Scientists, Literature and Logic
January 27 marks the day of Charles Ludwig Dodgson’s birthday! For those who don’t know who that is, Dodgson’s pen name was Lewis Carroll. Dodgson’s contributions to literature and mathematics were impressive! He wrote 14 bodies of literary works, some of which include his poetry. As a mathematician, he wrote 11 books along with the algorithm known as the Dodgson condensation.
Alice’s Adventures in Wonderland contains so many examples of good logic and deliberately bad logic. In my podcast, I read from the book from my favorite passage, the conversation at the Mad Hatter’s Tea Party.
“Then you should say what you mean,” the March Hare went on.
“I do,” Alice hastily replied, “at least—at least I mean what I say—that’s the same thing, you know.”
“Not the same thing a bit!” said the Hatter. “You might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”
“You might just as well say,” added the March Hare, “that ‘I like what I get’ is the same thing as ‘I get what I like’!”
“You might just as well say,” added the Dormouse, who seemed to be talking in his sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!”
“It IS the same thing with you,” said the Hatter, and here the conversation dropped, and the party sat silent for a minute, while Alice thought over all she could remember about ravens and writing-desks, which wasn’t much.
In this conversation, Carroll presents pairs of related statements between the Hare, the Hatter, and the Dormouse. Each pair of statements do not say the same thing.
The Hatter proposes the statement; “I see what I eat” is the same thing as “I eat what I see.”
This is not entirely true. For humans, we see our computers, or our cars, but we do not eat them.
The March Hare proposes, “I like what I get” is the same thing as “I get what I like.”
This is not entirely true as well, especially for those who received a really awful holiday gift last month and immediately re-gifted it! I am guilty of this one too!
Then, the Dormouse says, “I breathe when I sleep,” is the same thing as “I sleep when I breathe.”
When we turn the Dormouse’s argument into an “if-then” statement, we get:
If I sleep, then I breathe.
And
If I breathe, then I sleep.
In this setup, we apply the structure of logic with a conditional statement that takes on three forms:
Converse
Contrapositive
Inverse
In math, the first statement is a conditional statement. The conditional statement sets up the conditions by which the statement is either true or false.
So, using the conditional statement, we interchange the hypothesis (“sleep”) and the conclusion (“breathe”) to create a second statement. This is second statement then becomes a converse of the conditional statement.
The converse of a true statement can also be false, which means that a statement and its converse do not have the same meaning.
The contrapositive is a little like the converse only, instead of just interchanging the hypothesis and conclusion of the conditional statement, we also deny both the hypothesis and conclusion. So now, we get:
If I sleep, then I breathe.
and
If I do not breathe, then I do not sleep.
So, we have interchanged the hypothesis and conclusion and denied both of them.
Finally, if we take the converse of the original argument where the Dormouse says, “If I breathe, then I sleep,” and then invert the statement and deny both the hypothesis and conclusion. We then get the inverse, as follows:
If I breathe, then I sleep.
and
If I do not sleep, then I do not breathe.
As a result, the inverse of a conditional statement is when we interchange the hypothesis and the conclusion of the converse statement and then deny them both.
Euler Diagrams
Using Euler diagrams, it looks like this:
Each Euler diagram represents two complete statements. The first Euler diagram represents the two statements:
If I sleep, then I breathe.
and
If I do not breathe, then I do not sleep.
The second Euler diagram represents two statements as well:
If I breathe, then I sleep.
and
If I do not sleep, then I do not breathe.
A statement and its contrapositive are represented by the same diagram, which means that the statements are logically equivalent. What this means is that, unlike a converse statement, where one statement can be true and the other can be false, in a contrapositive statement, one statement cannot be true and the other statement cannot be false. They are either BOTH true OR they are either BOTH false.
In addition, the converse of the conditional statement and the inverse of the conditional statement are logically equivalent as well. This means that in both statements, they too are either BOTH true OR they are either BOTH false.
Mathematical Symbols
In mathematical symbols, the hypothesis can be represented with a, and the conclusion can be represented with b.
So, using symbols, the conditional statement is written as:
a → b
The converse is written as
b → a
The contrapositive is written as
not b → not a
The inverse is written as
not a → not b
And so, our logically equivalent statements are grouped as follows:
Scientists and their Literature!
If you are interested in reading the works of some of the authors that I listed on my podcast, I am providing links to either their Wikipedia pages or their own webpage, where you can find their books! These authors were and are incredibly talented and amazing, not just as authors, but as scientists as well! Happy reading!
E.T. Bell (also known as John Taine)
J. Goldenlane (also known as Júlia Goldman)
Rudy Rucker — his Amazon page is here
Neal Stephenson at https://www.NealStephenson.com