PROPOSITIONAL LOGIC

1. PROPOSITIONAL LOGIC • Logic : Language for reasoning • Propositional logic • Proposition is either true or false • Grass is green • 2 + 5 = 5 • Truth value: truth(T) or Falsity (F) of a propositon • Simple porpositions: single statement • Compound propositions: combination of simple propositions • Logical connectives: device to link pair of propositions • Truth value of compount proposition • Negation Logical connectives: • Conjuction (AND – p ^q) • True only when both p and q are true • Inclusive Disjuction (OR – p v q) • Atkeast one of the two must be true • Exclusive disjuction At most one of the two, not both, must be true • Conditional proposition (p  q ) • If p holds then q must be true, if p is false no requirement on q • Bidirectional proposition (p  q) • True if p and q have the same values.

2. Propositional Logic • Arguments • Consists of a set of propositions called premises and a conclusion supposed to follow logically from the premises. • Is valid if the P1 ^ : : : ^ Pn |- Q i.e., if the premises logically implies the conclusion • Inference Rules • Basic inference rule Modus Pones is based on tautology [p ^ (pq)]  q • i.e., if both PQ & P hold, then Q can be conluded • Inference rule Modus Tollens is based on the tautology [p ^ (pq)] q • Inference rules Addition, Simplification and Conjuction are fairly obvious. • Tautologies: • always true, • ex., • Contraciction: • Always false • ex., p ^ p • Logical Equivalence • P and q are logically equivalent if P Q is a tautology • Logical Implication • P logically implies Q if PQ is a tautology • The converse does not apply • Contigency • Neither a tautology nor a contradiction • Ex., P v Q