Modern Logic: The Boolean Period: Venn
MODERN LOGIC: THE BOOLEAN PERIOD: VENN
The logic of John Venn (1834–1923), sketched briefly in the Princeton Review (1880) and more fully elaborated in his Symbolic Logic (London, 1881), shows a greater understanding of George Boole's intentions and a better acquaintance with the historical background than had yet been displayed by anyone else. Though he did not suppose the new methods to have any great practical advantage over the old, he saw no reason, either, to suspect them of being anything more than a generalization of traditional practices, couched, for convenience, in a mathematical form. He therefore resisted the Jevonian simplifications and was at pains to bring out the logical significance of such operations as subtraction and division, though the latter is admitted to merit inclusion more on grounds of consistency than for any use made of it in the reasoning of everyday life.
Venn's own account of the matter proceeds from what he calls the "compartmental," or "existential," view of logic, whose purpose is to set out the possible ways in which the four classes designated by x, y, and their negatives, in combination, may have one or more of their components empty. Omitting the case where all four compartments are unoccupied, this yields fifteen forms of proposition, compared with the four that arise on the traditional, or predication, view, whereby an attribute is asserted or denied of a class, and the five that emerge from diagrammatic consideration of the ways in which two nonempty classes may include, exclude, or overlap one another. Each view has its merits, in Venn's opinion, the choice between them being ultimately a conventional one.
This leads Venn to the discussion of another vexed issue, the "existential import" of propositions. Traditional logic must in consistency assume that its classes have members and nonmembers alike, and its universal propositions are thereby rendered hypothetical. To Venn it was clearer what the universal denies than what it asserts, and he therefore proposed to write A, "All x is y," as xȳ = 0 and E, "No x is y," as xy = 0. These propositions are definite, yet they do not require members in x or y to make them true, since they deny only the existence of members in the common class. Particular propositions do, however, imply the presence of members in each class, since they contradict the universals; they are therefore to be written I, xy ≠ 0, and O, xȳ ≠ 0, respectively. This was an improvement on Boole's use of indefinite symbols and has since been generally adopted, though one consequence of it (also noted by Hugh MacColl) is that subalternation ceases to be valid and that the "syllogisms of weakened conclusion" which depend on it have therefore to be rejected.
Venn was not much enamored of the syllogism, but he deserves the gratitude of all beginners in the subject for what is probably his best-known contribution to logic, the diagrams that bear his name. These are, in effect, graphical representations of the algebraic processes introduced by Boole and mechanically illustrated in Jevons's alphabet: The partitioning of a universe in terms of the possible combinations of x, y, and so on, and the elimination of those subdivisions inconsistent with the premises given. For two terms a pair of intersecting circles (x and y ) on a ground give the four compartments xy + xȳ + x̄y + x̄ȳ = 1 (Figure 1). Three interlaced circles (Figure 2) depict the eight combinations of Jevons's table,
given earlier. The effect of a universal premise is to declare one or more compartments to be empty, shown by shading the area in question. A particular premise indicates that one or more compartments have occupants, shown by a cross (which may lie ambiguously on the boundary between two areas). The conclusion can then be read off, in various ways, by inspection. By the use of ellipses the same principle can be employed for up to five terms, but it then becomes unwieldy, especially in the "inverse problem" of formulating the outcome, so that one or another of the square diagrams devised by later authors is at that stage generally preferable. With suitable modifications the method can also be extended to the calculus of propositions. Though Venn did not carry this extension far, he was led by it to an early realization of the truth-functional character of the relation of material implication.
The merit of Venn's work lies not in its original departures, which are few, but rather in the light it throws on the obscurities of Boole's procedure and in its very careful and fair discussion of opposing views.
P. L. Heath (1967)