Principle of Explosion and Reductio ad Absurdum

The Principle of Explosion (sometimes called Ex Falso Sequitor Quodlibet, Ex Falso Quodlibet, Principle of Pseudo-Scotus and even (mainly by Italians) Scoto’s Law)1 is a law of classical logic according to which any statement follows from a contradiction. Symbolically it’s written as:

P‏ ∧ ¬P ⊢ Q

and read as “we can derive Q from P and it’s negation ¬P (i.e. a contradiction)” where Q is any proposition. For instance if we can prove that “it’s raining” (this will be P) and “it’s not raining” (this will be ¬P) we can as well prove that “Bertrand Russell was a frog” (a possible Q), but we can also prove that “Bertrand Russell wasn’t a frog” (another Q). Due to examples like that many people says that a contradictions implies all the, infinite, others. This is obviously incredibly important for mathematical logic since it tell us that if a system can derive a contradiction then it’s worthless since it can basically derive an infinite amount of nonsensical stuff. For instance let’s imagine that a contradiction arises in a formal system supposed to work with natural numbers: this system is then able to prove falsehoods like “1+2=42” or “1 is a prime” and must be abandoned for a better one.

But why is the Principle of Explosion true? How it is possible to obtain Q from P and ¬P?

The standard way of proving it consist in applying some basic rules of classical logic (namely conjunction elimination, disjunction introduction, disjunctive syllogism and conditional proof) but this is not what we’re going to do. We’d rather like to prove it using the powerful tool that the Reduction ad Absurdum (RAA) is. For those who does not know how a RAA works: if you want to prove R then assume ¬R (it’s negation) to be true and prove that this implies a contradiction, so ¬R can’t be true and so it must be the case that R is. Let’s now rewrite our hypothesis and our thesis:

Hypothesis: P‏ ∧ ¬P

Thesis: Q

Let’s proceed via RAA: so we’re assuming ¬Q. But look: P‏ ∧ ¬P, our hypothesis, is a contradiction! So assuming ¬Q (no matter what the actual content of Q is) leads to a contradiction (symbolically: ¬Q ⊢ ⊥) and thus, by RAA, Q must be true and P‏ ∧ ¬P ⊢ Q clearly holds. q.e.d.

In conclusion: what’s the sense of this? Well, I’m supposed to answer with something on the lines of “that’s to show that most of the times there are different solution for a single problem etc. etc.” but that really isn’t it. I just wanted to share this “new” proof that randomly popped in my mind while I was showering, that all.3

Notes:

  1. We’re going to call it Principle of Explosion so that the featured image does not lose its memetic powers.
  2. The careful reader may notice that we can prove in the same way P‏ ∧ ¬P ⊢ ¬Q but this is not a problem since Q stands for any proposition and so the two (P‏ ∧ P ⊢ ¬Q and P ∧ ¬P ⊢ ¬Q) are equivalent.
  3. Yes, this is the worst conclusion ever.
Advertisements

2 thoughts on “Principle of Explosion and Reductio ad Absurdum

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s