Propositional Logic 7) Not Copyright 2008, Scott Gray

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

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

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

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 3-5 ~I Copyright 2008, Scott Gray

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

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

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

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