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