1. Inference in First-Order Logic Inference Rules with Quantifiers The three new inference rules are as follows: Universal Elimination: For any sentence , variable v and ground term g:  v  _____________ SUBST({v/g},  ) For example, from  x Likes(x,IceCream), we can use the substitution {x/Ben} and infer Likes(Ben, IeeCream).

2. Existential Elimination: For any sentence a, variable v and constant symbol k that does not occur elsewhere in the knowledge base:  v  _____________ SUBST({v/k},  ) For example, from  x Kill(x, Victim), we can infer Kill(Murderer, Vctim), as long as Murderer does not appear elsewhere in the knowledge base.

3. Existential Introduction: For any sentence a, variable v that does not occur in a and ground term g that does occur in a :  _______________  v SUBST({v/g},  ) For example, from Likes(Jerry, IceCream) we can infer  x Likes(x, IceCream).

4. An Example Proof One can imagine following situation: The law says that it is a crime to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles and all of it missiles were sold by Colonel West, who is American. Next it is proved that West is a criminal: 1.  x, y, z American(x)  Weapon(y)  Nation(z)  Hostile(z)  Sells(x, y, z)  Criminal(x) “The law says that it is a crime for an American to sell weapons to hostile nations” “Nono … has some missiles”  x Owns(Nono, x)  Missile(x) “All of its missile were sold to it by colonel West”  x Owns(Nono, x)  Missile(x)  Sells(West, Nono, x) We will also need to know that missiles are weapons:  x Missile(x)  Weapon(x) And that an enemy of america counts as “hostile”  x Enemy(x, America)  Hostile(x) “West, who is American …..” 6. American(West)

5. “The country Nono…..” 7. Nation(Nono) “Nono, an enemy of America …” 8. Enemy(Nono, America) 9. Nation(America) The proof consists of a series of applications of the inference rules: From (2) and Existential Elimination: 10. Owns(Nono, M1)  Missile(M1) From (10) and And-Elimination: 11. Owns(Nono, M1) 12. Missile(M1) From (4) and Universal Elimination : 13. Missile(M1)  Weapon(M1) From (12), (13), and Modus Ponens: 14. Weapon(M1)

6. From (3) and Universal-Elimination : 15. Owns(Nono, M1)  Missile(M1)  Sells(West, Nono, M1) From (15), (10), and Modus Ponens : 16. Sells(West, Nono, M1) From (1) and Universal Elimination (three times): 17. American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono)  Sells(West, Nono, M1)  Criminal(West) From (5) and Universal Elimination : 18. Enemy(Nono, America)  Hostile(Nono) From (8), (18), and Modus Ponens : 19. Hostile(Nono) From (6), (7), (14), (16), (19), and And- Introduction: 20.American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono)  Sells(West, Nono, M1) From (17), (20), and Modus Ponens : 21. Criminal(West)