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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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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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Hasty Generalization
Hasty generalization is an informal fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence—essentially making a rushed conclusion without considering all of the variables. In statistics, it may involve basing broad conclusions regarding the statistics of a survey from a small sample group that fails to sufficiently represent an entire population.[1] Its opposite fallacy is called slothful induction, or denying a reasonable conclusion of an inductive argument (e.g
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Deduction Theorem
In mathematical logic, the deduction theorem is a metatheorem of propositional and first-order logic.[1] It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results
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Turnstile (symbol)
In mathematical logic and computer science the symbol ⊢ displaystyle vdash has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails"
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Validity
In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.[1] It is not required that a valid argument have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.Contents1 Arguments 2 Valid formula 3 Statements 4 Soundness 5 Satisfiability 6 Preservation 7 See also 8 References 9 Further readingArguments[edit] An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid. Under such conditions it would be self-contradictory to affirm the premises and deny the conclusion
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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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First-order 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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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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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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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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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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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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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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