# Sound and Fury

Signifying nothing

## My favourite tautologies

So today I was reading Artificial Intelligence and Scientific Method and Gillies mentions that $(p \rightarrow q) \vee (q \rightarrow p)$ is a logical truth; a tautology in classical logic (but not in intuitionistic logic). I was interested by that and check that it is indeed the case by writing out the truth table. I didn’t bother to actually find a proof, but contented myself with the fact that the truth table + the fact that the propositional calculus is sound and complete is kind of a proof. Then I thought what are some other cool tautologies. Here are a couple I remembered.

• $\neg p \rightarrow (p \rightarrow q)$
• $( ( p \rightarrow q) \wedge (\neg p \rightarrow q ) ) \rightarrow q$

Then I thought. Hmm. I’m a geek. But I’m still enamoured of this ability to put $\LaTeX$ code in my wordpress posts so I thought I’d write about it…