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The second pattern of potentially globally redundant proofs appearing in global redundancy definition is related to the well-known [further explanation needed] notion of regularity [further explanation needed]. Informally, a proof is irregular if there is a path from a node to the root of the proof such that a literal is used more than once as ...
In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. [1] [2] The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution. Literals can be divided into two types: [2] A positive literal is just an atom (e.g., ).
The resolution rule in propositional logic is a single valid inference rule that produces a new clause implied by two clauses containing complementary literals. A literal is a propositional variable or the negation of a propositional variable.
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof.
A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
Proof theory is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques.
A graphic representation of the deduction system. In a Hilbert system, a formal deduction (or proof) is a finite sequence of formulas in which each formula is either an axiom or is obtained from previous formulas by a rule of inference.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science.Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language.