Lewis Carroll ((1832-1898), born in Daresbury, Cheshire, is best known as the author of Alice in Wonderland and Alice Through the Looking Glass. His real name was Charles Dodgson. His father, the Reverend Charles Dodgson, instilled in his son a love of mathematics from an early age. Lewis studied at Oxford, and later taught there as a Mathematics Lecturer. He wrote several mathematics books, including Euclid and his Modern Rivals in which he defended the use of Euclid's Elements to teach geometry. His works on logic include The Game of Logic (1887) and Symbolic Logic (1896).
One of his favorite hobbies was photography, and he especially liked to photograph children. In addition to photographing them, he enjoyed entertaining his child subjects with stories, at the same time illustrating them with pencil or ink drawings. These stories were eventually polished and published as the Alice books.
Lewis Carroll died suddenly after contracting a cold which developed into a serious chest problem.
Like John Venn, Carroll was interested in using diagrams to analyze logical arguments. However, Carroll was mainly interested in using logic diagrams as a pedagogical tool. In fact, he wrote a book called The Game of Logic which was intended to teach logic to children. His "game" consisted of a card with two diagrams, together with a set of counters, five grey and four red. The two diagrams were Carroll's version of a two-set and a three-set Venn diagram. They are shown below:
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As you can see, Carroll's logic diagrams are quite different from Venn's. Carroll used a big square to represent his universe. He then drew a horizontal line dividing the square into two equal halves. The top half was his first set $X$, and the bottom half then was the complement $X'$.
Next, Carroll drew a vertical line that again divided his original square into two equal halves. The left half was his second set $Y$ and the right half then was the complement $Y'$. So now his original square has been divided into four smaller squares, as shown in Figure . The four regions labeled $0$, $1$, $2$, and $3$ correspond to the four sets $X'Y'$, $XY'$, $X'Y$, and $XY$ respectively.
To add a third set $M$, Carroll drew a square at the center of the large square, thus dividing his original square into 23=8 regions. Each region consists of a set of the form $ABC$ where $A$ is either $X$ or $X'$, $B$ is either $Y$ or $Y'$, and $C$ is either $M$ or $M'$. For example, region 5 in Figure corresponds to $XY'M$.
The counters that came with The Game of Logic were used to illustrate logical statements. A grey counter was used to denote that a region was empty, and a red counter was used to denote that a region was occupied. A red counter on the division line between two regions was used to denote that one (or both) of the two regions was occupied.
Carroll also drew a four-set diagram shown below. The third set has been elongated into a rectangle, and the fourth set is a similar rectangle rotated by 90 degrees.
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Lewis Carroll's Logic Puzzles
Lewis Carroll used logic diagrams and the Game of Logic to teach children how to determine what valid conclusions can be drawn from a pair of given premises involving at most three statements. He then extended such problems to more than three statements and more than two premises. Carroll's book Symbolic Logic contains literally dozens of such puzzles. He believed heartily that children would enjoy learning mathematics if they could be enticed by amusing stories and puzzles.
As an example, consider the following three premises:
(1) Every one who is sane can do Logic.
(2) No lunatics are fit to serve on a jury.
(3) None of your sons can do Logic.
What conclusion, if any, can we draw from these premises?
The following dictionary is proposed: Universe "persons"; $A=$ able to do Logic; $B=$ fit to serve on a jury; $C=$ sane; $D=$ your sons.
We can then translate the premises using modern symbolism:
(1) $C\Rightarrow A$.
(2) $C'\Rightarrow B'$.
(3) $D\Rightarrow A'$.
Now statement (3) is equivalent to its contrapositive $A\Rightarrow D'$. So from statements (1) and (3) we obtain that $C\Rightarrow D'$ (4). Statement (2) is equivalent to its contrapositive $B\Rightarrow C$. So from statements (4) and (2) we obtain that $B\Rightarrow D'$.
At this point, we have used all three premises, so we are done. The conclusion, $B \Rightarrow D'$, translates to
"If a person is fit to serve on a jury then he is not your son." or equivalently "None of your sons is fit to serve on a jury."
The basic solution method, once the premises have been translated into modern symbolism, is as follows:
- Select two premises of the form $X\Rightarrow Y$ and $Y\Rightarrow Z$, and draw the conclusion $X\Rightarrow Z$. Keep in mind that $X\Rightarrow Y$ is equivalent to $Y'\Rightarrow X'$.
- Find a premise among the ones not yet used that can be combined with the most recent conclusion to draw the next conclusion.
- Proceed in this way until all premises have been used.
- Translate the last conclusion back.
- If you get stuck, try beginning with a different pair of premises.
Lewis Carroll loved puzzles. As a teacher of logic he designed many interesting and entertaining puzzles to teach children the art of systematic reasoning. The puzzles in the following exercises are taken from Lewis Carroll's Symbolic Logic.
Exercises
1. Carroll was not very successful in drawing a nice symmetrical five-set diagram.
(a) Draw a five-set diagram by adding a fifth cross-shaped set to the four-set diagram in Figure 1. Note that each time you add a set, each of the old regions should get divided into exactly two new regions (the part inside the new set and the part outside). A five-set diagram should have 25=32 distinct regions, so you should be able to number the regions from 0 to 31.
(b) Extra Credit: Draw a six-set diagram.
2. Find the conclusion.
(a) Babies are illogical;
(b) Nobody is despised who can manage a crocodile;
(c) Illogical persons are despised.
Univ. “persons”; $A=$ able to managa a crocodile; $B=$ babies; $C=$ despised; $D=$ logical.
3. Find the conclusion.
(a) No interesting poems are unpopular among people of real taste.
(b) No modern poetry is free from affectation.
(c) All your poems are on the subject of soap-bubbles.
(d) No affected poetry is popular among people of real taste.
(e) No ancient poem is on the subject of soap-bubbles.
Univ. “poems”; $A=$ interesting; $B=$ modern; $C=$ your; $D=$ popular among people of real taste; $E=$ affected; $F=$ on the subject of soap-bubbles.
4. Find the conclusion.
(a) No kitten, that loves fish, is unteachable;
(b) No kitten without a tail will play with a gorilla;
(c) Kittens with whiskers always love fish;
(d) No teachable kitten has green eyes;
(e) No kittens have tails unless they have whiskers.
Univ. “kittens”; $A=$ green-eyed; $B=$ loving fish; $C=$ tailed; $D=$ teachable; $E=$ whiskered; $F=$ willing to play with a gorilla.
For more examples, see http://www.math.hawaii.edu/~hile/math100/logice.htm.
5. Find the conclusion.
(a) The only animals in this house are cats.
(b) Every animal is suitable for a pet, that loves to gaze at the moon.
(c) When I detest an animal, I avoid it.
(d) No animals are carnivorous, unless they prowl at night.
(e) No cat fails to kill mice.
(f) No animals ever take to me, except what are in this house.
(g) Kangaroos are not suitable for pets.
(h) None but carnivora kill mice.
(i) I detest animals that do not take to me.
(j) Animals, that prowl at night, always love to gaze at the moon.
Univ. “animals”;
$A=$ in this house;
$B=$ cats;
$C=$ suitable for a pet;
$D=$ loves to gaze at the moon;
$E=$ detested by me;
$F=$ avoided by me;
$G=$ carnivorous;
$H=$ prowl at night;
$I=$ kills mice;
$J=$ take to me;
$K=$ kangaroos.
