1. Propositional Logic
8) Iff
Copyright 2008, Scott Gray

2. Just What is “Iff”? Iff == if and only if This is a biconditional
It is the equivalent of the conjunction of two conditionals
Example:
I will play a game of pinball if and only if I have a quarter
(P → Q) & (Q → P)
P ↔ Q
Copyright 2008, Scott Gray

3. Synonymns if and only if is a necessary and sufficient condition for
just in case
exactly if (only common in logic)
Note that casual usage sometimes finds “if” but means “if and only if”
I will make the flight if I rush
I will make the flight if and only if I rush
Copyright 2008, Scott Gray

4. Iff Commutative A ↔ B is equivalent to B ↔ A
We will prove this once we learn the double arrow in and double arrow out rules
Copyright 2008, Scott Gray

5. Double Arrow In Rule From C → D and D → C derive C ↔ D
Justification includes both conditional lines, e.g. 2,4 ↔I
Double arrow in depends on both conditional’s dependencies
Copyright 2008, Scott Gray

6. Double Arrow Out Rule From C ↔ D derive either C → D or D → C
The justification is the line containing the iff, e.g. 4 ↔O
The dependencies are the same as the iff’s dependencies
Copyright 2008, Scott Gray

7. Double Arrow Strategies
If a goal line is an iff, look for the two conditionals needed. If not found, the conditionals become a goal line themselves.
If a premise is an iff, use the double arrow out rule one or two times.
Copyright 2008, Scott Gray

8. Example 1 M ↔ G ∴ G ↔ M 1 (1) M ↔ G A 2 (2) M → G 1 ↔O
3 (3) G → M 1 ↔O
4 (4) G ↔ M 3,2 ↔I
This proves iff is commutative
Copyright 2008, Scott Gray

9. Example 2 V ↔ ~L, ~L ↔ ~D, C → ~D ∴ C → V 1 (1) V ↔ ~L A
2 (2) ~L ↔ ~D A
3 (3) C → ~D A
4 (4) C PA
3,4 (5) ~D 3,4 →O
2 (6) ~D → ~L 2 ↔O
2,3,4 (7) ~L 6,5 →O
1 (8) ~L ↔ V 1 ↔O
1,2,3,4 (9) V 8,7 →O
1,2,3 (10) C → V 4-9 → I
Copyright 2008, Scott Gray

10. Assignments Read Chapter 6 Do all of the exercises
Be sure to ask me questions if you don’t understand something or can’t solve a problem
Copyright 2008, Scott Gray