leibniz's complete propositional logic

Leibniz's Complete Propositional Logic Hector-Neri Castafieda

I. Background and main claims

Leibniz is, according to general consensus, "one of the greatest logicians of all times. ''1 Nevertheless he did not publish his logic works. Hence, he is regarded as the first symbolic 2 or mathematical 3 logician, but not as the founder of symbolic or mathematical logic. It is also widely held that Leibniz accomplished practically nothing in logic. As Parkinson has put it so succinctly (P lix):

The question now arises, Why Leibniz published so little. The short answer is that he left very little that was publishable.

Thus, historians of logic have raised the question: Why did Leibniz accomplish so little? 4 This questioning assumes that Leibniz was a great logician. The question, not to mention the proposed answers, manifests a desire to make excuses for his failures. Why? (Because we are all beyond the 17th century European nationalisms and feel that he was unfairly treated by the Newtonians? Are we atoning for them?) Whatever the answer to this question may be, perhaps we should ask: Why is Leibniz regarded as one of the great logicians of all times?

In the end Leibniz is praised for having practiced a style, the symbolic style. The style embodies the twofold idea that logic could be treated mathematically and that there is a non-numerical, non-quantitative mathemat- ics. 5 Le style est l'homme rn~me, said Buffon. Leibniz is, thus, regarded as an outstanding example of acknowl- edged greatness for a great idea mounted on an almost total failure of execution.

Here I argue that the concluding round of logical theorizing in Leibniz's Generales Inquisitiones, Sections 1 9 5 - - 2 0 0 , 6 is a brilliant short essay. It contains a significant logical accomplishment. With minor editing, it should have been published in 1686 or soon there- after. Our current standards of rigor would require more editing, but no revision of principles, except a clarification about the association of conjunction.

The GI is a manifold of philosophical reflections un-

dergirded by a series of logical investigations. Leibniz's goal is to construct a general logical calculus that encompasses traditional syllogistics. He makes attempt after attempt at developing the envisaged calculus. The consensus is that in the end he gives up in failure. Parkinson, apparently understating it, has vented the general judgment:

The Generales Inquisitiones ends with the assertion that the fundamentals of logical form are contained 'in these few propositions' (his paucis), which seem from the context to refer to the last set of principles stated -- that is, those of par. 198. Leibniz makes little attempt at substantiating his c l a i m . . . Leibniz, then, can hardly be said to have substantiated his claim to have stated the 'fundamentals of logical form'; but the more modest claim that in this paper he has made 'excellent progress' seems amply justified.

I submit that Leibniz is completely justified in claiming that the principles he has given are fundamentals of logical form. (Since Latin does not have articles, it is not clear that he meant the fundamentals, or simply some fundamentals.) At any rate, a set of principles for a certain logical domain may be complete; yet there are always equivalent alternative characteriza- tions of the same domain of valid inferences. Hence, the issue of uniqueness is not so important as whether the principles under consideration are valid and of central value. The next important question is whether the set of such principles is in an appropriate sense complete.

Here I want to argue for two major theses. First, the principles Leibniz formulates in GI 195--200 are valid, important, and exciting. Second, with the adjunction of the principle of associativity of conjunction, which Leibniz does not mention, although he uses it all over, the set of principles he formulates explicitly includes a redundant complete formulation of the standard propositional calculus. Moreover, these principles also contain a full characterization of the truth-table semantics of propositional logic. Furthermore, that calculus includes a correct Boolean treatment of syllogistics.

I won't quibble with Parkinson about Leibniz's not

Topoi 9: 15--28, 1990. �9 1990 KluwerAcademic Publishers. Printed in the Netherlands.

16 H E C T O R - N E R I CA ST A lqE D A

providing a substantiation of his claim. Undoubtedly, Leibniz would have gained credence and even self- assurance had he bothered to derive some twenty or more theorems and meta-theorems. Certainly, he could have discussed applications of those theorems. Not- withstanding, he proposed there a rich and adequate theory of propositional and syllogistic logical truths and deduction.

II. Leibniz's final syllogistico-propositional system in the GI

Preamble. The ensuing Section 1 is a developmental and exegetical introduction to Leibniz's concluding logical system in GI 195--200. It attempts to re-live, vicariously and with hindsight, Leibniz's momentous discovery of material equivalence and the material conditional. This discovery culminates his dramatic generalization of syllogistics to propositional logic. Section 2 is a translation of the main part of GI 195--200. Klaus Jacobi has suggested that a direct presentation of this translation, unmediated by Section 1, may be more effective. He may be right. Perhaps the reader interested in seeing Leibniz's logical calculus right away may prefer this Jacobian alternative experience. Then Section 1 may be read as an appendix. 7

1. Leibniz's logical investigations in the GI

The GI contains about twelve attempts at formulating the classical Aristotelian syllogism as a system of the inclusion [altematively, co-inclusion], conjunction or overlapping, and negation of terms. (See P xxxii--xlix, S 159--209.) Here terms are initially properties or concepts thereof. Soon Leibniz generalizes his calculus to treat also propositions, using first the word 'concept' as the genus, but then using the expression 'term' to cover propositions as well:

Any letter, such as A, B, L, etc., means for me either some integral term or a different integral proposition. [Just before the numbered sections; see also (4) and (13).]

(32) I take a concept [notionem] to be incomplex or complex. Terms are incomplex categorematic [concepts]. [Propositions are complex concepts.]

Leibniz is thus committing himself to viewing his rules and axioms for incomplex categorematic terms or con-

cepts (that is, syllogistics) as also applicable to complex concepts, i.e., propositions. This generalization of his calculus is precisely our topic here.

The results of Leibniz's efforts at producing a generalized syllogistics vary. This is due in part to the varying assumptions underlying his different formulations. Among these assumptions is the topic of Leibniz's study: initially concepts and their necessary connections, but in the end also their co-exemplifications. He takes up other issues that lead him to some theoretical vacillations -- or explorations. Among these are the existential import of the syllogistic terms, and whether his logic deals with extensions or with the concepts themselves. I prefer NOT to regard any of those formulations or their underlying motivations as final. I see them as experimental theoretical exercises. From a Darwinian perspective (see Note 6) more weight should be assigned to the last stage of those investigations.

Like everybody else, I place a great value on the general objectives of Leibniz's logical investigation. Above all is his chief goal of developing a comprehensive logical calculus. As remarked, he meant this to include syllogistics, not as one alternative interpretation, but as the nuclear component. Next in line is his desire to develop an algebra of thoughts. Algebra in his day was a theory of numerical equations.

Hence, Leibniz's plan to develop an equational system of logic is central to his project of building a generalized syllogistics that includes a propositional logic. This plan pivots on a form of sameness, or equality, as the basic type of predication. This is his copula coincidence, or virtual sameness (iidem virtu- aliter) (C 362, P 53, S 20--21). In the GI he represents 'coincides with' by an equal sign '--'. Later on he used instead ,oo'. This post-GI change of notation signals an emphasis on the fact, already underscored by his use of predicative expression 'coincides with', that coincidence is not identity.

Leibniz's principle of substitution of coincidentals is a crucial rule of inference. On this he has been fully honored through the well established practice of calling 'Leibniz Law' the conflation of identity with indiscerni- bility. Yet his principle of substitution of coincidentals is much more general. It includes the 'substitution of material equivalents in all truth-functional compounds. This type of substitution is the keystone of his propositional logic. Clearly, for this development he had to discover the notion of truth-functional connectives, in particular material equivalence and material conditionality. This discovery, properly recorded early in the GI, I

L E I B N I Z ' S C O M P L E T E P R O P O S I T I O N A L L O G I C 17

wish to highlight in this preliminary developmental section.

The logical discussion in the GI starts with a necessary sameness or equivalence between concepts. This sameness is nicely illustrated by the properties (concepts) trilateral and triangle, which constitute Leibniz's second example (C 362, P 53, S 20--21). It is easily extended to cover the sameness between man and rational animal, a steady example pair in the GI. In these cases it is patently logically or semantically impossible that something instantiates one concept without instantiating its coincidental. This restriction is obviously a drastic limitation inimical to the idea of a generalized syllogistics. 8

Interestingly enough, Leibniz's very first example of sameness is Alexander the Great is the same as [coincides with] the King of Macedonia who conquered Darius. This is a paradigm example of a contingent proposition. His choice of the first example may signal that he is seeking after a generalized syllogistics applicable to both necessary and contingent propositions. If so, behind his example the idea must have lurked of a general concept of sameness or coincidence that encompasses both necessary and contingent predication. 9 Here we do not pursue Leibniz's theory of predication, except to the extent that it applies to propositions. Clearly, the distinction between contingent and necessary sameness or coincidence extends to those propositions having as components, not incomplex concepts, but complex ones (propositions). This is precisely the generalization of his coincidence as a form of predication to material equivalence.

For Leibniz a non-contingent subject-predicate proposition is necessarily true if its subject includes its predicate. Thus there is a necessary, as well as a contingent, proposition of the form Alexander the Great was the same as the king of Macedonia who conquered Darius. Hence, most likely when Leibniz offered this as his first example he may have been thinking of the necessary proposition of that form. Nevertheless, in none of these cases is there a literal or strict identity of concepts. In fact in the very same passage (C 362, P 53, S 20--21) Leibniz contrasts the virtual sameness his coincidence captures with the formal sameness or literal identity, which is absolutely fundamental.

Some recent authors have argued that Leibniz's principle of substitution of coincidentals is a principle or criterion of concept identity. 1~ This interpretation mis- construes Leibniz's view of concepts and propositions. It attributes to him a confusion between equivalence

and identity. To illustrate, the concepts trilateral and triangle are different, even though it is necessary that whatever is trilateral be a triangle, and vice versa. 11 These necessities are grounded, not on a literal or formal identity, but on their necessary coincidence. The interpretation of coincidence as identity of concepts distorts Leibniz's logical accomplishment. It neglects his efforts at fulfilling his profound desire to erect a general logical calculus that applies to all propositions, whether necessary or contingent.

Leibniz tackles contingent and existential propositions. Thus, he generalizes his view of coincidence, or virtual sameness, as the form of copulation that forms propositions out of terms. Contingently co-extensional, but necessarily different properties are coincident or the same. To use my own example, we have the sameness between men in this room and philosophy teachers or students.

Contingent sameness seems initially in the GI to bother Leibniz somewhat. Yet he promptly appreciates what has turned up in his hands. He immediately acknowledges that the modal terminology he has been using for coincidence is not adequate for contingent propositions. He accepts with equanimity the required generalization as follows (where the capitals express my own exegetical emphasis):

(32bis) B not-B is impossible,THAT IS: if B not-B = C, C will be impossible. [Note:] In the case of incomplex terms [i.e., terms that are not, or do not represent, propositions] impossible is NOT-BEING, in the case of complex terms [i.e., propositions] FALSE.

This is a great moment of insightful generalization. We shall see its pivotal role in the final stage of Leibniz's logical investigation in GI 198 principles 4 and 6. Undoubtedly, the form B not-B is itself impossible in the sense of being contradictory. Yet a concept C coinciding with B not-B need not involve a contradiction. If concept C is a property, then its coinciding with B not-B may be just its not being, i.e., its not having instances. If concept C is a proposition, its coinciding with B not-B is simply tantamount (that is, is materially equivalent) to its being false. Let's savor this fully. Clearly, if coincidence were literal identity, each contradiction would coincide with itself and nothing else, not even another contradiction. If coincidence were logical equivalence, then all contradictions would coincide with each other, but with nothing else. If coincidence were a conceptually or semantically necessary

18 H E C T O R - N E R I CASTAIq EDA

equivalence, all conceptually or semantically impossible propositions would coincide with one another, in particular with contradictions; but they would not coincide with any contingent falsehood. Leibniz is, however, extending farther than this his notion of coincidence. Part of what he is claiming in GI 32bis is this: A proposition C coinciding with an obvious contradiction B not-B is not sufficient to make C either a contradiction or a conceptually necessary falsehood. It is sufficient to make it simply impossible -- but in a special sense. Then he hurries to explain that by 'impossible' he means merely, in the case of propositions, false!

In sum, in (32bis) Leibniz is generalizing his copula coincides with in its propositional role as a connective to be, not a modal connective, but a truth-functional one. He executes this generalization by approaching coincidence through the modal interpretation suitable for his early examples. He quickly changes this modal interpretation. According to (32bis) the propositional impossibility suitable for general coincidence is just falsehood. Hence, any falsehood may be taken as a representative of falsehood. Of course a safe choice is an obviously necessary falsehood, like B not-B. Here is one half of material equivalence. The other half pertains to truth. This is trivial half if one accepts bivalence.

Palpably, then, Leibniz's uses of modal terms after GI 32bis must be handled very gingerly. In particular, we should expect Leibniz himself to use 'impossible' in the more general sense of necessarily not the case. This may be, yet perhaps is not, the case in:

(33) Hence, if A -- not B, A B will be impossible.

Anyway, in line with this extended use of 'impossible', the word 'inconsistent' in GI 200 does not mean there self-contradictory, or logically incompatible. Rather, it has its etymological sense of not standing as, or existing or being, true together, whether because of logical reasons or because of contingent causes.

In GI 32bis Leibniz has his glimpse of his generalized coincidence both as a general contingent copula and, in the special case of propositional coincidence, as one half of material equivalence. Then he develops the idea. He moves from modality to deducibility and from material equivalence to the material conditional:

(35) A false proposition is one which contains that A B contains not-B (assuming that A and B are possible). I understand B and Y [A] to be terms or propositions.

Let's concentrate on the case of propositions. By (33) and bivalence, which Leibniz has endorsed at (1), we must interpret 'possible' as not-impossible, hence as not-false, therefore, as true. Thus, (35) is reporting that: Given that A is true and B is true, the proposition A B contains not-B is false, and this falsity can be adopted as the containment-paradigm of falsehood. Of course, if A is true and B is true, A B is true. Hence, (35) reports that:

(35*) A proposition C whatever [whether necessary or contingent] is false, if it contains (a given truth whatever [whether necessary or contingent] contains a given falsehood [whether necessary or contingent]).

Clearly, such a generalized propositional containment can be no other than material conditionality. Doubtless, (35) is too cumbersome to deliver a useful paradigm of falsehood. We can see, however, that Leibniz is on the verge of discovering that in classical propositional logic C ~ F implies not-C, where F is a falsehood, and '--" stands for the material conditional.

After some deductions Leibniz lives a dramatic experience. At GI 38 Leibniz poses a profound question. Obviously, all propositions of the form Everything that is both A and B is B, or, in terms of concepts or properties, A B is [contains] B, are true, indeed necessarily true. Leibniz wonders about their ground of truth. He finds nothing. He proposes that A B is B is like a definition. He had already at GI 16 proposed to analyze A is B as A -- A B and pointed out the presupposition that B ---- BB. Hence, A B is B coincides with A B = ABB, which reduces to A B = AB. These transforma- tions provide a proof of that truth. This is of course superficial and inconclusive. It transfers the problem to the ground of the truth A -- A, or, as he himself remarks at GI 16, of the presupposed A -- AA. Patently, this truth has no ground. It is an ultimate truth. In GI 38 the example provokes the question about the ground of truth of those ultimate propositions. The actual example is immaterial. The issue is metaphysical.

Leibniz, however, sees a logical side to that issue. The idea of "definition" is not here that of a reduction schema for the elimination of a complex. It is, rather, the idea of a general principle that can serve as a universal formal criterion of that which is allegedly "defined." This is a sort of internal logical pragmatism that justifies the truth of (Every) A B is B. What can that justifying logical function of this form may be? After a little reflective interlude in GI 39, in which he uses this form,


the answer dawns immediately upon Leibniz. At GI 40 he proposes: the justifying logical function of (Every) AB is B is to serve as a general formal criterion of truth:

(40)

(41)

A 'true proposition' is one which coincides with AB is B, or, which can be reduced to this primary truth. (I think that this can also be applied to non-categorical propositions.)

Therefore, since a false proposition is one which is not true (by 3), it follows (by 40) that a false proposition is the same as a proposition which does not coincide with AB is B.

This is a great moment in Leibniz's reflections. He does not refer back to GI 32bis, or (35). Nevertheless, in (40) he continues the line of reflection registered in (35). He is turning the topic of (32bis) upside down: putting modality aside and approaching it from its proof-theoretical side. He now focuses on truth. He is now consolidating in focal vision the glimpse of (32bis) and the insight of (35).

In (40)--(41) Leibniz records his full discovery of material equivalence on the basis of bivalence. Negation partitions semantically the domain of propositions into two subdomains: the false ones and the true ones. This dichotomy is equivalent to the logical dichotomy produced by the property coinciding with: AB is B and (bivalent) negation. It is also equivalent to the dichotomy inflicted by coinciding with: B not-B and (bivalent) negation. This is a powerful generalization in several directions: (i) coincidence is the most comprehensive propositional equivalence relation; (ii) propositions function as terms in propositions, or have terms corresponding to them that represent them in other propositions; (iii) coincidence is both the basic form of copulation, i.e. a proposition-forming connection, which links terms, and a crucial propositional connective, which links propositions. Leibniz has taken a powerful stride toward his comprehensive calculus that embrac- ing syllogistics.

The discovery of material equivalence is a major turning point in the logical investigation in the GI. By extending his copula coincidence to a universal propositional connective he sees other connections. In particular, he explores an intimate connection between contingent sameness and existential import. This has to be overcome. He also explores the connection between truth-value and proof. Once coincidence is adopted as a general connective, the distinction between necessary and contingent truths has to be reconsidered. All these

issues and others lead to several rounds of discussion of the syllogistic premises. Those explorations are very interesting in their own right. They lie, however, beyond the scope of this study.

Leibniz has now fully endorsed the extensionality of standard truth-functional propositional logic. Yet his terms continue to be as fine-grained, or intensional, as ever. Nevertheless, now that coincidence is a connection subsuming material equivalence, substitution of coincidentals needs a restriction. Substitution of contingent coincidentals is not valid in modal or psychological propositions. Indeed substitution of necessary equivalents is not valid in psychological contexts. Leibniz is aware of this, but not fully:

(19) . . . (Except in the case of propositions that could be called formal, in which one of the coincidentals is taken formally in such a way that it is distinguished from the others. Actually these are reflexive, and do not speak so much about a thing, as about our way of conceiving it -- in which there is certainly a distinction.)

This is not the time to subject GI 19 to careful exegesis. In GI Leibniz is not always formulating general principles about natural language. One of his main endeavors is to develop a formal propositional-syllogistic calculus. Within this calculus he has no need to restrict his rule of coincidental substitution. The calculus is a closed system that allows as formulas only those that conform to its formation rules. With this under- standing, let's proceed to Leibniz's last attempt at such a calculus in GI.

2. Leibniz's final syllogistico-propositional system in the GI

In GI 195--200 Leibniz sums up his chief results and insights. Most impressive! Here is the full essay in Parkinson's translation with some noted revisions in capital letters, my additions and comments within braces:

(195)

(196)

A proposition is that which states what term is or is not contained in another. So a proposit ion can also state some term to be false, {FOR EXAMPLE} if it says that Y not-Y is contained in it, and to be true if it denies this. A proposit ion is also that which says whether or not some term coincides with another; for those terms which coincide are contained in each other reciprocally.

A proposition is false if it contains opposite propositions,

20 H E C T O R - N E R I C A S T A I q E D A

such as *and not-*. (197) A proposition itself can be conceived as a term: thus, 'Some A is B', i.e. 'AB is a true term' is a term -- namely, 'AB true'. Again, we have 'Every A is B', i.e. 'A not-B is false', i.e. 'A not-B false' is a new term; and again 'No A is B', i.e. 'AB is false', i.e. 'AB false' is a new term.

{Here I subs t i t u t e the in i t ia l i nde f in i t e a r t ic le ' A ' for

P a r k i n s o n ' s 'The ' , r e s to re L e i b n i z ' s ' n e w ' i n s t ead of P ' s

r e p l a c e m e n t ' t rue ' , a n d r e a r r a n g e P 's q u o t a t i o n marks.}

(198) Principles {I h a v e r e s t o r e d L ' s A r a b i c o r d i n a l s

in p l ace o f P 's E n g l i s h o rd ina l s . I t r ans l a t e L 's u n q u a l i -

f ied 'est ' , n o t as 'exists ' l ike P, b u t as 'IS T H E C A S E ' .

M y h e r m e n e u t i c a l ( theore t i co -exege t i ca l ) r e a s o n is tha t

jus t as L e x t e n d e d the m e a n i n g of ' i m p o s s i b l e ' in G I

32bis , he n e e d s the r ev e r s e e x t e n s i o n for 'est ' . H e r e

'exists ' is t oo emp i r i c a l a n d c o n t i n g e n t to do the job .

T h e r e is a lso a n exege t i co - t ex tua l r ea son . I n G I 144 L

goes t h r o u g h grea t t r o u b l e to d i s t i ngu i sh a n essen t ia l

f r o m a n exis ten t ia l copu la , a n d he uses a qua l i f i ed 'est '

for the lat ter , n a m e l y : 'es t vel existit ' . A s I see it, 'exis ts '

w o n ' t do for a gene r i c 'es t ' tha t has 'exis ts ' as a species .)

(1) Coincidentals can be substituted for one another. (2)AA =A. (3) not-not-A ~ A. (4) That term is false, i.e. not true, which contains A not-A;

that term is true which does not contain it. (5) A proposition is that which adds to a term that it is true or

false; as, for example, if A is a term and there is ascribed to it A's being true or A's being not true. It is also often said simply that A is the case or that A is not the case.

(6) The addition of 'true' or 'is the case' leaves things as they were, but the addition of 'false' or 'is not the case' changes them into their opposite. So if it is said to be true that something is true or false, it remains true or false; but if it is said to be false that it is true or false, it becomes false from being true and true from being false.

(7) A proposition itself becomes a term if 'true' or 'false' is added to the term. Thus, let A be a term, and 'A is the case' or 'A is true' be a proposition; then 'A true,' or 'that A is true', or, 'that A is the case' will be a new term, from which a new proposition can in turn be made.

(8) That a proposition follows from a proposition is nothing else than the consequence is contained in the antecedent, as a term in a term. By this method we reduce inferences to propositions, and propositions to terms.

(9) That A contains l is the same as that A = xl.

{I omi t Le ibn i z ' s m a r g i n a l n o t e a b o u t syllogisms.}

(199) The particular affirmative proposition: AB is the case. The particular negative: A not-B is the case. The universal affirmative: A not-B is not the case ({EVEN ]2} assuming that A and B are the case). The universal negative: AB is not the case.

From this it is evident at once that there are no more than these, and what are their oppositions and conversions. For a particular affirmative and a universal negative are opposed, as are a particular negative and a universal affirmative. It is also evident that in the proposition 'AB is the case' or 'AB is not the case' EACH OF THE TERMS OCCURS IN THE SAME WAY, {Parkinson translates: 'each term is in the same relation'} and so conversion simpliciter is valid. 'Not-A not-B is the case' or 'Not-A not-B is not the case' could be added, but this is in no way different from 'LM is the case' or 'LM is not the case', assuming that not-A is L and not-B is M. The universal affirmative, i.e. 'A not-B is not the case', is the same as 'A contains B'. For that A does not contain B is the same as that A not-B is true. Therefore, that A contains B is the same as that A not-B is not true.

{I o m i t h e r e the s e c o n d hal f o f Sec t ion 199.}

(200) If I say 'AB is not the case,' this is the same as if I were to say 'A contains not-B', or 'B contains not-A,' i.e. 'A and B are inconsistent'. Similarly, if I say 'A not-B is not the case', this is the same as ifI were to say 'A contains not-not-B', i.e. 'A contains B', and similarly 'not-B contains not-A'.

In these few principles, therefore, the fundamentals of [logical I form [formae] are contained.

{Here I have subs t i t u t ed ' p r inc ip le s ' for P 's ' p r o p o s i -

t ions ' as the impl ic i t ca tegor ia l n o u n a b s o r b e d in the

n o m i n a l i z e d Yew' [paucis] in the last s en t ence . Th i s

h a n k e r s b a c k to (198) , w h e r e L e i b n i z uses ' p r i nc ip i a ' as

the h e a d i n g of the p r inc ip les . I differ f r o m P a r k i n s o n in

c o n s t r u i n g the a n t e c e d e n t of the d e m o n s t r a t i v e p h r a s e

' this few [pr inciples] ' to b e n o t jus t the p r inc ip l e s l i s ted

in (198) , b u t all t hose g iven in the who le s u m m a r y

( 1 9 5 ) - - ( 2 0 0 ) . P e r h a p s L e i b n i z t hough t tha t the p r i n -

c iples i n (199) a n d (200) fo l low f r o m those in (198) .

Th i s is immate r i a l . E v e n if he was m i s t a k e n a b o u t that

de r iva t ion , the fact is tha t he he ld all of them.}

I I I . F o r m a l i z a t i o n of Leibniz's syllogistico- propositional calculus

1. Preliminary exegesis of Leibn&'s fundamental

principles of logic

Let ' s r u n d o w n t h r o u g h the text to disti l l the po in t s

L e i b n i z is m a k i n g at eve ry s tep of his s u m m a r y .

Sec t ion (195) is a m i x t u r e of two sub-pro jec t s .

O s t e n s i b l y L e i b n i z is p r o v i d i n g s o m e rules of f o r m a t i o n

for s e n t e n c e s tha t express t r u t h - v a l u e d p ropos i t i ons . H e

is a lso fu rn i sh ing a list o f p r imi t i ve exp re s s ions of his

calculus . H e is say ing o r i m p l y i n g the fo l lowing:

LEIBNIZ'S COMPLETE PROPOSITIONAL LOGIC 21

A. Primitive signs are the following: 1. A class of primitive terms: Presumably capital

letters of the Latin alphabet (These letters have as intended interpretation to stand for concepts.) Leibniz, as he explains at GI 26, uses letters, mostly capitals, but occasionally small letters, too, essentially as meta- variables ranging over terms (whether simple or complex) of the calculus he is constructing:

We must note something else about this calculus which we should have stated earlier. What is generally asserted or concluded, not as an hypothesis, about any letters that have not heretofore been used is to be understood as valid for any other letters. Thus, if A = AA is asserted, it will also be possible to assert B = BB.

2. The logical connectives: 'contains' (followed by predicative expressions, meaning inclusion or containment), 'not-' (meaning negation or concept-complemen- tation), ,2, (meaning coincidence), juxtaposition (meaning conjunction of concepts or propositions).

3. Semantic terms: T (denoting truth), and F denoting falsehood)

ABBREVIATION: We shall abbreviate Leibniz's primitive connective 'contains' as _ .

B. Terms are sequences of primitive signs of the following forms:

1. Incomplex or simple terms are the primitive signs of kind A.1 and of kind A.3.

2. If X is a term, so is not-X.

3. If X and Y are terms, not necessarily distinct, (X ___ Y) and X = Y are also terms.

*4. If X1, X 2 , . . . , X , are terms, so is X I X 2 . . . X~, for any natural number n = 2, 3 . . . . .

Remark L Leibniz shows in his example that he takes juxtaposition of terms to denote a dyadic term-formation operator. It is not clear, however, that he takes this operation to be just dyadic. His reason in (199) for allowing "simple conversion" of A B into BA is this: in these formulas "EACH OF THE TERMS OCCURS IN THE SAME WAY" (utrumque terminum eodem modo habere). Clearly the way in question pertains to the juxtaposition form of the formulas. I presume that his using associativity so casually, as a matter of course without bothering to list an axiom or principle is another aspect of his view of juxtaposition. I submit that Leibniz took the operation denoted by juxtaposition to be of an indefinite variable rank: from 2 (perhaps 1) to any natural number. That is, the argument of this operation is, so to speak, a set, an un-ordered set, of terms. Hence, any way of listing them is just the same self-identical set.

Remark II. On this interpretation the laws of associativity and commutativity of the juxtaposition operator are deeply inserted in the formation rule for juxtaposition complexes. These laws would of course be theorems, derived not from the axioms of the calculus but from a fundamental formation rule like B.4*.

Remark III. I concede that in proposing this interpretation and rule B.4* I may be attributing to Leibniz, not merely a rule he obscurely envisaged, but a rule he never envisioned at all.

Remark IV. To be sure, the above interpretation attributes to Leibniz a logical principle that runs against well-entrenched views. Variable rank for logical connectives or for predicates is a taboo deeply-ingrained and thoroughly widespread in the logical and philosophical communities. This is not of course a reason not to recognize in Leibniz some such a view if the text recommends it. As I have indicated, the text does seem to recommend that attribution to Leibniz. If the violation of the taboo is an error, we must not refuse Leibniz (or any other historical logician) the right to commit that error. Or any other error, for that matter. On the other hand, that violating the taboo is an error has to be established.

Remark V. We may of course attribute to Leibniz the standard view that the juxtaposition operation is dyadic. Then we should expect him to deliver axioms and rules from which to derive the commutativity and the associativity of juxtaposition. Commutativity is easily derivable from the last principles in GI 200. However, Leibniz is on this occasion not interested in such a derivation. He has already given his reason for claiming commutativity. It has nothing to do with those last principles.

Let's continue with the formation rules in GI 195. C. Propositions are sequences of primitive signs of

the following forms:

1. X _ X and X = Y, where X and Y are terms 2. not-X if X is a proposition.

*3. If X1, X 2 . . . . , X n are propositions, so is X1X 2 . . . x.,

for n = 1 . . . infinity. Remark V. Rule *3 is meant to match B.*4. Remark VI. In building his formal system of logic

Leibniz has two desiderata based on ontologically important semantic distinctions. (It has often been said that Leibniz's metaphysics was drastically influenced by his logic. I demur. I see his metaphysics drastically influencing his theory of logic.) First, he conceives of the

22 HECTOR-NERI CASTAIqEDA

semantic terms T and F and the terms exists, or, better, it is the case, and its complement as mere terms. They are not propositions. They do not stand alone as premises or conclusions. Hence, the following formulas are not theorems: T, Existence, Ens, It is the case, not-F. This feature is reflected in the above rules. Second, Leibniz conceives the main difference between the two pairs of terms as follows. The term T applies to propositions to form further propositions. It is the case applies to non-propositional terms to form propositions. The pure syllogistic core of his system includes It is the case (Ens) and its complement (Non-Ens) and not T or F.

The above rules are inexact in not representing the difference between the two pairs of semantic terms. Here I am not concerned with that semantico-ontological difference. The reason is that our objective is to evaluate Leibniz's propositional logic. We are, for this purpose only, disregarding the syllogistic core of that logic.

Remark VII. Many contemporary logicians have included a general propositional symbol as a theorem. Similarly, Leibniz could have liberalized his formation rules and allowed both 'T' and 'F ' to be propositional symbols. Then his axioms and rules of inference in GI (198)--(200) can deliver T and not-F as theorems. This would follow from any theorem and (198).6. Because of the key role played by the rule of substitution, (198).1, no computational simplification is gained by "defining" one of these symbols as the negation of the other. The axiom in (198).4 is philosophically sound: it does not promote the reduction of F to T. An alternative development is to introduce a canonical paradigmatic logical truth, as Leibniz does in GI 40, say, A -- AA.

Let's return to GI 195. First of all, we adopt the standard definitions of 'deduction', 'derivation', 'theorem', etc. We use the standard sign ' t - ' to denote derivability of the formula after the sign from the formulas before it; when there is no formula before ' ~-' the formula following is signaled to be a theorem.

The final sentence of (195) describes a principle that may be an axiom of the form:

I- (A ---- B) _ ((A _G B) (B ~ A)).

Or perhaps this twofold rule of inference:

B = A ~- (A c_ B) (B C A )

A = B ~ (A c_ B) (B C_A)

These are all derivable from the principles in GI 198. The derivations pivot on (198).9.

In Section (196) Leibniz states this rule of reductio ad absurdum:

(A _ B), (A c_ not-B) F- A --F.

The 'absurdum' in question is the contradiction in the premises. What is established is merely the falsity of the statement A. This is consonant with the general sense of 'impossible' recorded above in (32bis). This principle follows from the principles in (198)--(200).

Perhaps Leibniz was thinking of an axiom like this:

((A ___ B) (A C not-B)) _ (A ---- F)

An axiom or a rule, what was really in Leibniz's mind? These uncertainties are of little consequence. Principle (198).8 is the so-called deduction theorem. Hence, the axiom formula can be derived from its corresponding rule. Conversely, by modus ponens the axiom can be used to derive the rule. Leibniz does not include modus ponens in his list of principles in (198)--(200). How- ever, as we shall see modus ponens follows from those principles. (He formulates it (55).)

An initial puzzle in both interpretations is that in the antecedent the term "A c B" must be a proposition. Indeed, the term A itself must be a proposition in the consequent "A -- F." Leibniz proceeds to dissolve this puzzle by immediately expanding his formation rules in (197):

B. Terms:

5. Propositions are terms.

Let's continue. The principles listed in (198) are of the greatest importance. They represent a good distilla- tion of previous sets of principles. For instance, Leibniz no longer posits as an axiom the principle of self- coincidence, t- A ---- A. This follows from principles (198).1 and (198).2.

Principle (198).1 is a powerful principle of substitution. Let F(X) be any proposition containing the term X zero or more times, and F(X/Y) be the proposition that results from F(X) by substituting occurrences of Y for occurrences of X zero or more times. Leibniz is asserting both:

(S*I) A =B, F(A) f- F(A/B)

(S'2) A = B, F(B) ~- F(B/A).

However, one of these rules is deducible from the other by deducing A ---- B ~- B --- A. (Thus: Axiom 2: ~- AA = A. By (S*.1) taking A as B, and AA as A yields: b- A


--- A. Hence by (S*.1) from the hypothesis A -- B, taking B for (the second) A: B ---- A. Hence: A -- B t- B = A.)

The principle of Double Negation, (198).3, is derivable from the principles given in Section (200).

Principle (198).4 contains one axiom and two rules:

Ax4.1. t- F - - not- T Rule 4.2. (B c A not-A) ~- (B --- F ) Rule 4.3. not(B c_ A not-A) t-- (B --- T)

Rule 4.2 and Rule 4.3 are derivable from the previous and later principles.

Principle (198).6 raises an exegetical question concerning the meaning of 'addition'. As the examples show, this addition is a predication. Leibniz does not say "contains." In an intensional sense to say that A is true is to say that A is identical with a truth, similarly for falsehoods. Recall the foregoing discussion of GI 40. There Leibniz equated the predication 'is true' with 'coincides with [the earmarked truth] A B is B. In (198).6, Leibniz is generalizing GI 40--41. Recall his parenthetical "2 think that this can also be applied to non-categorical propositions." Now the archetype truth T and the archetype falsehood F play jointly the role of the paradigm logical truth A B = B and negation in GI 40--41. This implies that any truth once it is established as such can function as the paradigm.

We have these two axioms:

Ax6.1. ~- (A = T)----A

Ax6.2 t- (A = F) ----- not-A.

Obviously, we need principles for the other connectives, for instance:

( A F ) = F

~- ( A T ) = A

~- (A C_ T)= T

f- (A _C F ) = not-A

These are, however, derivable from Ax6.1, Ax6.2 and the other principles. In fact, the rules and axioms suffice to derive all the truth-table properties of classical negation, conjunction, conditional, and biconditional.

Paragraph (19).7 iterates the formation rule that propositions are terms and form with the semantic terms further propositions. Here we find the distinction between T and it is the case.

Principle (198).8 is a most important rule of inference. It is the so-called deduction theorem, or Rule of

Conditional Proof (RCP). Let's see how. First he says: in a logical implication, say A t- B, the antecedent contains the consequent just as a term, that is, an incomplex term contains a term: A __. B here. This containment is of course logical, non-contingent. Hence, if A ~- B, then ~- A c_ B. Next Leibniz generalizes to all inferences. This generalization can be interpreted in different ways:

RCP.1. I rA 1 . . . . . An, An+ 1 ~- B, then f- A~ _C ( . . . (An _C (An+l _C B ) ) . . . )

RCP.2. If A~ . . . . , An, An + ~ ~- B,

then t-- (A1. �9 A,rAn + 1) C_ B

RCP.3. I f A 1 , . . . , A n , An+ 1 ~- B,

then A ~ , . . . , A n ~ An + 1 C_ B

Which one did Leibniz intend? Perhaps RCP.1 or RCP.2. The best choice is perhaps RCP.1. Yet it does not very much matter. Under exportation RCP.1 and RCP.2 are equivalent. RCP.3 follows from RCP.1 and modus ponens.

Leibniz seems to be proud of this rule. He hails it as a major generalization of his equational work on syllogistics. He claims that the RCP allows him a reduction of inferences to propositions and these to terms. To us it is clear that RCP unifies the deductive structure of a logical system. 13 There is in Leibniz's claim the hint that another branch of logic dealing with inferences as terms can be erected. Indeed a form of modal propositional logic seems to be lurking behind. Given the transitivity of deduction in his system that modal logic is at least of an $4 type. Given that theoremhood in syllogistics and in propositional logic is decidable, an $5 structure can be erected on his calculus.

Principle (198).9 is very important. Yet the present formulation is not as good as one he has given many times before in the GI, the first time in the note to GI 16 mentioned above. In GI 83 he refers back to (16). More significantly, he also gives the better formulation in (199). That formulation is, with my name:

Cond.Bicond. t- (A c_ B)-= (A = AB) .

This is an equivalence between containment (and, a fortiori, conditionality) and coincidence (and material equivalence). It could be used as a "definition" of containment. A special case of it deserves to be called "Leibniz's Law for Conditionals": (p --" q) ~ (p ~ p & q).

The system of rules described above is a satisfactory

24 HECTOR-NERI CASTAIqEDA

regimentation of propositional logic. In saying this I am assuming that the above rules of formation for juxtaposition include both the commutativity and associativity of the juxtaposition operator. As remarked above, in (199) Leibniz seems to think that this is certainly the case for commutativity. Now, from the rules in (200) and the other rules Double Negation, Commutation and the Rules 4.2 and 4.3 can be derived. Hence, associativity is the only one in need of special recognition:

I-- (A(BC)) = ( (AB)C) .

In GI (199) Leibniz offers a general and valid formulation of the syllogistic premises. The existential import of the terms has been set aside. Then he proceeds to discuss other rules. These pertain to the connection between containment and juxtaposition, i.e., between conditionality and conjunction. The formulas are all correct.

In GI (200) Leibniz summarizes and generalizes the main points he makes in (199). He presents the coin- cidences between the connectives in a general way, independently of any application to syllogistics. Three of them deserve special mention:

(200).1. F- (not-(A not-B) = T) = (A ~ B)

(200).2. F- (not-(A not-B) - T) -- (A E not-not-B)

(200).3. F- (not-(A not-B) -- T) -~ (not-B c not-A).

With the help of these three axioms Commutation and Double Negation are derivable. From Commutation, Double Negation, and (200.1) we can derive (200).2 and (200).3.

2. General comment

Leibniz's propositional logic deals very well with negation, conjunction, conditional, and biconditional. It leaves out disjunction. This is in the present system not a serious lacuna. It is a standard practice for logicians to construct formal systems with few primitive connectives and then "define" the others. The lacuna in Leibniz's system is the peripheral one of not having "defined" disjunction. Nonetheless, this can be easily corrected. Many avenues are open. The simplest one is this:

Def. A V B- -no t -A ___ B.

Now, Leibniz takes the connective _c to form propositions. Negation and conjunction form terms that need not be propositional. Thus, the best course would be to

introduce 'A V B' by means of De Morgan's equivalence. In this regard Parkinson's comment about Leibniz's not apprehending De Morgan is interesting.

If the formation rules for juxtaposition (conjunction) with variable rank are not adopted, then Leibniz did have a difficulty with the associativity of conjunction.

Except for my precisifications the above principles are Leibniz's very own at the end of the GI. They are most important. They provide a definite characterization of the basic logical form of propositions. He could have done with fewer principles. Some of them are redundant. This does not, however, minimize his claim about having provided the fundamentals of logical form.

I proceed now to show that a subset of those principles provides a complete system of propositional logic.

IV. The completeness of Leibniz's propositional logic

The following argument is straightforward. From Leibniz's principles we can deduce J. Barkley Rosser's axioms and rule of inference for propositional logic. This has been shown to be complete. 14 Hence, Leibniz's system is complete for propositional logic. Since Leibniz's system contains its truth-table semantics, some internal meta-theorems about them are forthcoming.

1. Rosser's axiomatization of the classical propositional calculus

Rosser's calculus is very economical. It has just three axiom schemes and one rule of inference.

Ax io m schemes: Axioms are well-formed formulas of the calculus that have at least one of the following forms, where we use the same meta-variables used by Leibniz for his system, and use '--" for Rosser's horseshoe:

R1. A --, AA

R2. A B --" A

R3. (A --, B) -" ( - ( B C ) --" - (CA) )

Modus Ponens: A --, B, A F-- B.

2. Some theorems in Leibniz's system

We adopt for the present system of Leibniz's the standard definitions of 'theorem', 'proof', 'derived rule',


L1.

L2.

L3.

L4.

L5.

L6.

L7.

L8.

etc. We translate Rosser's horseshoe and Leibniz's __. into ' ~ ', Leibniz's 'not-' into Rosser's ' - ' , and Rosser's dyadic juxtaposition into the dyadic part of Leibniz's juxtaposition. Rosser's sign for the material biconditional translates into Leibniz's sign ('ffi') for coincidence.

To derive Rosser's axioms and rules we need the following lemmata, easily derivable from Leibniz's principles.

~ A - -A

~- (A ffi B) = (B -- A)

( (AB)C) = (A(BC))

~- ( A B ) = (BA)

~- - - A ~ - A

~- - ( A B ) = ( A -~ - B )

~- (TB) = B

~- ( T = B ) f B

These can be derived very simply from a subset of Leibniz's principles.

3. Derivation of Rosser's axioms and rule

Recall that (S*I) is only one half of the rule of substitution at (198).1, and that Cond.Bicond is the rule formulated at (198).9.

R1.1. ~- AA - A (198).2 2. ~- A - - A A 1, L2; (S'1) 3. ~ A -- A(AA) 2, 2; (S'1) 4. ~- A ~ AA 3, Cond.Bicond; (S'1)

R2.1. F- (AB) = (AB) L1 2. ~- ( A B ) - - ((AA)B) 1, R1.2; (S*I) 3. F- (AB) -- (A (AB) ) 2, L3; (S*I) 4. ~ ( A B ) = ( (AB)A) 3, L4; (S'1) 5. F- A B ~ A 4, Cond.Bicond; (S'1)

R3.1. A ~ B Hypothesis 2. - ( BC) Hypothesis 3. A = A B 1, Cond.Bicond; (S'1) 4. B --" - C 2, L6; (S*I) 5. B -- B - C 4, Cond.Bicond, (S*I) 6. A • A ( B - C ) 3, 5; (S*I) 7. A = ( A B ) - C 6, L3; (S*I) 8. A • A - C 7, 3; (S*I) 9. A -* - C 8, Cond.Bicond; (S'1) 10. - (AC) 9 ;L6 ; (S '1 )

11. - ( C A ) 10, L4;(S*I) 12. A ~ B t- -(BC)--* - ( C A ) 2 - - 1 1 ; R C P a t

(198).8 13. F- (A ~ B ) ~ ( - ( B C ) ~ - ( C A ) ) 1 - - 1 2 ;

RCP

Modus ponens

1. A ~ B Hypothesis 2. A Hypothesis 3. A ffi A B 1, Cond.Bicond; (S'1) 4. A = T 2, L7; (S '1) 5. T = (TB) 3,4;(S*1) 6. T - - B 5, L8;(S*1) 7. B 6, L2; (S '1) 8. A --" B , A ~ B 1- -7 ;RCP

At this point we can use within Leibniz's system the derived rules and theorems of Rosser's calculus.

4. The truth-table conditions

1. TT----T Axiom schemaAx2 2. FF • F Ax2 3. A F - - F A ---- AA; A --' A [Cond.Bicond];

- (A -A) ffi T [(200).1]; hence: - - (A -A) ffi - T; ( A - A ) = F [(198).3, (198).4.11; A F -- A ( A ~A); - - (AA) -A [Assoc]; - - ( A - A ) [Ax2]; -- F [preceding step].

4. T F = F [from 3] 5. F T - - F [fromCom] 6. A T = A A T = A - ( A - A ) ; = - - ( A - ( A - A ) ) [Double Neg];

-- -(m --, (A-A) ) [(200).11;-- - (A --" F)[Ax6l; -- - (A -- AF) [Cond.Bic];

-- - (A ffi F) [2 above]; - ( - A ) [Ax6]; -- A

[Double Neg.].

The truth-table conditions for ' --" follow just as easily.

V. Conclusion

I have shown (to my satisfaction) that Leibniz's final attempt at a generalized syllogistico-propositional calculus in the Generales Inquisitiones was pretty success- ful. The calculus includes the truth-table semantics for the propositional calculus. It contains an unorthodox view of conjunction. It offers a plethora of very important logical principles. These deserve to be called a set


of fundamentals of logical form. Aside from some imprecisions and redundancies the system is a good systematization of propositional logic, its semantics, and a correct account of general syllogistics. For 1686 it was quite an accomplishment. It is a pity that Leibniz himself did not fully appreciate what he had achieved. It does seem to me that this was due in part, as the Kneales urge (Note 4), to his having kept the focus of his attention on traditional syllogistics. It is a great pity that he did not polish GI 195--200 for publication. The publication of GI 195,198, and 200 would have most likely promoted further research.15

N o ~ s

I. M. Bocheriski, A History of Formal Logic (New York: Chelsea Publishing Company, transl, and edited by Ivo Thomas, 1970), p. 266. Similar words occur in William Kneale and Martha Kneale, The Development of Logic (Oxford: The Clarendon Press, 1962), p. 320. 2 This is the word used by Parkinson in his Introduction to G. H. R. Parkinson, ed., Leibniz: Logical Papers (Oxford: Clarendon Press, transl, by Parkinson himself, 1966), p. lvii, hereafter P. 3 This is Bocheriski's word. 4 Different answers have been given. Leibniz was stuck with syllogistics and had a fixation with the subject-predicate dogma and the "analytic" view of truth (Kneale). Leibniz's neglect of disjunction may be why "Leibniz failed to state such laws as De Morgan's" (Parkinson). Parkinson takes issue (Note 5 to p. lxi) with Couturat's claim that Leibniz discovered the De Morgan laws. See Louis Cou- turat (ed.), Opuscules et fragments in~dits de Leibniz (Hildesheim: Georg Olms Verlagsbuchhandlung, reprint of the 1903 original, 1966), p. 425, Note 2. After the recent work on the logic of terms by Sommers perhaps the complaint that Leibniz was too addicted to the subject-predicate form needs some revision. See Fred Sommers, 'The Calculus of Terms', Mind LXXIX (1970), 1--39, and his The Logic of Natural Language (Oxford: The Clarendon Press, 1980). 5 The Kneales explain very well how Leibniz's major accomplishment was to produce a meager calculus of containment that is abstract, purely formal, and allows of different interpretations. 6 Generales Inquisitiones de Analysi Notionum et Veritatum, hereafter GI. I have consulted the Latin text in Couturat's Opuscules 356--399, to be referred to as C followed by page number, and also the Latin text and its German translation in Franz Schupp (ed.), G. W. Leibniz: Allgemeine Untersuchungen fiber die Analyse der Begriffe und Wahrheiten (Hamburg: Felix Meiner Verlag, transl, and long commentary by Schupp, 1982), to be cited as S followed by page number. I have made special use of Parkinson's translation in Leibniz: Logical Papers 47--87, referred to as P followed by page number. With several revisions it is the text I exegesize here.

In the present study I continue a type of investigation began in 'Leibniz's Syllogistico-Propositional Logic Calculus', Notre Dame Journal of Formal Logic XVII (1976), 481--500. In this earlier

paper I examine some middle sections of the GI where Leibniz seems to be constructing a generalized syllogistics that can be taken as a monadic predicate calculus. His treatment of Existence as a term like any other term leads to trouble. Then concerning propositional logic I turned from the GI to Leibniz's The Primary Bases of a Logical Calculus (August 1, 1690), C 235--237, and The Bases of a Logical Calculus (August 2, 1690), C 421--423. I neglected the final round of discussion in GI, and so apparently did Leibniz.

My two studies are grounded in what I have called the Darwinian approach to the history of ideas. On this methodology an author's views develop, fail, are abandoned, even contradict other of his/her views. On the Darwinian approach there is no assumption about the system the studied author has propounded. Claims about con- sistency, unity, and duration of views have to be established by piecemeal textual interpretation and slow building of those interpretations. Views have a place in the thinker's development. It is an error to pick and choose from anywhere in a philosopher's corpus to construct his or her lifetime view. That approach contrasts with the widely practiced approach, which I call Athenian. On this approach a writer takes the whole corpus as presenting one view -- as if this view had sprung into the philosopher's head whole and mature as Athena sprung out of Zeus's head. That is why in my early study I focused on two papers writted on consecutive days, thus presuming continuity of thought. In this study I focus on the final attempt at a logical system in the GI. For more on the Darwinian-Athenian contrast see Hector-Neri Castafieda, 'Leibniz's Meditation on April 16, 1676, About Existence, Dreams, and Space', Studia Leibnitiana, Sup. Vol. XVIII (1978): Leibniz a Paris (1672--1676), Tome III: 91--129. This includes a detailed illustration of the Darwinian methodology in the exegesis of a metaphysical text by Leibniz. 7 Jacobi's suggestion is grounded on very good reasons. In particular, Section 1 selects for discussion just some of the principles Leibniz considers before GI 195--200. Section 1 is really the beginning of another, developmental study of GI. Both points are well taken. s Leibniz was already aware that he needed a general theory of logic that applies to contingent and empirical propositions. A drastic limitation of his Ars combinatoria was that it applied only to "eternal truths." See P 5f, Paragraph (83), translated from GotO*n'ed Wilhelm Leibniz: Siimtliche Schriften und Briefe (Darmstadt and Berlin: Academy edition. 1923--), VI, I, 199. 9 It is worth-noting that (40) has nothing to do with Leibniz's celebrated theory of truth. On the standard interpretation, according to this theory a proposition is true if and only if the (concept of the) subject includes the (concept of the) predicate. Obviously, this principle is valid only for subject-predicate propositions. It would be valid for all propositions if all were of, reducible to, the subject- predicate form. Although Leibniz thought this reducible feasible, that reduction is not at issue in GI 40. The standard interpretation raises a serious question about Leibniz's view of the difference between contingent and necessary truths. Leibniz dealt with this question explicitly. In fact in GI 58--74, 130--136, 144--151 he discusses the distinction. As I read him, he ends up proposing to distinguish between necessary coincidence (sameness) and contingent coincidence (sameness). This requires that the inclusion of the predicate in the subject be only a necessary condition for truth in general, but sufficient for necessary truth. Yet all these matters are


broached but not taken up in the logical calculus at the end of the GI. This calculus is very general. Metaphysically speaking Leibniz's coincidence copula '= ' is a sameness schema, interpretable as necessary coincidence or as contingent coincidence, depending on the application to which the calculus us put. 10 Outstanding examples are Hid6 Ishiguro, Leibniz's Philosophy of Logic and Language (Ithaca: Cornell University Press, 1972), and Fred Feldman, 'Leibniz and Leibniz's Law', The Philosophical Review 79 (1970), 510--522. Ishiguro's claims are subjected to a detailed critical examination in Hector-Neri Castafieda, 'Leibniz's Concepts and Their Coincidence Salva Veritate', Nous $ (1974), 381--398. Feldman's thesis is discussed in Notes 3, 5, and 8 (pp. 397f). 11 The concepts trilateral and triangle composed the first example in his earlier discussion of 'coincidence' in Elements of a Calculus dated April, 1679. See C 52, P 20. He is clear that they are different concepts that are necessarily equivalent. The difference pertains to their identity as entities and, because of this, to their being thinkable content. The equivalence pertains to their instantiation, to their extensions. The essay does not, however, reveal Leibniz has decided to have a general logic that applies both to necessary and to contingent propositions. In the GI this decision comes to fruition. They seem to be the example at the back of his mind when he writes the parenthetical exclusion of (19) quoted below. In 1704 he uses the same pair again to insist on conceptual distinctions. New Essays, IV, vii, 2: " . . . as if one wished to say, the trilateral (or that which has three sides) is not a triangle, because, in fact, trilateraliy is not triangularity." 12 Klaus Jacobi has suggested that I explain why I insert 'even' here. It looks more like a correction of Leibniz than a translation of his words. My first suspicion that it was needed was the very fact that Leibniz felt the need to put the preface ETposito A et B esse to his formalization of the universal affirmative proposition. Why not in the other types? The preceding two formulas are positive. The universal affirmative is given a negative formula: A non-B non est. Leibniz undoubtedly wants to draw attention to a contrast. That A non-B does not exist when no A exists is immediately clear. So obvious is this that I cannot conceive Leibniz going through the trouble of ruling it out! The formula holds even in not so obvious cases in which A's exist. This justifies taking some trouble to underscore it. Given that Leibniz wants a general calculus, it is most likely that he means to generalize. The sentence has a syntactic ambiguity. The word Et can be taken as a conjunction meaning 'and' that link the sentence it begins to other sentences. Perhaps to the preceding formalizations. This is, however, somewhat strange. Why should Leibniz put 'and' to announce the third item of a list, rather than the last one? The other interpretation is more attractive: "Et' is an adverb modifying the verb 'posito'. My Latin-English dictionaries tell me that 'et' as an adverb means 'even' and 'also'. Perhaps to be taken in the sense of even 'et' requires a different grammar, e.g., a subjunctive form of the verb. This is beyond my parvularian knowledge of Latin. Thus, I have in favor of putting 'even' in the translation: (i) the contrastive style of the items on the list; (ii) the strange presence of 'et' in the midst of the listing; (iii) the semantic need to justify a very restrictive interpretation; (iv) a grammatical reason. If the grammar does not suffice to equate Leibniz's 'et' with my 'even', the alternative is not to translate 'et' altogether, as both

Parkinson and Schupp do. This leaves two minor mysteries behind. Why did Leibniz take the trouble not to include the obviously true cases? Why did Leibniz place an unneeded adverb 'et' before 'posito'? That the adverb "et' is unnecessary is suggested by (35), where 'posito' enters without 'et'?

In brief, if the grammar of the sentence Et posito A et B esse . . . does not justify my translating 'et' as 'even', then I am correcting Leibniz's grammar to capture in English his thought. 13 Unquestionably, in (198).8 Leibniz has envisioned a major meta-rule of propositional logic. Its meta-character shines through his talk of reduction. He is ostensively aware of its enormous signifi- cance. He writes as it were his own discovery. Yet it seems to have been known before. If Leibniz knew the Port Royal Logique by Nicole and Arnaud, probably he read in it a germ of RCP. His protracted correspondence with Arnaud suggests that he may have read this book perhaps when he was in Paris (1672--1676). If this is the case, what be read may have not had serious impact on him. The context of and the amount of the discussion of RCP in the Port Royal Logique is very different from that of Leibniz's discussion. My basis is the following passage from the Kneales (p. 320, again capitals signal my emphasis):

A small point of some interest in their treatment of REASON- ING is a REMARK that, when only one premiss of a valid syllogism is known to be true, the other premiss may be introduced as a condition to the conclusion. And when neither premiss is known to be true, we may even present them both as conditions in hypothetical statement. 1 [1. Logique, iii, 13.] HERE WE HAVE IN EFFECT A VERSION OF THE PRINCIPLE OF CONDITIONALIZATION (or 'deduction theorem') [i.e., RCP] which WAS TAKEN FOR GRANTED BY ARISTOTLE AND EXPLICITLY USED BY THE STOICS.

Obviously, there is a great difference between providing a rule for a special use of syllogisms and formulating a principle of logic, which grounds that use. Nicole and Arnaud's rule is practical and limited. Leibniz's concern is theoretical. His principle is very general and is a key element in a systematization of a comprehensive logic.

Concerning the Stoics the Kneales say (p. 170, with capitals conveying my added emphasis):

It seems VERY PROBABLE, therefore, that in such contexts Chrysippus allowed himself the use of supplementary premisses which had been derived from already accepted moods by the PRINCIPLE OF CONDITIONALIZATION. t Jl. This most important step in the RECONSTRUCTION was suggested by Benson Mates in a doctoral dissertation of 1948 on The Logic of the OM Stoa and elaborated in his Stoic Logic (University of California Publications in Philosophy, vol. 16). The suggestion that the principle of conditionalization MIGHT be the fourth thema was considered by O. Becker in Zwei Untersuchungen zur antiken Logik ( Klassisch-Philologische Studien, Heft. 17), p. 43, but not adopted by him.

It appears, then, that Leibniz had to carry out a Mates-type of research to know that RCP was already proposed, theoretically, by the Stoics. Obviously, too, the context of the Stoic theorization and Leibniz's are very different. The Stoics were investigating propositional logic in itself, entirely separate from syllogistics. Leibniz is in


the GI building propositional logic in the middle of syllogistics. Leibniz writes as if he never knew anything of Stoic logic. 14 j. Barkley Rosser, Logic for Mathematicians (New York: McGraw-Hill Book Company, Inc, 1953), Ch. 4. Rosser's system is readily comparable with Leibniz's because it uses negation and conjunction as primitives and these together with the conditional in the formulation of its axioms and rules. Yet we could have chosen any other axiomatizations. Systems with disjunction as a primitive sign require a definition of disjunction in Leibniz's system. That is, however, a trivial detour. On the other hand, systems containing a paradigmatic tautology, like T, or a paradigmatic contradiction, say, F, are somewhat richer than Leibniz's. We could extend Leibniz's system as indicated above. For some such systems see Alonzo Church, Introduction to Mathematical Logic, Vol. 1 (Princeton: Princeton University Press, 1956).

15 This paper was conceived in a Seminar on the Generales Inquisitiones offered by Professor Klaus Jacobi at the University of Freiburg during the 1987 winter semester. I am grateful to him for having allowed me to participate in that exciting seminar. I am grateful to all the seminar participants, especially to Professor Jacobi, Professor Klaus Erich Kaehler, Doctor Helmut Pape, and Herr Hans-Peter Engelhart for sustained and illuminating discus- sions of some passages of the GI. Jacobi was extremely kind in reading the second version of this paper with a highly refined comb. I am most grateful to him for having pointed out typos, stylistic infelicities, and conceptual obscurities. He also provided advice on the translation, and, most generously and cooperatively, offered suggestions for improving the exposition and the arguments.

leibniz's complete propositional logic

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