is an assertion that can either be true or false.
statement
Statements which do not use connectives are called
atomic or
simple statements
statements which use connectives are
called
compound statements
most common connectives and their symbols
and/but ∧
or ∨
If, then ⇒
if and only if ⇔
Simple Statements
Assertion p p is true
Negation ¬p ¬p is false
Compound Statements
Conjunction p ∧ q p and q
Disjunction p ∨ q p or q
Conditional p ⇒ q If p, then q
Biconditional p ⇔ q p if and only if q
A compound statement of the form If p, then q, written symbolically as p ⇒ q, is called
conditional statement
A compound statement of the form p if and only if q, or ”p if q” written symbolically as p ⇔ q, is called a
biconditional statement
Two statements are said to be equivalent whenever they have ”identical truth tables”
Logical Equivalence
is a statement whose truth value is true regardless of the truth value of the atomic statements.
tautology
is a statement whose truth value is false regardless of the truth value of the atomic statements.
contradiction
i.) [ p ∧ (p ⇒ q)] ⇒ q
ii.) [ (p ⇒ q) ∧ p] ⇒ q
Law of Detachment
i.) (p ∧ q) ⇒ q
ii.) (p ∧ q) ⇒ p
Law of Simplification
i.) p ⇒ (p ∨ q)
ii.) q ⇒ (p ∨ q)
Law of Addition
[ (p ∨ q) ∧ ¬p] ⇒ q
Modus Tollendo Ponens
¬(¬p) ⇔ p
Law of Double Negation
i.) (p ∧ q) ⇔ (q ∧ p)
ii.) (p ∨ q) ⇔ (q ∨ p)
Commutative Laws
i.) [(p ∧ q) ∧ r] ⇔ [p ∧ (q ∧ r)]
ii.) [(p ∨ q) ∨ r] ⇔ [p ∨ (q ∨ r)]
Associative Laws
[(p ⇒ q) ∧ (q ⇒ r)] ⇒ (p ⇒ r)
Transitivity of Implication
i.) [ p ∧ (q ∨ r)] ⇔ [(p ∧ q) ∨ (p ∧ r)]
ii.) [ p ∨ (q ∧ r)] ⇔ [(p ∨ q) ∧ (p ∨ r)]
Distributive Laws
(p ⇒ q) ⇔ (¬q ⇒ ¬p)
Law of Contrapositive
i.) [(p ⇒ q) ∧ (p ⇒ ¬q)] ⇔ ¬p
ii.) [ p ⇒ (q ∧ ¬q)] ⇔ ¬p
Law of Absurdity
¬(p ∧ ¬p)
Law of Contradiction
p ∨ ¬p
Law of Excluded Middle
