If Φ⊢¬(ϕ→ψ) , show that Φ,¬ϕ and Φ, ψ are both inconsistent.
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So far I have shown that Φ, ψ is inconsistent:
If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ)
By the axiom ψ→ϕ→ψ and Modus Ponens, Φ, ψ⊢ϕ→ψ.
So Φ, ψ is inconsistent.
Could anyone help me to prove that Φ,¬ϕ is inconsistent?
propositional-calculus
asked Nov 18 at 8:03
mdryizk
153
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up vote
-1
down vote
favorite
So far I have shown that Φ, ψ is inconsistent:
If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ)
By the axiom ψ→ϕ→ψ and Modus Ponens, Φ, ψ⊢ϕ→ψ.
So Φ, ψ is inconsistent.
Could anyone help me to prove that Φ,¬ϕ is inconsistent?
propositional-calculus
asked Nov 18 at 8:03
mdryizk
153
I do not get the gist of your question: what is $Phi$, $phi$ and $psi$? Which conditions hold for them? $Phi,psivdashlnot(phitopsi)$ is the only condition you have assumed?
– Hanul Jeon
Nov 18 at 8:06
I missed a step here. If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ).
– mdryizk
Nov 18 at 8:25
Hint: $Phivdash philand lnotpsi$.
– Hanul Jeon
Nov 18 at 8:26
Thank you for the hint. I think i know what you are trying to tell me. But i'm looking for a proof using only ¬ and →.
– mdryizk
Nov 18 at 8:30
1
@mdryizk If you're looking for a proof in a particular formal system you should indicate this in your question.
– spaceisdarkgreen
Nov 18 at 8:48
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
So far I have shown that Φ, ψ is inconsistent:
If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ)
By the axiom ψ→ϕ→ψ and Modus Ponens, Φ, ψ⊢ϕ→ψ.
So Φ, ψ is inconsistent.
Could anyone help me to prove that Φ,¬ϕ is inconsistent?
propositional-calculus
asked Nov 18 at 8:03
mdryizk
153
So far I have shown that Φ, ψ is inconsistent:
If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ)
By the axiom ψ→ϕ→ψ and Modus Ponens, Φ, ψ⊢ϕ→ψ.
So Φ, ψ is inconsistent.
Could anyone help me to prove that Φ,¬ϕ is inconsistent?
propositional-calculus
propositional-calculus
asked Nov 18 at 8:03
mdryizk
153
asked Nov 18 at 8:03
mdryizk
153
edited Nov 18 at 9:11
asked Nov 18 at 8:03
mdryizk
153
asked Nov 18 at 8:03
mdryizk
153
asked Nov 18 at 8:03
mdryizk
153
153
I do not get the gist of your question: what is $Phi$, $phi$ and $psi$? Which conditions hold for them? $Phi,psivdashlnot(phitopsi)$ is the only condition you have assumed?
– Hanul Jeon
Nov 18 at 8:06
I missed a step here. If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ).
– mdryizk
Nov 18 at 8:25
Hint: $Phivdash philand lnotpsi$.
– Hanul Jeon
Nov 18 at 8:26
Thank you for the hint. I think i know what you are trying to tell me. But i'm looking for a proof using only ¬ and →.
– mdryizk
Nov 18 at 8:30
1
@mdryizk If you're looking for a proof in a particular formal system you should indicate this in your question.
– spaceisdarkgreen
Nov 18 at 8:48
add a comment |
I do not get the gist of your question: what is $Phi$, $phi$ and $psi$? Which conditions hold for them? $Phi,psivdashlnot(phitopsi)$ is the only condition you have assumed?
– Hanul Jeon
Nov 18 at 8:06
I missed a step here. If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ).
– mdryizk
Nov 18 at 8:25
Hint: $Phivdash philand lnotpsi$.
– Hanul Jeon
Nov 18 at 8:26
Thank you for the hint. I think i know what you are trying to tell me. But i'm looking for a proof using only ¬ and →.
– mdryizk
Nov 18 at 8:30
1
@mdryizk If you're looking for a proof in a particular formal system you should indicate this in your question.
– spaceisdarkgreen
Nov 18 at 8:48
I do not get the gist of your question: what is $Phi$, $phi$ and $psi$? Which conditions hold for them? $Phi,psivdashlnot(phitopsi)$ is the only condition you have assumed?
– Hanul Jeon
Nov 18 at 8:06
I do not get the gist of your question: what is $Phi$, $phi$ and $psi$? Which conditions hold for them? $Phi,psivdashlnot(phitopsi)$ is the only condition you have assumed?
– Hanul Jeon
Nov 18 at 8:06
I missed a step here. If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ).
– mdryizk
Nov 18 at 8:25
I missed a step here. If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ).
– mdryizk
Nov 18 at 8:25
Hint: $Phivdash philand lnotpsi$.
– Hanul Jeon
Nov 18 at 8:26
Hint: $Phivdash philand lnotpsi$.
– Hanul Jeon
Nov 18 at 8:26
Thank you for the hint. I think i know what you are trying to tell me. But i'm looking for a proof using only ¬ and →.
– mdryizk
Nov 18 at 8:30
Thank you for the hint. I think i know what you are trying to tell me. But i'm looking for a proof using only ¬ and →.
– mdryizk
Nov 18 at 8:30
1
1
@mdryizk If you're looking for a proof in a particular formal system you should indicate this in your question.
– spaceisdarkgreen
Nov 18 at 8:48
@mdryizk If you're looking for a proof in a particular formal system you should indicate this in your question.
– spaceisdarkgreen
Nov 18 at 8:48
add a comment |
1 Answer
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From the same reasoning you used for the first part, $$ Phi,lnotphi vdash lnot psi to lnot phi.$$ By the contrapositive axiom (theorem? you haven't told us which system you are using... it seems clear that it's Hilbert style from context, but there are several commonly used versions of Axiom 3), $$ Phi,lnot phivdash (lnotpsito lnot phi)to(phito psi), $$ so by MP $$Phi,lnot phivdash phitopsi $$
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
From the same reasoning you used for the first part, $$ Phi,lnotphi vdash lnot psi to lnot phi.$$ By the contrapositive axiom (theorem? you haven't told us which system you are using... it seems clear that it's Hilbert style from context, but there are several commonly used versions of Axiom 3), $$ Phi,lnot phivdash (lnotpsito lnot phi)to(phito psi), $$ so by MP $$Phi,lnot phivdash phitopsi $$
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
add a comment |
up vote
0
down vote
accepted
From the same reasoning you used for the first part, $$ Phi,lnotphi vdash lnot psi to lnot phi.$$ By the contrapositive axiom (theorem? you haven't told us which system you are using... it seems clear that it's Hilbert style from context, but there are several commonly used versions of Axiom 3), $$ Phi,lnot phivdash (lnotpsito lnot phi)to(phito psi), $$ so by MP $$Phi,lnot phivdash phitopsi $$
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
From the same reasoning you used for the first part, $$ Phi,lnotphi vdash lnot psi to lnot phi.$$ By the contrapositive axiom (theorem? you haven't told us which system you are using... it seems clear that it's Hilbert style from context, but there are several commonly used versions of Axiom 3), $$ Phi,lnot phivdash (lnotpsito lnot phi)to(phito psi), $$ so by MP $$Phi,lnot phivdash phitopsi $$
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
From the same reasoning you used for the first part, $$ Phi,lnotphi vdash lnot psi to lnot phi.$$ By the contrapositive axiom (theorem? you haven't told us which system you are using... it seems clear that it's Hilbert style from context, but there are several commonly used versions of Axiom 3), $$ Phi,lnot phivdash (lnotpsito lnot phi)to(phito psi), $$ so by MP $$Phi,lnot phivdash phitopsi $$
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
answered Nov 18 at 9:01
spaceisdarkgreen
31.7k21552
31.7k21552
add a comment |
add a comment |
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I do not get the gist of your question: what is $Phi$, $phi$ and $psi$? Which conditions hold for them? $Phi,psivdashlnot(phitopsi)$ is the only condition you have assumed?
– Hanul Jeon
Nov 18 at 8:06
I missed a step here. If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ).
– mdryizk
Nov 18 at 8:25
Hint: $Phivdash philand lnotpsi$.
– Hanul Jeon
Nov 18 at 8:26
Thank you for the hint. I think i know what you are trying to tell me. But i'm looking for a proof using only ¬ and →.
– mdryizk
Nov 18 at 8:30
1
@mdryizk If you're looking for a proof in a particular formal system you should indicate this in your question.
– spaceisdarkgreen
Nov 18 at 8:48