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Problems teaching introductory logic. Is this a statement? “If x is an integer, then…â€
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Consider the claim, "If $x$ is an integer, then $x^3>0$". Is this a statement?
My text defines a statement as "a declarative sentence which is true or false, but not both."
At first it seemed clear to me that the claim was a false statement. But I imagine a skeptical student saying that the sentence is not a statement because it is sometimes true and sometimes false, depending on the value of $x$. How would you respond?
Is "$x$ is an integer" a statement? It only has a truth value if we set a value for $x$. I am confusing myself. Help?
logic first-order-logic predicate-logic quantifiers
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
asked Aug 6 at 23:22
PlatonicTradition
514
add a comment |Â
up vote
10
down vote
favorite
Consider the claim, "If $x$ is an integer, then $x^3>0$". Is this a statement?
My text defines a statement as "a declarative sentence which is true or false, but not both."
At first it seemed clear to me that the claim was a false statement. But I imagine a skeptical student saying that the sentence is not a statement because it is sometimes true and sometimes false, depending on the value of $x$. How would you respond?
Is "$x$ is an integer" a statement? It only has a truth value if we set a value for $x$. I am confusing myself. Help?
logic first-order-logic predicate-logic quantifiers
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
asked Aug 6 at 23:22
PlatonicTradition
514
8
A statement is not allowed to have free variables (in this case $x$). The problem is that in natural language one might take this sentence to have an implied quantifier, so that it means the same as "For every integer $x$, $x^3 > 0$".
– Robert Israel
Aug 6 at 23:31
2
I guess OP is teaching some high school levels students, to whom it's not very trivial to explain bound and free variables. Answers in a less jargonized manner might be more helpful.
– Cave Johnson
Aug 6 at 23:43
3
@CaveJohnson The answers are directed at the OP, not the OP's students, and we're assuming that the OP does understand the basic concepts like "universal quantifier" etc. (Note that the OP isn't asking how to explain something.)
– Noah Schweber
Aug 6 at 23:50
1
Implicit in natural language is: For all $x$, if $x$ is an integer then $x^3 > 0$. This is false since $-1$ is an integer and $(-1)^3 < 0$. So, we have a a false statement. In a formal proof (not in natural language), whether a formula is a statement or not is not an important distinction.
– Dan Christensen
Aug 7 at 14:59
add a comment |Â
up vote
10
down vote
favorite
up vote
10
down vote
favorite
Consider the claim, "If $x$ is an integer, then $x^3>0$". Is this a statement?
My text defines a statement as "a declarative sentence which is true or false, but not both."
At first it seemed clear to me that the claim was a false statement. But I imagine a skeptical student saying that the sentence is not a statement because it is sometimes true and sometimes false, depending on the value of $x$. How would you respond?
Is "$x$ is an integer" a statement? It only has a truth value if we set a value for $x$. I am confusing myself. Help?
logic first-order-logic predicate-logic quantifiers
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
asked Aug 6 at 23:22
PlatonicTradition
514
Consider the claim, "If $x$ is an integer, then $x^3>0$". Is this a statement?
My text defines a statement as "a declarative sentence which is true or false, but not both."
At first it seemed clear to me that the claim was a false statement. But I imagine a skeptical student saying that the sentence is not a statement because it is sometimes true and sometimes false, depending on the value of $x$. How would you respond?
Is "$x$ is an integer" a statement? It only has a truth value if we set a value for $x$. I am confusing myself. Help?
logic first-order-logic predicate-logic quantifiers
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
asked Aug 6 at 23:22
PlatonicTradition
514
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
edited Aug 9 at 3:32
Taroccoesbrocco
3,51441431
3,51441431
asked Aug 6 at 23:22
PlatonicTradition
514
asked Aug 6 at 23:22
PlatonicTradition
514
asked Aug 6 at 23:22
PlatonicTradition
514
514
8
A statement is not allowed to have free variables (in this case $x$). The problem is that in natural language one might take this sentence to have an implied quantifier, so that it means the same as "For every integer $x$, $x^3 > 0$".
– Robert Israel
Aug 6 at 23:31
2
I guess OP is teaching some high school levels students, to whom it's not very trivial to explain bound and free variables. Answers in a less jargonized manner might be more helpful.
– Cave Johnson
Aug 6 at 23:43
3
@CaveJohnson The answers are directed at the OP, not the OP's students, and we're assuming that the OP does understand the basic concepts like "universal quantifier" etc. (Note that the OP isn't asking how to explain something.)
– Noah Schweber
Aug 6 at 23:50
1
Implicit in natural language is: For all $x$, if $x$ is an integer then $x^3 > 0$. This is false since $-1$ is an integer and $(-1)^3 < 0$. So, we have a a false statement. In a formal proof (not in natural language), whether a formula is a statement or not is not an important distinction.
– Dan Christensen
Aug 7 at 14:59
add a comment |Â
8
A statement is not allowed to have free variables (in this case $x$). The problem is that in natural language one might take this sentence to have an implied quantifier, so that it means the same as "For every integer $x$, $x^3 > 0$".
– Robert Israel
Aug 6 at 23:31
2
I guess OP is teaching some high school levels students, to whom it's not very trivial to explain bound and free variables. Answers in a less jargonized manner might be more helpful.
– Cave Johnson
Aug 6 at 23:43
3
@CaveJohnson The answers are directed at the OP, not the OP's students, and we're assuming that the OP does understand the basic concepts like "universal quantifier" etc. (Note that the OP isn't asking how to explain something.)
– Noah Schweber
Aug 6 at 23:50
1
Implicit in natural language is: For all $x$, if $x$ is an integer then $x^3 > 0$. This is false since $-1$ is an integer and $(-1)^3 < 0$. So, we have a a false statement. In a formal proof (not in natural language), whether a formula is a statement or not is not an important distinction.
– Dan Christensen
Aug 7 at 14:59
8
8
A statement is not allowed to have free variables (in this case $x$). The problem is that in natural language one might take this sentence to have an implied quantifier, so that it means the same as "For every integer $x$, $x^3 > 0$".
– Robert Israel
Aug 6 at 23:31
A statement is not allowed to have free variables (in this case $x$). The problem is that in natural language one might take this sentence to have an implied quantifier, so that it means the same as "For every integer $x$, $x^3 > 0$".
– Robert Israel
Aug 6 at 23:31
2
2
I guess OP is teaching some high school levels students, to whom it's not very trivial to explain bound and free variables. Answers in a less jargonized manner might be more helpful.
– Cave Johnson
Aug 6 at 23:43
I guess OP is teaching some high school levels students, to whom it's not very trivial to explain bound and free variables. Answers in a less jargonized manner might be more helpful.
– Cave Johnson
Aug 6 at 23:43
3
3
@CaveJohnson The answers are directed at the OP, not the OP's students, and we're assuming that the OP does understand the basic concepts like "universal quantifier" etc. (Note that the OP isn't asking how to explain something.)
– Noah Schweber
Aug 6 at 23:50
@CaveJohnson The answers are directed at the OP, not the OP's students, and we're assuming that the OP does understand the basic concepts like "universal quantifier" etc. (Note that the OP isn't asking how to explain something.)
– Noah Schweber
Aug 6 at 23:50
1
1
Implicit in natural language is: For all $x$, if $x$ is an integer then $x^3 > 0$. This is false since $-1$ is an integer and $(-1)^3 < 0$. So, we have a a false statement. In a formal proof (not in natural language), whether a formula is a statement or not is not an important distinction.
– Dan Christensen
Aug 7 at 14:59
Implicit in natural language is: For all $x$, if $x$ is an integer then $x^3 > 0$. This is false since $-1$ is an integer and $(-1)^3 < 0$. So, we have a a false statement. In a formal proof (not in natural language), whether a formula is a statement or not is not an important distinction.
– Dan Christensen
Aug 7 at 14:59
add a comment |Â
4 Answers
4
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up vote
13
down vote
This gets into a bit of a murky situation, actually. Let's start with a simpler claim: the expression "If $x$ is an integer, then $x^3>3$" is not a sentence, but rather a formula - the issue being the free variable $x$.
This brings us to a subtle point: the difference between a formula and its universal closure. Outside of logic, "If $x$ is an integer, then $x^3>3$" would often be treated as an abbreviation for "For all $x$, if $x$ is an integer then $x^3>3$" (the latter is the universal closure of the former: we take the original formula and add "for all"s to the beginning to account for all the free variables). However, within logic there are actually very good reasons (which I won't go into here) to distinguish between a formula and its universal closure, and in particular to be very careful about natural-language conventions like implicit universal quantifiers. This even extends to formulas like "$x=x$," which are tautologically true regardless of what value one assigns to $x$; within logic, "$x=x$" is a formula but not a sentence.
Because of formulas like "$x=x$," where we have a free variable but it's somehow not importantly free, I find language like "a statement is "a declarative sentence which is true or false, but not both"" to be quite confusing. For example, if within this context we say that "$x=x$" is a statement since it's obviously true regardless of what $x$ is, then the Boolean combination of two non-statements could be a statement: "$x>0$" and "$x<0$" are both of indeterminate truth value, but "$x>0$ and $x<0$" is false regardless of what value we assign to $x$. By contrast, the notions of well-formed formula and sentence are perfectly rigorous and easy to work with, and to my mind one of the major points of logic is that it's valuable to shift away from natural-language reasoning about truth and towards more rigorous (if limited and less intuitive) grounds.
So what's the answer to your question? Well, I would say that the expression is not a statement. One might disagree and say that when it comes to evaluating the truth of a formula, we implicitly choose to work with its universal closure instead, but I think this is problematic for a number of reasons. Because of this, my advice to you personally is to pick a convention and stick to it (but first read the book carefully and tell if it is already making such an implicit convention!). After all, once a convention is settled on everything will work itself out well (even if the "wrong" convention is chosen), but there may be some disagreement over what the convention should be.
answered Aug 6 at 23:45
Noah Schweber
111k9140264
3
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
add a comment |Â
up vote
5
down vote
"If x is an integer, ..." means "For all integers x, ..." The universal quantifier is hidden here, but this a convention of mathematical English. The student needs to understand this point- it's very important.
answered Aug 6 at 23:29
Brian Borchers
5,07711119
8
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
4
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
2
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
1
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
2
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
 |Â
show 4 more comments
up vote
3
down vote
This is depends on the convention, but without context I would say that $(xin S)implies(P(x))$ is not a statement!
A statement has only bound variable, in some cases the language you are talking can imply some quantifier but we have to be precise, try to write it formally, if $x$ is free then it is not a statement
answered Aug 6 at 23:38
Holo
4,2662629
1
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
add a comment |Â
up vote
1
down vote
If you like programming, formulate your problem like this:
IsPosCube(x) := x^3 > 0
NumCheck(x) := if IsInt(x) then IsPosCube(x) else True
Which I think is nice, because by making functions you can see that your expressions can't be evaluated without variable assignments, making them formulas and not statements. However, NumCheck(-3) is easily seen to be a false statement. But interestingly, NumCheck(3.3) is a true statement! This is why your formula isn't equivalent to the statement "The cube of any integer is positive."
answered Aug 7 at 2:11
Björn Lindqvist
324212
4
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
1
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
13
down vote
This gets into a bit of a murky situation, actually. Let's start with a simpler claim: the expression "If $x$ is an integer, then $x^3>3$" is not a sentence, but rather a formula - the issue being the free variable $x$.
This brings us to a subtle point: the difference between a formula and its universal closure. Outside of logic, "If $x$ is an integer, then $x^3>3$" would often be treated as an abbreviation for "For all $x$, if $x$ is an integer then $x^3>3$" (the latter is the universal closure of the former: we take the original formula and add "for all"s to the beginning to account for all the free variables). However, within logic there are actually very good reasons (which I won't go into here) to distinguish between a formula and its universal closure, and in particular to be very careful about natural-language conventions like implicit universal quantifiers. This even extends to formulas like "$x=x$," which are tautologically true regardless of what value one assigns to $x$; within logic, "$x=x$" is a formula but not a sentence.
Because of formulas like "$x=x$," where we have a free variable but it's somehow not importantly free, I find language like "a statement is "a declarative sentence which is true or false, but not both"" to be quite confusing. For example, if within this context we say that "$x=x$" is a statement since it's obviously true regardless of what $x$ is, then the Boolean combination of two non-statements could be a statement: "$x>0$" and "$x<0$" are both of indeterminate truth value, but "$x>0$ and $x<0$" is false regardless of what value we assign to $x$. By contrast, the notions of well-formed formula and sentence are perfectly rigorous and easy to work with, and to my mind one of the major points of logic is that it's valuable to shift away from natural-language reasoning about truth and towards more rigorous (if limited and less intuitive) grounds.
So what's the answer to your question? Well, I would say that the expression is not a statement. One might disagree and say that when it comes to evaluating the truth of a formula, we implicitly choose to work with its universal closure instead, but I think this is problematic for a number of reasons. Because of this, my advice to you personally is to pick a convention and stick to it (but first read the book carefully and tell if it is already making such an implicit convention!). After all, once a convention is settled on everything will work itself out well (even if the "wrong" convention is chosen), but there may be some disagreement over what the convention should be.
answered Aug 6 at 23:45
Noah Schweber
111k9140264
3
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
add a comment |Â
up vote
13
down vote
This gets into a bit of a murky situation, actually. Let's start with a simpler claim: the expression "If $x$ is an integer, then $x^3>3$" is not a sentence, but rather a formula - the issue being the free variable $x$.
This brings us to a subtle point: the difference between a formula and its universal closure. Outside of logic, "If $x$ is an integer, then $x^3>3$" would often be treated as an abbreviation for "For all $x$, if $x$ is an integer then $x^3>3$" (the latter is the universal closure of the former: we take the original formula and add "for all"s to the beginning to account for all the free variables). However, within logic there are actually very good reasons (which I won't go into here) to distinguish between a formula and its universal closure, and in particular to be very careful about natural-language conventions like implicit universal quantifiers. This even extends to formulas like "$x=x$," which are tautologically true regardless of what value one assigns to $x$; within logic, "$x=x$" is a formula but not a sentence.
Because of formulas like "$x=x$," where we have a free variable but it's somehow not importantly free, I find language like "a statement is "a declarative sentence which is true or false, but not both"" to be quite confusing. For example, if within this context we say that "$x=x$" is a statement since it's obviously true regardless of what $x$ is, then the Boolean combination of two non-statements could be a statement: "$x>0$" and "$x<0$" are both of indeterminate truth value, but "$x>0$ and $x<0$" is false regardless of what value we assign to $x$. By contrast, the notions of well-formed formula and sentence are perfectly rigorous and easy to work with, and to my mind one of the major points of logic is that it's valuable to shift away from natural-language reasoning about truth and towards more rigorous (if limited and less intuitive) grounds.
So what's the answer to your question? Well, I would say that the expression is not a statement. One might disagree and say that when it comes to evaluating the truth of a formula, we implicitly choose to work with its universal closure instead, but I think this is problematic for a number of reasons. Because of this, my advice to you personally is to pick a convention and stick to it (but first read the book carefully and tell if it is already making such an implicit convention!). After all, once a convention is settled on everything will work itself out well (even if the "wrong" convention is chosen), but there may be some disagreement over what the convention should be.
answered Aug 6 at 23:45
Noah Schweber
111k9140264
3
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
add a comment |Â
up vote
13
down vote
up vote
13
down vote
This gets into a bit of a murky situation, actually. Let's start with a simpler claim: the expression "If $x$ is an integer, then $x^3>3$" is not a sentence, but rather a formula - the issue being the free variable $x$.
This brings us to a subtle point: the difference between a formula and its universal closure. Outside of logic, "If $x$ is an integer, then $x^3>3$" would often be treated as an abbreviation for "For all $x$, if $x$ is an integer then $x^3>3$" (the latter is the universal closure of the former: we take the original formula and add "for all"s to the beginning to account for all the free variables). However, within logic there are actually very good reasons (which I won't go into here) to distinguish between a formula and its universal closure, and in particular to be very careful about natural-language conventions like implicit universal quantifiers. This even extends to formulas like "$x=x$," which are tautologically true regardless of what value one assigns to $x$; within logic, "$x=x$" is a formula but not a sentence.
Because of formulas like "$x=x$," where we have a free variable but it's somehow not importantly free, I find language like "a statement is "a declarative sentence which is true or false, but not both"" to be quite confusing. For example, if within this context we say that "$x=x$" is a statement since it's obviously true regardless of what $x$ is, then the Boolean combination of two non-statements could be a statement: "$x>0$" and "$x<0$" are both of indeterminate truth value, but "$x>0$ and $x<0$" is false regardless of what value we assign to $x$. By contrast, the notions of well-formed formula and sentence are perfectly rigorous and easy to work with, and to my mind one of the major points of logic is that it's valuable to shift away from natural-language reasoning about truth and towards more rigorous (if limited and less intuitive) grounds.
So what's the answer to your question? Well, I would say that the expression is not a statement. One might disagree and say that when it comes to evaluating the truth of a formula, we implicitly choose to work with its universal closure instead, but I think this is problematic for a number of reasons. Because of this, my advice to you personally is to pick a convention and stick to it (but first read the book carefully and tell if it is already making such an implicit convention!). After all, once a convention is settled on everything will work itself out well (even if the "wrong" convention is chosen), but there may be some disagreement over what the convention should be.
answered Aug 6 at 23:45
Noah Schweber
111k9140264
This gets into a bit of a murky situation, actually. Let's start with a simpler claim: the expression "If $x$ is an integer, then $x^3>3$" is not a sentence, but rather a formula - the issue being the free variable $x$.
This brings us to a subtle point: the difference between a formula and its universal closure. Outside of logic, "If $x$ is an integer, then $x^3>3$" would often be treated as an abbreviation for "For all $x$, if $x$ is an integer then $x^3>3$" (the latter is the universal closure of the former: we take the original formula and add "for all"s to the beginning to account for all the free variables). However, within logic there are actually very good reasons (which I won't go into here) to distinguish between a formula and its universal closure, and in particular to be very careful about natural-language conventions like implicit universal quantifiers. This even extends to formulas like "$x=x$," which are tautologically true regardless of what value one assigns to $x$; within logic, "$x=x$" is a formula but not a sentence.
Because of formulas like "$x=x$," where we have a free variable but it's somehow not importantly free, I find language like "a statement is "a declarative sentence which is true or false, but not both"" to be quite confusing. For example, if within this context we say that "$x=x$" is a statement since it's obviously true regardless of what $x$ is, then the Boolean combination of two non-statements could be a statement: "$x>0$" and "$x<0$" are both of indeterminate truth value, but "$x>0$ and $x<0$" is false regardless of what value we assign to $x$. By contrast, the notions of well-formed formula and sentence are perfectly rigorous and easy to work with, and to my mind one of the major points of logic is that it's valuable to shift away from natural-language reasoning about truth and towards more rigorous (if limited and less intuitive) grounds.
So what's the answer to your question? Well, I would say that the expression is not a statement. One might disagree and say that when it comes to evaluating the truth of a formula, we implicitly choose to work with its universal closure instead, but I think this is problematic for a number of reasons. Because of this, my advice to you personally is to pick a convention and stick to it (but first read the book carefully and tell if it is already making such an implicit convention!). After all, once a convention is settled on everything will work itself out well (even if the "wrong" convention is chosen), but there may be some disagreement over what the convention should be.
answered Aug 6 at 23:45
Noah Schweber
111k9140264
answered Aug 6 at 23:45
Noah Schweber
111k9140264
answered Aug 6 at 23:45
Noah Schweber
111k9140264
answered Aug 6 at 23:45
Noah Schweber
111k9140264
111k9140264
3
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
add a comment |Â
3
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
3
3
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
In non-mathematical contexts the universal closure is often implicitly assumed to be part of the sentence. E.g. if you said " If a man is an NHL (National Hockey League) player then he earns over $100K/yr.", most people would consider this to mean "For all men, if a man is an NHL player then...(etc.)." And would also assume that an existential quantifier is included somewhere, as they don't expect you would say it if you weren't sure that the NHL has more than $0$ players............+1
– DanielWainfleet
Aug 7 at 9:36
add a comment |Â
up vote
5
down vote
"If x is an integer, ..." means "For all integers x, ..." The universal quantifier is hidden here, but this a convention of mathematical English. The student needs to understand this point- it's very important.
answered Aug 6 at 23:29
Brian Borchers
5,07711119
8
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
4
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
2
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
1
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
2
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
 |Â
show 4 more comments
up vote
5
down vote
"If x is an integer, ..." means "For all integers x, ..." The universal quantifier is hidden here, but this a convention of mathematical English. The student needs to understand this point- it's very important.
answered Aug 6 at 23:29
Brian Borchers
5,07711119
8
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
4
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
2
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
1
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
2
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
 |Â
show 4 more comments
up vote
5
down vote
up vote
5
down vote
"If x is an integer, ..." means "For all integers x, ..." The universal quantifier is hidden here, but this a convention of mathematical English. The student needs to understand this point- it's very important.
answered Aug 6 at 23:29
Brian Borchers
5,07711119
"If x is an integer, ..." means "For all integers x, ..." The universal quantifier is hidden here, but this a convention of mathematical English. The student needs to understand this point- it's very important.
answered Aug 6 at 23:29
Brian Borchers
5,07711119
edited Aug 6 at 23:32
answered Aug 6 at 23:29
Brian Borchers
5,07711119
answered Aug 6 at 23:29
Brian Borchers
5,07711119
answered Aug 6 at 23:29
Brian Borchers
5,07711119
5,07711119
8
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
4
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
2
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
1
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
2
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
 |Â
show 4 more comments
8
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
4
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
2
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
1
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
2
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
8
8
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
I personally disagree; I think we need to distinguish between formulas and their universal closures. In this case I would say the expression is not a statement.
– Noah Schweber
Aug 6 at 23:31
4
4
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
Context matters: a universal quantifier might be considered implicit when the sentence is clearly talking about an arbitrary $x$, but it may be refering to a particular one. Since we don't have the context, we cannot simply presume it.
– Graham Kemp
Aug 6 at 23:50
2
2
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
@GrahamKemp And the recognition of the importance of that context is why we generally make this distinction in logic.
– Noah Schweber
Aug 6 at 23:51
1
1
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
As much as you might dislike this construction, it's commonly enough used in mathematical writing that students should have it explained to them.
– Brian Borchers
Aug 7 at 0:05
2
2
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
See the discussion on page 623 and following of A Handbook of Mathematical Discourse, by Charles Wells, which cites examples. case.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf
– Brian Borchers
Aug 8 at 3:32
 |Â
show 4 more comments
up vote
3
down vote
This is depends on the convention, but without context I would say that $(xin S)implies(P(x))$ is not a statement!
A statement has only bound variable, in some cases the language you are talking can imply some quantifier but we have to be precise, try to write it formally, if $x$ is free then it is not a statement
answered Aug 6 at 23:38
Holo
4,2662629
1
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
add a comment |Â
up vote
3
down vote
This is depends on the convention, but without context I would say that $(xin S)implies(P(x))$ is not a statement!
A statement has only bound variable, in some cases the language you are talking can imply some quantifier but we have to be precise, try to write it formally, if $x$ is free then it is not a statement
answered Aug 6 at 23:38
Holo
4,2662629
1
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
add a comment |Â
up vote
3
down vote
up vote
3
down vote
This is depends on the convention, but without context I would say that $(xin S)implies(P(x))$ is not a statement!
A statement has only bound variable, in some cases the language you are talking can imply some quantifier but we have to be precise, try to write it formally, if $x$ is free then it is not a statement
answered Aug 6 at 23:38
Holo
4,2662629
This is depends on the convention, but without context I would say that $(xin S)implies(P(x))$ is not a statement!
A statement has only bound variable, in some cases the language you are talking can imply some quantifier but we have to be precise, try to write it formally, if $x$ is free then it is not a statement
answered Aug 6 at 23:38
Holo
4,2662629
edited Aug 6 at 23:42
answered Aug 6 at 23:38
Holo
4,2662629
answered Aug 6 at 23:38
Holo
4,2662629
answered Aug 6 at 23:38
Holo
4,2662629
4,2662629
1
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
add a comment |Â
1
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
1
1
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
@NoahSchweber Yes, thanks for pointing it out
– Holo
Aug 6 at 23:42
add a comment |Â
up vote
1
down vote
If you like programming, formulate your problem like this:
IsPosCube(x) := x^3 > 0
NumCheck(x) := if IsInt(x) then IsPosCube(x) else True
Which I think is nice, because by making functions you can see that your expressions can't be evaluated without variable assignments, making them formulas and not statements. However, NumCheck(-3) is easily seen to be a false statement. But interestingly, NumCheck(3.3) is a true statement! This is why your formula isn't equivalent to the statement "The cube of any integer is positive."
answered Aug 7 at 2:11
Björn Lindqvist
324212
4
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
1
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
add a comment |Â
up vote
1
down vote
If you like programming, formulate your problem like this:
IsPosCube(x) := x^3 > 0
NumCheck(x) := if IsInt(x) then IsPosCube(x) else True
Which I think is nice, because by making functions you can see that your expressions can't be evaluated without variable assignments, making them formulas and not statements. However, NumCheck(-3) is easily seen to be a false statement. But interestingly, NumCheck(3.3) is a true statement! This is why your formula isn't equivalent to the statement "The cube of any integer is positive."
answered Aug 7 at 2:11
Björn Lindqvist
324212
4
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
1
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If you like programming, formulate your problem like this:
IsPosCube(x) := x^3 > 0
NumCheck(x) := if IsInt(x) then IsPosCube(x) else True
Which I think is nice, because by making functions you can see that your expressions can't be evaluated without variable assignments, making them formulas and not statements. However, NumCheck(-3) is easily seen to be a false statement. But interestingly, NumCheck(3.3) is a true statement! This is why your formula isn't equivalent to the statement "The cube of any integer is positive."
answered Aug 7 at 2:11
Björn Lindqvist
324212
If you like programming, formulate your problem like this:
IsPosCube(x) := x^3 > 0
NumCheck(x) := if IsInt(x) then IsPosCube(x) else True
Which I think is nice, because by making functions you can see that your expressions can't be evaluated without variable assignments, making them formulas and not statements. However, NumCheck(-3) is easily seen to be a false statement. But interestingly, NumCheck(3.3) is a true statement! This is why your formula isn't equivalent to the statement "The cube of any integer is positive."
answered Aug 7 at 2:11
Björn Lindqvist
324212
answered Aug 7 at 2:11
Björn Lindqvist
324212
answered Aug 7 at 2:11
Björn Lindqvist
324212
answered Aug 7 at 2:11
Björn Lindqvist
324212
324212
4
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
1
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
add a comment |Â
4
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
1
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
4
4
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
Is the 'if ... then' of the OP equivalent to an "if-then-else" construct? That seems doubtful.
– Doug Spoonwood
Aug 7 at 3:09
1
1
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
Yes, when evaluated!
– Björn Lindqvist
Aug 7 at 14:46
add a comment |Â
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8
A statement is not allowed to have free variables (in this case $x$). The problem is that in natural language one might take this sentence to have an implied quantifier, so that it means the same as "For every integer $x$, $x^3 > 0$".
– Robert Israel
Aug 6 at 23:31
2
I guess OP is teaching some high school levels students, to whom it's not very trivial to explain bound and free variables. Answers in a less jargonized manner might be more helpful.
– Cave Johnson
Aug 6 at 23:43
3
@CaveJohnson The answers are directed at the OP, not the OP's students, and we're assuming that the OP does understand the basic concepts like "universal quantifier" etc. (Note that the OP isn't asking how to explain something.)
– Noah Schweber
Aug 6 at 23:50
1
Implicit in natural language is: For all $x$, if $x$ is an integer then $x^3 > 0$. This is false since $-1$ is an integer and $(-1)^3 < 0$. So, we have a a false statement. In a formal proof (not in natural language), whether a formula is a statement or not is not an important distinction.
– Dan Christensen
Aug 7 at 14:59