introduction to logic - discreet maths

8/4/2019 Introduction to Logic - Discreet Maths

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INTRODUCTION TO LOGIC

Chapter Objectives

In this chapter you will learn:

what logic is; about the basis of logic: the simple

statement;

how to construct a truth table; about the five basic logic connectives and

there truth tables;

the rules of logic; what makes a statement a tautology; what makes a statement a contradiction; what logical equivalence is; what a proposition is; what makes an argument valid;


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1.1 WHAT IS LOGIC?

Logic is the study of the principles and methodsused in distinguishing valid arguments from

those that are invalid.

Logic is also known as propositional calculus.1.1.1 SIMPLE STATEMENTS

The basic building block in logic is thestatement, also referred to as a proposition.

A statement is a declarative sentence, which canonly be either true or false.

Statements are represented by letters such asp, q,and r, . . .

Example: Jakarta is a city in Indonesia


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1.1.2 TRUTH TABLES

Since a statement can only be true orfalse, thevalues of a statement can be represented by a

truth table.

Variablesp, q, r, . . . are used to represent basicstatements.

T and F stand for true and false.p q

T T

T FF T

F F

In a truth table the number of truth-values (rows)is n2 , where n is the number of basic statements(variables)


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1.1.3 COMPOUND STATEMENTS

The combination of two or more simplestatements is a compound statement, or

compound proposition.

Example: 2 + 1 = 5 and 6 + 2 = 8

Example: The sky is clear or It is raining today

The variablesp, q, r, . . . denote simplestatements in the compound proposition ,,, rqpP , where P is a proposition.

1.2 BASIC LOGIC CONNECTIVES

Compound statements are connected usingmainly five basic connectives: conjunction,

disjunction, negation, conditional, and

biconditional.


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1.2.1 CONJUNCTION

Any two statements can be combined by theword and to form a composite statement which

is called the conjunction of the original

statements.

The connection of the statementsp and q issymbolically represented bypq

The truth values ofpq in a truth table:p q p q

T T T

T F F

F T F

F F F

Ifp is true and q is true thenpq is true;otherwisepq is false.

Example 12: Sidney is in Australia and 2 + 2 = 4

Example 13: Sidney is in Australia and 2 + 2 = 5

Example 14: Sidney is in Malaysia and 2 + 2 = 4

Example 15: Sidney is in Malaysia and 2 + 2 = 5

Only example 12 contains two simple statementswhich are true, the others all contain simple

statements in which at least one of them is false,

so only example 12 is true.


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1.2.2 DISJUNCTION

Any two statements can be combined by theword or (in the sense of and/or, called the

inclusive or), to form a new statement which is

called the disjunction of the original two

statements.

The connection of the statementsp or q issymbolically represented bypq


T T T

T F T

F T TF F F

Ifp is true or q is true or bothp and q are true,thenpq is true; otherwisepq is false.

Example 16: Sidney is in Australia or 2 + 2 = 4

Example 17: Sidney is in Australia or 2 + 2 = 5

Example 18: Sidney is in Malaysia or 2 + 2 = 4

Example 19: Sidney is in Malaysia or 2 + 2 = 5

Only example 19 is false. Each of the othercompound statements is true since at least one of

its simple statements is true.


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1.2.3 NEGATION

Given any statementp, another statement, calledthe negation ofp, can be formed by writing It is

false that . . . beforep or, if possible, by

inserting inpthe word not

Negation can be symbolically represented by ~p,or

p

The truth values of ~p in a truth table:p ~p

T F

F T

Ifp is true then ~p is false; ifp is false, then ~p istrue.

Example 20: Ifpis Sidney is in Australia, then Sidney

is not in Australia is the negation ~p

Example 21: Ifpis 2 + 2 = x, then 2 + 2 x is the

negation ~p


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1.2.4 CONDITIONAL

Many statements, especially in mathematics, areof the form Ifp then q or p implies q. Such

statements are called conditional statements.

Conditional statements are symbolicallyrepresented aspq


T T T

T F F

F T T

F F T

The conditionalpq is true unlessp is true andq is false.

Example 22: If Sidney is in Australia then 2 + 2 = 4

Example 23: If Sidney is in Australia then 2 + 2 = 5

Example 24: If Sidney is in Malaysia then 2 + 2 = 4

Example 25: If Sidney is in Malaysia then 2 + 2 = 5

By the conditionalpq only example 23 isfalse. But how can this be as clearly 2 +2 = 4 is

true and 2 + 2 =5 is clearly false? This is the

case as once we know that the if is false we no

longer care if the that is true or not.


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Exercise 1: Construct the truth table for ~pq

p q ~p ~p qT T F F

T F F F

F T T T

F F T F

Exercise 2: Construct the truth table for(pq) (pq)

p q p q p q (p q) (p q)

T T T T T

T F T F F

F T T F F

F F F F T


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1.2.5 BICONDITIONAL

Another common statement called a bicon-ditional statementis of the form p iffq

Biconditional statements are symbolicallyrepresented aspq

The truth-values ofpq in a truth table:p q p q

T T T

T F F

F T F

F F T

Ifp and q have the same truth value, thenpqis true;

if p and q have opposite truth values, thenpqis false.

Example 26: Sidney is in Australia iff 2 + 2 = 4

Example 27: Sidney is in Australia iff 2 + 2 = 5

Example 28: Sidney is in Malaysia iff 2 + 2 = 4

Example 29: Sidney is in Malaysia iff 2 + 2 = 5

By the conditionalpq examples 26 & 29 aretrue, examples 27 & 28 are false.

pq can be represented as pqqp


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1.3 PROPOSITIONS AND TRUTH TABLES

By repetitive use of the logical connectives (, ,~, , and, ), we can construct compoundstatements that are more involved.

In the case where the substatementsp, q, . . . of acompound statement, P(p, q, ...) are variables, the

compound statement is called a proposition.

The truth-value of a proposition is known oncethe truth-values of its variables are known. A

simple concise way to show this relationship is

through a truth table.

Example 30: The truth table of the proposition

qp ~~ is:

p q ~q p ~q ~(p ~q)

T T F F T

T F T T F

F T F F T

F F T F T

Observe that the first columns of the table are forthe variablesp, q, . . . and there are enough rows

in the table to allow for all possible combinations

of T and F for these variables, i.e. the number of

rows =

n

2


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Example 31: Construct the truth table for pqqp

p q p q q p (p q) (q p)

T T T T T

T F F T F

F T T F F

F F T T T

Example 32: Construct the truth table for

~(pq) ~(qp)

p q p q q p ~(p q) ~(q p) ~(p q)

~(q p)

T T T T F F F

T F F F T T T

F T F F T T T

F F F T T F T


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1.3.1 EXCLUSIVE DISJUNCTION

In addition to the inclusive or there is anothermeaning for or in English called the exclusive

or, which means either one or the other, but not

both. In Logic exclusive or is referred to as

exclusive disjunction.

In mathematics or in logic, or always meansinclusive or i.e., or always refers to thedisjunction connective.

Exclusive or can be expressed symbolically as~(pq)

Exercise 3: Complete the truth table for

~(pq)

p q p q ~(p q)

T T T F

T F F T

F T F T

F F T F


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1.4.1 TAUTOLOGY

A compound proposition that is always TRUE iscalled a tautology.

Example 33: pp ~ is a tautology as all entries in

the last column are Ts

p ~p p ~p

T F T

F T T

1.4.2 CONTRADICTIONA compound proposition that is always FALSE

is called a contradiction.

Example 34: pp ~ is a contradiction as all

entries in the last column are Fs

p ~p p ~p

T F FF T F


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1.4.3 PRINCIPLE OF SUBSTITUTION

If ),,(

qpP is a tautology, then ),,( 21

PPP isalso a tautology.

Example 35: We have shown thatp ~p is a

tautology so by the principle of

substitution, substituting qrforp

we obtain the proposition

(qr) ~(qr), which is also a

tautology.

q r q r ~(q r) (q r) ~(q r)

T T T F T

T F F T T

F T F T TF F F T T

1.4.4 LAW OF SYLLOGISMA fundamental principle of logical reasoning,

called the Law of Syllogism, states: Ifp implies

q and q implies r, thenp implies r.

In other words the proposition[(pq) (qr)] (pr) is a tautology.


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Example 36:

Show that [(pq) (qr)] (pr)

is a tautology.

p q r p q q r (p q)

(q r)

p r [(p q)

(q r)]

(p r)

T T T T T T T T

T T F T F F F T

T F T F T F T T

T F F F T F F T

F T T T T T T T

F T F T F F T T

F F T T T T T T

F F F T T T T T

1.5 LOGICAL EQUIVALENCE

Two propositions ,...),( qpP and ,...),( qpQ aresaid to be logically equivalent if the final

columns in their truth tables are the same.

Logical equivalence is denoted with


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Exercise 4: Show thatpq and ~q ~p are

logically equivalent.

p q ~p ~q p q ~q ~p

T T F F T T

T F F T F F

F T T F T T

F F T T T T

Exercise 5: Show that (pq) (qp) pq

p q p q q p (p q) (q p) p q

T T T T T T

T F F T F F

F T T F F F

F F T T T T

Since the last two columns in the truth tables arethe same the statements are logically equivalent.

1.5.1 DeMORGANS LAWSDeMorgans Laws are simply:

a. qpqp ~~~ b. qpqp ~~~

DeMorgans Laws are important both in logicand in set theory.


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1.5.2 LOGICALLY TRUE STATEMENTS

A statement is said to be logically true if it isderivable from a tautology.

Example 37: It is raining or it is not raining.

Example 37 is logically true since it is derivablefrom the tautologyp ~p, wherepis It is

raining.

1.5.3 LOGICALLY EQUIVALENTSTATEMENTS

Statements are logically equivalent if thepropositions that make up the statement arelogically equivalent to each other.

Example 38: Since qpqp ~~~ , thestatement It is not true that roses arered and violets are blue is logically

equivalent to the statement Roses are

not red or violets are not blue.

Wherepis roses are red, and q is

violets are blue.


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1.6 ARGUMENTS

An argument is a relationship between a set ofpropositions, nPPP ,,, 21 , called premises, and

another proposition Q, called the conclusion.

An argument is denoted by nPPP ,,, 21 QAn argument nPPP ,,, 21 Q is said to be valid

iff QPPP n )...( 21 is a tautology.

Example 39: The argumentp,pq q is valid,

since [p (pq)] q is a

tautology.

p q p q p (p q) [p (p q)]q

T T T T T

T F F F T

F T T F T

F F T F T

Example 40: The argumentpq, q p is a

fallacy, since [(pq)q] p isnot a tautology.

p q p q (p q) q [(p q) q] p

T T T T T

T F F F T

F T T T F

F F T F T


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Example 41: Analyse the following argument.

1S : If a man is a bachelor, he is unhappy.2S : If a man is unhappy, he dies young.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

S: Bachelors die young.

First we must define our terms:

p: he is a bachelor q: he is unhappy

r: he dies young

and the argument ispq, qr pr

Show that [(pq) (qr)] (pr) is a

tautology.

p q r p q q r [(p q)

(q r)]

p r [(p q)

(q r)] (p r)

T T T T T T T T

T T F T F F F T

T F T F T F T T

T F F F T F F T

F T T T T T T TF T F T F F T T

F F T T T T T T

F F F T T T T T

Since [(pq) (qr)] (pr) is a

tautology the argument is valid.


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Another type of argument does not differentiatebetween the statements and the conclusion by the

use of a line; it uses the word therefore instead.

Example 42: Is this argument valid? An interesting

teacher keeps me awake. I stay awake

in CS218 class. Therefore, my CS218

teacher is interesting.

First define the terms:

t: my teacher is interesting

a: I stay awake m: I am in CS218 class

and the argument is: ta,am mt

Show that [(ta) (am)] (mt) is a

tautology.t a m t a a m (t a)

(a m)

m t [(t a)

(a m)]

(m t)

T T T T T T T T

T T F T F F F T

T F T F F F T TT F F F F F F T

F T T T T T F F

F T F T F F F T

F F T T F F F T

F F F T F F F T

Since [(ta) (am)] (mt) is not a

tautology, the argument is not a valid argument.


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Example 43: Is this argument valid? Sean is either a

carpenter or a plumber (but not both).

If he carries a wrench, hes a plumber.Sean is a carpenter. Therefore, he does

not carry a wrench.

First define the terms:

c: Sean is a carpenter

p: Sean is a plumber

w: Sean carries a wrench

and the argument is ~(cp), wp, c ~w

So we need to show that

[~(cp) (wp) c] ~w is a tautology.

c p w ~(c

p) w

p ~(c

p) (w p)

c

~w [~(c

p) (w p)

c] ~w

T T T F T F F T

T T F F T F T T

T F T T F F F T

T F F T T T T T

F T T T T F F T

F T F T T F T T

F F T F F F F T

F F F F T F T T

Since [~(cp) (wp) c] ~w is a

tautology, the argument is valid.


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1.7 LOGICAL IMPLICATION

A proposition P(p, q, . . .) is said to logicallyimply a proposition Q(p, q, . . .), written

P(p, q, . . .) Q(p, q, . . .) if Q(p, q, . . .) is true

whenever P(p, q, . . .) is true.

Example 44: Consider the truth table below.

Observe thatp is true in cases (lines)

1 and 2, and in these casespq isalso true. In other words, p logically

impliespq

p q p q

T T T

T F TF T T

F F F

1.8.1 BASIC LOGIC CONNECTIVESConnective Meaning Symbolized byconjunction and disjunction or negation not , ~

conditional If ... then ..., Implies

biconditional If and only if


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REMARKS

The number of truth values (rows) is

n2

, where nis the number of basic statements (variables).

The biconditional statement can be defined as pqqp

The truth table of a proposition consists preciselyof the columns under the variables and thecolumn under the proposition.

If P(p, q, . . .) is a tautology then ~P(p, q, . . .) is acontradiction, and if ~P(p, q, . . .) is a tautology

then P(p, q, . . .) is a contradiction.

An argumentnPPP ,,,

21 Q is valid if Q is

true whenever all the premises nPPP ,,, 21 are

true.

The argument nPPP ,,, 21 Q is valid iff nPPP 21 Q is a tautology.

For any propositions P(p, q, . . .) and Q(p, q, . . .),the following statements are equivalent:i) P(p, q, . . .) logically implies Q(p, q, . . .)ii) The argument P(p, q, . . .) Q(p, q, . . .) is

valid.

iii)The proposition P(p, q, . . .)

Q(p, q, . . .) isa tautology.

introduction to logic - discreet maths

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