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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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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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Logic
Logic
Logic
(from the Ancient Greek: λογική, translit. logikḗ[1]), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth,[2] and is now generally held to consist of the systematic study of the form of valid inference. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. (In ordinary discourse, inferences may be signified by words like therefore, hence, ergo, and so on.) There is no universal agreement as to the exact scope and subject matter of logic (see § Rival conceptions, below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, the study of inference, including fallacies, and the study of semantics, including paradoxes
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Willard Van Orman Quine
Willard Van Orman Quine
Willard Van Orman Quine
(/kwaɪn/; known to intimates as "Van";[2] June 25, 1908 – December 25, 2000) was an American philosopher
American philosopher
and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century."[3] From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University
Harvard University
in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978
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John Etchemendy
John W. Etchemendy (born 1952 in Reno, Nevada) was Stanford University's twelfth Provost. He succeeded John L. Hennessy
John L. Hennessy
to the post on September 1, 2000 and stepped down on January 31, 2017.Left to right: John L. Hennessy, Susan Rice, and John Etchemendy, June 2010John Etchemendy John Etchemendy
John Etchemendy
received his bachelor's and master's degrees at the University of Nevada, Reno
University of Nevada, Reno
before earning his PhD in philosophy at Stanford in 1982. He has been a faculty member in Stanford's Department of Philosophy since 1983, prior to which he was a faculty member in the Philosophy Department at Princeton University
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Jon Barwise
Kenneth Jon Barwise (/ˈbɑːrwaɪz/; June 29, 1942 – March 5, 2000)[1] was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used.Contents1 Biography 2 Works 3 See also 4 References 5 External linksBiography[edit] Born in Independence, Missouri
Independence, Missouri
to Kenneth T. and Evelyn Barwise, Jon was a precocious child. A pupil of Solomon Feferman at Stanford University, Barwise started his research in infinitary logic. After positions as assistant professor at the Universities of Yale and Wisconsin, during which time his interests turned to natural language, he returned to Stanford in 1983 to direct the Center for the Study of Language and Information. He began teaching at Indiana University in 1990
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Quantification (logic)
In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula. The two most common quantifiers mean "for all" and "there exists". For example, in arithmetic, quantifiers allow one to say that the natural numbers go on for ever, by writing that for all n (where n is a natural number), there is another number (say, the successor of n) which is one bigger than n. A language element which generates a quantification (such as "every") is called a quantifier. The resulting expression is a quantified expression, it is said to be quantified over the predicate (such as "the natural number x has a successor") whose free variable is bound by the quantifier. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted
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Fitch Notation
Fitch notation, also known as Fitch diagrams (named after Frederic Fitch), is a notational system for constructing formal proofs used in sentential logics and predicate logics. Fitch-style proofs arrange the sequence of sentences that make up the proof into rows. A unique feature of Fitch notation is that the degree of indentation of each row conveys which assumptions are active for that step.Contents1 Example 2 See also 3 References 4 External linksExample[edit] Each row in a Fitch-style proof is either:an assumption or subproof assumption. a sentence justified by the citation of (1) a rule of inference and (2) the prior line or lines of the proof that license that rule.Introducing a new assumption increases the level of indentation, and begins a new vertical "scope" bar that continues to indent subsequent lines until the assumption is discharged
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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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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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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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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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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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Existential Quantifier
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". Some sources use the term existentialization to refer to existential quantification.[1] It is usually denoted by the turned E (∃) logical operator symbol, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)")
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