The method of constructing a three-valued logic is determined by the idea
that themetalanguageofthe language consideredcanbeconceivedasbelong- ingtoatwo-valuedlogic. Wethus considerstatementsoftheform"Ahas the truth value T" as two-valued statements. The truth tables of three-valued logic then can be constructed in a way analogous to the construction of the tables oftwo-valued logic. Theonlydifferenceisthatin theverticalcolumns to theleftofthedoubleline we mustassumeallpossiblecombinationsofthe three valuesT,7,F.
The numberof definable operationsis muchgreater inthree-valuedtables thanintwo-valuedones.Theoperations definedcanbeconsideredas generali- zations of the operations of two-valued logic; we then, however, shall have various generalizations of each operation of two-valued logic. We thus shall obtain variousforms of negations, implications, etc. We confine ourselves to thedefinition ofthe operations presentedintruthtables4A and4B.1Asbefore, three-valuedpropositionswillbewrittenwithcapitalletters.
The negation is an operation which applies to one proposition; therefore only one negation exists in two-valued logic. In three-valued logic several operations applyingtoonepropositioncan beconstructed. Wecallallofthem negations because theychangethetruth valueofaproposition. Itisexpedient to consider the truthvalues, in the order Ty /,F, asrunning fromthehighest
1Mostofthese operations havebeendefinedby Post, withthe exceptionofthecom- plete negation, the alternative implication, the quasi implication, and the alternative equivalence, which we introduce here for quantum mechanical purposes. Post defines sqmefurther implicationswhichwedonotuse.OurstandardimplicationisPost's impli- cation D withm * 3andM= 1,i.e.,fora three-valuedlogicand M= t\ =* truth.
32.
value Tto the lowestvalue F. Using this terminology, we may say that the cyclicalnegationshiftsatruthvaluetothenext lowerone,exceptforthe case ofthelowest, which isshifted tothe highestvalue. We thereforeread the ex- pression~ Aintheformnext-A. Thediametrical negationreverses T and F, butleaves / unchanged. This correspondsto the functionofthe arithmetical
TABLE 4A
TABLE 4B
minussignwhenthe valueIisinterpretedasthenumber0;and we therefore call the expression Athenegative ofA,readingitasminus-A. Thecomplete
negation shifts a truth value to the higherone of the other two. We read A
asnon-A.Theuseof thisnegationwillbecomeclear presently.
Disjunctionand conjunctioncorrespondto the homonymousoperations of two-valuedlogic.Thetruthvalueofthe disjunctionisgivenbythe higherone ofthe truth valuesoftheelementarypropositions;thatofthe conjunction,by
the lowerone.
There are many ways ofconstructing implications. We shall use only the
152
three implications defined intable 4B. Our first implication is a three-three operation,i.e.,itleadsfromthree truth valuesoftheelementarypropositions to three truth values of the operation. We call it standard implication. Our second implication is a three-two operation, since it has only the values T
and Finitscolumn;wetherefore callitalternativeimplication. Ourthirdim- plicationiscalledquasiimplicationbecauseitdoesnotsatisfyalltherequire- mentswhichare usuallymadefor implications.
What we demandinthefirstplaceofanimplicationisthatitmakespossible the procedureofinference, whichisrepresentedby therule: IfA istrue, and
AimpliesBistrue,thenBistrue.In symbols:
A
AlB (1)
B
Allour three implicationssatisfythisrule; sowillevery operationwhichhas a
Tinthefirstlineand no Tinthesecondandthirdlineofitstruthtable.In the secondplace,weshalldemandthatifA istrueand Bisfalse,the implication
isfalsified; this requires an F inthethird line & conditionalso satisfied by our implications. These two conditions are equally satisfied by the "and", and wecanindeedreplacethe implicationin (1) bythe conjunction. Ifwe do notconsider the "and" as animplication, this is owing to the fact that the
"and" says too much. If the second line of (1) is A.B, the firstline can be dropped,and theinferenceremainsvalid. Wethus demandthat the implica- tionbesodefinedthatwithoutthefirstlinein (1) theinferencedoes not hold;
this requires that there are some T's in the lines below the third line. This requirement is satisfied by the first and second implication, though not by the quasiimplication.Afurtherconditionforanimplicationisthataimpliesa
isalwaystrue.Whereasthefirstandsecondimplicationsatisfy this condition, the quasi implication does not. Thereason for consideringthis operation, in spite of these discrepancies, as some kind of implication will appear later
(cf. 34).
Itisusuallyrequired thatA impliesBdoesnotnecessarily entailBimplies
A, i.e., that the implication isnonsymmetrical. Our three implications fulfill this requirement. The latter condition distinguishes an implication from an equivalence(andisalsoafurther distinctionfromthe"and").Theequivalence
isanoperationwhichstates equalityoftruth valuesof A andjB; ittherefore
must have a T7 inthe first, the middle, and the last line. Furthermore, it is requiredtobe symmetricalinA andB, suchthat withA equivalentB wealso haveB equivalentA. Theseconditions are satisfied byourtwo equivalences.
Since these conditions leave thedefinition ofequivalenceopenwithinacertain frame, further equivalences couldbedefined;weneed, however, only thetwo giveninthetables.
3*. 153 Tosimplify our notation we use thefollowing rule ofbindingforce for our
"
strongestbindingforce completenegation
cyclicalnegation \ lf ~
-,. , . , ,. > equalforce diametrical negation )
^ conjunction
disjunction V
quasi implication -s-
standard implication D
alternativeimplication ->
standard equivalence =
alternativeequivalence =
weakestbinding force Ifseveralnegations ofthe diametrical orcyclicalformprecedealetterA, we convenethat the one immediately preceding A has the strongestconnection with Aj and so oninthe same order. The line ofthe completenegation ex- tendedovercompoundexpressionswillbe usedlikeparentheses.
Ourtruth valuesare sodefined that only a statementhavingthe truth value
Tcan be asserted.When wewishto state thata statement has a truth value otherthan T,thiscanbedoneby meansofthenegations.Thustheassertion
~~A (2)
statesthatAisindeterminate.Similarly, eitheroneoftheassertions
~A -A (3)
statesthatAisfalse.
This use of the negations enables us to eliminatestatements inthe meta- language about truth values. Thus the statement of the object language next-next-A takes theplace ofthe semanticalstatement"A isindeterminate".
Similarly, the statement ofthe metalanguage "A is false" istranslated into one ofthe statements (3) ofthe object language, andthen ispronounced, re- spectively, "next-A", or"minus-A". Wethuscancarrythroughtheprinciple thatwhatwewishtosayissaidinatruestatement ofthe objectlanguage.
Asintwo-valuedlogic, aformulaiscalledtautologicalifithas only T's inits column;contradictory,ifithas onlyF's;andsynthetic, if ithasatleastone T
initscolumn, butalso atleastoneother truthvalue.Whereasthestatements oftwo-valued logic divide into these three classes, we have a more compli- cateddivisioninthree-valuedlogic. Thethreeclassesmentioned existalso in the three-valued logic, but between synthetic and contradictory statements
wehave aclassofstatementswhicharenevertrue,but notcontradictory;they have only I's and F's in theircolumn, or even only /'s, and may be called
asyntheticstatements. Theclassofsynthetic statements subdividesintothree categories. The first consists of statements which can have all three truth
154
values;weshall callthem,fully synthetic statements.Thesecond containsstate- ments which can beonly true orfalse;theymaybecalledtrue-falsestatements, or plain-syntheticstatements. They are synthetic in the simple sense of two- valuedlogic. Theuseof these statementsinquantummechanicswill beindi-
cated on p. 159. The third category contains statements which can be only true or indeterminate. Of the two properties of the synthetic statements of two-valuedlogic, theproperties ofbeingsometimestrue andsometimesfalse, these statements possess only the first; they will therefore be called semi- syntheticstatements.
The cyclical or the diametrical negation ofa contradictionis a tautology;
similarly, thecompletenegation ofanasyntheticstatementis atautology. A
synthetic statement cannot be madea tautology simply bythe addition ofa negation.
Allquantummechanicalstatementsaresyntheticinthe sensedefined. They
assert something about the physical world. Conversely, if a statement is to beasserted,itmusthaveat leastonevalue Tinits columndeterminedbythe truth tables. Asserting a statement means stating that one of its T-cases holds. Contradictory and asynthetic statements are therefore unassertable.
Onthe other hand,tautologiesandsemisynthetic statementsare indisprovable;
theycannot befalse. Butwhereastautologiesmustbetrue,thesamedoesnot follow for semisynthetic statements. When a semisynthetic statement is asserted, this assertionhasthereforea content, i.e.,is not emptyas in thecase of a tautology. For this reason we include semisynthetic statements in the synthetic statements;allsynthetic statements,andonlythese, haveacontent.
Theuniqueposition ofthe truthvalue Tconfers to tautologies ofthethree- valued logic the same rank which is held by these formulae in two-valued
logic. Such formulaeare always true, since they havethe value Tfor every combinationofthe truth valuesoftheelementarypropositions.Asbefore,the proof of tautological character can be given by case analysis on the base of the truth tables; this analysiswillincludecombinationsin whichtheelemen- tafy propositions have the truth value /. We nowshall present some ofthe moreimportant tautologies ofthree-valued logic, following the orderused in the presentationofthetwo-valuedtautologies (1)-(12), 31.
Therule ofidentityholds, of course:
A ^ A (4)
Therule of doublenegation holdsforthe diametrical negation:
A = --A (5)
Forthecyclicalnegationwehave arule oftriplenegation:
A m A (6)
32. 155 Forthecompletenegation therule ofdoublenegation holdsintheform
3 = 2 (7)
It should be noticed that from (7) the formula A = A cannot be deduced, sinceit is notpermissible to substitute A for A;and thisformula, infact, is not atautology. We shallthereforesay that the rule ofdouble negation does not holddirectly.Apermissible substitutionisgivenbysubstitutingA forA
inthisWay we can increase the number of negation signs in (7) correspond- inglyoh bothsides. Thispeculiarity ofthe completenegationisexplainedby the fact that a statement above which the line of this negation is drawn is
thusreduced toa semisynthetic statement;furtheradditionofsuchlines will
make the truthvalue alternate only between truth andindeterminacy.
Betweenthecyclicalandthecompletenegation thefollowing relationholds:
A 55 ~AV~~A (8)
The tertium non datur does not hold for the diametrical negation, since
A V A is synthetic. Forthecyclicalnegationwehaveaquartum nondatur:
A V~AV~~A (9)
Thelast two terms of this formula can be replaced by A, according to (8);
wethereforehaveforthe completenegation aformulawhichwe calla pseudo tertiumnondatur:
AVA (1ftt
This formula justifies the name "completenegation" and, at the same time,
reveals the reason why we introduce this kind of negation; the relation (8),
whichmakes(10) possible,maybeconsideredasthedefinition ofthecomplete negation.The namewhichwegive tothisformulaischoseninorderto indicate that theformula (10) does nothavetheproperties ofthe tertiumnondaturof two-valuedlogic. Thereasonisthat thecompletenegation doesnot have the properties of an ordinary negation: It does not enable us to infer
the_truth valueofA ifwe knowthat Aistrue. Thisisclearfrom (8);ifwe knowA, we know only that A is either false or indeterminate. This ambiguity finds a further expressioninthefactthatforthecompletenegationnoconverse opera- tioncan be defined,i.e.,no operation leadingfromAtoA. Suchanoperation
is impossible,because its truth tableswouldcoordinate to the value Tof A, sometimesthe value/ofAyandsometimesthevalueFofA.
Therule of contradictionholdsinthefollowingforms:
ZI (11)
(12) (13)
156
TherulesofDe Morganhold onlyforthe diametrical negation:
-4V r (14)
-A.-B (15)
The twodistributive rulesholdinthesameformas intwo-valuedlogic:
A.(BVC)^A.BVA.C (16)