2023-2 Modal Logic (Segment 2)
Q)
UG, MP. E! is primitive, ∼ , ∨ are primitive and we have the usual definitions.
Sketch proofs of:
① ├FL (∀x)( E!x ≡ (∃y)(x = y)).
② ├FL (∀y) ((∀x) ϕx ⊃ ϕy).
Now consider the system FL* which is just like the above except that (∀y) ((∀x) ϕx ⊃ ϕy) replaces axiom schemas 2 and 3, and E! is dropped as a primitive sign.
Sketch proofs of:
③ ├FL* (∀x) (∃y)(x = y)
④ ├FL* (∀x)Bx & (∃y)(t = y) .⊃. Bt, where t is free x in B.
⑤ Why doesn’t Free Logic embrace ϕ(tz)(ϕz)?
Hint: (tz)(z≠z) ≠ (tz)(z≠z)
