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Lecture Notes 8. CS1502. Example Proof. A (B C) (A B) (A C). Valid Argument. P 1 P 2 … P n Q Q is a tautological (logical) consequence of P 1 , P 2 , …, P n (P 1 P 2 … P n ) Q is a tautology (logical necessity). NEW IDEA. Valid Argument. P. Example.
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Lecture Notes 8 CS1502
Example Proof A (B C) (A B) (A C)
Valid Argument • P1 P2 … Pn Q • Q is a tautological (logical) consequence of P1, P2, …, Pn • (P1 P2 … Pn) Q is a tautology (logical necessity). NEW IDEA Valid Argument
P Example • Show P is a tautological consequence of (P Q). • Methods of attack: • Boole • Show P is a tautological consequence of (P Q). • Show (P Q) P is a tautology. • Fitch • Show (P Q) is a valid argument
P Example • Show P is not a tautological consequence of (P Q). • Method of attack: • Boole • Show P is not a tautological consequence of (P Q). • Show (P Q) P is not a tautology. • Build a world • Show (P Q) is an invalid argument
Build a World • Let P be assigned true and Q false.(P Q) is true while P is false. conclusion premises
Example • Show the following argument is valid. Cube(b)(Cube(c) Cube(b)) Cube(c)
Every non-spurious row is true! In fact, every row is true, so a Tautology!! Logical Necessity
Non-consequence • Show the following argument is invalid. Cube(a) Cube(b)(Cube(c) Cube(b)) Cube(c)
Inference Patterns • Modus Ponens P Q P Q
Elimination • P Q … P … Q Elim
Introduction • P … Q P Q Intro
Elimination • P Q … P … Q Elim
Introduction • P … Q Q … P P Q Intro
Inference Patterns • Modus Tollens P Q Q P
