McColl and Minimization

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McColl and MinimizationFrank Markham Browna

a Department of Electrical and Computer Engineering, Air Force Institute of Technology, Dayton,Ohio, USA

Online publication date: 27 October 2010

To cite this Article Markham Brown, Frank(2010) 'McColl and Minimization', History and Philosophy of Logic, 31: 4, 337— 348To link to this Article: DOI: 10.1080/01445340.2010.517387URL: http://dx.doi.org/10.1080/01445340.2010.517387

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McColl and Minimization

FRANK MARKHAM BROWN

Department of Electrical and Computer Engineering, Air Force Institute of Technology,Dayton, Ohio, USA

Received 27 January 2010 Revised 24 July 2010 Accepted 14 August 2010

In 1952, Quine showed that the problem of reducing a propositional formula to a simplest normalequivalent can be solved in two steps, viz., (i) express the given formula, F, equivalently as the disjunctionof all its prime implicants, and (ii) find all non-redundant disjunctions of the latter that are equivalent to F(Quine 1952). However, it seems not generally known that an ingenious form of the same two-step processwas published by Hugh McColl in 1878.

1. Introduction

In 1877, motivated by problems concerning the limits of multiple integrals, andnot having read Boole or Jevons, Hugh McColl (1837–1909) created the first calculusof propositions (McColl 1877a).1 Curiously, McColl did not seem initially toconsider his calculus applicable to logic.

Reading McColl’s early work, one is struck by the aptness of his notation and bythe number of truth-functional properties he discovered. Bochenski (1961, p. 309)writes, ‘In a way [McColl 1877b] marks the highest level of mathematical logic beforeFrege’. Peckhaus (1998, p. 17) notes that ‘some of the particulars of his ‘‘Calculus ofEquivalent Statements’’ are regarded as ingenious anticipations of innovations whichwere only much later introduced to logic’.

Seemingly unremarked among such anticipations is one concerning theminimization of truth-formulas. Quine (1952) solved a problem which, he wrote,‘has proved curiously stubborn; viz., the problem of devising a general mechanicalprocedure for reducing any formula to its simplest equivalent’. Quine’s procedurecomprises two steps: (1) express the given formula, F, equivalently as the disjunctionof all its prime implicants, and (2) find all primitive forms2 (non-redundantdisjunctions of the prime implicants) equivalent to F.

The objective of this article is to show that in 1878 Hugh McColl published aningenious form of the same two-step process. To that end, we discuss the syllogisticforms3 equivalent to F (disjunctive formulas comprising among their clauses allprime implicants of F); we review Quine’s well-known minimization procedure; and

1 ‘McColl’ is the spelling used in the works of interest for this paper. After April, 1885, he began spelling his name

‘MacColl’. See Astroh, Grattan-Guinness, and Read (2001) for a survey of McColl’s life. For an overview of his logical

work, together with a reprinting of his main publications in logic, see Rahman and Redmond (2007).2 McColl’s terminology.3 Blake’s theory of syllogistic forms (1937), which anticipates both Quine’s concept of prime implicant and all of the main

methods for generating prime implicants, seems little known.

HISTORY AND PHILOSOPHY OF LOGIC, 31 (November 2010), 337–348

History and Philosophy of Logic ISSN 0144-5340 print/ISSN 1464-5149 online ª 2010 Taylor & Francishttp://www.informaworld.com DOI: 10.1080/01445340.2010.517387

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we outline the initial form of McColl’s calculus. Last, we analyze McColl’s laconicstatement of his minimization procedure.

2. Syllogistic forms and prime implicants

Following Quine (1952), we say that the formulas of the propositional calculusare to be built up of the statement letters ‘p’, ‘q’, ‘r’, . . . by the notations of negation,conjunction, and disjunction, viz., ‘�p’, ‘pq’, and ‘p_ q’, to any degree of iteration.Letters and negations of letters will be spoken of collectively as literals. A singleliteral, or a conjunction of literals in which no letter appears twice, will be called aclause (in McColl’s notation, a term).

Any alternation of clauses will be called a normal formula. Such a formula will besaid to be developed if all of its letters appear in each of its clauses, e.g. pq�r_ p�qr.A normal formula will be called irredundant if it has no superfluous clauses and noneof its clauses has a superfluous literal. A clause, f, will be called a prime implicant of aformula F if f implies F and f ceases to do so if any of its literals are deleted.

2.1. Syllogistic formsBlake (1937) presented a generalized and comprehensive treatment of Boolean

formulas involving prime implicants. Translating from Blake’s Boolean terminology,we say that a normal form F1 formally implies a normal form F2 in case every clauseof F1 implies (i.e. has all the literals of) some clause of F2. Thus, p�q_ �p�qr formallyimplies p�q_ �pr. A normal form will be called syllogistic if every normal form thatimplies it also formally implies it. The formula p�q_ �pr, for example, is not syllogistic(�qrs implies it but does not formally imply it); however, p�q_ �pr_ �qr is syllogistic.

Suppose clauses f1 and f2, belonging to a normal formula F, are such that f1

implies f2. If f1 is deleted, the resulting normal formula remains equivalent to F; wesay in this case that f1 is absorbed by f2, and denote by absorption the process ofremoving f1. If all possible absorptions are carried out in any order on a normalformula F, the resulting normal formula, which we follow Blake in calling absorptive,is uniquely determined by F (Blake 1937, Theorem 8.5). A syllogistic form remainssyllogistic under absorption (Blake 1937, Theorem 9.6).

McColl’s approach to minimization is based on what we shall call themultiplication theorem, viz., if a conjunction of syllogistic forms is expanded bymeans of the distributive laws into normal form, the result is syllogistic (Blake 1937,Theorem 10.9).

2.2. The Blake canonical and primitive formsAn absorptive syllogistic formula equivalent to F is the disjunction of all of the

prime implicants of F (Brown 2003, Theorem A.4.1). We call this formula (unique towithin the order of its clauses) the Blake canonical form4 for F and symbolize it byBCF(F). Although absorptive and syllogistic, BCF(F) may not be irredundant. Theformula p�q_ �pr_ �qr, for example, is in Blake canonical form; however, �qr issuperfluous. An irredundant syllogistic formula equivalent to F will be called aprimitive form5 for F. Each clause of a primitive form for F is a clause of BCF(F).

4 Blake (1937, p. 26) calls this the ‘simplified’ canonical form.5 The term ‘primitive form’ is due to McColl (1878b, p. 18).

338 Frank Markham Brown


Blake showed that BCF(F) can be constructed from a normal form, F, in twoefficient ways, which we shall call iterated consensus (Blake 1937, p. 32) andmultiplying(p. 36 ff.) The first was employed by Quine in two forms; the second by McColl.

2.2.1. Iterated consensus. If a normal formula contains two clauses such thatexactly one letter, x, appears negated in one clause and not negated in the other, thenthe consensus6 of the two clauses is a new clause comprising all the literals, except xand �x, in either of the original clauses, omitting literals made superfluous byrepetition.

The Blake canonical form for a normal formula F is attained, beginning with F ,by persisting in the the following steps until neither is applicable: (1) if the currentformula contains two clauses generating a consensus not implying a clause alreadypresent, adjoin the consensus; (2) if clause f1 in the current formula implies anotherclause, f2, in the formula, delete f1.

Example. Let F be defined by

F ¼ rs _ qr s _ p rs _ p qrs: ð1Þ

Typical stages in the development of BCF(F) by iterated consensus are shownbelow:

rs _ p rs _ qr s _ p qrs

rs _ p rs _ p qrs _ qs

rs _ p rs _ qs _ p qr

rs _ p rs _ qs _ p qr _ pqr

The consensus of r�s and q�r �s in the first formula is q�s; thus q�s is adjoined in thesecond, absorbing q�r �s. The consensus of r�s and �p�qrs in the second formula is �p�qr,which absorbs �p�qrs. The consensus of �p�rs and q�s in the third formula is �pq�r. Finally,the consensus of �p�rs and �p�qr in the fourth formula is �p�qs, yielding

BCFðFÞ ¼ rs _ p rs _ qs _ p qr _ pqr _ p qs: ð2Þ

Any further consensus-clause is absorbed by a clause already present inEquation (2).

2.2.2. Multiplying. A disjunction of literals is clearly syllogistic. Thus, recallingthe multiplication theorem, cf. section 2.1, BCF(F) is constructed as follows: (1)convert F to a conjunctive normal form; (2) expand the latter into normal form byapplication of the distributive laws; (3) delete absorbed clauses.

This ‘multiplying’ method – first announced by McColl (1878b, Rule 21) andproved by Blake (1937, Theorem 10.9) – has been rediscovered several times, notably

6 The term ‘consensus’ is due to Quine (1955, p. 627). The consensus of two clauses was called by Blake (1937, p. 25) their

‘syllogistic result.’

McColl and Minimization 339


by Samson and Mills (1954) and Nelson (1955). Blake (1937, p. 36) states, ‘thismethod was known to Peirce and his students’, and adds, ‘It is mentioned atseveral places in Studies in Logic, by members of the Johns Hopkins University,1883’.7

3. Quine’s approach to minimization

It was shown by Quine (1952, Theorem 1) that under ‘reasonable standards ofsimplicity’, a simplest normal equivalent of a normal formula F is a primitive formfor F. Accordingly, both Quine and McColl construct a simplest normal equivalentfor F in two stages: first, reduce F to its Blake canonical form; second, use the latterformula to find the primitive forms for F. (Subsequent selection of a simplestformula among the primitive forms is assumed.)

3.1. Construction of BCF(F)3.1.1. Quine’s first method. Quine employs two forms of iterated consensus toconstruct BCF(F). His first method, given in (Quine 1952), requires F to be indeveloped normal form, f1_ . . . _fn, at the outset. The prime implicants of F areproduced by the following mechanical routine:

We make a growing list which [. . .] begins with f1, . . . ,fn and is extendedaccording to the following principle: whenever two entries can be found in the listwhich are related as wx and wx [. . .] add their common part w as a new entry in the

list. Check marks are to be applied to any entries wx and wx which thusgenerate new entries, but a check mark is not to be treated as disqualifying anentry from reuse [. . .]. When the list has been extended as far as possible [theprime implicants of F] are the entries which bear no check marks (Quine 1952,p. 525).

Example. Let us consider again the example formula F defined by Equation (1), viz.

rs _ qr s _ p rs _ p qrs:

The format shown in Table 1, suggested in McCluskey 1956, organizes Quine’smethod so that stages of consensus generation alternate in a fixed way with stages ofabsorption. The clauses of the developed normal form of F, each containing nliterals, are written in column 1. The clauses derived by consensus from column 1(containing n7 1 literals each) are written in column 2, after which the clauses incolumn 1 absorbed by clauses in column 2 are checked, indicating deletion. Thisprocess is carried out repeatedly; each column generates the succeeding column byconsensus, after which it may suffer absorption of some of its clauses by thesucceeding column. The clauses surviving unchecked (i.e. unabsorbed) are those ofBCF(F).

7 The only such mention we have found is one by Christine Ladd (1883, p. 23), citing (McColl 1878b): ‘If the reduction [of

an expression to simplest form] is not evident, it my be facilitated by taking the negative of the expression, reducing it,

and then restoring it to the positive form.’



The operations in each pass are on clauses of fixed length, and all operations(consensus and absorption) derive from the single rule wx _ wx ¼ w, where x is a singleletter and w is a clause. Thus the work involves simple operations organized inconvenient stages. Quine’s method is made simpler by grouping the clauses of thedeveloped normal form in column 1, as shown in Table 1, according to the number oftheir negated literals. The consensus operation is then possible only between clausesin vertically adjacent groups; this property is inherited by all subsequent columns, asshown in Table 1. Further simplification results from introducing binary notation(McCluskey 1956).

3.1.2. Quine’s second method. The computational advantages of Quine’s methodare offset by the necessity to begin with the developed form of F, and by theconsequent need to process a large number of clauses. In a subsequent paper, Quine(1955) announced ‘a speedy and direct method’, namely, the method of iteratedconsensus discussed above (section 2.2.1), which begins with any normal form. (In anote added in proof, Quine (1955, p. 627) states that the new procedure wasanticipated by Samson and Mills (1954). As noted in Brown 1968, however, iteratedconsensus was discussed extensively by Blake (1937, p. 32).)

3.1.3. Extraction of primitive forms. To find all primitive forms equivalent to F,Quine (1952) employs a table of prime implicants, Table 2.

Table 1. Organization of work for Quine’s method.

�pqr�s

��p�qrs �qr�s��pq�rs �pq�s

��pqr�s �pr�s�pq�r �s�p�qr�s

��p�q�ss �p�qs q�s

��p�qr�s �p�qr r�s

��pq�r �s �p�rs�pq�r��pr�s

��pq�s

�q�r �s��qr�s

Table 2. Table of prime implicants.

1 2 3 4 5 6 7 8 9

�p�q�rs �p�qrs �p�qr�s �pq�r�s �pq�rs �pqr�s pq�r�s pqr�s p�qr�s

q�s 6 6 6 6r�s 6 6 6 6�p�qr 6 6�p�qs 6 6�pq�r 6 6�p�rs 6 6



The rows of this table correspond to the clauses, c1, . . . ,cm, of BCF(F); thecolumns correspond to the clauses, f1, . . . ,fn, of the developed normal form for F.A cross is placed at the intersection of row i and column j in case fj implies ci. Asubset of {c1, . . . ,cm} is a cover of F if and only if the rows defined by the subsetproduce a cross in each column of the table. A minimal cover of F is a cover of Fnone of whose proper subsets is a cover of F. A primitive form for F is thedisjunction of the elements of a minimal cover of F.

Finding minimal covers using a table of prime implicants is simplified by thefollowing rule: if column ci has all the crosses of column cj, then cj is said to dominateci , and ci may be deleted. In Table 2, column 7 dominates columns 4, 6, and 8; also,column 9 dominates columns 3, 6, and 8. Thus columns 3, 4, 6, and 8 may be deleted.Table 3 shows the reduced result.

From Table 3, we infer that any minimal cover of F contains at least one elementfrom each of the following sets: {q�s}, {r�s}, {�p�qs, �p�rs}, {�p�qr, �p�qs}, and {�pq�r, �p�rs}. Thereare three minimal covers in this case, to which correspond the three possible primitiveforms:

qs _ rs _ p qs _ pqr

qs _ rs _ p qs _ p rs

qs _ rs _ p rs _ p qr

ð3Þ

Quine provides no means other than inspection for arriving at the primitive formsimplied by a reduced table. Algorithms for doing so were given by Petrick (1956) andMcCluskey (1956).

4. McColl’s calculus

Before discussing Hugh McColl’s approach to minimization, it will be useful tooutline his ‘symbolical or abbreviated language’, which appeared in The EducationalTimes of London (McColl 1877a).8 McColl’s language is a propositional calculustreated as a Boolean algebra:

DEF. 1.—The symbols A, B, C, &c. denote statements (or propositions) registeredfor convenience in a table. The equation A¼ 1 asserts that the statement A is true;the equation A¼ 0 asserts that the statement A is false; the equation A¼B assertsthat A and B are equivalent statements.DEF. 2.—The symbol A6B6C or ABC denotes a compound statement, inwhich A, B, C may be called the factors. The equation ABC¼ 1 asserts that allthree statements are true; the equation ABC¼ 0 asserts that all three statementsare not true, i.e. that at least one of the three is false. [. . .]DEF. 3.—The symbol AþBþC denotes an indeterminate statement of which A,B, C may be called the terms. The equation AþBþC¼ 0 asserts that all the threestatements are false; the equation AþBþC¼ 1 asserts that all the three are notfalse, i.e. that at least one of the three is true. [. . .]DEF. 4.—The symbol A0 is the denial of the statement A. The two state-ments A and A0 are so related that they satisfy the two equations AA0 ¼ 0 andAþA0 ¼ 1 . [. . .] The same symbol (i.e., a dash) [McColl’s term for a prime] will

8 See Tattersall (2009) for a survey of McColl’s contributions to the Educational Times.



convert any complex statement into its denial. For example, (AB)0 is the denial ofthe compound statement AB.

Additional definitions and several rules, e.g. De Morgan’s rules for negation,follow. The remainder of McColl 1877a applies McColl’s language to ‘determine thenew limits of integration which arise when we change the order of integration or thevariables in a multiple integral’.

Renamed ‘the calculus of equivalent statements’, McColl’s system is elaboratedand applied in a series of seven papers in the Proceedings of the London MathematicalSociety, of which the first three are of interest here. The first, McColl l877b, anexpanded version of McColl l877a, includes:

RULE 3. (AB)0 ¼AB0 þA0BþA0B0

RULE 4. (AþB)0 ¼A0B0; (AþBþC)0 ¼A0B0C0, and so on.

The second paper, McColl 1878a, belonging ‘more strictly to the province of logicthan to that of mathematics’, adds implication to McColl’s calculus:

DEF. 12. – The symbol A : B (which may be called an implication) asserts that thestatement A implies B; or that whenever A is true B is also true.

Note.—It is evident that the implication A : B and the equation A¼AB areequivalent statements.

and includes an additional rule of present interest:

RULE 9.—As an extension of Rule 3 we give the following:—(ABCD . . . )0 ¼A0 þB0 þC0 þD0þ . . .

McColl had by this time seen an outline of Boole’s (1854) Laws of Thought inBain’s Logic (1870, pp. 226–244). Bain includes a statement, with solution, of one ofBoole’s problems (Boole 1854, p. 118), omitting Boole’s method of solution. McCollshowed how his own implication rules can be used to solve the problem andcorrected mistakes in Bain’s discussion of it. In this paper, McColl 1878a, he alsoapplied his calculus to the syllogism.

In the third paper, McColl 1878b, in this series, he noted that he had now beenlent a copy of Boole’s Laws of Thought. This paper begins with a remarkable list ofidentities in his expanded language, among which the fifth,

Table 3. Reduced table of prime implicants.

1 2 5 7 9

�p�q�rs �p�qrs �pq�rs pq�r�s p�qr�sq�s 6r�s 6�p�qr 6�p�qs 6 6�pp�r 6�p�rs 6 6



ðaþ AÞðaþ BÞðaþ CÞ . . . ¼ aþ ABC . . . ; ð4Þ

is applied by McColl to reduce the labor of minimization9 (see the example insection 5.2).

5. McColl’s approach to minimization

McColl makes no attempt to justify his minimization procedure, other than tosupply some examples. We verify it by showing that he carried out the same two stepsas Quine, albeit in a different and ingenious way10. To minimize F, that is, heconstructs BCF(F) and uses it to find all primitive forms for F.

5.1. Removal of redundant termsAn important part of the method in McColl 1878b is the deletion of superfluous

clauses, using the following rules:

RULE 20. – To discover the redundant terms of any indeterminate statement.

(a) These redundant terms are easily detected by mere inspection when theyimply (or are multiples of) single co-terms, as in the expression

a0bgþ a0bþ abg0 þ bg0:

But when they do not imply single co-terms, but the sum of two or more co-terms, they cannot generally be thus detected by inspection. They can always,however, be discovered by the following rule, which includes all cases:—

(b) Any term of an indeterminate statement may be omitted as redundant whenthis term, multiplied by the denial of the sum of all its co-terms gives theproduct zero; if the product is not zero, the term cannot be cancelled asredundant. [We introduce the headings ‘(a)’ and ‘(b)’ in order to distinguishthese essentially different rules.]

RULE 22. – Let f(x) denote any unconditional statement involving x. Thenxf(x)¼ xf(1) and x0f(x)¼ x0f(0).

Although not stated by McColl explicitly, it is clear that Rule 20a, easily carriedout by inspection, is to be applied before 20b. McColl detects redundancies in thestatement

CD0 þ C0Dþ B0C0 þ B0D0 ð5Þ

by combining Rules 20b and 22:

9 Identity (4) is implied by Proposition II of Boole (1854, Chapter IX), applicable to interpretable formulas having

positive coefficients. In summary, translating from Boole’s polynomial algebra to McColl’s calculus: Let p, q, and r be

clauses such that r is absorbed by (has all the literals of) q. Then (pþ qþA)(pþ rþB)¼ pþ rþ (qþA)B. Boole (1854,

p. 131) gives the following example: (xþ yþ z)(xþ ywþ t)¼ xþ ywþ (yþ z)(t).10 McColl does not, for example, require a preliminary transformation to developed form.



Note.—The process for discovering redundant terms may be much abbreviatedby means of Rule 22. Thus, CD0 (C0DþB0C0 þB0D0)0 ¼CD0(0þ 0þB0)0 ¼BCD0,and, again, B0D0 (CD0 þC0DþB0C0)0 ¼B0D0(Cþ 0þC0)0 ¼ 0.

We summarize his extended analysis of redundant terms in Equation (5) using thefollowing format:

CD0 þ C0D þ B0C0 þ B0D0

CD0 ð 0 þ 0 þ B0 Þ0 ¼ BCD0

C0D ð 0 þ B0 þ 0 Þ0 ¼ BC0D

B0C0 ð 0 þ D þ D0 Þ0 ¼ 0

B0D0 ð C þ 0 þ C0 Þ0 ¼ 0

ð6Þ

Thus, either B0C0 or B0D0 may be deleted. If B0C0 is deleted, we see as follows thatB0D0 may not be deleted:

B0D0ðCD0 þ C0DÞ0 ¼ B0D0ðCþ 0Þ0 ¼ B0C0D0:

Likewise, if B0D0 is deleted, B0C0 may not be deleted. The given statement hastherefore two primitive forms, namely, CD0 þC0DþB0D0 and CD0 þC0DþB0C0.

5.2. Finding the primitive formsThe following is McColl’s spare description (1878b, p. 18) of his method for

minimization:

RULE 21. – To reduce an indeterminate statement to its primitive form:—Developthe denial of the statement by the formula

ðAþ Bþ C . . .Þ0 ¼ A0B0C0 . . . ðsee Rule 4Þ;

multiply together the indeterminate factors that the application of the formula

ðABC . . .Þ0 ¼ A0 þ B0 þ C0 þ � � � ðsee Rule 9Þ

will give rise to; omit the redundant terms in the product (see Rule 20).Then develop the denial of this product, and go through the sameprocess as before. The result will be the primitive form of the original givenstatement.

When dissected, McColl’s Rule 21 reduces to eight steps. We illustrate these stepsby returning to the normal form F defined in Equation (1). Translating to McColl’snotation,

F ¼ rs0 þ qr0s0 þ p0r0sþ p0q0rs: ð7Þ

Example. Beginning with F, the steps implied by McColl’s Rule 21 are asfollows:



1. Deny. Using the De Morgan laws, express the negation of F in conjunctivenormal form:

ðr0 þ sÞðq0 þ rþ sÞðpþ rþ s0Þðpþ qþ r0 þ s0Þ

2. Develop. Use the distributive laws, augmented by McColl’s identity (4), toachieve a disjunctive normal form:

ðsþ q0r0Þðpþ s0 þ qrÞpsþ qrsþ pq0r0 þ q0r0s0

3. Apply rule 20a. Delete absorbed terms:

psþ qrsþ pq0r0 þ q0r0s0 ðno changeÞ

4. Apply rule 20b. Delete other superfluous terms:

psþ qrsþ q0r0s0

‘and go through the same process as before’

5. Deny. Using the De Morgan laws, express the negation of the current formulain conjunctive normal form:

ðp0 þ s0Þðq0 þ r0 þ s0Þðqþ rþ sÞ

6. Develop. Use the distributive laws, augmented by McColl’s identity (4), toachieve a disjunctive normal form:

ðs0 þ p0q0 þ p0r0Þðqþ rþ sÞqs0 þ rs0 þ p0q0rþ p0q0sþ p0qr0 þ p0r0s

7. Apply rule 20a. Delete absorbed terms. The result is BCF(F):

qs0 þ rs0 þ p0q0rþ p0q0sþ p0qr0 þ p0r0s ðno changeÞ

8. Apply rule 20b. Delete other superfluous terms in all possible ways. The resultsare the primitive forms for F, which agree with those (3) produced by Quine’smethod.

qs0 þ rs0 þ p0q0sþ p0r0s

qs0 þ rs0 þ p0q0sþ p0qr0

qs0 þ rs0 þ p0r0sþ p0q0r

Steps 3 and 7 (deletion of absorbed terms) produce no change in the presentexample because identity (4), applied in the preceding steps, happens in both cases toavert the production of terms subsequently absorbed. Steps 3 and 4 may reducesubsequent labor, but are not required.

Step 8 requires further discussion. McColl (1878b, Rule 20b) shows how todetermine the superfluous clauses in a normal formula, and provides an example (5)of a formula possessing two primitive forms. It would be useful in more complexcases to have a systematic procedure for finding all sets of deletions leading toprimitive forms. One such procedure is to restrict sequences of deletions to those in



which the successor of a deleted clause, f, occurs later than f in the sequence ofclauses expressing BCF(F). Candidates for deletion, at any stage, are found usingtables of the form (6). Such a table, for the present example, is shown below.

qs0 þ rs0 þ p0q0r þ p0q0s þ p0qr0 þ p0r0s

qs0 ð r þ 0 þ 0 þ p0r0 þ 0 Þ0 ¼ pqr0s0

rs0 ðq þ p0q0 þ 0 þ 0 þ 0 Þ0 ¼ pq0rs0

p0q0r ð0 þ s0 þ s þ 0 þ 0 Þ0 ¼ 0p0q0s ð0 þ 0 þ r þ 0 þ r0 Þ0 ¼ 0p0qr0 ðs0 þ 0 þ 0 þ 0 þ s Þ0 ¼ 0p0r0s ð0 þ 0 þ 0 þ q0 þ q Þ0 ¼ 0:

ð8Þ

The first superfluous clause in Equation (8) is p0q0r; thus a partial deletionsequence is

ðp0q0rÞ: ð9Þ

Removing p0q0 r from BCF(F) leads to the next stage:

qs0 þ rs0 þ p0q0s þ p0qr0 þ p0r0s

p0q0s ð0 þ 0 þ 0 þ r0 Þ0 ¼ p0q0rsp0qr0 ðs0 þ 0 þ 0 þ s Þ0 ¼ 0p0r0s ð0 þ 0 þ q0 þ q Þ0 ¼ 0:

ð10Þ

The extended deletion sequence,

ðp0q0r; p0qr0Þ: ð11Þ

leads to the reduced stage

qs0 þ rs0 þ p0q0s þ p0r0s

p0r0sð0 þ 0 þ q0 Þ0 ¼ p0qr0s:ð12Þ

Thus (11) cannot be extended. Returning to stage (10), we see that (9) can beextended as

ðp0q0r; p0r0sÞ: ð13Þ

The latter sequence cannot be extended. Returning to stage (8) and proceeding in thesame way, exactly one further sequence of deletions can be formed:

ðp0q0s; p0qr0Þ: ð14Þ

Deletion sequences (11), (13), and (14) lead to the primitive forms shown in Step 8above.



6. Conclusion

Analysis of the minimization method in McColl (1878b) shows it to be aningenious variant of the two-step mechanical procedure given in Quine 1952 forsimplifying truth-functional formulas.

References

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