TRADITIONAL LOGIC AS a LOGIC of DISTRIBUTION-VALUES Coiwyll

Home , Categorical proposition, Obversion

TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES

COIWyll WILLIAMSON

Even today, something called Traditional Logic is applied to the minds o f a great many students in a great many uni- versities. I f the tru th is admitted, th is traditional lo g ic is a poor th in g , sca rce ly mo re th a n a piecemeal co lle ctio n o f fragments f ro m a system the greater part o f wh ich remains permanently submerged. Students acquire a kind of familiarity with the square o f opposition, conversion, obversion, contraposition, moods, figures, a n d s o o n , b u t t h e y a re ra re ly permitted a glimpse o f the t ru ly systematic properties o f the logic behind these devices. It ma y occur to one who suffers traditional logic and later learns something of propositional logic that there is a consid- erable difference in the methods used in these two branches of what is supposed to be the same discipline. I t is not ju st that the standards o f exactitude in the older logic seem slip- shod; the real problem is that it does not appear to embody a precise conception of validity. The unfortunate student who is t ryin g t o fin d house-room f o r th e rules o f equipollence, quantity, quality and distribution, the four figures, sixty-fo u r moods and 256 syllogisms, has n o t ime t o b e troubled b y anything so out o f the wa y as a simple and straightforward technique for testing validity. The comparison with propositional logic is instructive. The student of propositional calculus generally has a t his disposal two methods f o r testing validity, one based o n constructing truth-tables, t h e o th e r in vo lvin g deductive techniques. Th e axiomatic treatment of traditional logic has been extensively developed in recent times, notably b y Lukasiewicz and Boch- efiski; and it is natural to ask whether it might not be possible also to employ in this field techniques analogous to the truth- table calculations o f propositional logic. I t is common know- 730 COLWYN WILLIAMSON ledge that traditional logic does n o t a t present have such a method. T h e f irs t t a sk, therefore, i s t o ma k e g o o d t h is deficiency. In the case o f propositional logic, techniques o f this kin d provide imp o rta n t insights in t o t h e general character a n d systematic properties o f truth-functions. Simila rly, m y p rin - cipal objective in discussing the decision procedures of traditional logic is to arrive at a better understanding of that logic as an autonomous and exhaustive system. Failing that, it may still be possible to perform a more mundane service; f o r the provision of an easy mechanical test will surely simp lify the task of those obliged to instruct students in the eccentricities of traditional logic. My design, then, is to inquire whether, taking the inferences of traditional logic as they are, and principles o f va lid ity as they ma y be made, i t is possible to establish some ju st and certain rules fo r the administration of logical order.

I. The doctrine of distribution probably represents the near- est t h a t traditional lo g ic e v e r comes t o a simp le testing technique. Th e doctrine purports t o p ro vid e a method f o r determining va lid ity together wit h a semantical th e o ry that somehow explains wh y the method works. But with regard to both o f these pretentions i t has fallen o n hard times. Peter Geach has sh o wn th a t th e semantical notions u p o n wh ich the doctrine is based are confused, and he has also pointed out that it will not wo rk as a test of va lid ity ( Ithen, the doctrine does not stand at all. The question is, can )a.n ytAhings b e sialvtage d sfrotma then rudinss ? ,I t is best to begin b y trying to understand the defects that brought d o wn the o ld rules of distribution. Rules o f distribution are o f course based o n the idea that each term in a categorical proposition is either distributed o r undistributed. The reasoning b y wh ich logicians have hoped to ju stify the classification o f terms in this wa y will not be

( "D1is tribution: A Las t word", Th e Philos ophic al Rev iew, 1960. ) P e t e r G E A C I I , R e f e r e n c e a n d G e n e r a l i t y , I t h a c a , N . Y . , 1 9 6 2 , a n d TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 731 discussed here. Fo r present purposes, i t is sufficient t o sa y that a te rm is distributed o r undistributed according to th e type o f proposition it appears in and the place it occupies in that proposition. A, E, I and 0 will be used as term-operators corresponding to th e f o u r traditional fo rms o f categorical proposition, s o that A ma y be read as A l l — are —", E as " No — are —", I as "Some — are — '', and 0 as "Some — are not —". a, b and c are term-variables standing fo r plural nouns o r expres- sions plausibly construed as the equivalents o f plural nouns. The first t e rm o f a proposition is it s subject, th e second its predicate. The idea of distribution will be expressed b y intro- ducing the notion of a distribution-value and saying that any term has a distribution-value of 1 or O. One o f the mnemonics f o r distribution is SUPN, "Subjects of Universals, Predicates o f Negatives". Thus, t h e subject term o f a universal proposition (A o r E) a n d the predicate term o f a negative proposition (E o r 0 ) have a distribution- value o f 1, and remaining terms have a distribution-value o f O. That is,

Eab: a =1 , b=1. Aab: a =1 , b = 0 Oab: a =0 , b = 1 Jab: a =0 , b = 0

If p and q are categorical propositions wit h common terms, then p q (an immediate inference) signifies that q ma y be inferred f ro m p , a n d p q th a t each proposition ma y b e inferred from the other. When p q is true, p and q are equipollent. N signifies propositional-negation; n signifies t e rm- negation. The standard ru le o f distribution fo r immediate inferences is that it is not permissible to pass from a proposition in which a term is undistributed to one in which it is distributed. Thus, to g ive a usual example, universal affirmatives a re said t o be incapable of simple conversion because, in Aab A b a , b 732 COLWYN WILLIAMSON is distributed i n t h e conclusion a n d undistributed i n t h e premiss. Let us consider where the rule goes astray. It is evident that the rule cannot by itself specify necessary and sufficient conditions o f validity. To g ive o n ly the most obvious example, b o th te rms o f a universal negative a re distributed and both terms o f a particular affirmative are undistributed, and the rule therefore permits any inference from the fo rme r o r t o t h e latter, though ma n y such inferences (e.g. Eab--- > lab) a re cle a rly n o t valid. So th e standard ru le can aspire to provide o n ly necessary conditions o f validity. However, even this modest project proves to be too ambi- tious. Consider the notorious case o f inversion. The inference Aab---> Onab must be va lid because i t is the summation o f a series o f inferences that are separately recognized as valid: Aub—> Eanb---> Enba--> Anbna--> Inbna--> Inanb—> °nab. On the other hand, Aab ---> Onab ought not to be va lid if the rule o f distribution is sound, because the predicate o f Onab is distributed and that of Aab is not. This argument, borrowed from Geach, shows th a t the ru le cannot manage to specify even necessary conditions. I wa n t to show that the predicament o f the formal theory of distribution is in one sense fa r worse th a n Geach's argument indicates, i n another sense much better. But things must be allowed to get worse before they can get better. Let me propose a certain p o licy concerning the effect o f negation o n distribution-value. Propositional-negation, I w i l l say, reverses the distribution-value o f both the terms o f the proposition o n wh ich i t operates. Thus, since a and b both have a value of 0 in lab, they both have a value of 1 in Nlab. And, in general:

N ( 1 1 ) N(10)=-01 ,N ( 0 1 ) N(00)=11 0, 0 In order to desc1ri0be the rô le o f term-negation, I w i l l first adopt the convention of saying that in propositions like Aab, Enab, lanb, Onanb, and so on, a and b are the only constituent terms. I n o th e r words, t h e u su a l practice o f speaking o f TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 3 3

"negative te rms" is se t aside i n fa vo u r o f regarding te rm- negation as part of the term-operator. The terminological, and other, difficulties behind this decision w i l l n o t be discussed here. Now, term-negation, I w i l l say, reverses the value o f the term on which it operates:

n(1)=0 n(0)=1

What this means is that, for example, since a has a value of 0 in lab, it has a value of 1 in mob. The idea th a t negation alters distribution-value is n o t a familiar one, though the ro le assigned to N flo ws n a tu ra lly from t h e recognition that, a s Keynes p u t s i t , " a n y ',term distributed i n a proposition i s undistributed i n it s co n tra - dictory". An d despite the novelty of regarding term-negation in this wa y it is easy to see that some such procedure must be adopted. I f it is said that term-negation has n o effect on distribution, then even a straightforward case like lanb ---> Oab (obversion) i s rendered invalid. No w, t h e problem ma y b e avoided fo r obversion b y confining the rule to subject terms. (This is the method of some authors. Others prefer simp ly to keep quiet about distribution when they deal wit h obversion and contraposition.) But restricting the rule to one term is an unhappy solution, because t h e sa me d ifficu ltie s a ris e i n sharper fo rm elsewhere. I f n fails to alter distribution-value, neither of the following contrapositions is acceptable:

Aab —› Anbna, O a b —>• Onbna.

And it is plain that any attempt to confine the rule to subject or predicate must founder on one o r other of these inferences. Once we a llo w that n reverses distribution-values both a re acceptable. (I t ma y seem strange, b y the way, that the lo g i- cians w h o present ru le s o f d istrib u tio n a n d t h e n g o o n immediately to describe obversion and contraposition appear not to notice that the rules d o n o t a p p ly to obversion and contraposition.) 734 COLWYN WILLIAMSON

The re a l weakness o f distribution a s a mechanical test, and this is what Geach's criticism does n o t bring out, is the lack of a policy for coping with negation. Consider again the problem o f inversion. Each o f the steps in the series Aab Eanb —* Enba —* Anbna I n b n a —› Inanb —* Onab i s supposed to be permitted b y the rule o f distribution. But this can o n ly appear t o b e th e case because o f certain subterfuges. Th e obversions are handled by confining the rule to subject terms, and a conversion lik e Eanb —>• Enba is regarded a s ju st a n instance o f Eal3 —) Etki (which conforms t o th e ru le ) w i t h a "negative t e rm" substituted f o r (3. I f w e t u rn a b lin d e ye on these manoeuvres, the problem arises o n ly fo r inversion. But wh a t is re a lly problematic about inversion can also be found in other segments of the chain, for example:

Aab —> Enba, Eanb —* Inbna, A n b n a —* °nab.

The ordinary rule of distribution cannot tell us anything about cases such as these. Th is is th e point: i t is n o t that these cases violate the ru le b u t that there is n o wa y o f applying the rule to them. To be sure, we might invent a special t rick for coping with inferences like the first one. Aati ---> Enba, we might say, is a cobversion; and the special feature o f cobversions is that the rule of distribution applies to the subject of the first proposition and the predicate o f the second. This device, as good as many others in traditional logic, parallels the t ric k f o r handling obversions. (I t is perhaps surprising that no-one discovered cobversions.) But no such patching-up will help us with the second two inferences: the standard rule just does n o t a p p ly t o th e m a t a ll. Th e supposed problem of inversion is in reality only a symptom of the general failure of the ru le o f distribution t o a p p ly t o inferences in vo lvin g "negative terms". What is needed, then, is a clear p o licy o n term-negation, and th is has been provided. Bu t the difficulties d o n o t end there. The predicament of inversion is no better than before: that n reverses distribution-value does nothing t o imp ro ve the standing of Aab --> °nab. Indeed, the immediate effect o f TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 3 5 the policy is to make matters worse: the number of rejected valid inferences increases despairingly. Thus, to demonstrate that in ve rsio n i s sound w e passed through t h e inference Anbna -* Inbnct; and, applying the convention fo r term-negation, b violates the standard ru le o f distribution. So f a r as inversion is concerned, then, t h e standard ru le i s a t least consistent: i t rejects the inversion o f A propositions d ire ctly and it also rejects one of the steps by which inversion is shown to be valid. But this is hardly consoling, since both rejected inferences are (recognized as) valid. We are now in a position to get to the heart of the difficulties. I t has been said that the problem of inversion is located in the lack of a policy on term-negation; but it must now be added t h a t inversion b rin g s in t o t h e o p e n a n e ve n mo re serious deficiency in the standard rule of distribution. The text- book rule is really an ill-judged attempt to impose an appear- ance of uniformity on inferences that are importantly different. The traditional to p ic o f Immediate Inference covers, f irst , inferences passing from a universal proposition to a universal proposition, o r from a particular to a particular, and second, those wh ich pass fro m a universal to a particular. The f irst kind I w i l l ca ll non-reductive, t h e second reductive. No w, inversion, like superimplication, conversion per accidens, and so on, is an instance o f reductive inference; and the fact is that the standard rule of distribution has o n ly the most acci- dental bearing o n the wh o le sphere o f reductive inference. When it is allowed that n reverses distribution-value (wh ich must b e allowed i f other inferences a re t o pass muster), a large number of inferences, a ll of them reductive, fa il to meet the requirements of the standard rule. Ananb--> Inanb, Anab---> Inab a n d Enanb—> Onanb wo u ld a l l b e regarded a s v a lid superimplications (a superimplication being of the form A O ---> / 4 o r Eafi -->Oati), but they are rejected b y the rule. Let me summarize the difficulties. A number of valid inferences show tha t th e ru le cannot be made t o wo rk u n t il a policy o n term-negation is introduced. Wh e n such a p o licy is introduced, the rule applies to non-reductive inferences (at least to the extent o f providing necessary conditions), b u t it 736 COLWYN WILLIAMSON will not wo rk for reductive inferences. Even for non-reductive inferences, th e ru le does n o t provide sufficient conditions; and, as will emerge shortly, it works for these inferences only in a w a y mo st misleadingly expressed b y t h e traditional formulation. We have arrived at the stage at which the ru le of distribution itself must be reformed.

2. Wh e n p and q are propositions both o f wh ich have a and (3 as their constituent terms, and when both a and r, have the same distribution-value i n p and q , th e n p a n d q a re equivalent in distribution-value. A n d when t wo propositions are both universal o r both particular, they are equivalent in quantity. E, A, NO and NI are the universal operators; 0 , I, NE a n d N A a re t h e p a rticu la r operators. (Equivalence i n quantity is o f course the ma rk o f non-reductive inference.) Consider a n example o f propositions t h a t a re equivalent both i n quantity and distribution-value. Granted certain o b - vious presuppositions th a t w i l l n o t b e discussed n o w, t h e following is a n exhaustive list o f propositions that are u n i- versal and such that both their terms have a distribution-value of 1:

Eab, Aanb, NOurib, Nlab, Eba, Abna, NObna, Nlba.

Since any proposition must be universal or particular, and any proposition must have a distribution-value of 11, 10, 01 o r 00, it is evident that there are eight such groups o f propositions equivalent both in quantity and distribution-value; and each group contains eight propositions. Now, i t is easy to show b y obversion, simple conversion and the principle o f contradiction depicted in the square o f opposition (i.e. A -the eight groups, th e remaining seven propositions ma y be NinfOerr,e d fro m any member o f the group. I n short, these are NgrouAps oOf equipollents: any member of a group is equipollent ,wit h the rest, and no member of a group is equipollent with a EmembNer of aI nother group. Hence, the formula for equipollence , N E = I ) t h a t , i n e a c h o f TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 3 7 is: equivalence in quantity plus equivalence in distribution- value. We ma y therefore formulate t h e fo llo win g simp le ru le o f equipollence: p = q if and o n ly if p and q are equivalent in quantity a n d distribution-value. S imila rly, t h e fo llo win g i s the correct ru le o f distribution f o r non-reductive immediate inferences : p—>q if and o n ly if p and q are equivalent in distribution-value. This ru le provides necessary and sufficient conditions f o r the validity of all non-reductive immediate inferences. Further- more, the rule covers the greater part of the traditional theory of immediate inference: most aspects of that theory are in fact concerned wit h equipollences o f one k in d o r another. Th e elaborate doctrines o f simple conversion, obversion, contraposition, rules of equipollence, and so on, are here reduced to one elementary principle, t h a t o f distribution-equivalence. (This p rin cip le has been presented as a mechanical test o f validity, but, o f course, it may also be seen as a rule fo r the mechanical construction of equipollents.) How does th e n e w rule stand wit h regard to the o ld ? A s it happens, the old rule works for all inferences between equipollents, provided o u r p o licy o n term-negation is accepted; but i t wo rks i n a wa y misleadingly expressed b y th e ru le itself. A ll such inferences are between distribution-equivalents, and th at a te rm cannot be distributed i n th e conclusion i f undistributed i n t h e premiss simp ly fo llo ws f ro m t h e f a ct that t h e distribution-value cannot change. T h e d istin ctio n made in the traditional ru le between consequent and antecedent is quite irrelevant. A n d where the o ld ru le hopes (and fails) t o p ro vid e o n ly necessary conditions, t h e n e w ru le provides completely necessary and sufficient conditions. Final- ly, it is a great advantage of the new rule that, unlike the old, it clearly marks out the distinctive logical properties of equipollence. The co rre ct ru le o f distribution f o r reductive immediate inferences i s simp ly this: p --> q i f and o n ly i f one o f th e constituent propositions h a s a n u n mixe d distribution-value (11 o r 00) and the other a mixed value (10 o r 01). I f we say 738 COLWYN WILLIAMSON that a te rm wit h the same value throughout an inference is single-valued, and a term which changes its value is double- valued, the same rule becomes: a reductive inference is valid if and o n ly if one o f its terms is single-valued. Simila rly: a non-reductive inference is valid if and only if both terms are single-valued. It may easily be shown that this rule is thoroughly effective. We have seen that the propositions o f traditional lo g ic f a ll into eight groups o f equipollents. Fo r the purpose o f testing, therefore, a single representative of each of the eight groups will suffice. O f th e sixteen inferences produced, e ig h t a re valid and eight in va lid ; a n d th e d ivisio n conforms e xa ctly to the rule. It ma y be noted that, as in the case o f non-reductive inference, th e distinction made i n th e traditional ru le between consequent a n d antecedent p la ys n o p a rt i n o u r decision procedure. In the light of the two correct rules of distribution, it should be easy t o see h o w th e single traditional ru le ma y h a ve appeared almost to wo rk as a kin d o f compromise between the two varieties o f immediate inference. A s we have seen, none o f the inferences between equipollents can violate the old ru le ; and, o f the eight varieties o f reductive inference distinguishable a cco rd in g t o distribution-value, o n l y h a l f (those fro m 00 to 10, fro m 00 to 01, fro m 10 t o 11, and fro m 01 to 11) violate the rule. But there is no room for compromise in logic. The sometimes tortuous w a y i n wh ic h p ro p e r ru le s o f distribution have been a rrive d a t should n o t be allowed to distract attention f ro m the essential simp licity o f th e ru le s themselves. Granted our method fo r determining distribution- value, the entire topic of validity in immediate inference may be disposed of in two o r three sentences. A n immediate inference is va lid o n ly i f i t is non-reductive o r reductive. No n - reductive inferences are va lid if and o n ly if both terms are single-valued; reductive inferences a re va lid i f and o n ly i f one and only one term is single-valued. Whereas the traditional rule could aspire to providing o n ly TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 3 9 necessary conditions, and fe ll short even o f that sma ll aim, the revised ru le s p ro vid e conditions sufficient a s w e l l a s necessary for the va lid ity of all immediate inferences without exception. B u t i t st ill remains t o be shown that procedures of this kin d are applicable to syllogistic inferences.

A method f o r determining the distribution-value o f terms has been described, a n d effective ru le s o f distribution f o r immediate inference h a ve been formulated. B u t t h e t o p ic of immediate inference occupies a comparatively small place in traditional logic, and it still needs to be shown that the theory of distribution-values is applicable to the compound inferences called syllogisms.

I. I t is best to begin b y setting aside a number of inconve- niences that stand in the wa y o f an easy understanding o f syllogistic theory. Th e doctrine o f moods and figures i s o f little importance. The magic number 256 (the supposed number of syllogisms) ma y b e forgotten. Th e eight-or-so ru le s f o r syllogism provided that (i) the inference contains three distinct terms, and (ii) p and sq, p and r, and q and r each have a term in common. The term common to the premisses (p and q) is the traditional logic and should be given up in our more sceptical age. First le t us define wh a t is to count as a syllogism. I f p, and r are categorical propositions, then p q --->r (signifying that r ma y be inferred f ro m p and q taken together) i s a syllogism provided that (i) the inference three distinct terms, and (ii) p and q, p and r, and q and r each have a te rm in common. The term common to the premisses (p and cl) is the middle te rm; th e remaining t wo a re extreme terms. A s i n connection with immediate inference, the practice of speaking of "negative terms" is abandoned: a and na do not count as separate terms. It ma y be objected th at the conception o f syllogism e m- 740 COLWYN WILLIAMSON bodied in the above definition is different to the traditional conception, p a rt icu la rly w i t h re g a rd t o t h e admission o f negation (N) and term-negation (n). I t follows that formubm positions constructed out of terms (a, b, c) and term-operators (A, E, I, 0 ) together with any number of signs of propositional- negation (N) and term-negation (n). I t follows that formulae like A a b & Enbc —> Onanc a n d Nia b & Abc ---> NEac (t o t a ke two examples almost a t random) a re counted a s perfectly respectable syllogisms, even though many traditional logicians would have thought them blasphemous. The point is that the syllogism as it appears in traditional logic is an artifice based on disregarding most o f the forms o f proposition recognized in the standard treatments of immediate inference. In connection with, fo r instance, obversion, contraposition and rules o f equipollence, propositions involving term-negation and propositional-negation are admitted into the system: it is therefore inconsistent and a rtificia l t o o mit them fro m a discussion o f compound inference. A logical system must always be, with in its terms of reference, exhaustive. I t should be stressed, however, that, although the scope of the syllogism is considerably widened b y o u r definition, everything traditionally counted as a syllogism is incorporated in passing. A n d anyone hide- bound enough to wish to restrict the actual wo rd "syllogism" to the old forms is at liberty to do so. There a re t w o ru le s o f distribution f o r syllogisms. Th e first, which is similar to the old rule for immediate inferences, is that no te rm ma y be distributed in the conclusion if it is undistributed in the premisses. This rule is generally presented in the textbooks as two fallacies, " illicit process of the major" and " illicit process o f the minor"; b u t this wa y o f putting it simply represents a needless distinction between t h e case where th e so-called " ma jo r term'*, and th e case wh e re th e "minor term", violates the rule. The second ru le is found in the famous "undistributed mid d le " fa lla cy; a n d wh a t t h is amounts to is that the middle term of a va lid syllogism must be distributed at least once. Whereas there are definite violations of the rule of distribution for immediate inferences, no such violations can be found TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 4 1 among th e twenty-four syllogisms u su a lly said t o b e va lid — the reason being that "negative terms" are excluded fro m the orthodox syllogism. I f "negative terms" a re admitted, i t soon becomes apparent that the standard rules will not work. Thus,

Anab & Abc —> mac violates the ru le fo r extreme terms (granted the method fo r calculating distribution-value), since a is distributed i n th e conclusion a n d undistributed i n t h e premisses. B u t i t ca n easily be seen that Anab & Abc---> lnac is merely Barbari with one te rm negated throughout, and therefore valid. Similarly,

Anba & Anbc--> lac is valid, since it is Darapti wit h a negated middle term; but the middle te rm has a distribution-value o f 0 in both occur- rences. If we confine o u r attention to the standard 256 syllogisms, it is apparent that the accepted rules fa il to specify sufficient conditions of validity:

Eab & Ebc -3 Aac is invalid, b u t it violates neither of the rules o f distribution. Of cource, a " ru le o f quality" (n o va lid syllogism has t wo negative premisses) ma y be adduced to eliminate the above example. But rules of quality are just an embarrassment when term-negation is employed: Eab & Ebc--->Onac is valid, despite its negative premisses. I n fact, the idea o f " q u a lit y d- roippse d ebnteirelsy. t As ma y be suggested b y the uneasy wording o f the ru le for middle terms (they must be distributed " a t least once"), the rules o f distribution f o r syllogisms, lik e th e traditional rule fo r immediate inferences, attempt to impose false u n ity on distinct varieties o f inference. W e mu st therefore begin by discovering what distinctions need to be made. 742 COLWYN WILLIAMSON

Some traditional logicians distinguish three kinds o f syllogisms: weakened, strengthened and fundamental syllogisms. A weakened syllogism is one where a particular conclusion is drawn from premisses that would have justified a universal conclusion. Thus, Cesar° is weakened because, although we may conclude Oac fro m Ecb and Aab, these same premisses justify the conclusion Eac (Cesare). A strengthened syllogism is one where a particular conclusion is drawn fro m universal premisses a n d t h e conclusion s t i l l f o llo ws i f o n e o f t h e premisses i s replaced b y a p a rticu la r proposition. Th u s, Celaront is strengthened: although we may conclude Oac from Ebc and Aab, the same conclusion follows when Aa b is re - placed by lab (Ferio). A fundamental syllogism is neither weak nor strong, but just right. In the textbooks, the notions of strengthened and weakened syllogisms a re based o n a further assumption wh ich is n o t always made explicit. To speak of a syllogism being strengthened o r weakened is re a lly to indicate a relation between it and some fundamental syllogism; a n d wh a t i s assumed i s that this fundamental syllogism must be in the same figure as its weakened o r strengthened version. This assumption, when it is made explicit, is expressed b y speaking o f "subaltern moods". Since figures should b e g ive n up, th is p a rt o f the meaning o f "strengthened" a n d "weakened" w i l l a lso b e discarded. Are the three traditional categories soundly based ? I f we think o n ly o f th e ir quantity, va lid syllogisms a re o f three kinds: universal syllogisms (as th e y ma y b e called), wh ich contain nothing b u t universal propositions; particular syllo - gisms, wh ich have a particular conclusion and one particular premiss; and syllogisms wit h universal premisses and a particular conclusion. Using U and P to signify quantity, the three kinds are: U & U --> U U & P P U —>• P

The first two kinds, universal and particular, together consti- TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 4 3 tute th e fundamental syllogisms o f traditional logic. I w i l l call t h e m non-reductive syllogisms. T h e t h ird kin d , w i t h universal premisses and a particular conclusion, will be called reductive syllogisms. Traditional logic, i t has been explained, distinguishes t wo varieties o f reductive syllogism, strengthened and weakened — though some syllogisms (Barbon, f o r example) a re both strengthened and weakened. But the category "strengthened" may be given up, because it is based on the erroneous assumption that some reductive syllogisms are strengthened, others not. To say that a syllogism is strengthened is re a lly to say that it is of the form

U" & U 2 - and that there is some p>a rticular syllogism wit h a common premiss and the same conPclusion, i.e. a syllogism of the fo rm 3 Ul & 1 32 In reality, t h is p r-o-p e rty is possessed b y a ll reductive s y l- logisms; a n d the >onPe reductive syllogism (Camenop) wh ich appears i n t h e st3a ndard accounts n o t t o b e strengthened seems this way onoly because there is no corresponding particular syllogism i nr th e same fig u re — unless, th a t is, t e rm- negation is permittPed. Nevertheless, ev' en if the notion of a strengthened syllogism is given up, it mus&t be r ecognized that there are two varieties of reductive sylloUgism, namely those th a t would, and those that would not, ju2stify a universal conclusion. The kin d that would ju stify sucPh a conclusion already have a name: th e y are weakened sy3llogisms. Th e o t h e r k in d m a y b e ca lle d simply non-weakened. This turns o u t to be the o n ly distinction that matters. A syllogism of the form

U t is weakened only if there& i s a corresponding syllogism of the form U 2 - - > P 3 744 COLWYN WILLIAMSON

Ul & U 2 When there is no such co—rr>es ponding universal syllogism the original reductive syllo gUism" i s non-weakened. ("Correspon- ding" means here: h a vin g th e same status w i t h regard t o validity.) From th is p o in t o f vie w, then, th e re a re three kin d s o f syllogism:

(i) Non-reductive (ii) Reductive Weakened (iii) Reductive Non-weakened

It o n ly remains to describe the mechanical tests appropriate to each of these forms.

2. The method for calculating the distribution-value of terms has already been explained. A non-reductive syllogism, whether universal o r particular, is valid if and only if its middle term is double-valued (disti- buted once and only once) and its extreme terms are single- valued (have the same distribution-value thoughout the syllogism.) Consider the following examples, the first (Barbara) a u n i- versal syllogism, the second (Ferio) a particular syllogism:

Aab & Abc—> , JaAb &a Ebcc O a c

In the first case, the middle te rm b has a value o f 0 in the first premiss and I i n the second premiss; the extreme te rm a has a value of I in its premiss and in the conclusion; and the extreme te rm c has a value o f 0 in both its occurences. I n the second case, th e middle te rm b has a value o f 0 in the first premiss and I in the second; c has a value of 1 throughout; and a has a value o f 0 throughout. Thus, i n both cases the middle te rm is double-valued and the extreme terms single- valued. Th e same holds t ru e o f a ll possible non-reductive syllogisms. TRADITIONAL LOGIC AS A LOGIC OF DISTRIBUTION-VALUES 7 4 5

A reductive weakened syllogism is va lid if and o n ly if its middle te rm and one and o n ly one o f its extreme terms are double-valued. Take Braman tip as an example:

Aba & Acb ---> lac is va lid because the middle te rm b has a value o f 0 in one premiss and 1 in the other, the extreme term a has the same value (0) throughout, and the extreme term c is double-valued — it has a value of I in its premiss and 0 in the conclusion. It should be easy, remembering the definition of a weakened syllogism, to see the connection between the rule of distribution f o r syllogisms o f th is kin d and th e ru le f o r reductive immediate inferences. A reductive immediate inference i s valid only if one and o n ly one of its terms is double-valued. Hence, i n a weakened syllogism, t h e p a rticu la r conclusion must contain one term wit h a value different to the value it would have had in the universal conclusion that might have been drawn. To take a simple example, the premisses Aab and Abc justify the conclusion Aac (Barbara); and they also justify the conclusion lac (Barbart). But the distribution-value of Aac is 10, th e distribution-value o f la c is 00, a n d M c ma y b e inferred d ire ctly fro m Aac. I t w i l l be evident that th is ru le always holds. The ru le o f distribution f o r non-weakened syllogisms i s simply this: such syllogisms are va lid if and o n ly if a ll th e ir terms are single-valued. Thus, Darapti, Aba & Abc ---> lac is va lid because a = 0, b = I and c 0 throughout the syllogism. These three ru le s o f distribution a p p ly t o a n y syllo g ism that can be constructed out o f the propositions o f traditional logic. It ma y be noted that it is in connection with non-weakened syllogisms that the o ld fallacy o f undistributed middle goes by the board. The middle te rm o f non-weakened syllogisms must b e single-valued, b u t th e va lu e need n o t b e I . Thus, 746 COLWYN WILLIAMSON

Aab & Acb-->Onac is va lid although it has an undistributed middle; f o r a ll that matters is that the middle term should be completely distributed or undistributed. Similarly, t h e weakened syllogisms demonstrate t h e u n - soundness of the old illicit process fallacies:

Aba & Acb--> Inane is valid, despite the fact that a is undistributed in the premisses and distributed in the conclusion. This, then, is the simple procedure wh ich ma y be used b y anyone obliged to test the va lid ity o f syllogistic inferences. The f irs t t h in g t o b e settled i s wh e th e r t h e inference i s reductive o r non-reductive; that is, whether it proceeds fro m universal premisses to a universal conclusion, from a universal premiss and a particular premiss to a particular conclusion, o r from universal premisses to a particular conclusion. (If it does none of these things, the inference is invalid without reference to the rules of distribution.) I f the inference is non-reductive, it is va lid if and o n ly if the middle te rm is distributed once and the extreme terms are single-valued. I f it is reductive, it is va lid if and o n ly i f either a ll terms are single-valued o r the mid d le t e rm i s distributed once a n d o n ly o n e o f th e extreme terms is double-valued. These rules provide entirely sufficient and necessary conditions o f va lid ity in syllogistic reasoning, and therefore replace all the traditional rules, most of wh ich a re i n a n y case unreliable. Procedures based o n these rules also serve to solve a problem which perplexed some of the classical exponents o f traditional logic, th e apparent lack o f "harmony" i n th e th e o ry o f the syllogism. B u t th e light cast b y the logic o f distribution-values on the true harmony and completeness o f syllogistic theory deserves to be discussed separately.

University College of Swansea C O I W y l l WILLIAMSON