Meaning of this statement (logic)

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In summary, the statement [S ⋁ (G ⋀ ¬S)] ⋁ ¬G is a tautology that can be translated to "either Steve is happy or not, George is happy or not". It also implies that if George is not unhappy, then Steve is unhappy and it is uncertain if "Steve is happy" implies that George is also happy.
  • #1
autodidude
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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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  • #2
hi autodidude! :smile:
autodidude said:
a) [S ⋁ (G ⋀ ¬S)] ⋁ ¬G

either draw a venn diagram, or use the distributive law :wink:
 
  • #3
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
 

Related to Meaning of this statement (logic)

1. What is the purpose of logic in this statement?

The purpose of logic in this statement is to provide a rational and systematic approach to understanding and evaluating the meaning of the statement. It helps to identify the underlying principles and reasoning behind the statement.

2. How does logic play a role in interpreting the meaning of this statement?

Logic plays a crucial role in interpreting the meaning of this statement by breaking it down into smaller, more manageable parts and analyzing the relationships between them. By applying logical principles, we can determine the validity and soundness of the statement.

3. Can logic be used to prove or disprove the meaning of this statement?

Yes, logic can be used to prove or disprove the meaning of this statement. By using logical reasoning and valid arguments, we can determine whether the statement is true or false.

4. Are there different types of logic that can be applied to this statement?

Yes, there are different types of logic that can be applied to this statement, such as deductive, inductive, and modal logic. Each type of logic has its own set of rules and principles that can be used to analyze and interpret the meaning of the statement.

5. How does understanding the logic behind this statement contribute to its overall meaning?

Understanding the logic behind this statement is crucial in fully comprehending its meaning. By analyzing the logical structure and reasoning behind the statement, we can gain a deeper understanding of its implications and significance.

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