Mixing
formalize in predicate logic: All students join some course that they don't like.
$begingroup$
I came up with two formulas and have no Idea whether their correct.
1. ∀x(S(x)∧¬C(x))
2. ∀x∃y(S(x)∧C(y)→DoesNotLike(x,y))
And then I have one a bit more complicated, that I can think of multiple ways how to formalize but have no way of knowing if correctly:
Every student attends a class taught by a teacher that they dont like.
P.s. this is not my homework, just preparation for an exam.
logic predicate-logic logic-translation
asked Jan 24 at 18:05
Martin LMartin L
61
$endgroup$
add a comment |
$begingroup$
I came up with two formulas and have no Idea whether their correct.
1. ∀x(S(x)∧¬C(x))
2. ∀x∃y(S(x)∧C(y)→DoesNotLike(x,y))
And then I have one a bit more complicated, that I can think of multiple ways how to formalize but have no way of knowing if correctly:
Every student attends a class taught by a teacher that they dont like.
P.s. this is not my homework, just preparation for an exam.
logic predicate-logic logic-translation
asked Jan 24 at 18:05
Martin LMartin L
61
$endgroup$
add a comment |
$begingroup$
I came up with two formulas and have no Idea whether their correct.
1. ∀x(S(x)∧¬C(x))
2. ∀x∃y(S(x)∧C(y)→DoesNotLike(x,y))
And then I have one a bit more complicated, that I can think of multiple ways how to formalize but have no way of knowing if correctly:
Every student attends a class taught by a teacher that they dont like.
P.s. this is not my homework, just preparation for an exam.
logic predicate-logic logic-translation
asked Jan 24 at 18:05
Martin LMartin L
61
$endgroup$
I came up with two formulas and have no Idea whether their correct.
1. ∀x(S(x)∧¬C(x))
2. ∀x∃y(S(x)∧C(y)→DoesNotLike(x,y))
And then I have one a bit more complicated, that I can think of multiple ways how to formalize but have no way of knowing if correctly:
Every student attends a class taught by a teacher that they dont like.
P.s. this is not my homework, just preparation for an exam.
logic predicate-logic logic-translation
logic predicate-logic logic-translation
asked Jan 24 at 18:05
Martin LMartin L
61
asked Jan 24 at 18:05
Martin LMartin L
61
asked Jan 24 at 18:05
Martin LMartin L
61
asked Jan 24 at 18:05
Martin LMartin L
61
asked Jan 24 at 18:05
Martin LMartin L
61
61
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You need to specify carefully what you're taking the predicates $ S $ and $ C $ to mean.
Presumably, $ Sleft(xright) $ means "$ x $ is a student". I think your first formulation would then be synonymous with the sentence given in the title if $ Cleft(xright) $ were taken to mean "$ x $ likes all courses he or she has ever joined, or will ever join", and the conjunction were to be replaced with implication:$^{mathbf dagger}$
$$
forall xleft(Sleft(xright)rightarrow neg Cleft(xright)right)
$$
However, I can't think of any interpretation of $ Cleft(yright) $ that would make your second formulation similarly synonymous with that sentence.
It looks like $ Cleft(yright) $ in the second formulation was intended to mean "$ y $ is a course", but if that were so, there wouldn't appear to be anything in that formulation to specify that the course $ y $ is one that student $ x $ has joined or will join at some time. This could be partly rectified by making $ C $ a two-place predicate for which $ Cleft(x,yright) $ means "$ y $ is a course which $ x $ has joined or will join at some time", but even that's not enough to make the second formulation synonymous with the target sentence.
A second problem with the second formulation is the placement of the predicate $ C $ before, rather than after, the implication.$^{mathbf *}$ As written, the sentence
$$
forall xexists yleft( Sleft(xright) wedge Cleft(x,yright)rightarrowmbox{DoesNotLike}left(x,yright)right)
$$
merely says that for every student there is some course that the student wouldn't like if he or she were to join it. It does not assert that every student will, in fact, join such a course at some time. This can be rectified by placing the predicate $ C $ after the $mbox{implication:}^{mathbf *}$
$$
forall xleft( Sleft(xright) rightarrowexists yleft( Cleft(x,yright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
As now formulated the predicate $ C $ represents an existential statement which could be further spelt out by putting $ Cleft(x,yright) = exists t,Jleft(x,y,tright) $, where $ Jleft(x,y,tright) $ is taken to mean "$ y $ is a course which $ x $ joined, or will join, at time $ t $. The last formulation above would then become:
$$
forall x left( Sleft(xright) rightarrowexists yexists tleft( Jleft(x,y,tright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
* Edit: Thanks to Daniel Schepler, in his comment below, for pointing out an error in my original attempt to rectify this deficiency.
$mathbfdagger$ Further edit: I forgot that the first formulation suffered from the same problem as my first attempted rectification of the second.
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
$endgroup$
1
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
add a comment |
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$begingroup$
You need to specify carefully what you're taking the predicates $ S $ and $ C $ to mean.
Presumably, $ Sleft(xright) $ means "$ x $ is a student". I think your first formulation would then be synonymous with the sentence given in the title if $ Cleft(xright) $ were taken to mean "$ x $ likes all courses he or she has ever joined, or will ever join", and the conjunction were to be replaced with implication:$^{mathbf dagger}$
$$
forall xleft(Sleft(xright)rightarrow neg Cleft(xright)right)
$$
However, I can't think of any interpretation of $ Cleft(yright) $ that would make your second formulation similarly synonymous with that sentence.
It looks like $ Cleft(yright) $ in the second formulation was intended to mean "$ y $ is a course", but if that were so, there wouldn't appear to be anything in that formulation to specify that the course $ y $ is one that student $ x $ has joined or will join at some time. This could be partly rectified by making $ C $ a two-place predicate for which $ Cleft(x,yright) $ means "$ y $ is a course which $ x $ has joined or will join at some time", but even that's not enough to make the second formulation synonymous with the target sentence.
A second problem with the second formulation is the placement of the predicate $ C $ before, rather than after, the implication.$^{mathbf *}$ As written, the sentence
$$
forall xexists yleft( Sleft(xright) wedge Cleft(x,yright)rightarrowmbox{DoesNotLike}left(x,yright)right)
$$
merely says that for every student there is some course that the student wouldn't like if he or she were to join it. It does not assert that every student will, in fact, join such a course at some time. This can be rectified by placing the predicate $ C $ after the $mbox{implication:}^{mathbf *}$
$$
forall xleft( Sleft(xright) rightarrowexists yleft( Cleft(x,yright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
As now formulated the predicate $ C $ represents an existential statement which could be further spelt out by putting $ Cleft(x,yright) = exists t,Jleft(x,y,tright) $, where $ Jleft(x,y,tright) $ is taken to mean "$ y $ is a course which $ x $ joined, or will join, at time $ t $. The last formulation above would then become:
$$
forall x left( Sleft(xright) rightarrowexists yexists tleft( Jleft(x,y,tright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
* Edit: Thanks to Daniel Schepler, in his comment below, for pointing out an error in my original attempt to rectify this deficiency.
$mathbfdagger$ Further edit: I forgot that the first formulation suffered from the same problem as my first attempted rectification of the second.
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
$endgroup$
1
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
add a comment |
$begingroup$
You need to specify carefully what you're taking the predicates $ S $ and $ C $ to mean.
Presumably, $ Sleft(xright) $ means "$ x $ is a student". I think your first formulation would then be synonymous with the sentence given in the title if $ Cleft(xright) $ were taken to mean "$ x $ likes all courses he or she has ever joined, or will ever join", and the conjunction were to be replaced with implication:$^{mathbf dagger}$
$$
forall xleft(Sleft(xright)rightarrow neg Cleft(xright)right)
$$
However, I can't think of any interpretation of $ Cleft(yright) $ that would make your second formulation similarly synonymous with that sentence.
It looks like $ Cleft(yright) $ in the second formulation was intended to mean "$ y $ is a course", but if that were so, there wouldn't appear to be anything in that formulation to specify that the course $ y $ is one that student $ x $ has joined or will join at some time. This could be partly rectified by making $ C $ a two-place predicate for which $ Cleft(x,yright) $ means "$ y $ is a course which $ x $ has joined or will join at some time", but even that's not enough to make the second formulation synonymous with the target sentence.
A second problem with the second formulation is the placement of the predicate $ C $ before, rather than after, the implication.$^{mathbf *}$ As written, the sentence
$$
forall xexists yleft( Sleft(xright) wedge Cleft(x,yright)rightarrowmbox{DoesNotLike}left(x,yright)right)
$$
merely says that for every student there is some course that the student wouldn't like if he or she were to join it. It does not assert that every student will, in fact, join such a course at some time. This can be rectified by placing the predicate $ C $ after the $mbox{implication:}^{mathbf *}$
$$
forall xleft( Sleft(xright) rightarrowexists yleft( Cleft(x,yright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
As now formulated the predicate $ C $ represents an existential statement which could be further spelt out by putting $ Cleft(x,yright) = exists t,Jleft(x,y,tright) $, where $ Jleft(x,y,tright) $ is taken to mean "$ y $ is a course which $ x $ joined, or will join, at time $ t $. The last formulation above would then become:
$$
forall x left( Sleft(xright) rightarrowexists yexists tleft( Jleft(x,y,tright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
* Edit: Thanks to Daniel Schepler, in his comment below, for pointing out an error in my original attempt to rectify this deficiency.
$mathbfdagger$ Further edit: I forgot that the first formulation suffered from the same problem as my first attempted rectification of the second.
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
$endgroup$
1
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
add a comment |
$begingroup$
You need to specify carefully what you're taking the predicates $ S $ and $ C $ to mean.
Presumably, $ Sleft(xright) $ means "$ x $ is a student". I think your first formulation would then be synonymous with the sentence given in the title if $ Cleft(xright) $ were taken to mean "$ x $ likes all courses he or she has ever joined, or will ever join", and the conjunction were to be replaced with implication:$^{mathbf dagger}$
$$
forall xleft(Sleft(xright)rightarrow neg Cleft(xright)right)
$$
However, I can't think of any interpretation of $ Cleft(yright) $ that would make your second formulation similarly synonymous with that sentence.
It looks like $ Cleft(yright) $ in the second formulation was intended to mean "$ y $ is a course", but if that were so, there wouldn't appear to be anything in that formulation to specify that the course $ y $ is one that student $ x $ has joined or will join at some time. This could be partly rectified by making $ C $ a two-place predicate for which $ Cleft(x,yright) $ means "$ y $ is a course which $ x $ has joined or will join at some time", but even that's not enough to make the second formulation synonymous with the target sentence.
A second problem with the second formulation is the placement of the predicate $ C $ before, rather than after, the implication.$^{mathbf *}$ As written, the sentence
$$
forall xexists yleft( Sleft(xright) wedge Cleft(x,yright)rightarrowmbox{DoesNotLike}left(x,yright)right)
$$
merely says that for every student there is some course that the student wouldn't like if he or she were to join it. It does not assert that every student will, in fact, join such a course at some time. This can be rectified by placing the predicate $ C $ after the $mbox{implication:}^{mathbf *}$
$$
forall xleft( Sleft(xright) rightarrowexists yleft( Cleft(x,yright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
As now formulated the predicate $ C $ represents an existential statement which could be further spelt out by putting $ Cleft(x,yright) = exists t,Jleft(x,y,tright) $, where $ Jleft(x,y,tright) $ is taken to mean "$ y $ is a course which $ x $ joined, or will join, at time $ t $. The last formulation above would then become:
$$
forall x left( Sleft(xright) rightarrowexists yexists tleft( Jleft(x,y,tright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
* Edit: Thanks to Daniel Schepler, in his comment below, for pointing out an error in my original attempt to rectify this deficiency.
$mathbfdagger$ Further edit: I forgot that the first formulation suffered from the same problem as my first attempted rectification of the second.
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
$endgroup$
You need to specify carefully what you're taking the predicates $ S $ and $ C $ to mean.
Presumably, $ Sleft(xright) $ means "$ x $ is a student". I think your first formulation would then be synonymous with the sentence given in the title if $ Cleft(xright) $ were taken to mean "$ x $ likes all courses he or she has ever joined, or will ever join", and the conjunction were to be replaced with implication:$^{mathbf dagger}$
$$
forall xleft(Sleft(xright)rightarrow neg Cleft(xright)right)
$$
However, I can't think of any interpretation of $ Cleft(yright) $ that would make your second formulation similarly synonymous with that sentence.
It looks like $ Cleft(yright) $ in the second formulation was intended to mean "$ y $ is a course", but if that were so, there wouldn't appear to be anything in that formulation to specify that the course $ y $ is one that student $ x $ has joined or will join at some time. This could be partly rectified by making $ C $ a two-place predicate for which $ Cleft(x,yright) $ means "$ y $ is a course which $ x $ has joined or will join at some time", but even that's not enough to make the second formulation synonymous with the target sentence.
A second problem with the second formulation is the placement of the predicate $ C $ before, rather than after, the implication.$^{mathbf *}$ As written, the sentence
$$
forall xexists yleft( Sleft(xright) wedge Cleft(x,yright)rightarrowmbox{DoesNotLike}left(x,yright)right)
$$
merely says that for every student there is some course that the student wouldn't like if he or she were to join it. It does not assert that every student will, in fact, join such a course at some time. This can be rectified by placing the predicate $ C $ after the $mbox{implication:}^{mathbf *}$
$$
forall xleft( Sleft(xright) rightarrowexists yleft( Cleft(x,yright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
As now formulated the predicate $ C $ represents an existential statement which could be further spelt out by putting $ Cleft(x,yright) = exists t,Jleft(x,y,tright) $, where $ Jleft(x,y,tright) $ is taken to mean "$ y $ is a course which $ x $ joined, or will join, at time $ t $. The last formulation above would then become:
$$
forall x left( Sleft(xright) rightarrowexists yexists tleft( Jleft(x,y,tright)wedgembox{DoesNotLike}left(x,yright)right)right) .
$$
* Edit: Thanks to Daniel Schepler, in his comment below, for pointing out an error in my original attempt to rectify this deficiency.
$mathbfdagger$ Further edit: I forgot that the first formulation suffered from the same problem as my first attempted rectification of the second.
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
edited Jan 25 at 2:02
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
answered Jan 24 at 23:54
lonza leggieralonza leggiera
1,07928
1,07928
1
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
add a comment |
1
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
1
1
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Should you also change $S(x) wedge cdots$ to $S(x) rightarrow cdots$ so the statement doesn't require that everything in the domain of discourse (including also each course - unless you're using a multi-sorted logic) is a student?
$endgroup$
– Daniel Schepler
Jan 24 at 23:59
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
$begingroup$
Yes, of course you're right. I have now amended the answer.
$endgroup$
– lonza leggiera
Jan 25 at 0:34
add a comment |
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