# Translate these English statements into Predicate Logic

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

• Both are correct. $\forall$ needs $\to$ while $\exists$ needs $\land$. Jun 4, 2019 at 15:31
• See similar post. Jun 4, 2019 at 15:33

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 $$\forall x (R(x) \implies \neg P(x))$$.

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 $$\forall x$$.

I understand the difference between conjunction and implication can be confusing at first. Just remember almost all translations with the existential claims (∃x) use conjunction, most universal claims (∀x) use implication.

∃x(P(x)∧Q(x))
translates to:
"There exists a clear explanation, and it is satisfactory."

But
∃x(P(x)-->Q(x)) which we know is equivalent to ∃x(~P(x) V Q(x))
would translate to:
"There exists a (member of the universe) such that if it were a clear explanation, it would be satisfactory." This sentence basically means there may or may not be a clear explanation that is satisfactory, which is a very weak claim, so it is kinda rare to see implication being used with existential claims.

also:
∀x(R(x) --> ¬P(x))
translates to:
"All excuses are not clear explanations."
Try to think of it like an if-then statement. If there is an excuse, then it is not a clear explanation.

but if the formula was:
∀x(R(x) ∧ ¬P(x))
It would mean every (member in the universe) is an excuse, and not a clear explanation.

So basically you'll use ∧ with ∃, and --> with ∀ like 99% of the time. If you have not yet learned about defining a universe, or universal set, then you will probably use use ∧ with ∃, and --> with ∀ 100% of the time, because statements like ∀x(R(x) ∧ ¬P(x)) and ∃x(P(x)-->Q(x)) wouldn't make sense.

Hope this helped. Feel free to ask any other questions, I just took discrete math last semester :)