Proofs (4) Free Logic

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)