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Can you help us with this Critical Thinking problem?


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This problem is from Critical Thinking Book 2 (the Critical Thinking Company)

 

My daughter needs to figure out if this is a tautology by proving if the argument is valid.

 

[P->(Q->R)]->[(P->Q)->R]

 

She is working independantly through this book so I have NO IDEA what all of this means.

Thanks in advance!

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This problem is from Critical Thinking Book 2 (the Critical Thinking Company)

 

My daughter needs to figure out if this is a tautology by proving if the argument is valid.

 

[P->(Q->R)]->[(P->Q)->R]

 

She is working independantly through this book so I have NO IDEA what all of this means.

Thanks in advance!

 

Using the logical implication truth table from Wikipedia as a guide, consider the following:

 

Suppose P, Q, and R are all FALSE.  Then:

Since P is FALSE, [P->(Q->R)] is TRUE, regardless of what value is assigned to (Q->R)

(P->Q) = TRUE, therefore (P->Q) -> R is FALSE (since R is assumed FALSE)

 

Thus, [P->(Q->R)]->[(P->Q)->R] is equivalent to [TRUE -> FALSE] which is FALSE.

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