Propositional Logic: Statements, Connectives, Truth Tables

Proposition (Statement)

A proposition (or statement) is a declarative sentence that has exactly one truth value: it is either True (T) or False (F), but not both.

In simple terms:

If a sentence makes a definite claim that can be classified as true or false, then it is a proposition.

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Examples of Propositions

Consider the following sentences:

1. New Delhi is the capital of India.
   This is a proposition because it makes a definite claim. Its truth value is True (T).

2. $7 + 5 = 15$
   This is also a proposition. The claim is incorrect, so its truth value is False (F).

Notice that a proposition does not have to be true. A false statement is still a proposition as long as it has a definite truth value.

Therefore, both

$$2 + 3 = 5$$

and

$$2 + 3 = 8$$

are propositions.

The first is true, while the second is false.

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What Is Not a Proposition?

Not every sentence is a proposition.

Consider:

$$x + 4 > 10$$

We cannot determine whether this expression is true or false until we know the value of $x$.

For example:

• If $x = 8$, then

$$8 + 4 > 10$$

which is True.

• If $x = 2$, then

$$2 + 4 > 10$$

which is False.

Therefore,

$$x + 4 > 10$$

is not a proposition because its truth value depends on the value assigned to $x$.

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Truth Value

The truth value of a proposition tells us whether the proposition is true or false.

There are only two possible truth values:

Since every proposition has one of these two truth values, this system is also known as two-valued logic.

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Identifying Propositions

Consider the following sentences:

Key Point

Questions, commands, and exclamations are not propositions because they do not make claims that can be assigned a truth value.

A statement about an unknown or future event can still be a proposition if the statement itself must ultimately be either true or false.

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Propositional Variables

Instead of repeatedly writing complete propositions, we usually represent them using lowercase letters such as

$$p,\ q,\ r,\ s,\ldots$$

These are called propositional variables.

For example, let

$$p = \text{``The Earth has two moons.''}$$

Since this statement is false, we can write

$$p = F$$

Similarly, if

$$q = \text{``The Earth revolves around the Sun.''}$$

then

$$q = T$$

A propositional variable such as $p$ or $q$ acts as a symbol representing a proposition. Its value can only be one of:

$$T \quad \text{or} \quad F$$

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Remember

Proposition = Declarative sentence + Definite truth value

A sentence is a proposition only when it can be classified as True (T) or False (F), but not both.

Atomic and Compound Propositions

Propositions can be classified as atomic propositions or compound propositions depending on how they are formed.

Atomic Proposition

An atomic proposition is a proposition that represents a single statement and cannot be broken down into simpler propositions.

Atomic propositions are also called simple, primary, or primitive propositions.

For example, consider:

The number 8 is even.

This is an atomic proposition because it expresses a single idea.

We can represent it using a propositional variable:

$$p = \text{``The number 8 is even.''}$$

Similarly,

$$q = \text{``15 is divisible by 3.''}$$

is another atomic proposition.

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Compound Proposition

A compound proposition is formed by combining one or more propositions using logical connectives.

For example, suppose

$$p = \text{``8 is even''}$$

and

$$q = \text{``8 is greater than 5''}$$

We can combine them to form:

8 is even and 8 is greater than 5.

Symbolically, this can be written as:

$$p \land q$$

This is a compound proposition because two propositions have been joined using the logical connective AND.

A compound proposition can also be created from a single proposition by negating it.

For example, if

$$p = \text{``8 is even''}$$

then

$$\neg p$$

means:

8 is not even.

Therefore, compound propositions can be formed either by combining propositions or by negating a proposition.

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Logical Connectives

Logical connectives are words or symbols used to connect or modify propositions to create compound propositions.

There are five basic logical connectives:

For example, let

$$p = \text{``I study''}$$

and

$$q = \text{``I pass the examination.''}$$

Using different logical connectives, we can construct statements such as:

$$p \land q$$

I study and I pass the examination.

$$p \lor q$$

I study or I pass the examination.

$$\neg p$$

I do not study.

$$p \rightarrow q$$

If I study, then I pass the examination.

$$p \leftrightarrow q$$

I study if and only if I pass the examination.

Truth Tables

A truth table shows the truth value of a compound proposition for all possible combinations of the truth values of its propositions.

For example, consider the conjunction:

$$p \land q$$

The proposition $p \land q$ is true only when both $p$ and $q$ are true. Otherwise, it is false.

For two propositions $p$ and $q$, there are four possible cases:

1. $p$ is True and $q$ is True
2. $p$ is True and $q$ is False
3. $p$ is False and $q$ is True
4. $p$ is False and $q$ is False

Example

Construct the truth table for each of the following compound propositions:

1. $p \land (\neg q \lor q)$
2. $\neg(p \lor q) \lor (\neg p \land \neg q)$

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(i) $p \land (\neg q \lor q)$

First, find $\neg q$, then evaluate $\neg q \lor q$, and finally evaluate the complete expression $p \land (\neg q \lor q)$.

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(ii) $\neg(p \lor q) \lor (\neg p \land \neg q)$

First, calculate $\neg p$ and $\neg q$. Then evaluate $p \lor q$, $\neg(p \lor q)$, and $\neg p \land \neg q$. Finally, evaluate the complete expression.