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.

Noun

1. proof of a logical theorem