What would be the negation of these statements?












0












$begingroup$


What would be the negation of



"No dogs have three legs".



I think "Some dogs do not have three legs"



"Some animals don't eat meat"



I think "All animals eat meat"



"I make the bread, or she does not make the bread"



I think "I do not make the bread, and she does make the bread"



Am i right here?






logic







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$endgroup$












  • $begingroup$
    Note that "No dog has three legs" and "One dog has four legs more than no dog" does not imply "One dog has seven legs" :)
    $endgroup$
    – Hagen von Eitzen
    Jul 18 '13 at 16:04
















0












$begingroup$


What would be the negation of



"No dogs have three legs".



I think "Some dogs do not have three legs"



"Some animals don't eat meat"



I think "All animals eat meat"



"I make the bread, or she does not make the bread"



I think "I do not make the bread, and she does make the bread"



Am i right here?






logic







share|cite|improve this question











$endgroup$












  • $begingroup$
    Note that "No dog has three legs" and "One dog has four legs more than no dog" does not imply "One dog has seven legs" :)
    $endgroup$
    – Hagen von Eitzen
    Jul 18 '13 at 16:04














0












0








0





$begingroup$


What would be the negation of



"No dogs have three legs".



I think "Some dogs do not have three legs"



"Some animals don't eat meat"



I think "All animals eat meat"



"I make the bread, or she does not make the bread"



I think "I do not make the bread, and she does make the bread"



Am i right here?






logic







share|cite|improve this question











$endgroup$




What would be the negation of



"No dogs have three legs".



I think "Some dogs do not have three legs"



"Some animals don't eat meat"



I think "All animals eat meat"



"I make the bread, or she does not make the bread"



I think "I do not make the bread, and she does make the bread"



Am i right here?






logic




logic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jul 18 '13 at 16:08







MethodManX

















asked Jul 18 '13 at 16:01









MethodManXMethodManX

594112135




594112135












  • $begingroup$
    Note that "No dog has three legs" and "One dog has four legs more than no dog" does not imply "One dog has seven legs" :)
    $endgroup$
    – Hagen von Eitzen
    Jul 18 '13 at 16:04


















  • $begingroup$
    Note that "No dog has three legs" and "One dog has four legs more than no dog" does not imply "One dog has seven legs" :)
    $endgroup$
    – Hagen von Eitzen
    Jul 18 '13 at 16:04
















$begingroup$
Note that "No dog has three legs" and "One dog has four legs more than no dog" does not imply "One dog has seven legs" :)
$endgroup$
– Hagen von Eitzen
Jul 18 '13 at 16:04




$begingroup$
Note that "No dog has three legs" and "One dog has four legs more than no dog" does not imply "One dog has seven legs" :)
$endgroup$
– Hagen von Eitzen
Jul 18 '13 at 16:04










3 Answers
3






active

oldest

votes


















3












$begingroup$

The negation of the first statement would be "Some dogs have three legs." You are correct on the second and the third.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you i just realized that
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:03










  • $begingroup$
    My friend can you check my last question please
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:08










  • $begingroup$
    Yes, your last one is correct.
    $endgroup$
    – Adrian Keister
    Jul 18 '13 at 16:11



















3












$begingroup$

Tip: a good way to start when negating any proposition $P$ is to assert $lnot P$, i.e., if we have a sentence "P", we can negate it simply by writing "It is not the case that P". Then, if we want to "translate" further from loglish to more customary natural language, we can do so if we choose. I personally find it helpful to translate the initial statement into logic. Then I negate it, perhaps distribute the negation, and "read off" the result back in natural language. I do that simply because negating natural language can be easily side-tracked or awkward to directly negate.



You did fine, overall, except for the first assertion:



(1) We start with the assertion $P$: No dogs have three legs.




  • Then we negate it by stating $lnot P:$ It is NOT the case that (No dogs have three legs).

  • If we translate further, we see that $lnot P$ can be expressed as "There exists one or more dogs with three legs": I.e., "Some dogs have three legs".


If we were to translate the initial statement to "logic" first, with the domain being "dogs": and $T(x)$ meaning $x$ has three legs, then the initial first statement can be espressed as $P: forall x lnot T(x)$



The negation would be $lnot P$: $$lnot [forall x lnot T(x)] iff exists x(lnot lnot T(x)) iff exists x T(x)$$ $$;;text{Negated sentence: Some dogs have three legs}$$




$(2)$ Your second negation is just fine.



$(3)$ Yes, you used DeMorgan's correctly, and your translation is correct.



P: I make the bread;
Q: she makes the bread



Given $P lor lnot Q$, its negation is $lnot (Plor lnot Q) equiv lnot P land Q$, which gives us "I don't make the bread and she makes the bread."






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks, @Amzoti! Appreciated!
    $endgroup$
    – Namaste
    Jul 19 '13 at 0:52



















2












$begingroup$

For a sanity check, try to imagine someone saying these sentences in conversation. What would you have to do to prove that they are lying?



For "no dogs have three legs", you would just have to prove the existence of a dog with three legs (hopefully you can find such a dog, rather than having to produce your own example...)






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Sanity check for a police inspector?
    $endgroup$
    – Ilya
    Jul 18 '13 at 16:09











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3 Answers
3






active

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3 Answers
3






active

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active

oldest

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active

oldest

votes









3












$begingroup$

The negation of the first statement would be "Some dogs have three legs." You are correct on the second and the third.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you i just realized that
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:03










  • $begingroup$
    My friend can you check my last question please
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:08










  • $begingroup$
    Yes, your last one is correct.
    $endgroup$
    – Adrian Keister
    Jul 18 '13 at 16:11
















3












$begingroup$

The negation of the first statement would be "Some dogs have three legs." You are correct on the second and the third.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thank you i just realized that
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:03










  • $begingroup$
    My friend can you check my last question please
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:08










  • $begingroup$
    Yes, your last one is correct.
    $endgroup$
    – Adrian Keister
    Jul 18 '13 at 16:11














3












3








3





$begingroup$

The negation of the first statement would be "Some dogs have three legs." You are correct on the second and the third.






share|cite|improve this answer











$endgroup$



The negation of the first statement would be "Some dogs have three legs." You are correct on the second and the third.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jul 18 '13 at 16:12

























answered Jul 18 '13 at 16:02









Adrian KeisterAdrian Keister

5,27371933




5,27371933












  • $begingroup$
    Thank you i just realized that
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:03










  • $begingroup$
    My friend can you check my last question please
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:08










  • $begingroup$
    Yes, your last one is correct.
    $endgroup$
    – Adrian Keister
    Jul 18 '13 at 16:11


















  • $begingroup$
    Thank you i just realized that
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:03










  • $begingroup$
    My friend can you check my last question please
    $endgroup$
    – MethodManX
    Jul 18 '13 at 16:08










  • $begingroup$
    Yes, your last one is correct.
    $endgroup$
    – Adrian Keister
    Jul 18 '13 at 16:11
















$begingroup$
Thank you i just realized that
$endgroup$
– MethodManX
Jul 18 '13 at 16:03




$begingroup$
Thank you i just realized that
$endgroup$
– MethodManX
Jul 18 '13 at 16:03












$begingroup$
My friend can you check my last question please
$endgroup$
– MethodManX
Jul 18 '13 at 16:08




$begingroup$
My friend can you check my last question please
$endgroup$
– MethodManX
Jul 18 '13 at 16:08












$begingroup$
Yes, your last one is correct.
$endgroup$
– Adrian Keister
Jul 18 '13 at 16:11




$begingroup$
Yes, your last one is correct.
$endgroup$
– Adrian Keister
Jul 18 '13 at 16:11











3












$begingroup$

Tip: a good way to start when negating any proposition $P$ is to assert $lnot P$, i.e., if we have a sentence "P", we can negate it simply by writing "It is not the case that P". Then, if we want to "translate" further from loglish to more customary natural language, we can do so if we choose. I personally find it helpful to translate the initial statement into logic. Then I negate it, perhaps distribute the negation, and "read off" the result back in natural language. I do that simply because negating natural language can be easily side-tracked or awkward to directly negate.



You did fine, overall, except for the first assertion:



(1) We start with the assertion $P$: No dogs have three legs.




  • Then we negate it by stating $lnot P:$ It is NOT the case that (No dogs have three legs).

  • If we translate further, we see that $lnot P$ can be expressed as "There exists one or more dogs with three legs": I.e., "Some dogs have three legs".


If we were to translate the initial statement to "logic" first, with the domain being "dogs": and $T(x)$ meaning $x$ has three legs, then the initial first statement can be espressed as $P: forall x lnot T(x)$



The negation would be $lnot P$: $$lnot [forall x lnot T(x)] iff exists x(lnot lnot T(x)) iff exists x T(x)$$ $$;;text{Negated sentence: Some dogs have three legs}$$




$(2)$ Your second negation is just fine.



$(3)$ Yes, you used DeMorgan's correctly, and your translation is correct.



P: I make the bread;
Q: she makes the bread



Given $P lor lnot Q$, its negation is $lnot (Plor lnot Q) equiv lnot P land Q$, which gives us "I don't make the bread and she makes the bread."






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks, @Amzoti! Appreciated!
    $endgroup$
    – Namaste
    Jul 19 '13 at 0:52
















3












$begingroup$

Tip: a good way to start when negating any proposition $P$ is to assert $lnot P$, i.e., if we have a sentence "P", we can negate it simply by writing "It is not the case that P". Then, if we want to "translate" further from loglish to more customary natural language, we can do so if we choose. I personally find it helpful to translate the initial statement into logic. Then I negate it, perhaps distribute the negation, and "read off" the result back in natural language. I do that simply because negating natural language can be easily side-tracked or awkward to directly negate.



You did fine, overall, except for the first assertion:



(1) We start with the assertion $P$: No dogs have three legs.




  • Then we negate it by stating $lnot P:$ It is NOT the case that (No dogs have three legs).

  • If we translate further, we see that $lnot P$ can be expressed as "There exists one or more dogs with three legs": I.e., "Some dogs have three legs".


If we were to translate the initial statement to "logic" first, with the domain being "dogs": and $T(x)$ meaning $x$ has three legs, then the initial first statement can be espressed as $P: forall x lnot T(x)$



The negation would be $lnot P$: $$lnot [forall x lnot T(x)] iff exists x(lnot lnot T(x)) iff exists x T(x)$$ $$;;text{Negated sentence: Some dogs have three legs}$$




$(2)$ Your second negation is just fine.



$(3)$ Yes, you used DeMorgan's correctly, and your translation is correct.



P: I make the bread;
Q: she makes the bread



Given $P lor lnot Q$, its negation is $lnot (Plor lnot Q) equiv lnot P land Q$, which gives us "I don't make the bread and she makes the bread."






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks, @Amzoti! Appreciated!
    $endgroup$
    – Namaste
    Jul 19 '13 at 0:52














3












3








3





$begingroup$

Tip: a good way to start when negating any proposition $P$ is to assert $lnot P$, i.e., if we have a sentence "P", we can negate it simply by writing "It is not the case that P". Then, if we want to "translate" further from loglish to more customary natural language, we can do so if we choose. I personally find it helpful to translate the initial statement into logic. Then I negate it, perhaps distribute the negation, and "read off" the result back in natural language. I do that simply because negating natural language can be easily side-tracked or awkward to directly negate.



You did fine, overall, except for the first assertion:



(1) We start with the assertion $P$: No dogs have three legs.




  • Then we negate it by stating $lnot P:$ It is NOT the case that (No dogs have three legs).

  • If we translate further, we see that $lnot P$ can be expressed as "There exists one or more dogs with three legs": I.e., "Some dogs have three legs".


If we were to translate the initial statement to "logic" first, with the domain being "dogs": and $T(x)$ meaning $x$ has three legs, then the initial first statement can be espressed as $P: forall x lnot T(x)$



The negation would be $lnot P$: $$lnot [forall x lnot T(x)] iff exists x(lnot lnot T(x)) iff exists x T(x)$$ $$;;text{Negated sentence: Some dogs have three legs}$$




$(2)$ Your second negation is just fine.



$(3)$ Yes, you used DeMorgan's correctly, and your translation is correct.



P: I make the bread;
Q: she makes the bread



Given $P lor lnot Q$, its negation is $lnot (Plor lnot Q) equiv lnot P land Q$, which gives us "I don't make the bread and she makes the bread."






share|cite|improve this answer











$endgroup$



Tip: a good way to start when negating any proposition $P$ is to assert $lnot P$, i.e., if we have a sentence "P", we can negate it simply by writing "It is not the case that P". Then, if we want to "translate" further from loglish to more customary natural language, we can do so if we choose. I personally find it helpful to translate the initial statement into logic. Then I negate it, perhaps distribute the negation, and "read off" the result back in natural language. I do that simply because negating natural language can be easily side-tracked or awkward to directly negate.



You did fine, overall, except for the first assertion:



(1) We start with the assertion $P$: No dogs have three legs.




  • Then we negate it by stating $lnot P:$ It is NOT the case that (No dogs have three legs).

  • If we translate further, we see that $lnot P$ can be expressed as "There exists one or more dogs with three legs": I.e., "Some dogs have three legs".


If we were to translate the initial statement to "logic" first, with the domain being "dogs": and $T(x)$ meaning $x$ has three legs, then the initial first statement can be espressed as $P: forall x lnot T(x)$



The negation would be $lnot P$: $$lnot [forall x lnot T(x)] iff exists x(lnot lnot T(x)) iff exists x T(x)$$ $$;;text{Negated sentence: Some dogs have three legs}$$




$(2)$ Your second negation is just fine.



$(3)$ Yes, you used DeMorgan's correctly, and your translation is correct.



P: I make the bread;
Q: she makes the bread



Given $P lor lnot Q$, its negation is $lnot (Plor lnot Q) equiv lnot P land Q$, which gives us "I don't make the bread and she makes the bread."







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jul 18 '13 at 17:54

























answered Jul 18 '13 at 16:04









NamasteNamaste

1




1












  • $begingroup$
    Thanks, @Amzoti! Appreciated!
    $endgroup$
    – Namaste
    Jul 19 '13 at 0:52


















  • $begingroup$
    Thanks, @Amzoti! Appreciated!
    $endgroup$
    – Namaste
    Jul 19 '13 at 0:52
















$begingroup$
Thanks, @Amzoti! Appreciated!
$endgroup$
– Namaste
Jul 19 '13 at 0:52




$begingroup$
Thanks, @Amzoti! Appreciated!
$endgroup$
– Namaste
Jul 19 '13 at 0:52











2












$begingroup$

For a sanity check, try to imagine someone saying these sentences in conversation. What would you have to do to prove that they are lying?



For "no dogs have three legs", you would just have to prove the existence of a dog with three legs (hopefully you can find such a dog, rather than having to produce your own example...)






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Sanity check for a police inspector?
    $endgroup$
    – Ilya
    Jul 18 '13 at 16:09
















2












$begingroup$

For a sanity check, try to imagine someone saying these sentences in conversation. What would you have to do to prove that they are lying?



For "no dogs have three legs", you would just have to prove the existence of a dog with three legs (hopefully you can find such a dog, rather than having to produce your own example...)






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Sanity check for a police inspector?
    $endgroup$
    – Ilya
    Jul 18 '13 at 16:09














2












2








2





$begingroup$

For a sanity check, try to imagine someone saying these sentences in conversation. What would you have to do to prove that they are lying?



For "no dogs have three legs", you would just have to prove the existence of a dog with three legs (hopefully you can find such a dog, rather than having to produce your own example...)






share|cite|improve this answer









$endgroup$



For a sanity check, try to imagine someone saying these sentences in conversation. What would you have to do to prove that they are lying?



For "no dogs have three legs", you would just have to prove the existence of a dog with three legs (hopefully you can find such a dog, rather than having to produce your own example...)







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jul 18 '13 at 16:04









citedcorpsecitedcorpse

2,019818




2,019818








  • 1




    $begingroup$
    Sanity check for a police inspector?
    $endgroup$
    – Ilya
    Jul 18 '13 at 16:09














  • 1




    $begingroup$
    Sanity check for a police inspector?
    $endgroup$
    – Ilya
    Jul 18 '13 at 16:09








1




1




$begingroup$
Sanity check for a police inspector?
$endgroup$
– Ilya
Jul 18 '13 at 16:09




$begingroup$
Sanity check for a police inspector?
$endgroup$
– Ilya
Jul 18 '13 at 16:09


















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