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This is a list of rules of inference. Image File history File links Portal. ...
In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ...
Introduction
Rules of inference are syntactical transformation rules which one can use to infer a conclusion from a premise to create an argument. 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.
Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation
 indicates such a subderivation from the temporary assumption  to  .
In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...
Rules for negations - Reductio ad absurdum (or Negation Introduction)
  - Reductio ad absurdum (related to the law of excluded middle)
  - Noncontradiction (or Negation Elimination)
 - Double negation elimination
 - Double negation introduction
  Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption...
The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ...
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In logic and the propositional calculus, double negative elimination is a rule that states that double negatives can be removed from a proposition without changing its meaning: means the same as: Formally: ¬ ¬ A ∴ A The rule of double negative introduction states the converse, that double negatives can be added without...
In logic and the propositional logic, the inference rules double negative elimination (also called double negation elimination, double negative introduction, double negation introduction, or simply double negation) allow deriving the double negative equivalent by adding (for double negative introduction) or removing (for double negative elimination) a pair of negation signs. ...
Rules for conditionals - Deduction theorem (or Conditional Introduction)
  - Modus ponens (or Conditional Elimination)
 - Modus tollens
 In mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i. ...
Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...
In logic, modus ponens (Latin: mode that affirms; often abbreviated MP) is a valid, simple argument form. ...
In logic, Modus tollens (Latin for mode that denies) is the formal name for indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT. It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such...
Rules for conjunctions - Adjunction (or Conjunction Introduction)
  - Simplification (or Conjunction Elimination)
 In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. ...
In logic, simplification is a valid simple argument of the form: A and B, therefore A or A and B, therefore B or in logical operator notation: or The argument has one premise, namely a conjunction, and one often uses simplification in longer arguments to make one of the conjuncts...
Rules for disjunctions - Addition (or Disjunction Introduction)
  - Separation of Cases (or Disjunction Elimination)
   - Disjunctive syllogism
  Addition is a valid, simple argument form in logic: A. Therefore, A or B. or in logical operator (sequent) notation: The argument form has one premise. ...
Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately. ...
A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: P or Q Not P Therefore, Q In logical operator notation: ¬ where represents the logical assertion. ...
Rules for biconditionals - Biconditional introduction
  - Biconditional Elimination
 Biconditional introduction is the inference that, if B follows from A, and A follows from B, then A if and only if B. For example: if Im breathing, then Im alive; also, if Im alive, then Im breathing. ...
In the following rules,  is exactly like except for having the free term  everywhere has the free variable  . First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...
- Universal Introduction (or Universal Generalization)
  Restriction: does not occur free in or in any non-discharged assumption. Generalization is an inference rule of predicate calculus which states that: If is true (valid) then so is . ...
- Universal Elimination (or Universal Instantiation)
  - Existential Introduction (or Existential Generalization)
 - Existential Elimination (or Existential Instantiation)
 Restriction: does not occur free in  , in or in any non-discharged assumption. In logic Universal instantiation (UI) is an inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. ...
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