1. For Wednesday, read Chapter 4, section 3 (pp. 100-101)
Nongraded Homework: Exercises on p. 102.
Graded Homework #4 is due on Friday at the beginning of class.

2. Natural Deduction Proofs (proofs)
You are given a valid argument and you must reason, in a step-by-step fashion, from the premises to the conclusion.
Each step in the argument must be justified by a rule of inference. Section 2 introduces the first five.

3. --Premises are put on a vertical list; lines are numbered (numbers in parentheses);
--add new lines to the numbered list until you reach the conclusion;
--every line must have a justification to its right. This is either ‘Premise’, ‘Assumption’, or the statement of a rule of inference used to generate that line, along with the numbers of the previous lines to which the rule of inference was applied.

4. --each line must also have a listing of dependence numbers; these are off to the far left; these are not necessarily the lines to which a rule has been applied; they are the lines from which the new line was ultimately derived;
--a proof has been completed when, and only when, one reaches a line that matches the argument’s conclusion and that depends only on the argument’s premises.

5. Simple version of ampersand introduction (&I)q
\ p & q
Forbes states these rules in a precise schematic form, which makes clear which dependence lines to include on your new line. For the new line, include all of the dependence numbers (without redundancy) of the lines to which the rule that generated your new line was applied.

6. Official version of ampersand introduction (&I):
a1,...,an (j) p .
b1,...,bu (k) q
a1,...,an, b1,...,bu (m) p & q j, k &I
Be sure you eventually understand what EVERY variable in this schema refers to.

7. Simple version of ampersand elimination (&E):
p & q
\ p OR
\ 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.

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

9. 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.

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

11. 1 (1) (A & E) → F Premise
2 (2) E Premise
3 (3) F → ~ C Premise
4 (4) A Premise \ ~ C

12. 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 your goal-arrow (derived from the assumption line plus the line on which the consequent of your goal-arrow appears)
--eliminate the assumption’s dependence number from the new line’s dependence numbers

13. Problems on p. 99