introductory logic phi 120

Introductory LogicPHI 120

Presentation: “Double Turnstile Problems"

Homework• Proofs: 1.5.1 (A/H, p.29-30)

– S21 – S24 (v ->)

– S25 – S27 (the dilemmas)

– S44 (Imp/Exp)

• External Web Pages: – “R. Smith Guide: Proofs without tears”• available through class web page

->I and RAA

Internalize These Strategies

->I1. Assume antecedent of

the conclusion

2. Solve for the consequent

3. Apply ->I rule

RAA1. Assume the denial of

what you’re solving for

2. Derive a contradiction

3. Apply RAA rule

P v Q ~P -> Q⊣⊢Double Turnstile Problems

P v Q ⊣⊢ ~P -> Q

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢ ~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢ ~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A

(2)

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P ⊢ -> Q1 (1) P v Q A

(2) ??

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P ⊢ -> Q1 (1) P v Q A

(2) ??

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ⊢ ~P -> Q1 (1) P v Q A

(2) ??

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A

(2) ??

~P -> Q P v Q⊢

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ⊢ ~P -> Q1 (1) P v Q A2 (2) ~P A

~P -> Q P v Q⊢

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ~P⊢ -> Q1 (1) P v Q A2 (2) ~P A

~P -> Q P v Q⊢We now have too many assumptions!

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ~P -> ⊢ Q1 (1) P v Q A2 (2) ~P A

(3) ??

~P -> Q P v Q⊢

Phase II: Solve for consequent

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A

(3) ??

~P -> Q P v Q⊢

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A

(3) Q 1,2 vE

~P -> Q P v Q⊢

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE

~P -> Q P v Q⊢

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE

(4) ??

~P -> Q P v Q⊢

Strategy of ->I1. Assume the antecedent of the conclusion2. Solve for the consequent (as a conclusion)3. Apply ->I rule.

Phase III: Apply ->I rule

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE

(4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE

(4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE

(4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢

1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A

(2)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P ⊢ v Q1 (1) ~P -> Q A

(2)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A

(2) ??

Look at the premise in relation to the conclusion?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q ⊢ P v Q1 (1) ~P -> Q A

(2) ??

Look at the premise in relation to the conclusion?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A

(2) A

Assume what?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A

The antecedent of (1)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A

(3)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A

(3) Q 1,2 ->E

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A

(3) Q 1,2 ->E

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q ⊢ P v Q1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E

(4) ??

Make the wedge (i.e., the conclusion)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI

1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI

Too many assumptions!!!!

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI

To discharge assumptions:->I or RAA?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI

(5) A

Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI

(5) ~(P v Q) A

Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6)Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6) 4,5 RAA(?) Strategy of RAA1. Assume the denial of the conclusion2. Derive a contradiction3. Use RAA to deny/discharge an assumption

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6) 4,5 RAA(?) Which assumption should you discharge first?• 1, 2, or 5

Assumptions

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6) 4,5 RAA(?)

not [1]

Which assumption should you discharge first?• 1, 2, or 5

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6) 4,5 RAA(?) not [5]

Which assumption should you discharge first?• 1, 2, or 5

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6) 4,5 RAA(2)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A

(6) P 4,5 RAA(2)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2)

(7) ??

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vILook at your assumptions

1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI

(8) ??

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI

(8) 5,7 RAA(5)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI

(8) P v Q 5,7 RAA(5)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI

(8) P v Q 5,7 RAA(5)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)

1. Is the final line the main conclusion?2. Are the assumptions correct on this final line?

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)

Typical Structure for multiple RAA

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)

m,n RAA (k)

P v Q ⊣⊢ ~P -> QP v Q ~P -> Q⊢1 (1) P v Q A2 (2) ~P A1,2 (3) Q 1,2 vE1 (4) ~P -> Q 3 ->I(2)

~P -> Q P v Q⊢1 (1) ~P -> Q A2 (2) ~P A1,2 (3) Q 1,2 ->E1,2 (4) P v Q 3 vI5 (5) ~(P v Q) A1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI1 (8) P v Q 5,7 RAA(5)

Homework• Proofs: 1.5.1 (A/H, p.29-30)

– S21 – S24 (v ->)

– S25 – S27 (the dilemmas)

– S44 (Imp/Exp)

• External Web Pages: “R. Smith Guides”– available through class web page