Logical Equivalences in Descartes’ Joke

I recently told the old Rene Descartes joke to a math class: Rene Descartes walks into a bar. The bartender asks, “Would you like a beer?” Descartes pauses for a moment and then replies, “I think not.” Then poof – he disappears.

Of course I naively I assumed my students had been exposed to the Descartes quote, “I think, therefore I am.” Philosopher and mathematician Rene Descartes wrote this in French in 1637, and later in Latin as cogito, ergo sum.”

After explaining the Descartes quote, I think the students understood the joke. Well, maybe it’s not that funny.

But perhaps funnier to math people than you realize, is: this joke is logically flawed because the punchline is the inverse to
the original conditional statement, and an inverse is not logically equivalent to the original statement.

Let P and Q be declarative sentences that can be definitively classified as either true or false. Then define:

• Conditional Statement: If P, then Q.
• Converse: If Q, then P.
• Inverse: If not P, then not Q.
• Contrapositive: If not Q, then not P

Two conditional statements are defined as logically equivalent when their truth values are identical for every possible combination of truth values for their individual declarative sentences.

The above table shows statement and contrapositive have the same truth values in columns 3 and 6, and so are logically equivalent. Statement and inverse are not logically equivalent.

The Descartes quote is, “If I think, therefore I am”, or “If P then Q”. The punchline is, “If I don’t think, therefore I am not”, or “If not P, then not Q”. The punchline is the inverse, and is not logically equivalent to the quote. If P is false, then “if P then Q” is true regardless of the value of Q. So Q can be either true or false.

Occasionally on television someone, often a police detective, will make a statement where they confuse a statement with its converse or inverse, and I have been known to yell at the television.

Descartes is known for developing analytic geometry, which uses algebra to describe geometry. Descartes’ rule of signs counts the roots of a polynomial by examining sign changes in its coefficients.

And before someone else feels the need to say this, I will: “Don’t put Descartes before the horse.” This is perhaps the punchline to changing the original joke to “A horse walks into a bar … ”

The following is R code to create truth tables. Logical is a variable type in R. Conditional statements in R are created
using the fact that “If P then Q” is equivalent to “Not P or Q”. I am defining the logic rules for statement, converse, inverse, contrapositive, but I could have defined the rules for more complicated statements as well.

# Define the possible values for P and Q
P <- c(TRUE, TRUE, FALSE, FALSE)
Q <- c(TRUE, FALSE, TRUE, FALSE)

# Calculate the 4 logical rules: statement, converse, inverse, contrapositive
# (Note that “if P then Q” is equivalent to “Not P or Q”.)
P_implies_Q <- !P | Q
Q_implies_P <- !Q | P
not_P_implies_not_Q <- P | !Q
not_Q_implies_not_P <- Q | !P

# expand.grid(P, Q) would also be a good start, but I wanted a specific ordering
# Create a data frame to display the truth table
truth_table <- data.frame(
P = P,
Q = Q,
`P -> Q` = P_implies_Q,
`Q -> P` = Q_implies_P,
`!P -> !Q` = not_P_implies_not_Q,
`!Q -> !P` = not_Q_implies_not_P
)

# Print the truth table
colnames(truth_table) <- c("P", "Q", "statement", "converse", "inverse", "contrapositive")
print(truth_table)

P_variable <- "I think"
Q_variable <- "I am"

colnames(truth_table) <- c(P_variable, Q_variable, "statement", "converse", "inverse", "contrapositive")
print(truth_table)

End