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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 its complement, then the antecedent is a contradiction.[1] [2] Formal notation[edit] This can be written as: ( P → Q ) ∧ ( P → ¬ Q ) ↔ ¬ P displaystyle (Prightarrow Q)land (Prightarrow neg Q)leftrightarrow neg P An example of its use would be an attempt to prove two contradictory statements from a single fact
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Predicate Logic
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic
First-order logic
uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.[2] A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007
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Rule Of Inference
In 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 (or conclusions). For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e
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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 A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.[1] Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,[2] but it is disallowed by intuitionistic logic.[3] The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: ∗ 4 ⋅ 13 .     ⊢ .   p   ≡   ∼ ( ∼ p ) displaystyle mathbf *4cdot 13 . vdash
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De Morgan's Laws
In propositional logic and boolean algebra, De Morgan's laws[1][2][3] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician
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Transposition (logic)
In propositional logic, transposition[1][2][3] 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 negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely.[4][5] It is very closely related to the rule of inference modus tollens
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Material Implication (rule Of Inference)
In propositional logic, material implication [1][2] is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated
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Exportation (logic)
Exportation[1][2][3][4] 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 vice versa in logical proofs
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Universal Generalization
In predicate logic, generalization (also universal generalization or universal introduction,[1][2][3] GEN) is a valid inference rule. It states that if ⊢ P ( x ) displaystyle vdash P(x) has been derived, then ⊢ ∀ x P ( x ) displaystyle vdash forall x,P(x) can be derived.Contents1 Generalization with hypotheses 2 Example of a proof 3 See also 4 ReferencesGeneralization with hypotheses[edit] The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume Γ is a set of formulas, φ displaystyle varphi a formula, and Γ ⊢ φ ( y ) displaystyle Gamma vdash varphi (y) has been derived
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Tautology (rule Of Inference)
In propositional logic, tautology is one 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
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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 depend on it. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations such as the multiplication and addition of numbers are commutative, was for many years implicitly assumed
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Universal Instantiation
In predicate logic universal instantiation[1][2][3] (UI, also called universal specification or universal elimination, and sometimes confused with Dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom. It is one of the basic principles used in quantification theory. Example: "All dogs are mammals. Fido is a dog
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Existential Generalization
In predicate logic, existential generalization[1][2] (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 quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier (∃) in formal proofs. Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail." In the Fitch-style calculus: Q ( a ) →   ∃ x Q ( x ) displaystyle Q(a)to exists x ,Q(x) Where a replaces all free instances of x within Q(x).[3] Quine[edit] Universal instantiation and Existential Generalization are two aspects of a single principle, for instead of saying that "∀x x=x" implies "Socrates=Socrates", we could as well say that the denial "Socrates≠Socrates"' implies "∃x x≠x"
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Existential Instantiation
In predicate logic, existential instantiation (also called existential elimination)[1][2][3] is a valid rule of inference which says that, given a formula of the form ( ∃ x ) ϕ ( x ) displaystyle (exists x)phi (x) , one may infer ϕ ( c ) displaystyle phi (c) for a new constant symbol c. The rule has the restriction that the constant c introduced by the rule must be a new term that has not occurred earlier in the proof. In one formal notation, the rule may be denoted ( ∃ x ) F x :: F a , displaystyle (exists x) mathcal F x:: mathcal F a, where a is a new constant symbol that has not appeared in the proof. See also[edit]existential fallacyReferences[edit]^ Hurley, Patrick. A Concise Introduction to Logic
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