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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.
Thanks for your help.

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) ] $.

sam
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2 Answers2

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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!

Bram28
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  • 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 03 '19 at 00:51
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    @sam Yes, perfect! You're getting it! – Bram28 Dec 03 '19 at 01:01
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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.

Keshav Srinivasan
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  • 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 03 '19 at 00:22