Meaning of this statement (logic)

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The logical expression [S ⋁ (G ⋀ ¬S)] ⋁ ¬G, derived from Velleman's 'How To Prove It', represents a tautology, meaning it always evaluates to true regardless of the truth values of S ('Steve is happy') and G ('George is happy'). The interpretation indicates that either Steve is happy, or George is happy while Steve is unhappy, or George is unhappy. This leads to the conclusion that the happiness of Steve does not necessarily imply the happiness of George, as the expression encompasses all possible truth scenarios.

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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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hi autodidude! :smile:
autodidude said:
a) [S ⋁ (G ⋀ ¬S)] ⋁ ¬G

either draw a venn diagram, or use the distributive law :wink:
 
autodidude said:
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
 

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