How to make an English sentence from a first-order logic formula with unbound variables?
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I'll first quote the problem from 'Computability and Logic' by Boolos, Burgess, Jeffrey.
9.3 Consider a language with a two-place predicate P and a one-place predicate F, and an interpretation in which the domain is the set of persons, the denotation of P is the relation of parent to child, and the denotation of F is the set of all female persons. What do the following amount to, in colloquial terms, under that interpretation?
(a) $exists zexists uexists v(u neq v land mathbf Puy land mathbf Pvy land mathbf Puz land mathbf Pvz land mathbf Pzx land lnot mathbf Fy )$
I left out b, since it's quite similar to a.
Now, I understand how to make a colloquial English sentence from a statement like this: $forall a forall b (mathbf Pab implies mathbf lnot mathbf Pba)$. For example, this could be 'No one is a parent of his or her parent.' But in the problem above, x and y are unbound variables. How do I deal with this? Are they perhaps implicitly meant to be universally bounded? Thanks in advance.
logic first-order-logic
asked Nov 13 at 18:11
Steven Wagter
555
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up vote
0
down vote
favorite
I'll first quote the problem from 'Computability and Logic' by Boolos, Burgess, Jeffrey.
9.3 Consider a language with a two-place predicate P and a one-place predicate F, and an interpretation in which the domain is the set of persons, the denotation of P is the relation of parent to child, and the denotation of F is the set of all female persons. What do the following amount to, in colloquial terms, under that interpretation?
(a) $exists zexists uexists v(u neq v land mathbf Puy land mathbf Pvy land mathbf Puz land mathbf Pvz land mathbf Pzx land lnot mathbf Fy )$
I left out b, since it's quite similar to a.
Now, I understand how to make a colloquial English sentence from a statement like this: $forall a forall b (mathbf Pab implies mathbf lnot mathbf Pba)$. For example, this could be 'No one is a parent of his or her parent.' But in the problem above, x and y are unbound variables. How do I deal with this? Are they perhaps implicitly meant to be universally bounded? Thanks in advance.
logic first-order-logic
asked Nov 13 at 18:11
Steven Wagter
555
How would you phrase it if you used $y$ as the name of a specific person, and you were saying something about $y$?
– Malice Vidrine
Nov 13 at 18:17
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'll first quote the problem from 'Computability and Logic' by Boolos, Burgess, Jeffrey.
9.3 Consider a language with a two-place predicate P and a one-place predicate F, and an interpretation in which the domain is the set of persons, the denotation of P is the relation of parent to child, and the denotation of F is the set of all female persons. What do the following amount to, in colloquial terms, under that interpretation?
(a) $exists zexists uexists v(u neq v land mathbf Puy land mathbf Pvy land mathbf Puz land mathbf Pvz land mathbf Pzx land lnot mathbf Fy )$
I left out b, since it's quite similar to a.
Now, I understand how to make a colloquial English sentence from a statement like this: $forall a forall b (mathbf Pab implies mathbf lnot mathbf Pba)$. For example, this could be 'No one is a parent of his or her parent.' But in the problem above, x and y are unbound variables. How do I deal with this? Are they perhaps implicitly meant to be universally bounded? Thanks in advance.
logic first-order-logic
asked Nov 13 at 18:11
Steven Wagter
555
I'll first quote the problem from 'Computability and Logic' by Boolos, Burgess, Jeffrey.
9.3 Consider a language with a two-place predicate P and a one-place predicate F, and an interpretation in which the domain is the set of persons, the denotation of P is the relation of parent to child, and the denotation of F is the set of all female persons. What do the following amount to, in colloquial terms, under that interpretation?
(a) $exists zexists uexists v(u neq v land mathbf Puy land mathbf Pvy land mathbf Puz land mathbf Pvz land mathbf Pzx land lnot mathbf Fy )$
I left out b, since it's quite similar to a.
Now, I understand how to make a colloquial English sentence from a statement like this: $forall a forall b (mathbf Pab implies mathbf lnot mathbf Pba)$. For example, this could be 'No one is a parent of his or her parent.' But in the problem above, x and y are unbound variables. How do I deal with this? Are they perhaps implicitly meant to be universally bounded? Thanks in advance.
logic first-order-logic
logic first-order-logic
asked Nov 13 at 18:11
Steven Wagter
555
asked Nov 13 at 18:11
Steven Wagter
555
asked Nov 13 at 18:11
Steven Wagter
555
asked Nov 13 at 18:11
Steven Wagter
555
asked Nov 13 at 18:11
Steven Wagter
555
555
How would you phrase it if you used $y$ as the name of a specific person, and you were saying something about $y$?
– Malice Vidrine
Nov 13 at 18:17
add a comment |
How would you phrase it if you used $y$ as the name of a specific person, and you were saying something about $y$?
– Malice Vidrine
Nov 13 at 18:17
How would you phrase it if you used $y$ as the name of a specific person, and you were saying something about $y$?
– Malice Vidrine
Nov 13 at 18:17
How would you phrase it if you used $y$ as the name of a specific person, and you were saying something about $y$?
– Malice Vidrine
Nov 13 at 18:17
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
If you have unbound (free) variables, then you should just refer to those variables.
That is, just like we define predicates by sayhing something like:
$P(x,y)$: '$x$ is a father of $y$'
we can likewise express complex formulas with free variables, e.g.
$exists y(P(x,y) land P(y,z))$: 'There is someone who has $x$ as a father, and who is a father of $z$'
answered Nov 13 at 18:34
Bram28
58.2k44185
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
If you have unbound (free) variables, then you should just refer to those variables.
That is, just like we define predicates by sayhing something like:
$P(x,y)$: '$x$ is a father of $y$'
we can likewise express complex formulas with free variables, e.g.
$exists y(P(x,y) land P(y,z))$: 'There is someone who has $x$ as a father, and who is a father of $z$'
answered Nov 13 at 18:34
Bram28
58.2k44185
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
add a comment |
up vote
1
down vote
accepted
If you have unbound (free) variables, then you should just refer to those variables.
That is, just like we define predicates by sayhing something like:
$P(x,y)$: '$x$ is a father of $y$'
we can likewise express complex formulas with free variables, e.g.
$exists y(P(x,y) land P(y,z))$: 'There is someone who has $x$ as a father, and who is a father of $z$'
answered Nov 13 at 18:34
Bram28
58.2k44185
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If you have unbound (free) variables, then you should just refer to those variables.
That is, just like we define predicates by sayhing something like:
$P(x,y)$: '$x$ is a father of $y$'
we can likewise express complex formulas with free variables, e.g.
$exists y(P(x,y) land P(y,z))$: 'There is someone who has $x$ as a father, and who is a father of $z$'
answered Nov 13 at 18:34
Bram28
58.2k44185
If you have unbound (free) variables, then you should just refer to those variables.
That is, just like we define predicates by sayhing something like:
$P(x,y)$: '$x$ is a father of $y$'
we can likewise express complex formulas with free variables, e.g.
$exists y(P(x,y) land P(y,z))$: 'There is someone who has $x$ as a father, and who is a father of $z$'
answered Nov 13 at 18:34
Bram28
58.2k44185
edited Nov 13 at 18:45
answered Nov 13 at 18:34
Bram28
58.2k44185
answered Nov 13 at 18:34
Bram28
58.2k44185
answered Nov 13 at 18:34
Bram28
58.2k44185
58.2k44185
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
add a comment |
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
So your example and the problem above are propositions whose truth depends on which variables you put in, or in which way you bind the variables, unlike my example?
– Steven Wagter
Nov 13 at 18:37
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
@StevenWagter Yes, as long as you have free variables, the formula is not a sentence, i.e. not something you can evaluate and assign a truth-value to. It is only when you quantify (bind) all the free variables that it becomes a sentence with a truth-value. Your formula is no different. It has free variables $x$ and $y$, and hence is not a sentence; just a formula. But you can still express that formula.
– Bram28
Nov 13 at 18:44
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
Thank you very much.
– Steven Wagter
Nov 13 at 18:49
add a comment |
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How would you phrase it if you used $y$ as the name of a specific person, and you were saying something about $y$?
– Malice Vidrine
Nov 13 at 18:17