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Problem is :

Let p and q be propositions. Using only the Truth Table, decide whether

                     p↔q does not imply p→¬q

is True or False.

I try to explain :

Rule of inference "if the premises hold, then the conclusion holds."

Premise#1 Premise#2 ... Premise#n
(then) Conclusion

P→Q ,and Q→P then P→(Q)' (conclusion). It is not tautology , so "does not" should be True.

What I'm asking: Is my approach correct ? If not , then what is solution ?

Edited : 

p   q   p↔q p→¬q    (p↔q)→(p→¬q)
0   0   1   1           1
0   1   0   1           1
1   0   0   1           1
1   1   1   0           0

So, "imply" is FALSE making "does not imply" TRUE.

ypercubeᵀᴹ
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Mithlesh Upadhyay
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1 Answers1

1

I think your solution (after edited) is correct (and I think it's a good approach to include partial's of the expression under investigation). The point one have to take is that you have to consider the meaning of the phrase "does not imply":

The first thought maybe that "$A$ does not imply $B$" means $\neg(A\Rightarrow B)$, but that's normally not what it means. It normally means that $A\Rightarrow B$ is not always true. It's the latter meaning you have used in your truth table.

Perhaps one should not categorically state that $(p \Leftrightarrow q) => (p \Rightarrow\neg q)$ is false, becase the truth tables states that under some circumstances it's true.

skyking
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