Propositional Logic

Propositional Logic

Negation • Given a proposition p, negation of p is the ‘not’ of p

Conjunction • Representing ‘and’ between propositions. • Given two propositions p and q, conjunction of p and q is true only when both of the propositions are true

Disjunction • Representing ‘or’ between proposition • Given two propositions p and q, disjunction of p and q is true when either or both of the propositions are true

Implication • Representing ‘if …then…’ to connect propositions • Given two propositions p and q, we say that “p implies q” which is the implication of q by p , the result is true in all cases except where p is true and q is false

Equivalence • Representing “if and only if” • Two propositions are equivalent if and only if they have the same truth value

Some rules • Disjunction is an associative and a commutative truth function • p  ( q  r )  (p  q )  r  p  q  r • p  q  q  p • Conjunction is a commutative and associative truth function • Distributive • p  ( q  r )  (p  q)  (p  r) • p  ( q  r )  (p  q)  (p  r) • Implication is not commutative p  q is not the same as q  p • is not associative p ( q  r ) is not the same as (p q ) r • Is transitive p q and q  r , p r

Exercise • Let s, t, and u denote the following atomic propositions: s : Sally goes out for a walk. t : The moon is out u : It is windy Write a possible translation for each of the following statements: • If the moon is out and it is not windy, then Sally goes out for a walk • If the moon is not out, then if it is not windy Sally goes out for a walk

Propositional Logic