1. No new reading for Monday or Wednesday Exam #2 is next Friday, and we’ll review and work on proofs on Monday and Wed.

2. 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)

3. →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

4. 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.

5. 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.

6. 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).

7. 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. 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.