# What kinds of proofs can be given for axioms, e.g. the modal axiom S5?

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?

[W,R]

t ∈ W

tA

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 tA then t⋄A; ∀t(tRt) Reflexive Property

⋄A → □⋄A; ∀t∀Δ∃Θ(ΘRΔ ∧ ΔRt → ΘRt) Transitive Property

Q.E.D

Perhaps a better question could be: What proofs, if any, can there be given for the S5 Axiom?

https://math.stackexchange.com/questions/3399637/what-could-be-said-of-the-legitimacy-if-you-will-regarding-this-supposed-pr

• This is a little cryptic. What is the justification for each line? Which ones are assumptions and which ones are inferences? – Eliran Oct 19 '19 at 2:15
• To your final question, the authors already state "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." What more are you looking for? – Eliran Oct 19 '19 at 2:18
• That wasn't a useless comment - since the original MSE question linked to in your post is deleted, a link should instead be provided to its new MSE incarnation. – Noah Schweber Oct 21 '19 at 23:31