PRECURSORS OF MODERN LOGIC: LEIBNIZ

Gottfried Wilhelm Leibniz (1646–1716) was distinguished in many fields, but in none more than in logic. There, however, his worth was not fully appreciated until the twentieth century. He early began to investigate Aristotelian syllogistic and never completely escaped from the syllogistic point of view. In 1666 he wrote a Dissertatio de Arte Combinatoria, a juvenile work that was not free of mistakes, as he later realized, but that showed a new, high sense of organization and a genuine feeling for formal logic, very rare at the time. In one part of this book Leibniz worked out for himself the calculations of Hospinianus (1560) relative to the possible and the valid moods of syllogism. He differed from Hospinianus in making singular propositions equivalent to universal ones, as did Wallis and Euler. He arrived at twenty-four strictly Aristotelian syllogisms, six in each of four figures, which he arranged in a neat tableau suggestive of certain deductive relationships. Leibniz's standard method of proof in this context was reductio ad absurdum, as suggested to him by his teacher Jakob Thomasius (1622–1684), author of Erotemata Logica (Leipzig, 1670), but he also recognized the need for conversion. He wrongly credited Ramus with a method actually known in the thirteenth century, the device of proving laws of conversion and subalternation by means of syllogism and the laws of identity "All a is a " and "Some a is a."

Leibniz often returned to syllogistic and was periodically vexed by semantic considerations, namely whether to think of the matter in extension or in intension—whether in "All a is b " it is the a 's which are said to be contained in the b 's or the property a which contains the property b. Leibniz had something of a fixation on the intensional approach, although he often suspected that extension was more effective and logically satisfactory. One thing that pushed him in the direction of extensionality was a fondness for experimenting with spatial interpretations. Thus, we find several attempts at diagrammatic representation, some using ruled and dotted lines and some using circles. He found it impossible to carry through such interpretations when thinking in intension.

theory of combinations

The theory of combinations is highly relevant to logic. Chrysippus is reported to have shown some interest in combinations, Kilwardby and others in the thirteenth century repeatedly made combinatory summaries of assertoric and modal syllogistic, and semantic interpretations of logical formulas in finite domains employ the theory. Besides the syllogistic computations described, Leibniz considered how many predicates can be truly asserted of a given subject or how many subjects set under a given predicate. Such problems need some preliminary arrangements, and Leibniz supposed that a composite concept is analyzable into a number of ultimate simples, just as an integer is uniquely decomposable into its prime factors. Correlating the simple concepts with prime numbers, we can say that a predicate is truly attributable to its subject if the product associated with the predicate divides that associated with the subject. The essentials of this idea have been used in modern times to obtain a decision procedure for syllogistic, and unique decomposition into primes plays an essential part in Gödel numbering.

universal language

The idea of decomposing concepts into "prime factors" suggested to Leibniz the possibility of following up the initial steps toward a universal language taken by John Wilkins (1668), Jean Joachim Becher (1661), George Dalgarno (1661), Athanasius Kircher (1663), and others. He wanted such a language not merely to be practically or commercially useful, as were many of the pioneer efforts, but to be logically constructed so as to have general scientific import. Leibniz later distinguished a universal language from a logical calculus and desired to base his language on a thorough analysis of the communicative function of the various parts of speech, tenses, suffixes, and so on (an anticipation of modern theories of syntactical categories), and at one point (Analysis Linguarum, 1678) he envisaged a basic Latin rather in the style of C. K. Ogden and I. A. Richards's basic English.

In saying that nouns express ideas and verbs express propositions Leibniz radically altered the Aristotelian basis of the distinction and gave, in germ, the concept of a propositional function. Such reflections led him to a reductionist program, with adverbs reduced to (derived from) adjectives and adjectives to nouns, and with the copula taken as the only fundamental verb. He recognized that particles, connectives, and prepositions are of especial importance to linguistic structure. In taking us out of the syllogistic area this theory recalls the medieval doctrine of syncategorematic terms and Thomas Aquinas's analysis of many prepositions, while it adumbrates the logic of truth-functional connectives and of relations. Leibniz knew that not all arguments are syllogistic, in this matter acknowledging a debt to Jung, but the dominance of a syllogistic point of view in Leibniz's thought is shown by his curious distinction between syllogistic and "grammatical" consequences.

This part of Leibniz's thought constitutes a distinct chapter in the history of the relations between grammar and logic. Grammar had been influential in the constitution of scholastic logic, but in the interregnum it had yielded to the third member of the medieval trivium, rhetoric, as a dominant power. In the projects for a universal and rational language we see grammar reasserting itself. But Leibniz was not content to confine logic to the "trivial" arts.

logical calculus

The idea that logic might be quadrivial, and notably mathematical, was not new with Leibniz. Leibniz considered Aristotle to have been, in his logic, the first to write mathematically outside mathematics (letter to Gabriel Wagner, 1596). Roger Bacon (thirteenth century)—who also wished to reduce the trivial art of grammar to the quadrivial one of music—stated in his Opus Maius that "all the predicaments depend on the knowledge of quantity, with which mathematics deals, and therefore the whole of logic depends on mathematics." It is in the light of this that one should read the statement in his Communia Mathematica that "the mere logician cannot accomplish anything worthwhile in logical matters" (nihil dignum potest purus logicus in logicalibus pertractare ). William of Ockham had been of the opposite opinion, and in De Sacramento Altaris he described mathematicians as among those less skilled in logic. Ramón Lull had written a combinatorial work, Ars Magna (which captured Leibniz's imagination, though he soon came to understand its deficiencies), and Thomas Hobbes had elaborated suggestively, if ineffectively, on the theme "by ratiocination I mean computation" ("Computatio Sive Logica," in De Corpore ).

There is little doubt, however, that Leibniz's ideas, which far outstripped in detail and understanding any earlier hints, were his own spontaneous creation. "While I was yet a boy with a knowledge only of common logic, and without instruction in mathematics, the thought came to me, I know not by what instinct, that an analysis of ideas could be devised, whence in some combinatory way truths could arise and be estimated as though by numbers" (Elementa Rationis ). He was thereafter constantly occupied with such notions and attempted to contrive an alphabet of thought, or characteristica universalis, which would represent ideas in a logical way, not things in a pictorial way, and would be mechanical in operation, unambiguous, and nonquantitative; this alphabet of thought would be a means of discovery, a support to intuition, and an aid in ending disputes.

Leibniz regarded his great invention of the infinitesimal calculus (1675) as emerging from such researches, and the calculus led him to reflect still more intently on the properties desirable in such a characteristic. Exactly what he meant by "mechanical" and "calculation" is still in question, and he no doubt underestimated the task he set himself, but the imaginative fervor with which he always wrote of it reveals, as we can now appreciate, a true prophetic instinct. He often used an image from mythology to summarize his intentions, saying that his method was to be a filum Ariadnes, a thread of Ariadne. Many authors had long envisaged logic as a Cretan maze in need of such a clue—and that this should be so in an age when logic was scarcely existent does them little credit—but from the pen of Leibniz the allusion was more than a literary elegance and condensed a program of "palpable demonstrations, like the calculations of arithmeticians or the diagrams of geometers." (For Leibnizian references to the filum, see Louis Couturat, La Logique de Leibniz, pp. 90–92, 124; for other authors, see Ivo Thomas, "Medieval Aftermath.")

encyclopedia

One may ask what the theory of combinations was meant to combine, what the logical calculus was meant to calculate with, or where the analyses presupposed by the unified language of science were to be found. Leibniz was not content to leave such analysis in the state of a general project. The enormous range of his knowledge and interests, which included unity in religion, international relations, cooperation among scientists and scholars, and jurisprudence, as well as the not unrelated ordering of thought, prompted his lasting interest in the construction of an encyclopedia. T. Zwinger's Theatrum Vitae Humanae (1565) and Johann Heinrich Alsted's Encyclopaedia (1608) provided Leibniz with a basis for early schematisms, and sketches and fragments from about 1668 to the end of his life show an unceasing interest in the plan, which he believed had failed of completion through his own distractions and the lack of younger assistants. Appeals to monarchs and to learned societies met with little response. The project was, of course, a gigantic one, impossible of immediate fulfillment, but it should not be supposed that Leibniz thought it could be perfected quickly. Rather, its elaboration was to proceed gradually, along with that of the universal language and a calculus of logic. In later drafts this calculus took an ever more prominent place.

structure of the calculus

The main stages (1679, 1686, 1690) of Leibniz's many experiments in logical algebra have often been expounded and commented on. Here only some laws which were constant features will be mentioned.

1. a is a ;
2. If a is b and b is c, then a is c.

Propositions of the form "a is b " are intended as universal affirmatives, "All a is b," which Leibniz normally thought of as meaning that the property a contains the property b. Sometimes he wrote "a contains b " instead of "a is b." Accordingly, rule (1) is one of the syllogistic laws of identity which, as was said above, he used from the start in syllogistic demonstrations, and rule (2) is the Barbara syllogism. Today we know that by means of the calculus of quantifiers and some definitions all asserted laws of the syllogistic can be obtained from rules (1) and (2) alone. Leibniz lacked those aids, but he admitted negative terms that obey the laws

• (3) a is interchangeable with not-not-a ;
• (4) a is b if and only if not-b is not-a.

Rule (4) is the law of contraposition familiar to the Scholastics and, for Leibniz, most recently given prominence by Jung. From rules (1) to (4), with some definitions and Leibniz's favorite method of reductio ad absurdum, the whole syllogistic can be obtained. Leibniz did not use exactly that method but adopted at one time a rather similar one based on a restatement of rule (1), a = aa, and rule (5), below. Identity has the substitutive property described below; "a is b " is made equivalent to "a = ab "; and "Some a is b " is written "Sa = b." Compound terms such as ab were thought of as signifying the addition of properties a and b. They obey the laws

• (5) ab is a ;
• (6) ab is b ;
• (7) If a is b and a is c, then a is bc.

It has been pointed out by Karl Popper that if rules (5) and (6) are made the premises of the mood Darapti, we have the conclusion "Some a is b." This does not render the system inconsistent, but it does show that the system is already more extensive and more trivial than Leibniz presumably intended. From rules (1), (2), (5), (6), and (7) it is easy to deduce, as Leibniz did,

(8) If a is bc, then a is b, and a is c, which is the converse of (7), and

(9) If a is b, then ac is bc (using rules 2, 5, 6, and 7);

(10) If a is b and c is d, then ac is bd (using rule 9 twice and then rule 2).

Rule (10), which was known to Abelard in the twelfth century, Leibniz called praeclarum theorema, a very notable theorem.

Identity of terms was introduced in various ways, but always so that it was equivalent to the conjunction of "a is b " and "b is a " and so that identical terms could be substituted for one another in all contexts of the calculus. The first definition in the Non Inelegans Specimen Demonstrandi in Abstractis, for instance, posits that a = b holds if and only if a and b can be substituted for each other without altering the truth of any statement. The "only if" part is commonly called the principle of the identity of indiscernibles; for its converse W. V. Quine has suggested "the indiscernibility of identicals." As a principle of general application it has given rise to much discussion, although it is normally accepted in logic. While it is commonly attributed to Leibniz, Aristotle presented it in essentials in the Topics (VII, 1, 152a31 ff.) and De Sophisticis Elenchis (Ch. 24, 179a37 ff.).

An algebraic calculus requires that substitution for variables be possible, and Leibniz explicitly recognized this, in what was certainly the clearest statement in logic of the principle up to his time. Some medievals—Albert the Great, for instance—had shown their understanding of the generality conferred by variables when they called them "transcendental terms." Three more laws important for the calculus were known to Leibniz, following from rules (1), (5), (6), and (7):

• (11) ab is ba (using 5, 6, and 7);
• (12) a is aa (using 5);
• (13) aa is a (using 1 and 7).

In the course of his experiments Leibniz came to see that particular propositions have existential import, whereas universals may not, and it was a puzzle to him what the existential import might be—factual existence or logical possibility—and whether it was built into his system or had to be further provided for. This problem had been raised by medieval logicians from the time of Abelard. One of Leibniz's solutions—that subalternation is invalid if the universal states a relation of concepts and the particular states a matter of fact but holds if we stay in one of those domains—is essentially that of Paul of Venice, who required the subjects of both propositions to have the same suppositio.

At a late stage Leibniz used the addition sign in place of, and with the sense of, multiplication; that is, he used a + b instead of ab. But he knew that such expressions could be interpreted as logical disjunctions, and there is also an early hint that the calculus could be interpreted propositionally, the antecedent of a conditional being said to contain the consequent. This hint may serve as a summary indication of Leibniz's position in the history of logic. Aristotle had used "antecedent" and "consequent" for "subject" and "predicate"; among medievals (such as Abelard and Kilwardby) it is often hard to tell whether the words were used of propositions or of terms; Leibniz offered a glimpse of the two domains as distinct but analogous. If his work had not gone long unpublished (we still have no complete edition), we might not have had to wait so long for the full light of Boolean day.

Bibliography

Baylis, C. A. Review of various articles on the identity of indiscernibles. Journal of Symbolic Logic 21 (1960): 86.

Couturat, Louis. La Logique de Leibniz d'après des documents inédits. Paris: Alcan, 1901; reprinted, Hildesheim, 1961.

Dürr, Karl. Leibniz' Forschungen im Gebiet der Syllogistik. Leibniz zu seinem 300. Geburtstag. Berlin, 1949.

Kauppi, Raili. Über die Leibnizsche Logik mit besonderer Berücksichtigung des Problems der Intension und der Extension. Helsinki, 1960.

Rescher, Nicholas. "Leibniz's Interpretation of His Logical Calculi." Journal of Symbolic Logic 19 (1954): 1–13. Reviewed by M. A. E. Dummett in Journal of Symbolic Logic 21 (1960): 197–199.

Ivo Thomas (1967)