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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 (\exists x) \phi(x), one may infer \phi(c) for a new constant symbol ''c''. The rule has the restrictions that the constant ''c'' introduced by the rule must be a new term that has not occurred earlier in the proof, and it also must not occur in the conclusion of the proof. It is also necessary that every instance of x which is bound to \exists x must be uniformly replaced by ''c''. This is implied by the notation P\left(\right), but its explicit statement is often left out of explanations. In one formal notation, the rule may be denoted by :\exists x P \left(\right) \implies P \left(\right) where ''a'' is a new constant symbol that has not appeared in the proof. See also * Existential fallacy * Universal instantiation In predicate logic, universal instantiation (UI; also called universal specification or universal eliminat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as ''modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prentice Hall
Prentice Hall was a major American publishing#Textbook_publishing, educational publisher. It published print and digital content for the 6–12 and higher-education market. It was an independent company throughout the bulk of the twentieth century. In its last few years it was owned by, then absorbed into, Savvas Learning Company. In the Web era, it distributed its technical titles through the Safari Books Online e-reference service for some years. History On October 13, 1913, law professor Charles Gerstenberg and his student Richard Ettinger founded Prentice Hall. Gerstenberg and Ettinger took their mothers' maiden names, Prentice and Hall, to name their new company. At the time the name was usually styled as Prentice-Hall (as seen for example on many title pages), per an orthographic norm for Dash#Relationships and connections, coordinate elements within such compounds (compare also ''McGraw-Hill'' with later styling as ''McGraw Hill''). Prentice-Hall became known as a publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as ''modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Fallacy
The existential fallacy, or existential instantiation, is a formal fallacy. In the existential fallacy, one presupposes that a class has members when one is not supposed to do so; i.e., when one should not assume existential import. Not to be confused with the 'Affirming the consequent', as in "If A, then B. B. Therefore A". One example would be: "''Every unicorn has a horn on its forehead''". It does not imply that there are any unicorns at all in the world, and thus it cannot be assumed that, if the statement were true, somewhere there is a unicorn in the world (with a horn on its forehead). The statement, if assumed true, implies only that if there were any unicorns, each would definitely have a horn on its forehead. Overview An existential fallacy is committed in a medieval categorical syllogism because it has two universal premises and a particular conclusion with no assumption that at least one member of the class exists, an assumption which is not established by the prem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 schema. It is one of the basic principles used in quantification theory. Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal." Formally, the rule as an axiom schema is given as : \forall x \, A \Rightarrow A\, for every formula ''A'' and every term ''t'', where A\ is the result of substituting ''t'' for each ''free'' occurrence of ''x'' in ''A''. \, A\ is an instance of \forall x \, A. And as a rule of inference it is :from \vdash \forall x A infer \vdash A \ . Irving Copi noted that universal instantiation "... follows from variants of ru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Rules Of Inference
This is a list of rules of inference, logical laws that relate to mathematical formulae. Introduction Rules of inference are syntactical transform 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 : \varphi \vdash \psi indicates such a subderivation from the temporary assumption \varphi to \psi. Rules for propositional calculus Rules for negations ;Reductio ad absurdum (or '' Negation Introduction''): : \varphi \vdash \psi : \underline : \lnot \varphi ;Reductio ad absurdum (related to the law of excluded middle): : \lnot \varphi \vdash \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rules Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as '' modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argument forms involving lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
