1. Semantics of FO Formulas

2. Semantics Given a vocabulary R1, …, Rk A model is D = (D, R1D, …, RkD)
D = a set, called domain, or universe
RiD  D  D  ...  D, (ar(Ri) times) i = 1,...,k

3. Semantics D |= [s] Given:
A model D = (D, R1D, ..., RkD)
A formula 
A substitution s : {x1, x2, ...}  D
We define next the relation: meaning “D satisfies with s”
D |= [s]

4. Semantics D |= (R(t1, ..., tn)) [s] D |= (t= t’) [s]
If (s(t1), ..., s(tn))  RD
D |= (t= t’) [s]
If s(t) = s(t’)

5. Semantics D |= (  ’) [s] D |= (  ’) [s] D |= (’) [s]
If D |= ()[s] and D |= (’) [s]
D |= (  ’) [s]
If D |= ()[s] or D |= (’) [s]
D |= (’) [s]
If not D |= ()[s]

6. First Order Logic: Semantics
D |= (x.) [s]
If for all s’ s.t. s(y) = s’(y) for all
variables y other than x, D |= ()[s’]
D |= (x.) [s]
If for some s’ s.t. s(y) = s’(y) for all
variables y other than x, D |= ()[s’]

7. FO and Databases FO: a sentence  is true in D if D |= 
Databases: a formula  with free variables x1, ..., xn defines the query: (D) = {(s(x1), ..., s(xn)) | D |= [s]}