How to decide if a proposition is true

• Write a proof

1. To prove a → b, assume a and present an argument for b (→I)

2. If a is proved, and if a → b is proved, then b is proved (→E)

• For example, let's prove (a → b) → (b → c) → (a → c)

1	Assume a → b
2	Assume b → c
3	Assume a
4	From (3) and (1), conclude b	(→E)
5	From (4) and (2), conclude c	(→E)
6	From (3) and (5), conclude a → c	(→I)	(discharges 3)
7	From (2) and (6), conclude (b → c) → (a → c)	(→I)	(discharges 2)
8	From (1) and (7), conclude (a → b) → (b → c) → (a → c)	(→I)	(discharges 1)

• Could we stop at step 7 and conclude (b → c) → (a → c)?

• No, because assumption 1 is still undischarged