Propositional Logic Reference Sheet Page 2
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ROPOSITIONAL
OGIC
EFERENCE
HEET
Forms of Conditional Statements
•
Contrapositive of p → q:
~q → ~p
(logically equivalent)
•
Converse of p → q:
q → p
(not logically equivalent)
•
Inverse of p → q:
~p → ~q
(not logically equivalent)
Necessary and Sufficient Conditions
•
“r → s”
“r is a sufficient condition for s”
means
•
“~r → ~s”
(or “s → r” by contraposition)
“r is a necessary condition for s”
means
•
“r is a necessary and sufficient condition for s” means “r ↔ s”
Rules of Inference
Modus Ponens
Modus Tollens
p → q
p → q
p
~q
∴ q
∴ ~p
Generalization (a.k.a. ∨ Introduction)
Specialization (a.k.a. ∧ Elimination)
p ∧ q
p ∧ q
p
q
∴ p ∨ q
∴ p ∨ q
∴ p
∴ q
Elimination (better, ∨ Elimination)
Transitivity
p ∨ q
p ∨ q
p → q
~q
~p
q → r
∴ p
∴ q
∴ p → r
Cases
Contradiction
p ∨ q
~p → c
p → r
∴ p
q → r
∴ r
UIC CS 151
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