Principle of Explosion
Principle of Explosion
[Cueball's Cueball-like friend is talking to him.] Friend: If you assume contradictory axioms, you can derive anything. It's called the principle of explosion. Cueball: Anything? Lemme try.
[Cueball is writing on a piece of paper on a desk.]
[Cueball is holding up a piece of paper to his friend, while holding a phone.] Cueball: Hey, you're right! I started with P∧ ¬ P and derived your mom's phone number! Friend: That's not how that works.
[The friend is looking at the piece of paper, while Cueball is talking to someone on a phone. The desk from before can be seen to the right.] Cueball: Mrs. Lenhart? Friend: Wait, this is her number! How— Cueball: Hi, I'm a friend of— Why, yes, I am free tonight! Friend: Mom! Cueball: No, box wine sounds lovely!

Cueball's friend (who also looks like Cueball) explains the principle of explosion , a classical theorem of logic, which shows that if within a system of logic you can use the axioms and rules of deduction to derive (prove) a contradiction, it then becomes possible to derive any statement at all within that system (whether it’s actually true or not). In particular, if you start by assuming a self-contradictory statement, you can derive anything.
Cueball then proceeds to misinterpret (perhaps intentionally) that you can derive any fact about the physical world. His formula of propositional logic in the third panel reads " P and not P ", where ∧ is the formal logic symbol for "and" and ¬ is the symbol for "not". P stands for a proposition. As " P and not P " is shorthand for " P is both true and false", this forms a contradiction from which the principle of explosion can begin. Humorously and to his friend's bewilderment he then successfully manages to 'derive' the phone number for his friend's mom.
An example from math : If you assume that √2 is a rational number, you can 'prove' things that are obviously false, such as the fact that some numbers must be both even and odd. Consequently, you can draw the conclusion that √2 must be an irrational number (provided such a thing exists at all! - luckily, it does and obeys the same calculation rules as for rational numbers; this is how proof by contradiction works.)
This can be seen in a Truth Table :
P ¬P P ∧ ¬P P ∧ ¬P ⇒ Q T F F T F T F T
The formula P ∧ ¬P ⇒ Q is true in every possible interpretation. No matter what propositions are substituted for P and Q the implication is true. So if a single example of a contradiction were found, then every proposition would be true, (and simultaneously false).
After deriving the phone number Cueball instantly calls his friend's mom, who turns out to be Mrs. Lenhart . She asks Cueball out, without any preamble, to his friend's vexation. It does not get better when it is obvious that she wishes to drink "cheap" boxed wine with him, and Cueball is free tonight! There is definitely a hint of Mrs. Robinson over Mrs. Lenhart here.
In the title text we hear more of Cueball's (one-sided) conversation with Mrs. Lenhart. She asks him to pick up waffle cones, a variety of ice cream cone . And when he sounds bewildered by this she explains that it is for drinking the wine. This is probably not a very good idea, since waffles are typically not water proof and would also dissolve into the wine. But it could also be considered kinky; something Mrs. Lenhart's son would not like to hear about. The rest of the title text is just more of the main comic's derivation joke, since Cueball will use a second to derive her son's credit card number, so he can buy the cones at his expense.
In reality, Cueball really could start with the principle of explosion and "prove" a statement such as "Mrs. Lenhart's phone number is 867-5309 ", but the same could be said of any conceivable phone number, most of which don't actually belong to Mrs. Lenhart, and because his axiom system is inconsistent, he has no way of knowing which is correct. Likewise for his friend's credit card number. Much like The Library of Babel , an axiom system which can prove any statement might as well prove nothing. Perhaps Cueball already knows these phone and credit card numbers, and is just talking about the principle of explosion to mess with his friend.
[Cueball's Cueball-like friend is talking to him.] Friend: If you assume contradictory axioms, you can derive anything. It's called the principle of explosion. Cueball: Anything? Lemme try.
[Cueball is writing on a piece of paper on a desk.]
[Cueball is holding up a piece of paper to his friend, while holding a phone.] Cueball: Hey, you're right! I started with P∧ ¬ P and derived your mom's phone number! Friend: That's not how that works.
[The friend is looking at the piece of paper, while Cueball is talking to someone on a phone. The desk from before can be seen to the right.] Cueball: Mrs. Lenhart? Friend: Wait, this is her number! How— Cueball: Hi, I'm a friend of— Why, yes, I am free tonight! Friend: Mom! Cueball: No, box wine sounds lovely!
