1. Deduction In addition to being able to represent facts, or real- world statements, as formulas, we want to be able to manipulate facts, e.g., derive new facts from a set of statements. For example, if we know that If it is raining, my dog is wet. If my dog is wet, there will be water on the floor. There is no water on the floor. We should be able to conclude that it is not raining.

2. Deduction (cont'd) In propositional form, this argument would look like: P -> Q Q -> R ⌐ R ⌐ P

3. Natural Deduction Is there a formal (a systematic, unambiguous way) of deriving new facts? We could have a set of rules that are apply to formulas that can be used to create (derive) new formulas which follow the original ones. One such rule is Modus Ponens: P -> Q P Q

4. Another Rule Another such rule is called Modus Tollens: P -> Q ⌐ Q ⌐ P We could use two applications of Modus Tollens to show that it is not raining.

5. Conjunction Rules In fact, there are other rules, some very obvious, so not so obvious, that we use to derive new facts from old ones. Three obvious ones deal with conjunction: P P ∧ Q P ∧ Q Q P Q P ∧ Q

6. Disjunction Rules Two simple rules deal with disjunction: P Q P ∨ Q and one not-so-simple rule deals with proof-by-cases, also involving disjunction. If you have P ∨ Q, you can proceed by cases. If you assume that P is true and you can show R, or, alternatively, if you assume that Q is true and you can still prove R, then R must be true in either case.

7. Boxes In the previous slide, the blue box introduces an additional assumption which is at the top of the box. The scope of the assumption is within the box. After the box, the formula is no longer assumed to be true and may not be used. There may be additional formulas inside the box between the first and last lines. Also, boxes may be nested to introduce several assumptions.

8. Implication Rules Two rules involve implication. The second is MP: P P Q P -> Q P -> Q Q

9. Other Rules P P false ⌐P false ⌐⌐P ⌐ P false P P

10. Rule Names In general, for each operator there are two (sets of) rules: an introduction rule which introduces the operator, that is, the operator appears in the conclusion, and an elimination rule which removes an operator, that is, the operator appears in one of the hypotheses. The is one conjunction introduction rule, and two conjunction elimination rules, one disjunction elimination rule (proof by cases) and two disjunction introduction rules.

11. Rules Names (cont'd) Implication elimination is the Modus Ponens rule and there is a corresponding implication introduction (assume that P is true, prove Q, and you have the implication P->Q. The other four rules are negation introduction, negation elimination, contradiction elimination, and double negation elimination.

12. Natural Deduction In all, there are twelve rules (Modens Tollens can be derived from the other rules). These rules codify one approach that people take toward deduction (hence the name Natural Deduction). A natural deduction proof is a sequence of steps where each step is either a hypothesis (formula assumed to be true) or is derived from previous formulas by one of the rules.

13. Simple Proof Here is a proof that if we have hypothesis P->Q, and Q->R, we can derive the formula P->R: 1. P ->Q Hyp. 2. Q-> R Hyp. 3. P Assume 4. Q MP 5. R MP 6. P -> R Implication intro

14. Modus Tollens Modus Tollens is a derived rule, that is, any proof which uses Modus Tollens could be done by using the other rules instead. To see this, consider: 1. P -> Q 2. ⌐ Q 3. P Assume 4. Q -> Elim: 3,1 5. false ⌐ Elim: 4, 2 6. ⌐ P ⌐ Intro : 3-5

15. Derivability Since we have a system of natural deduction, we have a way to derive new facts from a set of given facts (hypotheses). We construct a proof which each line is either a hypothesis or derived from previous lines by a proof rule. The last line of the proof is the conclusion. If such a proof exists, we say that the conclusion C can be derived from the hypotheses H, H ⊢ C (read “H derives C.”)

16. Semantic Entailment We also have a way of relating a formula to a set of formulas which we assume to be true (the hypotheses). If every interpretation which makes the hypotheses true (satisfies the hypotheses), also makes the conclusion true, i.e., whenever the hypotheses are true, the conclusion is also true, we say the the hypotheses semantically entails the conclusion, H ⊨ C.

17. Tautologies and Contradictions A formula that is true under every interpretation is said to be valid and is also called a tautology. A formula that is false under every interpretation is call a contradiction and said to be unsatisfiable. A formula that is true under some interpretation is said to be satisfiable.

18. Proof vs. Truth In short, we have two notions: Proof and Truth. We have facts that we can prove from the set of hypotheses (using the rules of natural deduction) and we have facts that we know are true given that the hypotheses are true (which can check by constructing the truth table). It turns out that for the natural deduction method for proposition calculus, these two notions coincide.

19. Soundness and Completeness If we can prove formula C from the set of hypotheses H, then H must also semantically entail C. In other words, if we can prove it, it must be true. This is known as soundness. If C is semantically entailed from H, then there is a proof of C from hypotheses H. In other words, if it is true, we can prove it. This is known as completeness.

20. Are All Systems Sound and/or Complete? No. Consider a system with one rule: P that is, you can prove anything. Such a system is complete (if it's true you can prove it because you can prove anything), but it is not sound (you can prove things that aren't true). To construct a system that is not sound, delete one of the twelve rules of natural deduction.