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 Tautology (rule Of Inference) In propositional logic, tautology is one of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs [...More...] "Tautology (rule Of Inference)" on: Wikipedia Google Yahoo Parouse Proposition The term proposition has a broad use in contemporary analytic philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity [...More...] "Proposition" on: Wikipedia Google Yahoo Parouse picture info Theorem In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises [...More...] "Theorem" on: Wikipedia Google Yahoo Parouse Tautology (logic) In logic, a tautology (from the Greek word ταυτολογία) is a formula or assertion that is true in every possible interpretation. A simple example is "(x equals y) or (x does not equal y)" (or as a less abstract example, "The ball is green or the ball is not green"). Philosopher Philosopher Ludwig Wittgenstein Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921. (It had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternative sense.) A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent [...More...] "Tautology (logic)" on: Wikipedia Google Yahoo Parouse picture info Logical Disjunction In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as ∨ or +. "A or B" is true if A is true, or if B is true, or if both A and B are true. In logic, or by itself means the inclusive or, distinguished from an exclusive or, which is false when both of its arguments are true, while an "or" is true in that case. An operand of a disjunction is called a disjunct. Related concepts in other fields are:In natural language, the coordinating conjunction "or". In programming languages, the short-circuit or control structure. In set theory, union. In predicate logic, existential quantification.Contents1 Notation 2 Definition2.1 Truth table3 Properties 4 Symbol 5 Applications in computer science5.1 Bitwise operation 5.2 Logical oper [...More...] "Logical Disjunction" on: Wikipedia Google Yahoo Parouse picture info Logical Conjunction In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true [...More...] "Logical Conjunction" on: Wikipedia Google Yahoo Parouse Formal Proof A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof. The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof [...More...] "Formal Proof" on: Wikipedia Google Yahoo Parouse picture info Idempotence Idempotence Idempotence (UK: /ˌɪdɛmˈpoʊtəns/, US: /ˌaɪdəm-/) is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application [...More...] "Idempotence" on: Wikipedia Google Yahoo Parouse Metalogic Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations [...More...] "Metalogic" on: Wikipedia Google Yahoo Parouse picture info Symbol (formal) A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern.[citation needed] Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term "symbol" refers to the idea, and the marks are considered to be a token instance of the symbol.[dubious – discuss] In logic, symbols build literal utility to illustrate ideas.Contents1 Overview 2 Can words be modeled as formal symbols? 3 References 4 See alsoOverview Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses) [...More...] "Symbol (formal)" on: Wikipedia Google Yahoo Parouse Well-formed Formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.Contents1 Introduction 2 Propositional calculus 3 Predicate logic 4 Atomic and open formulas 5 Closed formulas 6 Properties applicable to formulas 7 Usage of the terminology 8 See also 9 Notes 10 References 11 External linksIntroduction A key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated [...More...] "Well-formed Formula" on: Wikipedia Google Yahoo Parouse Semantic Consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e [...More...] "Semantic Consequence" on: Wikipedia Google Yahoo Parouse Sequent In mathematical logic, a sequent is a very general kind of conditional assertion. A 1 , … , A m ⊢ B 1 , … , B n . displaystyle A_ 1 ,,dots ,A_ m ,vdash ,B_ 1 ,,dots ,B_ n . A sequent may have any number m of condition formulas Ai (called "antecedents") and any number n of asserted formulas Bj (called "succedents" or "consequents"). A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true [...More...] "Sequent" on: Wikipedia Google Yahoo Parouse Logical Consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e [...More...] "Logical Consequence" on: Wikipedia Google Yahoo Parouse Formal System A formal system or logical calculus is any well-defined system of abstract thought based on the model of mathematics. A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements [...More...] "Formal System" on: Wikipedia Google Yahoo Parouse picture info Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (/ˈrʌsəl/; 18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life, Russell considered himself a liberal, a socialist and a pacifist, but he also admitted that he had "never been any of these things, in any profound sense". Russell was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British "revolt against idealism". He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics [...More...] "Bertrand Russell" on: Wikipedia Google Yahoo Parouse
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