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Posts tagged ‘Logic’

## The Liberal Paradox

As it turns out, it is not logically consistent to both support personal freedom, and efficiency. Ethics is a complicated topic.

## Why I believe that 1≠2 more than I do the existence of God

Well, personally I am agnostic/atheist, but I’ve heard from several people that believe in God, that they believe “he exists” more than they believe that 1≠2.

Please put aside for one moment the exact meaning of the sentence “God exists”. All knowledge is, to an extent, subjective. I know that 1≠2 because it is a mathematical theorem, but also is a theorem that Zorn’s Lemma is provable from the Axiom of Choice – my level of confidence in these two theorems being true is rather different.

I claim that anyone (if he understands logic), even if he believes that “God exists”, should place more confidence in the statement “1≠2” than in his belief in God *. Why? Let’s assume 1=2, and prove God does not exists:

1. Assume 1=2, and suppose by negation that God exists
2. 1≠2 (from standard arithmetic axioms and inference rules)
3. If God exists, then 1=2 and 1≠2 (from 1 and 2)
4. Therefore, God does not exist (from 3, by Proof by Contradiction)

Q.E.D

So, one can prove from 1=2 that God exists, but I doubt you’ll find a proof that 1=2 from the assumption God exists. So believing 1=2 requires more faith, and provides more information, than believing God exists (because it implies God exists, but the other way around does not hold).

* Of course, there is at least one flaw in the above argument. Can you find it?

## A Cartoon Proof of Löb’s Theorem

Recommended for hardcore logicians only (or just people who like to think).

Löb’s Theorem apparently states that "if it is provable that ‘if X is provable, then X’, then X itself is provable". The cartoon helps you (just a bit) to avoid getting trapped in an endless loop of logic.

There’s also a bonus exercise at the end, quite refreshing.

## 2008 Turing Awards

Just thought to tell everyone that the 2008 Turing Award will be given to two guys that invented Model Checking, that has a very loose connection to my own master thesis. How loose? Both relate to Temporal Logic, that’s it.