In description logics, how do you define a concept?
$begingroup$
In some texts I read I see a concept defined as a subset and in other texts I see it defined with equivalence.
Eg 1: Person is a subset of hasMother.Person and hasFather.Person
Eg 2: Mother is equivalent to Woman and hasChild.Person
When is it appropriate to use subset and when is it appropriate to use equivalence? I can't seem to find a distinction between the two.
logic first-order-logic
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
asked Jan 14 at 10:50
DeeDee
1184
$endgroup$
add a comment |
$begingroup$
In some texts I read I see a concept defined as a subset and in other texts I see it defined with equivalence.
Eg 1: Person is a subset of hasMother.Person and hasFather.Person
Eg 2: Mother is equivalent to Woman and hasChild.Person
When is it appropriate to use subset and when is it appropriate to use equivalence? I can't seem to find a distinction between the two.
logic first-order-logic
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
asked Jan 14 at 10:50
DeeDee
1184
$endgroup$
add a comment |
$begingroup$
In some texts I read I see a concept defined as a subset and in other texts I see it defined with equivalence.
Eg 1: Person is a subset of hasMother.Person and hasFather.Person
Eg 2: Mother is equivalent to Woman and hasChild.Person
When is it appropriate to use subset and when is it appropriate to use equivalence? I can't seem to find a distinction between the two.
logic first-order-logic
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
asked Jan 14 at 10:50
DeeDee
1184
$endgroup$
In some texts I read I see a concept defined as a subset and in other texts I see it defined with equivalence.
Eg 1: Person is a subset of hasMother.Person and hasFather.Person
Eg 2: Mother is equivalent to Woman and hasChild.Person
When is it appropriate to use subset and when is it appropriate to use equivalence? I can't seem to find a distinction between the two.
logic first-order-logic
logic first-order-logic
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
asked Jan 14 at 10:50
DeeDee
1184
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
asked Jan 14 at 10:50
DeeDee
1184
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
edited Jan 14 at 12:04
Mauro ALLEGRANZA
68.3k449117
68.3k449117
asked Jan 14 at 10:50
DeeDee
1184
asked Jan 14 at 10:50
DeeDee
1184
asked Jan 14 at 10:50
DeeDee
1184
1184
add a comment |
add a comment |
1 Answer
1
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$begingroup$
The second one is a defintion; so, it is appropriate to use equivalence.
The concept $text {Mother}$ is defined as the intersection of the two concepts : $text {Woman}$ and $text {hasChild.Person}$.
Reagrding the first one, we have that $text{Person}$ is a concept, whose meaning is the class of all humans. Also $text {hasMother.Person}$ is a concept and its meaning is again the class of all humans, because every human has a mother.
Thus the two concepts have the same meaning, and this implies that, given an interpretation $mathfrak I$ with domain $Delta$ containing all human beings :
$(text{Person})^{mathfrak I} = (text {hasMother.Person})^{mathfrak I} subseteq Delta$.
In this sense, we have both :
$(text{Person})^{mathfrak I} subseteq (text {hasMother.Person})^{mathfrak I} text { and } (text {hasMother.Person})^{mathfrak I} subseteq (text{Person})^{mathfrak I}$.
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
$endgroup$
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
add a comment |
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$begingroup$
The second one is a defintion; so, it is appropriate to use equivalence.
The concept $text {Mother}$ is defined as the intersection of the two concepts : $text {Woman}$ and $text {hasChild.Person}$.
Reagrding the first one, we have that $text{Person}$ is a concept, whose meaning is the class of all humans. Also $text {hasMother.Person}$ is a concept and its meaning is again the class of all humans, because every human has a mother.
Thus the two concepts have the same meaning, and this implies that, given an interpretation $mathfrak I$ with domain $Delta$ containing all human beings :
$(text{Person})^{mathfrak I} = (text {hasMother.Person})^{mathfrak I} subseteq Delta$.
In this sense, we have both :
$(text{Person})^{mathfrak I} subseteq (text {hasMother.Person})^{mathfrak I} text { and } (text {hasMother.Person})^{mathfrak I} subseteq (text{Person})^{mathfrak I}$.
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
$endgroup$
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
add a comment |
$begingroup$
The second one is a defintion; so, it is appropriate to use equivalence.
The concept $text {Mother}$ is defined as the intersection of the two concepts : $text {Woman}$ and $text {hasChild.Person}$.
Reagrding the first one, we have that $text{Person}$ is a concept, whose meaning is the class of all humans. Also $text {hasMother.Person}$ is a concept and its meaning is again the class of all humans, because every human has a mother.
Thus the two concepts have the same meaning, and this implies that, given an interpretation $mathfrak I$ with domain $Delta$ containing all human beings :
$(text{Person})^{mathfrak I} = (text {hasMother.Person})^{mathfrak I} subseteq Delta$.
In this sense, we have both :
$(text{Person})^{mathfrak I} subseteq (text {hasMother.Person})^{mathfrak I} text { and } (text {hasMother.Person})^{mathfrak I} subseteq (text{Person})^{mathfrak I}$.
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
$endgroup$
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
add a comment |
$begingroup$
The second one is a defintion; so, it is appropriate to use equivalence.
The concept $text {Mother}$ is defined as the intersection of the two concepts : $text {Woman}$ and $text {hasChild.Person}$.
Reagrding the first one, we have that $text{Person}$ is a concept, whose meaning is the class of all humans. Also $text {hasMother.Person}$ is a concept and its meaning is again the class of all humans, because every human has a mother.
Thus the two concepts have the same meaning, and this implies that, given an interpretation $mathfrak I$ with domain $Delta$ containing all human beings :
$(text{Person})^{mathfrak I} = (text {hasMother.Person})^{mathfrak I} subseteq Delta$.
In this sense, we have both :
$(text{Person})^{mathfrak I} subseteq (text {hasMother.Person})^{mathfrak I} text { and } (text {hasMother.Person})^{mathfrak I} subseteq (text{Person})^{mathfrak I}$.
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
$endgroup$
The second one is a defintion; so, it is appropriate to use equivalence.
The concept $text {Mother}$ is defined as the intersection of the two concepts : $text {Woman}$ and $text {hasChild.Person}$.
Reagrding the first one, we have that $text{Person}$ is a concept, whose meaning is the class of all humans. Also $text {hasMother.Person}$ is a concept and its meaning is again the class of all humans, because every human has a mother.
Thus the two concepts have the same meaning, and this implies that, given an interpretation $mathfrak I$ with domain $Delta$ containing all human beings :
$(text{Person})^{mathfrak I} = (text {hasMother.Person})^{mathfrak I} subseteq Delta$.
In this sense, we have both :
$(text{Person})^{mathfrak I} subseteq (text {hasMother.Person})^{mathfrak I} text { and } (text {hasMother.Person})^{mathfrak I} subseteq (text{Person})^{mathfrak I}$.
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
answered Jan 14 at 12:04
Mauro ALLEGRANZAMauro ALLEGRANZA
68.3k449117
68.3k449117
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
add a comment |
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
Thank you. I still seem to be missing something critical though. How would one know whether they ought to use equivalence or subset? What is the ultimate deciding factor? I am trying to formulate my own axioms for a project and I'm still stumped...
$endgroup$
– Dee
Jan 14 at 16:37
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
$begingroup$
@Dee : $subseteq$ is used with sets and sets are the interpretation of concepts. Concepts are rnelations are aprt of the language and we use definitions (which needs equivalence) to add new "names" to the language.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 16:48
add a comment |
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