Mixing
what do we define definitely true?
$begingroup$
I make the following statements p q r :
p:if a pig has horns,then it can breathe fire.
q:if a pig can breathe fire, then it has wings.
r: if a pig has wings, then is has horns.
Each statement is either true or false,but i don't know which.
i then see a pig with wings breathing fire. It has no horns.
Which statements, if any,can i now conclude are definitely true or definitely false?
my answer is r only,since given one statement we can only figure out which one is definitely false.my question is how do we define definitely true, do we need three cases to be true to make the statements definitely true? (i.e.
1. if a is true,then b is true
2. if a is not true,then b is true
3. if a is not true,then b is not true)
propositional-calculus
asked Jan 2 at 14:39
KevinKevin
143
$endgroup$
|
show 2 more comments
$begingroup$
I make the following statements p q r :
p:if a pig has horns,then it can breathe fire.
q:if a pig can breathe fire, then it has wings.
r: if a pig has wings, then is has horns.
Each statement is either true or false,but i don't know which.
i then see a pig with wings breathing fire. It has no horns.
Which statements, if any,can i now conclude are definitely true or definitely false?
my answer is r only,since given one statement we can only figure out which one is definitely false.my question is how do we define definitely true, do we need three cases to be true to make the statements definitely true? (i.e.
1. if a is true,then b is true
2. if a is not true,then b is true
3. if a is not true,then b is not true)
propositional-calculus
asked Jan 2 at 14:39
KevinKevin
143
$endgroup$
$begingroup$
As you remark (I think, it's a bit hard to follow) a single example can disprove a claim but it can not prove it. Here, all we know about statements $p,q$ is that they might be true or they might be false. We have a fire breathing pig with wings, so we can't disprove $q$. We don't have a pig with horns so we don't even have a test case for $p$. We can disprove $r$ however as we have an example of a winged pig without horns.
$endgroup$
– lulu
Jan 2 at 14:42
1
$begingroup$
@MauroALLEGRANZA I'm not sure that's appropriate. Is "a pig" the unique identifier of the only pig in the world? If not, then I think the given implications all have implicit universal quantifiers.
$endgroup$
– David K
Jan 2 at 14:56
$begingroup$
@MauroALLEGRANZA but if p is not true , whatever the q is true or false the statement could also be true, so can we say the p---> q true (sry havent learn to type logic sympol yet) with only one case?
$endgroup$
– Kevin
Jan 2 at 15:07
1
$begingroup$
If you can prove that there is no pig anywhere that has horns, then it is certainly true that "if a pig has horns then it can breathe fire." Finding one pig without horns is not a proof that no pig has horns.
$endgroup$
– David K
Jan 2 at 15:13
1
$begingroup$
It is possible that you could disprove all three statements by finding just three pigs. (The other two pigs you need are a horned pig that does not breath fire and a fire-breathing pig without wings.) Proving them true is another matter altogether.
$endgroup$
– David K
Jan 2 at 15:17
|
show 2 more comments
$begingroup$
I make the following statements p q r :
p:if a pig has horns,then it can breathe fire.
q:if a pig can breathe fire, then it has wings.
r: if a pig has wings, then is has horns.
Each statement is either true or false,but i don't know which.
i then see a pig with wings breathing fire. It has no horns.
Which statements, if any,can i now conclude are definitely true or definitely false?
my answer is r only,since given one statement we can only figure out which one is definitely false.my question is how do we define definitely true, do we need three cases to be true to make the statements definitely true? (i.e.
1. if a is true,then b is true
2. if a is not true,then b is true
3. if a is not true,then b is not true)
propositional-calculus
asked Jan 2 at 14:39
KevinKevin
143
$endgroup$
I make the following statements p q r :
p:if a pig has horns,then it can breathe fire.
q:if a pig can breathe fire, then it has wings.
r: if a pig has wings, then is has horns.
Each statement is either true or false,but i don't know which.
i then see a pig with wings breathing fire. It has no horns.
Which statements, if any,can i now conclude are definitely true or definitely false?
my answer is r only,since given one statement we can only figure out which one is definitely false.my question is how do we define definitely true, do we need three cases to be true to make the statements definitely true? (i.e.
1. if a is true,then b is true
2. if a is not true,then b is true
3. if a is not true,then b is not true)
propositional-calculus
propositional-calculus
asked Jan 2 at 14:39
KevinKevin
143
asked Jan 2 at 14:39
KevinKevin
143
asked Jan 2 at 14:39
KevinKevin
143
asked Jan 2 at 14:39
KevinKevin
143
asked Jan 2 at 14:39
KevinKevin
143
143
$begingroup$
As you remark (I think, it's a bit hard to follow) a single example can disprove a claim but it can not prove it. Here, all we know about statements $p,q$ is that they might be true or they might be false. We have a fire breathing pig with wings, so we can't disprove $q$. We don't have a pig with horns so we don't even have a test case for $p$. We can disprove $r$ however as we have an example of a winged pig without horns.
$endgroup$
– lulu
Jan 2 at 14:42
1
$begingroup$
@MauroALLEGRANZA I'm not sure that's appropriate. Is "a pig" the unique identifier of the only pig in the world? If not, then I think the given implications all have implicit universal quantifiers.
$endgroup$
– David K
Jan 2 at 14:56
$begingroup$
@MauroALLEGRANZA but if p is not true , whatever the q is true or false the statement could also be true, so can we say the p---> q true (sry havent learn to type logic sympol yet) with only one case?
$endgroup$
– Kevin
Jan 2 at 15:07
1
$begingroup$
If you can prove that there is no pig anywhere that has horns, then it is certainly true that "if a pig has horns then it can breathe fire." Finding one pig without horns is not a proof that no pig has horns.
$endgroup$
– David K
Jan 2 at 15:13
1
$begingroup$
It is possible that you could disprove all three statements by finding just three pigs. (The other two pigs you need are a horned pig that does not breath fire and a fire-breathing pig without wings.) Proving them true is another matter altogether.
$endgroup$
– David K
Jan 2 at 15:17
|
show 2 more comments
$begingroup$
As you remark (I think, it's a bit hard to follow) a single example can disprove a claim but it can not prove it. Here, all we know about statements $p,q$ is that they might be true or they might be false. We have a fire breathing pig with wings, so we can't disprove $q$. We don't have a pig with horns so we don't even have a test case for $p$. We can disprove $r$ however as we have an example of a winged pig without horns.
$endgroup$
– lulu
Jan 2 at 14:42
1
$begingroup$
@MauroALLEGRANZA I'm not sure that's appropriate. Is "a pig" the unique identifier of the only pig in the world? If not, then I think the given implications all have implicit universal quantifiers.
$endgroup$
– David K
Jan 2 at 14:56
$begingroup$
@MauroALLEGRANZA but if p is not true , whatever the q is true or false the statement could also be true, so can we say the p---> q true (sry havent learn to type logic sympol yet) with only one case?
$endgroup$
– Kevin
Jan 2 at 15:07
1
$begingroup$
If you can prove that there is no pig anywhere that has horns, then it is certainly true that "if a pig has horns then it can breathe fire." Finding one pig without horns is not a proof that no pig has horns.
$endgroup$
– David K
Jan 2 at 15:13
1
$begingroup$
It is possible that you could disprove all three statements by finding just three pigs. (The other two pigs you need are a horned pig that does not breath fire and a fire-breathing pig without wings.) Proving them true is another matter altogether.
$endgroup$
– David K
Jan 2 at 15:17
$begingroup$
As you remark (I think, it's a bit hard to follow) a single example can disprove a claim but it can not prove it. Here, all we know about statements $p,q$ is that they might be true or they might be false. We have a fire breathing pig with wings, so we can't disprove $q$. We don't have a pig with horns so we don't even have a test case for $p$. We can disprove $r$ however as we have an example of a winged pig without horns.
$endgroup$
– lulu
Jan 2 at 14:42
$begingroup$
As you remark (I think, it's a bit hard to follow) a single example can disprove a claim but it can not prove it. Here, all we know about statements $p,q$ is that they might be true or they might be false. We have a fire breathing pig with wings, so we can't disprove $q$. We don't have a pig with horns so we don't even have a test case for $p$. We can disprove $r$ however as we have an example of a winged pig without horns.
$endgroup$
– lulu
Jan 2 at 14:42
1
1
$begingroup$
@MauroALLEGRANZA I'm not sure that's appropriate. Is "a pig" the unique identifier of the only pig in the world? If not, then I think the given implications all have implicit universal quantifiers.
$endgroup$
– David K
Jan 2 at 14:56
$begingroup$
@MauroALLEGRANZA I'm not sure that's appropriate. Is "a pig" the unique identifier of the only pig in the world? If not, then I think the given implications all have implicit universal quantifiers.
$endgroup$
– David K
Jan 2 at 14:56
$begingroup$
@MauroALLEGRANZA but if p is not true , whatever the q is true or false the statement could also be true, so can we say the p---> q true (sry havent learn to type logic sympol yet) with only one case?
$endgroup$
– Kevin
Jan 2 at 15:07
$begingroup$
@MauroALLEGRANZA but if p is not true , whatever the q is true or false the statement could also be true, so can we say the p---> q true (sry havent learn to type logic sympol yet) with only one case?
$endgroup$
– Kevin
Jan 2 at 15:07
1
1
$begingroup$
If you can prove that there is no pig anywhere that has horns, then it is certainly true that "if a pig has horns then it can breathe fire." Finding one pig without horns is not a proof that no pig has horns.
$endgroup$
– David K
Jan 2 at 15:13
$begingroup$
If you can prove that there is no pig anywhere that has horns, then it is certainly true that "if a pig has horns then it can breathe fire." Finding one pig without horns is not a proof that no pig has horns.
$endgroup$
– David K
Jan 2 at 15:13
1
1
$begingroup$
It is possible that you could disprove all three statements by finding just three pigs. (The other two pigs you need are a horned pig that does not breath fire and a fire-breathing pig without wings.) Proving them true is another matter altogether.
$endgroup$
– David K
Jan 2 at 15:17
$begingroup$
It is possible that you could disprove all three statements by finding just three pigs. (The other two pigs you need are a horned pig that does not breath fire and a fire-breathing pig without wings.) Proving them true is another matter altogether.
$endgroup$
– David K
Jan 2 at 15:17
|
show 2 more comments
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$begingroup$
As you remark (I think, it's a bit hard to follow) a single example can disprove a claim but it can not prove it. Here, all we know about statements $p,q$ is that they might be true or they might be false. We have a fire breathing pig with wings, so we can't disprove $q$. We don't have a pig with horns so we don't even have a test case for $p$. We can disprove $r$ however as we have an example of a winged pig without horns.
$endgroup$
– lulu
Jan 2 at 14:42
1
$begingroup$
@MauroALLEGRANZA I'm not sure that's appropriate. Is "a pig" the unique identifier of the only pig in the world? If not, then I think the given implications all have implicit universal quantifiers.
$endgroup$
– David K
Jan 2 at 14:56
$begingroup$
@MauroALLEGRANZA but if p is not true , whatever the q is true or false the statement could also be true, so can we say the p---> q true (sry havent learn to type logic sympol yet) with only one case?
$endgroup$
– Kevin
Jan 2 at 15:07
1
$begingroup$
If you can prove that there is no pig anywhere that has horns, then it is certainly true that "if a pig has horns then it can breathe fire." Finding one pig without horns is not a proof that no pig has horns.
$endgroup$
– David K
Jan 2 at 15:13
1
$begingroup$
It is possible that you could disprove all three statements by finding just three pigs. (The other two pigs you need are a horned pig that does not breath fire and a fire-breathing pig without wings.) Proving them true is another matter altogether.
$endgroup$
– David K
Jan 2 at 15:17