GUIDE TO LOGIC

Statements I

Contents

1. Logic
2. Statements
3. Symbols for Statements
4. Negation: Logical NOT
5. Double Negative
6. Logical AND
7. Commutative Law for Logical AND
8. Associative Law for Logical AND
9. Contradiction
10. Logical OR
11. Commutative Law for Logical OR
12. Associative Law for Logical OR
13. Excluded Middle
14. Negation Law for Logical AND
15. Negation Law for Logical OR
16. First Distributive Law
17. Second Distributive Law

1. Logic

Logic is the science of clear thinking and correct reasoning. It is used to determine the relations between statements, and to obtain conclusions from true statements.

2. Statements

Logic is about statements of fact, such as:

The garden is behind the house. The sky is blue.

Statements of fact are different from questions asking for information, such as:

What is the name of this town? How many days are there in April?

Statements of fact are also different from instructions that tell people what to do, such as:

Open the book. Turn left at the crossroads.

EXERCISE

Say whether each of the following sentences is a statement, a question, or an instruction.

(a) Switch the light on.
(b) January is the first month of the year.
(c) The trees shade the house.
(d) When does the sun set?

3. Symbols for Statements

Statements may be represented by letters, such as p, q, r. For example, we can put:

p = Bangkok is in Thailand.
q = The temperature falls in winter.
r = A rose is a flower.

4.Negation: Logical NOT

Simple statements are negated by using the word not, and the symbol not-. For example, let

p = Paris is in England.

Then the negation of p is:

not-p = It is not true that Paris is in England = Paris is not in England.

EXERCISE

Give the negations of the following statements.

(a) The garden is behind the house.
(b) The sky is green.

5. Double Negative

The negation of a negative statement is the original positive statement. For example, let

p = The picture is beautiful.

Then

not-p = The picture is not beautiful,

and

not-(not-p) = It is not true that the picture is not beautiful = The picture is beautiful = p.

The law for a double negative statement is summarized in symbols as follows:

not-(not-p) = p.
This law is true whatever p represents.

EXERCISE

Give the negations of the following negative statements.

(a) The walls around the city are not old.
(b) There is nothing in the box.

6. Logical AND

Two statements may be connected by the word and to make a composite statement:

p and q.

This indicates that the statements p and q are both true at the same time. For example, let

p = John is a man,

and

q = Mary is a woman.

Then

p and q = John is a man and Mary is a woman.

EXERCISE

Use the word and to form a composite statement from the following two statements.

The square root of 4 is 2.
Half of 4 is 2.

7. Commutative Law for Logical AND

When two statements are connected by the word and, it does not matter which statement is first. For example, the composite statement

There are 25.4 millimetres in one inch, and twelve inches make one foot

has the same meaning as

Twelve inches make one foot, and there are 25.4 millimetres in one inch.

This fact is called the commutative law for the logical AND. It is summarized in symbols as follows:

p and q = q and p.

8. Associative Law for Logical AND

We can make a composite statement having three statements connected by the word and as follows:

(p and q) and r.

These symbols indicate that we first take the statement (p and q) and then connect the statement r to it by the word and.

Another way of making a composite statement having three statements connected by the word and is as follows:

p and (q and r).

Here we connect the statement p by the word and to the statement (q and r).

Both of these composite statements have the same meaning. This fact is called the associative law for the logical AND. It is summarized in symbols as follows:

(p and q) and r = p and (q and r).
As a result of this law we may omit the brackets and write

p and q and r.

For example, let

p = Human beings are animals
q = Elephants are animals
r = Birds are animals.

Then

p and q and r = Human beings are animals, and elephants are animals, and birds are animals.

In ordinary writing we shorten this by omitting unnecessary words as follows:

Human beings, elephants, and birds are animals.

EXERCISE

Make the composite statement symbolized by

p and q and r,

where

p = The melting point of silver is 962 degrees Celsius
q = The boiling point of water is 100 degrees Celsius
r = The freezing point of water is zero degrees Celsius.

9. Contradiction

When any statement is connected to its negation by the word and the resulting composite statement is false. For example, the composite statement

They have a motorcar, and they do not have a motor car

is false.

We summarize this fact in symbols as follows:

p and not-p = F,
where F stands for a composite statement that is always false. This is called the law of contradiction.

EXERCISE

Say whether or not each of the following composite statements is a contradiction:

(a) The time is after twelve o'clock, and the time is before twelve o'clock.

(b) Today is a holiday, and tomorrow is not a holiday.

10. Logical OR

Two statements may be connected by the word or to make a composite statement

p or q.

This logical OR statement indicates three possibilities: (1) p is true but q is false, (2) both p and q are true, (3) p is false but q is true.

For example, let

p = John is a man
q = John is a woman.

Then

p or q = John is a man or John is a woman.

Here the statement p is true, but the statement q is false.

As another example, let

p = The weather is wet
q = The weather is windy.

Then

p or q = The weather is wet or the weather is windy.

Here the statement p or q includes all three possibilities: (1) the weather is wet, but not windy, (2) the weather is wet and windy, (3) the weather is not wet, but it is windy.

EXERCISE

Use the word or to form a composite statement from the following pairs of statements, and say whether or not both statements can be true at the same time.

(a) This woman is a teacher
This woman is a mother.

(b) Tomorrow is Sunday
Yesterday was Saturday.

11. Commutative Law for Logical OR

When two statements are connected by the word or it does not matter which statement is first. For example, the composite statement

He is not at home, or he is asleep

has the same meaning as

He is asleep, or he is not at home.

This fact is called the commutative law for the logical OR. It is summarized in symbols as follows:

p or q = q or p.

12. Associative Law for Logical OR

We can make a composite statement having three statements connected by the word or as follows:

(p or q) or r.

These symbols indicate that we first take the statement (p or q) and then connect the statement r to it by the word or.

Another way of making a composite statement having three statements connected by the word or is as follows:

p or (q or r).

Here we connect the statement p by the word or to the statement (q or r).

Both of these composite statements have the same meaning. This fact is called the associative law for the logical OR. It is summarized in symbols as follows:

(p or q) or r = p or (q or r).
As a result of this law we may omit the brackets and write

p or q or r.

For example, let

p = They like tea
q = They like coffee
r = They like milk.

Then

p or q or r = They like tea, or they like coffee, or they like milk.

In ordinary writing we shorten this by omitting unnecessary words as follows:

They like tea, coffee, or milk.

EXERCISE

Make the composite statement symbolized by

p or q or r,

where

p = The shoes are black
q = The shoes are brown
r = The shoes are white.

13. Excluded Middle

When any statement is connected to its negation by the word or the resulting composite statement is true. For example, the statement

They have a motor-car, or they do not have a motor-car

is true.

We summarize this fact in symbols as follows:

p or not-p = T,
where T stands for a composite statement that is always true. As there is no other possibility apart from p and not-p, this is called the law of excluded middle.

EXERCISE

Say whether or not each of the following composite statements is true by the law of excluded middle.

(a) He came before 20 March, or he came after 20 March.
(b) Tomorrow is a holiday, or tomorrow is not a holiday.

14. Negation Law for Logical AND

The negation of a logical AND statement is written in symbols as follows:

not-(p and q).

This means It is not true that p and q. This has the same meaning as not-p or not-q. For example, let

p = The weather is wet
q = The weather is cold.

Then

not-(p and q)

means

It is not true that the weather is wet and cold ,

in other words:

The weather is not wet, or the weather is not cold .

We summarize this fact in symbols as follows:

not-(p and q) = not-p or not-q.
This is the first of two laws called De Morgan's laws.

EXERCISE

Give the negation of the statement:

They are rich and they are intelligent.

15. Negation Law for Logical OR

The negation of a logical OR statement is written in symbols as follows:

not-(p or q).

This means It is not true that p or q. This has the same meaning as not-p and not-q. For example, let

p = The castle is on the hill
q = The castle is on the north side of the town.

Then

not-(p or q)

means

It is not true that the castle is on the hill or the castle is on the north side of the town.

In other words:

The castle is not on the hill and not on the north side of the town.

We summarize this fact in symbols as follows:

not-(p or q) = not-p and not-q.
This is the second of De Morgan's laws.

EXERCISE

Give the negation of the statement:

They play football or they go swimming.

16. First Distributive Law

We may connect another statement to a logical AND statement by the word or as follows:

p or (q and r).

For example, let

p = The bird is in the cage
q = The cage is open
r = The bird has escaped.

Then

p or (q and r) = The bird is in the cage, or the cage is open and the bird has escaped.

This means the same as: The bird is in the cage or the cage is open, and the bird is in the cage or the bird has escaped.

The general law is:

p or (q and r) = (p or q) and (p or r).
This is one of the distributive laws for statements. It is true whatever the statements p, q, and r may be.

EXERCISE

Simplify the following statement using this distributive law.

He was sick or he left home early, and he was sick or he arrived in the office at 8 a.m.

17. Second Distributive Law

We may connect another statement to a logical OR statement by the word and as follows:

p and (q or r).

For example, let

p = You may choose rice
q = You may choose fish
r = You may choose meat.

Then

p and (q or r) = You may choose rice, and you may choose fish or meat.

This means the same as: You may choose rice and fish, or you may choose rice and meat.

The general law is:

p and (q or r) = (p and q) or (p and r).
This is the other distributive law for statements. It is true whatever the statements p, q, and r may be.

EXERCISE

Simplify the following composite statement using this distributive law.

He saves money and buys new clothes, or he saves money and buys new shoes.

Forward links: Statements II   Quantifiers I   Quantifiers II   Sets I   Sets II   Relations I   Relations II

By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.
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