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Lewis Carroll
Symbolic Logic
Part I: Elementary
Book I: Things and their Attributes
Chapter III: Division
§1 Introductory
[Image credits at the end.]
Previously we discussed subclassification. We first considered the class of Greek monsters. Then we picked out an attribute, "made-up of many animals." Our examples were the chimera and the sphinx. The chimera's body is a lioness, whose tail is a serpent. A goat head emerges from her spine.
The sphinx has the body of a lion, the head of a human, and the wings of a bird.
Let's say, then, that "made-up of many animals" means at least three component creatures. We know then, there are Ancient Greek monsters that only involve one creature each, for example, the cyclops and Cerberus.
Cyclops is a giant with one eye.
And Cerberus is a dog with three heads.
But there are monsters also to consider which are neither single-creatured nor many-creatured. For they are made-of two creatures, for example, the centaur and the satyr.
The centaur is half man and half horse.
And the satyr is half man, and half goat.
So all these instances above are examples of subdivision. And "division" is the mental process that we use in order to accomplish to create these subgroupings. Division may produce two or more smaller classes. Carroll offers three examples:
1) we divide the class "books" into "bound books" and "unbound books," or
2) we divide the class "books" into three classes: "books priced at less than a shilling," "shilling-books," and "books priced at more than a shilling."
3) we divide the class "books" into twenty-six subgroups: "books whose names begin with A," "books whose names begin with B," and so on.
§2 Dichotomy
So we subdivided the class of Greek monsters first into the subclass 'many-creatured.' We found that this did not leave only one other subclass. For "not-many-creatured" is ambiguous. It could either be not-many because it is two, or not many because it is one. Hence this is not a dichotomous division, for there are more than two groups produced by the subdivision. Later we will want dichotomies. For, if something is not one type, it must be the other. But in the case of our current monster subdivisions, if it is not the one, it would be one of two others. So nothing can be concluded by negating one class.
Now consider instead if we divided the monsters up into "part-human" and "animal only." Then we would only have two groups. We could title these also, "part-human" and "not-part-human," or the same would be "not-animal only" and "animal only." For, "part-human" means the same as "not-animal-only." And, "not-part-human" means the same as "animal only". [Unless there are Greek monsters that are neither animal nor man, for example a monster made of elements, like a rock monster.] Carroll writes:
'Dichotomy' is Division into two Classes. If a certain Class of Things be divided into two smaller Classes, one containing all the Members of it which possess a certain Attribute (or Set of Attributes), and the other all which do not possess it, we may regard each of these two Classes as having its own peculiar Attribute -- the first having the Attribute (or Set of Attributes) used in the Process of Division, and the second having the same, with the word "not" prefixed. This Process is called 'Dichotomy by Contradiction.' (4c)
Then Carroll gives the following examples:
For example, we may divide "books" into the two Classes, "books that possess the Attribute 'old'," and "books that do not possess it," that is, into the two Classes, whose peculiar Attributes are "old" and "not-old." (4d)
Let's consider our Greek monsters example. If someone introduced us to the monster 'minotaur,' and said to us that it is not an "animal only" monster, we would know automatically that it is part human. Which it is. Minotaur is half bull and half man.
Now consider Carroll's example of old and not-old books. Some books might clearly be old or not-old. But many could be either. For this reason, we often must stipulate some rule for determining a Thing's classification.
Thus, in dividing "books" into "old" and "not-old," we may say "Let all books printed before A.D. 1801, be regarded as 'old,' and all others as 'not-old '." (5a)
So from this point forward, Carroll will remain consistent regarding dichotomies by contradiction:
if a Class of Things be divided into two Classes, whose peculiar Attributes have contrary meanings, each Attribute is to be regarded as equivalent to the other with the word " not " prefixed. (5b)
For example
if "books" be divided into "old" and "new," the Attribute "old " is to be regarded as equivalent to "not-new," and the Attribute "new" as equivalent to "not-old." (5bc)
So, we may divide a Class into two contradictory dichotomous sub-Classes. And yet, we may sub-divide those sub-classes yet again. Then again. And so on. Every time we further subdivide, we double the number of classes. And we may repeat this division for as long as our collections permit.
For example, we may divide "books" into "old" and "new" (i.e. "not-old"): we may then sub-divide each of these into "English" and "foreign" (i.e. "not-English"),thus getting four Classes, viz.(1) old English;(2) old foreign;(3) new English;(4) new foreign.If we had begun by dividing into "English " and "foreign," and had then sub-divided into "old" and "new," the four Classes would have been(1) English old;(2) English new ;(3) foreign old;(4) foreign new.The Reader will easily see that these are the very same four Classes which we had before. 95c.d)
Images from the text [click on image for an enlargement]:
From:
Carroll, Lewis. Symbolic Logic. London: MacMillan and Co., 1869.
Available online at:
The chimera image taken from:
sphinx:
centaur:
satyr:
Cerberus:
cyclops:
minotaur:
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