TITLE: Simplifying ambiguous statements
QUESTION [5 upvotes]: I have been working on the following question from Velleman's How to prove book:

Let S stand for the statement “Steve is happy” and G for “George is
  happy.” What English sentences are represented by the following
  expressions? (a) (S ∨ G) ∧ (¬S ∨ ¬G). (b) [S ∨ (G ∧ ¬S)] ∨ ¬G. (c) S ∨
  [G ∧ (¬S ∨ ¬G)].

Now, this is how I'm solving each of these:
(a)

(S ∨ G) = Steve is happy or George is happy
(¬S ∨ ¬G) = Steve is not happy or George is not happy

(S ∨ G) ∧ (¬S ∨ ¬G) = Steve is happy or George is 
and Steve is not happy or George is not happy.

(b)

¬G = George is not happy
S ∨ (G ∧ ¬S) = Steve is happy or George is happy and steve ins't happy

[S ∨ (G ∧ ¬S)] ∨ ¬G = Either George isn't happy or Steve is or George
is happy and steve isn't.

(c)

[G ∧ (¬S ∨ ¬G)] = George is happy and steve isn't.
S ∨ [G ∧ (¬S ∨ ¬G)] = Either Steve is happy or George is and steve isn't.

But the problem which I feel that are present in the solutions are that are ambiguous and doesn't really present a concise information to the readers ? Is my solutions correct or more simplification needed for it ? How do logicians provide statements for that ?

REPLY [6 votes]: You can try simplifying the statements as far as possible before translating to English:
(a)
\begin{align*}
(S \lor G) \land (\neg S \lor \neg G)
&\equiv (S \land \neg S) \lor (S \land \neg G) \lor (G \land \neg S) \lor (G \land \neg G) \\
&\equiv (S \land \neg G) \lor (G \land \neg S) \\
&\equiv \text{"Either Steve is happy or George is happy (but not both)."}
\end{align*}
(b)
\begin{align*}
[S \lor (G \land \neg S)] \lor \neg G
&\equiv [(S \lor G) \land (S \lor \neg S)] \lor \neg G \\
&\equiv (S \lor G) \lor \neg G \\
&\equiv \top \\
&\equiv \text{"George is either happy or not happy."} \\
&\equiv \text{"2 + 3 = 5."}
\end{align*}
(c)
\begin{align*}
S \lor [G \land (\neg S \lor \neg G)]
&\equiv S \lor [(G \land \neg S) \lor (G \land \neg G)] \\
&\equiv S \lor (G \land \neg S) \\
&\equiv (S \lor G) \land (S \lor \neg S) \\
&\equiv S \lor G \\
&\equiv \text{"Steve is happy or George is happy (or both)."}
\end{align*}