Question

Is (p>q)>[(p>q)>q] a tautology? Why or why not?

Answer 1

Let us show that the formula $(p\to q)\to[(p\to q)\to q]$ is not a tautology. Taking into account that for $(p_0,q_0)=(0,0)\in\{0,1\}^2$ we have that $|(p_0\to q_0)\to[(p_0\to q_0)\to q_0]|=(0\to 0)\to[(0\to 0)\to 0]=1\to [1\to 0]=1\to 0=0,$ we conclude that the formula is not a tautology.