Distinguish 'the only f are g' from 'only f are g' in predicate logic
$begingroup$
I'm trying to teach myself some predicate logic by reading Howard Pospesel's Predicate Logic and doing the exercises.
In chapter 2, exercise 7 contains some optional, challenging exercises, and one of them has me a bit stumped. In its entirety, the exercise is this:
(e) (newspaper) "The only senators ever (e)xpelled were those found guilty of (t)reason." (Ex = x is an expelled senator, Tx = x is found guilty of treason) (Note: Distinguish "The only 𝓕 are 𝓖" from "Only 𝓕 are 𝓖." Can you formulate a translation principle for statements such as (e)?)
The newspaper in brackets in the beginning is just the original source of the text. That's how the book writes the problems. Likewise, the bracketed letters are suggestions for the predicate symbol, which in this exercise is further expanded to an explicit dictionary.
It's the note that confuses me. The book has previously formulated a translation principle for statements of the type only 𝓕 are 𝓖:
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕
I understand the note to ask for a similar translation principle.
I'm not sure that I entirely understand how only 𝓕 are 𝓖 are to be distinguished from the only 𝓕 are 𝓖. When considering the above statement about senators, I might symbolise it like this:
∀x(Tx → Ex)
Meaning that for all x, if x was found guilty of treason, then x was expelled.
I have doubts about this answer, though, because that's the exact same way that one symbolises only 𝓕 are 𝓖 statements, and the note seems to indicate that it's not the same.
English is my second language, so it may be that there's some linguistic subtlety that escapes me.
If anyone can make this clearer for me, I'd appreciate it.
predicate-logic
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
$endgroup$
add a comment |
$begingroup$
I'm trying to teach myself some predicate logic by reading Howard Pospesel's Predicate Logic and doing the exercises.
In chapter 2, exercise 7 contains some optional, challenging exercises, and one of them has me a bit stumped. In its entirety, the exercise is this:
(e) (newspaper) "The only senators ever (e)xpelled were those found guilty of (t)reason." (Ex = x is an expelled senator, Tx = x is found guilty of treason) (Note: Distinguish "The only 𝓕 are 𝓖" from "Only 𝓕 are 𝓖." Can you formulate a translation principle for statements such as (e)?)
The newspaper in brackets in the beginning is just the original source of the text. That's how the book writes the problems. Likewise, the bracketed letters are suggestions for the predicate symbol, which in this exercise is further expanded to an explicit dictionary.
It's the note that confuses me. The book has previously formulated a translation principle for statements of the type only 𝓕 are 𝓖:
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕
I understand the note to ask for a similar translation principle.
I'm not sure that I entirely understand how only 𝓕 are 𝓖 are to be distinguished from the only 𝓕 are 𝓖. When considering the above statement about senators, I might symbolise it like this:
∀x(Tx → Ex)
Meaning that for all x, if x was found guilty of treason, then x was expelled.
I have doubts about this answer, though, because that's the exact same way that one symbolises only 𝓕 are 𝓖 statements, and the note seems to indicate that it's not the same.
English is my second language, so it may be that there's some linguistic subtlety that escapes me.
If anyone can make this clearer for me, I'd appreciate it.
predicate-logic
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
$endgroup$
add a comment |
$begingroup$
I'm trying to teach myself some predicate logic by reading Howard Pospesel's Predicate Logic and doing the exercises.
In chapter 2, exercise 7 contains some optional, challenging exercises, and one of them has me a bit stumped. In its entirety, the exercise is this:
(e) (newspaper) "The only senators ever (e)xpelled were those found guilty of (t)reason." (Ex = x is an expelled senator, Tx = x is found guilty of treason) (Note: Distinguish "The only 𝓕 are 𝓖" from "Only 𝓕 are 𝓖." Can you formulate a translation principle for statements such as (e)?)
The newspaper in brackets in the beginning is just the original source of the text. That's how the book writes the problems. Likewise, the bracketed letters are suggestions for the predicate symbol, which in this exercise is further expanded to an explicit dictionary.
It's the note that confuses me. The book has previously formulated a translation principle for statements of the type only 𝓕 are 𝓖:
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕
I understand the note to ask for a similar translation principle.
I'm not sure that I entirely understand how only 𝓕 are 𝓖 are to be distinguished from the only 𝓕 are 𝓖. When considering the above statement about senators, I might symbolise it like this:
∀x(Tx → Ex)
Meaning that for all x, if x was found guilty of treason, then x was expelled.
I have doubts about this answer, though, because that's the exact same way that one symbolises only 𝓕 are 𝓖 statements, and the note seems to indicate that it's not the same.
English is my second language, so it may be that there's some linguistic subtlety that escapes me.
If anyone can make this clearer for me, I'd appreciate it.
predicate-logic
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
$endgroup$
I'm trying to teach myself some predicate logic by reading Howard Pospesel's Predicate Logic and doing the exercises.
In chapter 2, exercise 7 contains some optional, challenging exercises, and one of them has me a bit stumped. In its entirety, the exercise is this:
(e) (newspaper) "The only senators ever (e)xpelled were those found guilty of (t)reason." (Ex = x is an expelled senator, Tx = x is found guilty of treason) (Note: Distinguish "The only 𝓕 are 𝓖" from "Only 𝓕 are 𝓖." Can you formulate a translation principle for statements such as (e)?)
The newspaper in brackets in the beginning is just the original source of the text. That's how the book writes the problems. Likewise, the bracketed letters are suggestions for the predicate symbol, which in this exercise is further expanded to an explicit dictionary.
It's the note that confuses me. The book has previously formulated a translation principle for statements of the type only 𝓕 are 𝓖:
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕
I understand the note to ask for a similar translation principle.
I'm not sure that I entirely understand how only 𝓕 are 𝓖 are to be distinguished from the only 𝓕 are 𝓖. When considering the above statement about senators, I might symbolise it like this:
∀x(Tx → Ex)
Meaning that for all x, if x was found guilty of treason, then x was expelled.
I have doubts about this answer, though, because that's the exact same way that one symbolises only 𝓕 are 𝓖 statements, and the note seems to indicate that it's not the same.
English is my second language, so it may be that there's some linguistic subtlety that escapes me.
If anyone can make this clearer for me, I'd appreciate it.
predicate-logic
predicate-logic
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
asked Jan 14 at 15:16
Mark SeemannMark Seemann
237214
237214
add a comment |
add a comment |
2 Answers
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$begingroup$
Long comment
Why
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕 ?
Because "Only 𝓕 are 𝓖" means that there is no 𝓖 that is not an 𝓕 (only women are mothers, because there is no mother which is not a woman).
And "there is no 𝓖 that is not an 𝓕" is $lnot exists x (Gx land lnot Fx)$.
But the last one is equivalent to : $forall x lnot (Gx land lnot Fx)$ and in turn $lnot (P land lnot Q)$ is equivalent to $(P to Q)$.
Conclusion :
"Only 𝓕 are 𝓖 " can be symbolized ad : $forall x (Gx to Fx)$, that reads : "All 𝓖 are 𝓕".
What about "The only 𝓕 are 𝓖" ?
It is dubtful how to read it... but we can try to follow the author's suggestion to distinguish it form the previous one.
"The only senators ever (e)xpelled were those found guilty of (t)reason" means that the senators expelled (if any) were those guilty of treason (which does not implies that every senator guilty of treason has been expelled...)
This is :
$forall x (Ex to Tx)$
that, by the same equivalences above, is (as you said) is equivalent to : $lnot exists x (Ex land lnot Tx)$.
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
1
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
add a comment |
$begingroup$
Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:
-∃x(Ex & -Tx)
Translating back into words, no x exists such that x was expelled, and x wasn't found guilty of treason.
Does that seem reasonable?
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
Long comment
Why
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕 ?
Because "Only 𝓕 are 𝓖" means that there is no 𝓖 that is not an 𝓕 (only women are mothers, because there is no mother which is not a woman).
And "there is no 𝓖 that is not an 𝓕" is $lnot exists x (Gx land lnot Fx)$.
But the last one is equivalent to : $forall x lnot (Gx land lnot Fx)$ and in turn $lnot (P land lnot Q)$ is equivalent to $(P to Q)$.
Conclusion :
"Only 𝓕 are 𝓖 " can be symbolized ad : $forall x (Gx to Fx)$, that reads : "All 𝓖 are 𝓕".
What about "The only 𝓕 are 𝓖" ?
It is dubtful how to read it... but we can try to follow the author's suggestion to distinguish it form the previous one.
"The only senators ever (e)xpelled were those found guilty of (t)reason" means that the senators expelled (if any) were those guilty of treason (which does not implies that every senator guilty of treason has been expelled...)
This is :
$forall x (Ex to Tx)$
that, by the same equivalences above, is (as you said) is equivalent to : $lnot exists x (Ex land lnot Tx)$.
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
1
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
add a comment |
$begingroup$
Long comment
Why
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕 ?
Because "Only 𝓕 are 𝓖" means that there is no 𝓖 that is not an 𝓕 (only women are mothers, because there is no mother which is not a woman).
And "there is no 𝓖 that is not an 𝓕" is $lnot exists x (Gx land lnot Fx)$.
But the last one is equivalent to : $forall x lnot (Gx land lnot Fx)$ and in turn $lnot (P land lnot Q)$ is equivalent to $(P to Q)$.
Conclusion :
"Only 𝓕 are 𝓖 " can be symbolized ad : $forall x (Gx to Fx)$, that reads : "All 𝓖 are 𝓕".
What about "The only 𝓕 are 𝓖" ?
It is dubtful how to read it... but we can try to follow the author's suggestion to distinguish it form the previous one.
"The only senators ever (e)xpelled were those found guilty of (t)reason" means that the senators expelled (if any) were those guilty of treason (which does not implies that every senator guilty of treason has been expelled...)
This is :
$forall x (Ex to Tx)$
that, by the same equivalences above, is (as you said) is equivalent to : $lnot exists x (Ex land lnot Tx)$.
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
1
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
add a comment |
$begingroup$
Long comment
Why
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕 ?
Because "Only 𝓕 are 𝓖" means that there is no 𝓖 that is not an 𝓕 (only women are mothers, because there is no mother which is not a woman).
And "there is no 𝓖 that is not an 𝓕" is $lnot exists x (Gx land lnot Fx)$.
But the last one is equivalent to : $forall x lnot (Gx land lnot Fx)$ and in turn $lnot (P land lnot Q)$ is equivalent to $(P to Q)$.
Conclusion :
"Only 𝓕 are 𝓖 " can be symbolized ad : $forall x (Gx to Fx)$, that reads : "All 𝓖 are 𝓕".
What about "The only 𝓕 are 𝓖" ?
It is dubtful how to read it... but we can try to follow the author's suggestion to distinguish it form the previous one.
"The only senators ever (e)xpelled were those found guilty of (t)reason" means that the senators expelled (if any) were those guilty of treason (which does not implies that every senator guilty of treason has been expelled...)
This is :
$forall x (Ex to Tx)$
that, by the same equivalences above, is (as you said) is equivalent to : $lnot exists x (Ex land lnot Tx)$.
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
Long comment
Why
Only 𝓕 are 𝓖 = All 𝓖 are 𝓕 ?
Because "Only 𝓕 are 𝓖" means that there is no 𝓖 that is not an 𝓕 (only women are mothers, because there is no mother which is not a woman).
And "there is no 𝓖 that is not an 𝓕" is $lnot exists x (Gx land lnot Fx)$.
But the last one is equivalent to : $forall x lnot (Gx land lnot Fx)$ and in turn $lnot (P land lnot Q)$ is equivalent to $(P to Q)$.
Conclusion :
"Only 𝓕 are 𝓖 " can be symbolized ad : $forall x (Gx to Fx)$, that reads : "All 𝓖 are 𝓕".
What about "The only 𝓕 are 𝓖" ?
It is dubtful how to read it... but we can try to follow the author's suggestion to distinguish it form the previous one.
"The only senators ever (e)xpelled were those found guilty of (t)reason" means that the senators expelled (if any) were those guilty of treason (which does not implies that every senator guilty of treason has been expelled...)
This is :
$forall x (Ex to Tx)$
that, by the same equivalences above, is (as you said) is equivalent to : $lnot exists x (Ex land lnot Tx)$.
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
edited Jan 14 at 16:12
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
answered Jan 14 at 15:55
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
67.4k449116
1
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
add a comment |
1
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
1
1
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
$begingroup$
I just found out that the CD-ROM that accompanies the book comes with a program that, among other things, can check answers to the exercises, and that program has this as the correct answer.
$endgroup$
– Mark Seemann
Jan 16 at 15:29
add a comment |
$begingroup$
Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:
-∃x(Ex & -Tx)
Translating back into words, no x exists such that x was expelled, and x wasn't found guilty of treason.
Does that seem reasonable?
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
$endgroup$
add a comment |
$begingroup$
Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:
-∃x(Ex & -Tx)
Translating back into words, no x exists such that x was expelled, and x wasn't found guilty of treason.
Does that seem reasonable?
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
$endgroup$
add a comment |
$begingroup$
Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:
-∃x(Ex & -Tx)
Translating back into words, no x exists such that x was expelled, and x wasn't found guilty of treason.
Does that seem reasonable?
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
$endgroup$
Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:
-∃x(Ex & -Tx)
Translating back into words, no x exists such that x was expelled, and x wasn't found guilty of treason.
Does that seem reasonable?
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
answered Jan 14 at 15:21
Mark SeemannMark Seemann
237214
237214
add a comment |
add a comment |
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