# Proposition into spoken language

Given:

$\sim( p \leftrightarrow (q \vee r) )$

$p:$ It's raining

$q:$ The sun is shining

$r:$ There are clouds in the sky.

Translate the proposition into spoken language.

Solution:

It's false that it's raining if and only if the sun is shining or there are clouds in the sky.

Would the solution above be okay?

• Pretty much; I'd add a comma after "that". You might also just say "It's not raining if and only if ...". – James Aug 22 '14 at 6:07
• @James Okay, I get it! However, you suggested "It's not raining if and only if..." but isn't the negation symbol affecting the whole proposition? I understand by writing that only $p$ is being negated. Am I wrong? – rickHdz Aug 22 '14 at 6:28
• @James: A comma after 'that' would definitely be ungrammatical. – TonyK Aug 22 '14 at 8:01

$\sim(A\leftrightarrow B)\equiv (A\leftrightarrow \sim B)$.
So $\sim(p\leftrightarrow(q\vee r)\equiv p\leftrightarrow (\sim q \wedge \sim r)$
[I wonder if your $r$ should be 'there are no clouds in the sky'. That would make more sense.]