Logic: Lecture 11, 8 September 2015
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A complete axiomatization
(A1) All tautologies of PC
(A2a) x = x
(A2b) x = y => phi(x) = phi(y), where phi(u) is an atomic formula
(A3) phi(t) => Ex phi(x)
alpha, alpha => beta
(MP) --------------------
beta
phi(x) => psi
(G) -------------------- x not in FV(psi)
Ex phi(x) => psi
(A1 at the level of prime formulas)
Define X |- phi as usual.
Can derive equality axioms (see notes)
Soundness: If X |- phi then X |= phi
Proof:
By induction. Check that G preserves validity
Assume phi(x) => psi is valid. For every I = (M,sigma),
I |= phi(x) => psi
Let I' = (M',sigma') such that I' |= Ex phi(x). Then, for some
a in S', I'[x |-> a] |= phi(x). Since phi(x) => psi is
valid, I'[x |-> a] |= psi. But x not in FV(psi), so
I' |= psi as well.
To show completeness
First use PC and (G) to show that the following hold
(1) If X |- phi -> psi and X |- ~phi -> psi then |- psi
(2) If X |- (phi -> theta) -> psi then X |- ~phi -> psi and
X |- theta -> psi
(3) If x not in FV(psi) and X |- (Ey phi(y) -> phi(x)) -> psi,
then X |- psi
Proof of (3)
Suppose X |- (Ey phi(y) -> phi(x)) -> psi,
By (2), X |- ~Ey phi(y) -> psi, X |- phi(x) -> psi
X |- phi(x) via (G) plus renaming gives X |- Ey phi(y) -> psi
Use (A3) of PC to get X |- psi
Now we have
Completeness : If X |= phi then X |- phi
Proof:
X u {~phi} not satisfiable. Some finite subset Y of
X U Phi_H U Phi_Q U Phi_Eq such that Y U {~phi} not satisfiable.
List Y as {a1,..,am,b1,..,bk} where ai are from X, Phi_Q and
Phi_Eq and b1 are from Phi_H in decreasing order of rank.
|= (a1 -> (a2 -> .... (b1 -> (b2 -> ... ) -> phi)
Obtain ai', bj' by replacing witnessing constants by distinct variables.
|= (a1' -> (a2' -> .... (b1' -> (b2' -> ... ) -> phi')
But phi' = phi, since phi is in L.
Hence, by completeness of PC
|- (a1' -> (a2' -> .... (b1' -> (b2' -> ... ) -> phi)
ai' are from X (so ai' = ai) or from Phi_Q (axiom) or Phi_Eq
(derivable), so eliminate by MP to get
|- (b1' -> (b2' -> ... ) -> phi)
Apply previous result part (3) m times to eliminate the bi's.
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Quick introduction to sequent calculus
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