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: 1.If the moon is out and it is not windy, then Sally goes out for a walk 2.If the moon is not out, then if it is not windy Sally goes out for a walk