Precursors of Modern Logic: Lambert and Ploucquet
PRECURSORS OF MODERN LOGIC: LAMBERT AND PLOUCQUET
Johann Heinrich Lambert (1728–1777), German physicist, mathematician, and astronomer, devoted a number of essays to the enterprise of making a calculus of logic, which he evidently thought of in connection with the tree of Porphyry. His standpoint is, as is usual with the early investigators, intensional. Let a and b be any concepts, a + b their combination into a compound concept, and ab their common part. The letters γ and δ can be multiplied with conceptual variables, so that aγ and aδ are read as "the genus of a " and "the difference of a. " The intended meaning suggests that γ and δ are descriptive operators; yet Lambert sometimes treated them as though they were placeholders for generic or differential concepts. At any rate Lambert, following an elementary intuition, posited a = aγ + aδ = a (γ + δ ). Wanting to descend the tree to subordinate species as well as to ascend to superordinate genera and differences, he used the notation aγ −1 or a /γ, which should mean "the genus under a. " Waiving the fact that a concept containing a may be an ultimate species, we reflect that although aγ is unique, aγ −1 may not be so. This accounts for the trouble that Lambert found in applying multiplication and division, for (a /γ )γ, "the genus of a species of a," is identical with a whereas (aγ )/γ, "a species of the genus of a," need not be a itself. Lambert used subtraction to obtain the removal of a concept. He did not account for the appearance of coefficients and, in general, did not question the logical appropriateness of the algebraic operations to which his basic intuitions gave rise. Boole met with similar difficulties but reflected on them.
In syllogistic Lambert started not from the Aristotelian relations but from the five that are now attributed to Gergonne. This is feasible, but Lambert failed to achieve a satisfactory notion for the mutual exclusion of two terms. His most promising innovation lay in his attention to the relative product, but he did not develop this in any practical way.
Lambert, like Leibniz, experimented with sets of ruled and dotted lines to illustrate the relationships of syllogistic terms, in part trying to correct the defect in Euler's circles of not allowing for a = b. Some stages of his investigations were criticized by his correspondents G. J. von Holland (whose extensional standpoint was remarkable for the time) and Gottfried Ploucquet (1716–1790), both of whom were making their own efforts to evolve a logical calculus. Ploucquet, who was a teacher of Hegel, claimed independence of Euler in his use of closed figures—he used squares (1759)—and seems to have been the first to base his syllogistic on thoroughgoing quantification of the predicate. One of his notations, "A 〉 B " for "No A is B," strangely, enjoyed some popularity.
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Ivo Thomas (1967)