Proofs (3) Quantified Modal Logic

2023-2 Modal Logic (Segment 2)

 

Q) Quantification theory may be formalized this way: 

Prove the following in the systems indicated.

1. ├ L(∀x)ϕx . ⊃. (∀x)Lϕx

2. ├ β (∀x)Lϕx . ⊃. L(∀x)ϕx (Recall that system β is T + p ⊃ LMp. )

3. ├ M(∃x)ϕx . ⊃. (∃x)Mϕx (The system is QML + (∀x)Lϕx . ⊃. L(∀x)ϕx)

4. ├ (∀x)Lα ⊃ L(∀x)α (The system is QML+ (∀x)L(α ⊃ β) .⊃. L((∀x)α ⊃ (∀x)β)