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7LOGICAL THEORY IN LEIBNIZ

INTRODUCTION: ELEMENTS OF LOGICAL THEORY IN LEIBNIZ’S PHILOSOPHY

After Aristotle, Leibniz’s advances in logic surpass those of any writer before the nine-teenth century and rival those of anyone up to Frege. Yet like so much of his pioneering intellectual work, Leibniz’s ideas in logic were not published and remained mostly unknown until the twentieth century. As a logician he is notable now for his prescience rather than for his influence, and while he can fairly be said to have discovered sym-bolic and mathematical logic, the actual lin-eage of contemporary logic goes back not to Leibniz’s work but to the later, independent discoveries of Boole, Schröder, Venn, Frege, Peano and others.

Still, readers of Leibniz’s philosophy will have come hard against the shoals of his logic in a few places even without exposure to his logical works. His intensional theory of truth and his doctrine of complete individuals con-cepts, for example, have enjoyed philosoph-ical acclaim, and notoriety, although both are in fact just consequences of more basic prin-ciples of his logic. In a characteristic and cel-ebrated passage Leibniz writes to Arnauld on 14 July 1686 about truth: ‘[I]n every true affirmative proposition, necessary or

contingent, universal or particular, the con-cept of the predicate is in a sense included in the concept of the subject. Praedicatum inest subjecto, or else I do not know what truth is’ (GP II, 56/M 63).

Take the universal statement ‘Every human is mortal’. It is true if and only if the concept expressed by the predicate term ‘mortal’ is included in the concept expressed by the sub-ject term ‘human’. As Leibniz indicates, the account extends also to particular propos-itions (e.g. ‘Some human is mortal’) and to singulars (‘Socrates is mortal’), and generally to all propositions of the basic subject-copula- predicate form ‘A is F’. This ultimately includes negative propositions as well, despite his restriction to ‘affirmative’ ones in the let-ter to Arnauld, a qualification he had crossed out in another, though earlier, formulation of the same claim (cf. A VI, iv, 223/AG 11). Fur-ther, it appears to be part of Leibniz’s larger view of language that all statements can be expressed or captured in this form, including relational statements, conditionals etc.; as he writes in a paper on language dated to 1687–8: ‘Everything in discourse can be ana-lysed into the noun substantive, Ens or Res, the copula or substantive verb est, adjectives and formal particles’ (A VI, iv, 886/LLP 16). So the reach of this analysis purports to be extremely wide. But even setting this aside,

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already just as an initial claim about the analysis of truth it is surprising to philoso-phers in two ways.

First, it has questionable consequences for the content of concepts in many instances. Perhaps mortality is included in the concept human, but creating a race of robot philoso-phers does not seem to belong to that con-cept even if it is true that humans should achieve such a breakthrough. Singular state-ments such as ‘Aristotle wore a beard’ like-wise yield counterintuitive results about the concepts of subject terms. And only a little reflection is required to see that, once an unrestricted principle of bivalence is adopted, the concept associated with a singular term, i.e. the concept of the individual denoted by that term, will include the concept of every predicate ever true of that individual. This doctrine of complete individual concepts is the one that so shocked Arnauld and trig-gered an avalanche of subsequent criticism and scholarship concerning Leibniz’s view of individuals, freedom, necessity and essentialism.

Second, the analysis of truth inverts what we would (now) typically take to be the nat-ural interpretation of truth conditions for statements, on which the truth of a statement is a matter of the relations among the exten-sions of its terms and not of their concepts or intensions. ‘Every human is mortal’ is true because the individuals to which the term ‘human’ applies belong to the class of indi-viduals to which the term ‘mortal’ applies. Also, it is not hard to see that this inversion is precisely what generates the counter-intuitive results about concepts. While it is astonishing to think that an individual’s con-cept might contain the concept of every predi-cate ever true of him or her, it is wholly unremarkable to suppose that an individual might belong to every class of individuals that

makes up an extension of a predicate true of him or her. Aristotle is among the humans, the beard wearers, the philosophers, the polit-ical advisers, etc., by virtue of the facts, known or unknown – even if, as we might think, his concept does not include each and every one of the corresponding concepts.

The embrace of an intensional theory of truth instead of an extensional one is not an accident on Leibniz’s part. He was well aware of the options and can be found spelling out the extensional analysis of truth in various texts and explicitly deploying an extensional approach to logic from time to time as well, as we shall see below. Indeed, it was Leibniz himself who coined the term ‘intension’ to contrast with ‘extension’ (cf. A VI, vi, 486), and he understood it as having the same meaning that ‘comprehension’ or ‘connota-tion’ had in such logic treatises of the day as the Port-Royal Logic (cf. Arnauld and Nicole 1996, p. 39). On this use, the intension of a term is the concept it expresses, while its extension consists in the (class of) individuals to which it applies. Concepts were taken to be composed of those sub-concepts they included and, if the account is pressed all the way to the limit, ultimately to be made up of simple conceptual ‘notes’ – primitive con-cepts – as basic elements of thought. In con-trast, extensions were understood as composed of the individuals (or classes of individuals) included in them, and the indi-viduals themselves are the basic elements in this scheme. Thus the intensional theory of truth has a relation of concept containment at its centre, while the extensional theory gives priority to a relation of class inclusion.

Leibniz’s preference for an intensional theory of truth is part of his more general intensional approach to logic as a whole, one he sees as natural and for certain rea-sons superior to an extensional approach,

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although he also considers the two to be formally equivalent, mere inverses of one another. As he writes in Elements of a Calcu-lus (1679), in noting his own favour for an intensional approach as against the exten-sional approach of the scholastics:

The Scholastics speak differently; for they consider not concepts but instances which are brought under universal con-cepts. So they say that metal is wider than gold, since it contains more species than gold, and if we wish to enumerate the individuals made of gold on the one hand and those made of metal on the other, the latter will be more than the former, which will therefore be contained in the latter as the part in the whole. By the use of this observation, and with suitable symbols, we could prove all the rules of logic by a calculus somewhat dif-ferent from the present one – that is, sim-ply by a kind of inversion of it (A VI, iv, 199/LLP 20)

Leibniz articulates the same thought that the two approaches are ‘inverses’ of one another even more clearly in a paper written nearly a decade later (A Study in the Calculus of Coincidents and Inexistents, 1686–7?):

Being quadrilateral is in parallelogram, and being a parallelogram is in rectangle (i.e. a figure every angle of which is a right angle). Therefore being quadrilateral is in rectangle. These can be inverted, if instead of concepts considered in themselves we consider the individuals [singularia] com-prehended under a concept. . . . For all rectangles are comprehended in the number of parallelograms, and all paral-lelograms are comprehended in the number of quadrilaterals; therefore, all rectangles are contained in quadrilaterals. In the same way, all men are contained in

all animals, and all animals in all cor-poreal substances; therefore all men are contained in all corporeal substances. On the other hand, the concept corporeal substance is in the concept animal and the concept animal is in the concept man; for being aman contains being ananimal. (A VI, iv, 838–9/LLP 136)

In fact Leibniz takes the intensional and exten-sional interpretations of logic and of the rela-tion expressed by the copula itself to be reciprocals: the concept expressed by a term A contains the concept expressed by a term B if and only if the class of Bs includes the class of As. Using the symbols ‘!’ for concept contain-ment and ‘!‘ for class inclusion (cf. Swoyer, 1995a; Mugnai, 2006), we may abbreviate this reciprocity principle with the formula:

(PR) (A ! B) iff (A ! B).

This is an important claim, and we shall con-sider it with more attention when we turn to the formal details of the logic. Also, we shall sometimes need a neutral term for the relation expressed by the copula in Leibniz’s account, and for this I shall use the label ‘inherence’ for Leibniz’s est, though in contexts in which there is no risk of unclearness I shall some-times just use the more natural ‘inclusion’ as the generic term for which concept contain-ment or class inclusion are specific interpret-ations. For now let us focus a little longer on the intensional and extensional interpret-ations and some broader philosophical issues linked with the choice between the two (cf. Adams, 1994, pp. 57–63).

Leibniz, like his predecessors, orders much of his work in logic around the traditional cat-egorical propositions from the theory of syllo-gism, the first four of which make up the ‘square of opposition’, given here with examples:

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The contrast between intensional and exten-sional interpretations is plainest in the case of the universal affirmative, as is Leibniz’s claim that the two are inverses of one another. On the intensional account, the predicate is included in the subject: the concept soldier is contained in the concept man. On the exten-sional account the subject is included in the predicate: the class of soldiers contains the class of men. The idea of inherence or inclu-sion as simple containment runs short after this case, however, and a more subtle treat-ment is required.

For the particular affirmative, ‘Some man is a soldier’, the extensional interpretation is still quite straightforward: the class of soldiers intersects the class of men (and the intersection is non-empty), though it may not contain the whole of the class of men. On the intensional interpretation, it is less evident what to say, as the analogy with intersection is harder to make out in the case of concepts. Leibniz’s proposal, as he puts it forward in Elements of a Calcu-lus, is that the particular affirmative will be true when something can be consistently added to the concept of the subject to include the predicate concept (A VI, iv, 198/LLP 19). So ‘Some man is a soldier’ is true if the concept soldier, or one including it, can be consistently added to the concept man. ‘A soldier-man is a soldier’ is clearly true using this device, and, therefore, so is ‘Some man is a soldier’.

An immediate objection to this ‘consistent addition’ proposal is that it counts too many

particular affirmatives as true. Although con-sistency may rule out some cases, it seems too lax a constraint. ‘Some philosopher is a ruler’ is true by Leibniz’s account, since ‘phil-osopher-ruler’ marks a consistent addition to ‘ruler’ and ‘A philosopher-ruler is a ruler’ will clearly come out as true on the intensional approach. But it may yet be that there are no philosopher-rulers, and counting the state-ment as true seems therefore to fly in the face of the facts. How can the particular affirma-tive be true without the existence of a subject satisfying the predicate? The objection assumes the standpoint of the extensional interpretation: the truth of the proposition should be a matter of the individuals to which the terms refer, not (only) the relations among the concepts. To that extent a defender of an intensional approach need not be dis-tressed by the objection. Still, if Leibniz’s principle of reciprocity (PR) between exten-sional and intensional interpretations is cor-rect, there should not be disagreement in the truth values the two interpretations assign to the same sentence. We shall want to ask whether Leibniz can answer the objection, a matter to which we shall return below.

Negative categorical propositions, univer-sal or particular, present a puzzle to both interpretations. The natural extensional read-ing of ‘No man is a soldier’ would be that the class of men and the class of soldiers do not intersect. Likewise, on the intensional inter-pretation, if we carry over Leibniz’s device of ‘consistent addition’, the result would be that on no consistent addition does the concept man include the concept soldier. In both cases this is to treat the relevant relation between subject and predicate not as a form of inclusion but as exclusion: class exclusion or concept exclusion. Leibniz is aware of this, and his preferred treatment is to read the negation as qualifying the predicate

universal affirmative: Every A is B.Every man is a soldier.

universal negative: No A is BNo man is a soldier.

particular affirmative: Some A is BSome man is a soldier.

particular negative: Some A is not BSome man is not

a soldier.

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rather than the copula. ‘No A is B’ then is understood as ‘Every A is not-B’; and ‘Some A is not B’ likewise is understood as ‘Some A is not-B’. Thus we have ‘Every man is a non-soldier’ and ‘Some man is a non-soldier’. We shall want to ask whether this is a satisfac-tory resolution to the problem, and again we shall return to it later.

In addition to the four categorical forms in the traditional square of opposition are the singular affirmative and singular negative, proposition types that include individual terms, here written as ‘X’:

Singular propositions present a case of spe-cial interest and are illuminating for Leibniz’s philosophy of logic. We shall consider two issues, one raised by each interpretation.

Extensionally interpreted, the expected analysis is that ‘Peter is a soldier’ is true just in case Peter is included in the class of sol-diers. In this case it can be asked whether ‘inclusion’ is containment or merely intersec-tion. Either answer seems to suffice, for the singular can effectively be treated equally as universal and as particular. As Leibniz notes, ‘Every Apostle Peter’ and ‘Some Apostle Peter’ coincide, since there is but a single individual (cf. GP VII, 211). So a class con-tains the class of (every) Peter if and only if it intersects the class of (every) Peter, thus yield-ing the desired equivalence.

There is a second fine point arising here concerning the inclusion relation. Are we to say that it is the class whose sole member is Peter which is included in the class of sol-diers, or Peter himself? This forces us to con-front two distinct notions of inclusion: (1) the relation of an element to a class, i.e. class

membership, and (2) the relation of a sub-class to a class. In modern terms we distin-guish these relations as x " y and x ! y. Standard set-theoretic treatments of quantifi-cation would assign the unit class or single-ton rather than the individual that is its single member as the object included in the exten-sion of the predicate. As scholars have noted, the relevant concepts are still embryonic in Leibniz’s work and, moreover, they are inter-twined with the part-whole relation (cf. Mugnai, 2006, p. 217), and the distinction between a singleton and its single member is not observed in Leibniz. So his extensional interpretation of logic slightly resists a direct translation into the usual contemporary set-theoretic framework.

It is possible with only small accommoda-tions to let (extensional) inclusion be inter-preted as the subclass relation, building in an epicycle to handle the case of singular prop-ositions (cf. Swoyer, 1995a, pp. 97f.). The mechanics matter little. What is of interest is that Leibniz’s own statements of the exten-sional interpretation are typically framed not with the device of classes at all but with plural terms, and this suggests a relation of inclusion that forgoes sets or classes entirely. ‘Every man is a soldier’ is true just in case all the men are included among the soldiers. ‘Some man is a soldier’ and ‘Peter is a soldier’ will be true just in case at least one man is among the soldiers or Peter himself is among the soldiers, respectively. This is no accident on Leibniz’s part. He consistently opts for plural phrasing rather than set-theoretic ones in his analyses of expressions for infinitely large classes precisely in order to avoid com-mitment to infinite sets (cf. A, VI, iii, 503; A VI, vi, 157). And in his most advanced work involving quantified statements, the plural forms are explicit as well (cf. GP VII, 215f., C 193f.). It would seem that the most natural

singular affirmative: X is B.

singular negative:X is not B.

Peter is a soldier. Peter is not a soldier.

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logical development of this phrasing is not by means of set-theoretic devices with an epi-cycle for singletons but with the logic of plurals, an area of study that has come into its own only recently (cf. Boolos, 1984; McKay, 2006; Nickel, forthcoming; see section 7 below). When the logic of plurals becomes more widespread and standard, replacing the usual artifice of set-theoretic analyses of natural language expressions for many purposes, as is my guess, Leibniz may again be regarded as having been ahead of his time and many of his successors.

On the intensional interpretation, singular propositions raise a similar question, and Leibniz’s answer produces one of his signa-ture metaphysical doctrines (cf. Kauppi, 1960, 213; Adams, 1994, pp. 62–3). The sin-gular affirmative ‘Peter is soldier’ is true, intensionally understood, just in case the individual concept Peter includes the concept soldier. But must the predicate concept be directly included in the subject or may it be included only by means of a consistent add-ition? That is, as in the extensional case, ought the singular to be treated as universal or as particular? Again the answer is both: ‘the singular is equivalent to a universal and a particular’ (GP VII, 211/LLP 115). Such an equivalence requires that the concept Peter includes the concept soldier if and only if a consistent addition to the concept Peter will include the concept soldier. It is clear enough that if the concept Peter includes the concept soldier, then so will a consistent addition to the concept Peter, for the concept soldier is already included in it. But what underpins the opposite implication that if the concept soldier is a consistent addition to the concept Peter, then the concept Peter includes it? This will be true only if every concept that can be consistently added to the concept Peter is already included in it: that is, the concept

Peter must be complete. This holds equally for affirmative and negative singular propos-itions, and Leibniz explicitly holds that for any opposed pair of singular propositions, one must be true and the other false: ‘If two propositions of precisely the singular subject are presented, one of which predicates one of a pair of contradictory terms and the other predicates the other, then necessarily one proposition is true and the other is false.’ (C 67) Therefore the concept of any singular term must likewise be ‘logically complete’ or ‘negation complete’ in the strict sense: for any individual term X, for every predicate P, either the concept of P or the concept of ~P is included in the concept of X.

Leibniz’s preference for the intensional approach over the extensional one is defended on grounds of naturalness (to Leibniz, any-way) and the wish to have the rules of logic free from a commitment to the existence of individuals. In Elements of a Universal Cal-culus, after noting the scholastics’ way of reading logic in terms of individuals that are instances of universal concepts, he writes: ‘I have preferred to consider universal con-cepts, i.e. ideas and their combinations, as they do not depend for their existence on individuals’ (A VI, iv, 199–200/P 20). The idea, presumably, is that the validity of the rules of logical inference should be prior to any contingent facts about what individuals there are, or whether there are any at all; so logic is best studied without the presuppos-ition of the existence of individuals. Another point in favour of an intensional approach is a familiar oddity involved in the extensional approach. The extensional approach faces a sort of dilemma about universal propos-itions. If ‘Every A is B’ requires that there exist some As to constitute the relevant extension of ‘A’, then it seems universal claims carry ontological commitment to individu-

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als. But sometimes we should like to allow some universal claims to be true even if they lack instances: for example, ‘Glass spheres are fragile’ is true even if there happen not to be any (though note the shift here to the generic rather than the universal proper). If, on the other hand, the extension of ‘A’ does not require the existence of any As but can stand empty as the null class, it seems to fol-low that in the case in which there are no As, both ‘Every A is B’ and ‘Every A is not B’ will be true, since the null class is automatically included as a subclass in any class. So, for example, it will be true that all philosopher-rulers are wise and that all philosopher-rulers are not wise. These are not strictly contra-dictory claims (we have been taught to accept), but it remains an uncomfortable result. And so too does the usual bromide that universal statements with no instances are always true but ‘only vacuously so’.

The intensional approach avoids those dif-ficulties. But as we have noted along the way, it has problems of its own. First, it seems to put too many sub-concepts into subject con-cepts, most spectacularly in the case of singu-lar propositions. Second, the ‘consistent addition’ analysis seems to count particular propositions as true so long as the predicate is consistent with the subject even without instances of the subject having the predicate. (This is in effect the inverse of the problem that the extensional interpretation faces with universal propositions.) It is counterintuitive in itself and also apparently at odds with Leib-niz’s claim that the intensional and extensional interpretations are equivalent, since, for instance, ‘Some philosopher-rulers are wise’ will be true intensionally interpreted but not extensionally interpreted. A third worry not yet mentioned concerns the identity of inten-sions. Suppose ‘Every A is B’ and ‘Every B is A’ are both true. In that case the intensions of ‘A’

and ‘B’ include each other. This implies that they contain all and only the same sub-con-cepts. If intensions are exhausted by their com-ponent concepts, then it seems that the concept of A and the concept of B are identical. (As we shall see below, Leibniz explicitly formulates this principle that co-inclusion implies iden-tity.) Even if the concepts should somehow contain further (non-concept) ingredients to distinguish them, they will nonetheless be at least equivalent and thus guaranteed to apply to exactly the same things. Either way this seems too strong, for even when coextensive the concept of A and the concept of B may seem distinct and to contain distinct sub-con-cepts; and it may be only by chance that every A is B, or that every B is A. Even if all and only senators are corrupt, it seems at least possible for the concepts senator and corrupt to involve different sub-concepts and to pick out distinct classes of individuals (overlapping or not). Yet on the intensional approach those possibilities seem ruled out. Again, that seems counter-intuitive, and it seems at odds with the idea of the equivalence of the intensional and exten-sional interpretations, since it seems extensions can coincide without necessarily coinciding, whereas intensions cannot.

The problems for the claimed equivalence between the intensional and extensional inter-pretations are mitigated once we see that Leibniz intends the domain of individuals for the extensional interpretation of logic to include not just actual individuals but all pos-sible individuals. This ensures the equivalence of the intensional and extensional readings of ‘Some As are Bs’, for there will be a consistent addition of the concept of B to the concept of A if and only if it is possible for an individual to be an A and a B. Likewise, ‘Every A is B’ and ‘Every B is A’ will both be true, extension-ally interpreted, only if the terms ‘A’ and ‘B’ are necessarily coextensive, and it is not

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entirely implausible to treat necessarily coex-tensive terms as conceptually equivalent. (Leibniz was aware of ‘hyperintensional’ dif-ferences between necessarily coextensive terms; cf. Mugnai, 2006, p. 218.) Still, the dif-ficulty remains for the intensional interpret-ation that it puts too much into subject concepts in its requirements of concept inclu-sion. It may help to recall that even in his logical works Leibniz inclines towards a divine view of intensions and the possibilities of things (cf. A VI, 4, 762), and perhaps God’s concepts of things really do include the whole truth about them. Also, if individuals are taken to be ‘world-bound’, it becomes easier to suppose that the concept of, say, this Peter (as opposed to any other possible individual) includes the concepts of everything that is ever true of him. If it did not, how would it be the concept of Peter himself rather than the concept of some other counterpart possible individual similar to Peter? Nonetheless, this spills over into a vast metaphysical debate about which it can fairly be doubted whether it belongs in the foundations of logic. Although Leibniz was fond of the ambitious claim that his metaphysics and logic were essentially two sides of the same system – as he wrote to Sophie, ‘the true Metaphysics is hardly different from the true Logic’ (GP IV, 292) – the right reply for Leibniz at this point may instead be an instrumentalist one: ‘It’s only a model, and it’s enough that it correctly identifies the valid inferences.’ In any case, we shall move on now to consider some details of the logical systems devised by Leibniz.

LEIBNIZ’S LOGICAL CALCULI

Leibniz’s writings on logic are scattered across many texts and topics. For our pur-

poses we can simplify the landscape by focus-ing on the formal logical calculi presented in four concentrated studies.

The first dates from 1679 and is presented concisely in the Specimen of a Universal Cal-culus and the Addenda to the Specimen (A VI, iv, N. 69 and 70). The second is worked out during the first half of the 1680s and cul-minates in Leibniz’s great 1686 work Gen-eral Inquiries About the Analysis of Concepts and Truths (A VI, iv, N.165). It incorporates the results of the earlier Specimen and expands on them significantly, yielding Leib-niz’s most extensive logical system, a full-blown ‘algebra of concepts’ if one that remains only quite roughly drafted. The third study of logical calculi comes in a series of papers written directly after the General Inquiries and is commonly referred to as the ‘studies in the calculus of real addition’, although this is slightly misleading as the key papers include both a calculus of real add-ition and a more comprehensive calculus of real addition and subtraction, the ‘plus-minus calculus’. The most notable documents here are A Study in the Calculus of Real Addition (1686/7, A VI, iv, N 177), and A not inelegant specimen of abstract proof, also known as A Study in the Plus-Minus Calculus (1687, A VI, iv, N. 178). These are more polished works than the General Inquiries, although narrower in focus and encompassing ‘weaker’ logical systems in that they do not have the resources to define all the key technical con-cepts of the General Inquiries. The fourth logical system is on display in On the Math-ematical Determination of Syllogistic Forms (C 410–16) and A Mathematics of Reason (C 193–206), dating from about 1705. It pro-vides the clearest notice of Leibniz’s full ana-lysis of categorical logic and the theory of syllogism as well as his most intriguing insights into the theory of quantification.

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Only a very limited report on any of these logical calculi can be offered here. I shall sketch the 1679 calculus most briefly and save the majority of the comments for the General Inquiries and the studies in real addition. We shall end with a short consid-eration of Leibniz’s efforts to capture syllo-gistic logic, his insights into quantification, and a fairly deep problem his preferred inten-sional approach faces concerning the concept of negation.

THE 1679 CALCULUS OF THE SPECIMEN

Leibniz’s system in the Specimen of a Univer-sal Calculus and the related papers arises out of his earlier attempts to give an arithmetical treatment of logic. In the calculus of the Specimen, Leibniz uses lower-case Roman letters for terms (a, b, c, etc.), a unary oper-ation of term negation (expressed by the Latin non), a binary conjunction-like operation (by juxtaposition: ab), and a binary predication relation between terms (by the Latin copula est) and its negation (non est), and relations of identity (eadem sunt) and distinctness (diversa sunt).

Leibniz reached no stable decision about which propositions were to be axioms and which were to be derived theorems, changing his mind about the membership of the axiom base across different expositions. This is characteristic of his logical studies, and we shall mostly ignore the distinction in consid-ering the propositions of his calculi here and elsewhere below. In presenting elements of the 1679 calculus, we adopt the usual con-vention of using the symbols ‘=‘ and ‘"’ for Leibniz’s Latin phrases for identity and dis-tinctness, and we use the metalinguistic terms

‘if’, ‘then’, ‘not’, ‘and’ and ‘iff’ in their usual meanings. Here then is a sample of the propos-itions of this calculus, grouped roughly by the logical concepts they highlight and not observing Leibniz’s order or numbering from the Specimen (cf. Rescher, 1954).

Predication: est, and negation: non-a

(i) a est a (ii) a est b iff non-b est non-a (iii) if a est b and b est c, then a est c (iv) if a est b and b est a iff a = b (v) a non est b iff not a est b (vi) a non est non-a (vii) if a est non-b, then a non est b

Conjunction- juxtaposition: ab

(viii) ab est a (ix) ab est b (x) a est bc iff a est b and a est c (xi) if a est b and b est c, then a est c (xii) if b est a and c est a, then bc est a (xiii) if a est b and c est d, then ac est bd

Identity

(xiv) aa = a (xv) ab = ba (xvi) If a = b and b = c, then a = c (xvii) a " b iff not a = b(xviii) if a = b, then b = a

Leibniz expressly states the principle of sub-stitutivity salva veritate for identicals: ‘Those terms are the same of which one can be sub-stituted in place of the other salva veritate.’ (A VI, iv, 282/LLP 34) He also makes it clear that he construes this calculus intensionally: ‘By term I understand, not a name, but a con-cept, i.e. that which is signified by a name; you could call it a notion, an idea.’ (A VI, iv, 288/LLP 39) It is in the Addenda that Leibniz states outright his idea that the est relation should be interpreted as that of container to content, whereas in the Specimen itself est is

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left unspecified but obviously is read as the ‘is’ of predication, without forcing a deeper analysis of the structure of that relation. Interestingly, Leibniz also makes clear that his calculus allows a systematic interpret-ation that reverses the content-container relation: ‘All this is easily proved from the one assumption that the subject is as it were a container and the predicate a simultaneous or conjunctive content; or, conversely, that the subject is as it were a content, and the predicate an alternative or disjunctive con-tainer.’ (A VI, iv, 291/LLP 42)

When ‘a est b’ holds by virtue of the fact that a contains b, we find that b is a conjunct of a, so that, in effect, a = by for some y (though y need not be distinct from b: for example, if a = b, since ‘a est a’ is always true as well). If, however, ‘a est b’ means that a is contained in b, then b turns out to be a ‘dis-junctive container’. The point is clearer if we consider an extensional interpretation on which ‘a est b’ is true just in case the exten-sion of a is contained in the extension of b. The term b now serves as a container of ‘alternatives’ or ‘disjuncts’ in the familiar way: its extension consists in a class of indi-viduals, say, x, y, z, etc., and ‘a est b’ will be true just in case a is (identical with) either x or y or z, etc. Still, the fact that the link between the predication relation and the operation of conjunction is not yet made explicit as a principle is a notable difference between the Specimen and Leibniz’s later calculi.

The 1679 calculus provides resources for rendering the traditional categorical forms for propositions, although in the Specimen and Addenda Leibniz says he is only going to address the universal affirmative, and he says he always assumes the subject letters to be prefixed with the ‘sign of universality’ (A VI, iv, 280). This time noting the traditional

letter names for the categorical types, the translations are:

With est indicating a container-content rela-tion, the categorical propositions come out true in much the way described earlier. Inten-sionally construed: (A) ‘a est b’ is true just in case the concept a contains the concept b; (E) ‘a est non-b’ is true just in case the con-cept a contains the concept not-b; (I) ‘a non est non-b’ is true just in case the concept a does not contain the concept not-b; (O) ‘a non est b’ is true just in case the concept a does not include the concept b. As noted earlier, it is not directly clear why we should accept this account of the particular affirma-tive and negative, I and O, and although Leibniz’s explanation, in terms of the ‘con-sistent addition’ account is in place in the same year in Elements of a Calculus, an explicit statement of the theory of complete individual concepts that this will require is still years away.

As Rescher (1954, p. 4) points out, this calculus will have to incorporate conditions of propriety for its terms requiring that they not be self-contradictory. In later develop-ments Leibniz expressly demands that a term is an Ens or a possible being, and says this is always ‘tacitly assumed’ (GP VII, 214; cf. A VI, iv, 783: ‘A non-A non est res’). This is not observed in the Specimen or the Addenda, however, and exposes the system to incon-sistency if we allow ‘b non-b’, for example, to

A: universal affirmative E: universal negativeEvery A is B No A is Ba est b a est non-b

I: particular affirmative O: particular negativeSome A is B Some A is not Ba non est non-b a non est b

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count as a term permitted for any principle of the calculus. Suppose, from (vi), that b non est non-b. Then by (viii), b non-b est non-b; by (vii) bnon-b est b; by (ii) non-b est non-(b non-b); and thus (b non-b) est non-(b non-b), apparently refuting (vi). Problems will also arise for the traditional inference rules of opposition and subalternation in the absence of a requirement of consistency of terms. The difficulties are precluded if it is required that for any ‘proper’ term a, there is no b such that a est b non-b.

THE CALCULUS OF THE GENERAL INQUIRIES

The General Inquiries provides a remarkable construction of an ‘algebra of concepts’ artic-ulated in 200 numbered entries. Leibniz evi-dently thought well of his results, and noted on the text: Hic egregie progressus sum (‘Here I have made great progress’, A VI, iv, 793n). Again, we shall select only a few items to consider and not follow Leibniz’s order or numbering.

Leibniz uses capital letters as terms for con-cepts or propositions. He then sets out a relation of inclusion expressed by the copula (sometimes continet but mainly est is used so that ‘A est B’ means ‘in A is B’ or ‘A includes B’); a conjunc-tion-like operation expressed again by the juxta-position of letters (e.g. AB); an operation of negation expressed by affixing the Latin non to any combination of letters; and the principle of substitutivity salva veritate. Again using the expedient of rendering Leibniz’s work here in semi-formal terms, updating some of his nota-tion (e.g. using ‘=’ for Leibniz’s ‘#’), and enrich-ing the formalism to note the more evolved state of the system of the General Inquiries, we may set out the elements of this calculus as follows (cf. Lenzen, 2004; Mugnai, 2006).

Capital letters A, B, C and so on will stand for non-logical terms. For logical symbols, we use the following:

A = B identity: A is identical (coincides) with B

AB conjunction-juxtaposition: AB (for example: ‘philosopher ruler’)

~A negation: non-AA ! B inclusion: A includes B (Leibniz’s

‘A est B’)

We shall help ourselves also to parentheses and the following metalinguistic symbols with their usual meanings: ‘&’, ‘#’ and ‘$’.

In the General Inquiries Leibniz considers but does not settle on a set of axioms from which to derive his many results. The principles he states, and often proves, are interesting inde-pendently of their status as theorems or assump-tions of his calculus. The selection of principles below, all of which he expressly states, gathers together a sample to help describe formal prop-erties of his logical terms. We also note relevant article numbers in the General Inquiries where the principles may be found.

INCLUSION

(1) A ! A [§37] (2) (A ! B) & (B ! C) # (A ! C) [§19] (3) (A ! B) iff (AB = A) [§83] (4) ((A ! B) & (B ! A)) iff (A = B) [§§30, 110, 119] (5) ((A ! B) & A) # B [§55] (6) (A ! BC) iff ((A ! B) & (A ! C)) [§35]

CONJUNCTION

(7) AA = A [§§18, 24, 26, 171] (8) AB ! A [§38] (9) AB ! B [§38](10) AB # BA [§6]

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IDENTITY

(11) A = A [§§10, 156, 171](12) ((A=B) & (B = C)) # (A = C) [§8](13) (A = B) # (AC = BC) [§102]

NEGATION

(14) ~A = ~A [§15](15) A " ~A [§11](16) ~~A = A [§96](17) (A ! B) iff (~B ! ~ A) [§77](18) (A = B) # (~A = ~B) [§§2, 9, 171](19) ((A ! ~B) & (A ! ~C)) # (A ! ~B~C) [§103]

It is a striking collection, even as a small sample, and to the eye of readers familiar with the celebrated nineteenth-century devel-opments in algebraic logic, several of Leibniz’s principles will jump out. Principles (1), (2) and (4) show that Leibniz’s inclusion relation ! is reflexive, transitive and anti-symmetric, respectively. This means that ! amounts to a partial ordering. But even more notable among his principles are (7) and (3), and, together with those two, (10). He regards principle (7),

AA = A,

as the characteristic law of his calculus, and it is precisely what would later become known as ‘Boole’s Law’. AA = A expresses the property of idempotence, and it is by Boole’s own account his main innovation in logic. Although the earlier 1679 calculus like-wise included aa = a, this takes on greater significance for the system of the General Inquiries in the light of Leibniz principle (3),

(A ! B) iff (AB = A),

which Leibniz regards as explaining the nature of intensional conjunction. Given his principle of reciprocity between extensional and intensional interpretations of inclusion,

(PR) (A ! B) iff (A ! B),

from (7) the following principle concerning extensional conjunction is an immediate consequence:

(*) (A ! B) iff (AB = A).

This tells us that extensional conjunction is, in effect, intersection, or the Boolean ‘meet’, an impression that is further confirmed in the obvious extensional interpretations of (5) and (6). Principle (10) expresses the commu-tativity of conjunction. Leibniz also assumes, though he does not state, the associativity of conjunction (i.e. A(BC) = (AB)C). With con-junction then defined as associative, commu-tative and idempotent, now adjoined to principle (*), plus the basic principles of negation, Leibniz’s formal algebra, inter-preted extensionally, precisely describes a system equivalent to the structure of a meet semilattice with complement.

In fact, as Lenzen has observed, principles (1), (2), (5), (6), (16), and Leibniz’s principle (stated at §200 of the General Inquiries and more perspicuously at C 407–8)

A~B is impossible iff A ! B,

taken together provide a complete axiomati-zation for an algebra of concepts that is iso-morphic to Boolean algebra: i.e. the algebra resulting from those axioms can be proved to be deductively equivalent to Boole’s (cf. Lenzen, 1984, p. 200).

This last principle just mentioned, con-cerning impossibility of a term, opens up a

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further layer of the logic of the General Inquiries. It is a modal system as well. Tied to Leibniz’s discussions of negation is the idea of the possibility or consistency of con-cepts or terms. In various places in the text Leibniz notes that a term A is taken to be possible (possibile) or a being (Ens). As he writes in a marginal note on §2: ‘A non-A is a contradictory term [contradictorium]. Pos-sible is that which does not contain a contra-dictory term, i.e. A non-A. Possible is what is not: Y non-Y.’ (A VI, iv, 749n8) Although he considers the idea of allowing terms to be contradictory and to correspond to no being (cf. §154), it is clearly regarded as a theoret-ical option outside his preferred framework. An interesting expression of the use of the concept of possibility for terms is on display in §55. Principle (5) above is stated there, and it looks at first glance simply to be the correlate for modus ponens for terms or propositions: ‘If A contains B and A is true, B also is true.’ (A VI, iv, 757) A modal com-ponent comes to light, however, as Leibniz then further explains: ‘By a false letter I understand either a false term (i.e. one which is impossible, i.e. is a non-being [non-Ens]) or a false proposition. In the same way, by true can be understood a possible term, or a true proposition.’ (Ibid.) Thus ‘true’ and ‘false’ applied to terms are indicators of pos-sibility and impossibility. Construed as denoting concepts, terms are true if the expressed concepts are consistent and false if they are contradictory. Construed as a denot-ing term for individuals, a term is true if it corresponds to a possible individual (or a class of them) and false if it does not. (Leib-niz makes no affordance for ‘impossible individuals’ to instantiate false ideas.) This fits smoothly with his view that the exten-sions of terms include all possible individ-uals and not merely actual ones.

The modal aspect of the account plays more than an incidental role in the logic of the General Inquiries. It serves to underpin a theory of propriety for terms – which seemed missing from the calculus of the Specimen – and, arguably, it shows that Leibniz’s propos-itional calculus, in its treatment of conditional statements, includes an account of strict implication.

Like Boole, Leibniz sees that his algebra of concepts can be used to construct a logical calculus for propositions. He is quite clear that his formal letters are meant to be inter-pretable either as terms (concepts) or as propositions. Even before the General Inquir-ies he had already noted that the contain-ment relation can be taken as a relation of implication between propositions, and he treats the relation of antecedent to conse-quent in a conditional as one of containment (cf. A VI, 4, 551). In the General Inquiries, Leibniz says that from the proposition A contains B can abstracted a single term, A’s being B, which itself can serve as a term to stand in a relation of inclusion (cf. §138). So, for example, the hypothetical proposition ‘If A is B, then C is D’, linking two categorical propositions, can then be represented as the containment of the consequent in the ante-cedent, so that A’s being B contains C’s being D. At §75 Leibniz writes: ‘If, as I hope, I can conceive all propositions as terms, and hypo-theticals as categoricals, and if I can treat all propositions in a very general way, this promises a wonderful ease in my symbolism and analysis of concepts, and will be a dis-covery of the greatest importance.’ (A VI, iv, 764/LLP 66) The simple principle that is the key to the account comes at §189: ‘Whatever is said of a term that contains a term can also be said of a proposition from which another proposition follows’ (A VI, iv, 785/LLP 85) The plan for a logic of propositions is then to

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treat categorical propositions as terms and to treat the relation of implication for condi-tionals by the same rules he uses to analyse categorical propositions in the theory of term-containment. He can thereby show that ‘absolute and hypothetical truths have one and the same laws and are contained in the same general theorems’ (A VI, iv, 777). The outline of the account is straightforward. Using the familiar notion from propositional logic, the rules for terms can be translated into rules for propositions by replacing ‘A ! B’ with ‘P # Q’, replacing ‘~A’ with ‘¬P’, and replacing ‘AB’ with ‘A & B’.

This yields a rich set of principles in prop-ositional logic, but Leibniz provides little more than the key schematic suggestions and does not develop propositional logic in detail here or elsewhere, although in passages across many texts he can be observed cor-rectly formulating various of the related rules. For all his interest in logic, he did not concern himself much with the propositional calculus. Still, the link between the modal theory of propriety for terms and the analysis of implication as containment holds his attention long enough for Leibniz to put his finger on the idea that the relevant notion of implication is that of entailment:

A contains B is a true proposition if A non-B entails a contradiction. This applies both to categorical and to hypo-thetical propositions, e.g. If A contains B, C contains D can be formulated as fol-lows: ‘A contains B’ contains ‘C contains D’; and thus, A containing B and at the same time C not containing D entails a contradiction. (C 407).

This accords precisely with his view that ‘A!B’ is equivalent not only to the falsity of ‘A~B’ but to its impossibility. Translated into propositional terms, this means that ‘P # Q’

corresponds not to ‘¬(P & ¬Q)’ of the material conditional but rather to ‘¬ Possibly ¬(P & ¬Q)’ of the strict conditional. It is a delicate matter to parse the meaning of the conditional occurring in an author’s logical texts prior to the full clarification of the dif-ference between the material and strict condi-tionals in the twentieth century. Leibniz does not discuss the distinction himself, and his formulations can be more ambiguous than the one just given. For example at A VI, 4, 656, he writes: ‘If L is true follows from M is true, this means that it cannot at the same time be supposed [non simul supponi potest] that L is true and M is false.’ Here the intended force of ‘cannot’ is not simply put beyond doubt. Given the careful deployment of the concept of possibility in the General Inquir-ies, the case for regarding Leibniz’s logic for propositions as including strict implication is quite plausible. Still, as Mugnai (2005, p. 178) observes, it would have been rather more rev-olutionary if Leibniz had given pre-eminence to the material conditional, as later authors of logical calculi did.

THE CALCULUS OF REAL ADDITION

Leibniz refines his work in logical calculus in essays written soon after the General Inquir-ies in which he elaborates his calculus of ‘real addition’ and the fuller ‘plus-minus calculus’. These are perhaps his most polished writings in logic, although they are by no means com-prehensive of his views and are less ambitious than the General Inquiries. In these papers Leibniz introduces a binary operation of con-junction denoted by the symbol ‘%’, taken to be analogous to but distinct from arithmet-ical addition, and a binary operation analo-gous to subtraction denoted by a long bar ‘–’

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infixed between terms. Leibniz also includes symbols for identity (or ‘coincidence’) and distinctness, along with the principle of sub-stitutivity salva veritate. He does not offer a study of negation. (The novel and difficult treatment of the idea of ‘real subtraction’ will have to be set aside here; for a development, see Lenzen, 2004, pp. 20–34.) Again, a few extracts may help to give the flavour of the account, this time taking directly from the language of the texts, in this case from the paper A Study in the Plus-Minus Calculus.

In this paper Leibniz uses the plain symbol ‘+’ with the same technical meaning as he does elsewhere with ‘%’; and even in other places he often uses ‘+’ for convenience once the new symbol ‘%’ is introduced. To high-light the intended meaning of the operation, I use ‘%’ throughout. Here, then, is a sample from the Plus-Minus Calculus:

Definition 1. Those terms are the same of which one can be substituted for the other salva veritate.

Definition 2. Those terms are different which are not the same, i.e. in which a substi-tution sometimes does not hold good.

Symbol 1. A = B means that A and B are the same, or coincident

Symbol 2. A " B, or ‘B " A’, means that A and B are different.

Definition 3. If several terms taken together coincide with one <term>, any one of those several is said to be in or to be contained in that one term, and the one term is said to be the container. Conversely, if some term is in another, it will be among several others which together coincide with that other term [ . . . and this term] is called an inexistent or content. . . . It can happen that container and content coincide, e.g. if one should have A % B = L, and A and L coincide; for then B

will contain nothing other than A, but if it does not signify A, it will signify Nothing.

Symbol 3. A % B = L means that A is in L, or is contained in it.

Definition 5. If A is in L, and some other term, N, should be produced, in which there remains everything which is in L except what is also in A (of which nothing must remain in N), A will be said to be subtracted or removed from L, and N will be called the remainder.

Symbol 4. If we have L – A = N, what is meant is that L is a container, of which the remainder is N if you subtract A from it.

Axiom 1. If the same term is taken with itself, nothing new is constituted; i.e. A % A = A.

Axiom 2. If the same term is added and sub-tracted, then whatever is constituted in another as a result of this coincides with Nothing. That is, A – A = Nothing.

Postulate 1. Several terms, whatever they may be, can be taken together to constitute one <term>; thus, if they are A and B there can be formed from these A % B, which can be called L.

(With omissions, from A VI, 4, 846–8/LLP 123–4)

Leibniz goes on to prove a little over a dozen theorems, including (Thm. I) if A = B and B = C, then A = C; (Thm. III) if A = B, then A % C = B % C; and (Thm. VII) if B is in A, then A % B = A, and its converse (call it ‘Thm. (VII*)’) if A % B = A, then B is in A. Those last two, (VII) and (VII*), together yield the principle that A contains B if and only if A % B = A, or again using ‘!’ for containment:

(#) A ! B iff A % B = A.

This is of course familiar from the General Inquiries already, having been noted as (3) in

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our discussion above, and it illuminates an important property of the conjunction oper-ation %. In this later essay, the remark in Leibniz’s Definition 3 shows a nice property of %. Leibniz considers the case in which A % B = L and A = L, and says that B either coin-cides with A or signifies Nothing. (As Mugnai [2006, p. 224] observes, Leibniz overlooks here the possibility that B is itself contained in A without coinciding with A.) Pursuing the latter alternative yields the result that A % Nothing = L, and then since A = L, it fol-lows that A %Nothing = A.

In his paper A Study in the Calculus of Real Addition, of the same year or perhaps the year before, we find very much the same treatment of the key concepts, though Leibniz stresses certain features of the % operation in a way that throws further light on the subject. For instance, the commutativity of % is given explicitly as an axiom, B % N = N % B, along-side the axiom expressing the characteristic law of idempotence A % A = A. In a scholium to those two axioms, the properties of % and aspects of Leibniz’s view of his logical calcu-lus in general are made even clearer:

As general algebra is merely the represen-tation and treatment of combinations by signs, and as various laws of combination can be discovered, the result of this is that various methods of computation arise. Here, however, no account is taken of the variation which consists in a change of order alone, and AB is the same for us as BA. Next, no account is taken here of repetition; i.e. AA is the same for us as A. Consequently, whenever these laws are observed, the present calculus can be applied. It is evident that this is observed in the composition of absolute concepts when no account is taken of order or repetition. Thus, it is the same to say ‘hot

and bright’ as to say ‘bright and hot’, and to speak of ‘hot fire’ or ‘white milk’, with the poets, is a pleonasm; ‘white milk’ is simply ‘milk’, and ‘rational man’ – i.e. ‘rational animal which is rational’ – is simply ‘rational animal’. It is the same when certain determinate things are said to exist in things: real addition of the same thing is vain repetition. When two and two are said to be four, the latter two must be different from the former. If they were the same, nothing new would result; it would be just as if, for a joke, I wanted to make six eggs out of three by first counting three eggs, then taking away one and counting the remaining two, and finally taking away one again and count-ing the remaining one. But in the calculus of numbers and magnitudes, A, B or other signs do not stand for a certain thing, but for any thing of the same number or con-gruent parts. (A VI, iv, 834/ LLP 142–3)

We now have the elements for a nice comple-ment to the results concerning logical con-junction from the General Inquiries in which Leibniz’s calculus (extensionally interpreted) yielded a meet-semilattice with conjunction equalling set-theoretic intersection. In the later papers, the principles of commutativity and idempotence are again expressly formu-lated for %, and again associativity (i.e. A % (B % C) = (A % B) % C) is assumed by Leibniz in his proofs but not stated. Adding now (#) A ! B iff A % B = A, and the principle that A %Nothing = A, the resulting system describes a structure equivalent to a join-semilattice in which ‘%’ denotes an operation that behaves formally like set-theoretic union (cf. Swoyer, 1995a and Mugnai, 2006).

Still, care must be taken in considering these results about the formal behaviour of Leibniz’s symbol ‘%’. To ‘read’ Leibniz’s

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algebraic logic extensionally – i.e. to inter-pret its symbols as implicitly defined by its laws and then consider a model of it in terms of sets of individuals – is one thing. To con-sider his theory of the extensions of terms in his preferred semantics is another. Thus, if we read inherence extensionally, so that ‘A est B’ means ‘the extension of A contains the exten-sion of B’, we find that ‘%’ denotes union and the formal structure is a join-semilattice. If we read inherence intensionally, so that ‘A est B’ means ‘the concept of A contains the con-cept of B’, ‘%’ denotes Leibniz’s relation of ‘real addition’. Yet certainly for Leibniz those two interpretations are not themselves equiva-lent; rather they are reciprocal. When Leibniz asserts (#) A ! B iff A % B = A, his meaning for the copula est (here ‘!’) is indexed to the notion of concept containment. What ‘%’ denotes, for him, is ‘real addition’ for con-cepts. His principle of reciprocity (PR) then tells us that the equivalent truthabout exten-sions is A ! B iff A % B = A. Under those conditions, of course, ‘%’ denotes intersec-tion and the relevant structure is a meet-sem-ilattice, as noted above in connection with the General Inquiries. (See Swoyer, 1995a, pp. 105ff. for further analysis.)

When the semantics comes apart from the abstract formal systems in this way, we encounter one of the deep, defining facts about logic: the same formal system of rules allows a plurality of different interpretations, yielding theories that may have in common only their abstract structure. This is a very ‘modern’ idea about logical systems, and one that Leibniz appears to have seen as well. As he points out, his inherence relation – ‘to be in’ or inesse – can be interpreted to stand for any number of different relations:

We say that concept of the genus is in the concept of the species, the individuals of

the species in the individuals of the genus, a part in the whole, and in the indivisible in the continuum. . . . In general this con-sideration extends very widely. We also say that inexistents are contained in those terms in which they are. Nor does it mat-ter here, with regard to this general con-cept, how those terms which are in something are related to each other or to the container. So our proofs hold even of those terms which compose something distributively, as all species together com-pose a genus. (A VI, iv, 832–3/LLP 141)

Leibniz was aware as well that his operator ‘%’ which he understood to express real add-ition, a relation of conjunction, would express instead a relation of disjunction if inherence is interpreted extensionally. This is what we noted earlier in connection with the Addenda to a Specimen of a Universal Calculus where Leibniz observed that ‘a est b’ could be read as ‘a contains b’ or ‘a is contained by b’, requiring only a switch from interpreting the content as a conjunct of the container to a disjunct or alternative within it (cf. A VI, iv, 291) – a choice of stances that naturally invokes either the intensional or extensional interpretation, respectively.

So in many respects Leibniz shows an atti-tude towards the study of logic far ahead of his time. O’Briant (1968, p. 25) suggests that Leibniz never published his works in logic in part because they lacked a suitable audience: ‘Who would have been his readers?’

This outpouring of ideas in the algebra of logic comes 150 years or more before the same ideas would be discovered with fanfare by the nineteenth-century algebraists. It is those sorts of results above all – though not only those results – that readers of the his-tory of logic have had in mind when retro-spectively giving Leibniz a position in the top

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rank of writers in logic. George Boole’s widow, Mary Everest Boole, reports that Boole himself felt ‘as if Leibniz had come and shaken hands with him across the centuries’ when he learned of Leibniz’s anticipations of his own work in algebraic logic (quoted in Latia 1976, p. 243; cf. Peckhaus, 2009). Boole was not alone in the sentiment. The mathematical logicians at the end of the nineteenth century, although their discover-ies were independent of Leibniz’s, immedi-ately recognized the importance of his results when they finally became known and accepted his priority.

There is much more in the details of the General Inquiries and the studies in real addition and subtraction than can properly be examined here. For the moment let us consider what the algebraic results already reviewed mean about Leibniz’s place in the canon of logic. As we noted above, principles can be extracted from Leibniz’s writings on logical calculus that together constitute a set of axioms and rules of inference sufficient to capture all Boolean algebra. Still, it is not clear how far Leibniz’s anticipations amount to a first discovery of Boole’s logic. We should want more than just the fact that key prin-ciples stated by Leibniz would amount to a complete axiomatization. The grasp of the relations between the propositions and the logical system itself, at least as reflected in the proof practices, should be central here as well.

On this score there are certainly gaps evi-dent between Leibniz’s work and standard formulations of Boolean algebra, probably the most striking of which is his relative silence about the operation of disjunction that is the logical dual of conjunction. Although conjunction and complement are together sufficient to define disjunction, and although in A Mathematics of Reason (c.

1705) Leibniz reserves the letter ‘v’ (short for the Latin vel) as a symbol for inclusive dis-junction (cf. GM VII, 57), Leibniz shows no interest in or evident awareness of the duality of conjunction and disjunction, whereas in modern logic this duality is central to the basic algebraic practice of logic. The passage from A VI, iv, 291, quoted above, noting how ‘%’ can be interpreted either as conjunction or as disjunction, is one of the very few men-tions in the texts, and the idea is not further developed. It is worth observing as well that the familiar equivalences between statements involving the two operations are normally effected by means of De Morgan’s rules, which themselves are quite neglected by Leib-niz, even in the few places where he notes or employs them in passing (cf. Mugnai, 2005, pp. 22f.). So although Leibniz’s principles can readily be codified in a way that yields a system equivalent to Boole’s, and Leibniz himself gives primacy to many of the key elem ents, it also seems clear that Leibniz did not share the grasp of a number of analytical relations among the terms of the calculus that would seem integral to the characteristic understanding of modern algebraic logic. Commentators through most of the twenti-eth century tended to underestimate the depth and breadth of Leibniz’s advances in logic, and often quite greatly. But it is not hard to exaggerate his accomplishment either, especially in algebraic logic where his papers are so rich with premonitory insights.

THE JUSTIFICATION OF SYLLOGISTIC LOGIC

As is widely recognized, one of Leibniz’s cen-tral aims in logic across his career was to develop a formal system that would justify

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the traditional theory of syllogism. It has often been suggested that his efforts fell short on exactly this point and, even, that he ‘never succeeded in producing a calculus which cov-ered even the whole theory of the syllogism’ (Kneale and Kneale, 1962, p. 337). This nega-tive verdict seems to be incorrect. Moreover, the logical papers in which Leibniz offers his most comprehensive treatment of the theory of syllogism are also where his most advanced work on a theory of ‘indefinite terms’ and other forays into the nature of quantification can be found.

The theory of syllogism concerns the trad-itional syllogistic inference forms involving the four types of categorical propositions, A, E, I and O, as well as the laws of ‘immediate’ inference: opposition, subalternation and conversion. Each syllogism is a three-part argument, with a major premise, a minor premise and a conclusion, where the premises share a common middle term that is elimi-nated in the inference to the conclusion. So, for example, we may have:

A Every C is DA Every B is CA Every B is D

This syllogism can be generically coded AAA for its constituent propositions; its trad-itional mnemonic name is Barbara. In a shorthand sometimes favoured by Leibniz, categorical propositions can be more pre-cisely coded by category letter, subject term and predicate term, so that ‘Every C is D’ becomes ACD, and a whole syllogism can be then expressed as a sequence of three propos-ition codes. For Barbara this yields the sequence: ACD ABC ABD.

There are 256 possible types of syllogisms, ignoring the order of major and minor premises, of which 24 are traditionally identified as valid

forms. The valid forms are further classified into four ‘figures’ with six ‘moods’ each. The moods of the first figure are Barbara, Celarent, Darii and Ferio, Barbari and Celaro, named after their constituent categorical forms: AAA, EAE, AII, AEO, AAI and EAO.

A logical theory that justifies the trad-itional theory of syllogism would need to prove the validity of the 24 valid forms and of the 8 immediate inferences:

Opposition: ¬ABC iff OBC, ¬EBC iff IBC

Subalternation: ABC # IBC, EBC # OBCConversion: EBC iff ECB, EBC # OCB,

ABC # ICB, IBC iff ICB

Leibniz discovers that the project of justifying traditional syllogistic theory can be reduced to proving the validity of Barbara, Celarent, Darii and Ferio, and the two rules of opposition. As he shows in the paper Of the Mathematical Determination of Syllogistic Forms (C 410–16), all the rest of the figures and moods can be derived from those six pieces plus a basic logical inference form he calls ‘regress’. In Leibniz’s approach there are five major elements. Bar-bari and Celaro follow from the first four moods plus the rules of subalternation. The rules of subalternation follow from Darii and Ferio. The moods of the second and third fig-ures can be reduced to those of the first by ‘regress’. The rules of conversion can be derived from the second and third figures. Leibniz says also that the moods of the fourth figure follow from the others by the rules of conversion, but he leaves only the words ‘FiguraQuarta’ at the bottom of the manuscript as a reminder for the promised analysis without carrying it out. In A Mathematics of Reason, however, Leibniz pro-duces that proof as well, and the reduction is complete (cf. Lenzen, 1988 and 2004).

In the Syllogistic Forms, Leibniz argues from first principles concerning containment

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relations for the validity of the four ‘primi-tive’ moods of the first figure (C 410f./LLP 105–6). In doing so he appeals to the mereo-logical concepts of part and whole in spelling out the ideas of containment, and having completed his justification he remarks: ‘These statements have no less geometrical certainty than if it were said that that which contains a whole contains a part of the whole, or that that from which a whole is removed has a part of that whole removed from it.’ (C 411/LLP 106) The general scheme of part and whole can be applied equally to intensional and extensional interpretations of the logic. For Leibniz, subconcepts are parts of con-cepts and subclasses are parts of classes, and in both cases the constituents can be resolved down to elements, whether primitive simple concepts or individuals. But as it is clear from the actual ‘direction’ of the containment rela-tions described in Syllogistic Forms, Leibniz is working here with an extensional inter-pretation: in the universal affirmative, for example, the individuals named by the sub-ject term are contained in those named by the predicate.

QUANTIFICATION AND PLURALS

Leibniz’s handling of ‘indefinite’ terms in various texts on logic is often reminiscent of contemporary uses of variables and quanti-fiers, and it shows that he was at least feeling his way towards a theory of quantification. Already by 1679 Leibniz had, in the context of a mathematical model of the syllogisms, worked out a formulation of the universal affirmative ‘a est b’ as ‘a = by’, where the intended reading of the identity is that a is the product of b together with some y (cf. C 57). The idea of expressing the categorical

propositions as algebraic equations is a last-ing thread of his formalism, and so too is the use of indefinite letters to indicate the idea of quantity. Across many later texts Leibniz uses capital letters, mostly from the end of the alphabet, as privileged symbols for quantity in his expressions of the categorical propos-itions and typically with an eye towards for-malizing syllogistic inferences.

In the General Inquiries, the use of a letter for an indefinite terms seems closer still to the existential quantifier when Leibniz explains his expression ‘A = BY’ this way:

[B]y the sign Y I mean something undeter-mined so that BY is the same as some B . . . , so A est B is the same as A coin-cides with some B, or, A = BY. (A VI, iv, 751/LLP 56)

Since ‘B’ itself is a definite term, the sign ‘Y’ appears to act as a quantified variable so that ‘some B’ means ‘something Y that is B’. Lenzen (2004, pp. 48–9) explicitly writes ‘&Y(A = BY)’ for this, and with some justice. There are various passages in the fragments on logic in which Leibniz’s own words strike directly the canonical contemporary phras-ing ‘there is a Y such that . . . . Y . . . . ’ (for example, at C 235/LLP 90: Si A = AB, assumi potestY tale ut sit A = YB; or C 261: dicere A non est B, idem esse ac dicere: datur Q tale ut QA sit non B). Leibniz can also be found appearing to formulate and apply instances of the rule for existential introduction in the General Inquiries at §§23, 24, 49 and 117.

Yet Leibniz often does not have elemen-tary aspects of the quantifiers sorted out in his use of indefinite terms, and in particular he does not see with clarity the quantifier negation rules ¬'XP iff &X¬P and ¬&XP iff 'X¬P. This puts into doubt the degree to which his treatment of indefinite terms

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effectively distinguishes between the exis-tential and universal quantifiers. Still, there are moments at least where he is strikingly trying to mark the difference, as for example in §112 of the General Inquiries where, struggling to sort the two out, he reserves a distinct notation ‘Ÿ’ expressly to mean ‘any Y’ in contrast to the plain sign ‘Y’ for ‘some Y’. And in a similar passage at C 271, he writes: ‘Let us see how X and ˆX differ, cer-tainly as some and any, but this occurs by accident and I want it to be X simpliciter.’ This might have been the start of a close study of the quantifiers and their rules, but he does not pursue the subject (cf. Lenzen, 2004, p. 53).

There are other provocative insights as well. As has been widely noted, in the later paper A Mathematics of Reason, Leibniz offers an interpretation of the categorical propositions that involves quantifying the predicate. The laws of syllogism, says Leib-niz, can be proved by an analysis given in terms of ‘same’ (euisdem) and ‘distinct’ (diversum), and he renders the categorical forms as follows (quotations excerpted from C 193/LLP 95):

A Every A is B: ‘Any one of those which are called A is the same as some one of those which are called B.’

E Some A is B: ‘Some one of those which are called A is the same as some one of those which are called B.’

I No A is B: ‘Any one of those which are called A is distinct from any one of those which are called B’.

O Some ‘Some one of those which are A is not B: called A is distinct from

any one of those which are called B’

Leibniz remarks, ‘Hence, by virtue of logical form, the predicate is particular in affirma-tive propositions and universal in negative propositions’ (ibid.).

Commentators have typically followed Couturat’s surprising claim that Leibniz puts this analysis forward in order to reject it (C 194 fn. 1). I agree here with Lenzen (2004, p. 65) that this opinion is ‘somewhat incom-prehensible’, as it is plainly Leibniz’s pre-ferred apparatus in the paper, and he puts it to use in the crucial derivation of the moods of the fourth figure. Nevertheless, quantifica-tion of the predicate admits of different inter-pretations, and it remains to consider how best to understand Leibniz’s treatment.

One way would be to use the familiar modern set-theoretic framework employed in interpreting second-order logic, so that, for example, the universal affirmative ‘Every A is B’ is taken to mean that for every mem-ber x of the set A, there is a member y of the set B such that y = x. With ‘"‘ indicating set membership and capital letters denoting sets, we put this into symbols as: ('x)(x" A # (&y)(x" B &y = x)) (cf. Lenzen, 2004, pp. 63f.). Thus Leibniz’s statements of the categorical propositions turn out to be ‘set theory in sheep’s clothing’.

But the set-theoretic device seems to be an unnecessary add-on to Leibniz’s own formula-tions, which read very naturally instead as quantification with plural terms. Just what are ‘those which are called A’ and ‘those which are called B’? The As and the Bs themselves. We do not need sets to express this or the correspond-ing categorical statements. The relation of sameness Leibniz isolates with the phrase ‘one of those which are called A’ can be stated more simply as ‘x is one of the As’. We then have:

('x)(x is one of the As # (&y) (y is one of the Bs &x = y)).

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This agrees quite closely with Leibniz’s own words, in Parkinson’s translation: ‘Any one of those [quemlibet eorem] which are called [qui dicuntur] A is the same as [eundeum esse cum] some one of those [aliquo eorum] which are called B.’ Leibniz’s Latin terms quemlibet and aliquo mark the number of the quantifiers as singular, but the under-lying logical ideas can be presented in number-neutral terms that allow a plural reading of the variables as well. A few easy pieces of formalism will help to draw this out. First, we use ‘variably polyadic’ varia-bles that admit of singular or plural read-ings (written here with a surface plural form): ‘xs’, ‘ys’, etc., and their correspond-ing quantifiers ‘('xs)’ and ‘(&ys)’. Second, we interpret the sign ‘=‘ for identity as not marked for number so that it can be taken as ‘is’ or ‘are’ as needed. Last, we introduce a variably polyadic relation of inclusion in a plurality denoted by ‘<‘ and rendered in English as ‘is one of’ in the singular or ‘are among’ in the plural (in fact ‘among’ is the more general, of which ‘one of’ is a special case: cf. McKay, 2006, pp. 57 and 135–7). The plural phrase ‘the As’ also admits of further analysis, but Leibniz does not parse it and so we shall leave it intact to mirror his constructions. With those in hand we can formulate Leibniz’s version of the uni-versal affirmative this way:

A ('xs)(xs< the As # (&ys) (ys< the Bs &xs = ys)).

Likewise, the other categorical forms are expressed as follows, with ‘"’ for the corresponding relation of being distinct from:

E (&xs)(xs< the As & (&ys) (ys< the Bs &xs = ys))

I ('xs)(xs< the As # ('ys) (ys< the Bs #xs " ys))

O (&xs)(xs< As & ('ys) (ys< the Bs &xs " ys))

This has the advantage of relying only on sameness and distinctness, plus (the general version of) Leibniz’s ‘is one of’ to capture all the key logical relations, which is clearly what Leibniz himself presents as the central innovation in his analysis in A Mathematics of Reason, as will be evident at the end of the next passage quoted below. It also has the advantage of forgoing the use of sets, and it provides a better treatment of a few remarks Leibniz makes just after explaining the cat-egorical forms. With the terms of his analysis in place, he observes that four less familiar forms of propositions are also readily con-structed. As he states them (numbers added):

It could also be that [1] every A is every B; i.e. that all those which are called A are the same as all those which are called B, i.e. that the proposition is reciprocal. But this is not in accordance with our linguistic usage. In the same way we do not say that [2] some As are the same as all Bs, for we express that when we say that all Bs are A. But it would be superfluous to say that [3] no A is some B, i.e. that any one of those which are called A is different from some one of those which are called B; for this is self-evident unless B is unique. Much more would it be superfluous to say that [4] some one of those which are called A is different from some one of those which are called B. So we see that logical theory is perfected by transferring matters from predication to identity. (C 194–5/LLP 95)

The logic of plurals handles the four alterna-tive statement forms comfortably without

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imposing set-theoretic devices. The first two fairly demand such treatment to avoid artificiality:

[1] ‘Every A is every B; i.e. that all those which are called A are the same as all those which are called B.’

[1*] ('xs)(xs = the As # ('ys)(ys = the Bs &xs = ys))

[2] ‘Some As are the same as all Bs.’[2*] (&xs)(xs< the As & ('ys)(ys = the Bs

#xs = ys))

Lenzen (2004, p. 64) notes (as a virtue) of the set-theoretic versions of [1] and [2] that they are not true unless sets A and B contain only a single element. This is plainly false of [1] and [2] if we take them as Leibniz does in plural terms (or we rely in terms that allow plural readings). All and only the As may be Bs, with no limit on how many As or Bs there are. Like-wise, it may be that although all the Bs are As, some As might still not be Bs, again with no limit on the numbers. The fact that the plural formulations of [1] and [2], but not the set-the-oretic ones, honour this is a clear advantage. Leibniz does regard [1] and [2] as out of step with common language – and they would be unusual to say, for us as well – but this hardly means that they are, or that he sees them as, false unless there is only one A and only one B.

By contrast, Leibniz says [3] and [4] are superfluous and ‘self-evident unless B is unique’.

[3] ‘No A is some B, i.e. that any one of those which are called A is different from some one of those which are called B.’

[4] ‘Some one of those which are called A is different from some one of those which are called B.’

They are self-evident if there are at least two Bs, since any A you pick will not be

identical with at least one of those Bs. But the corresponding plural (or variably poly-adic) formulations will not be self evident even when B is not unique, for it is not automatically true that just any As you pick will be distinct from some given Bs, even allowing that there are at least two Bs. Thus the plural readings would not capture Leib-niz’s intended meaning. On the other hand, the set-theoretic reading yields the ‘self evi-dent unless B is unique’ result as desired. But again the set theory is not necessary. If we provide a routine singular reading of the relevant variables for these sentences, Leib-niz’s result falls out neatly, still staying within the limits of the vocabulary of ‘=’, ‘"’ and ‘<’, as follows:

[3*] ('x)(x< the As # (&y)(y< the Bs #x " y))

[4*] (&x)(x< the As & (&y)(y< the Bs &x "y))

So it is evident by now that Leibniz’s quan-tification of the predicate does not need to be read as disguised set theory, and that his sug-gestion that predication can be transferred to identity is both correct and, again, prescient of a future development in logic. In this case it is a future development that is still under-way: second-order logic for predicate quanti-fication is lately finding a natural first-order replacement with expressions for plurals and inclusion in pluralities, and logical theory is being ‘perfected’ along the lines Leibniz him-self indicated.

The disposal of set theory for the inter-pretation of quantified statements involving plural terms in favour of a first-order logic for plurals has several motivations, of which two are especially notable for our purposes. First, ‘non-distributive’ predicates such as ‘form a circle by holding hands’ or ‘gathered in the yard’ are incorrectly handled by the set-theoretic analysis: when some children – A,

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B, C, say – form a circle by holding hands or gather in the yard, the set (A, B, C) does none of those things. Quantification draws out further contrasts here: all the children may have gathered in the yard, but it is not true that each child gathered in the yard (cf. Nickel, forthcoming). Leibniz is aware of such predicates. As he writes in A Study of the Calculus of Real Addition: ‘our proofs hold even of those terms which compose something distributively, as all species together compose a genus’ (A VI, iv, 833/LLP 141). All of the species together com-pose a genus, but it is not the case that each species forms a genus. It is not clear whether Leibniz recognizes the advantage of his later analysis for handling these cases, although the two fit naturally together. Second, the set-theoretic treatment gives contradictory results for sentences such as ‘Some sets are exactly the ones that are not members of themselves’: this statement is true, but there is no set of those sets. Leibniz was not aware of Russell’s paradox, but he was concerned to avoid a commitment to sets – in particular infinite sets – because of the paradoxes he saw associated with them, paradoxes that play the role for Leibniz that Russell’s para-dox plays for set theorists now, limiting the size on permissible collections. On the set-theoretic account, the sentence ‘Some inte-gers are squares of others’ would yield a result Leibniz explicitly regards as contra-dictory: the existence of infinite sets of inte-gers, in this case the set of squares and the set of all integers. The plural treatment, on the other hand, allows the sentence to be true without running afoul of Leibniz’s prin-cipled stance against infinite sets. So the con-temporary ‘pluralist’ position is one with a very strong natural affinity to Leibniz’s most matured analysis of predication in terms of identity and the ‘one-of’ relation.

Still, it must also be noted that [3*] and [4*] are not precisely the negations of [1*] and [2*], respectively, as Leibniz might have intended them to be in A Mathematics of Reason, if he was expecting a symmetry of [1], [2], [3] and [4] with A, E, I and O. It then may be concluded either that Leibniz was not seeing clearly how the relations between singular and plural terms would play out in those cases, or that he was not intending such a symmetry. (The set-theoretic versions do yield a symmetry here with [3] and [4] being the negations of [1] and [2]; this is pre-sumably why Lenzen appears to see it is a virtue of the set-theoretic versions of [1] and [2] that they are false unless A and B are unique.)

A PROBLEM ABOUT NEGATION

As we have seen, Leibniz understands his logi-cal systems to admit of intensional and exten-sional interpretations, and while he advocates the intensional approach as philosophically superior and most often writes with it in mind, in fact he works in both modes. His willing-ness to allow either approach is evidence of his conceiving of logical calculi in a quite modern way as abstract formal systems sepa-rable from intended interpretations, but no doubt it is due also to his principle (PR) of reciprocity of his own intensional and exten-sional readings. Is Leibniz’s reciprocity princi-ple correct? So long as the domain of individuals is taken to be that of all possible individuals and the operation of negation is not included, it seems that reciprocity will hold without trouble. Negation poses a prob-lem, however, and not an equal one: it calls into doubt the intensional interpretation itself in a fairly deep way.

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Extensionally construed, in Leibniz’s logi-cal calculi negation can be interpreted as the equivalent of set-theoretic complement. As before, ‘No A is B’ is rewritten by Leibniz as ‘Every A is non-B’, and this is true just in case the class of As is included in the class of non-Bs, i.e. in the complement of the class of Bs. (Equivalently, ‘No A is B’ is true just in case the intersection of the class of As and the class of Bs is empty.) But what does inten-sional negation mean? In the intensional con-strual, ‘Every A is non-B’ is true just in case the concept of A contains the concept of non-B. Yet how are we to understand the lat-ter, negative concept?

Leibniz regards the intension of a term as a concept built up of subconcepts. For a term like ‘man’, it may be plausible to suppose that the intension, i.e. the concept man, is the sum of the concepts rational and animal, and that those are further composed of sim-pler concepts, etc. What, then, is to be said about the negation of that concept? In §76 of the General Inquiries, Leibniz says that for A = AB, the negation of concept A contains the negation of the concept AB:

Not-A contains not-(AB), i.e., not-A = Y not-AB. Every not-man is a not-(rational man). (A,VI, iv, 764/LLP 67; cf. §§104–5)

This indicates that negative concepts will ‘involve’ particular contents in their ana-lysis. But it does almost nothing to explain what the negation ‘not-A’ means because it leaves the same question unanswered for the term ‘not-AB’, and likewise for however many conjuncts are further elicited in the analysans.

Dummett (1956, p. 199), who first raised the problem of intensional negation for

Leibniz’s logic, mentions a possible response. It is not too hard to suppose that content of a given concept C will correspond to the combination of those properties that all Cs have in common. Likewise, the concept not-C may consist in subconcepts corre-sponding to the properties common to all non-Cs, i.e. to all possible individuals that do not satisfy concept C. This manoeuvre seems technically sound but, as Mugnai (2005, p. 174) notes, it will provide a very thin intension for the term ‘not-C’, since the class of all possible non-Cs will be extremely diverse and its members will have few prop-erties of interest in common. By contrast, when we say ‘No elm tree is a man’, it seems we have quite rich intensions in mind: fully constituted positive ideas of men and elm trees, for example. If ‘Every elm tree is a non-man’ is to serve as a suitable translation, its terms should not be poorer in intensional content than those of the original.

The problem of understanding negation is an ancient one. If it is no surprise that Leib-niz’s logic encounters it, there may yet be some lesson in seeing the particular difficulty for Leibniz’s account. If we are to have a general concept corresponding to the logical operation of negation, it is not going to help us to try to construct it from ‘negative con-cepts’. Of course the extensional approach is not free from the issue either. Considering negation to be explained by its interpret-ation as complement presupposes that the concept of complement is perfectly general as well, which it is not: not every set has a complement. In fact, no set does if we con-sider an unrestricted universe as our domain and take complements of sets to be sets. This is a lesson of Russell’s paradox. But once we fall back on the idea of a ‘proper class’ as the complement of a given set, in order to

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characterize the elements of the particular proper class it seems we will inevitably call upon the concept of negation to do so. The advantage of the extensional approach is only that it asks us to draw upon a concept we take ourselves to have – that of negation – without

being able to offer a more basic explanation, whereas the intensional approach asks us to adopt as primitive an idea we do not clearly have, the negative concept.

Samuel Levey

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