logical proof

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logical proof

A student writes a logical proof on the whiteboard.

Definition

Noun: - A formal demonstration of the validity of a logical theorem: A logical proof is a sequence of logical statements, each justified by an axiom or a previously established rule of inference, that establishes the truth of a given proposition within a formal logical system.

Usage
  • A logical proof is a core concept in mathematics, philosophy, and computer science, used to demonstrate that a conclusion necessarily follows from a set of premises.
  • It is constructed using a formal language and a defined set of inference rules, ensuring the argument is free from ambiguity.
Examples
Advanced Usage
  • Within a Formal System: A logical proof is considered valid only within the specific axiomatic system for which it is constructed. A proof in one system (e.g., propositional logic) may not be valid in another (e.g., predicate logic).
  • Proof Theory: This is the subfield of mathematical logic that studies logical proofs as formal mathematical objects, analyzing their structure and properties.
Variants and Related Words
  • Proof (n): A more general term for evidence or argument establishing a fact or the truth of a statement. A logical proof is a specific, formal type of proof.
  • Theorem (n): A statement that has been proven on the basis of previously established statements, such as other theorems, axioms, and rules of inference.
  • Deduction (n): The process of reaching a conclusion through logical reasoning from given premises; a logical proof is a formalized deduction.
  • Derivation (n): Often used synonymously in formal logic to mean a sequence of formulas that constitutes a proof.
Synonyms
  • Formal proof
  • Deductive proof
  • Demonstration (in a formal, logical context)
Related Phrases
  • Construct a proof: To methodically build the sequence of logical steps.
  • Proof by contradiction: A specific method of logical proof where one assumes the negation of what is to be proven and shows this leads to a contradiction.
  • Proof system: The set of axioms and inference rules used to generate logical proofs.
logical proof

A student writes a logical proof on the whiteboard.

Noun
  1. proof of a logical theorem