Download Introduction to Logic What is Logic? Simple Statements Which one is

What is Logic?
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Introduction to Logic
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Logic is the study of reasoning
It is specifically concerned with whether reasoning
is correct
Logic is also known as Propositional Calculus
Peter Lo
CS218 © Peter Lo 2004
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CS218 © Peter Lo 2004
Simple Statements
Which one is statement?
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Simple Statement is the basic building block of
Logic.
Simple Statement is referred as a proposition.
A statement is a declarative sentence that either
True or False.
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Today is Friday.
u This is a Statement
How to celebrate the Mid-Autumn?
u This is not a Statement
Let’s go for dinner together after lesson.
u This is not a Statement
1+1=3
u This is a Statement
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Compound Statement
True Table
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Compound Statement is the combination of two
or more Simple Statement.
Example:
u “Today is Friday” and “Tomorrow is holiday”
CS218 © Peter Lo 2004
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The value of a statement can be represented by a
Truth Table.
Only True and False is appear in a Truth Table
Example:
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Basic Logic Connectives
Conjunction
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Compound statements are connected using mainly
five basic connectives:
u Conjunction
u Disjunction
u Negation
u Conditional
u Biconditional
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Conjunction is the combination of statements
using AND.
The conjunction of two statement is True only if
each component is True.
Represented as p ^ q.
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Disjunction (Inclusive OR)
Negation
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Disjunction is the combination of statements using
OR.
The conjunction of two statement is True if either
one component is True.
Represented as p ∨ q.
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Negation is the NOT of a simple statement.
The Truth value of the statement negation of a
statement is the opposite of the truth value of the
original statement.
Represented as ~p.
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Conditional
Biconditional
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Conditional Statement is the statement in the form
“If p, then q” or “p implies q”.
The conditional p à q is True unless p is True and
q is False.
Represented as p à q
(Do the Ex. 1 & 2)
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Biconditional Statement is the statement in the
form “p if and only if q” or “p iff q”.
If p and q have the same value, p ↔ q is True,
otherwise will be False.
Represented as p ↔ q
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Propositions
Truth Table
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When the sub-statement of a compound statement
are variables and represented in logical
connectives, the compound statement is called
Proposition.
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The truth value of a proposition depends
exclusively upon the truth values of its variables.
The truth value of a proposition is known once the
truth values of its variables are known.
(See E.g. 30 - 32)
CS218 © Peter Lo 2004
Exclusive Disjunction (Exclusive OR)
Tautologies and Contradictions
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Exclusive Disjunction means “either one or the
other, but not both”.
The conditional of a exclusive disjunction is True
when p and q are not the same.
Exclusive Disjunction can be expressed using
basic connectives ~ (p ↔ q)
(Do the Ex. 3)
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Tautologies
u A Compound statement that is always True is
called Tautologies
u (See E.g. 33)
Contradictions
u A Compound statement that is always False is
called Contradictions
u (See E.g. 34)
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Principle of Substitution
Law of Syllogism
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If P(p, q, … ) is a Tautology, then P(P1 , P2 , … ) is
also a Tautology.
(See E.g. 35)
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A fundamental principle of logical reasoning,
called the Law of Syllogism, states “If p implies q
and q implies r, then p implies r”
[(p → q) ∧ (q → r)] → (p → r) is a Tautology
(See E.g. 36)
CS218 © Peter Lo 2004
Logical Equivalence
DeMorgan’s Laws
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Two propositions P and Q are said to be logically
equivalent if the final columns in their truth table
are the same.
Represented as ≡
(Do the Ex. 4 & 5)
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DeMorgan’s Laws show that:
u ~ (p ∨ q) ≡ ~p ∧ ~q
u ~ (p ∧ q) ≡ ~p ∨ ~q
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Logically True & Logically Equivalent
Argument
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Logically True Statement
u A statement is said to be Logically True if it is
derivable from a Tautology.
Logically Equivalent Statement
u Statement of the form P(p 0 , q 0 , ..) and Q (p 0 ,
q 0 , ..) are said to be Logically Equivalent if the
propositions P(p, q, … ) and Q(p, q, ..) are
logical equivalent.
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An argument is a relationship between a set of
proposition, P1 , P2 , … Pn , called Premises and
other proposition Q, called the Conclusion.
An argument is denoted by P1 , P2 , … Pn | Q
An argument is said to be Validif the premises
yield the conclusion.
An argument is said to be Fallacy if that is not
valid.
(See E.g. 39 & 40)
CS218 © Peter Lo 2004
Logical Implication
Example
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A proposition P(p, q, … ) is said to Logical Imply
a proposition Q(p, q, … ), written P(p, q, … ) =>
Q(p, q, … ) if Q(p, q, … ) is true whenever P(p, q, )
is True.
(See E.g. 44)
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Consider the following argument: if I am not in
Malaysia, then I am not happy; if I am happy, then
I am singing; I am not singing; therefore I am not
in Malaysia. Using the translation
show that this argument is valid, or explain why it
is not.
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CS218 © Peter Lo 2004
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Answer
Laws of Algebra of Propositions
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Laws of Algebra of Propositions
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Distributive Laws
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p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
u p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Identity Laws
CS218 © Peter Lo 2004
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p ∨F≡p
p ∧T ≡p
p ∨T ≡T
u p ∧F ≡ F
(T = True, F = False)
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Laws of Algebra of Propositions
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Idempotent Laws
u p ∨p ≡ p
u p ∧p ≡ p
Associative Laws
u (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
u (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Communicative Laws
u p ∨q≡ q ∨p
u p ∧q≡ q ∧p
u (p ↔ q) ≡ (q ↔ p)
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Complement Laws
u p ∨ ~p ≡ T
u p ∧ ~p ≡ F
u ~T ≡ F
u ~F ≡ T
(T = True, F = False)
Involution Law
u ~ ~p ≡ p
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Laws of Algebra of Propositions
Laws of Algebra of Propositions
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DeMorgan’s Laws
u ~ (p ∨ q) ≡ ~p ∧ ~q
u ~ (p ∧ q) ≡ ~p ∨ ~q
u (p ∨ q) ≡ ~ (~p ∧ ~q)
u (p ∧ q) ≡ ~ (~p ∨ ~q)
Contrapositive
u (p → q) ≡ (~q → ~p)
Implication
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(p ∧ q) ≡ ~(p → ~q)
(p → r) ∧ (q → r) ≡ (p ∨ q) → r
u (p → q) ∧ (p → r) ≡ p → (q ∧ r)
Equivalence
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CS218 © Peter Lo 2004
(p → q) ≡ (~q ∨ p)
(p → q) ≡ ~(p ∧ ~q)
(p ∨ q) ≡ (~p → q)
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(p ↔ q) = (p → q) ∧ (q → p)
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Laws of Algebra of Propositions
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Exportation Law
u (p ∧ q) → r ≡ p → (q → r)
Absorbtion Law
u (p ∨ q) ∧ (p ∧ q) ≡ p ∨ q
u (p ∧ q) ∧ (p ∨ q) ≡ p ∨ q
Reductio ad adsurdum
u (p → q) ≡ (p ∧ ~q) → F
(T = True, F = False)
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