logical implication

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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

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