The considerations of the preceding section have shown that, if we regard
statements about values ofunobserved entities as meaningless, we must in- clude meaningless statements of this kind inthe language of physics. Ifwe
*Mr.Strauss*informsmethatheisplanningtopublish anewandsomewhatmodified presentationofhis conceptions.
30. 145 wishtoavoidthisconsequence,we mustuseaninterpretationwhichexcludes such statements, not from the domain of meaning, but from the domain of assertability.We thusare led toathree-valuedlogic,whichhasa special cate- goryfor thiskindofstatements.
Ordinarylogicistwo-valued;it isconstructedintermsofthe truth values truth andfalsehood. It is possible to introduce an intermediate truth value whichmay becalled indeterminacy, andto coordinatethistruth valueto the group of statements which inthe Bohr-Heisenberg interpretation are called meaningless.Several reasons can beadducedfor suchaninterpretation. If an
entity which can be measured under certain conditions cannot be measured underotherconditions,itappears naturaltoconsideritsvalueunderthelatter conditionsasindeterminate. Itisnot necessaryto crossoutstatementsabout
thisentity fromthe domainofmeaningfulstatements;allweneedisa direc- tionthatsuchstatementscan bedealtwith neither as true noras falsestate- ments. This is achieved with the introduction of a third truth value of indeterminacy. The meaning of the term "indeterminate"mustbecarefully distinguished from the meaning of the term "unknown''. The latter term applieseventotwo-valuedstatements,sincethe truth valueofa statement of ordinarylogic can beunknown; we then know, however, that the statement
is either true or false. The principle of the tertium non datur, or of the excluded middle, expressedin thisassertion, isone ofthepillars of traditional logic. If, onthe other hand,wehave athirdtruthvalueofindeterminacy, the tertium nondatur is no longer a valid formula; there is atertium, a middle value, representedby thelogicalstatus indeterminate.
The quantum mechanical significance of the truth-value indeterminacy is
made clearbythe following consideration. Imagine a general physicalsitua- tions,in whichwe makea measurementoftheentityq;in doingso wehave onceandforeverrenouncedknowing whatwould haveresultedifwe had made
a measurementofthe entity p. It isuseless to make a measurement of p in thenewsituation, sincewe know that themeasurementofq haschangedthe situation. It is equally useless to construct another system with the same
situationsas before,andtomakeameasurementofpin thissystem. Since the result of a measurement of p is determined only with a certain probability, this repetition ofthe measurement mayproduce a value different fromthat whichwewould have obtainedinthefirstcase. Theprobability characterof
quantummechanical predictions entailsanabsolutismoftheindividual case;
itmakesthe individual occurrence unrepeatable,irretrievable.Weexpressthis fact by regarding the unobserved value as indeterminate, this word being takeninthe senseofathirdtruthvalue.
Thecaseconsideredisdifferent,withrespecttological structure,from acase ofmacrocosmicprobabilityrelations. Let usassumethatJohnsays, "IfIcast thedie in the next throw, I shallget 'six* ". Peterstates, "IfI cast the die, instead, Ishallget'five'". LetJohnthrowthedie,andlet 'four'behis result;
146
wethenknowthat John'sstatementwasfalse.Asto Peter'sstatement, how-
ever, weareleftwithoutadecision.Thesituationresembles thequantum me-
chanical caseso far as we haveno meansofdetermining the truth of Peter's statement by having Peter cast the die after John's throw; since his throw thenstarts in anew situation, itsresult cannot informus about what would have happened ifPeter hadcastthe die inJohn's place. Since casting a die,
however, isa macrocosmicaffair, wehavein principleother meansof testing Peter's statement after John has thrown the die, or even before either has thrownthedie.Weshouldhavetomeasureexactly theposition ofthedie,the status of Peter's muscles, etc., and then could foretell the result of Peter's throw with as high a probability as we wish; or letus better say, since we cannot do it, Laplace's superman could. For us the truth value of John's
TABLE 2
statementwillalways remainunknown; butit isnotindeterminate, since itis possible in principle to determineit, and only lackof technical abilities pre- ventsusfromso doing. Itisdifferent withthe quantum mechanical example considered.Afterameasurementofqhas beenmadeinthe generalsituations,
even Laplace'ssupermancouldnotfindoutwhat wouldhavehappenedifwe hadmeasured p. We express this factby giving to a statement aboutp the
logicaltruth-valueindeterminate.
Theintroduction of the truth-value indeterminate in quantum mechanical language can beformally representedbytable2, whichdetermines the truth valuesofquantummechanicalstatementsasafunctionofthe truth values of statements ofthe observational language. We denote truth by T, falsehood
by Fy indeterminacyby 7. The meaningofthe symbols mu, u, and Uis the sameasexplainedonpage141.*
Let us add some remarks concerning thelogical position of the quantum mechanical languageso constructed. When wedividetheexhaustiveinterpre- tations intoacorpusclelanguageandawavelanguage,thelanguageintroduced bytable2maybeconsideredasaneutral language, sinceitdoesnotdetermine oneofthese interpretations. It istrue that we speakofthe measured entity
1Astothe useofthefunctor,"the valueoftheentity,"cf.fn. 4,p. 159.
30. 147 sometimesasthepathofaparticle,sometimesasthepathofneedleradiation;
orsometimesastheenergyofaparticle, sometimesasthe frequencyofawave.
This terminology, however, is only aremainder derivingfrom the corpuscle orwavelanguage. Since the valuesofunobservedentitiesarenot determined, thelanguageof table2leavesitopen whetherthemeasuredentitiesbelongto waves or corpuscles; we shall therefore use a neutral term and say that the measured entities represent parameters of quantum mechanical objects. The
differencebetweencalling such a parameter anenergy or a frequencythenis
only adifferencewithrespect toafactorhinthe numerical valueoftheparam-
eter.This ambiguityinthe interpretation ofunobservedentitiesismade pos-
sible throughthe use ofthe categoryindeterminate. Sinceit isindeterminate whetherthe unmeasuredentityhas the value u\
9 oru2, oretc., it isalso inde- terminatewhetherithasthe valuesu\,andu<i, andetc.,atthesametime;i.e.,
it is indeterminatewhetherthequantum mechanical objectisa particleor a wave (cf.p. 130).
The nameneutral language, however, cannot beapplied tothe language of table1,page142.Thislanguage doesnotincludestatementsaboutunobserved
entities, since it callsthem meaningless; it is therefore not equivalentto the exhaustive languages, butonlytoa partofthem. Thelanguageof table2, on thecontrary,isequivalenttothese languagesto theirfullextent;tostatements about unmeasured entities of these languages it coordinates indeterminate statements.
Constructionsofmultivaluedlogicswerefirstgiven,independently,byE.L.
Post2and byJ. LucasiewiczandA. Tarski.3Since thattime, suchlogicshave beenmuchdiscussed,andfieldsof applicationshave beensoughtfor;theorigi- nalpublicationsleftthe questionofapplicationopenandthewriters restricted themselvestotheformal constructionofacalculus.Theconstructionofalogic of probability, inwhich a continuousscaleoftruth values isintroduced, has been given by the author.4 This logic corresponds more to classical physics thantoquantummechanics. Since, init, every proposition hasadeterminate probability, it hasno room for a truth valueofindeterminacy; a probability ofiisnotwhatismeant bythe categoryindeterminate of quantum mechan-
icalstatements. Probabilitylogicisageneralization oftwo-valuedlogicforthe caseofakindoftruth possessing a continuousgradation.Quantummechanics
isinterested insuch alogiconlyso farasageneralization ofitscategories true and/aZseisintended,whichisnecessaryin thisdomaininthesamesense as in classicalphysics;theuseofthe"sharp" categoriestrueand/afeemustbecon- sidered in both cases as an idealization applicable only in the sense of an
2E. L. Post, 'Introduction to a General Theory ofElementaryPropositions/' Am.
Journ.of Math.,XLIII(1921), p. 163.
8J. Lucasiewicz, Comptesrendus Soc. d. Sciences Varsovie, XXIII (1930), Cl. Ill, p.
51;J.LucasiewiczandA.Tarski, op.cit.,p.1.ThefirstpublicationbyLucasiewiczofhis ideaswasmadeinthePolish journalRuchFilozoficzny, V (Lwow,1920), pp. 169-170.
4H.Reichenbach, "Wahrscheinlichkeitslogik," Ber.d.Preuss. Akad.,Phys,-Math.Kl.
(Berlin, 1932).
148
approximation.The quantummechanical truth-value indeterminate,however, representsatopologically different category.Theapplication ofathree-valued logic to quantum mechanics has been frequently envisaged; thus, Paulette F^vrier8haspublished theoutlines ofsuchalogic. Theconstructionwhichwe
shall present hereisdifferent, and isdetermined by the epistemological con- siderationspresentedinthe precedingsections.