Translating Nested Quantifiers to English sentences
$begingroup$
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) $forall xexists y lnot P(x, y)$
(b) $exists xforall y lnot P(x, y)$
(c) $forall xexists y lnot P(y, x)$
(d) $exists xforall y lnot P(y, x)$
(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?
predicate-logic
edited Sep 13 '17 at 19:58
Shaun
9,226113684
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
$endgroup$
add a comment |
$begingroup$
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) $forall xexists y lnot P(x, y)$
(b) $exists xforall y lnot P(x, y)$
(c) $forall xexists y lnot P(y, x)$
(d) $exists xforall y lnot P(y, x)$
(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?
predicate-logic
edited Sep 13 '17 at 19:58
Shaun
9,226113684
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
$endgroup$
$begingroup$
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.
$endgroup$
– Git Gud
Apr 21 '14 at 19:10
$begingroup$
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.
$endgroup$
– Steve Kass
Apr 21 '14 at 19:14
$begingroup$
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? :)
$endgroup$
– snulty
Apr 21 '14 at 19:23
add a comment |
$begingroup$
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) $forall xexists y lnot P(x, y)$
(b) $exists xforall y lnot P(x, y)$
(c) $forall xexists y lnot P(y, x)$
(d) $exists xforall y lnot P(y, x)$
(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?
predicate-logic
edited Sep 13 '17 at 19:58
Shaun
9,226113684
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
$endgroup$
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) $forall xexists y lnot P(x, y)$
(b) $exists xforall y lnot P(x, y)$
(c) $forall xexists y lnot P(y, x)$
(d) $exists xforall y lnot P(y, x)$
(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?
predicate-logic
predicate-logic
edited Sep 13 '17 at 19:58
Shaun
9,226113684
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
edited Sep 13 '17 at 19:58
Shaun
9,226113684
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
edited Sep 13 '17 at 19:58
Shaun
9,226113684
edited Sep 13 '17 at 19:58
Shaun
9,226113684
edited Sep 13 '17 at 19:58
Shaun
9,226113684
9,226113684
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
asked Apr 21 '14 at 19:06
harsefhfharsefhf
161
161
$begingroup$
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.
$endgroup$
– Git Gud
Apr 21 '14 at 19:10
$begingroup$
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.
$endgroup$
– Steve Kass
Apr 21 '14 at 19:14
$begingroup$
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? :)
$endgroup$
– snulty
Apr 21 '14 at 19:23
add a comment |
$begingroup$
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.
$endgroup$
– Git Gud
Apr 21 '14 at 19:10
$begingroup$
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.
$endgroup$
– Steve Kass
Apr 21 '14 at 19:14
$begingroup$
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? :)
$endgroup$
– snulty
Apr 21 '14 at 19:23
$begingroup$
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.
$endgroup$
– Git Gud
Apr 21 '14 at 19:10
$begingroup$
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.
$endgroup$
– Git Gud
Apr 21 '14 at 19:10
$begingroup$
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.
$endgroup$
– Steve Kass
Apr 21 '14 at 19:14
$begingroup$
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.
$endgroup$
– Steve Kass
Apr 21 '14 at 19:14
$begingroup$
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? :)
$endgroup$
– snulty
Apr 21 '14 at 19:23
$begingroup$
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? :)
$endgroup$
– snulty
Apr 21 '14 at 19:23
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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.
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
$endgroup$
add a comment |
$begingroup$
Try the equivalent formulas in (1) and (3)
1) $negexists x forall y P(x,y)$:
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) $negexists x forall y P(y,x)$: 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.
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
$endgroup$
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
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oldest
votes
$begingroup$
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.
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
$endgroup$
add a comment |
$begingroup$
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.
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
$endgroup$
add a comment |
$begingroup$
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.
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
$endgroup$
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.
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
edited Apr 21 '14 at 19:33
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
answered Apr 21 '14 at 19:20
Peter SmithPeter Smith
40.8k340120
40.8k340120
add a comment |
add a comment |
$begingroup$
Try the equivalent formulas in (1) and (3)
1) $negexists x forall y P(x,y)$:
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) $negexists x forall y P(y,x)$: 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.
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
$endgroup$
add a comment |
$begingroup$
Try the equivalent formulas in (1) and (3)
1) $negexists x forall y P(x,y)$:
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) $negexists x forall y P(y,x)$: 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.
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
$endgroup$
add a comment |
$begingroup$
Try the equivalent formulas in (1) and (3)
1) $negexists x forall y P(x,y)$:
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) $negexists x forall y P(y,x)$: 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.
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
$endgroup$
Try the equivalent formulas in (1) and (3)
1) $negexists x forall y P(x,y)$:
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) $negexists x forall y P(y,x)$: 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.
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
answered Sep 13 '17 at 22:38
LoMaPhLoMaPh
8111916
8111916
add a comment |
add a comment |
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$begingroup$
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.
$endgroup$
– Git Gud
Apr 21 '14 at 19:10
$begingroup$
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.
$endgroup$
– Steve Kass
Apr 21 '14 at 19:14
$begingroup$
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? :)
$endgroup$
– snulty
Apr 21 '14 at 19:23