Category: Rules of inference

Resolution (logic)
In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order l
List of rules of inference
This is a list of rules of inference, logical laws that relate to mathematical formulae.
Double negation
In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition
Existential generalization
In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quanti
Negation as failure
Negation as failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive (i.e. that is assumed not to hold) from failure to derive . Note that can be different from
Hypothetical syllogism
In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises. An example in English: If I do not wake up, then I cann
Free choice inference
Free choice is a phenomenon in natural language where a linguistic disjunction appears to receive a logical conjunctive interpretation when it interacts with a modal operator. For example, the followi
Simplification of disjunctive antecedents
In formal semantics and philosophical logic, simplification of disjunctive antecedents (SDA) is the phenomenon whereby a disjunction in the antecedent of a conditional appears to distribute over the c
Modus tollens
In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument
Tautology (rule of inference)
In propositional logic, tautology is either of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. Th
Biconditional introduction
In propositional logic, biconditional introduction is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements. The rule makes it possible to introduce a b
Structural rule
In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgment or sequents directly. Structural rules often mimic intended
Negation introduction
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus. Negation introduction states that if a given antecedent implies both the consequent and it
List of valid argument forms
Of the many and varied argument forms that can possibly be constructed, only very few are valid argument forms. In order to evaluate these forms, statements are put into logical form. Logical form rep
Absorption (logic)
Absorption is a valid argument form and rule of inference of propositional logic. The rule states that if implies , then implies and . The rule makes it possible to introduce conjunctions to proofs. I
Conjunction introduction
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic. The rule makes it possible to int
Existential instantiation
In predicate logic, existential instantiation (also called existential elimination) is a rule of inference which says that, given a formula of the form , one may infer for a new constant symbol c. The
Modus ponendo tollens
Modus ponendo tollens (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollendo ponens.
Universal generalization
In predicate logic, generalization (also universal generalization or universal introduction, GEN) is a valid inference rule. It states that if has been derived, then can be derived.
Contraposition (traditional logic)
In traditional logic, contraposition is a form of immediate inference in which a proposition is inferred from another and where the former has for its subject the contradictory of the original logical
Disjunction elimination
In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disj
Universal instantiation
In predicate logic, universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth abo
Biconditional elimination
Biconditional elimination is the name of two valid rules of inference of propositional logic. It allows for one to infer a conditional from a biconditional. If is true, then one may infer that is true
Associative property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, asso
Commutativity of conjunction
In propositional logic, the commutativity of conjunction is a valid argument form and truth-functional tautology. It is considered to be a law of classical logic. It is the principle that the conjunct
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustu
No description available.
Exportation (logic)
Exportation is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and
Constructive dilemma
Constructive dilemma is a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum,
Modus ponens
In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is
Modus non excipiens
In logic, modus non excipiens is a valid rule of inference that is closely related to modus ponens. This argument form was created by Bart Verheij to address certain arguments which are types of modus
Admissible rule
In logic, a rule of inference is admissible in a formal system if the set of theorems of the system does not change when that rule is added to the existing rules of the system. In other words, every f
Cut rule
In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. Its meaning is that, if a formula A appears as a conc
Disjunctive syllogism
In classical logic, disjunctive syllogism (historically known as modus tollendo ponens (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjuncti
Conjunction elimination
In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) is a valid immediate inference, argument form and rule of inference which makes the infe
Distributive property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality is always true in elementary algebra.For example, in elementary arithme
Rule of replacement
In logic, a rule of replacement is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of
Rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion
SLD resolution
SLD resolution (Selective Linear Definite clause resolution) is the basic inference rule used in logic programming. It is a refinement of resolution, which is both sound and refutation complete for Ho
Transposition (logic)
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both n
Material implication (rule of inference)
In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states
Destructive dilemma
Destructive dilemma is the name of a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either Q is false or S is false, then either P or R mu
Disjunction introduction
Disjunction introduction or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjun