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For this entire question, please use the following propositional function: $P(x,y)$: $x$ has sent a postcard to $y$. Translate the following quantified propositions to English sentences. Try to use sentences as natural as possible.

(a) $\forall x\exists y \lnot P(x, y)$

(b) $\exists x\forall y \lnot P(x, y)$

(c) $\forall x\exists y \lnot P(y, x)$

(d) $\exists x\forall y \lnot P(y, x)$

(a) Every $x$ has some $y$ to whom he has not sent a postcard.

(b) Some $x$ has not sent a postcard to every $y$.

(c) Every $x$ has some $y$ from whom he has not received a postcard.

(d) Some $x$ has not received a postcard from every $y$.

Do you think my answers are correct?

Shaun
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harsefhf
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  • I think part of 'as natural as possible' is removing the variables from the natural language. For instance, b) could be translated as: Someone has never sent a postcard. - This is true, by the way, either due to still birth or because I have never sent a post card. – Git Gud Apr 21 '14 at 19:10
  • These sentences are not very natural. That is, they're not things a person would typically say out loud. Here's a suggestion for a): No matter who you are, there is always someone you've never sent a postcard to. And your answer to b) is not quite right. b) means there is someone who has never sent a postcard to anyone at all. – Steve Kass Apr 21 '14 at 19:14
  • For (c),(d) if $x$ didn't receive a postcard from $y$, does it mean $y$ hasn't sent it? What if the postcard was lost, or is caught up in the mailing system? :) – snulty Apr 21 '14 at 19:23

2 Answers2

1

Try the equivalent formulas in (1) and (3)

1) $\neg\exists x \forall y P(x,y)$: There is no one that has sent a postcard to every person.

2) There is a person that has not sent a postcard to anyone.

3) $\neg\exists x \forall y P(y,x)$: There is no one that has received a postcard from every person.

4) There is a person that has not received a postcard from anyone.

LoMaPh
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What language does the expression

For every $x$ there is some $y$ such that $x$ has not sent a postcard to $y$.

belong to? If anything, it belongs to that logic-English mix which some call Loglish. Using Loglish is a very useful step on the way towards producing a translation into natural language (see my Introduction to Formal Logic, §24.1 for lots of examples of Loglish being used this way). So I'm all for using Loglish. But you can't stop there. For Loglish certainly isn't English. Just try saying the displayed expression to any native speaker outside the logic classroom and see what reaction you get!

You are being asked to translate into natural English, and that means go on to drop the $x$s and $y$s -- which, at a first step, typically involves replacing them by natural-language pronouns. You make a rather fumbled first step when you write

Every $x$ has some $y$ to whom he has not sent a postcard.

where you've replaced the second '$x$' with 'he', and the 'to $y$' with 'to whom'. But you shouldn't do this piecemeal, ditching some of the $x$s say and not the others: it won't even be good LogLish! You need to lose all the related $x$s in one go. And then all the $y$s. That way you will end up, as you are asked to do, with natural English, without any $x$s and $y$s.

Peter Smith
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