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How to write formulae in propositional logic with infinite domain
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As an assignment I have had couple of weeks ago placing 5 different groups M,R,T,H,G of people in 25*25 festival site , the formulae that were asked were the following :
a) There is one group placed in each parcel :
enter image description here
b) Ms and Rs can't stand Ts to be their neighbors :
enter image description here
c) Hs are happy iff they are not placed at a corner and they don't have any Ms as their neighbours
d) There is one parcel which is occupied by Gs and around this parcel (not diagonal neighbours but the neighbours above ,below , left and right there are only Gs
This week however we need to write these formulae for a site plan with infinite*infinite parcels. How is the transition from finite formulae to infinite sets of formulae?
One of the questions asked to explain why it is not possible to write d for the new festival-site , I would really appreciate if someone could also explain this.
logic
asked yesterday
Ali Bektas
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As an assignment I have had couple of weeks ago placing 5 different groups M,R,T,H,G of people in 25*25 festival site , the formulae that were asked were the following :
a) There is one group placed in each parcel :
enter image description here
b) Ms and Rs can't stand Ts to be their neighbors :
enter image description here
c) Hs are happy iff they are not placed at a corner and they don't have any Ms as their neighbours
d) There is one parcel which is occupied by Gs and around this parcel (not diagonal neighbours but the neighbours above ,below , left and right there are only Gs
This week however we need to write these formulae for a site plan with infinite*infinite parcels. How is the transition from finite formulae to infinite sets of formulae?
One of the questions asked to explain why it is not possible to write d for the new festival-site , I would really appreciate if someone could also explain this.
logic
asked yesterday
Ali Bektas
41
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I think this needs some more context. For example, are you working in a framework where a theory can have infinitely many axioms as long as each of those axioms is a finite formula?
– Henning Makholm
yesterday
@HenningMakholm The only thing that I didn't mention is that there are now infinite amount of propositional-logic symbol H_{i,j} , G_{i,j} , M_{i,j} so on. Unfortunately I couldn't even understand your question.
– Ali Bektas
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
As an assignment I have had couple of weeks ago placing 5 different groups M,R,T,H,G of people in 25*25 festival site , the formulae that were asked were the following :
a) There is one group placed in each parcel :
enter image description here
b) Ms and Rs can't stand Ts to be their neighbors :
enter image description here
c) Hs are happy iff they are not placed at a corner and they don't have any Ms as their neighbours
d) There is one parcel which is occupied by Gs and around this parcel (not diagonal neighbours but the neighbours above ,below , left and right there are only Gs
This week however we need to write these formulae for a site plan with infinite*infinite parcels. How is the transition from finite formulae to infinite sets of formulae?
One of the questions asked to explain why it is not possible to write d for the new festival-site , I would really appreciate if someone could also explain this.
logic
asked yesterday
Ali Bektas
41
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
As an assignment I have had couple of weeks ago placing 5 different groups M,R,T,H,G of people in 25*25 festival site , the formulae that were asked were the following :
a) There is one group placed in each parcel :
enter image description here
b) Ms and Rs can't stand Ts to be their neighbors :
enter image description here
c) Hs are happy iff they are not placed at a corner and they don't have any Ms as their neighbours
d) There is one parcel which is occupied by Gs and around this parcel (not diagonal neighbours but the neighbours above ,below , left and right there are only Gs
This week however we need to write these formulae for a site plan with infinite*infinite parcels. How is the transition from finite formulae to infinite sets of formulae?
One of the questions asked to explain why it is not possible to write d for the new festival-site , I would really appreciate if someone could also explain this.
logic
logic
asked yesterday
Ali Bektas
41
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Ali Bektas
41
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Ali Bektas
41
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Ali Bektas
41
asked yesterday
Ali Bektas
41
41
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Ali Bektas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I think this needs some more context. For example, are you working in a framework where a theory can have infinitely many axioms as long as each of those axioms is a finite formula?
– Henning Makholm
yesterday
@HenningMakholm The only thing that I didn't mention is that there are now infinite amount of propositional-logic symbol H_{i,j} , G_{i,j} , M_{i,j} so on. Unfortunately I couldn't even understand your question.
– Ali Bektas
yesterday
add a comment |
I think this needs some more context. For example, are you working in a framework where a theory can have infinitely many axioms as long as each of those axioms is a finite formula?
– Henning Makholm
yesterday
@HenningMakholm The only thing that I didn't mention is that there are now infinite amount of propositional-logic symbol H_{i,j} , G_{i,j} , M_{i,j} so on. Unfortunately I couldn't even understand your question.
– Ali Bektas
yesterday
I think this needs some more context. For example, are you working in a framework where a theory can have infinitely many axioms as long as each of those axioms is a finite formula?
– Henning Makholm
yesterday
I think this needs some more context. For example, are you working in a framework where a theory can have infinitely many axioms as long as each of those axioms is a finite formula?
– Henning Makholm
yesterday
@HenningMakholm The only thing that I didn't mention is that there are now infinite amount of propositional-logic symbol H_{i,j} , G_{i,j} , M_{i,j} so on. Unfortunately I couldn't even understand your question.
– Ali Bektas
yesterday
@HenningMakholm The only thing that I didn't mention is that there are now infinite amount of propositional-logic symbol H_{i,j} , G_{i,j} , M_{i,j} so on. Unfortunately I couldn't even understand your question.
– Ali Bektas
yesterday
add a comment |
1 Answer
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If d) means 'at least one parcel', then you can just do:
$$bigvee_{i,j in mathbb{Z}} (G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1}) $$
... so, I gather that d) must be meant as saying *exactly * one parcel
And yeah, with infinite propositions, that cannot be done with a propositional logic formula: you need to be able to say that there is some $i,j$ pair for which $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $, but also that for all other $i$ and $j$, we do not have $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $
And as long as the number of parcels is finite, you can do that: you can specify that, say, $12,17$ is that special parcel, while the others are not, and 'the others' is just one long-ass formula. But when you move to infinity, you are going to have to use quantifiers to do this.
answered 20 hours ago
Bram28
58k44185
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
If d) means 'at least one parcel', then you can just do:
$$bigvee_{i,j in mathbb{Z}} (G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1}) $$
... so, I gather that d) must be meant as saying *exactly * one parcel
And yeah, with infinite propositions, that cannot be done with a propositional logic formula: you need to be able to say that there is some $i,j$ pair for which $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $, but also that for all other $i$ and $j$, we do not have $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $
And as long as the number of parcels is finite, you can do that: you can specify that, say, $12,17$ is that special parcel, while the others are not, and 'the others' is just one long-ass formula. But when you move to infinity, you are going to have to use quantifiers to do this.
answered 20 hours ago
Bram28
58k44185
add a comment |
up vote
0
down vote
If d) means 'at least one parcel', then you can just do:
$$bigvee_{i,j in mathbb{Z}} (G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1}) $$
... so, I gather that d) must be meant as saying *exactly * one parcel
And yeah, with infinite propositions, that cannot be done with a propositional logic formula: you need to be able to say that there is some $i,j$ pair for which $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $, but also that for all other $i$ and $j$, we do not have $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $
And as long as the number of parcels is finite, you can do that: you can specify that, say, $12,17$ is that special parcel, while the others are not, and 'the others' is just one long-ass formula. But when you move to infinity, you are going to have to use quantifiers to do this.
answered 20 hours ago
Bram28
58k44185
add a comment |
up vote
0
down vote
up vote
0
down vote
If d) means 'at least one parcel', then you can just do:
$$bigvee_{i,j in mathbb{Z}} (G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1}) $$
... so, I gather that d) must be meant as saying *exactly * one parcel
And yeah, with infinite propositions, that cannot be done with a propositional logic formula: you need to be able to say that there is some $i,j$ pair for which $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $, but also that for all other $i$ and $j$, we do not have $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $
And as long as the number of parcels is finite, you can do that: you can specify that, say, $12,17$ is that special parcel, while the others are not, and 'the others' is just one long-ass formula. But when you move to infinity, you are going to have to use quantifiers to do this.
answered 20 hours ago
Bram28
58k44185
If d) means 'at least one parcel', then you can just do:
$$bigvee_{i,j in mathbb{Z}} (G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1}) $$
... so, I gather that d) must be meant as saying *exactly * one parcel
And yeah, with infinite propositions, that cannot be done with a propositional logic formula: you need to be able to say that there is some $i,j$ pair for which $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $, but also that for all other $i$ and $j$, we do not have $G_{i,j} land G_{i-1,j} land G_{i+1,j} land G_{i,j-1} land G_{i,j+1} $
And as long as the number of parcels is finite, you can do that: you can specify that, say, $12,17$ is that special parcel, while the others are not, and 'the others' is just one long-ass formula. But when you move to infinity, you are going to have to use quantifiers to do this.
answered 20 hours ago
Bram28
58k44185
answered 20 hours ago
Bram28
58k44185
answered 20 hours ago
Bram28
58k44185
answered 20 hours ago
Bram28
58k44185
58k44185
add a comment |
add a comment |
Ali Bektas is a new contributor. Be nice, and check out our Code of Conduct.
Ali Bektas is a new contributor. Be nice, and check out our Code of Conduct.
Ali Bektas is a new contributor. Be nice, and check out our Code of Conduct.
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I think this needs some more context. For example, are you working in a framework where a theory can have infinitely many axioms as long as each of those axioms is a finite formula?
– Henning Makholm
yesterday
@HenningMakholm The only thing that I didn't mention is that there are now infinite amount of propositional-logic symbol H_{i,j} , G_{i,j} , M_{i,j} so on. Unfortunately I couldn't even understand your question.
– Ali Bektas
yesterday