclude that it is only in sense (b), not sense (a), that statements, assertions, remarks, hypotheses, and
theories can be the bearers of truth-values. We may put it like this: beliefs, statements, etc., are true or
false just when that which is believed, stated, etc., is true or false.

EXERCISES

For each of the following, explain in which sense of "belief" (act of belief or object of belief) the claim
could be true.

7. His belief that it was raining was fleeting; he glanced again out the window and saw that it was
sleeting.

2. The belief that it is raining is inconsistent with the belief that it is not.

3. That a person believes that it is raining is consistent with his also believing that it is not.

4. Her belief that she was prime minister was induced by a mushroom omelet.

5. Her belief that she was prime minister was false.

6. No one's beliefs antecede his birth (or at least the moment of his conception).

*****

5. When we come to Thesis 5 we will discover that Quine, who eschews propositions, does not abandon the
thesis that the number of truth-value bearers must be infinite. In arguing that the number of truth-value bearers
is infinite, Quine is in argreement with us.

§ 2 The Bearers of Truth-Values 71

Thesis 3: That which is believed, stated, etc., is what is true or false.

Objection to Thesis 3: Talk about 'that which is believed' is unclear. What sorts of things can be
believed? In particular, are these things sentences or propositions?

The distinction we have just made between acts, statements, or dispositions of believing, asserting, etc.,
and that which is believed or asserted is important and valuable. But the matter can hardly end there.
The trouble is that our talk — in the manner of (b) — of that which is believed, stated, asserted,
remarked, hypothesized, and theorized is itself ambiguous. Consider the so-called theory that the earth
is flat, and call it E. Plainly we may not only speak of the theory that E , but also of the belief that E ,
the statement that E, the assertion that E, the remark that E, and the hypothesis that E. But what
exactly is the status of E itself? Here philosophical opinion divides between those who say that " E " is
the name of a sentence, viz., "The earth is flat", and those who say that " E " is the name of what they
call a proposition, viz., the proposition that the earth is flat — a proposition which is typically
expressed in English by the sentence "The earth is flat" but which is not to be identified with that or
any other sentence. Accordingly, our talk of those things which are believed, stated, etc., and which can
be bearers of truth-values, is equivocal between: (b') talk of sentences; and (b") talk of propositions.
Let us examine the case for each.

Thesis 4: Sentences are the bearers of truth-values.

Objections to Thesis 4: Only certain kinds of sentences are plausible candidates for the bearers of
truth-values. In addition there is an ambiguity in the notion of "sentence". It is necessary to
distinguish between sentence-tokens and sentence-types.

Much of the impetus for saying — in the manner of (b') — that it is sentences which have truth-values
(whether or not these sentences are believed, stated, etc., by anyone), comes from an aversion to talk
about propositions. It is not that talk of sentences' being true or false is any more natural than the rival
piece of philosophers' jargon; for it may well be argued that when, on occasion, persons speak of a
sentence as true or false they are merely speaking elliptically of the truth or falsity of that which is
expressed by the sentence (of that which, on the rival theory, is to be called a proposition). Rather it is
claimed that propositions, unlike sentences, cannot easily, if at all, be individuated (distinguished from
one another and from other things in such a way that we can identify one and the same individual); or
again it is claimed that there is simply no need to populate the universe with such obscure entities
when all our practical and philosophical purposes are as well, if not better, served by talk of sentences.

Let us see, then, how well the thesis that sentences can be thought to be the bearers of truth and
falsity fares under a careful examination.

All is not plain sailing for those who prefer sentences. For a start, a number of refinements are
needed. We might begin, for instance, by pointing out that not all sentences, but only those which we
call declarative — as opposed to, say, interrogative and imperative sentences — can on any ordinary
interpretation, be said to be true or false. But even this restriction of the class of truth-valued sentences
will not suffice. In extraordinary circumstances a declarative sentence may be used in such a way that
neither truth nor falsity is attributable to it (as, for example, if members of a secret society were to use
"John Doe is ill now" as a password); and in quite ordinary circumstances something true or false
may be conveyed by a nondeclarative sentence or even by something other than a sentence (as, for
example, when John Doe utters the word "Yes" in answer to the interrogative "Are you ill?" or when
he merely nods his head). Moreover, it is clear that John Doe can have a true or false belief, e.g., that
he is ill, without his either uttering or inscribing any sentence at all. The answer that would be given,

72 P R O P O S I T I O N S

however, is that although John Doe may not engage in any verbal performance of utterance at all, he
must, insofar as he has the belief that he is ill, be either disposed to engage in the utterance of the
sentence "I am ill now" or at least disposed to accept the sentence "I am ill now" as true of himself.
Such refinements of the sentence-theory can be, and have been made. But we will not pursue them,
since as we shall soon see, other more serious difficulties have yet to be faced.

Before turning to these more serious difficulties in the sentence-theory, however, let us pause to
recognize an ambiguity in the very term "sentence".

The ambiguity can be brought to light by asking how many sentences occur in the box below:

John Doe is ill on Christmas Day 1973.
John Doe is ill on Christmas Day 1973.

Two answers could be given, with equal plausibility.
(i) We might say that there is only one sentence in the box, and that it has there been inscribed

twice; or
(ii) We might say that there are two sentences — albeit of the same type — inscribed in the box.

If we choose to answer in the manner of (i), we are thinking in terms of what philosophers call
"sentence-types"; while if we choose to answer in the manner of (ii), we are thinking in terms of what
philosophers call "sentence-tokens". This distinction between sentence-types and sentence-tokens may
be illuminated by asking, somewhat analogously, how many colors occur in the box below:

Again, two answers could be given with equal plausibility, (i') We might say that there is one color in
the box, viz., the color black, (ii') Alternatively, we might say that there are two color patches in the
box, both black. If we choose to answer in the manner of (i'), we are thinking in terms of what
philosophers call universals; while if we choose to answer in the manner of (ii'), we are thinking in
terms of what philosophers call instances. Plainly, sentence-tokens stand to sentence-types in much the
same way as color-patches stand to colors: they are instances of universals.

Which is it, then, that are to count as vehicles of truth-values: sentence-tokens or sentence-types?

§ 2 The Bearers of Truth-Values 73

EXERCISES

1. The rectangle below

Some athletes are students.

Some students are athletes.

contains

A. two sentence-tokens of the same type;
B. two sentence-types of the same token;
C. two sentence-tokens of different types;
D. two sentence-types of different tokens;
E. none of the above.

2. The rectangle in question 7 above contains

A. two logically consistent sentences;
B. 8 word-tokens instancing 4 word-types;
C. 8 word-tokens instancing 8 word-types;
D. 4 word-tokens instancing 8 word-types;
E. 4 word-tokens instancing 4 word-types;
F. 8 word-types instancing 4 word-tokens;
G. 4 word-types instancing 8 word-tokens.

3. As well as distinguishing sentence-tokens from sentence-types, as was done in the preceding
discussion, and word-tokens from word-types, as was done implicitly in question 2, do we also have
to distinguish numeral-tokens from numeral-types? Explain.

*****

Thesis 5: Sentence-tokens are the bearers of truth-values.

Objection to thesis 5: The same difficulty arises as with Thesis 2, viz., there are more truths than
there are sentence-tokens.

Anyone who abjures the abstract and takes comfort i n the concrete w i l l be likely to opt for the thesis
that it is sentence-tokens w h i c h are truth-valued. F o r the criteria for individuating sentence-tokens are
reassuringly straightforward: whether they take the form of written inscriptions or speech episodes,
they can be individuated i n the standard ways appropriate to physical objects and physical events
(sound-producing movements), respectively. So there are no problems on this score. But how about
their credentials as truth-vehicles? Is it really the case that for every truth there actually exists a
sentence-token?

74 P R O P O S I T I O N S

Although the number of actual sentence-tokens is staggeringly large, it is nonetheless far, far smaller
than the number of truths. It is reasonable to suppose that the class of truths extends well beyond the
sentential expression of these truths. There are many truths of mathematics and logic, for instance,
which neither have been, nor (we may suppose) ever will be, encapsulated in sentential form. As just
one instance recall our earlier example (Objection to Thesis 2) of the infinitude of noncontingent truths
which ascribe the property of having a successor to each of the natural numbers. Then, too, there are
vast numbers of contingent truths about the physical universe which have not yet, and possibly never
will be, discovered. As a practical matter, just consider how many contingent truths there are about the
actual world, truths which no one has asserted, is now asserting, or ever will assert. Just reflect on the
series of sentences which begins: "(1) There is more than one atom of oxygen in the Atlantic Ocean;
(2) There are more than two atoms of oxygen in the Atlantic Ocean; (3) There are more than three
atoms of oxygen in the Atlantic Ocean; etc., etc., etc." We have good reason to believe that even if
every person in the entire history and future of the world were to spend his lifetime adding more
sentences to this one set, the set would never be completed in the anticipated lifespan of the physical
universe. And this is just one set; there are countless others sets of this sort besides.

One of the chief contemporary sponsors of the sentence-theory, Quine, argues explicitly that there
are quite enough sentence-tokens for them to be identified as the bearers of truth and falsity.6 His
argument is that the class of sentence-tokens need not be thought of as restricted merely to a subset of
the class of word strings which persons may happen to inscribe or utter. If, instead, we conceive of
sentence-tokens as mathematical sequences of those word-tokens which are inscribed or uttered at some
time or other, then we may conclude that there are as many sentence-tokens as there are truths and
falsities.

Against this ingenious theory we offer two main objections. First, there is something highly
implausible about the suggestion that actual sentence-tokens are merely mathematical sequences of
word-tokens. Ought they not rather to be identified with temporally and/or spatially ordered
sequences of word-tokens? Suppose — contrary to fact — that the only words ever to be uttered or
inscribed occurred in the utterance or inscription of the sentences "The cat is on the mat" and "Henry
stood in front of the door." Then it would seem to be just plain false to say that the world numbers
among its items the sentence-token "The mat is in front of the door." Yet this is what Quine's theory
commits us to. Secondly, Quine's theory makes the existence of any truths dependent upon an
historical accident, viz., the invention of language. It leads directly to the uncomfortable conclusion that
had words never been invented there never would have been any truths. We shall return to this
objection later, when dealing with Thesis 9, in our dialectical argument.

It seems clear, then, that actual sentence-tokens simply do not exist in the profusion that the theory
calls for, and that we must therefore — if intent on preserving the theory — allow that it is either
actual or possible sentence-tokens which carry truth and falsity. But once we say this, the advantages
of this version of the sentence-theory over the rival proposition-theory have been narrowed almost to
the point of nonexistence. For sentence-tokens which exist only as possibilia lack that concreteness and
ease of individuation which first recommended them to us. Possibilia are just as abstract as
propositions and admit of no easier individuation.

Thesis 6: Sentence-types are the bearers of truth-values.

Objection to Thesis 6: This thesis leads to ascribing contradictory assertions to persons who have not
contradicted themselves.

6. W.V.O. Quine, Word and Object, M.I.T., Technology Press, and New York, Wiley, 1960, pp. 194-195.

§ 2 The Bearers of Truth-Values 75

It will not do to say simply that a sentence-type, such as that which corresponds to "I am ill now", is
that which is true or false. For suppose that a token of this type were to be uttered by John Doe, while
a token of its grammatical denial, viz., "I am not ill now", were to be uttered by his sister, Jill Doe.
Then, if it were the sentence-type which was true or false, we should, in accordance with the dictates
of logic and commonsense alike, be obliged to say that John and Jill were contradicting each other and
that only one of them could be saying what is true. Or again, suppose the sentence-token "I am ill
now" were uttered by John Doe immediately after overindulging his appetite at Christmas, and that
the grammatical denial of the latter, the sentence "I am not ill now", were to be uttered by him after
he has recovered from all illness. Then, if it were the sentence-type corresponding to "I am ill now"
which is true (or false as the case may be), we should, once more in accordance with the dictates of
logic and commonsense, have to conclude that he was contradicting himself and that only one of his
utterances was true. After all, two people contradict each other if one says that something is true and
the other says that the very same thing is false; and likewise a single person contradicts himself if he
says both that something is true and that the very same thing is false.

Plainly something has gone wrong here. The simple fact of the matter is that what John Doe said,
when he uttered the sentence "I am ill now", and what Jill Doe said, when she uttered the sentence "I
am not ill now", were both true; and again that what John Doe said at Christmas and what he said
some time later were both true. There must be something seriously amiss with a theory which commits
us to imputing inconsistency where none exists. At the very least an amendment is called for.

Thesis 7: Context-free sentences are the bearers of truth-values.

Objection to thesis 7: It is necessary to distinguish context-free sentence-types from context-free
sentence-tokens.

The amendment which sentence-theorists tend to favor is that of saying that the sentences, or
utterances of sentences, of which truth and falsity are best predicated are what they call context-free
sentences.1 The basic idea of a context-free sentence is really very simple. If we consider a sentence
like, "At one atmosphere of pressure, water freezes at 32°F.", we are not at all tempted to suppose —
as we might in the case of "I am ill now" — that its truth-value varies with the special circumstances
of its utterance or inscription: with who uttered it, or when he uttered it, or the like. No matter who
utters it, or when he utters it, the sentence "At one atmosphere of pressure, water freezes at 32 °F."
has a constant truth-value, viz., truth. It is context-free in the way that the sentences of mathematics,
physics, and the sciences generally are. Plainly enough context-free sentences cannot generate
inconsistencies in the same way as do sentences like "I am ill now". If, then, some way could be found
of transforming context-dependent sentences into context-free ones, the theory that sentences are the
primary bearers of truth-values would have some chance of salvation. How might such a
transformation be effected? The proposal is that pronouns like "I" are to be replaced by names,
temporal references like "now" are replaced by dates, and tenses are cancelled altogether. Thus a
context-dependent sentence such as "I am ill now", when uttered by John Doe on Christmas Day
1973, is transformed into the context-free sentence "John Doe is [in a tenseless sense] ill on Christmas
Day 1973"; while the denial of this context-dependent sentence, when uttered by John Doe a week
later, is rendered by the context-free sentence "John Doe is not ill on January 1, 1974." These
context-free paraphrases of the two utterances of the context-dependent sentences "I am ill now" and
"I am not ill now" not only express more precisely what it was that John Doe took himself to be
asserting on each of the occasions when he uttered them, but also reveal — what we knew all along —

7. Quine and some others, somewhat less perspicuously, call context-free sentences "eternal sentences".

76 PROPOSITIONS

that he wasn't really contradicting himself at all when he first uttered the one and then, at a different
time, uttered the other.

But with the concept of a context-free sentence now in hand, we must ask whether it is context-free
sentence-types or context-free sentence-tokens which are supposed to be the bearers of truth and
falsity.

EXERCISE

Read the following sentence aloud: "I started reading this page five minutes ago." Now paraphrase
what you have just said so as to generate a context-free sentence.

Thesis 8: Context-free sentence-tokens are those things to which truth and falsity may be attributed.

Objection to Thesis 8: Again, there are more truths than there are sentence-tokens, context-free or
otherwise.

Context-free sentence-tokens form a proper subclass of the class of sentence-tokens. And if there are
too few sentence-tokens to satisfy the theory that sentence-tokens are to be identified as the bearers of
truth and falsity, then a fortiori, there must be too few context-free sentence-tokens to do the job. In
the whole history of mankind there have probably been only a few thousand, a few million at most,
context-free sentence-tokens offered as reconstructions of context-dependent ones.

EXERCISE

One of the objections to Thesis 2 was that a consequence of holding that actual belief acts are the bearers
of truth-values is that this makes the claim that every proposition has a contradictory both contingent
and actually false. Show how this same criticism can be brought to bear both on Thesis 5 and on
Thesis 8.

Thesis 9: Context-free sentence-types are those things to which truth and falsity may be attributed.

Objections to Thesis 9: First, the criteria for the individuation of context-free sentence-types are
obscure. Secondly, because persons sometimes use words with different meanings, they will
express different things even though the context-free sentence-types associated with their
utterances are identical. Thirdly, this account cannot do justice to the fact that persons
lacking a language can, nonetheless, hold true beliefs.

How about the supposition that it is context-free sentence-types which are truth-valued? Note that this
proposal does not fall victim to the objection leveled against its immediate predecessor. Context-free
sentence-types, unlike context-free sentence-tokens, do exist in the profusion required by the theory.
For a sentence-type exists even when no token of that type exists. And one can plausibly argue that to
every truth (and falsehood) there 'corresponds' a context-free sentence-type. But this is not to say that

§ 2 T h e Bearers of Truth-Values 77

these context-free sentence-types ought to be identified as the bearers of truth-values. There are
difficulties in such a hypothesis.

In the first place, if we adopt this view we immediately abandon whatever advantages actual
sentence-tokens have over propositions. For sentence-types — context-free or otherwise — are quite
unlike sentence-tokens in that they are not physical objects or events locatable in space and time; and
they cannot therefore be individuated i n the way that inscriptions and utterances can. Strictly
speaking, the answer to our earlier question as to how many sentences there were i n the box cannot be
— i f we are thinking of sentence-types — that there is one. F o r sentence-types, as distinct from
sentence-tokens, are not the sorts of things w h i c h can be in a box, or on a page, or on the tip of
someone's tongue, or anywhere else. There could have been one or two, or more, sentence-tokens in the
box; but there could not be any sentence-types in the box.

M o r e precisely, we should say rather that one sentence-type was instanced by the two
sentence-tokens in the box.

The box example, however, turns out to be a fairly simple one. F o r the two sentence-tokens which,
we said, were instances of one and the same sentence-type were typographically identical. But suppose
they had not been typographically identical. Suppose, for instance, that one were to be italicized (e.g.,
"John Doe is ill on Christmas Day 1973") and the other not; or that one were to be written i n capital
letters (eg., " J O H N D O E IS I L L O N C H R I S T M A S D A Y 1973") and the other in a mixture of
capital and lowercase letters (e.g., " J o h n Doe is i l l on Christmas D a y 1973"). W o u l d we still want to
say of these typographically different sentence-tokens that they were instances of the same
sentence-type? Probably we w o u l d ; for it seems reasonable to lay down as a sufficient condition of two
sentence-tokens being of the same type that they be composed of the same words in the same order.
But now suppose that one or more words i n one or more of the sentence-tokens were to be misspelled.
W o u l d we still want to say that they were tokens of the same type? O u r criteria for being of the same
type now begin to look fuzzy. After a l l , the change of just one letter (or digit) might convert a
sentence-token instancing one sentence-type into a sentence-token instancing another. Consider:

John Doe is ill on Christmas Day 1973.

John Doe is ill on Christmas Day 1975.

A n d finally consider this case:

John Doe is honored on Christmas D a y 1973.

John Doe is honoured on Christmas Day 1973.

H o w many sentence-types are instanced by the two sentence-tokens in this last box?
F r o m what has already been said it is apparent not only that sentence-types — context-free ones

included — are abstract entities, but also that the criteria for their individuation are troublesome.
Yet so far we have considered only those context-free sentence-types of w h i c h context-free

sentence-token instances actually exist. H o w about those which are instanced by the non-actual,.
merely possible, context-free sentence-tokens which we earlier felt obliged to postulate i n order to
accommodate unexpressed and undiscovered truths about logic, mathematics, and the universe at
large? If the context-free sentence-types which are instanced by actual context-free sentence-tokens are
abstract i n the first degree, those w h i c h are instanced only by possible (and hence abstract) ones must
be abstract i n the second degree; and the criteria for their individuation must be correspondinglv even
more elusive.

78 PROPOSITIONS

Problems of abstractness and of the elusiveness of identity-conditions for context-free sentence-types
are not the end of it: there are problems about their truth and falsity as well. These latter problems are
not unique to the context-free sentence-type account. They arise for any version of the theory that it is
sentences which are the primary bearers of truth-values, whether these sentences are context-free or
context-dependent, types or tokens, or any combination of these. For this reason we shall state them in
quite general terms.

In the first place, one and the same sentence (of whatever kind) is subject to different semantic
construals by different people or at different times: the same sentence may - as we commonly say -
"mean different things". Change of meaning, of words or lengthier expressions, is a familiar-enough
fact of comparative linguistics. Thus, for example, one might observe that many North American
speakers of English are disposed to use "disinterested" and its cognates to mean what English speakers
throughout most of the rest of the world mean by "uninterested". Suppose, then, that speakers from
each of these two classes of language-users are asked to consider the sentence "John Doe shows
complete disinterest over the question of who was responsible for the oil spill" and to say whether it is
true or false. Members of one class of speakers may say it is true, on the grounds that John Doe,
though very interested in ecological issues, is able to treat questions about responsibility for ecological
disturbance with the full impartiality that becomes one who is a judge; while members of the other
class of speakers may say it is false simply on the grounds that John Doe, though of undoubted
impartiality, is far from showing lack of interest in issues of ecology. Are they contradicting each
other? If it were precisely the same thing which members of the one class asserted and members of the
other class denied, we should have to say that they are, and hence that only members of one class could
be saying what is true. Yet plainly, since the members of these two different classes mean different
things by "disinterested", what some are asserting is quite different from what the others are denying.
So there is no inconsistency after all. The case reminds one of the apparent conflict, discussed earlier,
between John Doe and his sister Jill. However, in this case, unlike the earlier one, we cannot resolve
the seeming conflict by resorting to the notion of a context-free sentence and saying that the
context-free sentence asserted by one is different from the context-free sentence denied by the other.
For this time the conflict has arisen over what is itself already a context-free sentence. So differences in
meaning - for different people or even for the same people at different times - must be taken into
account; and we can avoid inconsistency only by allowing that it is not, after all, context-free sentences
simpliciter (whether tokens or types) which are the trouble-free bearers of truth and falsity; but rather
that it is sentences along with their meanings which are the bearers of truth and falsity. But once we
start talking thus about the meanings of sentences (context-free or context-dependent, types or tokens),
we are well into the realm of abstractions which sentence-theorists tend to regard as forbidden
territory.

In the second place, any version of the sentence-theory - type, token, context-free,
context-dependent - must run into difficulties over prelinguistic times. For even if there is no clear
sense in which our grunting forebears can be said to have made statements, assertions, or the like to
each other, nonetheless we must surely allow that they had beliefs and that certain of their beliefs were
true while others were false. But if, ex hypothesi, these were prelinguistic times, then these believers
lacked a grammar and a semantics. But for a sentence to be intelligible, to be a content of a belief, the
would-be believer must comprehend the grammar and semantics of that sentence, i.e. that sentence
must belong to a language which the believer has mastered, or must be translatable into a language
which the believer knows. But if the would-be believer lacks all language, then there is no sentence
which he believes. Nor will it do to argue in this case, in the sort of way that we did earlier when we
tried to accommodate the sentence-theory to John Doe's unexpressed belief that he was ill, that although
our forebears did not actually either utter or inscribe sentences like "That animal is dangerous", they
must, nevertheless, if they believed that a particular animal was dangerous, have been disposed to
assent to such a sentence. For the logic of the case we are envisaging is that our forebears on occasion
believed that a particular animal was dangerous when it was indeed dangerous; so that what they
believed was true; and yet that there was no sentence expressing this belief which they could - given