Where to start on a basic derivation?

Question

I have a problem I've been banging my head against for this derivation, I'm not really sure where to begin:

$P\rightarrow Q, R\rightarrow S \vdash (Q\rightarrow R) \rightarrow (P\rightarrow S) $

I'm not really sure how I can derive down the first two premises out of $\rightarrow$

Answer 1

Good thing I saw your earlier posts ...

The key rule is $\rightarrow I$, which (I assume .. I don't know how the rule is exactly defined in the system you work with ...) is defined as follows:

$\rightarrow I$: If you have a statement $\psi$ with Scope the set of sentences $\Gamma$ and $\varphi \in \Gamma$, then you can derive $\varphi \rightarrow \psi$ with Scope $\Gamma \setminus \{ \varphi \}$

I also assume that your $A$ (Assumption) rule is defined as follows:

$A$: At any point you can put down any statement $\varphi$ with Scope $\{ \varphi \}$

Applied to your proof:

\begin{array}{llll} Scope & Step & Derivation & Rule\\ 1 & (1) &P & A\\ 2 & (2) &P\rightarrow Q& A\\ 3 & (3) &Q\rightarrow R & A\\ 4 & (4) &R\rightarrow S & A\\ 1,2 & (5) &Q & 1,2 \rightarrow E \\ 1,2,3 & (6) &R & 3,5 \rightarrow E \\ 1,2,3,4 & (7) &S & 4,6 \rightarrow E \\ 2,3,4 & (8) &P \rightarrow S & 7 \rightarrow I \\ 2,4 & (9) &(Q \rightarrow R) \rightarrow (P \rightarrow S) & 8 \rightarrow I \\ \end{array}

Answer 2

Assume the following two statements are true:

$P\rightarrow Q$

and

$R\rightarrow S$

To show $(Q\rightarrow R) \rightarrow (P\rightarrow S)$

assume the premise, $Q\rightarrow R$ is true. Applying the If-Then Transitive Property, to

$P\rightarrow Q$ and $Q\rightarrow R$, we get $P\rightarrow R$

Applying the If-Then Transitive Property again, to

$P\rightarrow R$ and $R\rightarrow S$, we are forced to conclude that

$P\rightarrow S$ is true.

But this is what we wanted.