Ramble, Part 52: Beyond Barbara
Jun. 13th, 2008 03:31 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowHaving discussed nineteenth-century developments in geometry and algebra, we should next proceed to the third great branch of mathematics, analysis. However, those developments wove in and out with developments in what mathematicians call "foundations": logic, set theory, and their relatives. So, I'm going to spend the next few Rambles ranging over both of those areas. I'm also going to throw in a little linguistics, just 'cause I can.
As we saw earlier, Gottfried Leibniz made an initial stab at an algebra of logic. What he constructed was a symbolic representation of concepts, and the results were a bit muddled. It was soon recognized that representation of classes of things, rather than concepts, would be more effective. Several mathematicians developed the idea of diagrammatic representations of the classical (A-E-I-O) statements of syllogistic; this line extends from Euler through Augustus DeMorgan and (of all people) Lewis Carroll to John Venn, the last of whom introduced his well-known diagrams in the late nineteenth century.
Common to all of these diagrammatic representations was the following recognition. Given two classes X and Y, there are four possibilities: something may be in both classes; it may be in X but not Y, or vice versa; or it may be in neither. As regards each of these possibilities, it might be that the possibility is in fact void - e.g., there may be nothing which is in both classes - or that the possibility is in fact not void - e.g., there may be something which is in X but not in Y. Thus, for example, the classical type-A statement ("All X is Y") is the declaration that there is nothing which is in X but not in Y.
If you're alert, you may notice (and DeMorgan, among others, did notice) that this implies eight sorts of statement, not the four of the medieval syllogism. In the list below, I'll use capital letters (X, Y) to indicate things in each class, and lower case letters (x, y) for things not in the given class.
XY: empty ("No X is Y" = E), nonempty ("Some X is Y" = I)
Xy: empty ("All X is Y" = A), nonempty ("Some X is not Y" = O)
xY: empty ("All Y is X" = A), nonempty ("Some Y is not X" = O)
xy: empty ("Everything is either X or Y"), nonempty ("Something is neither X nor Y")
The statement types A and O, thus, each split in two: "All X is Y" is different from "All Y is X". (Types E and I are symmetric, and thus do not split.) In addition, we have two more types of statement, as in the last line. DeMorgan investigated this new set of eight types of statement, and derived a new and more complete list of valid syllogisms. (As an aside: some twenty years ago, I was unaware of DeMorgan's work, and I set one of my Masters' students precisely this task. I'll have to see if I can find his paper, to compare it to what DeMorgan did.)
This line of attack, though interesting and productive, was still a little premature; a couple of deeper foundational issues needed to be addressed first. DeMorgan was to be a major player in this development, but the heavy lifting was done by George Boole. We'll look at that in the next Ramble.
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As we saw earlier, Gottfried Leibniz made an initial stab at an algebra of logic. What he constructed was a symbolic representation of concepts, and the results were a bit muddled. It was soon recognized that representation of classes of things, rather than concepts, would be more effective. Several mathematicians developed the idea of diagrammatic representations of the classical (A-E-I-O) statements of syllogistic; this line extends from Euler through Augustus DeMorgan and (of all people) Lewis Carroll to John Venn, the last of whom introduced his well-known diagrams in the late nineteenth century.
Common to all of these diagrammatic representations was the following recognition. Given two classes X and Y, there are four possibilities: something may be in both classes; it may be in X but not Y, or vice versa; or it may be in neither. As regards each of these possibilities, it might be that the possibility is in fact void - e.g., there may be nothing which is in both classes - or that the possibility is in fact not void - e.g., there may be something which is in X but not in Y. Thus, for example, the classical type-A statement ("All X is Y") is the declaration that there is nothing which is in X but not in Y.
If you're alert, you may notice (and DeMorgan, among others, did notice) that this implies eight sorts of statement, not the four of the medieval syllogism. In the list below, I'll use capital letters (X, Y) to indicate things in each class, and lower case letters (x, y) for things not in the given class.
XY: empty ("No X is Y" = E), nonempty ("Some X is Y" = I)
Xy: empty ("All X is Y" = A), nonempty ("Some X is not Y" = O)
xY: empty ("All Y is X" = A), nonempty ("Some Y is not X" = O)
xy: empty ("Everything is either X or Y"), nonempty ("Something is neither X nor Y")
The statement types A and O, thus, each split in two: "All X is Y" is different from "All Y is X". (Types E and I are symmetric, and thus do not split.) In addition, we have two more types of statement, as in the last line. DeMorgan investigated this new set of eight types of statement, and derived a new and more complete list of valid syllogisms. (As an aside: some twenty years ago, I was unaware of DeMorgan's work, and I set one of my Masters' students precisely this task. I'll have to see if I can find his paper, to compare it to what DeMorgan did.)
This line of attack, though interesting and productive, was still a little premature; a couple of deeper foundational issues needed to be addressed first. DeMorgan was to be a major player in this development, but the heavy lifting was done by George Boole. We'll look at that in the next Ramble.
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Ramble Contents
