An Extended Procedure in Quantificational Logic

An Extended Procedure in Quantificational LogicAuthor(s): Robert StanleySource: The Journal of Symbolic Logic, Vol. 18, No. 2 (Jun., 1953), pp. 97-104Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268939 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 18, Number 2, June 1953

AN EXTENDED PROCEDURE IN QUANTIFICATIONAL LOGIC1

ROBERT STANLEY

1. Object. The aim of this paper is to present a decision procedure which seems to be as easy to use as other available procedures in quantification theory, but which is considerably stronger than the others, providing a mechanical test for a sub-species of polyadic validity which is very much broader than monadic validity.2 Of course, a test for polyadic validity in general is out of the question, but the present test's limits, short of polyadic validity, are not known. That is, of all the polyadically valid schemata which have been tested, none has failed to yield decisions under this method. The material thus examined includes schemata corresponding to the cases of all absolute metatheorems in ML's3 Chapters II and III, and five polyadic samples4 from MeL. The handiness of this test is illustrated by sample applications in ? 3 below.

2. Terminology. The logical symbolism used here follows much the same familiar pattern as that of MeL. Thus 'p' through 't' represent, or stand in place of, statements, and 'F', 'G', 'H', 'J', and 'K' similarly represent predicates.5 'ip', 'q', 'Gx', 'Hx', 'Hxyz', and so on, are schemata, and the results of putting schemata in the blanks of '( . )' or '( V )', or after I',

or after quantifiers such as '(x)', '(y)', and so on, are schemata in turn. Schemata of the last four kinds are conjuncts, disjuncts, negations, and quanti/ications, respectively. Then 'D', and other familiar propositional connectives are defined as usual, in terms of these three propositional symbols. Thus '(p D q)' abbreviates '(u.'p V q)', and '(p = q)' abbreviates '(p D q qD :P5)'. As usual, '(3x)', '(3y)' and the like abbreviate ' .'(x) A', '-(y)'-, and so on, correspondingly.

A quantifier-occurrence is said to cover every part of the schema-occurrence to which it is prefixed. A quantifier-occurrence binds its own variable-occurrence, and also any unbound occurrences of its variable which

Received June 11, 1951. 1 This procedure copies, essentially, a slightly weaker one which was presented in

A basis for logic in natural deduction, a thesis submitted toward the doctoral degree at Harvard University. I am indebted to Professor Quine for many helpful suggestions.

2 The monadically valid schemata, of course, including many polyadic cases besides the monadic ones, considerably outreach the valid, monadic schemata.

3 See the appended bibliography for the key to abbreviations of titles. 4 Those problems; namely, which are posed on pages 120, 121, 121, 179, and 185

of MeL, and proofs of which are treated on pages 175, 176, 176, 179, and 185, correspondingly.

5 'Predicate' is used here with the sense which is defined in section 23 of MeL, and section 4 of LQ.

97



98 ROBERT STANLEY

are covered by it, but by no other occurrence of the same quantifier in that quantification. When occurrences of a variable are not bound in a given schema, both the occurrences and that variable are called free in that schema. Schemata are variants just in case they have corresponding, though not necessarily identical, sets of bound variable-occurrences, while being alike in all other respects. In particular, any schema is thus a variant of itself.

The result of substituting one variable for another in a schema, for example 'x' for 'y', may be obtained as follows. Replace each free 'y'-occurrence in the given schema by an occurrence of 'x'. If the new 'x'-occurrences are all free, the new schema is the desired result. Otherwise, there is at least one new X'-occurrence which is bound by an occurrence of '(x)'. Replace the variable in all such complicating quantifiers by the alphabetically earliest variable present in none of the schemata they cover, and replace the 'x'- occurrences which were originally bound by the replaced quantifiers with occurrences of this new variable. The new schema is called the result of substituting 'x' for 'y' in the initial schema. For example, the result of substituting 'y' for 'z' in '(x)(y)Gxyz' is thus '(x)(z)Gxzy'.

Schema-occurrences which are linked by '.' to form a conjunct, or by 'V' to form a disjunct, are called clauses of that conjunct or disjunct. Trans- itively and reflexively, moreover, clauses of clauses of a schema are them- selves clauses of that schema, and any schema is a clause of itself. In other words, schema-occurrences which are neither overlaid by ', nor covered by any quantifier are clauses. A schema-occurrence which is contained in no larger schema-occurrence will be called a test-line. A schema is a con- tradiction just in case it is a conjunctive compound of various clauses, one of which is the negation of another's variant.

3. Procedure. The general method of the process at hand is one of reductio ad absurdum, in which one hypothesizes initially the negation of the schema in question, and then eliminates alternatives one by one, as they result from that assumption and are found to be contradictory. The cyclically-applied rules under which these consequences develop automatically may be set forth in some detail.

The primary steps, after the initial negative hypothesis, are taken under the propositional rules. These rules correspond to familiar principles of truth-functional analysis. They provide that one may transform clauses of test-lines by dropping initial occurrences of 'mu', and by distributing initial occurrences of 'a' through '.' and 'V' under DeMorgan's laws. Steps under these rules are thus no less intuitively obvious and acceptable than the principles, for example, which validate assigning of values in truth-table analysis. These propositional rules are to be applied repeatedly to the initial negation, as long as they have points of application.



AN EXTENDED PROCEDURE IN QUANTIFICATIONAL LOGIC 99

Next come steps of instantiation. A quantification or negated quantification is instantiated by a given variable when its foremost quantifier is dropped, and the given variable is then substituted in the resulting schema for the variable of that quantifier. Thus, when instantiated by 'y', the schema '(x)( .. .x. . .)' gives way to '( .. .y.. .)'; analogously, '-(x) (.. . ... ) '

yields .'(. . y. ..)'. Clauses which are quantifications may be instantiated validly without restriction, but clauses which are negated quantifications do not enjoy this freedom. For example, from the valid formula ',(x) (x _ y)' one could not validly infer ',.'(y y)', though '(3z),--(z = y)' follows.

Thus, where any clause of a test-line is a negated quantification, that quantification may be interpreted as failing to hold in that context, and as having, therefore, at least one counter-instance. In practice, such a situation is often signalled by instantiating with some noncommittal term, like 'a', which thus pretends to designate that unspecified entity for which the schema fails. In the present procedure a variable other than the free variables at hand will be used in such a case. The newness of the variable guarantees, as is essential, that there is no unfounded identifying of the counter-instance with any antecedently considered entity.

On the other hand, where a quantification occurs as a clause of a test-line, and is thus to be interpreted as holding, its main covered schema must hold for every instance whatever. In particular, then, this schema must hold for all entities referred to by free variables which have already appeared.

There are two vital points of order, as regards steps of instantiation, both of which stem from one fundamental consideration.

First, it is eminently desirable to make the weaker instantiations, of negated quantifications, namely, as early as possible. Quantification instantiations can "copy" validly the variables of previous steps, but instantiations of negated quantifications must not. Plainly, the more there is of such valid "copying," the better the chances for interrelating the results to obtain contradictions, and thus proofs. Where there are various instantiable clauses at hand, then, one should instantiate negations first.

Second, however, and more subtly, new negated quantifications may emerge as instantiable clauses, after various quantifiers have been peeled off in the course of instantiating quantification-occurrences. Thus, some quantifications are so constructed that if certain of their quantifiers are dropped, and the propositional rules are applied exhaustively to the prim- itive expansion, there results a schema some clause of which is a negated quantification. Such clauses will be called important, since they should be instantiated before other quantifications, though after negated quantifications, in order to give negated quantifications as much precedence as possible over quantifications.

The manner in which these waves of instantiations will lay open deeper and deeper levels of the schema at hand requires that there be a cyclical



100 ROBERT STANLEY

recapitulation of all previous stages, after each new one is carried out. For example, a propositional stage comes first. Next, one will discard any contradictory alternatives in the resulting test-line. Third will come instantiations of clauses which are negated quantifications. After these instantiations, however, the newly-revealed propositional structures must be broken up by a repetition of the propositional stage, then newly ex- traneous contradictory alternatives may be dropped in another second stage, and new negated quantifications may appear as clauses for another third stage, before one can proceed to the fourth level, of important instantiations, or further. Indeed, there may be any finite number of third stages before the fourth can be entered, and many fourth ones before the fifth, although the schemata actually met in practice involve very little complication of this sort. So it will go throughout, then, with cycles, cycles of cycles, and so on, until the meat of the schema in question is wholly consumed, leaving either an empty platter and a decision, or non- quantificational bones that resist further carving.

A process of the above sort, then, including also a device which guarantees consideration of all possible orders of attack on important clauses, is provided by the course and contents of the specific stages below. In order to test a given schema under the present procedure, carry out the following operation-cycles on its closure.6

(o) Replace all occurrences of particular quantifiers and propositional connective abbreviations by their definitional equivalents in terms of universal quantifiers, '.', 'V' and I'; replace '(3x)' and 'p -- q' for example, by '-.'(x)' and '(up V q) . (Isq V p)' respectively.

(oo) Carry out all possible steps of exportation7 on the result of (o). (ooo) Prefix I' to the result of (oo). In the steps that follow, where any clause of a test-line is unaffected by operations which are being carried out on other clauses of that test-line, simply duplicate the untouched material correspondingly, in the succeeding test-line. Wherever a clause of a test-line is reached, which clause is a quantification of a conjunct, distribute the quantifier through the conjunct before proceeding on. On the other hand, wherever such a clause is the negation of a conjunct in which quantification-occurrences are conjoined, "un-distribute" the quantifiers as follows. First obtain a variant of the right quantification which has the same variable in its quantifier that occurs in the quantifier of the left quantification. Then drop the right-hand quantification-occurrence, and conjoin the schema which is covered in the new

6 'Closure' means, as in ML, the result of prefixing quantifiers to the schema, in alphabetical order, for all of its free variables.

7 This process of exportation is the one which is characterized rigorously in section 4 of LQ.




variant to the schema which is covered in the left-hand quantification- occurrence.

(a) Working from left to right, apply the propositional rules repeatedly to the clauses of the test-line at hand, until these rules have no further point of application.

(b) Trim out the clauses of the resulting test-line (or lines) and check for termination as follows:

(ba) Drop any contradictory clauses, along with their linked 'V'-occurrences, if any;

(bb) Distribute '.' through 'V' in the left-most clause of appropriate form.

(c) Working from left to right through clauses which are negations of quantifications, instantiate each one by the earliest variable which is free neither in the whole given test-line, nor to the left of the instantiating schema-occurrence, in the succeeding test-line.

Repeat (a) through (c) as long as the cycle brings change. (d) Operate on important clauses as follows:

(da) Segregate (by commas, say) k occurrences of the test-line at hand, where k clauses of that line are important;

(db) In the ith such segregated test-line, where 'i' is varied from '1' through 'k', let that line's ith important clause from the left give way to a conjunct of m different clauses, where m is the number of different variables which are free in the whole upper test-line, and where each such clause is the result of instantiating that ith important clause by some one or other of those variables. In case m = 0, instantiate once by '.

Re-apply (a) through (d) repeatedly, until no more change ensues. (e) Replace each remaining instantiable clause by a conjunct of m'

different clauses, in which clauses the given quantification has been instantiated, as in (db), by the m' different variables which are free in the upper test-line, or by 'x', if m' 0.

Repeat (a) through (e) until nothing changes. If any one of the columns of segregated test-lines has a lowest test-line

which vanishes altogether in a (b)-stage, then the test terminates with that disappearance, and the schema under investigation is certified thereby to be valid. Otherwise, at the foot of every column there will remain some test-line or other, in which the propositional and quantificational super- structure has been stripped completely off every clause, with the result that steps (a) through (e) can have no further effect. In such a case, this procedure cannot establish that the given schema is valid, and the next step is to put the negation of that schema through the same process. If it too is left undecided, then no decision, either pro or con, can be made about that schema's validity, by means of the procedure at hand.



102 ROBERT STANLEY

In practice, fortunately, cases rarely call up much, if any, of the complexity which is prescribed in (da) and (db). These provisions do seem to be necessary and sufficient, however, at the very least, to ensure that all monadically valid schemata yield decisions directly, under this procedure.

The soundness of these steps is attested by the fact that they can be interpreted as inference-steps which conform, respectively, to various valid inference rules of the propositional calculus, and to valid quantificational rules comparable with UI and El of MeL. This interpretation can be effected by prefixing particular quantifiers to each whole test-line, corresponding to all of its free variables.

By way of illustration, this process is applied below to the schemata corresponding to (I) of LQ, and the two-case of *181 in ML. For brevity, where a stage is in order which would effect no change on the presented test-line, the duplicating line and that stage's citation are simply omitted.

(I) (3x) (y)Fxy D (y) (3x)Fxy. (o), (ooo) _'((x)>-(y)Fxy V (y)-(x)--Fxy).

(a) -_(x) .(y)Fxy . (y) .(x) Fxy, (c) r---(y)Fxy . --(x),-Fxy. (a) (y)Fxy. (x)--Fxy. (e) Fxx . Fxy. -Fxy . 'Fyy. (This vanishes under (b).)

*181,2. (Y) -(X) (Xey-Y-(Zip) (Z2) -(XF-Zl ZlEZ2 * Z2EX))s

Let 'Px' abbreviate '(zl) (z2) r-..(xFzl* ZlEZ2 * z2Ex)

(o), (ooo) --i(y) (x) (xEy . V Px: -.Px V . xFy).

(c) (by) (yX. Py : S-py V. yex).

(a) (y) (yex V PA) . (y) (-Py V. yex). (d) (Y) (y-x -V PA) . (A-PE V. xFx) .

(b) (Y) (yex -V Py) . -,Pox .V- (y) (yFeX X PA) . xex. (c) (y) (yZx .V Py) . (xcy .yz zcx) .V. (y) (yFx .V Py) . x.

(a) (y) (yex .V Py) (xy yez .zex) .V. (y) (yex .V Py) . XEX. Let 'Qw' abbreviate

I(wx . xw . Dw) -.. (wcx . XEy . yew) . (wax xCz . zfw).

'(wcy . yeX . Xew) -(wcy . ycy . yew) . (wcy yEZ . zFw).

r-/(wez . zex. xew) . -..(wez . ZEy yw) . '(wez . ZEZ. zcw)'.

(e) (xix .V Qx) . (y"x .V Qy) (zx .V Qz) . (xey . ycz . zcx) .V. (xEx .V Qx) . (y'x .V Qy) . (zix .V Qz) . (xEx). (This vanishes under (b).)

4. Strength of Procedure. In the first place, any monadic schema will be decided by application of this procedure. This can be shown by a demonstration that every monadic schema which tests under the procedure of LQ must test also under the present procedure.

Steps (o) and (oo) are essentially the same as preparatory steps in LQ. There is no need here, as will be seen, to rewrite any of the variables as x.




Steps (ooo), (a), and (b) will leave a schema which is intimately related to that portion of the truth-table which would arrive at step (a) in an LQ-test of the same original schema. This relationship may be seen as follows. Construe the truth-table rows, in the LQ material, as disjoined conjuncts in one large schema. These two differently derived schemata will be tautologically equivalent to the negation of the schema under test, and to each other, except for differences of variants. On that account, the steps of distribution under (bb) will leave alternatives which answer to conditions (a) through (c) of LQ, except for these variances.

If the schema under test is medadically valid, so that no such rows are left, under LQ, then and only then the repeated applications of (ba) and (bb) in the present test will wipe out all clauses before any (c)-steps are taken. If some rows remain, under LQ, and some alternatives remain correlatively in the present test, then a decision will be reached here as follows: (a) Where the alternative is of the sort described in (a) of LQ, in it is

conjoined a schema which begins with ', which is followed by a quantifier, which is followed by a medadically valid schema. In the next stage, (c), this clause will be instantiated by some appropriate variable, to yield 'a' followed by a schema which may be different, but which must still be medadically valid. Plainly everything which develops below this alternative must vanish, in the next (a)-(b) cycle of the procedure.

(b) Where the alternative qualifies with respect to (b) of LQ, in it are conjoined quantifications such that, if a single variable were put for the various variables which are bound in these different quantifications by the main quantifiers, then the negation of the conjunct of the resulting schemata would be medadically valid. In the course of the first (a)-(e) cycle, these conjoined clauses will all be instantiated by 'x', at least, in conjoined schemata of a large, corresponding alternative. Under the next (a)-(b) cycle, then, these conjoined schemata must effect the disappearance of this whole alternative, since their main schemata are like the hypothesized ones, whose conjunct would be medadically contravalid.

(c) Where the clause falls under (c) of LQ, in it are conjoined certain schemata, of which one is a negated quantification, and the rest are quantifications, such that the conditional whose antecedent is the conjunct of the quantifications' appropriate schemata, and whose consequent is the negation's appropriate schema, is medadically valid. The "appropriate" schemata, of course, are like the largest covered ones of the pertinent conjoined clauses, except for having free variables in common, as in the case of (b). Under the next (c)-stage, this negation clause will be instantiated by some suitable variable, and in the (e)- stage which soon follows, each of these quantification clauses will be



104 ROBERT STANLEY

instantiated by that same variable. As in the (b)-case, then, the next (a)-(b) cycle after this (e)-stage will wipe out the whole alternative which corresponds in this test-line to the original alternative. This is guaranteed, as before, by the fact that these schemata are of the sort hypothesized, where the relevant conditional, in this case, is medadically valid.

Thus every alternative which survives the first (b)-stage, where the given schema tests under LQ, must give way to alternatives which vanish altogether in some later (b)-stage. This procedure gives a decision, therefore, for any monadic schema that can be decided under LQ, and consequently for any monadic schema.

The complexity invoked by stage (d) seems to guarantee that monadically valid schemata will give tests directly, under application of this cyclical procedure. Since no simple argument to this conclusion presents itself, it suffices to remark that any given monadically valid schema can be decided by applying the present procedure to all monadic schemata of which the given, non-monadic schema is a substitution-instance.8 So far, however, no monadically valid schema, nor indeed any polyadically valid one either, has been tested which requires any such multiple, indirect treatment; all have yielded to a single, direct application of the prescribed (o)-(e) process.

BIBLIOGRAPHY

ML QUINE, W. V., Mathematical logic, New York, 1940. LQ QUINE, W. V., On the logic of quantification, this JOURNAL, Vo1. 10 (1945), pp.

1-12. MeL QUINE, W. V., Methods of logic, New York, 1950.

UNIVERSITY OF BRITISH COLUMBIA

8 This device of testing all "superstitution-instances" was suggested by Professor

Quine.



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An Extended Procedure in Quantificational Logic

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