Proposition in logic: meaning and examples

A proposition is a statement that is either true or false: “the door is open”, “7 is prime”. Propositional logic studies how the truth of compound statements depends on the truth of the simple ones they are built from.

A proposition is the unit of meaning in logic, the thing a truth value belongs to. Simple, or atomic, propositions are represented by letters, traditionally p, q, r, and compound propositions are built from them with connectives: p AND q, NOT r, p -> q. Questions, commands and sentences with unresolved references (“it is raining”, without a place and time) are not propositions, because they have no truth value. In Boolean algebra the same letters are called variables and the same compounds expressions.

The artefact on this page is p AND (q OR NOT r). The engine's reading says it holds exactly when p is true and q is true or r is false. The 8-row table reads true on rows 4, 6 and 7, all rows where p is true except row 5, where q is false and r is true. The satisfiable check holds with the witness p = TRUE, q = FALSE, r = FALSE, and the tautology check is violated with the counterexample p = FALSE, q = FALSE, r = FALSE.

Propositions are the vocabulary of introductory logic, where the first exercise is to translate sentences into letters and connectives, and of requirements writing, where each atomic proposition becomes a condition a system can check. A rule's glossary descriptions in the engine are propositions in prose, one per input. The term is confused with a sentence (the words rather than what they assert), with a predicate (a proposition with a free variable), and with a proposal.

Example

f := p AND (q OR NOT r) · output

Exact checks

Always true (tautology)? No Counterexample: p = F, q = F, r = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: p = T, q = F, r = F · Exhaustively checked and solver cross-checked · 5 rows

Can be true? Yes Example assignment: p = T, q = F, r = F · Exhaustively checked and solver cross-checked · 5 rows

Truth table (8 rows)

Proposition in logic: meaning and examples
#	p	q	r	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE
2	FALSE	TRUE	FALSE	FALSE
3	FALSE	TRUE	TRUE	FALSE
4	TRUE	FALSE	FALSE	TRUE
5	TRUE	FALSE	TRUE	FALSE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when both of the following hold:
1. p is true
2. q is true or r is false (or both)
In every other case, f is false.
About the operators:
- AND is true only when all of its parts are true.
- OR is inclusive: it is true when at least one part is true, including when several are.
- NOT reverses a value: true becomes false and false becomes true.

Intermediatef is true exactly when both of the following hold:
1. p is true
2. q is true or r is false (or both)
In every other case, f is false.

Advancedf is true exactly when both of the following hold [conjunction ∧]:
1. p is true
2. q is true or r is false (or both) [inclusive disjunction ∨]
In every other case, f is false.
Formal notation: p ∧ (q ∨ ¬r)

The common slip

Assigning a letter to a phrase that is not a proposition. “Unless the user is banned” is not true or false on its own; the proposition is “the user is banned”, and “unless” is a connective. The pitfall page on “unless” shows where the mistranslation leads.

Questions

Why p, q, r rather than A, B, C?

Convention only. Philosophy texts use lower-case letters from p onward for propositions; engineering texts use capitals for signals. The engine accepts either, and the letters change nothing about the table or the checks.

Guides

What is discrete math?

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.