Meaning of this statement (logic)

This is from Velleman's 'How To Prove It' book (not homework! Reading through it myself)

Let S stand for the statement 'Steve is happy' and G for 'George is happy'. What English sentences are represented by the following:

a) [S ⋁ (G ⋀ ¬S)] ⋁ ¬G

I interpret it as saying 'either Steve is happy or George is happy and Steve is unhappy, or George is unhappy'.

But I'm more interested what it actually means...

...if George is NOT unhappy, meaning he IS happy, then Steve is unhappy. If he is not unhappy, can Steve be happy? Does 'Steve is happy' alone imply that George is also happy? Or can we not say anything about that?

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tiny-tim
Homework Helper
hi autodidude!
a) [S ⋁ (G ⋀ ¬S)] ⋁ ¬G
either draw a venn diagram, or use the distributive law

This is from Velleman's 'How To Prove It' book (not homework! Reading through it myself)

Let S stand for the statement 'Steve is happy' and G for 'George is happy'. What English sentences are represented by the following:

a) [S ⋁ (G ⋀ ¬S)] ⋁ ¬G

I interpret it as saying 'either Steve is happy or George is happy and Steve is unhappy, or George is unhappy'.

But I'm more interested what it actually means...

...if George is NOT unhappy, meaning he IS happy, then Steve is unhappy. If he is not unhappy, can Steve be happy? Does 'Steve is happy' alone imply that George is also happy? Or can we not say anything about that?

The statement in (a) is a tautology as its true-false table gets only true values. Thus, we could perhaps

translate it into common language as Steven is happy or not, George is happy or not...

DonAntonio