Logical Form and Logical Equivalence
M260 2.1
Logical Form Example 1
• If the syntax is faultyor execution results in division by zero,then the program will generate an error message.
• Thereforeif the computer does not generate an error messagethen the syntax is correctand the execution does not result in division by zero.
Logical Form Example 2
• If x is a Real number such that x<-2 or x>2,then x2>4.
• Thereforeif x24,then x-2 and x2.
Logical Form Example 1
• If (the syntax is faulty)or (execution results in division by zero),then (the program will generate an error message).
• Thereforeif (the computer does not generate an error message)then (the syntax is correct)and (the execution does not result in division by zero).
Logical Form Example 1
• If (p)or (q),then (r).
• Thereforeif (not r)then (not p)and (not q).
Logical Form Example 2
• If (x<-2) or (x>2),then (x2>4).
• Thereforeif (x24),then (x-2) and (x2).
Logical Form Example 2
• If (p) or (q),then (r).
• Thereforeif (not r),then (not p) and (not q).
Logical Form vs Content
• Examples 1 and 2 have the same form:If p or q, then r.therefore if not r, then not p and not q.
• These examples have different values for the propositional variables p and q.
Formal Logic Goals
• Avoid Ambiguity
• Obtain Consistency
• Elucidate Proof Mechanisms
Mathematical Vocabulary
• New terms are defined using previously defined terms.
• Initial terms remain undefined.
• Undefined terms in logic: sentence, true, false.
Logic Symbols ~
• ~ denotes “not”
• Negation of p is ~p.
Logic Symbols ~
denotes “and”
• Conjunction of p and q is p q. denotes “or”
• Disjunction of p and q is p q.
• Precedence: first ~ then and (unordered)
Truth Values
• True
• False
Precedence Examples
• ~p q• ~p ~q
• ~ (p q)
Let p, q and r be 0<x, x<3, and x=3
• Rewrite x 3
• q r• Rewrite 0<x<3
• pq• Rewrite 0<x3
• p(q r)
Negation Truth Table
p ~p
T F
F T
Conjunction Truth Table
p q pq
T T T
T F F
F T F
F F F
Disjunction Truth Table
p q p q
T T T
T F T
F T T
F F F
Statement Form
• Statement variables
• Logical connectives
• Truth table
Exclusive Or
• p or q but not both
• (p q) ~(p q)
• Do a truth table
Exclusive Or Truth Table
p q p q p q ~(p q)(p q) ~(p q)
Exclusive Or Truth Table
p q p q p q ~(p q)(p q) ~(p q)
T T
T F
F T
F F
Exclusive Or Truth Table
p q p q p q ~(p q)(p q) ~(p q)
T T T T F
T F T F T
F T T F T
F F F F T
Exclusive Or Truth Table
p q p q p q ~(p q)(p q) ~(p q)
T T T T F F
T F T F T T
F T T F T T
F F F F T F
Logical Equivalence
• Statement Forms are logically equivalent if, and only if, they have the same truth tables.
• P Q
Logical Equivalence Examples
• 6>2 2<6
• p q q p• p ~(~p)
De Morgan’s Laws
• ~(p q) ~p ~ q
• ~(p q) ~p ~ q
• Do truth tables
~(p q) ~p ~ q
p q ~p ~q p q ~(p q) ~p ~q
~(p q) ~p ~ q
p q ~p ~q p q ~(p q) ~p ~q
T T
T F
F T
F F
~(p q) ~p ~ q
p q ~p ~q p q ~(p q) ~p ~q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T
Practice Negations
• John is six feet tall and weighs at least 200 pounds.
• John is not six feet tall or he weighs less than 200 pounds.
Practice Negations
• The bus was late or Tom’s watch was slow.
• The bus was not late and Tom’s watch was not slow.
Jim is tall and thin.
Logical And and Or are only allowed between statements.
Tautologies and Contradictions
• A tautology is a statement form that is always true regardless of the values of the statement variables.
• A contradiction is a statement form that is always false regardless of the values of the statement variables
Logically Equivalent Forms
• Commutative laws• Associative laws• Distributive laws• Identity laws• Negation laws• Double negative law
• Idempotent laws• De Morgan’s laws• Universal bound laws• Absorption laws• Negations of
tautologies and contradictions
Logical Equivalences• pq _________ pq ________• (pq)r _______ (pq)r _______ • p(qr) ______ p(qr) _______• pt __________pc __________• p~p _________p~p _________• ~(~p) ________• pp __________pp __________• ~(pq ) _______ ~(pq ) _______• pt __________ pc __________• p(pq) ______ p(pq) ______• ~t ___________~c ___________
Logical Equivalences• pq qp pq qp• (pq)r p(qr) (pq)r p(qr) • p(qr) (pq) (p r)• p(qr) (pq) (p r) • pt p pc p• p~p t p~p c• ~(~p) p• pp p pp p• ~(pq ) ~p~q ~(pq ) ~p~q • pt t pc c• p(pq) p p(pq) p• ~t c~c t