From John Bigelow and Robert Pargetter's book, titled, 'Science and Necessity', they assert the following:
. . . . The resulting system, S5, contains all the theorems of S4 and all the theorems of B and nothing more. Given axiom T plus axiom S5, we can prove S4 and B; but conversely, too, given T plus both S4 and B, we can prove S5. Yet T plus S4 without B does not enable us to prove S5; and T plus B without S4 does not enable us to prove S5. (Axiom S5 is intuitively `If something is possibly true, then it must be true that it is possibly true'.)
On the basis of this statement, it would appear as if there could be a 'proof' of the S5 Axiom. Prior to this statement they sketched 'a fairly orthodox axiomatic system' which introduced items such as: definitions, rules of inference, expositions of Axiom K, Axiom D, Axiom T, etc. and shared a diagram visualizing the system they had just exposited.
Here's the question: there was a classmate of mine who presented a 'proof' of the S5 Axiom, what could be said of the 'legitimacy', if you will, regarding this supposed 'proof' of the S5 Axiom?
t ∈ W
t ⊩ A
ΔRt; Δ is an arbitrary variable accessible to t
Δ ⊩ □A; ∀Δ∀t(ΔRt) Serial Property
ΘRΔ; Θ is an arbitrary variable accessible to Δ
Θ ⊩ ⋄A; ∀Δ∃Θ(ΘRΔ) Serial Property
If t ⊩ A then t ⊩ ⋄A; ∀t(tRt) Reflexive Property
⋄A → □⋄A; ∀t∀Δ∃Θ(ΘRΔ ∧ ΔRt → ΘRt) Transitive Property
Perhaps a better question could be: What proofs, if any, can there be given for the S5 Axiom?
Link to Original Question: