# translate the following into quantified statements

M(x) = "x is male"
F(x) = "x is female"
S(x,y) = "x is scared of y"
O(x) = "x is open-minded"

Translate the following:
a) Some open-minded females fear some closed-minded males.
b) No female fears all males.
c) Some males are females.
d) All males are scared of all females.

I could find the answer for c) $$\exists x[M(x) \land F(x)]$$ but could not figure out the S(x,y) ones.

Edit: After figuring out some answers: is this answer correct for part a) $$\exists x [ [F(x) \land O(x)] \land \exists y [[M(y) \land ¬O(y) ] \land S(x,y) ]$$.

• You may need to rethink things, because your answer for c) is wrong. – Keshav Srinivasan Dec 3 at 0:09
• Can you tell what to do for c please – sam Dec 3 at 0:10
• You're right to start with there exists an x, but think about what properties that x should have. – Keshav Srinivasan Dec 3 at 0:11
• Thanks! I saw the mistake. – sam Dec 3 at 0:12

Hint: Whenever you see the word "some", put $$\exists x$$ (or some other letter), and think about what property that $$x$$ that exists should have. Whenever you see the word "all", put $$\forall x$$ and think about what property every $$x$$ should have. And whenever you see "no" or "none", put $$\neg\exists x$$ and think about what property no $$x$$ should have.

If there are multiple occurrences of these kinds of words in a sentence, then you will need multiple quantifiers, and each quantifier should use a different letter for the variable.

• I tried but I am absolutely stumped with S(x,y). I can't figure out where to put M(x) or F(x) in S(x,y). – sam Dec 3 at 0:22

Keep following the patterns:

'ALl P are Q': $$\forall x (P(x) \to Q(x))$$

'Some P are Q': $$\exists x (P(x) \land Q(x))$$

'NO P are Q': $$\forall x (P(x) \to \neg Q(x))$$

Now, c) nicely follows that pattern. The others may seem like they do not, but when you break it up, they do as well. For example, let's take the second: 'no female fears all males'

This is still of the basic pattern 'No P are ...', and so it starts with:

$$\forall x (F(x) \to \neg ...)$$

Now, what should come after the $$\neg$$?

Well, here you want to say that '$$x$$ fears all males', i.e. '$$x$$ is scared of all males'

Ok, but that can be paraphrased into one of the common patterns:

'All males are scaring $$x$$'

which translates as :

$$\forall y (M(y) \to S(x,y))$$

OK, so plug that into the first formula after the $$\neg$$ and you get:

$$\forall x (F(x) \to \neg \forall y (M(y) \to S(x,y)))$$

Done!

The moral: divide and conquer!

• Thanks so much!! I figured out some things. Is this answer correct for a) $\exists x [ [F(x) \land O(x)] \land \exists y [[M(y) \land ¬O(y) ] \land S(x,y) ]$. – sam Dec 3 at 0:51
• @sam Yes, perfect! You're getting it! – Bram28 Dec 3 at 1:01