Question: Given: 

* P(x) = "x is a clear explanation"
* Q(x) = "x is satisfactory"
* R(x) = "x is an excuse

I need to translate 

* a) Some clear explanations are satisfactory.
* b) No excuses are clear explanations. (All excuses are not clear explanations)

I have a lot of difficult distinguishing when to use a conjunction and when to use an implication. Here are the translations I came up with:

* a) ∃x(P(x)∧Q(x))
* b) ∀x(R(x) --> ¬P(x))

Are these correct? Regardless of my correctness, can you provide an explanation why I was right/wrong in using the implication over the conjunction and vice versa. I would like to get their correct uses straight in my head.

Thanks

Answer: 
Welcome to Stack Exchange!

Your translations are correct. Here's a simple rule for translating phrases like "some clear explanations" and "no excuses":

* The sentence "some A's are B's" translates as $\exists x (A(x) \land B(x))$. (Some things are both A's and B's.)
* The sentence "all A's are B's" translates as $\forall x (A(x) \implies B(x))$. (Everything, if it is an A, is also a B. In other words, everything is either not-an-A, or a B.)

So, "Some clear explanations are satisfactory" translates as $\exists x (P(x) \land Q(x))$.

How about "No excuses are clear explanations"? Well, we can rephrase that as "All excuses are things that are not clear explanations". So this is $\forall x (R(x) \implies \neg P(x))$.
Your answers for both parts are correct.

For part a), "some clear explanations are satisfactory" means that "there exists a clear explanation which is satisfactory", which requires $x$ to be both a clear explanation ($P(x)$) and satisfactory ($Q(x)$). And "some" indicates $\exists x$.

For part b), "no excuses are clear explanations" means that "if we have an excuse, then it is not a clear explanation", which means for all $x$, $x$ becoming an excuse ($R(x)$) implies it is not a clear explanation ($\lnot P(x)$). And "no" indicates $\forall x$.