Logical Reasoning

Mathematics SSS 2 Second Term

WEEK 1

Logical Reasoning

Performance Objectives

Students should be able to;

1. Give the meaning of simple and compound statement
2. List five logical operations and their symbols
3. Write the truth table of a compound statement of any of the five logical operations

Logic

Content

Introduction

Logic is a branch of philosophy that is mainly concerned with the evaluation of arguments based on fundamental laws of scientific reasoning and thinking.

The term "logic is derived from the Greek word logos, which means thought, reason or law. It deals with the ability to argue and convince.

Human thoughts and feelings can only be communicated through the application of a suitable language. To communicate expression effectively, we use grammatical expressions, sentences and statements. Logic enables us to express ideas clearly and concisely, and to view arguments intelligently and critically.

Simple statements

A simple statement is a sentence which is true or false but is not both. It is a proposition that is either true or false. This statement may be verbal or written.

Example 1

The following are logical simple statements that are either true or false.

1.  All men are born by women.

2.  Algebra is a branch of mathematics.

3. 5 is greater than7.

4.  Men can be pregnant. e. t. c

Compound statements

There are certain words that are being used with their proper meaning, but which clearly have a special role in the logical structure of a proposition. Words or phrases like for, any, such that, and, or, implies, and if and only if have precise usage in grammar or writing propositions.

We have considered simple statements which contain simple ideas or propositions.

However, two or more simple statements that contain more than one idea, and that can be separated into two or more statements by means of a certain group of words called connectives are referred to as compound statements.

Some statements are composite, that is, they are composed of substatements and various Connectives. Such composite statements are called compound statements.

Example 2

Consider the following composite statements.

a) Mark is fat and Joy is slim', ' This is a compound statement with substatements 'Mark is fat' and 'Joy is slim'.

b) Bidemi is rich but Adeleke is poor'. This is a compound statement with substatements 'Bidemi is rich' and 'Adeleke is poor.

The basic property of a compound statement is that its truth value is completely determined by the truth value of its substatements together with the way in which they are connected to form the compound statements.

Logical Operations for solving compound statement

There are certain rules for solving compound statements and this involves some connectives used in combining the simple statements together to form compound statements.

The words which combine simple statements to form compound statements are called logical connectives or simply connectives.

The rules include;

 

S/N	CONNECTIVE	WORD	SYMBOLS
	Conjunction A	‘And’	˄
	Disjunction	'or'	V
	Negation	'not’	˜
	Implication	‘Implies’	=>
	Bi-implication	‘if and only if’	ó
	Equivalence	‘Is’	Ξ

 

1. Conjunction:

 This is a compound statement formed by joining two simple statements with 'and'. The symbol is '˄".

Example

Form a compound statement from the following;

P: John is tall

Q: John is handsome

Solution

Simple statements can be denoted by any of the letters P, Q, R, etc.

If P is John is tall and Q is he is handsome,

then P˄Q = John is tall and handsome.

Truth table for Conjunction

P	Q	P˄Q
T	T	T
T	F	F
F	T	F
F	F	F

 

2. Disjunction

Disjunction is a compound statement formed by joining two simple statements with 'or’ and with a symbol ‘˅’.

Example

If "P' is a statement: Pastor preaches' and

Q is another statement: 'Pastor prays', then,

P v Q = ‘Pastor preaches or prays' since he cannot do both at the same time.

Symbolically, P v Q denotes the disjunčtion of the statements P and Q, it is read as P or Q. P v Q is false if P and Q are both false, otherwise they are true.

Note that the English word 'or' is commonly used in two distinct ways. Sometimes, it is used in the sense of 'P or Q or both,' and sometimes, it is used as P or Q but not both. The former is called inclusive dis-junction while the latter is called an exclusive disjunction.

1. Inclusive disjunction

In a statement: ‘It is cold or it is raining’, the or is used in the inclusive sense, because it is possible to be cold as well as rain.

In this text, we use 'or' in the inclusive sense.

The truth value of the compound statement P v Q Inclusive disjunction

 

P	Q	P v Q
T	T	T
T	F	T
F	T	T
F	F	F

 

2. Exclusive disjunction

Consider the statement, Jide will go to Abuja or London. In this statement, the 'or is used in an exclusive sense, because it is not possible for Jide to go to Abuja and at the same time be in London.

The truth value of P v Q for Exclusive Disjunction

P	Q	P v Q
T	T	F
T	F	T
F	T	T
F	F	F

 

3. Implicative (or conditional) statements

In mathematics, many statements are of the form 'If P, then Q’. Such statements are called conditional statements or implications.

When two simple statements are combined by 'If ...... then', such that the first statement implies the second. They are denoted by P =» Q. The if clause statement i.e. P) is sometimes called the antecedent while the then clause (Q) is called the consequent.

The conditional statement P =» Q can also be read as:

a) P implies Q

b) Q only if P

c)P is sufficient for Q

d) P is necessary for Q

 

Example

1.  if P stands for 'Pastors preach' and Q stands for ‘Pastors talk a lot’.

Then P =» Q is the statement ‘If Pastor Preach, then he talks a lot’

 

 

The truth value of P =» Q for Implicative (or conditional) statements

P	Q	P =» Q
T	T	T
T	F	F
F	T	T
F	F	T

 

Properties of P =» Q

1. Converse of P =» Q is Q =» P
2. Inverse of P =» Q is ˜P =» ˜Q
3. Contra-positive of  P =» Q is ˜Q =»  ˜P

Note: the P =» Q column is false only when P is true and Q is false.

 

 

4. Bi-implicative statement or Bi-Conditional Statement

If P statement is =» Q  statement and Q statement is =» P,  such a relation implies equivalence.

Example

If P stands for ‘john speaks English fluently’ and Q stands for ‘john was born and bred in England’, then P ó Q.

Truth table for Bi-implicative statement or Bi-Conditional Statement

P	Q	P =» Q
T	T	T
T	F	F
F	T	T
F	F	T

 

5. Tautology

This is a compound statement or proposition whose end result is true despite the value of its substatements. Tautology is represented by T.

This is a situation where some compound statements remain true no matter the truth values of the substatements. 

TRUTH TABLE FOR TAUTOLOGY

Q	˜Q	Q v ˜Q
T	F	T
F	T	T

 

We have seen that a simple and concise way of determining the truth value of a compound statement is usually by constructing the truth table of the statement.