an introduction to symbolic logic - susanne k. langer

7/26/2019 An Introduction to Symbolic Logic - Susanne K. Langer

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n intodutionto

susanne k.Zanger


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N INVESTIGATION OF THE LAWS THOUGHT

by Georg Bool

{erg Bol was n f th atst mathmaticians th 19thntuy, and n h mst inuntia thnks a tm. Ntny dd h mak imptant cntibutins dintia quatinsand cacuus nit dincs h as was h discv invaiants, and th und mdn symbic ic. Accdint and Russ "Pu matmatics was dscvd by G in hs wk pubshd in 854"

THE IVESTIATIO OF H LAWS OF THOUHT is th st xniv statmnt th md vw that mathmatics s a pudductiv scic hat can b appid t vaus situatins st shwd hw cassca ic cud b tatd with abaictminy and patins and hn pcdd t a na symbc mthd ica innc; h as attmptd t dvis acacuus pbabiitis whch cud b appd t situains

hitht cnsdd bynd invstiatinTh nmus an his wk can b sn m chap hadins: Natu and Dsin This Wk; Sins and hi Laws; Divtn Laws; Divisn Ppsitins; Pincips SymbicaRasnn; Intptat; Emnatin; Rductin; Mtds Abbviatin; Cnditins a Pct Mthd; Scday Ppsitins;Mthds in Scnday Ppsitins; Cak and Spinza; Anaysis;Astian Lic; Thy Pbabiitis; na Mthd n Pbabitis; Emntay ustatins; Statistica Cnditins; Pbms\n Causs; Pbabity Judmnts; Cnstitutin th ntcthis ast chapt Cnstitutin h Intct 'is a vy sinicantanaysis th psychy dscvy a d !cntic mthd.

"A cassc pu mathmacs and symbic ic h pubsh is to thankd main it ava SCIETIFIC AMERI-CA. . '

Unabdd uctin 854 dti, wth cctins ma inth txt xviii + 424pp. 5% x 8 S8 Papbund $2.00


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T O YMBOLC LGC

angel itI '' " tdy l cearest book ever wrien on smboic ogic for

l

'" ' ra scienist and aman. I wi be particuar

1 t1 l y ho who ave been rebuffed b oher inroducorI f ucien mahematica aining No specia know

u tHtl s s required Even if ou have forgoten mos of

t !l ebra, you can ear to use maematica ogic

tll 1 1 l ir cios in this book You sta wi e simpes 1 lions and end up with a remarkabe grasp of

I O!


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AN INTRODUCTIO TOSYMBOLIC LOGIC

by SUSANNE K LANER

SEND EDIIN

(REVISED

' 625tmw,Q If'!0

Iv.Augs


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R UBLD

E 195

YRH 195 Y AE LER

) i


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T M MTHEREE U NAUTH


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CONTENTS

REFACE To the First EditionREFE o e Seond Edition

INTRDUCON

CHAPTER I

THE SUDY OF FORMS

I he Importance of Form

2. Logical Form

3 Structue4 Form ad Cotent

5 he Value of Aalogy

6. Abtracton

7 Concept8. Itepetation

' he Field of Logic10. Logc and Philoophy

CHAPTER

THE ESSELS OF LOGCAL STRUCTURE

I Relatos and Elements

2 erm and Degee

3 Propotions4 atural Language and Logical Symbolim

5 Some Picipe Goveig Symbolic Expresion

6 he Power of Symbol

CHAPT

PAGE

I3

1 7

21

45

HE ESSEALS OF LOGCAL SRCTRE (continued) 64

1 . Cotext2. Cocept and oceptions

3 Fomal otexthe niverse of DiscourseContituent Relations


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6 INTRODUCTION TO SBOIC OIC

4 TruthValue Related Propoton n a Formal Conex

6 Conttuent Relaton and ogcal elaton

7 SytemDeducteInductexed

CHATER ENEIZATION

I Regularte of a Sytem2 Variable3 Value

Propotonal Form5 The Quanter () and )6 eneral Propoton

7 The Economy of eneral Propoton8 The Formaly of eneral Propoton

g Quanted Term n Natural Dcoure

CHATER VCASSES

Inddual and Clae2 emberhp n a Cla3 Concept and Clae4 "Denng Form of Clae5 Clae and SubClae6 The Noton of a "Unit Cla7 The Noton of a "Null Cla8 The Noton of a "Unere Cla

g Identty of Clae10. The Unuene of I and "o

CHAPTER

PRINCIPA REATIONS AON CASSESI The Relato of ClaIncluon neuence of the Deniton of


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CONTENTS

4 Jont Incluon or Djuncton of Clae The Prncple of Dchotomy: A and A

6. The Importance of Dchotomy: Negaton7 The Ubuty of the Null Ca

8 Complement of Sum and Product

g Eualent Expreon

CHATER V

THE UNIVERSE OF CASSES

I Relaton and Predcate

2 Clae a Indpenable Contrct a Sytem3 Clae a "Prmte Concept n a Sytem

4 The eneraled Sytem of Clae A Conenence of Symbolm Logcal Punctuon

CHATER V

THE DEDUCTIVE SSTE OF CASSES

The ClaSytem a a Deducte Sytem2 Potulate and Theorem3 Truth and Valdty

4 Potulate for the Sytem of Clae K b


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AN INTRODUCTION TO SYMBOLIC LOGIC6 The Denition of


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CNTENTS

CHAPR XLOGISTICS

The Puose of Logistics2 ThePritive Ideas of Mathematics3 Functions

4 ssumptio for a Calculus of General Propositions

5 The Denitis of Class and Membeship

6. The Deiti of Relation

The Stuctureof Pinipia Mathematica

8 The Value of Lgic for Science and Philosophy

9AG

32

SYMBOLIC LOGIC AND T OGIC OF THE SYLLOGISM 340

PROOFS OF THEOREMS AND b

APPNX C

HE CONSTRUION AND USE OF TRUTTABLES

SUGGESTIONS FOR FURTHER STUDY

NDEX

30

352

36

36


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PREACE TO THE FIRS EDN

SYBLC Cis a reativey new subject, and the easiest

methods of a?poach have not yet been detemined npoint of aan!ement, theefoe, this ntrtn has nopredecesso. h\t is just why it was written: the need ofsome systematic gide fom the state of pefect innocenceto a possibe un


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PRE O E SOD EDIO

WH HIS boo was wrtten there was no systematctetbook of symolc logc n Englsh ecept the bgic of Lews ad Langford Perhaps Cotrat's tteSmmary The Algbra of Logc cod also be regarded as tet n that t se forth a system developed by varospersons notably De {organ Boole and Schroeder makngno clam for tself to nalty bt that \as after all anotlne rather than an trodcton. For the rest everyeposton was stll n nnecton wth a contrbtonLews's urvey f ymblic gic had been wrtten manlyto propose the system of st ct mplcaton Rssell rasedmany qestons he dd not cl m to settle Qne was almostcompletely orgnal and eve the most dgested work

(Lews and Langford) entered o no elementary dscssonof the basc logca notons \n ralization, abstractonrelaton form system. Sch fnamental concepts eretaken for granted.

They have alays been taken for granted n that paragonof pre scences mathematcs. t s the eceptonal stdentof mathematcs who knows why + s classed s an

operaton and' =''

as a relaton or ndeed why hs eercsesare called eamples or who nderstands even at a farlyadvanced stage of techncal procency jst how algebra sreated to arthmetc s tranng s entrely n technqesand ther applcaton to problems arthmetc geometry andalgebra are related for hm only by the fact that ther severaltechnqes may converge on ne and the same probem that s to say they are related practcally bt not ntellec

tally. The mathematcs books sed n schoos dwe almost ecsvely on rles of operaton.

The prestge of mathematcs s so great that ogc nemlatng ts method tended to follow ts pedagogy as well.T en tetbook o o at av ppared dn


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4 AN INTRODUCTION TO SBOIC OGICe las feen years (see e spplemenay n ls appendedo "Sggesons for frer readng ed smlarly o

mpasze ecnqesransformaon ference conssencyess decsonprocedreswo ealed eplanaons of e conceps nvolved Now maemacs as sucbvos praccal ses a o learn s rcks wonderandng er sgncance s no enrely slly bmay e same be sad of symbolc logc s e manplaonf s symbols of sc praccal mrance a sdens

sold lea o perform logcal oraons even woknowng or qesonng er co epal fondaons seems o me a despe s pr cal ses wc are slcomng o lg n nepeced q rers s cef vale s concepal Trog e nenc of e varos posvscscools of plosopy wc e ceranly e mos promnen and peraps e mos romsng scools oday or

scolars and edcaos are s mbed w meodology aey vale e new logc p marly for s codcaon of erles of nference. B only one of s conrbons oman og and een o scence meod an be ovempaszed and ends o be so n or nellecal lfe

Symbolc logc s an nsrmen of ac og bonalyc and consrcve s msson accordngly s noonl o valdae scenc meods b also o clarfy e

semanc confsons a bese e poplar nd as well ase professonal plosoper a e presen me. Semancs(blssd ord s n dre need of responsble analyss andsklfl andlng and symbolc logc s e mos eecveprparaon can nk of for a fronal aack on e paecmddles of modern plosopcal og blass naalmsconcepons evey move no by a process of de

bnkng b by prposefl and lcd consrcon of deas.Becase s book seeks o presen n clear sepwse

fason e elemenary conceps of logc canno encompasss mc ecncal maeral as oer ebooks do. B oda s no a seros embaassmen becae may be


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PR}AE O E SEO EIO spplemened 'Y a sandard ex One of e mos sefulevces now geerally ag-e conscon of "rableso es elegmacyof consrcs na rvalesysems ncled n s edon as Appendx C Teeadng ls fo fer sdy as been bog p odaeFo eresore'sonsavebeenmadeexcepocoecors

especally


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N T R O D U C T O N

THE rst thng that strkes the stdent of symbol logcs that t has developed along severa apparently nreatedlnes () the symboc expresson of tradtonal ogc ; (z hnventon of varos algebras sch as the algebra of logc,and wth ths the stdy of postatonal technqe; 3) thedervaton of mathematcs from a set of partally or whollylogcal postlates the nvestgaton of the laws ofthoght and formal dedcton of logcal prncpes themselves Every worker n the eld drfts nto one or anotherof these specalzed endeavors. To a novce, t s hard tosee what are the ams prncples and procedres of logcper se and what relatons the several branches bear to tand to each other

Underlyng them a s the prncple of enerality whch

clmnates n the attanment of abstractins Th ebrac of oc are n s pt Theams of ogcal research ay y wt th terets ofderent nvestgatorsone may be nterested n the valdtyof Arstotean ogc, another n that of mathematcs a thrdn the canons of scence, a forth n the relaton betweenmathematcs and scence or mathematcs and classcal logc

etc., etc.bt the procedre s everywhere the same t sprogressve systematzaton and generalzaton. Lkewse thecrteron of sccess s the same : t s the dscovery of abstractforms The atter nterest dstngshes logc from natralscence whch s n search of general bt not abstractformlae .e formlae for concree facts

n order to prepare the reader for frther stdy n anybranch of symbolc logc, have ndertaken to dscss thecharacterstcs of logcal scence as sch and show precseyhow the branches sprng from a parent trnk; to lay downthe prncples goveng symbolc sage and to ntrodcehm speccally to the most mportant systems of symboz


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IRODUCTIO1

R Whtehead and othes o ths end tephaszessse , ' b d

the priniples of logial onsrution; the

poss 1 tes an

mtsoffomazaton ;eqvaent conc:ptn, fnde

ta

types of fomae ctea of caty smpcty, eneaty,

andaboveathedierenebetweenfeundand

stelenons.n thsconnectont seeks tosh

owthebeangofogn

natascenceandphosophyofnate

Athoghts ntended asa tet fo a cose symboc

ogc (say a second cose, thogh t pepposes no

evos tanng t shod aso seve as todct

oy

eadng fo acosenphosophyof nate, o a coseofgenea

phosophy fo ppe assmen,

o fo gadestdents seekng a key to th

e motantmonogap

teate of symboc ogc andogstc t ams to tke no

technca knowedge of ogc,scence, o mathematcs fo

anted,bttodeveopeveydeafotheeveofcommong

that t maybe compehensbe and sef aso to

sense, so . fthethenteestedaymanwhodesesageneadscssoo

ppose, appoach technqe andestsofsymbocogc


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AN NTRODUCON TO SYMBOLC LGC

A T E R I

TE STUDY OF FORMS

H MOTA O OMAll knwledge, all cence and a av e gnnn

n he ecgnn a dnay, amla n may akn deen m O eale epeence nae ng ac nce we ee wae eezng n a anlcen lck, h nw wc e m heaven cangn wae efe vey eye A lle eecn make awae he change Whee d nw and an cmem Fm he vap mae we call cld, e w

m ha a n he a, and a he nw an decende cld dlve. They have ed wae, whe ake Whee dd he cld cme m Tey amad y me pce f anfman m e waef h eah Deen fm he ame hng may e wdely dvee n appeaance ha had nk hem a eenally he ame ance

All cence e edce dvey f ng n wld mee deence appeaance, and ea amany hng a ple a vaan e ame When Benjamn Fankn nd a lghnng nm eleccy, e mad a cenc dcvey, wcpved e a ep n a vey gea cnc, anamazng nme ng can edcd h amendamenal meng pan ance calledeleccy : e fc ha hld e mamen gehe, cackle a a , he ea n a an heckeng Nen Aa Eleccy ne hena ng n e wld ha can ake n a va vaey


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28 A NTRODUCTION TO SMBOC LOGIC

scae and magine t transposed one halftone higher sothat t reads: C# D#, # F# G# A# B# high C# Not asngle note n the C#-major scale gures n the scale of

natura* Yet the two have exactly the same fomwhich s commony called "the major scale It is a pecuiarfact that n music forms are easer to recognze than contents; most people can tell whether a given successon oftones is a major scale or notthey will mark any deviatonfrom the formbut very few can tell whether the given

major scale s C C#, or any other particular key A norma

ear w apprehend the form but only persons blessed withsocaled absolute pitch can dentify the content.

Furthermore, two dierent cntents or the same formmay vary so widely that they belong to entrely derentdepartments of human experience Suits of cloth or paperare, after all, equaly tangible, physical contents for a geo

metrc form. Tones in a major scale are al equally auditory

contents However we vary our material, as rom C to #D to D# etc. our scale is still a musical orm, and its contents some sort of sound. But why is the standard arrangementof these tones called a "scale? "Scale means "ladderThe fact s tat ordinay, common sense sees a similartyoform between the order o successive tones, each new tonebeing a little higher than ts predecessor, and te successiverungs of a ladder, each a ttle higher than the one beforet The word "sale or "ladder is transerred rom one tothe other In this way what was once the name of a certainkind of object has become the name of a certainform. Anyseries whose separate parts are arranged so that each iseither higher or lower than any other part s a "scaleThus we speak of "going up in the social scale or ca acertain series of successively "higher spiritual experiences

* On the pano the two scaes appear to have certan tones ncommon, becase E# and F B# and C, respectivey fal on thesame key; bt that s de to the naccracy o a "temperednstment On oln they are dstngushable.


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THE STUDY O ORMS 29

"the ladder of faith without any danger of being msunderstood, and being thought to refer to a series of tonesor a wooden contraption with steps or rungs. Everybodyadits the proprety of our usage by anaogy; ad

na1ogyi nothing but the recognition of a commo form m ernthings.

5 THE VALUE OF NAOGY

Whenever we draw a diagam say the groundplan of

ahouse, or a street-planto showthelocation ofts site or

a map, or an isographic chart, or a "curve represe?ting

the uCtuations of the stock-market we are drag a"logical picture o something A "logical picturediers

from an ordinarypictureinthatitneednotlooktheleast

bitlikeits object. Itsrelationtothe objectisnotthat of a

copy, but o analogy.We donottrytomake an architec's

drawing look as much as possible like the house; that s

eveniftheooris tobebrown theoorplansnotcon

sidered any better for being drawn n brown; nd if hehouseistobe largetheplanneednotconvey anmpressOn

ofvastnessAthattheplanmustdostocopyexactlythe

proportions oflength andwdth, the arrangement ofrooms

hals and stairs, doors andwindows. Thenarow dashthat

represents a window is not intended to looke one; tresemblestheobject forwhichit standsonlybytslocatOn

ntheplan whichmuste analogousto theocaton ofthe

windowntheroom.'

The dissimilaity in appearane between a "logical

picture andwhatitrepresentsis evenmoremarkednthe

caseof agraph.Supposingagaphnthenewspaperconveys

toyou thegrowth, accelerationclimax and declne of an

epidemic. Thegraphis spataits forms a shape,butthe

series of events doesnothave shapein aliteral sense The

graph is apicture of events only nlogical sensetscon

stituents, which are lttle squares on paper are arranged

nthe sameproportonsto each otheras theconstituents of


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32 AN INTRODUCTION TO YMBOLIC LOGIC

In every proposition and every inference there is,

esides the particular subect-matter concerned, a certain

om a way in which the constituents of the proposition or

inference are put together If I say, 'Socrates is mortal,'Jones is angry, 'The sun is hot, there is something in

ommon in these three cases, something indicated by the

word 'is. What is in common is the form of the proposition,not an actual constituent. If I say a number of things aboutSocratesthat he was an thenian, that he married

Xantippe, that he drank the hemlockthere is a common

constituent, namely Socrates, in all the propositions Ienunciate, but they have diverse forms. If, on the other

hand, I take any one of these propositions and replace its

constituents, one at a time, by other constituents, the

form remains constant, but no constituent remains. Take

(say) the series of propositions, 'Socrates drank the hemlock, Coleridge drank the hemlock, 'Coleridge drank

opium, 'Coleridge ate opium. The form remains unchanged

throughout this series, but all the constituents are altered

Thus form is not another constituent, ut is the way theconstituents are put together. We might understand

all the separate words of a sentence without understanding

the sentence : if a sentence is long and complicated, this is

apt to happen. In such a case we have knowledge of theconstituents, but not of the form We may also have

knowledge of the form without having knowledge of theconstituents. If I say, 'Rorarius drank the hemlock,

those among you who have never heard of Rorarius(supposing there are any) will understand the form, without

having knowledge of all the constituents In order to

understand a sentence, it is necessary to have know

ledge both of the constituents and of the particular in

stance of the form. Thus some kind of knowledgeof logical forms, though with most people it is not ex

plicit, is involved in all understanding of discourse It is

the usiness of philosophical logic to extract this know


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THE TUDY OF FORMS 33

ledge from its concrete integuments, and to render it ex

plicit and pure*

The great value of analogy is that by it, and it alone, we

are led to seeing a single logical form" in things whichmay be entirely discrepant as to content The power ofrecognizing similar forms in widely various exemplications,i.e. he power of discovering analogies, is logcal intuition

Some people have it by nature ; others must develop it (and

I believe all normal minds can develop it) , and certainly

all may sharpen the precision of their understanding, by a

systematic study of the principles of structure

6. BAON

The consideration of a form, which several analogous

things may have in common, apart from any contents, orconcrete integuments, is called abstraction. If we speakof the major scale apart from any particular key, we are

treating it as an abstracted form. If we note what is commonto a couple of days, a pair of gloves, a brace of partridges,and a set of twins, we are abstracting a form which each of

these items exhiits, namely its numerosity, two. If wespeak simply of a couple, without reference to any content,

or simply of two-ness" or two," we are treating of this

form in abstracto r again, if we consider the order inwhich hours of a day follow each otheralways one after

another, never two at once following the ae predecessorand then regard the order of inches on a ruler, or rngs

on a ladder, or the succession of volumes of the EncyclopaediaBritannica, or the sequence of Presidents of the UnitedStates, we see at once that there is a common form in all

these progressions They are all analogous, all dierent

contents for a pattern which is a section of the ordinal

number series: rst, second, third, etc. It is easy t see thtit is but a hort ste fro the nio o oge, or

Logc as the Esec of Philosophy" in Our Knowldg f hExteal World, London, 914.


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A IDUCI YMBLIC LGIC

would omit al reference to skyscrapers and ddle-stringsand teeh, and dscribe it, probably, as rhymic motionto and fro, or in some such terms that would onnote only

the sot o moton we are talking abou and not the soto tn tt moves Probably, each of us has leaed themeaning of oscillation through a dierent medium; butwhether we gathered our rst idea of it from the shaking ofGrandpa's palsied handsor from the quiver of a tuningforkor from the vibration of a parked auomobile wih themotor runninghowever our mentl pctues may dier

from each oher, they have one thing in common: they areall derived from some rhythmic motion to and fro Thethings exemplifying this type of moion are no necessarilyalike in oher respecs he swaying skyscraper and thevibraing violinsring are certainly no alike in appearance,origin, or purpose But their motions have the commonproperty of going rhythmically to and fro This propery is

he locl om of their moions, and so we may call allthese moions diverse insances of the ame formWhen we consider the common form of various hings,

or various evens, and call i by a name tha dos notsugges any paricular hing or event, or commi us to anymenal picturefor instance, when we consider this comonform of various movemens, and cal it by a name such asosciain"we are consciously, delberaely absracing

the form from things which have it. Such an absracedform is caled concept From our concree experiences weform the cncept o ocllton

The fact tha so many hings in nature e(emplify hesame forms makes it possible for us to collect our enormouslyvariegaed experiences of nature under relaively fewconceps If this were no the case, we could have no science

If there were no fundamental conceps such as oscillaion,gravitaion, radiation, etc., exemplied in nature over andover again, we could have no formulae of physics and discover no laws of nature Scientists roceed y absracting


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HE UDY F FM 37

morandmor funmenl fors (ofen seen imilaitiesamon absr or or concepts, themselves, andhus gah


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A DUC SYMBLC LGCmtter Interpretatin is te reverse strctin; teprcess f stractin eins wit rel tin nd deres

frm t te are frm, r cncept, wereas te prcess fnterprettn eins wit an empty cncept nd seekssme rel tin wic emdies it In te scences as nrdinary lie, we re interested in rms nly n s ar stey re te patterns certin tins tt cncern usMst f te stract cncepts we emply are anded dwn tusn lanuae, like l ur simple adjectival nd dverlcncepts r y delierate trinin, like ur cmmn knwlede f matematics, mecanics, and s rt We lerntem as tey apply t certain tins; we lea numer ycuntin tins, sapes y ttin jects teter, qualitiesy cmprin varius aricles, rules cnduct y raduallycllectin and judin instances d and evl. Teesiest way t teac a rmula is t present several nstancesnd pint ut teir cmmn rmal prperties But tis is

nt te esiest way t discver new patterns, wic n nepints ut t us Tere re, essentially, tw ways in wicnew frms tins are discvered (r) y astractinfrm nstnces wic nature appens t cllect r uste pwer t recnie a cmmn rm in suc cancecllectin s cientic enius ; (z y interprettin fempty rms we ave quite stractly cnstructed Te

latter wy is usually te easier, ecause i we knw retvariety pssile rms, we ave at least n ide wtwe are lkin r. Tecnical inventins re discveres ftis srt te inventr rarely, i ever, prpunds newprincple, ie new fundamental cncept pysics, utcleverly cmines te principles e as learned int newinterestin pattern ; e may ten cnstruct pysical tin,r cul model tat pattern His astract frmtecalculatin e makes n paperis a matematicl eoemrm purely cnceptual prnciples; is applicatin f t tte relm pysics is an nepeaion nd is mdel sn nn f ts interpreted special rm (nd t


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HE SUDY F FMS 39prnciples it cmines He s cncerned essentially utinterpretatins, and is astract wrk is ll perrmed fr

te ske f nin pysiclly nterpretale rms TE ED OF OG

ut f we wuld ld lf wle rm ny specil sciencend relly ain insit nt te ret streuse rmswc may e interpretle pysicaly, r psycically, rfr ny realm experience wtever, we must cnsider

strcted patterns s sucte rders in wic ny tnswtever may e arrned, te mdes under wic nyt watevr may present tsel t ur understandnTs sunds ike n utterly mpssile nd elusve tsk t csul server it certainly seems s tu tere muste s many incmmensurale frms in te wrld s terere derent kinds nd departments f experence ut

appily fr ur restricted intellects, tis is nt te cseMany tins wic lk utterly unlike n experencermre unlike tan te mtins skyscapers nd lnss, r tps nd planetsare really made up in erysmilr wys, nly t reqres a d del f prctice t seetis Orderliness nd system, said Jsi Ryce, remuc te same in teir mst enerl caracters, weter

tey ppear in Platnic dialue, r in a mde tetk f tany, r in te cmmercial cnduct usinessm, r in te rranement and disciplne f n rmy, rm leal cde, r in wrk f rt, r even n a dnce r inte plannin dinner Order is rder System is systemmdst all te vriatins f systems nd rders, certineneral types and carctestic reltins cn e cedTe tracin suc types nd relatins mn strctdrms, r cncepts, is te usness lic

he Prncp o

Log n Wndelband and ugeEncycl-

pdza th Phsphzcal Scincs vol , p 8 rther volmesdd no appear


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AN INDUCIN SYMBLC GIC

OGI A PHIOSOH

Wa is e use f a siene f pure frms? Te use fi emes apparen wen we nsider wa jus ne ranf is siene, pursued fr is wn sake, as added uman knwlede! mean e ran f li wi weall maemais. I as made all e quaniaive sienespssile y swin us wa relains my ld amnquaniies, ie. y ellin us wa lk fr Sme mae

maial rs praly were disvered in e urse fpraial apliain, y far e reaer numer andamn em all e mre sle and elusive (and, limaely,ms useful maemaial relains, were found y peplew ave eir ndivided aenin e sudy f asramaemaial frms. Nw, wa emais is enaral sienes li, e mre neral sdy f frms, is pils, e me eerl nrsandin f ewrld h aim f isy is ee all ins in e wrldin rri ea er in sme rder, .e. se realiya a sysem, r a leas any par i as elnin smesyem. B efre we an nd rder and ssem, we mknw smein a em, s as knw wa we arekin fr r we sall n renize em wen we ndem Sysemi paerns, weer are s mu eaier

sudy n bco wiu e nfusin irrelevanieof any pariular ase, a alu every siene maye said deal wi is i, li is e siene f rderp ecellence Pilspy and siene deal wi inerpreedpaes i is n ard see is wrk is realyfailiaed y a r mnd f e aa sienef frms In e ase f ilspy espeially, li is a well

ni indispensale l I illminaes prlems a aveeen sre fr undreds and even usands f years,y swin us a pssile new frmulain f sme rulymuddled quesin i des away wi innumerale ninsa are merely dieren names fr ne and e same n


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HE SUDY MS

ep i reveals innsisenies in r ms erisdus, and sess remarkale eneralizains f ideas

a seemed quie lal in eir apliain.Li is e pilser wa e elese is easrnmer an insrumen f visin If we wuld emeasrnmers, we ms learn e use f e elespenimpaienly, an e a mre impran wrk, ussemaially ndin a erain inrinsi ineres in emyseries f e insrumen iself. Grea sieni dis

veries ave fen rewarded men w laured ver sliimprvemens f eir insrumens. Te same is rue frli'; if we ry lea i in a rsry fasin rdin eime and u we wuld raer p n meapysialprlems, nsanly wnderin weer we ave naquired enu li fr r purpses, we sall never seee wrld in is lear li We mus wrk wi a enineineres in ur resried asra sje if we wan i lead us narally pilspial pis, as indeed i will prlems f episemly, meaysis and even eis.Li aplies everyin in e wrld u we musundersand is pwers and dilies rly efrewe an se i. elese des n f iself nd e jewe wis see, nr des i sw us any in a wi wedire i, nless we knw reisely w adjs e fs

a is say, nless we are rly familiar wi e insrmen. Li, likewise, emes sefl and impran epilser nly afer e as really rased is eniqe.ny enique seems ard e nvie ease a rs eis lmsy in is ways; js as a learner a e ypewrierfeels a is wrk is made needlessly ard y e eains e -sysem, s e einner in symli li is

ap ink e se f symls is a silly devie makeinkin mre dil. Bu as e ypis sn learns appreiae e remends advanae f a med adisenses wi si, s e liian e use f symlssn ems an inesimale aid in reasnin. erefre


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A DUC YMBLIC LGCe shuld try rm the very einni t perate ithsymls nt t write ln sentences rst and then translate

them int the required symlism Clear expressin is areexin clear cncepts and clear cncepts are thel lic

UMMAThe rst step in scientic thinkin is the reaizatin that

ne sustance may take many oms Many unlike thins

may e understd as dissimilar rms the samematerial T understand a thin is nt merely t knwhave sensry knwlede it ut t have knwlede boutit t kn hw it is made up and hat ther rms itmiht take

Frm des nt necessarily mean shape oclom" means stuctue" r the way a thin is puttether

The many synnms r rm shw that we use it in ahihly eneral sense Structure aain des nt mean adelierate puttin tether r even any actual cnsecutivecminatin parts that rst are iven separately Itmeans an rderly arranement parts that may e undin nature as well as in arteacts

cnstruct is nt necessarily made ut physical sustance Therere rm shuld e distinuished nt rmmatter ut rm content The cntent a icarm may e psychical musical tempral r in sme theay nnphysical just as well physical

T thins which have the same ica rm renloous The value analy is that a thin hich hasa certain lical rm may e epesented y anther whichhas the same structure ie which is anaus t it The

mst imprtant analy is that etween thuht andnuae anuae cpies the pattern thuht andtherey is ale t represent thuht T understand lanuaeequires sme apprehensin lical rm

stton is he cnsideratin lical rm apart


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HE UD F FM 4

rm cntent The reasn why peple distrust astractinsis simply that they d nt knw hw t make and use them

crrectly s that astract thuht leads them int errrand ewilderment stractin is perhaps the mst pwerful instrument human understandin

stracted rms are called concepts Because nature isll analies we can understand certain parts it inthe liht a ew very enera cncepts ll thins t whichthe same cncept applies are analus Scientists attempt

(1 t nd many rms r a iven cntent(z

t astracta cmmn frm rm diverse cntents and s rm conepts() t apply their cncepts t mre and mre kinds thini.e. nd new cntents r their astracted rms

Findin pssile cntents r an empty rm is calledntepettion The sciences deal nly with rms hichmay e interpreted r their special suject-matter

Lic deals with any rms whatever withut reerencet cntent knwlede rms and their relatins reatlyacilitates any study their pssile applicatins. Licis a tl philsphical thuht as mathematics is a tl physics

QUESTIONS FOR REVIEW

Wat s mant by transformaton Wy s ts otomportant or scnc Wat s t drnc btwn knowdg a tg and

knowdg u t you tnk a dog as bot kds oknowdg

3 Wat s a ogca pctur How dos t dr from ordnary pctur

4 Wat s a construct s t aways somtg tat as bnput togtr Is a coud a construct

Wat s mant by contt at s ts raton to forto stu May two orms av t sam contnt May oform av drnt contnts

6 Wat s mant by cang two tgs aaogous Wat st mportnc of aogy


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4 AN INDUCIN YMBIC LGIC

7 Wy s e grmmc srucure of nguge mporn ?W f nyng ve e Lords ryer nd e erNoser n common? W re er respecve consuens ?Hve ey ny common consuen(s ?

8 W s srcon?g W s e oppose of srcon ?

I. W re oes srcon py n scence? Do we ever usesrc noons n common-sense nkng ? Hve srcons prcc vue ?

II . Are scenss neresed prmry n srcos ? If sowy?

2. W s e reon of ogc o scence? Do you nk s ny vue for everydy fe ?

SUGGESIONS FOR CLASS WOR

(VOI EXME GVEN N TE TEXT)

I . Nme ree ngs my pper n deren forms

2. Gve wo exmpes of ogc form oer n spe.3 Gve n exmpe of consruc no deerey mde4 N e wo ngs one mer nd one mmer, wc

re nogous n form W nme woud you gve o e srucure exemped y

( srngure, (2 group of rdgepyers, (3) good-uck cover?

Gve wo deren nerpreons of e src concepprogresson. o w respcve eds of experence (eg

dere scences do ey eong ?7 Mke up roposon w deren conen u e smeform s o ws e ecer of Arsoe


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C H A P E I I

THE ESSENTILS OF LOGICL STRUCTURE

ATIONS AN MNSIn te previus capter it was said tat te lical

r a tin depends upn its structure, r te way it isput teter tat is t say, upn te way its several parts

areelated

t eac terWitut addin r sutractin any te actrs in tecpsitin a tin, we ay utterly cane te caracter tat tin y canin te elaton te varius actrst eac ter Fr instance, cnsider te tree naes,"RONA "ROAN and "ARNO tey cntain exactlyte sae letters ut te relative psitins tese letters

tat is t say, teir utualelaton

"ere and"ater, r "rit and "letare dierent in eac caseand utterly dierent wrds result We ay relate tesesix letters dierently yet, and spel te nae an Enlispet, "ANOR. Wat as een caned is nt any teletters, ut te relatins "ere and "ater antese letters

Nt nly in artcial devices like te letters tat spell awrd, ut in pysical tins as well, te iprtance relatins ay e seen in te very cnstitutin jectsteselves Cnsider, r instance, te caracter a cralck Its actrs are te sells illins liliputian indi

uals, all in denite relatins t eac ter Eac tinyhe ceres wit se ter, and tat wit anter and

s Ultiately tey all cere teter in an iense

lup S ln as teir elaton to eac ote te acttat ne is to te t anter, r on anter, r nea it,a om it, etc,an under n cane tey r aslid island But let se nstrus rindin rce caneteir relatins s tat shes whic were t te


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A IDUCI SBLIC LGIC

ohes e nw o e le shels which wee ne ech ohenow e dsn nd s hnd wh hs hppened?

The ock is n me Pehps n single ne is coshs been desoyed bu i hey no onge cohee nd huskeep hei spil elins o ech ohe gidy unleedhe ock is desyed mee shing pile cos isobiously n ck The ey nue o he objec dependedon he coeson o is ps nd cohesin is n isel co he ock bu is elon mon e co

I you wch yung child plying wh blocks you wilsee him expemening wih he ious elions ino whichhe blcks my ene wih one nohe nd eing oideniy he oms which esul He piles one upon nhend hid upon his nd uh upon h by usingonly he snge elion we cl upon he consucs column He ses one blck o e o nohe nd pus hid one coss he w he esul is he simpes s lnel nd so he poceeds o wl huse pymidHe does n chnge he blocks in ny wy he meely chngesh llimpon hng ced sucue by chnging heelons mng his bcks

Two hings ey dese me my s I heledy pined u he he sme om This is becusemny deen ns my ene no e sme elons o

only blocks my be se one boe nohe bu lso bxessones boks o housnd ohe hings column neso pn is us s much cumn s one o bicks becusehe sme elins hld mng he especie ps o bohsucues When we spek o column s such we equie ndeen o he nue o is me being conceed only wih ns ned bove o below ec oe

boe nd below e he esseni elions which lclumn sucues he n common nd by ue whichhey possess cmmn om

Suppose hwee column o be peecly hmogeneousmde o single sne o eeunk o cemen csng


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HE ESSEIALS LCAL SCU Hw c we sy hen h i is mde ps ngedne e he he I hs n ps in he sense in which

clumn lcks ks hs ps Thee e n diidinglies Wh is suppsed o be be wh? Well onehing he p is be he bse ls i we dw plleigs ud he clum ny such ing wil be e belowny he The igs diide he cumn in imginy pswhich e nged in de We my dw n innie numbe ings bu his elin will lwys in mng hem

In he peius chpe I pined ou h he elemens in sucue need n be physicy sepe s exisinglne nd hen bugh in cmbinin They mus onybe concepully dsnusble d ide ps e geomeiclly disinguishle hugh hey be se o omech ohe by icil een puely imginy linesTheee een hugh cumn e cmpsed piecesse ne upn nhe i des llw us cnceie is

possble secons s nged in his shi nd i is smplyn he bsis such cncepin h we cll n upighbem o n belisk een he i in chimney hemecuy in hemmee ue clumn We cn disnguish ps mng which he geing elon o hecolumnm hlds

Since he eled ps sucue my n be

ps bu my be physicy insepble quliiesspecs lcins wh n jus s wel s cul ingediens I shll n ee th s eled ps, bu selemens he sucue Thus in c-lumn, whee isnged be belw smehig else is n eemenEey sucue is les idelly cmpsed lemensIn musicl scle he elemens e nes in hgphy

lees n pennship hey wuld be he heigh cuuend sln o lnes which cmpse lees Ne h in hiss cse he elemens e n ps bu meely bscbleos ppeies no pins wiing Bu since hesoeies my be eled ech he since we my


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A TDCT T SBLC LC

speak he relaive heih ad widh he saci herudi ec leers hese characeisics ay e ake

as elemens i he srucure caled "a hadwrii Likewise i udi a usical isrue say a vii ecsiders he prris a ceai ariues issudhe ire clariy vlue ad s hraherha he relais idividual es each her Herehe varius equaliies are he elees a srucuewhich we call "he e he vii hey ae

searaleps

ayhi u ly disiuishale icceiNw he elees ale d csiue a srucue;

we us kw hese elees ad as he way hey arecied ha is we us kw he elons ha hlda he I ri a cceual icue a csrucas we d whe we "descrie i i lauae we us haveies i he lauae-icure sad r he eleesad as ies lauae eese he relais hereare s ay ways relai elees ha relais ushave aes I ac e ih say ha he cveyace relaiships a elees is he rea uci lauaeI ur ieres ceed eirely u hns we wuld eed he whle syse us ad adjecives vers adprepsiis which we cal a "lauae rus ad

desraive esures wud serve als al urses cuicai Bu relais are his we ca i hey ca e kw y ure acquaiace a kwlede hw he elees a hi e eher isalways a kwlede bou his hi ad equires a licalpicure such as a raaical wrdpicure r is expressi We us have sis elees ad sis r

hei relaishis d ha is jus wha we have i lauaeSuch wrds as "u " he rih "ear "reaerha are aes r relais hey d dee eleesi a csruc u he way hese elees are arraedhe ae is rue such wrds as "lves "haes "kws


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52 AN INTRODUCTION TO SYBOLIC LOGIC

To avoid such ambiguity as well as a great many otherdiculties, we replace words by arbitrary symbolsnumbers,

letters, or other signswhich are not subject to the vagariesof literary grammar and synta, but present a simpliedgrammar of logical structure. In such a symbolism, onecharacter stands for one term, another for another term,etc., and a symbol of dierent type entirely represents therelation. Let us suppose that A and B stand for Jones andfor his wife, respectively Then we would not denote therelation of g by K," which is another Roman capitalletter, but by some radically dierent symbol We mightuse an arbitrary sign an arrow or a curl or some dierenttype of letter, or combination of letters. Let us take acombination of lower case Roman letters, kd." Then theproposition symbolically presented reads

A k d

On the other hand, let C stan for Xanthippe and D forSocrates, and the relation is the wife of" be represented bywf" ; the second proposition is, then

C w f D

There is no danger of confusin wife" as a term withwife" as a relation, if one is called B" and the other wf"Our eye shows us the structural erence.

In this symbolism we do not distinguish between verbs,prepositions, adjectives, and nous which denote relations;for the present, we shall always merely name relations, andtake the function of the auxiliary verb for granted. Arelation symbol placed among symbols for elements shallalways have the force of the verb in a sentence, unlessotherwise specied eg., between" is a preposition, but if

we let bt" stand for the relation this preposition names,and take A B, and C for its terms, then

A bt B, C

is o be read A s between B and C"


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THE ESSENTIALS OF LOGICL STRUCTURE 53

Now, ordinary speech is rich in meanngs, is full ofimplicit ideas which we grasp by suggestion, by association,

by knowing the import of certain words, word-orders, orinections That is what gives natural language its colourvitality, speed, its whole emotional value and literaryadequacy A few words can convey very much But thissame wealth of signicance makes it unt for logical analysisMany apparently simple statements express, in telescopedform, more than one proposition Thus in Jones killed hiswife" the word which a grammarian would call the directobject does more than a direct object should, namely todenote the element to which Jones stood in the relation ofkilling it also conveys that this element stood in the relation wife" to Jones. In other words, Jones killed his wife"means more than A kd B," though that is its grammaticalform it signies

A kd B and B wf AHere we see how a noun comes to have the same name as arelation by the grammatical aid of the possessive pronounhis," it manages to represent a whole proposition Wherepropositions are thus telescoped into compact linguisticexpressions it is, of course, the easiest thing in the worldto miss their logical form completely

But confusion of elements and relations, and the contraction of several propositions into one, are not even theonly charges to be brought against natural language as arevealer of logical forms. The very nature of the relationmay be obscured A dyadic relation may look like a triadicone, as in the example adduced above: Montreal i Northof Albany and New York" Also, two relations of dierent

degree may be expressed in perfectly homologous waysFor example, consider these two statements

() I played bridge with my three cousins(2) I played chess with my three cousins


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A TDUCT T SBLC LGCThey seem t pesent eltns exctly sml stuctue.ut n fct, te eltn clled dgeplyng eques

fu tems e t s tetdc wees chessplyng sdydc. Lngustc gmm gves n ndctn whtevef ths dstnctn But f we pete wth symls thedeence f stuctue ecmes mmedtely ppentndeed, te secnd ppstn pves t e mguus, t emdes tw deent menngs Let stnd f thespeke B, C, D, the thee cusns espectvely, f the eltn dgeplyng c plyng cessThen the st sttement my e expessed

B C D

The secnd hweve, mens ete tt the speke plyedwt ech the tee cusns n tun, tt the teended tgete s ne ppnent vey supei plyeUpn te st ltetve, tee ppstns hve eencllected lngustclly nt ne sttement n te secndne f the ems f te eltn cess-plyng s cmpste tem ut te eltn s dydc, nne te less. I wechse the fme menng, te symlc endeng wuld e

c c

c DF the ltte we shuld wte

ch (B-CD*

But upn ethe ssumptn, t s evdent tt ndch e eltns deent degee s tt te tw ppstns whc n lnguge seem t de nly y the

* Th hyphn whch mgh alo b placd by xprs anpeaton whrby wo m a combnd no on b hssbc bong o a ar chapr o h poby o nngBCD ms hr b ak on ah


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THE ESSETALS F LGCAL STUCTUE 55

denttn f ne wd ctully epesent stutns fdeent lgcl stuctue In dny cnvestn tugh

es ccu tug mguty they e nt usully f seus ntue tey e ghted y explntns nd cmmnsense. But wen u whle pupse n qutng ppstnss te study te lgcl fms then medum whchscues such essentl deences s tht etween dydccnstuct nd tetdc ne s smply ndequte Wemust est t symlsm whch cpes the stuctue f

fcts me tully.Wen eltnsyml stnds n cnstuct te numef tems guped wt t evels te degee te eltnBut wen t s nt ctully used ut meely spken , ts smetmes cnvenent t hve sme wy f dentng tsdegee s my e dne y ddng numecl suscpt f exmle, kd2 mens tt kllng s dydc t3tht etween s tdc. e upst tetng the vepstns, (r) nd (z symlclly, s tht we ndwe ve etns deent degee, nmey c2 nd 4

Futeme, ntul lnguge hs stng tendencyt let ne wd emdy mny menngs. hs llcywhch we explt n mkng puns nd twstng gumentsseems, t st sgt t e nte f thse weknesses flnguge gnst whc we ugt t e guded y cmmn

sense n te vglnce f Engls teches We e supsedt knw tte mnng eks nd wth t eks my et,

nvlves n mplus use f te wd ekschnge m ne menng t nte nete wc, nths cse hppens t e te el ( litel, pmy

menng But mguty (nd wt t mply especllyn wds dentng eltns s muc me genel thn

dny, csul eexn wuld evel t us nd n lgct lest t my ve unsuspected dsstus eects Fewele e we tt they use s cmn nd mptnt


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AN NRDCN SYMBLC LGC

the ntepetatn which is to be attached to it Therefore itis adisable o state that a gien symbol equals by ntepetatn a certain term or relation This phrase is con

eniently abbreiated " int ' Thus in explaining thesymbolism in the four propositions aboe should heady discourse with the glossary

" int "identical with int membership in the class int entailment"E ! int "there exists

nd in the example of the cousins playing bridge and chessa complete notation of the propositions (accepting the rstalternatie for the meaning of " played chess with mythree cousins would be

" int "the speaer"B "C "D int "cousins (respectiely)

"br int "brige-playing"ch int "chess-playing

r br B C D(2) ch B

ch C c h D

The symbols here adopted are arbitrary We might as welhae written r, 2 4 for our elements or y ; and therelation "br might hae been expressed by a picture of aplaying card and "ch by a chess-pawn or any other signthat sgg

ested itself But certain general pncples

symbzatn should be borne in mind in the selection ofogica characters

) igns for elements and signs for relations should bedierent in ind

* When a reat ue the ubcpt omtted becae thntrt te reea the degee o the reaton


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HE ESSENALS F LGCAL SRUCURE 59

That is it would not do to use C and B for the relations nor be adisable to use een CH and BR asong as the elements are represented by Roman

capitals2 igns for relations should not strongly suggest relations which are not meant This principle wil beelucidated later when we classify relations by otherproperties than mere degree n eect it means thatone should not represent "aboe by a symbol suchas suggesting "within or "to right of by a

arrow pointing left or "br by a chess-pawn t isadantageous to choose symbols so that they suggesttheir meanings but

) uggestieness should neer be allowed to interferewith logical clarity or eleganceFor instance if we wanted to state that Chicago liesbetween New or and Dener we might well useC for Chicago and D for Dener but to use N for

New or would be confusing because if we useRoman capitals for elements the use of tw letterswould suggest some combination of two elementsf we wanted a more suggestie symbolism than B and C we might use C D and N or C D and but not C D and N

The assignment of arbitrary meanings to signs with

traditionally established uses should be aoidedThat is to say one should not use to mean chessplaying or for "to the right of because thesesigns are pre-empted for certain important relationsand generally used with those connotations

() igns should be easy to write and to recognize andbe as compact as possible chess-man unless it

were highly conentionalized would be a poor symbolfor "ch because it would be too elaborate lso ifwe did conentionalize it we should be careful notto let the resultant gure loo ie a 2 an i or any


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62 AN NDUCN SBLC LGC

Ieographic signs may be freely inente or aopte withinthe limits of the folowing principles of symbolic expression

Raical istinction between term signs an relationsigns() oiance of false suggestion3 Preceence of logical exactness oer any psychologica

aantages4 oiance of traitionaly pre-empte sgns(5) Due attention to istinctness, compactness an

typographical simplicity

goo symbolism leas not only to a cear unerstaningof o ieas, but often to the iscoery of new ones

QUESTIONS FOR REVIEWI Wt retion re inoled in te contitton o

net of tbe clter of wrming bee

row of fence picket repectely2. Wt contitent mt trcte e beide relton 3 Wt i te cef e of logic pictre 4 Wt i te mot impotnt kind of logicl pictre Wt ment by te term of eltion 6 Wt i ment by te degee of reltion7 Wt i dydic elton olydic 8 Wt i popoton Wt two fnction e performed

by te min eb in propoton

Wy do ogcn e ymbol o Wt dnge cn yo ee in te e of ntrl lnggeto peent propotion for logicl tdy

I I Wt princpe mt be obered in te coice o ymbol

SUGGESTIONS FOR CLASS WORI Nme contrct woe element e not pyicl prt

II In te foowng propoition ndeline te word denotingrelton:

) e Ayrin cme down lke wolf on te fold2 Te pen i mgtier tn te word3 Neer trole troble tl troble trble yo


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HE ESSENALS LCAL SRRE

II In te foowing popoiton ndeline te reton ndtte t degee

) MayMagaet,anJaneaesstes

z) James,JohnanJeyaetpets.(3 PasanBenaesmalethanLondon

4 TetoneCEandGfomatad

(5) Sheweeamongthentodenways

6) "Thegeatestoftheseschaty.

IV. UsingRoman captals for tems and sybolsof Y?'own

choce foreatons,statethefolowgpopos1tons :

(r) JessewasthefaheofDavd 2) KeatswasyoungethanWoswothandCoedge.

(3 New England s Mane, ew Hampshie, VemontMassahsetts,RoeIsan, andConnectc

4 S ge i og to Rod Sm ge dog bone


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C H A P T E R I I I

THE ESSENTIALS OF LOGCAL STRUCTURE(Continued)

1 CoNEXT

f you read in the paper a headline: Lived togetherten years without speaking," your imagination promptlysupplies suitable terms to the relation which alone is men

tioned. A man and his wife, or a mother and her daughter,in short : someone and someone else must have lived togetherten years without speaking. Again, if you open a cook-bookat random and read at the head of a page: these shouldbe rubbed together to a smooth paste, you know at oncethat these must refer to things like our, shortening,sugar, yolks of eggs, or the like You may say that you

have not the faintest idea what is to be rubbed together to asmooth paste, or who lived together ten years wthoutspeaking; but the mere fact that you say what and who,respectively, shows that you know something about themissing terms of such propositions. You would feel greatlydisconcerted to read that butter and sugar lived togetherten years without speaking or that mother and daughtershould be rubbed together to a smooth paste The reasonfor this is that you have understood the relation and youknow, in a general way, what is the range of its possibleapplications The same is true, of course, if a certain set ofterms is given and the relation left undetermined f youread, on either side of a blot on the back of a postcard We (blot) Niagara, you may not know whether the relationbetween your friends and Niagara is that they saw the

waterfall or that they missed it, liked or scoed, feared orphotographed it, but you do know that the lost verb cannotbe married or ate or strangled Such relations amongsuch terms would fail to make sense. Ou thoughts always


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THE ESSENTALS OF LOGIAL STRUCTURE 65

move within a certain range a great vaguely apprehendedclass of things which we feel could be related to each other

and we look only for relations which might conceialyhold among these things Such a rane of general subectmatterof terms nd elaion tha v entr togherino our thoughtsi calle a otext


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68 A IDUCI SBLIC LGIC

but we cannot ehaustively state such a contet and nowal ideas that are relevant to it

(A) Te nvese o DscouseThe otal collection of all those and only those eleenswhich belong t a formal contet is called a nvese oDscouse In ordinary conversation we assume the limitation of such a universe as when we say Everybody nowsthat another war is coming and assume that "everybodywill be properly understood to refer only to adults of normal

intelligence and European culture not to babies in theircribs idiots or the inhabitants of remote wildeessesor conversational purposes the tacit understanding willdo but if the statement is to be challenged ie if someonevolunteers to produce a person to whom it may be appliedbut of whom it is not true then it becomes important tonow just what the limits of its applicability really are

Arguments of this sort have their own techniue by whichthe opposition marshals contradictory casesin this eamplepersons who have no such nowledgeand the asseveratorrules them out as not meant by his statement Theuniverse of ordinary discourse is vague enough so that thisprocess can go on as long as the bellicosity of the twoadversaries lasts

Logicians and scientists however tae no pleasure incasuistry Their universe of discourse must be deniteenough to allow no dispute whatever about what does ordoes not belong to it or instance if an anthropologistmaes the statement that everybody is bo into somesocial group we understand him to mean the baby in itscrib and the head-hunter of Paagonia as well as theAmerican ambassador to the Court of St James because

the universe of discourse of his science is nown to includeall human beings Did it not he would be called upon tostate its limits

We may imit a universe of discourse as much as we ie


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HE ESSEIALS LGICAL SUCUE 6

so eaborate a structure as a natural science has a verygreat universe whereas a esser construct say the formal

set of rules for playing chess is limited to 6 elements calledsuares and 2 elements called men The chess gamedoes not tae in ll elements called chess-men ; its universeis limited to 2. In teaching chess to a novice the rst thingone does is to enumerate the elements involved call hisattention to the suares and the men before one mentionsthe relations which hold or fail to hod among them Nowthis is eactly the way to begin the rigorous analysis of any

structure The universe of discourse must be recognized andepressed Logicians usually denote it by the letter derived from the German Klsse a class and put iparentheses the notation chosen for its elements But sincewe have adopted the practice of using Roman capitaletters for elements I shall use an italic K for the total classof such elements because this is not itself an element and

should have a dierent ind of symboSuppose we select a very simple universe of discourse a

group of four houses whose relations to each other we aregoing to state We should then introduce our universe asfollows

K(A B C D) K =int houses

Here we have an enumeration of just four elements thesymbolic indication that they form a universe of discourseand the nepeon by vrtue of which the elements arenown to be houses

(B) onsuen RelonsA formal contet involves not only elements but therelations which connect such elements Obviously one cannot

introduce any relation at random into any universe whatever for instance one cannot say that 2 is older than or that one house is wiser than another Such statementswould be neither true nor false they would be simply


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72 AN INTRODUCTION TO SYBOLIC LOGC

Since nt is dyadic ie. combines its terms two at a time

there are sixteen possible ways o combining A B C and

D (counting the cases where the same letter stands beore

and ater the relation as A nt A B nt B etc.) . Each

possible combination by means o nt yields a proposition

which is either true or alse. Without any special inorma

tion about the our houses in question can we know the

truthvalues o any o these propositions ?

Well i nt2 is to mean North o then certainly no

element can have this relation to itsel; so we know o

hand thtA nt A ails*

B nt B ails

C nt C ails

D nt D ails

Here are our truth-values xed in advance ; none o the

houses is to the North o itsel. To express the act that arelation ails, i.e. that the proposition in which it unctions

is not true it is customary to enclose the proposition in

parentheses and prex to this whole expressin the sign "

Thus (A nt A) means A nt A ails and may be read

A s not North o A or "It alse that A nt A In ormalsymbolism then the our established propositions should

be written (A nt A)

, B nt B)

. (C nt C) (D nt D)

So the nature o the constituent relation determines the

truthvalue o our propositions any other proposition in

the ormal context however may be either true or alse.It seems that to know all the truthvalues o the sixteen

* A reaton is said to fail" whenever it is sgnicant, but doesnot hold among the stated terms.


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THE ESSENTALS OF LOGICAL STRUCTURE 73

possible propositions we would require twelve items o

pecial inormation.

Now let us assume just one such item namely that

A nt B is true. Immediately we know the truthvalue onot one but two propositions we know that

A nt B

and (B nt A)

The alsity o the latter ollows rom the truth o the ormer

as we recognize rom our common sense understanding o

"nt2. We know that this particular relation cannot connecttwo terms in both possible orders. The truth o one proposi

tion precludes the truth, or implies the falsity, o the other(Note that to be told A nt B is false would not give usany knowledge o B nt A since both might be alse

though not both can be true.)

Suppose now we know A nt B and are gien the

urther inormation B nt C. At once we realize these two

propositions together imply that C is also to the North o A.So the truth o two propositions determines the truth-valueso six

A n t B

B nt C

jointly asserted assure us o the ollowing acts :

A nt (B nt A)

(C nt A)

(C nt B)

So it appears that even in the case o the twelve undeter

mined propositions the truthvalues that might be assigned

to them are not entirely arbitrary and unrestricted The

truth of one propositon preclde that o another or thejoint assertion o two propositions implies a third. Thepossible truth-vlues that coud be attached to these twelve

constructs are relative to one another


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74 A ICI SBIC GIC

6. COUE ELAO AD OAL ELAO

In describing the formal contet for this discouse abouhouses the only elements to be used were given as A B D and the only relation as "nt. merely formulatingall the possible elementary popositions in this contecertainly no other constituents were employed But as sooas the propositions were formulated it as appaent thasome could not reasonably be asserted at all and even theothers could not all be asserted together indiscriminately.ertain ones were dependent upon cetain others by implication preclusion and the lie But implication o even themere oiningup of two propositions in one assertion arerelation so there appear to be relations operative in oudiscourse besides the relation "nt which is mentioned asa constituent of the formal contet

Such relations however hold mon popoition o te

dicoue not mon element The relations which holdamong elements form elementary propositions and areconstituents of those propositions and items in the formalcontet the relations which hold among opositions areno constituents of elementay propositions and are therefore not enumerated as materials of the formal contet shall call the latter ind loicl eltion to distinguish them

from the contituent eltion of the discourseThe constituent relations vary with the fomal contetin every discourse there must be constituen elations buwhat these are to be is arbitrary within the limits of wha"maes sense in the given universe. Logical relations toooccur in any discourse of more than one proposition buthey are always the same few relations They are sometimes referred to as the "logical constants of discourse

They hold among elementary propositions whatever theconstituents of those propositions may be

* he mpotae o ths dstto was rst poted t t meby roessor Sheer he termnology however s ot hs


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THE ESSENTIAS OF OGICAL STRUCTURE

The principal logical relations ae : (r) conjunction of

propositions or joint ertion ; this is expressedby the

word "and, or the traditional symbol ".. Thus, "A isto

theNorthof BandB isto theNorthof C" isexpressed

symbolically,(A nt B) . (B nt )

Diunction of propositions the assertion of one proposition o the other. This relation is taen in logic to mean"one or the other or both that is it means t let one

of the two propositions. or instance it is always the casethat eite A is not to the North of B o B is not to theNorth of A Either nt B or "B nt A must be false;they may in fact both be false. So using the acceptedsymbol V for "eitheror we may say

(A nt B) V (B nt A)

() mpliction of one proposition by another. This coversthe notion of pecluion which is the implication by oneproposition that another is false "A nt B precludes"B nt A i e. "A nt B implies " (B nt A) The symbolfor "implies is Sometimes a proposition is impliedby the oint assertion of two or moe others. Then you rstepress the counction bracet the whole epressionand use it as one poposition

(A nt B) (B nt ) (A nt )

(There are various ways of circumscribing and recastinghe appearance of the logical elations. Instead of we might have a symbol for preclusion since an impliedproposition is one whose falsity is precluded ust as aprecluded proposition is one hose falsity is implied (as in

" (A nt B) : (B nt A)) Or we might use and symbolizehe notion "neither . nor instead of "either . oru all these relations which may be used in place of " "V and eally cme to the same thing they epress


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76 AN INDUCIN SBLIC LGIC

the sae m the sae state of aais so we ay as weaide y the ones which ae in ost genea use)

7 STwo popositions which cnn bh be ue ae said to e

ncnssen Thus A nt B and B nt A ae inconsistentwith each othe; one ay e tue and the othe fase andogicay it is a atte of indieence whch is tue and whchis fase o they ay oth e fase (as they woud e if the

houses stood in the sae atitude) ut they cnn bothe tue iewise(A nt B) (B nt C) , (A nt C)

cannot e ointy asseted; if [(A nt B) (B nt C) (A nt C) then to asset the tuth of the st two and thefasity of the thid is inconsistent To assign inconsistenttuthvaues to the eeentay popositions within a foa

context esuts in a chaotic and absud discousef howeve we assign a tuthvaue to evey possibe

poposition so that each is consistent with a the othesointy asseted anyone wi eadiy see hat this is asysteatic desciption of a cetain state of aais (ea oiaginay) t is syteatic because the popositions aeinteeated ined to each othe by ogica eations uch

n deed dscuse whn ml cnex s clled sysemThee ae innueae ways of constucting systesDieent constituent eations aow of entiey dieentcobinations of eeentay popositions y eans of theogica eations dieent univeses of discouse oedieent possibiities of aangeent n soe systes thesevea popositions ae so cosey connected by mplcnthat if we now the tuthvaue of a vey few constucts wecan assign a the est uneuivocay The pocess of easoningfo one tuthvaue to anothe aong popositions isn

own as deduction; fo instance fo (A nt B) (B nt C)

one ay deduce A nt C A sste whein this is possie


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HE ESSENIALS FLGAL SUUE

77

l be of nown popositions deteines

so that a sa nu

atheestisadeducvesysem d

A fai exaple of such a syste ay be constuc

t

;dwithin the contextK(A B CD)ntz. Supposeyou ae tothat A nt B

B nt C

C nt D

. d A tC since

FothesttwoinconJunctonwede uce n '

[(A ntB) (Bnt C) A ntCLikewis (BntC) ( C nt

D) BntD

And ifwe tae this ast poposition togethewith thest

wehave :[(A ntB) (BntD)] A ntD

Hee ae thee popositions whose tuth is deducibe fo

that of the thee given ones Moeov

each of these tue

popositionsipliesthat its conveseSfalse

(AntB) (BntA)

(Ant C), (C ntB)

(A ntD), (DntA)

(Bnt C) (C ntB)

(BntD) (DntB)

(CntD),(DntB)

th tuth of Just thee of the tweve

So, by assumg e d popositions that coud e constucted

"undeterme

t t have een enaed to deduce a the est

our conex 'we

h 1 assuptions wee ad to

esta 1 aThe three oga

deductivesystem.t ae ovious Instead of

tweve constuctsts v ues d h

whose tuthvalues ust e abitaily assgne

Ve aveh ittenowedgeof ode eatednessshotony t ee a

d f meof cl srucure, seves

us instea o a cu eso '

itemizedknowledgeofmanyfacts


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AN INTRODCTION TO SBOLC LOGCSuppose, however, that we assume a derent formalcontext, say a collection of ve persons, whom I shalldenote as S,T,U,V,andW The constituent rel

ation slikes " I shall express it by the symbol lk w h 'th e aveen,

K(S,T,U,V,W)lk2 K=intpersons"lk2=intlikes"

Letusassume,SlkT

T UU lk V V lk W

What may be nferred from these statements ? Nothing atall Lkes" mayornot hold bothwaysbetween the samelements (SlkT) (TlkU) doesnottelluswhetherSlkus true or false ; we cannot evenguess whether anyone ofthee persons lkes hself,_ or not. The fact s, that anyelementary construct ths context s plasible in tselfnd i_s perfectly consistent wth every othr, or any conJncton of others If we want to assgn truth-values toall .the propostons we must make twenty-ve separateasnn:ents. Such a system I call inductive, in contra-dstncton from deductive Any systemn thegven co

ntet .K(S,T,U,V,W lk2 must be completely inductive ;thsIS duetothecharacterofitsconstituentrelatonMostsystemsexhbtamxtreofbothtypes ; e somepropostions imply others, but there are also somewh h Cemauntouched SuchasystemmaybeclassedasmixedMost scientic systems are of this hybrid sort ; they arepartly capable of deductve arrangement, but stll containalargenumberoffactsthatcoldneverhavebeeninferredbutaresmplykowntousasdata,arbtrarytruths"given'to us by expeence. The dream of every scientist s tond someformulation of al hs factswhereby they may


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THE ESSETAS OF OGCA STRUCTRE 79

e arraged a mpeey deduve sysem; ad f sdesirae ype we sha have a grea dea mre say

suseque hapers . We have w irdued a f he esseas f gasrre element ad elation amg eemes elementay popoition mpsed f hese smpe suseais amg eemeary prpss r logcal elanytem srued f eemeary prpss y measf hese ga reais hgher ga srures maye reaed as sysems r pars f sysems

UAEvery dsurse maer hw fragmeary r asa

mves a era context f ierreaed deas rdaryhg hs e is idee ad shfg

.The psyhga ex f r hughs argeyprvae ad persa Tw pepe

_

ag au he same

hig may piure i hemseves wdey dverge waysThey have he dere conception B if hey ersad eah her a a he her respeve epsemdy he same concept

Lg is ered eirey wh eps epi ga dsurse res u a prvae ad adeaaspes s ex mus e xed ad p T eemes

ad reas ha may eer is prpss mayherefre e emeraed advae These se hefrma ex f he dsrse

The a ei f eemes i a frma ex saed he uverse f disurse

The reas whih ai amg suh eemes areaed he se reas f he frma ex

The mias whh may e made u f he eemes a frma ex y eg a se reamie hem ardg s degree are he elementapopoition f he disurse Evey eemeary prs has a tuthvalue whh


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A RC SMBC GC

either tuth or flsity Value here has nothing to do withthe uality of bing valuable

The propositions constructible in a formal context maybe such that they cannot all be true or cannot all be falsethat is it may be that to x the truth-vales of some ofthem automatically assigns truth-values to others Thepropositions of such a context are interrelated

The relations which hod among elementary propositionsare not the constituent relations mentioned in the formal

context but are logicl relationsThe logical relations generally used in symbolic logic arecouctio ( disuctio V and iplcto Constituent relations may be freely chosen within the limitsof what the universe of discourse admits as maing sense ;logical relations are constants the same for every discourse.

total set of elementary propositions in a formal contextconnected by logical relations and jointly assertable without

inconsistency is a syste system wherein a small number of propositions nown

fro outside information to be true implies the truth orfalsity of all other elementary propositions is called adeductive syste

system wherein all truth-values must be separatelyassigned by pure assumption or outside information is an

nductiv syste system wherein some truthvalues may be deducedbut others neither imply anything nor are implied is amixed syste

The essentials of structure are eleets, and relationsamong elements or costituet eltios eleety popositios relations among elementary propositions orlogicl eltios and nally systes the higher forms ofstructure composed of related elementary propositionswithin a logical context


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HE ESSES F GC SCE 81

QSIOS O IW

I Whaimanbyhconofanorinaryconr

a_n.

?2.Whaihirncbwnaconcanaconc,n Wha i h inc bwn

ychologal con _an 3 formalcon Whichimorimoran conran .Whichiinlogic 14 \haarhnialofaformalcon

5 Whaianiroficor

l b6Howmayhmnaryrooiioninaforma con

rladoachohr ? d1 1

Whaihirncbwnconiunrlanan og

ca

7 raion8 Whaiaym ? ?9:Whaarhnialfacorologalrucur

SUGGESTIONS FOR CLSS WORK

G lofformalconx xr

ymbolically,I . veanexamp .

. '2 . alii

;nu

r

:niunrlaiononcanlogialrlaionw If w hav any ir, hn

ihr our ir la .t 1 ofullmn

anfulmlaourf '

wan orl a . 1 d t hrfor

anwanlaoain an

urf a ?a f

any ir la o ain Schonhaur argumn

or

imim

3 Exr hfollowing amnymbolical

!ca ial for lmn, ymbol of your own c o bocoiun rlaionor rlaion, an h cuomarymforlogicalrlaion :

John i olr han im an Tim ior han G

org ;

hrforJohniolrhanGorg.

p . b(b) If NwYork i colr ha

n Pari, hn a canno

colrhanNwYork

() Shlovm[or]Shlovmo

(d) I amyourbroranyouarm h(e) Neiherhe Wshington

nor heMuretmIS faet a

Bee

l


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H A T E R V

GENRALZATONI. EGULARTIES OF A YSTEMIn the foregoing chaptere dealtith a systemofjustfourelements ; insuchasystemtherearesixteenelementarypropositions When each one of these is either assertd odenied the system is completely and explicitly stated Ina formal context ith

a very small universe of discoursethis is perfecty practicable because sixteen proposiionsare easily perused Suppose hoever e choose a largeruniverse say one of ten elements ; for convenience let usagain takea dyadic relation and let the ne formalcontextbe

K(ABC,DEFGH, I ]fm2

K=int"personsfm2=int"fellomanof

Let the folloing trth-vales be assigned to propositionsofhesystem

A fm BB fm

fm DD fm E E fm F F fm G G fm H H fm fm ]

No, no one can fairly be called his on feloman :so e may add at once ten furthepropositions of estab-shedtruthvalue


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GENERAZATON

(A fm AB fm B

(] fm ]

urthermore in the natre of the constituent relationcetain impications folo from the nine originally grantedfacts. Thus (A fm B) (B fm A) (B fm ( fm B ndso forth so that e may assert the impcatn of the neconveres of the given propositions

(A fm B B fm A(B fm ( fm B)( fm D) D fm

( fm ]) (] fm )

A f "A fm B and "B fm are true then "A fm must

se\rue if A is a felloman of B and B of th Amust be a feoman of This gives us the proposon"A fm so by the same principle e further relate A toD to E to etc. Obviously in a universe of ten elementsthis gives us a sarming multitude of statements:

[(A fm B . (B fm (A fm (A fm ( fm D) (A fm D

[(A fm ( fm ]) (A fm ])

[(B fm ) ( fm D (B fm D)B fm D . (D fm E) (B fm E

[( fm D . (D fm E) ( fm E)


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AN NTRDCTN T SBC GCEach proposiion o he righ o he implicaionsign is anewlyesablshedelemenaryconsru+ andeachocourse,impliesisconverse,soweaugmenhelisbegnning :

AmB) (BmA)ec, ec ,

by (A m C) C mA)

(Hm ]) (] mH)

Alogeher, he explici saemen o his sysem requiresoo asserions.Many o hese asserins, however, look srikinglyalike.I have arranged hem above in hree liss. One conainssimple elemenary proposiions involving j us one elemenapiece ; he nex conains saemens o he eec ha anelemenarproposiion wih wo disinc erms implies is

converse e he proposiion which relaes he same woerms reverse?order; he hird lis consiss o pairs oproposns conJoed o imply a hird. In each lis heseveral asserions are analogous ; and by reason o his unmisakableanalogyI haveno roubledmysel owrie oueach iem, bu have merely sared each column, and lehe reader supplywha is deoed by dashes or by "ecWedono.havetolookveryfardownanylisttoappreciatehe repeusness o he logical relations expressed in i ;o recogze ha no maer which elemens we combineino an elemenary proposiion, he logical relaion o hisprposition to its converse is always " ; and no matterwhchthreeelementsweselect,ifwemaketwopropositionsou o hem, such ha he second erm o one is idenicalwh he rs erm o he

second, and relae hese woproposionsby ,henhis conjunchasherelaion "to a third proposition combining the rst term of the rstwih heseconderm ohesecond Wahe asserionsi


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GENERALZATN eac lis have in common, is he number o elemenaryproposiions involved in hem (2) he number o disicelemens involved in hese proposiions, () he locaion

o idenical and o disinc elemens, (4) he naure andlocaion o logical relaions conained in each oal asserin.These regulariies samp all he asserions in any one liswih he same logical orm, no maer which elemen, orwhich wo or hree elemens gure in any such asserion

. AABSI we wish o express symbolically he logical orm o awhole lis o proposiions we may do his by using whais called a vible sybol uch a symbol is no a eassigned o a cerain one o he elemens, bu means "Aor B or C, or J, whichever is chosen or his place.I is called a vible because i can mean all he elemens

in urn is meaning may vy rom A o ].To disinguish such symbols rom specic es, lieA, B C, ec, shall use lower case ialics or variablesThus,

(A m A)

means " is alse ha he elemen called A has he relaion

m o isel bu

means, by urns,

( m

(A m A), (B m B)

ec, ec

I may denoe any one o he en proposiions involving onlyone elemen. ikewise, we migh use a consruc o wovariables o mean, in urn, all proposiions involving juswo elemens Thus, m b may mean, in urn,


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86 AN NTROUCTON TO SYBOC LOGC

A m BA m C

B m CB m D

But it my just s we men "B m A s "A m B Tha nd b, respectivy, do not men "A nd "B, but "thrstmentioned term nd "the second-mentioned termI "B is the rst-mentioned term, then a mens "B I

"] is the secondmentioned trm, then b mens "].I, however, two eementry structurs re ogicy

reted, the whoe construct is one asseion; nd i vribeis given mening in one prt o tht ssertion it must kepit throughout. For instnce

my mn:

or:

(a m b) (b m a)

(C m D D m C

B m A A m B ;

in the frst instance, a means C and b means D in the'econd, a mens B nd b mens A; but in both cses, thefac that a ceain (s-menioned) tem has he elaion m oa ceain ohe (second) tem, impis tha the ohe tem has

that elaion o the s. Thereore, no mtter how " certinterm nd " certin othr term re chosenwhethrw choos B rst nd A second, or C rst nd D scond, orwht you wiwherever "tht rst term nd "tht secondterm recur, we must put in the one we chose rst, nd theone we chose nxt, respectivy So we my sy tht aaiable may mean any element bu whichee i does mean,

it must mean hat same one houghout he whole asseion.Th vribe a in (a m b) : (b m a) coud not men A inthe rst cse nd E in the cond. A m D D m Ais possib mening or (a m b) (b m a) nd o is


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8 A RODUCO O SOC OGCAn mnt whtv m b substitutd fo nd thsut w b poposition whichiseithertrue orflse. I f n xpssion invovs two vibs s foinstnc

a

thn th ng of this pi of vibs is th dydswhich cn b md in th fom contxt Rmmb thtth two tms of ddic tion m b th sm mnt tht is AfmA is ddic though th tms not distinct In th xpsion fm th is hg tot us ht nd ncssi diinct "th stmntiond mnt nd "th scondmntiond mntm v w b th sme mnt If th sm vrileoccsin both pcs thn wknowtht thtms dnotd idntic ; if dint vibs occu w do not knowwithout futh son whth thi vus idnticodistinct.

Th poposition "]fm] is thn possib mningfo th vib xpssion fm" So is "AfmB ]fmH, o n oth mnt poposition Th ntihundd ddic combintions which b md in thfom contxtf within th ng of ; tn ofthscompos th ng of" in th xpssion

a fm a

The process of assignin a spcc value to a variablem b cd pecction This is not th sm thing sintpttion fo whn w intpt smbo w com-mission it to mn certin kind of thing s K inthouss dtmins tht ABCD sh psnt houssnd not ts pop o ds. Intptton xs thsort of t invovd in th dscous But if w s : qus b spcction th mnt C (which m bbbvitd : =spC) this dos not t us tht mns th ortof thing cd hous but whichhous it


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GEERAZAO B

dnots t ous nmd C. T ng o ignicnc o vib tn is t css o tos nd o tos

mnts wic m b substitutd o it b specictnT notion o vibs is not pcu to ogc It sconstnt mpod in odin spc Ev ponoun s vib nd t us o spcction oundd t sm us tt w tugt to us in scoo gd tontcnts o ponouns. T o instnc t sntnc

I nw ou woud not do it.

T mning o suc sttmnt is indnit untiw now t mnings o I ou nd it. I tswods sid b on pson to not t two psonponouns v dint vus i. dint ntcdnts tt is usu t cs But mn wo s id in igsov o sistd gt tmpttion migt o dd

ims in tis sion. In tt cs t two vbs Ind ou woud dnot t sm mnt wo dintponouns usu but not ncssi v dnt ntcdnts. I owv on ss

ou nw woud not do it

tn ou must mn t sm pson in bot css i vib is onc spcid it must p t sm spcction tougout t nti sttmnt win it occuso in mo ii ngug ponoun must v t sntcdnt tougout t nti sttmnt And ts gis simp common sns. Moov ou m n npeson its ng is t css o psons. T sm t odso I. On t ot nd it s to somtg stn pson i.. it s dint ng om t ottwo vibs In contt w doing ms n snst univs must contin intepettion gnts nd ctsit m bcom speciction n on o t ctsws I nd ou m dnot spcic n two o


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92 AN INTRODUCTION TO SYBOLIC LOGIC

we have ten propositions about elements in relation to A

A fm A

A fm BA fm C

A fm ]

n these propositions the second ter varis fro A t ]

Suppose, then we use a variable in its place, and write 0 .

A fm

This is a propositional form of one variable It is neithertrue nor false one of its vues is a false proposition, whereasall the others are true ones

"Now

.here is also a list of propositions of the form

Bfma, and one of "Cfma and "D fma t S I1

' , e c o ongas onyoneelementisformaizedwened tenpropositionalfrs t

enumerate all the possible elementary propositionso t e dscourse But when we align these in a list :

A fm B f C f D f

] fm it is clear that he rst element too, could be foralizedthat we could write :

' '

b fm

nd let the two :erms vary over the entire range of both,Ie. over all possble dyadic combinations within K Th

h d

. ereare

one un red valus for this propositional form of two

vables Ten of these values are false propositions andmnetyaretrue.

The propositional form "Afma" is in one sense 1for "bf " b

a va uema ut not m the genuine sense in which


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GRZO 93

AfmE" would be "Afm is related tobfm by

prti speciction ; it still remains a propositional form

andI shallcalitarestrictedpropositionlform,

because th

specication of one element restricts its range to pro-

positions of the given rst term. A restricted propositional

formmightbecaledanmbiguousvluefortheunrestricted

form But all thesenames aremere conveniences the thing

to reember is that ssoon swereplcesingletermin

proposition vrible, we hve propositionl form

which has as any values as there are values for that

variabl; and t

an introduction to symbolic logic - susanne k. langer

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