Logic Made Easy by Deborah J. Bennett

Author:Deborah J. Bennett
Language: eng
Format: epub, pdf
Publisher: W. W. Norton & Company

Figure 37. De Morgan’s graphical representation of “Every X is Y.” The two solid Y-lines illustrate the two cases.

In an 1850 paper on logical syllogisms, De Morgan mentioned the graphical representations adopted by Euler, but like everyone else, De Morgan appeared to be unaware of Leibniz’s work in logic.11 De Morgan introduced a diagrammatic method that he felt was superior for illustrating propositions. Curiously, his diagrams bear an amazing resemblance to the line diagrams originating from Leibniz’s 1686 paper. De Morgan demonstrated “All X are Y” or “Every X is Y” as shown in Figure 37.

The solid part of the X line segment represented things that are X things, while the dotted part of that line segment represented things that are not-X. The same held for Y, but here De Morgan has illustrated the two possible cases that are the cause of much confusion: X-things could constitute all of the Ys (hence the solid segment that is the same length as the X segment) or only some of the Ys (the longer solid line segment). De Morgan went on to provide an example of a more complicated proposition involving a disjunction, “Some things are neither Xs nor Ys” as is seen in Figure 38.

Without a doubt, De Morgan thought that the laws of mathematics could be brought to bear in logic. He says, “Logic considers the laws of action of thought: mathematics applies these laws of thought to necessary matter of thought. . . . A generation will arise in which the leaders of education will know the value of logic, the value of mathematics, the value of logic in mathematics, and the value of mathematics in logic.”12 In this, he was influenced by the genius of George Boole, who had a brilliance and originality that De Morgan was among the first to encourage and recognize.

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