A statement (or proposition) is a declarative sentence that is either true (T) or false (F), but not both.

Simple statements cannot be broken into smaller statements. Compound statements are formed by combining two or more simple statements using logical connectives.

Examples

Sentences that are questions, commands, or contain variables (like $x + 5 = 10$) are not statements because their truth value cannot be determined.

Logical connectives are symbols used to combine propositions into compound statements.

1. Negation ($\sim$)

The negation of a statement has the opposite truth value.

Example: If $p$: "It is raining," then $\sim p$: "It is not raining."

2. Conjunction ($p \land q$)

The conjunction is true only when both statements are true. It uses the word "and."

Example: Let $p$: "2 is a prime number" (T), $q$: "4 is not a prime number" (T). Then $p \land q$ is true.

3. Disjunction ($p \lor q$)

The disjunction is true when at least one statement is true. It uses the word "or."

4. Conditional ($p \rightarrow q$)

"If $p$ then $q$." The conditional is false only when $p$ is true and $q$ is false.

Example: If $p$: "You get a degree," $q$: "You can get a job." The statement is $p \rightarrow q$.

5. Biconditional ($p \leftrightarrow q$)

"$p$ if and only if $q$." True when both statements have the same truth value.

When evaluating logical expressions, follow this order of dominance:

1. Negation ($\sim$) — highest priority
2. Conjunction ($\land$)
3. Disjunction ($\lor$)
4. Conditional ($\rightarrow$)
5. Biconditional ($\leftrightarrow$) — lowest priority

Always use parentheses when the order is unclear.

An argument consists of:

• Premises: Initial statements (before "therefore")
• Conclusion: The final statement (after "therefore")

Example:

• Premise 1: If I study, I will not fail the exam.
• Premise 2: I did not study.
• Conclusion: Therefore, I will fail the exam.

An argument is valid if whenever all premises are true, the conclusion must also be true. To test validity:

1. Write all premises and the conclusion in symbolic form
2. Form the compound statement: $[P_1 \land P_2 \land \ldots] \rightarrow C$
3. Check if this compound statement is a tautology (always true)

If the compound statement is a tautology, the argument is valid.

Electrical switches behave like logical propositions:

• A closed switch = True (current flows)
• An open switch = False (no current flows)

Series Connection (Conjunction)

When switches $p$ and $q$ are in series, current flows only when both are closed.

• Logical form: $p \land q$
• Equivalent to conjunction

Parallel Connection (Disjunction)

When switches $p$ and $q$ are in parallel, current flows when at least one is closed.

• Logical form: $p \lor q$
• Equivalent to disjunction

Simplifying Electrical Networks

To simplify a circuit:

1. Write the compound proposition for the network
2. Simplify using the laws of algebra of propositions

Example: Simplify $(p \land q) \lor (\sim p \land q)$

Using distributive law: $= (q \land p) \lor (q \land \sim p)$ $= q \land (p \lor \sim p)$ (Distributive) $= q \land T$ (Complement) $= q$ (Identity)

This means the simplified circuit needs only switch $q$.

In Tanzania, logic is used when making decisions in daily life. For example, a shopkeeper in Dar es Salaam might reason: "If a customer buys more than 10 items (p), then I give them a discount (q)." Using logical analysis, the shopkeeper can determine when the discount applies and verify if customer complaints about not receiving discounts are valid. Similarly, understanding electrical wiring in homes requires logic: the lights work only when both the main switch AND the room switch are closed (series = conjunction), while having both mains electricity AND a generator ensures power continuity (parallel = disjunction).