Logical connectives

Mathematical statement

For propositions P and Q, ¬P denotes negation, P∧Q says that both hold, P∨Q says that at least one holds, P→Q says that if P then Q, and P↔Q denotes the biconditional requiring both directions.

Representative formula (exact TeX)

P\land Q

Assumptions

• P and Q are propositions in a clearly specified propositional context.

Symbol scope

• P — left proposition
• Q — right proposition
• ∧ — conjunction connective

Intuition, definition, and misconceptions

The conjunction P∧Q holds only when both P and Q hold. The disjunction P∨Q is inclusive, so it permits both propositions to hold at the same time.

Material implication P→Q is a relation between propositions; it does not by itself assert temporal order or causation. The biconditional P↔Q requires implication in both directions.

Scope and parentheses change the structure of a compound proposition. Reading it correctly therefore begins by identifying the outermost connective.

Common mistakes include reading disjunction as exclusive-or, treating implication as causation, or proving only one direction and then claiming a biconditional.

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

• Proposition — Logical connectives form compound propositions from one or two propositions.

Verification boundary

Truth tables can check finite assignments of truth values, but they do not constitute a formal proof of every rule of inference.

not_formalized

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

Next obligation

Bind exact formal statements and evidence separately for Not, And, Or, implication, and Iff.

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.