1. Propositional Logic
7) Not
Copyright 2008, Scott Gray

2. Expressing “Not” The following are all equivalent:
It is not true that Mario will play for the Penguins this year.
It is false that Mario will play for the Penguins this year.
Mario will not play for the Penguins this year.
The Penguins will not have Mario play for them this year.
Copyright 2008, Scott Gray

3. Use of Parentheses
Use parentheses when the negation covers a non-simple statement
~(A & B)
No parentheses are needed to negate a simple statement
~A & ~B
Be sure to read the chapter for a more thorough discussion
Copyright 2008, Scott Gray

4. Tilde In
If from an assumption a standard contradiction can be derived, then derive the negation of the assumption
Standard contradiction: a conjunction whose right conjunct is the negation of the left conjunct
A & ~A
(B → C) & ~(B → C)
Not a standard contradiction: ~A & A
Copyright 2008, Scott Gray

5. Modus Tollens A → B, ~B ∴ ~A 1 (1) A → B A 2 (2) ~B A 3 (3) A PA
1,3 (4) B 1,3 →O
1,2,3 (5) B & ~B 4,2 &I
1,2 (6) ~A ~I
Copyright 2008, Scott Gray

6. Tilde In Dependencies
The statement derived by Tilda In depends on all of the assumptions the standard contradiction depends on, minus the assumption whose negation is derived
In the Modus Tollens example this is line #3
Copyright 2008, Scott Gray

7. Tilde Out The “inverse” of Tilde In
If from a negative assumption a standard contradiction can be derived, then derive the constituent of the negative assumption
Copyright 2008, Scott Gray

8. Tilde Out Example ~~R ∴ R 1 (1) ~~R A 2 (2) ~R PA
1,2 (3) ~R & ~~R 2,1 &I
1 (4) R ~O
Copyright 2008, Scott Gray

9. Assignments Read Chapter 5 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