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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

Absorption law

In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: a ¤ (a ⁂ b)

Import–export (logic)

In logic, import-export is a deductive argument form which states that . In natural language terms, the principle means that the following English sentences are logically equivalent. 1.
* If Mary isn

Monotonicity of entailment

Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this propert

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

Peirce's law

In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law

Nicod's axiom

In logic, Nicod's axiom (named after the French logician and philosopher Jean Nicod) is a formula that can be used as the sole axiom of a semantically complete system of propositional calculus. The on

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

Reductio ad absurdum

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that atte

Principle of explosion

In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequ

Contraposition

In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by co

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

Consensus theorem

In Boolean algebra, the consensus theorem or rule of consensus is the identity: The consensus or resolvent of the terms and is . It is the conjunction of all the unique literals of the terms, excludin

Idempotency of entailment

Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. This property can be captured by

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.

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

Consequentia mirabilis

Consequentia mirabilis (Latin for "admirable consequence"), also known as Clavius's Law, is used in traditional and classical logic to establish the truth of a proposition from the inconsistency of it

Proof by contradiction

In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contrad

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

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

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

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

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

Frege's theorem

In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, info

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

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

William of Soissons

William of Soissons was a French logician who lived in Paris in the 12th century. He belonged to a school of logicians, called the .

Law of noncontradiction

In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions ca

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

Law of excluded middle

In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of

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

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