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TRADITIONAL LOGIC AS a LOGIC of DISTRIBUTION-VALUES Coiwyll

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TRADITIONAL LOGIC AS a LOGIC of DISTRIBUTION-VALUES Coiwyll 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, contra- position, 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 tradi- tional 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 equi- pollent. 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 un- distributed, 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 predica- ment 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.
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