Proposition

Mathematical statement

This entry uses an introductory classical context: a proposition P may be true or false, but being a proposition does not mean that P has been proved true.

Representative formula (exact TeX)

P

Assumptions

• P is a well-formed, meaningful proposition in the specified logical context.
• The true-or-false dichotomy is limited to the introductory classical context used by this entry.

Symbol scope

• P — proposition variable

Intuition, definition, and misconceptions

A proposition is a statement that can be asserted or denied. Both true and false propositions remain propositions; truth value is a semantic state, not the sole prerequisite for classifying text as a proposition.

A proof supports a proposition as true, but “P is a proposition” does not mean “P already has a proof.” In Lean one may first have P : Prop and then separately ask whether a proof term of P exists.

An open sentence such as x>0 contains an unspecified free variable. It becomes evaluable in a closed context only after substitution, quantification, or an explicit contextual assignment.

Common mistakes include treating questions, commands, or unclosed open expressions as propositions, or confusing “is a proposition” with “has been proved true.”

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

Verification boundary

This entry is a conceptual candidate. Its research sources are not OCME Evidence, and no computational check or Lean formal proof has yet been registered.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Use P : Prop to establish a minimal Lean bridge between propositions and proof terms, while distinguishing any additional principles needed for classical truth semantics.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.