Definition

Noun: 1. A logical relation between two propositions, p and q, expressed as "if p then q": This relation, also called material implication or conditional, means that if proposition p (the antecedent) is true, then proposition q (the consequent) cannot be false. The truth of p guarantees the truth of q. The implication itself is considered false only when p is true and q is false.

Usage Examples

• Noun:
  • In the statement "If it is raining, then the ground is wet," the phrase "the ground is wet" is the logical implication of "it is raining."
  • Understanding logical implication is fundamental to constructing valid arguments and proofs in mathematics and philosophy.
  • The detective deduced the culprit by tracing the logical implications of the evidence.

Advanced Usage

• Truth-Functional Connective: In formal logic, implication is a truth-functional connective (often symbolized as → or ⊃). The compound statement "p → q" is defined to be false only when p is true and q is false; it is true in all other cases (when p is false, or when both are true).
• Distinction from Entailment: While often used informally as a synonym, some philosophical contexts distinguish logical implication (a semantic relation between statements based on truth conditions) from (a syntactic relation within a formal system where q can be derived from p using rules of inference).

Variants and Related Words

• Imply (verb): To express or involve something as a necessary consequence. (e.g., )
• Implicative (adjective): Tending to suggest or imply. (e.g., )
• Conditional (noun/adjective): Often used synonymously with implication as a noun; as an adjective, it describes a statement of this form. (e.g., )
• Material Implication (noun): A specific term for the truth-functional connective defining logical implication in propositional logic.

Synonyms

• Conditional
• Material Conditional (in formal logic)
• If-then statement

Related Phrases / Idioms

• "If p, then q": The standard linguistic form used to express a logical implication.
• "p implies q": A common alternative phrasing for the same logical relation.

Noun

1. a logical relation between propositions p and q of the form `if p then q'; if p is true then q cannot be false