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 8 hours ago
Later_72
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up vote
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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 8 hours ago
Later_72
61
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
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
7 hours ago
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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 8 hours ago
Later_72
61
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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 8 hours ago
Later_72
61
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Later_72
61
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Later_72
61
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Later_72
61
asked 8 hours ago
Later_72
61
61
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Later_72 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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1
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
7 hours ago
add a comment |
1
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
7 hours ago
1
1
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
7 hours ago
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
7 hours ago
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 6 hours ago
eyeballfrog
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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 6 hours ago
eyeballfrog
5,543528
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 6 hours ago
eyeballfrog
5,543528
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 6 hours ago
eyeballfrog
5,543528
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 6 hours ago
eyeballfrog
5,543528
answered 6 hours ago
eyeballfrog
5,543528
answered 6 hours ago
eyeballfrog
5,543528
answered 6 hours ago
eyeballfrog
5,543528
5,543528
add a comment |
add a comment |
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1
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
7 hours ago