2. Completeness and Consistency So far we have 11 rules. PA, and an In and an Out rule for each of 5 connectives. Call this system of 11 rules: P

3. Completeness and Consistency So far we have 11 rules. PA, and an In and an Out rule for each of 5 connectives. Call this system of 11 rules: P Worry: Do we have enough rules? Are there valid arguments that have no proofs in P?

4. Completeness and Consistency So far we have 11 rules. PA, and an In and an Out rule for each of 5 connectives. Call this system of 11 rules: P There is a reason to worry. Irvin Copi’s first book had 25 rules, but there are valid arguments that can’t be proven in it. Worry: Do we have enough rules? Are there valid arguments that have no proofs in P?

5. Completeness and Consistency So far we have 11 rules. PA, and an In and an Out rule for each of 5 connectives. Call this system of 11 rules: P Luckily we can prove that P is has all the rules its needs. If an argument is valid then it is provable in P. Worry: Do we have enough rules? Are there valid arguments that have no proofs in P?

6. Completeness and Consistency Definition of Completeness: A system is complete iff If an argument is valid then it can be proven in the system.

7. Completeness and Consistency Definition of Completeness: A system is complete iff If an argument is valid then it can be proven in the system. Luckily we can prove that P is complete: If an argument is valid then it is provable in P. Unfortunately Copi’s 25 rule system is not complete, despite its large number of rules.

8. Completeness and Consistency A second worry: Can you prove any invalid arguments in P? (That would be a disaster!)

9. Completeness and Consistency A second worry: Can you prove any invalid arguments in P? (That would be a disaster!) You might think we are safe: Each rule is clearly valid, and proofs use only valid arguments.

10. Completeness and Consistency A second worry: Can you prove any invalid arguments in P? (That would be a disaster!) You might think we are safe: Each rule is clearly valid, and proofs use only valid arguments. However more than once in history logicians have developed systems of rules that allowed one to prove everything!

11. Completeness and Consistency Definition of Consistency A system is consistent iff If an argument can be proven in the system, then it is valid.

12. Completeness and Consistency Definition of Consistency A system is consistent iff If an argument can be proven in the system, then it is valid. Fortunately it has been verified that P never proves invalid arguments.

13. Completeness and Consistency A system is consistent iff If an argument can be PROVEN in the system, then it is VALID. A system is complete iff If an argument is VALID then it can be PROVEN in the system.

14. Completeness and Consistency A system is consistent iff If an argument can be PROVEN in the system, then it is VALID. A system is complete iff If an argument is VALID then it can be PROVEN in the system. V>PV>P

15. Completeness and Consistency A system is consistent iff If an argument can be PROVEN in the system, then it is VALID. A system is complete iff If an argument is VALID then it can be PROVEN in the system. V>PV>P P>VP>V

16. Completeness and Consistency A system is consistent iff If an argument can be PROVEN in the system, then it is VALID. A system is complete iff If an argument is VALID then it can be PROVEN in the system. V>PV>P P>VP>V V <> P For more click here