Logic: Lecture 9, 1 September 2015 -------------------------------- Witnessing expansion L = (R,F,C) L_0 = L, C_0 = emptyset C_1 = {c_phi | Ex phi in Formulas(L_0)} L_1 is (R,F,C U C_1) C_2 = {c_phi | Ex phi in Formulas(L_1)} ... Assume we have L_n = (R,F,C U C_1 U ... U C_n) C_n+1 = {c_phi | Ex phi in Formulas(L_n) - Formulas(L_n-1)} L_n+1 is (R,F,C U C_1 U ... U C_n U C_n+1) C_H = C_1 U ... U C_n ... L_H = (R,F,C U C_H) Henkin axiom: Ex phi(x) => phi(c_phi(x)) Quantifer axiom: phi(t) => Ex phi(x), t closed Equality axioms: reflexitivy, symmetry and transitivity for all terms functions: equal arguments given equal results relations: equal tuples have equal membership properties Phi_H : all instances of Henkin axioms Phi_Q : all instances of quantifier axioms Phi_Eq : all instances of equality axioms Phi_H and Phi_Q are sentences. Phi_Eq are not sentences but are true under all assignments. ---------------------------------------------------------------------- FO satisfiability Theorem: Let X be a set of formulas over L. The following are equivalent. 1. There is an L-interpretation M = (S,i,sigma) that satisfies X 2. There is an L_H interprestructure M' = (S,i',sigma') that satisfies X 3. X U Phi_H U Phi_Q U Phi_Eq is propositonally satisfiable (2 => 1) Immediate (restrict i' over C U C_H to i over C) (1 => 2) Need to extend M = (S,i,sigma) to M' = (S,i',sigma') to cover L_H i'(c) = i(c) for c in C For c = c_phi in C_H, if M |= Ex phi, map i'(c_phi) to a s.t. I |= phi(a) otherwise map i'(c_phi) arbitrarily (need to do this in stages for C_1, C_2, ...) (2 => 3) From M' we derive v such that for every prime formula phi, v(phi) = tt iff M' |= phi It follows that v |= X U Phi_H U Phi_Q U Phi_Eq (3 => 2) We have a valuation v over all prime formulas in L_H such that v |= X U Phi_H U Phi_Q U Phi_Eq Need to define an L_H structure M = (S,i) that satisfies X First S: Define and equivalence relation on terms, t ~ t' iff v |= t = t' v |= Phi_Eq guarantees that ~ is an equivalence relation Elements of S are equivalence classes [c] for c in C U C_H Define i as follows: For C : i(c) = [c] For R : ([c1],...,[cn]) in i(r) iff v |= r(c1,...,cn) For F : Need to identify c such that i(f)([t1],...,[tn]) = [c] To show that such a constant c must exist Define phi(x) to be the formula f(c1,...,cn) = x If v |= Ey phi(y), then v |= f(c1,...,cn) = c_phi (Phi_H) If not, the quantifier axiom says phi(f(c1,...,cn)) -> Ey phi(y) Since not(v |= Ey phi(y)), we must have not(v |= phi(f(c1,...,cn))) But phi(f(c1,...,cn)) is just f(c1,...,cn) = f(c1,...,cn) which is an instance of the equality axiom t = t! The fact that interpretations of F and R are compatible with ~ follows from Phi_Eq (to be continued ...) ======================================================================