Symbolic Logic Question 
*

*Provide a translation for the below.

*Determine whether it is valid in LP.



Ralph is at the gardens, though Sam isn't. If the roses need pruning when the grass should be cut, then Ralph is at the gardens only provided that Sam is. So, the grass should be cut, but roses do not need pruning.

$$G \land \lnot{S}, (G \to R) \to G) \to S\ \vdash \ G \land \lnot{R}$$
I am not sure if my translation is correct so I need help verifying if it looks correct for the statement above. And if anyone can help me work on checking whether it is valid or invalid that would also be of great help.
 A: 
Ralph is at the gardens, though Sam isn't. 

You have $G \land \lnot S$ .   Not the letter choice I would have picked, but ... okay.

If the roses need pruning when the grass should be cut, then Ralph is at the gardens only provided that Sam is.

And so you run right into letter choice confusion.
You have $((G\to R)\to G)\to S)$ and appear to be confusing "Ralph is at the gardens" with "the grass should be cut". Rather, let us try with $C$ for "the grass should be cut." 
Also, that does not follow the structure of the sentence, which is: "If $R$ when $C$, then $G$ only if $S$". 
$$(C\to R)\to(G\to S)$$

So, the grass should be cut, but roses do not need pruning,

That would now be $C\land\lnot R$. 
$$G\land\lnot S, (C\to R)\to(G\to S)\vdash C\land\lnot R$$
The determination of the validity of the derivation is left to you.


Remark 
My letter choice would have been selected from people at the garden, and the actions needed to be performed.  Consistency helps.


*

*$R$alph is at the garden.

*$S$am is at the garden.

*the grass needs to be $C$ut.

*the roses need $P$runing.


$$R\land\lnot S, (C\to P)\to(R\to S)\vdash C\land\lnot P$$
