# How do I translate $[(P \Rightarrow Q) \land (Q \Rightarrow T)] \Rightarrow T$ into English?

In his book, Axioms & Set Theory, Robert Andre introduces logic with this statement:

If $Q$ is true whenever $P$ is true, and $T$ is true whenever $Q$ is true, then $T$ is true whenever $P$ is true.

He renders the statement symbolically:

$$[(P \Rightarrow Q) \land (Q \Rightarrow T)] \Rightarrow T$$

How is this so? The symbols seem to say:

If $Q$ is true whenever $P$ is true, and $T$ is true whenever $Q$ is true, then $T$ is true.

It seems to me that the English statement given by Andre would be rendered

$$[(P \Rightarrow Q) \land (Q \Rightarrow T)] \Rightarrow [P \Rightarrow T]$$

What am I missing?

• Looks like the book has a typo. You're right. – Mark S. Jun 12 '17 at 0:00
• You missed nothing, it is a (big) typo. Now go back to sleep , if it's 4:00am over there ;-) – magma Jun 12 '17 at 7:18
• “Fear is the path to the dark side. Fear leads to anger. Anger leads to hate. Hate leads to suffering."/Yoda – skyking Jun 12 '17 at 10:57
• Robert Andre is a sith... he wanted to say that all implications leads to the dark side... – Brethlosze Jun 12 '17 at 14:51

Your statement is correct since it reduces to $$[P \implies T] \implies [P \implies T]$$ which is a tautology(Always true),
Andre's statement however reduces to $$[P \implies T] \implies T ]$$ which is a false statement if both P and T are false.