How to use the deduction theorem on $vdash neg(arightarrow(a rightarrow b))rightarrowneg(arightarrow b)$ [closed]

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I'm having trouble using the deduction theorem because of the negation out side the parenthesis.



Is it correct to write this as $a,arightarrow(arightarrow b),arightarrow bvdash(arightarrow b)$ after cancelling the negation on both sides and using the deduction theorem?




logic propositional-calculus




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closed as off-topic by Mauro ALLEGRANZA, Xander Henderson, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:54


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Henrik, max_zorn, Mostafa Ayaz
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I don't understand. Do you want to prove $vdash lnot (a to (a to b)) to lnot (a to b)$ using deduction theorem?
    – Taroccoesbrocco
    Jul 24 at 9:11







  • 1




    In general, the fact that $A vdash B$ does not imply that $lnot A vdash lnot B$.
    – Taroccoesbrocco
    Jul 24 at 9:12











  • Yes i want to prove this ,but i want to end up with a set $A$ and a formula $varphi$ such that $Avdash varphi$ using the deduction theorem,then continuing to prove that $Avdash varphi$ using axioms (hilbert's system) instead of working with the original proposition and using the 3 axioms of hilbert's system.
    – user3133165
    Jul 24 at 9:17










  • Good to for to know that now @Taroccoesbrocco ,So what's the correct approach for creating the set $A$ here?
    – user3133165
    Jul 24 at 9:28










  • You have to prove : $ a, (a to b) vdash (a to b)$. Then use DT to get : $vdash (a to b) to (a to (a to b))$ and finally use Contraposition.
    – Mauro ALLEGRANZA
    Jul 27 at 7:40














up vote
1
down vote

favorite












I'm having trouble using the deduction theorem because of the negation out side the parenthesis.



Is it correct to write this as $a,arightarrow(arightarrow b),arightarrow bvdash(arightarrow b)$ after cancelling the negation on both sides and using the deduction theorem?




logic propositional-calculus




share|cite|improve this question











closed as off-topic by Mauro ALLEGRANZA, Xander Henderson, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:54


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Henrik, max_zorn, Mostafa Ayaz
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I don't understand. Do you want to prove $vdash lnot (a to (a to b)) to lnot (a to b)$ using deduction theorem?
    – Taroccoesbrocco
    Jul 24 at 9:11







  • 1




    In general, the fact that $A vdash B$ does not imply that $lnot A vdash lnot B$.
    – Taroccoesbrocco
    Jul 24 at 9:12











  • Yes i want to prove this ,but i want to end up with a set $A$ and a formula $varphi$ such that $Avdash varphi$ using the deduction theorem,then continuing to prove that $Avdash varphi$ using axioms (hilbert's system) instead of working with the original proposition and using the 3 axioms of hilbert's system.
    – user3133165
    Jul 24 at 9:17










  • Good to for to know that now @Taroccoesbrocco ,So what's the correct approach for creating the set $A$ here?
    – user3133165
    Jul 24 at 9:28










  • You have to prove : $ a, (a to b) vdash (a to b)$. Then use DT to get : $vdash (a to b) to (a to (a to b))$ and finally use Contraposition.
    – Mauro ALLEGRANZA
    Jul 27 at 7:40












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I'm having trouble using the deduction theorem because of the negation out side the parenthesis.



Is it correct to write this as $a,arightarrow(arightarrow b),arightarrow bvdash(arightarrow b)$ after cancelling the negation on both sides and using the deduction theorem?




logic propositional-calculus




share|cite|improve this question











I'm having trouble using the deduction theorem because of the negation out side the parenthesis.



Is it correct to write this as $a,arightarrow(arightarrow b),arightarrow bvdash(arightarrow b)$ after cancelling the negation on both sides and using the deduction theorem?





logic propositional-calculus





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 24 at 8:50









user3133165

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1618




closed as off-topic by Mauro ALLEGRANZA, Xander Henderson, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:54


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Henrik, max_zorn, Mostafa Ayaz
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Mauro ALLEGRANZA, Xander Henderson, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:54


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Henrik, max_zorn, Mostafa Ayaz
If this question can be reworded to fit the rules in the help center, please edit the question.











  • I don't understand. Do you want to prove $vdash lnot (a to (a to b)) to lnot (a to b)$ using deduction theorem?
    – Taroccoesbrocco
    Jul 24 at 9:11







  • 1




    In general, the fact that $A vdash B$ does not imply that $lnot A vdash lnot B$.
    – Taroccoesbrocco
    Jul 24 at 9:12











  • Yes i want to prove this ,but i want to end up with a set $A$ and a formula $varphi$ such that $Avdash varphi$ using the deduction theorem,then continuing to prove that $Avdash varphi$ using axioms (hilbert's system) instead of working with the original proposition and using the 3 axioms of hilbert's system.
    – user3133165
    Jul 24 at 9:17










  • Good to for to know that now @Taroccoesbrocco ,So what's the correct approach for creating the set $A$ here?
    – user3133165
    Jul 24 at 9:28










  • You have to prove : $ a, (a to b) vdash (a to b)$. Then use DT to get : $vdash (a to b) to (a to (a to b))$ and finally use Contraposition.
    – Mauro ALLEGRANZA
    Jul 27 at 7:40
















  • I don't understand. Do you want to prove $vdash lnot (a to (a to b)) to lnot (a to b)$ using deduction theorem?
    – Taroccoesbrocco
    Jul 24 at 9:11







  • 1




    In general, the fact that $A vdash B$ does not imply that $lnot A vdash lnot B$.
    – Taroccoesbrocco
    Jul 24 at 9:12











  • Yes i want to prove this ,but i want to end up with a set $A$ and a formula $varphi$ such that $Avdash varphi$ using the deduction theorem,then continuing to prove that $Avdash varphi$ using axioms (hilbert's system) instead of working with the original proposition and using the 3 axioms of hilbert's system.
    – user3133165
    Jul 24 at 9:17










  • Good to for to know that now @Taroccoesbrocco ,So what's the correct approach for creating the set $A$ here?
    – user3133165
    Jul 24 at 9:28










  • You have to prove : $ a, (a to b) vdash (a to b)$. Then use DT to get : $vdash (a to b) to (a to (a to b))$ and finally use Contraposition.
    – Mauro ALLEGRANZA
    Jul 27 at 7:40















I don't understand. Do you want to prove $vdash lnot (a to (a to b)) to lnot (a to b)$ using deduction theorem?
– Taroccoesbrocco
Jul 24 at 9:11





I don't understand. Do you want to prove $vdash lnot (a to (a to b)) to lnot (a to b)$ using deduction theorem?
– Taroccoesbrocco
Jul 24 at 9:11





1




1




In general, the fact that $A vdash B$ does not imply that $lnot A vdash lnot B$.
– Taroccoesbrocco
Jul 24 at 9:12





In general, the fact that $A vdash B$ does not imply that $lnot A vdash lnot B$.
– Taroccoesbrocco
Jul 24 at 9:12













Yes i want to prove this ,but i want to end up with a set $A$ and a formula $varphi$ such that $Avdash varphi$ using the deduction theorem,then continuing to prove that $Avdash varphi$ using axioms (hilbert's system) instead of working with the original proposition and using the 3 axioms of hilbert's system.
– user3133165
Jul 24 at 9:17




Yes i want to prove this ,but i want to end up with a set $A$ and a formula $varphi$ such that $Avdash varphi$ using the deduction theorem,then continuing to prove that $Avdash varphi$ using axioms (hilbert's system) instead of working with the original proposition and using the 3 axioms of hilbert's system.
– user3133165
Jul 24 at 9:17












Good to for to know that now @Taroccoesbrocco ,So what's the correct approach for creating the set $A$ here?
– user3133165
Jul 24 at 9:28




Good to for to know that now @Taroccoesbrocco ,So what's the correct approach for creating the set $A$ here?
– user3133165
Jul 24 at 9:28












You have to prove : $ a, (a to b) vdash (a to b)$. Then use DT to get : $vdash (a to b) to (a to (a to b))$ and finally use Contraposition.
– Mauro ALLEGRANZA
Jul 27 at 7:40




You have to prove : $ a, (a to b) vdash (a to b)$. Then use DT to get : $vdash (a to b) to (a to (a to b))$ and finally use Contraposition.
– Mauro ALLEGRANZA
Jul 27 at 7:40















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