Geometric Logic and Classifying Topoi

A first-order formula q5(x_l,x_n)is called“geometric”if it is built up from atomic formulas by using conjunction,disjunction,and existential quantification,Geometric logic is the logic of the implications between geometric formulas: 1$$\forall x(\phi (x) \to \psi (x))$$ where the arrow here is for “implication” and and z/) are geometric. Many mathematical structures can be axiomatized by formulas of this form (1).For instance, local rings are axiomatized by the usual equations for a commutative ring with unit, together with the axiom 2$$\forall x,y \in R(x + y = 1 \to \exists z(x \cdot z = 1) \vee \exists z(y \cdot z = 1))$$ which states that the ring is local; this axiom (2) is indeed of the form (1).