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

Question

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?

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 07:18
“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

score 0 · Answer 1 · answered Jul 11 '19 at 17:05

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.

As a side note this makes you right since proving logical correctness of a statement is the same as reducing it to a tautology

1 Answers1