Mixing
Axiomatic proof of $vdash p rightarrow ((prightarrow q) rightarrow q)$
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I'm trying to solve a question which asks me to prove $vdash p rightarrow ((prightarrow q) rightarrow q)$ using the axiomatic proof system with modus ponens as it's only rule, the axioms
PL1: $phirightarrow (psi rightarrow phi)$
PL2: $(phi rightarrow (psi rightarrow chi))rightarrow((phirightarrowpsi)rightarrow(phi rightarrowchi)) $
PL3: $(text{~}psi rightarrow text{~}phi)rightarrow((text{~}psi rightarrow phi)rightarrowpsi)$
using the deduction theorem for propositional logic (if $Gamma, phi vdash psi$ then $Gamma vdash phi rightarrow psi$).
I'm really struggling with this, so I'd appreciate any help you could offer.
logic propositional-calculus
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
asked Jan 28 at 16:00
yuyuyuyu
31
$endgroup$
add a comment |
$begingroup$
I'm trying to solve a question which asks me to prove $vdash p rightarrow ((prightarrow q) rightarrow q)$ using the axiomatic proof system with modus ponens as it's only rule, the axioms
PL1: $phirightarrow (psi rightarrow phi)$
PL2: $(phi rightarrow (psi rightarrow chi))rightarrow((phirightarrowpsi)rightarrow(phi rightarrowchi)) $
PL3: $(text{~}psi rightarrow text{~}phi)rightarrow((text{~}psi rightarrow phi)rightarrowpsi)$
using the deduction theorem for propositional logic (if $Gamma, phi vdash psi$ then $Gamma vdash phi rightarrow psi$).
I'm really struggling with this, so I'd appreciate any help you could offer.
logic propositional-calculus
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
asked Jan 28 at 16:00
yuyuyuyu
31
$endgroup$
$begingroup$
See math.stackexchange.com/questions/2470676/axiomatic-proofs/…
$endgroup$
– Bram28
Jan 31 at 16:36
add a comment |
$begingroup$
I'm trying to solve a question which asks me to prove $vdash p rightarrow ((prightarrow q) rightarrow q)$ using the axiomatic proof system with modus ponens as it's only rule, the axioms
PL1: $phirightarrow (psi rightarrow phi)$
PL2: $(phi rightarrow (psi rightarrow chi))rightarrow((phirightarrowpsi)rightarrow(phi rightarrowchi)) $
PL3: $(text{~}psi rightarrow text{~}phi)rightarrow((text{~}psi rightarrow phi)rightarrowpsi)$
using the deduction theorem for propositional logic (if $Gamma, phi vdash psi$ then $Gamma vdash phi rightarrow psi$).
I'm really struggling with this, so I'd appreciate any help you could offer.
logic propositional-calculus
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
asked Jan 28 at 16:00
yuyuyuyu
31
$endgroup$
I'm trying to solve a question which asks me to prove $vdash p rightarrow ((prightarrow q) rightarrow q)$ using the axiomatic proof system with modus ponens as it's only rule, the axioms
PL1: $phirightarrow (psi rightarrow phi)$
PL2: $(phi rightarrow (psi rightarrow chi))rightarrow((phirightarrowpsi)rightarrow(phi rightarrowchi)) $
PL3: $(text{~}psi rightarrow text{~}phi)rightarrow((text{~}psi rightarrow phi)rightarrowpsi)$
using the deduction theorem for propositional logic (if $Gamma, phi vdash psi$ then $Gamma vdash phi rightarrow psi$).
I'm really struggling with this, so I'd appreciate any help you could offer.
logic propositional-calculus
logic propositional-calculus
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
asked Jan 28 at 16:00
yuyuyuyu
31
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
asked Jan 28 at 16:00
yuyuyuyu
31
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
edited Jan 28 at 16:04
Mauro ALLEGRANZA
67.5k449117
67.5k449117
asked Jan 28 at 16:00
yuyuyuyu
31
asked Jan 28 at 16:00
yuyuyuyu
31
asked Jan 28 at 16:00
yuyuyuyu
31
31
$begingroup$
See math.stackexchange.com/questions/2470676/axiomatic-proofs/…
$endgroup$
– Bram28
Jan 31 at 16:36
add a comment |
$begingroup$
See math.stackexchange.com/questions/2470676/axiomatic-proofs/…
$endgroup$
– Bram28
Jan 31 at 16:36
$begingroup$
See math.stackexchange.com/questions/2470676/axiomatic-proofs/…
$endgroup$
– Bram28
Jan 31 at 16:36
$begingroup$
See math.stackexchange.com/questions/2470676/axiomatic-proofs/…
$endgroup$
– Bram28
Jan 31 at 16:36
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Hint
Use MP to prove :
$p, p to q vdash q$
and then apply Deduction Th twice.
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
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active
oldest
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oldest
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votes
$begingroup$
Hint
Use MP to prove :
$p, p to q vdash q$
and then apply Deduction Th twice.
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
$endgroup$
add a comment |
$begingroup$
Hint
Use MP to prove :
$p, p to q vdash q$
and then apply Deduction Th twice.
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
$endgroup$
add a comment |
$begingroup$
Hint
Use MP to prove :
$p, p to q vdash q$
and then apply Deduction Th twice.
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
$endgroup$
Hint
Use MP to prove :
$p, p to q vdash q$
and then apply Deduction Th twice.
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
answered Jan 28 at 16:04
Mauro ALLEGRANZAMauro ALLEGRANZA
67.5k449117
67.5k449117
add a comment |
add a comment |
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$begingroup$
See math.stackexchange.com/questions/2470676/axiomatic-proofs/…
$endgroup$
– Bram28
Jan 31 at 16:36