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LOGICAL REASONING

Statements

A statement in a logical context is a declaration, verbal or written that is either true or false but not both.

A true statement is said to have a truth value T, while a false statement is said to have a truth value F.

Example 1

The following are statements:

(a) Nigeria is an African country.

(b) The earth is conical in shape.

(c) If I run I shall not late.

(d) Japanese are hard working people.

Example 2

The following are not statements in the logical context.

(a) Who is he?

(b) What a lovely man!

(c) Take the pencil away.

(d) If I think of my family.

In general, questions, exclamations, commands and expressions of feelings which cannot be assigned a truth value T or F are not statements in the logical context.

By convention, we shall use letter P.q.r,… to denote statements.

Negation

Given a statement P, the negation ofP, written  ̴P is the statement; “it is false that P” or “nor P”.
If P is true,   ̴P is false and if P is false   ̴P is true. In other words, if P has the truth value T then   ̴P has the truth value F and if P has the truth value F then  ̴P has the truth value T.

The relationship between P and   ̴P can be summarizer in the following table.

If P is a statement: “Nigeria is a rich country” then   ̴P is the statement: “it is false that Nigeria is a rich country” or in a more reasonable English “Nigeria is not a rich country”.

Let q be the statement “some lawyers are honest people” then   ̴q is the statement” it is false that some lawyers are honest people”. In a more reasonable English, we can also write   ̴q as some lawyers are not honest people.

Let P be the statement 3 + 4 = 8 then   ̴P is the statement 3 + 4 ≠ 8.

Let P be the statement “The set of numbers 2, 4, 6, 8, … is a set of even numbers” then   ̴P is the statement “The set of numbers 2, 4, 6, 8, … is a set of odd numbers.

1. State which of the following are statements in the logical context:

(a) Caesar was a great leader.

(b) Stop talking to the boys.

(c) Decide whether you are going to the club’s meeting now.

(d) Oh Mansa Musa, you are wonderful!

(e) The Broking House in Ibadan, is a magnificent building.

2. State which of the following are statements in the logical context:

(a) As old as Methuselah.

(b) The set of numbers 3, 5 and 4 is not a Pythagorean triplet.

(c) Is he a serious teacher at all?

(d) If 6 is an odd number, then 3 + 5 = 10.

3. Write the negation of each of the following statements:

(a) He is a handsome man.

(b) It is very cold in Siberia.

(c) It is very hot in tropics.

(d) The sky is blue.

4. Write the negation of each of the following statements:

(a) The party leader will win the election.

(b) The football captain scored the first goal.

(c) Short cuts are dangerous.

(d) Honest men are very rare to come by.

5. Statement with reasons whether the statement  is a negation of the statement P in each of the following:

(a)        p. The line AB is parallel to the line CD,

q. The line AB is perpendicular to the line CD.

(b)        p. He is a good leader.

(c)        p. She is a good leader.

q. She is a good follower.

(6) Write the negation of each of the following avoiding the word ‘not’ as much as possible.

(a) The car is moving fast.

(b) He was present in school yesterday.

(c)The Equator is a Great circle.

(d) His friend is younger than my brother.

(7) Write the negation of each of the following avoiding the word ‘not’ as much as possible.

(a) He obtained the least mark in the examination.

(b) She is the shortest girl in the class.

(c) He is an ugly man.

(d) The hospital is in a bad state.

p implies q or

If p then q or

q is necessary for p or

p is sufficient for q or

p only if q or

p follows from p or

qifp.

Consider the following statements;

(a) If Cairo is in Africa then 8 is an even number.

(b) If Cairo is in Africa then 8 is an odd number.

(c)If Cairo is in Asia then 8 is an even number.

(d) If Cairo is in Asia then 8 is an odd number.

The statements (a) (c) and (d) are all true but the statement (b) is not true for the simple reason that the antecedent is true while the consequent is false. Note that although in statement (d) both the antecedent and consequent are false, yet the whole statement is true.

So a biconditional statement is true when the two substatements have the same truth value.

Consider the following four statements:

(a) Nyerere is an African name if and only if a, e, i, o, u are vowels;

(b) Nyerere is an African name if and only if a, e, i, o, u are consonants;

(c) Nyerere is a European name if and only if a, e, i, o, u are vowels;

(d) Nyerere is a European name if and only if a, e, i, o, u are are consonants.

The statements (a) and (d) are both true since the substatements of each have the same truth value. The statements (b) and (c) are false since the substatement of each have different truth values.

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