1. Reading: Chapter 4, section 4 Nongraded Homework: Problems at the end of section 4. Graded Homework #4 is due at the beginning of class on Friday. You will need to use the rule ~Introduction (from section) 4 for the last problem on the HW.

2. Simple version of ampersand elimination (&E): p & q  p p p pOR  q q q q Any well-formed formula can be put in for p or q (even the same formula for both), but assignments must be consistent throughout the application of the rule.

3. Simple version of arrow elimination (→E) p → q p  q q q q Rule of assumptions: Any wff can be entered on a new line with only its own line number as its dependence line.

4. Important restriction on the use of our rules of inference: They must be applied to entire lines; that is, for any line in the proof that is supposed to match up with a line in the rule (premise or conclusion), the entire line in the proof must match up with the entire line in the rule. In other words, only introduce or eliminate a main connective.

5. 1 (1) APremise 2 (2) A → (B → C)Premise 3 (3) B Premise  C 1,2 (4) B → C 1, 2 →E 1,2,3 (5) C 3, 4 →E On line four it would have been “illegal” to write C (thinking that you were applying →E).

6. 1 (1) (A & E) → DPremise 2 (2) EPremise 3 (3) D → ~ C Premise 4 (4) APremise  ~ C

7. Arrow Introduction →I: --Use the rule of assumptions to add a line that matches the antecedent of an arrow statement you would like to derive; --work until you reach a line that matches the consequent of your goal-arrow; --on a new line, write a new arrow-statement (the assumption line, as antecedent, plus the line on which the consequent of your goal-arrow appears, as consequent) --eliminate the assumption’s dependence number from the new line’s dependence numbers (this is called ‘discharging’ the assumption)

8. →Introduction j (j) p Assumption.. a 1,…,a n (k) q.. {a 1,…,a n }/j (m) p → q j, k →I j > k, j k, j < k, or j = k

9. What Do the Symbols Mean? To say that j > k or j = k is to say that the assumption can come after the line that becomes the consequent or that j and k can be the very same line. a 1,…,a n refers to the lines on which the thing that becomes the consequent depends. {a 1,…,a n }/j means “remove j from that set, if it’s in there” The line that becomes the antecedent is always an assumption. As an assumption, it depends only on itself.

10. Exercises on pp. 98-99, 13 and 15

11. Semantic vs. Deductive Consequences ‘p1…pn |= q’ says that it is impossible for p1…pn to be true while q is false. This double-turnstile says that the statement on the right is a semantic consequence of the statement(s) on the left. ‘p1…pn |- q’ (which is called a ‘sequent’) says that q can be derived from p1…pn using some particular natural deduction system (NK, in our case). It says that q is a deductive consequence of p1…pn.

12. Proving Theorems So, ‘|- q’, with no premises given on the left, means that q can be derived within our system from no premises at all. Statements that can be derived from no premises are the theorems of our natural deduction system. In sentential logic, the set of theorems is identical to the set of tautologies (assuming we have a complete natural deduction system).

13. How to Prove Theorems Always start by making an assumption. Let the conclusion (the theorem) be your guide. If the theorem is a conditional, start by assuming its entire antecedent. THIS ASSUMPTION IS NOT DERIVED FROM THE CONCLUSION. DO NOT TREAT THE THEOREM AS A PREMISE IN YOUR ARGUMENT. Then proceed with your proof, making other assumptions where necessary. When you arrive at the desired theorem, if you’ve done the proof properly, it should have no dependence lines listed off to its left.

14. Exercises on pp. 102, 5