Propositions and Logical Connectives

Why Logic?

Every time you make a decision, you use logic:

“If it’s raining, I’ll take an umbrella.”

“I can pay with cash or card.”

“The form is valid only if all fields are filled.”

Logic gives us a precise way to think about these statements. Instead of relying on intuition, we can:

1. Analyze statements systematically
2. Know for certain whether our reasoning is correct
3. Build complex arguments from simple pieces

This is the foundation of mathematics, programming, and clear thinking.

What is a Proposition?

A proposition is a statement that is either true or false.

That’s it. Not “maybe true” or “sometimes false” - it must be definitively one or the other.

These ARE propositions:

These are NOT propositions:

Key insight: A proposition must have a definite truth value, even if you don’t know what it is.

Example: “There is life on other planets”

• Is this a proposition? Yes.
• Do we know if it’s true? No.
• But is it definitely true or false? Yes.

Why Use Letters?

Writing full sentences gets tedious. So we use letters like $p$, $q$, $r$ to represent propositions.

Instead of writing:

“If it is raining then I will take an umbrella”

We write:

• Let $p$ = “It is raining”
• Let $q$ = “I will take an umbrella”
• Then: $p \to q$

This lets us focus on the structure of reasoning, not the specific words.

Logical Connectives

Now we can combine propositions to make more complex statements.

The tools for combining them are called connectives.

There are five basic connectives:

1. NOT - negation
2. AND - conjunction
3. OR - disjunction
4. IF-THEN - implication
5. IF AND ONLY IF - biconditional

Let’s go through each one.

NOT (Negation) - $\neg$

The simplest connective. It flips the truth value.

• If something is true, NOT makes it false
• If something is false, NOT makes it true

Example:

• $p$ = “The door is open”
• $\neg p$ = “The door is not open”

That’s all there is to NOT.

AND (Conjunction) - $\land$

The result is true only when both are true.

Think of it like a checklist - every item must be checked.

Example: Nuclear launch system

A nuclear launch requires two officers to turn their keys simultaneously.

• $p$ = “Officer A turns key”
• $q$ = “Officer B turns key”
• Launch happens when: $p \land q$

Only the first row is true. One officer alone cannot launch. Both must act together.

Real-world AND:

• Multi-factor authentication (password AND phone code)
• Safety systems (two-person rule)
• Form validation (all required fields filled)

OR (Disjunction) - $\lor$

The result is true when at least one is true.

Think of it as having options - any one will do.

Example: Payment at a store

• $p$ = “You have cash”
• $q$ = “You have card”
• Payment accepted when: $p \lor q$

Only the last row is false. No cash AND no card = can’t pay.

Note: This is inclusive or - having both cash and card still counts as true.

IF-THEN (Implication) - $\to$

This is the trickiest one. It connects a condition to a result.

$p \to q$ means: “If $p$ happens, then $q$ must happen.”

How to evaluate it:

“Did $p$ happen?”

• Yes $\to$ “Did $q$ happen?” $\to$ that’s your answer
• No $\to$ you’re good (automatically true)

Key insight: IF-THEN is only false when $p$ happens but $q$ doesn’t.

When $p$ doesn’t happen, the rule was never triggered, so it’s automatically true.

Example: Gender form validation

A form asks for gender: Male, Female, Other

The rule: “If user selects ‘Other’, they must fill the specify box.”

• $p$ = User selected ‘Other’
• $q$ = User filled the specify box

The form is only invalid when someone picks “Other” but doesn’t specify.

IF AND ONLY IF (Biconditional) - $\leftrightarrow$

Both sides must match. Either both true, or both false.

How to evaluate it:

“Are $p$ and $q$ the same?”

• Same $\to$ true
• Different $\to$ false

Example: Light switch

“The light is on if and only if the switch is up.”

• Switch up, light on = good
• Switch down, light off = good
• They don’t match = something is broken

Summary

These five connectives are the building blocks of all logical reasoning.