Propositions and Connectives

A proposition is a declarative sentence that is either true (T) or false (F), but not both. Before we can reason about anything, we need to be able to tell propositions apart from sentences that have no truth value.

What counts as a proposition

• “7 is a prime number.” → proposition (true)
• “Delhi is the capital of Nepal.” → proposition (false)
• “Close the door.” → not a proposition (a command)
• “Is it raining?” → not a proposition (a question)
• "$x + 2 = 5$" → not a proposition (an open sentence: its truth depends on $x$)

We use lowercase letters $p, q, r, \dots$ as propositional variables.

The five connectives

Compound propositions are built from simpler ones using connectives.

Symbol	Name	Read as	True when…
$\lnot p$	Negation	not $p$	$p$ is false
$p \land q$	Conjunction	$p$ and $q$	both are true
$p \lor q$	Disjunction	$p$ or $q$	at least one is true
$p \to q$	Implication	if $p$ then $q$	except when $p$ is T and $q$ is F
$p \leftrightarrow q$	Biconditional	$p$ if and only if $q$	both have the same value

¬pNOTflips the value

p ∧ qANDtrue only if both are true

p ∨ qORtrue if at least one is true

p → qIMPLIESfalse only for T → F

p ↔ qIFFtrue when both match

The five logical connectives at a glance

Worked example

Let $p$: “It is raining” and $q$: “I carry an umbrella”. Translate “If it is raining, then I carry an umbrella, and it is not raining.”

1. “If it is raining, then I carry an umbrella” is $p \to q$.
2. “it is not raining” is $\lnot p$.
3. The two parts are joined by “and”, so the result is $(p \to q) \land \lnot p$.

Common mistakes

• Treating “or” as exclusive. In logic, $p \lor q$ is also true when both are true.
• Thinking $p \to q$ is false when $p$ is false. A false premise makes the implication true (“vacuously true”).

Key takeaways

• A proposition has exactly one truth value: true or false.
• Questions, commands and open sentences are not propositions.
• The five connectives are $\lnot, \land, \lor, \to, \leftrightarrow$.
• $p \to q$ is false in exactly one case: $p$ true and $q$ false.

Test yourself

4 questions. Pick an answer to check it, or open "Show answer".

1. Which of the following is a proposition?

   • A“Close the door.”
   • B“7 is a prime number.”
   • C"$x + 2 = 5$"
   • D“Is it raining?”
   Show answer

   Answer: B. “7 is a prime number.”

   Explanation: It is a declarative sentence with a definite truth value (true). The others are a command, an open sentence whose truth depends on $x$, and a question.

2. When is $p \to q$ false?

   • AWhen $p$ is false and $q$ is true
   • BWhen both are false
   • CWhen $p$ is true and $q$ is false
   • DWhen both are true
   Show answer

   Answer: C. When $p$ is true and $q$ is false

   Explanation: An implication only fails when a true premise leads to a false conclusion. If $p$ is false, the implication is vacuously true.

3. If $p$ is true and $q$ is true, what is the value of $p \lor q$?

   • ATrue
   • BFalse
   • CIt depends on the context
   • DUndefined
   Show answer

   Answer: A. True

   Explanation: Logical “or” is inclusive: it is true when at least one part is true, including when both are. Choosing “False” confuses it with exclusive or (XOR).

4. Let $p$: “I study” and $q$: “I pass”. Which is the correct translation of “I pass only if I study”?

   • A$p \to q$
   • B$q \to p$
   • C$p \land q$
   • D$p \leftrightarrow q$
   Show answer

   Answer: B. $q \to p$

   Explanation: ”$A$ only if $B$” means $A \to B$. Here $A$ is “I pass” and $B$ is “I study”, so it is $q \to p$. The tempting answer $p \to q$ reverses the direction.