Translating a sentence into Symbolic Notation
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Let's say for example,
Every son of Trump has a mother but not every son of Trump has the same mother where Pr: x is a person; Fx: x is male; Pxy: x parents y; Tx: x is current Pres of the USA.
How would I go about this translation? I have not yet studied symbolic logic, but am interested and planning on it since I have a background in computer science
logic
asked Nov 14 at 5:58
Later_72
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add a comment |
up vote
1
down vote
favorite
Let's say for example,
Every son of Trump has a mother but not every son of Trump has the same mother where Pr: x is a person; Fx: x is male; Pxy: x parents y; Tx: x is current Pres of the USA.
How would I go about this translation? I have not yet studied symbolic logic, but am interested and planning on it since I have a background in computer science
logic
asked Nov 14 at 5:58
Later_72
61
2
Hi and welcome to the Math.SE. Please provide context for your question in order to help other members help you, and also have a look at the Math Jax tutorial if you want to produce graphically beautiful mathematical contents.
– Daniele Tampieri
Nov 14 at 6:40
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let's say for example,
Every son of Trump has a mother but not every son of Trump has the same mother where Pr: x is a person; Fx: x is male; Pxy: x parents y; Tx: x is current Pres of the USA.
How would I go about this translation? I have not yet studied symbolic logic, but am interested and planning on it since I have a background in computer science
logic
asked Nov 14 at 5:58
Later_72
61
Let's say for example,
Every son of Trump has a mother but not every son of Trump has the same mother where Pr: x is a person; Fx: x is male; Pxy: x parents y; Tx: x is current Pres of the USA.
How would I go about this translation? I have not yet studied symbolic logic, but am interested and planning on it since I have a background in computer science
logic
logic
asked Nov 14 at 5:58
Later_72
61
asked Nov 14 at 5:58
Later_72
61
asked Nov 14 at 5:58
Later_72
61
asked Nov 14 at 5:58
Later_72
61
asked Nov 14 at 5:58
Later_72
61
61
2
Hi and welcome to the Math.SE. Please provide context for your question in order to help other members help you, and also have a look at the Math Jax tutorial if you want to produce graphically beautiful mathematical contents.
– Daniele Tampieri
Nov 14 at 6:40
add a comment |
2
Hi and welcome to the Math.SE. Please provide context for your question in order to help other members help you, and also have a look at the Math Jax tutorial if you want to produce graphically beautiful mathematical contents.
– Daniele Tampieri
Nov 14 at 6:40
2
2
Hi and welcome to the Math.SE. Please provide context for your question in order to help other members help you, and also have a look at the Math Jax tutorial if you want to produce graphically beautiful mathematical contents.
– Daniele Tampieri
Nov 14 at 6:40
Hi and welcome to the Math.SE. Please provide context for your question in order to help other members help you, and also have a look at the Math Jax tutorial if you want to produce graphically beautiful mathematical contents.
– Daniele Tampieri
Nov 14 at 6:40
add a comment |
1 Answer
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Let's make a predicate Mxy for "y is x's mother". Well, a mother isn't male, and is a parent. So $Mx equiv Pxy land neg Fy$.
How about Sx for "x is a son of Trump"? Well that means x is male, there exists a person who is his parent, and that parent is Trump. So that's $Sx equiv Fx land exists y(Pxyland Ty)$.
So let's look at "Every son of Trump has a mother". This is saying that, for any person, if that person is a son of Trump, then there is a person who is that person's mother. Symbolically, this is $forall x[Sxrightarrowexists y (Mxy)]$. Expanding out the predicates gives $forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])$.
Now let's look at "every son of Trump has the same mother". That means there is a person y such that for every person x, if person x is a son of Trump, then person y is person x's mother. Symbolically, this is $exists y[forall x(Sxrightarrow Mxy)]$, and expanding out the predicates again gives $exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])]$.
Lastly, the whole statement "Every son of Trump has a mother but not every son of Trump has the same mother" is just saying "[Every son of Trump has a mother] $landneg$[every son of Trump has the same mother]", so the full logical statement is
$$
forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])land neg(exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])])
$$
If you want to practice manipulations on symbolic first-order logic, try simplifying this.
answered Nov 14 at 7:25
eyeballfrog
5,678528
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
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
Let's make a predicate Mxy for "y is x's mother". Well, a mother isn't male, and is a parent. So $Mx equiv Pxy land neg Fy$.
How about Sx for "x is a son of Trump"? Well that means x is male, there exists a person who is his parent, and that parent is Trump. So that's $Sx equiv Fx land exists y(Pxyland Ty)$.
So let's look at "Every son of Trump has a mother". This is saying that, for any person, if that person is a son of Trump, then there is a person who is that person's mother. Symbolically, this is $forall x[Sxrightarrowexists y (Mxy)]$. Expanding out the predicates gives $forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])$.
Now let's look at "every son of Trump has the same mother". That means there is a person y such that for every person x, if person x is a son of Trump, then person y is person x's mother. Symbolically, this is $exists y[forall x(Sxrightarrow Mxy)]$, and expanding out the predicates again gives $exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])]$.
Lastly, the whole statement "Every son of Trump has a mother but not every son of Trump has the same mother" is just saying "[Every son of Trump has a mother] $landneg$[every son of Trump has the same mother]", so the full logical statement is
$$
forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])land neg(exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])])
$$
If you want to practice manipulations on symbolic first-order logic, try simplifying this.
answered Nov 14 at 7:25
eyeballfrog
5,678528
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
add a comment |
up vote
1
down vote
Let's make a predicate Mxy for "y is x's mother". Well, a mother isn't male, and is a parent. So $Mx equiv Pxy land neg Fy$.
How about Sx for "x is a son of Trump"? Well that means x is male, there exists a person who is his parent, and that parent is Trump. So that's $Sx equiv Fx land exists y(Pxyland Ty)$.
So let's look at "Every son of Trump has a mother". This is saying that, for any person, if that person is a son of Trump, then there is a person who is that person's mother. Symbolically, this is $forall x[Sxrightarrowexists y (Mxy)]$. Expanding out the predicates gives $forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])$.
Now let's look at "every son of Trump has the same mother". That means there is a person y such that for every person x, if person x is a son of Trump, then person y is person x's mother. Symbolically, this is $exists y[forall x(Sxrightarrow Mxy)]$, and expanding out the predicates again gives $exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])]$.
Lastly, the whole statement "Every son of Trump has a mother but not every son of Trump has the same mother" is just saying "[Every son of Trump has a mother] $landneg$[every son of Trump has the same mother]", so the full logical statement is
$$
forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])land neg(exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])])
$$
If you want to practice manipulations on symbolic first-order logic, try simplifying this.
answered Nov 14 at 7:25
eyeballfrog
5,678528
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
add a comment |
up vote
1
down vote
up vote
1
down vote
Let's make a predicate Mxy for "y is x's mother". Well, a mother isn't male, and is a parent. So $Mx equiv Pxy land neg Fy$.
How about Sx for "x is a son of Trump"? Well that means x is male, there exists a person who is his parent, and that parent is Trump. So that's $Sx equiv Fx land exists y(Pxyland Ty)$.
So let's look at "Every son of Trump has a mother". This is saying that, for any person, if that person is a son of Trump, then there is a person who is that person's mother. Symbolically, this is $forall x[Sxrightarrowexists y (Mxy)]$. Expanding out the predicates gives $forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])$.
Now let's look at "every son of Trump has the same mother". That means there is a person y such that for every person x, if person x is a son of Trump, then person y is person x's mother. Symbolically, this is $exists y[forall x(Sxrightarrow Mxy)]$, and expanding out the predicates again gives $exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])]$.
Lastly, the whole statement "Every son of Trump has a mother but not every son of Trump has the same mother" is just saying "[Every son of Trump has a mother] $landneg$[every son of Trump has the same mother]", so the full logical statement is
$$
forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])land neg(exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])])
$$
If you want to practice manipulations on symbolic first-order logic, try simplifying this.
answered Nov 14 at 7:25
eyeballfrog
5,678528
Let's make a predicate Mxy for "y is x's mother". Well, a mother isn't male, and is a parent. So $Mx equiv Pxy land neg Fy$.
How about Sx for "x is a son of Trump"? Well that means x is male, there exists a person who is his parent, and that parent is Trump. So that's $Sx equiv Fx land exists y(Pxyland Ty)$.
So let's look at "Every son of Trump has a mother". This is saying that, for any person, if that person is a son of Trump, then there is a person who is that person's mother. Symbolically, this is $forall x[Sxrightarrowexists y (Mxy)]$. Expanding out the predicates gives $forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])$.
Now let's look at "every son of Trump has the same mother". That means there is a person y such that for every person x, if person x is a son of Trump, then person y is person x's mother. Symbolically, this is $exists y[forall x(Sxrightarrow Mxy)]$, and expanding out the predicates again gives $exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])]$.
Lastly, the whole statement "Every son of Trump has a mother but not every son of Trump has the same mother" is just saying "[Every son of Trump has a mother] $landneg$[every son of Trump has the same mother]", so the full logical statement is
$$
forall x([Fx land exists z(Pxzland Tz)]rightarrowexists y [Pxy land neg Fy])land neg(exists y[forall x([Fx land exists z(Pxzland Tz)]rightarrow [Pxy land neg Fy])])
$$
If you want to practice manipulations on symbolic first-order logic, try simplifying this.
answered Nov 14 at 7:25
eyeballfrog
5,678528
answered Nov 14 at 7:25
eyeballfrog
5,678528
answered Nov 14 at 7:25
eyeballfrog
5,678528
answered Nov 14 at 7:25
eyeballfrog
5,678528
5,678528
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
add a comment |
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
Excellent, thank you! Can you please show me the steps to: If Plato exists then someone authored The Republic. Px: x authored The Republic, Mx: x is a man? If you can show me, I'd really appreciate it. If possible also give me one to practice in addition to the task you provided. Thanks again! I'm watching tutorials currently.
– Later_72
Nov 14 at 20:01
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
In this case you'd need Px: x is Plato and Rx: x authored the Republic. This one is simple enough that I think you can handle it yourself. For another simple exercise, use the predicate Sxy "x shaves y" to form the barber predicate Bx: "x only shaves people who do not shave themselves". For a follow-up, form the statement "There is a barber who shaves everyone who does not shave himself." and see if you can reduce it to something interesting.
– eyeballfrog
Nov 14 at 21:36
add a comment |
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Hi and welcome to the Math.SE. Please provide context for your question in order to help other members help you, and also have a look at the Math Jax tutorial if you want to produce graphically beautiful mathematical contents.
– Daniele Tampieri
Nov 14 at 6:40