General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Purely formal derivations are true in the real world!
c○: Michael Kohlhase 182
2.6.4 Proof Theory for the Hilbert Calculus
We now show one of the meta-properties (soundness) for the Hilbert calculus H 0 . The statement
of the result is rather simple: it just says that the set of provable formulae is a subset of the set of
valid formulae. In other words: If a formula is provable, then it must be valid (a rather comforting
property for a calculus).
H 0 is sound (first version)
Theorem 315 ⊢ A implies |=A for all propositions A.
Proof: show by induction over proof length
P.1 Axioms are valid (we already know how to do this!)
P.2 inference rules preserve validity (let’s think)
P.2.1 Subst:
complicated, see next slide
P.2.2 MP:
P.2.2.1 Let A ⇒ B be valid, and ϕ: Vo → {T, F} arbitrary
P.2.2.2 then Iϕ(A) = F or Iϕ(B) = T (by definition of ⇒).
P.2.2.3 Since A is valid, Iϕ(A) = T = F, so Iϕ(B) = T.
P.2.2.4 As ϕ was arbitrary, B is valid.
c○: Michael Kohlhase 183
To complete the proof, we have to prove two more things. The first one is that the axioms are
valid. Fortunately, we know how to do this: we just have to show that under all assignments, the
axioms are satisfied. The simplest way to do this is just to use truth tables.
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