FORMALISINGBY SYMBOLIC

LIBRARYLOGIC

PROBLEMS

Atternpt s to Ior rna lise the ata t errient s in alibrary operation by rrieans of sylTlbo~ic logic forstr earn lining further data processing operation.Uses the notation of both the Principia and PolishsystelTl of Propositional Calculus. Illustrates the useof sy rnbolic logic to COmlTIOnlibrary search state-ments with the aid of truth values. Demonstrate thesimplication of a flow chart by truth table. Manipula-tions like formula conversion and reducUon were usedto show their help in streamlining library instructionmanuals. The application of logical possibilities fordefining an optimum path in a compl ex network oflibrary operations have been hinted. Statements ofvery simple. idealised and trivial problems are ana-lysed to dete r rn irie their logical structure using basiclogical concepts .and propositional calculus notationin step-by-step exposition.

The internal operation of a computer,the structure of computer languages andprogramming languages can only be reallyunderstood when we study symbolic logic. A.logical organisation of these statements andtheir symbolisation may help the librarian totranslate the verbal form of his problem into alanguage amenable to easy data processingoperations. One of such preliminary dataprocessing operations is the convenient flow-c;,.al'ting, The breaking up of the problem inc onv eni ent logical elements or the establish-ment of the logical flow in between these ele-m cnt s can be easily vi sua Ii s ed in flow charts.Y!'. the present article the value of expressinge tat erne nte of .th e problem in the library interms of the notation of the propositionalcalculus will be discussed. This notationand the logical cone epts and operations helpto see the flow chart operation in clearperspective and also to simplify them. Thisis most useful in those cases when thecomputer will make only very simple yes -riotype decisions. Other simplifications in thenumber of instructions in library manual mayalso be rna de , Some statements of verysimple. idealised and trivial problems will

Vol 16 No 3 & 4 Sept & Dec 1969

A R Chakraborty

Ins do c , New Delhi-12.

be analysed to determine their logical struc-ture. Even though problems in a real librarywhen stated are of much greater complexityit is hoped that they will be structurallysimilar to Some of the problems stated here.Since the programmer and the data-processorwill need clear and unambiguous flowchartsand statements of the problem in the minimumof possible instructions without any sup e r >

fluity, such logical formalisations will helpthe librarian to prepare his basic systemready for automation programmes in thelibrary. Therefore some very simple logicalconcepts and the notation of the propositionalcalculus has been introduced in a step-by-stepexposition and an endevour has been made toshow their relevancy to formalisation of dayto day language.

Statement and Propositions andTheir Truth-Value.

Any statement made about anythingmay be believed to be true or false. Thesestatements are expressed in the form ofsentences in the human language. Of coursein human language there are other types ofsentences, say, for example, the imperativeor the interjectionals but we can say withreasonable accuracy that the vast majorityof sentences are assertive or statementsentences or the negations to some otherstatement sentences. These again in theparlance of logic are said to be rep r e s enta> 'tions or expressions of logical propositionsin natural language with definite truth-values.

When one says that all propositions orstatements must be true or false it does notmean that this truth value should be or couldbe always determined from a correspondencewith the reality. A much more Irnpo r ta nt factis the concept that whatever statement isbelieved t.o be true is true and whatever is

131

CHAKRABORTY

not included in the list of such former state-ments is not true. If a verification of thestatement whose truth-value is to be deter-mined shows that it is incompatible w ichone's list of true statements then the state-ment in question is not true. Therefore suchquestion to the librarian as "Do you have thebook logic and algorithms by Robert Korfhagein your library? n can be viewed as a verifica-tion whether the statement "The book Logic andAlgorithm has been catalogued in the library"is true or not. A search through the cataloguelist (where all information of books arerecorded i. e. which is a list of true state-ments) now will prove the truth value of theabove statement. In an automatic library thecomputer will determine truth value of suchstatements from the list of many true state-ments about a work that are recorded in thepunch card or other recording media. Forexample, let us take two sentences likeLogic by Tarski is in the library and Logic byTarski and Logic by Suppes are in the library.These statements 1 and 2 may be representedby symbols P &t Q respectively. If all therequired information of the books in the libraryare recorded in punched cards the computercan read these cards and find out the truthvalues of P &: Q. This operation seems verysimple and trivial but is justified when thelibrary must scan information of a largenumber of books and large number of borrow-ers. Thus sentences such a s r-

1 "Have you ordered the book "Data process-ing in the library?"Z "Does the subscription data of "AmericanDocumentation" expires on 1 May 1969" ar~transformed into statements like "find whatis the truth value of the following sentences:

1 -The book "Data Processing" has beenordered in the library"2 "The subcription expiry date of AmericanDocumentation is 1 May 1969". If we re-present these two statements or sentences bythe symbols L, M respectively then thequestion will be to find the truth value ofL, M from our list of true statements. Ifthe truth values are represented by 1 and 0 orT and F for true and falsity then symbolicallywe can write

L = 1 or L :: 0 ?

L = T or L:: F etc.

132

Elementary Sy rnb ol isat ion of Co rnp ouridStatements: c onrre c tive not symbol ish e d .

We have up to now only referred tostatements ana propositions without referringto their complexities. Different simplepropositions are combined into bigger state-rn en t s or p r opo s iti on s and we bec om e con-c e r n ed with t he t r ut h of this compound pro-po s iti on to getb er , An we can investigate theprediction of truth or ialsity of the wholecompound proposition if we know the truth orfa l s itv of the constituent pcopo a itton s orstatements. Therefore representation of thecompound statements or propositions arenee e s sary showing the structure i. e. theconstituent propositions and the nature of theconnectives like 'and' 'or' etc. joining thesesimple propositions. And this structure canalso be svrri bo l is ed , Thus at an elementarylevel the Io l io w in g compound sentences ors ta t ern en ts :

The book "Catalytic cracking" has beenissued and is due by 3.9.69.

2 The book "Catalytic c r ac ki ng " has beenis au ed er is in the reading rOOIn.

3 Machines c ur r ent lv on tlie market caneither be r ent ed 01' purchased.

4 If there is no answer to the Ls t reminderwithin two weeks then a second reminderis is sued?

May be represented as(1)(Z)(3 )(4)

PAND QP OR Q

EITHER L OR MIF X THEN Y

Similarly before symbolising such asentence should be ana1ysed. The sentenceIIIf errors by a system study are c o r r ec te dthere will be savings from irrip ro ved opera-tion a part of which will pay for the effortinvolved in the study" may be analysed intothe following statements (1) "Errors ...•c or r ec te d." (2) "There .... operation (3) Apart. '" study and i.f the symbols P, Q &t Rrepresent these s ta te m ent.s respectively thenthe whole statement will look like thefollowing

IF P THEN Q AND R (5)

(Then, and etc. were omitted in the mainsentence)

Ann Lib Sci Doc

FORMALISING LIBRARY PROBLEMS BY SYMBOLIC LOGIC

Upper level symbolisation of compoundstatements: connectives symbolised.

Upto now only the constituent simplestatements or propositions of compoundsentences have been discussed. How theyshould be symbolised has also been noted.It might have been also noted that the connec-to rs like AND, OR, IF •••.. THEN, NOT, IF.AND ONLY IF etc. have not been symbolised.The next step would be symbolising themfurther both into the notations developed byBestrand Russel (i. e. Principia notation) andby Jan Lusiewicz (polish notation). Both thesenotations and the system of logic to a certainlevel is intel"t-ranslatable.

The previous formulas will look asfollows in Principia and Polish notationsrespectively:

The FORMULAS (1), (2)(4), (5) will then look likeon further symbolisationas shown here

OR

(1) PAND Q~P. Q 2 Kpq(2) P OR Q ::::;, PVQ ~ Apq(4) IF X THEN Y:::::::>-X:>Y Cxy(5) IF p THEN Q AND R

:::=:::} P )- (Q. R. ) . ;> CpKqr=!? (P >Q). R .::::;> Kc pq r

A few more symbolisations will clear theidea.

(1) Let the following sentences

"Logic by Tarski is in the Library""Logic by Suppes is in the Library"be represented by the symbols rand srespectively.

Then r, s = 'Logic' by Tarski and Suppesare in the Library.

r. vs = Logic by Tarski or Suppes is inin the Library.

r:::> s = If Logic by Tar ski is in theLibrary then Logic by Suppesis in the Library.

r--.-- s = Logic by Suppes is not in theLibrary.

Vol 16 No 3 &: 4 Sept &: Dec 1969

(2) Let the following sentences or statementsor propositions

"The acquisition Section communicateswith circulation Section"

and

"The circulation section communicatewith acquisition Section" be representedby P and Q respectively. Then thefollowing statements will be symbolisedas follows.

Acquisition and c ir cula ti on do notcommunicate with each other P. Q

Acquisition and c i r cul a ti on do notcommunicate with each other .• < "'--'P.,",-,Q

Acquisition transmit records tocirculation but circulation doesnot reciprocate

['But' is a specific and stronger conjunc-tion falling into the broader class of AND]

Neither Acquisition Nor circulationfail to communicate with eachother ... ('0'( ~ r. ".....Q)

[It is not the case that both acquisitionand circulation do not communicate witheach other]

Circulation can communicate withAcquisition on being informed byAcquisition P~ Q

Acquisition and circulation cancommunicate with each otheronly and only on a mutual basis P :: Q

Some other sentences are also symbolised.

(1 ) It is easier to train a Librarian in pro-gramming than it is to train a programmerin Librarianship ~ It is easy to train aLibrarian in programming but it is noteasy to train a programmer in Librarian-ship ~ It ..•....•.• programing ANDit ...•...•••. Librarianship => P. Q•••• (6)

(2) This is not to say that Books cannot behoth a container of information or an artiifaartifact of civilisation.~It is not truethat books cannot be both a container ofinformation or an artifact of ciViilisation

~""",,(,,",PV, •........Q) •••••• (7)

(3) If a library operation is not large eno}lgh1

to justify having A COMPUTER AT HAND

133

CHAKRABORTY

THEN THE USE OF SERYICE BUREAU IS2

A LOGICAL choice and few of them are----------3

especially if information retrieval is not4

involved.

Now symbolising l , 2, 3, 4, v v P,Q,R, S,

We have

IF NOT P THEN Q AND R(ESPECIALLY) IF xo r S

Reorganising according to the real intentit becomes

IF (......,p AND R AND NOT S) THEN Q

~(.-P. R."'-"S)::>Q ..•... (8)

(4 It is recognised that Library administra-tors will be the ones who ultimatelv decidewhether or not changes will be made in theexisting sv s terrr, but it should not beforgotten that it is the line stuff who willeither make or break the new systemthrough their attitude toward it and theirabHity to work with it.

Reformulating it in a more stiltedform it becomes, (It is true that) ~£):~i_~arv:

administrators decide THEN .ch'~~~~_illJ?~P

made in the e~isting system OF. Changes willQ

not be made_ in the syste~ ('or! in the cxc lu •.)'''--'''Q

s ive sense) AND [i t is true) IF the line s~u£f~

c:a:.:t:.:t1",-t:.:u",d=-e=-,p",e",r",t::.:a=-l~·:nc.::s=-:t;;:o,-=it AN D ~i th ey are wo r k edS L

with it THEN they will make a ~stem ORM

they will not make a system.M

:=} IF P THEN QY /"VQ AND (IF SS AND IF L) THEN (M OR "-'LM)(P:> (CY---Q) ). ::;(S. L) (MY""M»

•••... ~9)

Quine explains very clearly thenecessity of formalisation when he say!! thatlithe premisses and conclusions may treat ofany topics and are couched to begin with. inordinary language rather than in technicalIdeog r aphv of modern Logic. It is as an aid

134

to establishing implications that we thenproceed to mutilate and distort the statementsintroducing schematic letters in order to bringout relevant skeletal structures and t r a n s la tingvaried wo r ds into a few fixed symbols suchas 'and' in order to gain a manageable economyof structural elements, The task of tn u ssuitably paraphrasing a statement and isolatingthe relevant structure is just as essential tothe application of logic •.....•.• (P. 40).

~~mputer Language and SymbolicLang~~ge_~i P. ~.

Similarly translations can be madefrom different formal Languages like theprogramming language into the sylubolic lan-guage of P.C. This is because the symboliclanguage of P. C. itself with the minimum ofconnectives (i. e. the language of the BooleanAlgebra) is used as a base of computer lan-guages and pr og r arnrni ng languages. Thus anIf statement in FORTRAN like

IF (X-A) 10, 20,

can be translated in P. C. as (•.......p') 10).(P:> 20)

Where p;: X-A;: 0 10 ;: Statement numberedi o.

20 = Statement numbered20.

I"'-/p = X-A t- 0

Similarly it may be possible to investigatewhether formal language statement in U.D.C.or 'Colon I can be translated into the languageof Propositional Language.

Truth functions and T ruth Tables

The concept of truth function and truthtable has been used ill many fields. This canbe used also for rational construction of flowcharts. We have told before that the truth andfalsity of compound propositions depend on thetruth and falsity of the constituent propositions.In the language of mathematics then irc a n besaid that the truth value of a compound pro-position is the truth function of its constituentprepositions. A tabular way by which thesetruth function and t ruth dependence of theconstituent p r opo s iti on a a nd compound proposi-tions rnav be. represented are called the' truthtable.

Ann Lib Sci Doc

FORMALISING LIBRARY PROBLEMS BY SYMBOLIC LOGIC

T ruth Table and Flow chart

The flow charting of a number ofoperations or simply the algorithming fordoing a particular job can be rationalised iflogical analysis is made with the help of truthtable. A simple statement of a trivial libraryproblem has been presented here to illustratethis. The problem is stated as follows:

"A library having 100,000 books has tosend a list of books that are written by themodern American writers or the books thatare written by modern Negro writers to someclient. The library has an automated system(punched card system or computer) and infor-mation like author, author's nationality, title,date of publication etc. has been recorded foreach book. A list of all relevant books are tobe made. The problem is how to write mostrationally a flow chart for making thatlist. "

This problem. may be reformulated asfollows:

"To list the names of books that are writteneither by the Modern American writers or bythe m.odern Negro writers. n

If a books is written by a modernAmerican writer or If a book is writtenby a modern Negro writer then itsname is recorded in the list. -

~ If (p or q) then r~p V q) ) r ,

That means 'r' is true only when p v q is true.So whenever we will check from our recordswhether p vq are true we are to check alwaysfirst for p and then for q. We may try to findalso whether any shortcut is possible or not.Now if we see the disjunction truth tablewewill see that in case of horizontal colum~8land 2 if we find that p is true then we nolonger need the truth value of q for checking,since whenever p is true then p vq is trueirrespective of the fact whether q is true ornot. On the other hand in the 3.rd column evenif p is false p v q is together true. Therefore,in that q has to be searched for truth. On theother hand p v q is false only when q is fals e.In these two cases the truth value of p v qdepends only on the truth value of q. There-fore, our instruction will be first to find allthe cards where p is true; then we have tofind out those cards where p is false and outof these cards to find out the cards where qis true and to reject the fa l.se q ca re s ,

The whole proces s can be flow charted as follows:-

False

True

Cases 1, 2

Case 4

FLOW CHART OF (p v q) ::>r.

Voll6 No 3 8. 4 Sept 8. Dee !969 135

CHAKRABORTY

Now if we had n'f checked the basic truth table ourflow chart would have looked like the following:-

(l)

False

We can easily see that here are threedecision boxes instead of two in the previousflow chart. Machine time is shortened due tothe e1emination of the 2nd decision box. More-over, in the second box when q was false wewere returning those cards where p was h'-ue,on the other hand our l-i st should have recordsin all cases when p is true. This is definitelya wastage. A si~ilar such analysis can bemade for the expression [(p v q). (r v all /1or CKApqArsl

This expression means that a list isto be made whenever [pvq ), (rvs) or KApqArs istrue. A truth table is first constructed foranalysis. This truth table has been builtin Poli.h and the advarita ge of their use indrawing the flow chart will be clear.

Formula Conversion and FormulaReduction.

According to the basic tautologies orrules in Propositional Calculus most of theconnectors can be converted to a few basic

136

True True

False

True

connectors by suitable manipulation or as isusually said strings can be formed' in thelanguages of logic. Thus, sentences withimplication and equivalence connectivesmay be converted to sentences with only twoor three basic connectives like Negation,disjunction and conjunction. They can furtherbe reduced to only two i, e. either Negation(not) and disjunction (or) or Negation andConjunction which is nothing but the languageof the Boolean Algebra which the computer -understands. Thus, it might be possible toremove all 'if. ••..•..•.• then' or equivalenceconnectives in an ordinary language statementand to replace them by the 'not', 'and', 'or'connectives.

Propositional calculus provides us withrules by which smaller formulas can beexpanded and bigger formulas reduced. Thishas got an important application in rationalis-ing the flow diagram.

Thus an expression like (p vtp. q. )):;.1may be reduced to by a combined method

Ann Lib Sci Doc

FORMALISING LIBRARY PROBLEMS BY SYMBOLIC LOGIC

of conversion and truth-table analysis to( pvq) I which means significant simpli-cation in flow charting. This type of mani-pulation may be used for a very interestingaspect of ordinary language data processingin the library i. e., the weeding out of super-fluous sentences or propositions in an argu-ment formula. This may especially help tofind out from a set of instructions in arna nua l' or administrative rule book theunnecessary and extra instructions whichdo not serve the real purpose of the instruc-tions. This reduction method may then be usedas an efficient tool for rationalizing adminis-trative manuals. To give a taste of such typeof problems we can start with part of probableadr-u ni a rr a tiv e directive of a Librarian.

Minimisation of directives in libraryadministration.

In an human operated system the superfluitydoes not count much as the human brain canprocess and distil the really necessary andrelevant instructions and can operateeconomically with them. A relatively blunterman can go on following the rules strictlyand since human labour is not so costly insome countries we can afford to bear the costof an uneconomic set of instructions. Butthe problem really takes an enormous magni-tude when in a big organisation like the armyor a gigantic industrial complex a huge setof operations are to be performed mosteffeciently, most quickly, and where anenormous arnount of manual making isinvolved. Here then all superfluities are to beweeded out,and only a streamlined set ofinstructions. will be most effective in emer-gencysituations. The problem is aggravatedwhen machines are involved. Every stepsh ou.Id be specified for the machine and anyextra step due to cumbrous instructions orjumbled up set of rules will increase themachine-time thus increasing the cost factor.Any tool for the rationalising or correctinghastily formed manuals will then be ofinterest to the librarian standing on thethreshold of automation. Logic providessuch an analytical tool and to give a taste ofit the famous "Commissar and the SamovarIa cto r v " problem will be restated in a lan-guage nearer to the librarian.

Vol 16 No 3 &. 4 Sept &. Dec 1969

The librarian of a big library issuesthe following directives:

1) If a borrower keeps a book for more thentwo weeks then he must forthwith returnit.

2) If the borrower does not keep a book formore than two weeks then neither he mustreturn it forthwith nor he gets a reminder.

3) Eith.er he does not get a reminder or hedoes not keep the book for more than twoweeks.

After usual symbolisation of thecompound propositions and their furthercompounding the expression can be convertedto an expression with only negation, dis-junction and conjunction signs and by suitablemanipulation and conversion back to theimplication formula it can be proved that onlythe following directives were necessary:

)) If the borrower keeps a book for morethan two weeks then he must forthwithre tu r n it.

2) If the borrower returns the book forthwiththen he does not get a reminder.

Logical Possibilities and Network ofLibrary Operations.

When several statements are consi-dered together then there is a problem ofarranging them in hierarchical order asaccording to their importance, order of per-formance etc. In short each proposition ismade in relation to certain logical possibilitythe relative weights of the logical possibilitiesof different propositions are necessary fordetermination of the optimum path (with lesserrisk) in different types of communication net-works. In order to show logical independenceor dependence of different statements in acompound sentence such an analysis isnecessary. Tree diagrams are very usefulin such analysis of logical possibilities.Trees can be suitably utilised for visualisingthe steps in a mechanical classi!ictorydevice. Whenver, while performing a numberof operations in a time sequence, a decision isto be made at any step for determining theoptimal path towards a goal (so called 'mazeproblem ') then tree diagrams can be madewith p r cf it. It is reasonably hoped that. in alibra.ry such type of operations in a time asequence in a .hierarchical order are per-

137