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Logical Form and Logical Equivalence. M260 2.1. Logical Form Example 1. If the syntax is faulty or execution results in division by zero, then the program will generate an error message.
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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)
Statement Form • Statement variables • Logical connectives • Truth table
Exclusive Or • p or q but not both • (p q) ~(p q) • Do a truth table
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
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
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 Logically Equivalent Forms
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
