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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 ?

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1 Answer 1

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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*}

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  • 2
    $\begingroup$ This answer by Adriano is probably correct in a logical sense, but perhaps not appropriate in the context of the book, because manipulating (and thus simplifying) logical expressions hasn't been tackled at this point yet. $\endgroup$
    – Calculemus
    Jul 16, 2018 at 22:16

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