# Return to Question

2 replaced all greek letters with regular characters

# Prove the rule that proves 𝜒X(𝜙P) from 𝜒X(𝛼a) preserves derivability in modal system K

I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving 𝜒X(𝜙P) from 𝜒X(𝛼a) preserves derivability (i.e. if ⊢𝜒⊢X(𝛼a) then ⊢𝜒⊢X(𝜙P) in modal system K, where 𝛼a is a sentence letter which occurs zero or more times in an MPL-wff 𝜒X(𝛼a), and 𝜒X(𝜙P) is the result of replacing every instance of 𝛼a with the wff 𝜙P.

This would be relatively easy to do semantically and then use completeness of K to give the desired result, but the problem asks me to do this without making use of completeness.

In system K, I have the axioms

PL1: 𝜙→P→(𝜓→𝜙Q→P)

PL2: (𝜙→P→(𝜓→𝜒Q→R))→((𝜙→𝜓P→Q)→(𝜙→𝜒P→R))

PL3: (~𝜓→~𝜙~Q→~P)→((~𝜓→𝜙~Q→P)→𝜓→P)

K: ◻(𝜙→𝜓P→Q)→(◻𝜙→◻𝜓◻P→◻Q)

and the rules modus ponens and necessitation.

I really can't see how I can go from 𝜒X(𝛼a) to 𝜒X(𝜙P) using these, so I'd appreciate any help you could offer.

EDIT: Have replaced all greek letters with regular characters: note P, Q, R here represent arbitrary wffs and not just sentence letters

# Prove the rule that proves 𝜒(𝜙) from 𝜒(𝛼) preserves derivability in modal system K

I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving 𝜒(𝜙) from 𝜒(𝛼) preserves derivability (i.e. if ⊢𝜒(𝛼) then ⊢𝜒(𝜙) in modal system K, where 𝛼 is a sentence letter which occurs zero or more times in an MPL-wff 𝜒(𝛼), and 𝜒(𝜙) is the result of replacing every instance of 𝛼 with the wff 𝜙.

This would be relatively easy to do semantically and then use completeness of K to give the desired result, but the problem asks me to do this without making use of completeness.

In system K, I have the axioms

PL1: 𝜙→(𝜓→𝜙)

PL2: (𝜙→(𝜓→𝜒))→((𝜙→𝜓)→(𝜙→𝜒))

PL3: (~𝜓→~𝜙)→((~𝜓→𝜙)→𝜓)

K: ◻(𝜙→𝜓)→(◻𝜙→◻𝜓)

and the rules modus ponens and necessitation.

I really can't see how I can go from 𝜒(𝛼) to 𝜒(𝜙) using these, so I'd appreciate any help you could offer.

# Prove the rule that proves X(P) from X(a) preserves derivability in modal system K

I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving X(P) from X(a) preserves derivability (i.e. if ⊢X(a) then ⊢X(P) in modal system K, where a is a sentence letter which occurs zero or more times in an MPL-wff X(a), and X(P) is the result of replacing every instance of a with the wff P.

This would be relatively easy to do semantically and then use completeness of K to give the desired result, but the problem asks me to do this without making use of completeness.

In system K, I have the axioms

PL1: P→(Q→P)

PL2: (P→(Q→R))→((P→Q)→(P→R))

PL3: (~Q→~P)→((~Q→P)→P)

K: ◻(P→Q)→(◻P→◻Q)

and the rules modus ponens and necessitation.

I really can't see how I can go from X(a) to X(P) using these, so I'd appreciate any help you could offer.

EDIT: Have replaced all greek letters with regular characters: note P, Q, R here represent arbitrary wffs and not just sentence letters

1

# Prove the rule that proves 𝜒(𝜙) from 𝜒(𝛼) preserves derivability in modal system K

I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving 𝜒(𝜙) from 𝜒(𝛼) preserves derivability (i.e. if ⊢𝜒(𝛼) then ⊢𝜒(𝜙) in modal system K, where 𝛼 is a sentence letter which occurs zero or more times in an MPL-wff 𝜒(𝛼), and 𝜒(𝜙) is the result of replacing every instance of 𝛼 with the wff 𝜙.

This would be relatively easy to do semantically and then use completeness of K to give the desired result, but the problem asks me to do this without making use of completeness.

In system K, I have the axioms

PL1: 𝜙→(𝜓→𝜙)

PL2: (𝜙→(𝜓→𝜒))→((𝜙→𝜓)→(𝜙→𝜒))

PL3: (~𝜓→~𝜙)→((~𝜓→𝜙)→𝜓)

K: ◻(𝜙→𝜓)→(◻𝜙→◻𝜓)

and the rules modus ponens and necessitation.

I really can't see how I can go from 𝜒(𝛼) to 𝜒(𝜙) using these, so I'd appreciate any help you could offer.