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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.
One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination. Negation introduction states that if an absurdity can be drawn as conclusion from then must not be the case (i.e. is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from ...
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination; Conjunction introduction / elimination; Disjunction introduction / elimination; Disjunctive / hypothetical syllogism; Constructive / destructive dilemma; Absorption / modus tollens / modus ponendo tollens; Negation introduction; Rules of replacement
The rule allows one to introduce or eliminate a negation from a formal proof. The rule is based on the equivalence of, for example, It is false that it is not raining. and It is raining. The double negation introduction rule is: P P. and the double negation elimination rule is:
Classical logic is the standard logic of mathematics. Many mathematical theorems rely on classical rules of inference such as disjunctive syllogism and the double negation elimination. The adjective "classical" in logic is not related to the use of the adjective "classical" in physics, which has another meaning.
In propositional logic, tautology is either of two commonly used rules of replacement. [1] [2] [3] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are: The principle of idempotency of disjunction:
negation elimination A rule in natural deduction that allows the derivation of a conclusion by eliminating a negation, under certain conditions. negation introduction A rule in natural deduction that allows for the introduction of negation into a proof, typically by deriving a contradiction from the assumption that the negation is false.
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