Propositions & Truth Values

Overview

Welcome to Logic

Logic is the foundation of all mathematical reasoning and computer science. In this lesson you will learn to identify propositions - the basic building blocks of logical arguments. Everything in discrete math, from proofs to algorithms, rests on these ideas.

Definition

Proposition

A proposition is a declarative statement that is either true (T) or false (F), but not both. We typically use lowercase letters p, q, r, s to denote propositions.

For example, "2 + 3 = 5" is a proposition (its truth value is T), while "x + 1 = 3" is NOT a proposition because its truth value depends on x.

Example

Identifying Propositions

✓ "The Earth orbits the Sun" - proposition (T) ✓ "7 is an even number" - proposition (F) ✓ "Toronto is the capital of Canada" - proposition (F) ✗ "What time is it?" - not a proposition (question) ✗ "Close the door" - not a proposition (command) ✗ "x > 5" - not a proposition (open sentence with a variable) ✗ "This sentence is false" - not a proposition (paradox, no definite truth value)

Definition

Truth Value

The truth value of a proposition is T (true) or F (false). Every proposition has exactly one truth value. We sometimes write 1 for true and 0 for false, especially in computer science contexts.

Definition

Compound Propositions

A compound proposition is formed by combining one or more propositions using logical connectives (such as AND, OR, NOT, IF-THEN). For example, if p = "It is raining" and q = "I carry an umbrella," then "If it is raining, then I carry an umbrella" is a compound proposition built from p and q.

Note

Propositional Variables

We use letters like p, q, r, s (called propositional variables) to stand for specific propositions. This lets us study the structure of arguments without worrying about specific content. For instance, "p → q" represents any statement of the form "If ..., then ...".

Example

Assigning Truth Values

Let p = "3 is prime" (T) and q = "4 is prime" (F).

The compound proposition "3 is prime AND 4 is prime" is p ∧ q. Since q is false, p ∧ q is false.

The compound proposition "3 is prime OR 4 is prime" is p ∨ q. Since p is true, p ∨ q is true.

Note

Why This Matters

Propositions and truth values are used everywhere: in writing precise mathematical proofs, designing digital circuits, programming conditional statements (if/else), querying databases (SQL WHERE clauses), and building artificial intelligence systems. Mastering logic gives you the language to reason about all of these.